1000 Sentences With "ordinal" | Random Sentence Generator

The eight sides represent cardinal and ordinal directions on a compass.

In Chinese, age is traditionally written in an ordinal system, beginning with one.

RANK Rank is the ordinal number (first second third, etc.) that describes the position in a data set.

Phinney also noticed a small telltale: the use and misuse of the ordinal abbreviations "th" (as in 111th) and "st" (as in 1st).

Conversely, long-term debt ratings of individual securities or financial obligations of a corporate issuer address relative vulnerability to default on an ordinal scale.

It is possible that this lower ordinal diversity could potentially be a cause of why there is such high diversity in the New World.

But I found no other ordinal numerals in the mix, so I left that whole train of thought in the station (I was off track anyway, to muddle the metaphor, trying to come up with some sequence that wouldn't have worked).

Google does the same with Chrome using what it calls Randomized Aggregatable Privacy-Preserving Ordinal Response (RAPPOR), a differential privacy tool for analyzing and drawing insights from its browser that prevents sensitive info like personal browsing histories from being traceable.

Today, the children's shelves in bookstores, libraries and bedrooms are groaning with volumes that can seem to find more inspiration in the ordinal allure of wearing big, bright numbers on their spines than in the old exigencies of character, plot, perspective and suchlike.

In particular, the Court has held that &aposqualitative assessments of offerors&apos proposals, such as the adjectival rating assigned to an offeror&aposs past performance or an offeror&aposs ordinal ranking under one or another evaluation factor ... ha[ve] no bearing on the competitive process&apos and are not subject to redaction.

Every ordinal other than 0 is either a successor ordinal or a limit ordinal..

In set theory, the successor of an ordinal number α is the smallest ordinal number greater than α. An ordinal number that is a successor is called a successor ordinal.

ORCA (Ordinal Regression and Classification Algorithms) is an Octave/MATLAB framework including a wide set of ordinal regression methods. R packages that provide ordinal regression methods include MASS and Ordinal.

A regular ordinal is an ordinal which is equal to its cofinality. A singular ordinal is any ordinal which is not regular. Every regular ordinal is the initial ordinal of a cardinal. Any limit of regular ordinals is a limit of initial ordinals and thus is also initial but need not be regular.

Representation of the ordinal numbers up to ωω. Each turn of the spiral represents one power of ω. Limit ordinals are those that are non-zero and have no predecessor, such as ω or ω2 In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β < γ < λ.

Every ordinal number is either zero, or a successor ordinal, or a limit ordinal. For example, ω, the smallest ordinal greater than every natural number is a limit ordinal because for any smaller ordinal (i.e., for any natural number) n we can find another natural number larger than it (e.g. n+1), but still less than ω.

We will prove this by reductio ad absurdum. # Let \Omega be a set that contains all ordinal numbers. # \Omega is transitive because for every element x of \Omega (which is an ordinal number and can be any ordinal number) and every element y of x (i.e. under the definition of Von Neumann ordinals, for every ordinal number y < x), we have that y is an element of \Omega because any ordinal number contains only ordinal numbers, by the definition of this ordinal construction.

Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the order. Any well-ordered set having that ordinal as its order type has the same cardinality. The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, but most infinite ordinals are not initial.

A set is hereditarily ordinal definable if it is ordinal definable and all elements of its transitive closure are ordinal definable. The class of hereditarily ordinal definable sets is denoted by HOD, and is a transitive model of ZFC, with a definable well ordering. It is consistent with the axioms of set theory that all sets are ordinal definable, and so hereditarily ordinal definable. The assertion that this situation holds is referred to as V = OD or V = HOD.

In mathematics, the Bachmann–Howard ordinal (or Howard ordinal) is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as Kripke–Platek set theory (with the axiom of infinity) and the system CZF of constructive set theory. It was introduced by and .

In mathematics, the Ackermann ordinal is a certain large countable ordinal, named after Wilhelm Ackermann. The term "Ackermann ordinal" is also occasionally used for the small Veblen ordinal, a somewhat larger ordinal. Unfortunately there is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions".

In mathematics, the small Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. It is occasionally called the Ackermann ordinal, though the Ackermann ordinal described by is somewhat smaller than the small Veblen ordinal. Unfortunately there is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions".

A set A\, is called admissible if it is transitive and \langle A,\in \rangle is a model of Kripke–Platek set theory. An ordinal number α is called an admissible ordinal if Lα is an admissible set. The ordinal α is an admissible ordinal if and only if α is a limit ordinal and there does not exist a γ < α for which there is a Σ1(Lα) mapping from γ onto α. If M is a standard model of KP, then the set of ordinals in M is an admissible ordinal.

Because the class of ordinal numbers is well-ordered, there is a smallest infinite limit ordinal; denoted by ω (omega). The ordinal ω is also the smallest infinite ordinal (disregarding limit), as it is the least upper bound of the natural numbers. Hence ω represents the order type of the natural numbers. The next limit ordinal above the first is ω + ω = ω·2, which generalizes to ω·n for any natural number n.

The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal. In this case, it is traditional to identify the cardinal number with its initial ordinal, and we say that the initial ordinal is a cardinal. The \alpha-th infinite initial ordinal is written \omega_\alpha.

Each function exported by a DLL is identified by a numeric ordinal and optionally a name. Likewise, functions can be imported from a DLL either by ordinal or by name. The ordinal represents the position of the function's address pointer in the DLL Export Address table. It is common for internal functions to be exported by ordinal only.

Indeed, one can go far beyond this. So as an ordinal, an infinite initial ordinal is an extremely strong kind of limit.

The only von Neumann ordinals which can be shown to exist in NFU without additional assumptions are the concrete finite ones. However, the application of a permutation method can convert any model of NFU to a model in which every strongly cantorian ordinal is the order type of a von Neumann ordinal. This suggests that the concept "strongly cantorian ordinal of NFU" might be a better analogue to "ordinal of ZFC" than is the apparent analogue "ordinal of NFU".

The ordinal α is compact as a topological space if and only if α is a successor ordinal. The closed sets of a limit ordinal α are just the closed sets in the sense that we have already defined, namely, those that contain a limit ordinal whenever they contain all sufficiently large ordinals below it. Any ordinal is, of course, an open subset of any further ordinal. We can also define the topology on the ordinals in the following inductive way: 0 is the empty topological space, α+1 is obtained by taking the one-point compactification of α, and for δ a limit ordinal, δ is equipped with the inductive limit topology.

In statistics, ordinal regression (also called "ordinal classification") is a type of regression analysis used for predicting an ordinal variable, i.e. a variable whose value exists on an arbitrary scale where only the relative ordering between different values is significant. It can be considered an intermediate problem between regression and classification. Examples of ordinal regression are ordered logit and ordered probit.

An uncountable limit ordinal may have either cofinality ω as does ωω or an uncountable cofinality. The cofinality of 0 is 0. The cofinality of any successor ordinal is 1. The cofinality of any nonzero limit ordinal is an infinite regular cardinal.

Each ordinal associates with one cardinal, its cardinality. If there is a bijection between two ordinals (e.g. and ), then they associate with the same cardinal. Any well-ordered set having an ordinal as its order-type has the same cardinality as that ordinal.

There are other ordinal notations capable of capturing ordinals well past \varepsilon_0, but because there are only countably many strings over any finite alphabet, for any given ordinal notation there will be ordinals below \omega_1 (the first uncountable ordinal) that are not expressible. Such ordinals are known as large countable ordinals. The operations of addition, multiplication and exponentiation are all examples of primitive recursive ordinal functions, and more general primitive recursive ordinal functions can be used to describe larger ordinals.

The Feferman–Schütte ordinal can be defined as the smallest ordinal that cannot be obtained by starting with 0 and using the operations of ordinal addition and the Veblen functions φα(β). That is, it is the smallest α such that φα(0) = α.

The least ordinal associated with a given cardinal is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, and no other ordinal associates with its cardinal. But most infinite ordinals are not initial, as many infinite ordinals associate with the same cardinal. The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e.

In mathematics, the Feferman–Schütte ordinal Γ0 is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion. It is named after Solomon Feferman and Kurt Schütte. It is sometimes said to be the first impredicative ordinal,Kurt Schütte, Proof theory, Grundlehren der Mathematischen Wissenschaften, Band 225, Springer-Verlag, Berlin, Heidelberg, New York, 1977, xii + 302 pp.

In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample to several plausible-sounding conjectures. It is defined as the topological product of the two ordinal spaces [0,\omega_1] and [0,\omega], where \omega is the first infinite ordinal and \omega_1 the first uncountable ordinal. The deleted Tychonoff plank is obtained by deleting the point \infty = (\omega_1,\omega).

Heinz Bachmann (born 1924) is a mathematician who worked at the Eidgenössische Sternwarte (federal observatory) in Zürich. He introduced the Bachmann–Howard ordinal and ordinal collapsing functions.

Solomon Feferman, "Predicativity" (2002) though this is controversial, partly because there is no generally accepted precise definition of "predicative". Sometimes an ordinal is said to be predicative if it is less than Γ0. There is no standard notation for ordinals beyond the Feferman–Schütte ordinal. There are several ways of representing the Feferman–Schütte ordinal, some of which use ordinal collapsing functions: \psi(\Omega^\Omega), \theta(\Omega) or \phi_\Omega(0).

Obviously 1 is additively indecomposable, since 0+0<1. No finite ordinal other than 1 is additively indecomposable. Also, \omega is additively indecomposable, since the sum of two finite ordinals is still finite. More generally, every infinite initial ordinal (an ordinal corresponding to a cardinal number) is additively indecomposable.

Ordinal information is treated by a special joint distribution that preserves the ordinal information.Sousa R., Yevseyeva I., Pinto da Costa J.F., Cardoso J.S. (2013). Multicriteria models for learning ordinal data: A literature review. In Yang X.S. Artificial Intelligence, Evolutionary Computing and Metaheuristics: In the Footsteps of Alan Turing.

The 2-variable Veblen functions can be used to give a system of ordinal notation for ordinals less than the Feferman-Schutte ordinal. The Veblen functions in a finite or transfinite number of variables give systems of ordinal notations for ordinals less than the small and large Veblen ordinals.

The Howard ordinal (also known as the Bachmann–Howard ordinal) was named after him. He was elected to the 2018 class of fellows of the American Mathematical Society.

It can be shown that Wα is self-dual if and only if α is either 0, an even successor ordinal, or a limit ordinal of countable cofinality.

Conversely, any set S of ordinals that is downward-closed -- meaning that for any ordinal α in S and any ordinal β < α, β is also in S -- is (or can be identified with) an ordinal. There are infinite ordinals as well: the smallest infinite ordinal is \omega, which is the order type of the natural numbers (finite ordinals) and that can even be identified with the set of natural numbers. Indeed, the set of natural numbers is well-ordered—as is any set of ordinals—and since it is downward closed, it can be identified with the ordinal associated with it (which is exactly how \omega is defined). A graphical "matchstick" representation of the ordinal ω².

Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient portion of arithmetic to make statements about ordinal notations. The proof- theoretic ordinal of such a theory T is the smallest ordinal (necessarily recursive, see next section) that the theory cannot prove is well founded--the supremum of all ordinals \alpha for which there exists a notation o in Kleene's sense such that T proves that o is an ordinal notation. Equivalently, it is the supremum of all ordinals \alpha such that there exists a recursive relation R on \omega (the set of natural numbers) that well-orders it with ordinal \alpha and such that T proves transfinite induction of arithmetical statements for R.

Many behavioural scientists use the mean for ordinal data, anyway. This is often justified on the basis that the ordinal type in behavioural science is in fact somewhere between the true ordinal and interval types; although the interval difference between two ordinal ranks is not constant, it is often of the same order of magnitude. For example, applications of measurement models in educational contexts often indicate that total scores have a fairly linear relationship with measurements across the range of an assessment. Thus, some argue that so long as the unknown interval difference between ordinal scale ranks is not too variable, interval scale statistics such as means can meaningfully be used on ordinal scale variables.

Using the Von Neumann definition of ordinals, every ordinal is the well- ordered set of all smaller ordinals. The union of a nonempty set of ordinals that has no greatest element is then always a limit ordinal. Using Von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal.

Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The position of each element within the ordered set is also given by an ordinal number. In the case of a finite set, the basic operation of counting, to find the ordinal number of a particular object, or to find the object with a particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects. The size (number of elements, cardinal number) of a finite set is equal to the order type.

In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory.

In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even large cardinals (though they can be replaced with recursively large ordinals at the cost of extra technical difficulty), and then “collapse” them down to a system of notations for the sought-after ordinal. For this reason, ordinal collapsing functions are described as an impredicative manner of naming ordinals. The details of the definition of ordinal collapsing functions vary, and get more complicated as greater ordinals are being defined, but the typical idea is that whenever the notation system “runs out of fuel” and cannot name a certain ordinal, a much larger ordinal is brought “from above” to give a name to that critical point. An example of how this works will be detailed below, for an ordinal collapsing function defining the Bachmann–Howard ordinal (i.e.

Kurt Schütte (14 October 1909, Salzwedel - 18 August 1998, Munich) was a German mathematician who worked on proof theory and ordinal analysis. The Feferman–Schütte ordinal, which he showed to be the precise ordinal bound for predicativity, is named after him. He was the doctoral advisor of 16 students, including Wolfgang Bibel, Wolfgang Maaß, Wolfram Pohlers, and Martin Wirsing.

Nimbers have the characteristic that their Left and Right options are identical, following a certain schema, and that they are their own negatives, such that a positive ordinal may be added to another positive ordinal using nimber addition to find an ordinal of a lower value. The minimum excludant operation is applied to sets of nimbers.

This is a list of armies arranged by ordinal number.

Since there are only countably many notations, all ordinals with notations are exhausted well below the first uncountable ordinal ω1; their supremum is called Church–Kleene ω1 or ω1CK (not to be confused with the first uncountable ordinal, ω1), described below. Ordinal numbers below ω1CK are the recursive ordinals (see below). Countable ordinals larger than this may still be defined, but do not have notations. Due to the focus on countable ordinals, ordinal arithmetic is used throughout, except where otherwise noted.

Ordinal data can be considered as a quantitative variable. In logistic regression, the equation : logit[P(Y=1)] = \alpha + \beta_1 c + \beta_2 x is the model and c takes on the assigned levels of the categorical scale. In regression analysis, outcomes (dependent variables) that are ordinal variables can be predicted using a variant of ordinal regression, such as ordered logit or ordered probit. In multiple regression/correlation analysis, ordinal data can be accommodated using power polynomials and through normalization of scores and ranks.

For infinite groups, one can continue the lower central series to infinite ordinal numbers via transfinite recursion: for a limit ordinal λ, define Gλ = ∩ { Gα : α < λ}. If Gλ = 1 for some ordinal λ, then G is said to be a hypocentral group. For every ordinal λ, there is a group G such that Gλ = 1, but Gα ≠ 1 for all α < λ, . If ω is the first infinite ordinal, then Gω is the smallest normal subgroup of G such that the quotient is residually nilpotent, that is, such that every non-identity element has a non-identity homomorphic image in a nilpotent group .

Most of the mathematical criteria by which voting methods are compared were formulated for voters with ordinal preferences. Some methods, like approval voting, request information than can't be unambiguously inferred from a single set of ordinal preferences. The two-round system is such a method, because the voters are not forced to vote according to a single ordinal preference in both rounds. Since the two-round system requires more information from each voter than a single ordinal ballot provides, one can't fit the criteria that are formulated expressly for voters with ordinal preferences without making a generalization as to how the voters will behave.

The original versions suffered from the same problem as the joint-probability in that they treat the data as nominal and assume the ratings have no natural ordering; if the data actually have a rank (ordinal level of measurement), then that information in the measurements was not fully taken advantage of. Later extensions of the approach included versions that could handle "partial credit" and ordinal scales. These extensions converge with the family of intra-class correlations (ICCs), so there is a conceptually related way of estimating reliability for each level of measurement from nominal (kappa) to ordinal (ordinal kappa or ICC—stretching assumptions) to interval (ICC, or ordinal kappa—treating the interval scale as ordinal), and ratio (ICCs). There also are variants that can look at agreement by raters across a set of items (e.g.

The field of ordinal analysis was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε0. See Gentzen's consistency proof.

In thinking of the system of multitudinal quantity, we do not need to think about ordinal quantities, but we do need to attribute, to the objects we are thinking about, ordinal places in a series.

The ordinal utility concept was first introduced by Pareto in 1906.

This is a list of military corps arranged by ordinal number.

Gentzen's theorem spurred the development of ordinal analysis in proof theory.

Gentzen's proof is the first example of what is called proof-theoretical ordinal analysis. In ordinal analysis one gauges the strength of theories by measuring how large the (constructive) ordinals are that can be proven to be well-ordered, or equivalently for how large a (constructive) ordinal can transfinite induction be proven. A constructive ordinal is the order type of a recursive well-ordering of natural numbers. Laurence Kirby and Jeff Paris proved in 1982 that Goodstein's theorem cannot be proven in Peano arithmetic.

There are three usual operations on ordinals: addition, multiplication, and (ordinal) exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the operation or by using transfinite recursion. The Cantor normal form provides a standardized way of writing ordinals. It uniquely represents each ordinal as a finite sum of ordinal powers of ω.

One of his major contributions to psychometrics was the method for rotation of canonical components. Asserting that much of psychological data have only ordinal justification, Cliff also published various papers and a book on ordinal methods for research. On the one hand this included extensions to the established ordinal methods for correlating data (i.e. Kendall's tau, Spearman's rank correlation coefficient).

Tait (2005) gives a game-theoretic interpretation of Gentzen's method. Gentzen's consistency proof initiated the program of ordinal analysis in proof theory. In this program, formal theories of arithmetic or set theory are assigned ordinal numbers that measure the consistency strength of the theories. A theory will be unable to prove the consistency of another theory with a higher proof theoretic ordinal.

For dealing with infinite sets, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former gives the ordering of the set, while the latter gives its size. For finite sets, both ordinal and cardinal numbers are identified with the natural numbers. In the infinite case, many ordinal numbers correspond to the same cardinal number.

The cofinality of an ordinal α is the smallest ordinal δ which is the order type of a cofinal subset of α. The cofinality of a set of ordinals or any other well-ordered set is the cofinality of the order type of that set. Thus for a limit ordinal α, there exists a δ-indexed strictly increasing sequence with limit α. For example, the cofinality of ω² is ω, because the sequence ω·m (where m ranges over the natural numbers) tends to ω²; but, more generally, any countable limit ordinal has cofinality ω.

The smallest ordinal such that \varphi_\alpha(0) = \alpha is known as the Feferman–Schütte ordinal and generally written \Gamma_0. It can be described as the set of all ordinals that can be written as finite expressions, starting from zero, using only the Veblen hierarchy and addition. The Feferman–Schütte ordinal is important because, in a sense that is complicated to make precise, it is the smallest (infinite) ordinal that cannot be (“predicatively”) described using smaller ordinals. It measures the strength of such systems as “arithmetical transfinite recursion”.

The function f : Ord → Ord, f(α) = ωα is normal (see initial ordinal). Thus, there exists an ordinal θ such that θ = ωθ. In fact, the lemma shows that there is a closed, unbounded class of such θ.

For example, the cofinality of ω² is ω, because the sequence ω·m (where m ranges over the natural numbers) tends to ω²; but, more generally, any countable limit ordinal has cofinality ω. An uncountable limit ordinal may have either cofinality ω as does \omega_\omega or an uncountable cofinality. The cofinality of 0 is 0. And the cofinality of any successor ordinal is 1.

Nimber multiplication (nim-multiplication) is defined recursively by :. Except for the fact that nimbers form a proper class and not a set, the class of nimbers determines an algebraically closed field of characteristic 2. The nimber additive identity is the ordinal 0, and the nimber multiplicative identity is the ordinal 1. In keeping with the characteristic being 2, the nimber additive inverse of the ordinal is itself.

The use of ordinal data can be found in most areas of research where categorical data are generated. Settings where ordinal data are often collected include the social and behavioral sciences and governmental and business settings where measurements are collected from persons by observation, testing, or questionnaires. Some common contexts for the collection of ordinal data include survey research; and intelligence, aptitude, and personality testing.

For details on how the equation is estimated, see the article Ordinal regression.

For ordinal numbers (numbers indicating position) greater than ten the cardinal is used.

In computer data processing, ordinal ranking is also referred to as "row numbering".

In set theory, there is a more general notion of an enumeration than the characterization requiring the domain of the listing function to be an initial segment of the Natural numbers where the domain of the enumerating function can assume any ordinal. Under this definition, an enumeration of a set S is any surjection from an ordinal α onto S. The more restrictive version of enumeration mentioned before is the special case where α is a finite ordinal or the first limit ordinal ω. This more generalized version extends the aforementioned definition to encompass transfinite listings. Under this definition, the first uncountable ordinal \omega_1 can be enumerated by the identity function on \omega_1 so that these two notions do not coincide.

If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X merely means a function from α to X. This concept, a transfinite sequence or ordinal-indexed sequence, is a generalization of the concept of a sequence. An ordinary sequence corresponds to the case α = ω. If X is a topological space, we say that an α-indexed sequence of elements of X converges to a limit x when it converges as a net, in other words, when given any neighborhood U of x there is an ordinal β<α such that xι is in U for all ι≥β. Ordinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω1 (omega-one, the set of all countable ordinal numbers, and the smallest uncountable ordinal number), is a limit point of ω1+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω1-indexed sequence which maps any ordinal less than ω1 to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω1, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable.

More generally, one can call a subset of any ordinal \alpha cofinal in \alpha provided every ordinal less than \alpha is less than or equal to some ordinal in the set. The subset is said to be closed under \alpha provided it is closed for the order topology in \alpha, i.e. a limit of ordinals in the set is either in the set or equal to \alpha itself.

However, S may be bounded as subset of Rn with the lexicographical order, but not with respect to the Euclidean distance. A class of ordinal numbers is said to be unbounded, or cofinal, when given any ordinal, there is always some element of the class greater than it. Thus in this case "unbounded" does not mean unbounded by itself but unbounded as a subclass of the class of all ordinal numbers.

For every ordinal η, a cardinal κ is called η-extendible if for some ordinal λ there is a nontrivial elementary embedding j of Vκ+η into Vλ, where κ is the critical point of j, and as usual Vα denotes the αth level of the von Neumann hierarchy. A cardinal κ is called an extendible cardinal if it is η-extendible for every nonzero ordinal η (Kanamori 2003).

Gentzen's proof was published in 1943 and marked the beginning of ordinal proof theory.

In Chinese and Japanese, an ordinal number is prefixed by / ; for example, "first", "second".

He has also contributed to recursion theory (see admissible ordinal and Kripke–Platek set theory).

Any ordinal number can be made into a topological space by endowing it with the order topology (since, being well-ordered, an ordinal is in particular totally ordered): in the absence of indication to the contrary, it is always that order topology that is meant when an ordinal is thought of as a topological space. (Note that if we are willing to accept a proper class as a topological space, then the class of all ordinals is also a topological space for the order topology.) The set of limit points of an ordinal α is precisely the set of limit ordinals less than α. Successor ordinals (and zero) less than α are isolated points in α. In particular, the finite ordinals and ω are discrete topological spaces, and no ordinal beyond that is discrete.

Three well- orderings on the set of natural numbers with distinct order types (top to bottom): \omega, \omega+5, and \omega+\omega. Every well-ordered set is order- equivalent to exactly one ordinal number. The ordinal numbers are taken to be the canonical representatives of their classes, and so the order type of a well-ordered set is usually identified with the corresponding ordinal. For example, the order type of the natural numbers is .

Five judges scored each diver, giving two results. Each judge gave an ordinal placing for each diver in a group, with the five scores being summed to give a total ordinal points score. The judges also gave scores more closely resembling the modern scoring system.

In 1940 Wilhelm Ackermann published another consistency proof for Peano arithmetic, also using the ordinal ε0.

Correlation measures appropriate for two ordinal-scaled variables include Kendall's tau, gamma, rs, and dyx/dxy.

Ordinal linguistic personification normally co-occurs with other forms of synesthesia such as grapheme-color synesthesia.

If the categories are at least ordinal then a number of other indices may be computed.

Since the intersection of two closed unbounded classes is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may be empty, e.g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality. Rather than formulating these definitions for (proper) classes of ordinals, one can formulate them for sets of ordinals below a given ordinal \alpha: A subset of a limit ordinal \alpha is said to be unbounded (or cofinal) under \alpha provided any ordinal less than \alpha is less than some ordinal in the set.

It may not be obvious that it can be proven, without using AC, that there even exists a nonzero ordinal onto which there is no surjection from the reals (if there is such an ordinal, then there must be a least one because the ordinals are wellordered). However, suppose there were no such ordinal. Then to every ordinal α we could associate the set of all prewellorderings of the reals having length α. This would give an injection from the class of all ordinals into the set of all sets of orderings on the reals (which can to be seen to be a set via repeated application of the powerset axiom).

In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The name club is a contraction of "closed and unbounded".

Examples: (first grade (in elementary school)), (third edition), but . Furthermore, suffixes can be left out if the number obviously is an ordinal number, example: (3rd ed). Using a full stop as an ordinal indicator is considered archaic, but still occurs in military contexts. Example: (5th company).

The cofinality of an ordinal \alpha is the smallest ordinal \delta that is the order type of a cofinal subset of \alpha. Notice that a number of authors define cofinality or use it only for limit ordinals. The cofinality of a set of ordinals or any other well-ordered set is the cofinality of the order type of that set. Thus for a limit ordinal, there exists a \delta-indexed strictly increasing sequence with limit \alpha.

In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set which represents the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations.

The integrity of the von Neumann universe depends fundamentally on the integrity of the ordinal numbers, which act as the rank parameter in the construction, and the integrity of transfinite induction, by which both the ordinal numbers and the von Neumann universe are constructed. The integrity of the ordinal number construction may be said to rest upon von Neumann's 1923 and 1928 papers., . See also the English-language presentation of von Neumann's "general recursion theorem" by .

Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories is not known. These data exist on an ordinal scale, one of four levels of measurement described by S. S. Stevens in 1946. The ordinal scale is distinguished from the nominal scale by having a ranking. It also differs from interval and ratio scales by not having category widths that represent equal increments of the underlying attribute.

Formally, the definition is by transfinite induction: the \gamma-th element of the class is defined (provided it has already been defined for all \beta<\gamma), as the smallest element greater than the \beta- th element for all \beta<\gamma. This could be applied, for example, to the class of limit ordinals: the \gamma-th ordinal, which is either a limit or zero is \omega\cdot\gamma (see ordinal arithmetic for the definition of multiplication of ordinals). Similarly, one can consider additively indecomposable ordinals (meaning a nonzero ordinal that is not the sum of two strictly smaller ordinals): the \gamma-th additively indecomposable ordinal is indexed as \omega^\gamma. The technique of indexing classes of ordinals is often useful in the context of fixed points: for example, the \gamma-th ordinal \alpha such that \omega^\alpha = \alpha is written \varepsilon_\gamma.

The article provides links to lists of military divisions arranged by ordinal number, name, country or conflict.

Under certain conditions, an ordinal preference relation can be represented by a linear and continuous utility function.

The article provides links to lists of military corps arranged by ordinal number, name, country or conflict.

In 1943, another corps-level unit carrying the ordinal number 21 was created, the XXI Mountain Corps.

Ordinal regression turns up often in the social sciences, for example in the modeling of human levels of preference (on a scale from, say, 1–5 for "very poor" through "excellent"), as well as in information retrieval. In machine learning, ordinal regression may also be called ranking learning.

This "length" is called the order type of the set. Any ordinal is defined by the set of ordinals that precede it. In fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it. For example, the ordinal 42 is the order type of the ordinals less than it, that is, the ordinals from 0 (the smallest of all ordinals) to 41 (the immediate predecessor of 42), and it is generally identified as the set .

It may be clearer to apply Von Neumann cardinal assignment to finite cases and to use Scott's trick for sets which are infinite or do not admit well orderings. Note that cardinal and ordinal arithmetic agree for finite numbers. The α-th infinite initial ordinal is written \omega_\alpha, it is always a limit ordinal. Its cardinality is written \aleph_\alpha. For example, the cardinality of ω0 = ω is \aleph_0, which is also the cardinality of ω2 or ε0 (all are countable ordinals).

In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any \beta,\gamma<\alpha, we have \beta+\gamma<\alpha. Additively indecomposable ordinals are also called gamma numbers. The additively indecomposable ordinals are precisely those ordinals of the form \omega^\beta for some ordinal \beta. From the continuity of addition in its right argument, we get that if \beta < \alpha and α is additively indecomposable, then \beta + \alpha = \alpha.

The length of a branch is the ordinal that is order isomorphic to the branch. For each ordinal α, the α-th level of T is the set of all elements of T of height α. A tree is a κ-tree, for an ordinal number κ, if and only if it has height κ and every level has size less than the cardinality of κ. The width of a tree is the supremum of the cardinalities of its levels.

A simple normal function is given by f(α) = 1 + α (see ordinal arithmetic). But f(α) = α + 1 is not normal. If β is a fixed ordinal, then the functions f(α) = β + α, f(α) = β × α (for β ≥ 1), and f(α) = βα (for β ≥ 2) are all normal. More important examples of normal functions are given by the aleph numbers f(\alpha) = \aleph_\alpha which connect ordinal and cardinal numbers, and by the beth numbers f(\alpha) = \beth_\alpha.

For most Windows API functions only the names are preserved across different Windows releases; the ordinals are subject to change. Thus, one cannot reliably import Windows API functions by their ordinals. Importing functions by ordinal provides only slightly better performance than importing them by name: export tables of DLLs are ordered by name, so a binary search can be used to find a function. The index of the found name is then used to look up the ordinal in the Export Ordinal table.

The reason for this is that the rankings are ordinal scale numbers, and multiplication is not defined for ordinal numbers. The ordinal rankings only say that one ranking is better or worse than another, but not by how much. For instance, a ranking of "2" may not be twice as severe as a ranking of "1", or an "8" may not be twice as severe as a "4", but multiplication treats them as though they are. See Level of measurement for further discussion.

Stevens (1946) argued that, because the assumption of equal distance between categories does not hold for ordinal data, the use of means and standard deviations for description of ordinal distributions and of inferential statistics based on means and standard deviations was not appropriate. Instead, positional measures like the median and percentiles, in addition to descriptive statistics appropriate for nominal data (number of cases, mode, contingency correlation), should be used. Nonparametric methods have been proposed as the most appropriate procedures for inferential statistics involving ordinal data, especially those developed for the analysis of ranked measurements. However, use of parametric statistics for ordinal data may be permissible with certain caveats to take advantage of the greater range of available statistical procedures.

In Finnish orthography, when the numeral is followed by its head noun (which indicates the grammatical case of the ordinal), it is sufficient to write a period or full stop after the numeral: "In the competition, I finished in 2nd place". However, if the head noun is omitted, the ordinal indicator takes the form of a morphological suffix, which is attached to the numeral with a colon. In the nominative case, the suffix is for 1 and 2, and for larger numerals: "I came 2nd, and my brother came 3rd". This is derived from the endings of the spelled-out ordinal numbers: , , , , , , ... The system becomes rather complicated when the ordinal needs to be inflected, as the ordinal suffix is adjusted according to the case ending: (nominative case, which has no ending), (genitive case with ending ), (partitive case with ending ), (inessive case with ending ), (illative case with ending ), etc.

This definition extends the concept of indescribability to transfinite levels. A λ-shrewd cardinal is also μ-shrewd for any ordinal μ < λ. Shrewdness was developed by Michael Rathjen as part of his ordinal analysis of Π12-comprehension. It is essentially the nonrecursive analog to the stability property for admissible ordinals.

Here, "feminine" and "masculine" refers to grammatical gender. In Spanish, Portuguese, Galician and Italian, gender is usually distinguished by the suffixes -a and -o. These ordinal indicators are now distinct from the superior o and a characters. In the most of common available computer fonts today, ordinal indicators are not underlined.

An ordinal date is a calendar date typically consisting of a year and a day of the year ranging between 1 and 366 (starting on January 1), though year may sometimes be omitted. The two numbers can be formatted as YYYY-DDD to comply with the ISO 8601 ordinal date format.

In place of means and standard deviations, univariate statistics appropriate for ordinal data include the median, other percentiles (such as quartiles and deciles), and the quartile deviation. One-sample tests for ordinal data include the Kolmogorov-Smirnov one-sample test, the one-sample runs test, and the change-point test.

Another methodological limitations are the resulting ordinal, ipsatised scores that limit the type of useful analyses researchers can perform.

We now explain more systematically how the \psi function defines notations for ordinals up to the Bachmann–Howard ordinal.

Most typewriters for Spanish and other Romance languages had keys that could enter _o_ and _a_ directly, as a shorthand intended to be used primarily with ordinal numbers, such as 1. _o_ for first. In computing, early 8-bit character sets as code page 437 for the original IBM PC (circa 1981) also had these characters. In ISO-8859-1 Latin-1, and later in Unicode, they were assigned to and are known as U+00AA FEMININE ORDINAL INDICATOR (ª) and U+00BA MASCULINE ORDINAL INDICATOR (º).

The response categories represent an ordinal level of measurement. Ordinal level data, however, varies in terms of how closely it approximates interval level data. By using a numerical continuum as the response key instead of sentiments that reflect intensity of agreement, respondents may be able to quantify their responses in more equal units.

Numerals (or numbers) consist of two types: cardinal numerals and ordinal numerals. When occurring in noun phrases, cardinal and ordinal numerals occur in different syntactic positions with respect to the head noun. The article below only shows the native Vietnamese numerals, remember that Sino-Vietnamese numerals will be used in certain cases.

A class C of ordinals is said to be unbounded, or cofinal, when given any ordinal \alpha, there is a \beta in C such that \alpha < \beta (then the class must be a proper class, i.e., it cannot be a set). It is said to be closed when the limit of a sequence of ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function F is continuous in the sense that, for \delta a limit ordinal, F(\delta) (the \delta-th ordinal in the class) is the limit of all F(\gamma) for \gamma < \delta; this is also the same as being closed, in the topological sense, for the order topology (to avoid talking of topology on proper classes, one can demand that the intersection of the class with any given ordinal is closed for the order topology on that ordinal, this is again equivalent). Of particular importance are those classes of ordinals that are closed and unbounded, sometimes called clubs.

In written languages, an ordinal indicator is a character, or group of characters, following a numeral denoting that it is an ordinal number, rather than a cardinal number. In English orthography, this corresponds to the suffixes -st, -nd, -rd, -th in written ordinals (represented either on the line 1st, 2nd, 3rd, 4th or as superscript, 1st, 2nd, 3rd, 4th). Also commonly encountered are the superscript or superior (and often underlined) masculine ordinal indicator, , and feminine ordinal indicator, , originally from Romance, but via the cultural influence of Italian by the 18th century, widely used in the wider cultural sphere of Western Europe, as in 1º primo and 1ª prima "first, chief; prime quality". The practice of underlined (or doubly underlined) superscripted abbreviations was common in 19th-century writing (not limited to ordinal indicators in particular, and also extant in the numero sign ), and was also found in handwritten English until at least the late 19th century (e.g.

Then we can apply the axiom of union to that to get an ordinal greater than or equal to ω.

The selection methods can be split into three general types: fitness proportionate selection, ordinal based selection and threshold based selection.

In mathematics, Ψ0(Ωω) is a large countable ordinal that is used to measure the proof-theoretic strength of some mathematical systems. In particular, it is the proof theoretic ordinal of the subsystem \Pi_1^1-CA0 of second-order arithmetic; this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999).

Another approach is given by Rennie and Srebro, who, realizing that "even just evaluating the likelihood of a predictor is not straight-forward" in the ordered logit and ordered probit models, propose fitting ordinal regression models by adapting common loss functions from classification (such as the hinge loss and log loss) to the ordinal case.

Ordinal data analysis requires a different set of analyses than other qualitative variables. These methods incorporate the natural ordering of the variables in order to avoid loss of power. Computing the mean of a sample of ordinal data is discouraged; other measures of central tendency, including the median or mode, are generally more appropriate.

These sets are then taken to "be" cardinal numbers, by definition. In Zermelo-Fraenkel set theory with the axiom of choice, one way of assigning representatives to cardinal numbers is to associate each cardinal number with the least ordinal number of the same cardinality. These special ordinals are the ℵ numbers. But if the axiom of choice is not assumed, for some cardinal numbers it may not be possible to find such an ordinal number, and thus the cardinal numbers of those sets have no ordinal number as representatives.

If the index number is zero, then the NHI number is invalid and cannot be used. For the old format, the NHI Number contains a check digit. The algorithm for generating the digit is described below: Each alpha character is given a numeric representation equivalent to its ordinal position within the alphabet, starting at A through to Z. The letters I and O are omitted making the ordinal range 1 - 24. Each alpha character's numeric representation is multiplied by the inverse of its ordinal position within the NHI Number.

In mathematics, the nimbers, also called Grundy numbers, are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with nimber addition and nimber multiplication, which are distinct from ordinal addition and ordinal multiplication. Because of the Sprague–Grundy theorem which states that every impartial game is equivalent to a Nim heap of a certain size, nimbers arise in a much larger class of impartial games. They may also occur in partisan games like Domineering.

Recursive ordinals (or computable ordinals) are certain countable ordinals: loosely speaking those represented by a computable function. There are several equivalent definitions of this: the simplest is to say that a computable ordinal is the order-type of some recursive (i.e., computable) well-ordering of the natural numbers; so, essentially, an ordinal is recursive when we can present the set of smaller ordinals in such a way that a computer (Turing machine, say) can manipulate them (and, essentially, compare them). A different definition uses Kleene's system of ordinal notations.

An ordinal that is both admissible and a limit of admissibles, or equivalently such that \alpha is the \alpha-th admissible ordinal, is called recursively inaccessible. There exists a theory of large ordinals in this manner that is highly parallel to that of (small) large cardinals. For example, we can define recursively Mahlo ordinals: these are the \alpha such that every \alpha-recursive closed unbounded subset of \alpha contains an admissible ordinal (a recursive analog of the definition of a Mahlo cardinal). But note that we are still talking about possibly countable ordinals here.

The successor points and zero are the isolated points of the class of ordinal numbers, with respect to the order topology..

The rank among well-ordered sets is expressed by an ordinal number; for the natural numbers, this is denoted as (omega).

Thomas Norton (died 1513) was an English poet and alchemist best known for his 1477 alchemical poem, The Ordinal of Alchemy.

Cardinal methods (based on cardinal utility) and ordinal methods are two main categories of modern voting systems, along with plurality voting.

Cardinal and ordinal numbers must agree in gender (masculine or feminine; mixed groups are treated as masculine) with the noun they are describing. If there is no such noun (e.g. a telephone number or a house number in a street address), the feminine form is used. Ordinal numbers must also agree in number and definite status like other adjectives.

Ordinal-linguistic personification (OLP, or personification for short) is a form of synesthesia in which ordered sequences, such as ordinal numbers, days, months and letters are associated with personalities and/or genders (). Although this form of synesthesia was documented as early as the 1890s (; ) researchers have, until recently, paid little attention to this form (see History of synesthesia research).

This latter statement is proven by step 5. # But we have that no ordinal class is less than itself, including \Omega because of step 4 (\Omega is an ordinal class), i.e. \Omega less \Omega. We've deduced two contradictory propositions (\Omega < \Omega and \Omega less \Omega) from the sethood of \Omega and, therefore, disproved that \Omega is a set.

Furthermore, there is no injection from α into X, because if there were, then we would get the contradiction that α ∈ α. And finally, α is the least such ordinal with no injection into X. This is true because, since α is an ordinal, for any β < α, β ∈ α so there is an injection from β into X.

The essive ve- functions as a verbalizer, ordinal, relativizer, or indicator of inherent properties, depending on the context in which it appears.

Ordinal, multiplicative and multiple numerals declined and still decline in the same way as adjectives so they will not be discussed here.

The ordinal ε0 is still countable, as is any epsilon number whose index is countable (there exist uncountable ordinals, and uncountable epsilon numbers whose index is an uncountable ordinal). The smallest epsilon number ε0 appears in many induction proofs, because for many purposes, transfinite induction is only required up to ε0 (as in Gentzen's consistency proof and the proof of Goodstein's theorem). Its use by Gentzen to prove the consistency of Peano arithmetic, along with Gödel's second incompleteness theorem, show that Peano arithmetic cannot prove the well-foundedness of this ordering (it is in fact the least ordinal with this property, and as such, in proof-theoretic ordinal analysis, is used as a measure of the strength of the theory of Peano arithmetic). Many larger epsilon numbers can be defined using the Veblen function.

The version of the paradox above is anachronistic, because it presupposes the definition of the ordinals due to John von Neumann, under which each ordinal is the set of all preceding ordinals, which was not known at the time the paradox was framed by Burali-Forti. Here is an account with fewer presuppositions: suppose that we associate with each well-ordering an object called its order type in an unspecified way (the order types are the ordinal numbers). The order types (ordinal numbers) themselves are well- ordered in a natural way, and this well-ordering must have an order type \Omega. It is easily shown in naïve set theory (and remains true in ZFC but not in New Foundations) that the order type of all ordinal numbers less than a fixed \alpha is \alpha itself.

The sign is usually replaced with the abbreviations "n." or "nº", the latter using a masculine ordinal indicator, rather than a superscript 'O'.

Roman numerals above string instrument notes. Stackexchange (2017). The position can be indicated by ordinal numbers (e.g., "3rd") or a roman numeral (e.g.

Ordinal numbers (first, second, third, etc.) are formed by preceding the number with ic or inic.Andrews (2001): p. 452; Lockhart (2001): p. 50.

Matroid applications, 284–357, Encyclopedia Math. Appl., 40, Cambridge Univ. Press, Cambridge, 1992, Ordinal optimization has applications in the theory of queuing networks.

There are many different systems for ordinal notation introduced by various authors. It is often quite hard to convert between the different systems.

It is very tempting to read the informal description at the top of this article as claiming that the ∞-Borel sets are the smallest class of subsets of X containing all the open sets and closed under complementation and wellordered union. That is, one might wish to dispense with the ∞-Borel codes altogether and try a definition like this: : For each ordinal α define by transfinite recursion Bα as follows: :# B0 is the collection of all open subsets of X. :# For a given even ordinal α, Bα+1 is the union of Bα with the set of all complements of sets in Bα. :# For a given even ordinal α, Bα+2 is the set of all wellordered unions of sets in Bα+1. :# For a given limit ordinal λ, Bλ is the union of all Bα for α<λ : It follows from the Burali-Forti paradox that there must be some ordinal α such that Bβ equals Bα for every β>α. For this value of α, Bα is the collection of "∞-Borel sets".

Cantor then defines the addition and multiplication of the cardinal and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types. In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove Cantor's theorem: the cardinality of the power set of a set A is strictly larger than the cardinality of A. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined.

For example, if _α_ =(ω↦1) denotes the transfinite sequence with value 1 at ω and 0 everywhere else, then φ(ω↦1) is the smallest fixed point of all the functions ξ↦φ(ξ,0,…,0) with finitely many final zeroes (it is also the limit of the φ(1,0,…,0) with finitely many zeroes, the small Veblen ordinal). The smallest ordinal α such that α is greater than φ applied to any function with support in α (i.e., which cannot be reached “from below” using the Veblen function of transfinitely many variables) is sometimes known as the “large” Veblen ordinal.

Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable.

So the order type of all ordinal numbers less than \Omega is \Omega itself. But this means that \Omega, being the order type of a proper initial segment of the ordinals, is strictly less than the order type of all the ordinals, but the latter is \Omega itself by definition. This is a contradiction. If we use the von Neumann definition, under which each ordinal is identified as the set of all preceding ordinals, the paradox is unavoidable: the offending proposition that the order type of all ordinal numbers less than a fixed \alpha is \alpha itself must be true.

The notation is a finite string of symbols that intuitively stands for an ordinal number. By representing the ordinal in a finite way, Gentzen's proof does not presuppose strong axioms regarding ordinal numbers. He then proves by transfinite induction on these ordinals that no proof can conclude in a contradiction. The method used in this proof can also be used to prove a cut elimination result for Peano arithmetic in a stronger logic than first-order logic, but the consistency proof itself can be carried out in ordinary first-order logic using the axioms of primitive recursive arithmetic and a transfinite induction principle.

In this context using "zeroth" as an ordinal is not strictly correct, but a widespread habit in this profession. Other programming languages, such as Fortran or COBOL, have array subscripts starting with one, because they were meant as high-level programming languages, and as such they had to have a correspondence to the usual ordinal numbers which predate the invention of the zero by a long time. Pascal allows the range of an array to be of any ordinal type (including enumerated types). APL allows setting the index origin to 0 or 1 during runtime programatically.

In Italian, the circumflex is sometimes used in a similar manner to the ordinal indicator, most noticeably on tickets from Trenitalia, the primary operator of trains within Italy, and Rome's ATAC public transit system. On Trenitalia tickets, the travel class is often written as 1^ or 2^, meaning first class or second class respectively. This is due to the lack of the feminine ordinal indicator used in Italian in the (pre- Unicode) ISO Latin 1 character set (the masculine ordinal indicator is usually replaced by the degree sign when extended characters are not available or in less accurate typesetting).

In this case, the ordinals 0, 1, \omega, \omega_1, and \omega_2 are regular, whereas 2, 3, \omega_\omega, and ωω·2 are initial ordinals that are not regular. The cofinality of any ordinal α is a regular ordinal, i.e. the cofinality of the cofinality of α is the same as the cofinality of α. So the cofinality operation is idempotent.

Tetrapulmonata is a non-ranked supra-ordinal clade of arachnids. It is composed of the extant orders Thelyphonida (whip scorpions), Schizomida (short-tailed whip scorpions), Amblypygi (tail-less whip scorpions) and Araneae (spiders). It is the only supra-ordinal group of arachnids that is strongly supported in molecular phylogenetic studies. Two extinct orders are also placed in this clade, Haptopoda and Uraraneida.

Even native speakers sometimes find it difficult to exactly identify the ordinal suffix, as its borders with the word stem and the case ending may appear blurred. In such cases it may be preferable to write the ordinal word entirely with letters and particularly is rare even in the nominative case, as it is not significantly shorter than the full word .

324 & footnote (c): "This would appear more like a restitution of the old dignity than the creation of a new earldom"; Debrett's Peerage however gives the ordinal numbers as if a new earldom had been created. (Montague-Smith, P.W. (ed.), Debrett's Peerage, Baronetage, Knightage and Companionage, Kelly's Directories Ltd, Kingston- upon-Thames, 1968, p.353) and thus alternative ordinal numbers exist, given here.

If X is a compact Hausdorff space, then it coincides with its Stone–Čech compactification. Most other Stone–Čech compactifications lack concrete descriptions and are extremely unwieldy. Exceptions include: The Stone–Čech compactification of the first uncountable ordinal \omega_1, with the order topology, is the ordinal \omega_1 + 1. The Stone–Čech compactification of the deleted Tychonoff plank is the Tychonoff plank.

However, ordinal-indexed sequences are not powerful enough to replace nets (or filters) in general: for example, on the Tychonoff plank (the product space (\omega_1+1)\times(\omega+1)), the corner point (\omega_1,\omega) is a limit point (it is in the closure) of the open subset \omega_1\times\omega, but it is not the limit of an ordinal-indexed sequence.

324 & footnote (c): "This would appear more like a restitution of the old dignity than the creation of a new earldom"; Debrett's Peerage however gives the ordinal numbers as if a new earldom had been created. (Montague-Smith, P.W. (ed.), Debrett's Peerage, Baronetage, Knightage and Companionage, Kelly's Directories Ltd, Kingston-upon-Thames, 1968, p.353) and thus alternative ordinal numbers exist, given here.

But we can do this for systems far beyond Peano's axioms. For example, the proof-theoretic strength of Kripke–Platek set theory is the Bachmann–Howard ordinal, and, in fact, merely adding to Peano's axioms the axioms that state the well-ordering of all ordinals below the Bachmann–Howard ordinal is sufficient to obtain all arithmetical consequences of Kripke–Platek set theory.

The 1954 Theorems say, roughly, that every preference relation which is complete, transitive and continuous, can be represented by a continuous ordinal utility function.

In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to ask how much better it is or how good it is. All of the theory of consumer decision-making under conditions of certainty can be, and typically is, expressed in terms of ordinal utility. For example, suppose George tells us that "I prefer A to B and B to C". George's preferences can be represented by a function u such that: :u(A)=9, u(B)=8, u(C)=1 But critics of cardinal utility claim the only meaningful message of this function is the order u(A)>u(B)>u(C); the actual numbers are meaningless.

Ordinal numerals are formed by adding the suffix cəŋnə to the base number. For example, second is expressed as ənni-cəŋnə, meaning two- necessary suffix.

School 2 receives two first place scores and three second place scores. The ordinal score for School 2 is 8 (2+2+2+1+1).

Usually the ordinal (e.g. "50th") preceded it. Often the race was also advertised on the radio as the "Annual Memorial Day race," or similar variations.

ISO 2711 is an ISO standard describing formats for ordinal dates. It was issued in 1973, and was superseded by ISO 8601 in June 1988.

As with any machine- learning tool the Calculus of Concepts model implementation inputs can be either nominal or ordinal and depending on the particular case.

Cardinal Algebras studied algebras whose models include the arithmetic of cardinal numbers. Ordinal Algebras sets out an algebra for the additive theory of order types. Cardinal, but not ordinal, addition commutes. In 1941, Tarski published an important paper on binary relations, which began the work on relation algebra and its metamathematics that occupied Tarski and his students for much of the balance of his life.

The fundamental sequence for an ordinal with cofinality ω is a distinguished strictly increasing ω-sequence which has the ordinal as its limit. If one has fundamental sequences for α and all smaller limit ordinals, then one can create an explicit constructive bijection between ω and α, (i.e. one not using the axiom of choice). Here we will describe fundamental sequences for the Veblen hierarchy of ordinals.

Arabic numerals are kept as Arabic numerals: 635 fēnjī (, extension 635) ##According to 6.1.5.4, the dì () used in ordinal numerals is followed by a hyphen: dì-yī (, first), dì-356 (, 356th). The hyphen should not be used if the word in which dì () and the numeral appear does not refer to an ordinal number in the context. For example: Dìwǔ (, a Chinese compound surname).

In set theory, an honest leftmost branch of a tree T on ω × γ is a branch (maximal chain) ƒ ∈ [T] such that for each branch g ∈ [T], one has ∀ n ∈ ω : ƒ(n) ≤ g(n). Here, [T] denotes the set of branches of maximal length of T, ω is the ordinal (represented by the natural numbers N) and γ is some other ordinal.

In no case was the point-for-place result different for any round from the final scores, however, and ties in final scores led to an extra diver advancing to the semifinals as well as two bronze medals being awarded; the Official Report contains only the final scores and not the ordinal rankings from the judges.Official Report, p. 306. Indeed, it appears that "the ordinal scores are not recorded in any source" and "the individual judges' scores are not known, so it is not possible to reconstruct the ordinal placements."Sports-reference The competition was held over three rounds (first round, semifinals, and final).

The term "α-inaccessible cardinal" is ambiguous and different authors use inequivalent definitions. One definition is that a cardinal κ is called α-inaccessible, for α any ordinal, if κ is inaccessible and for every ordinal β < α, the set of β-inaccessibles less than κ is unbounded in κ (and thus of cardinality κ, since κ is regular). In this case the 0-inaccessible cardinals are the same as strongly inaccessible cardinals. Another possible definition is that a cardinal κ is called α-weakly inaccessible if κ is regular and for every ordinal β < α, the set of β-weakly inaccessibles less than κ is unbounded in κ.

Perianth position, whether hypogynous at the junction with pedicel, or epigynous at the fruit apex, or expanded to form wings, helps with familial and ordinal identification.

In mathematical logic and set theory, an ordinal notation is a partial function from the set of all finite sequences of symbols from a finite alphabet to a countable set of ordinals, and a Gödel numbering is a function from the set of well-formed formulae (a well-formed formula is a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each wff with a unique natural number, called its Gödel number. If a Gödel numbering is fixed, then the subset relation on the ordinals induces an ordering on well-formed formulae which in turn induces a well-ordering on the subset of natural numbers. A recursive ordinal notation must satisfy the following two additional properties: # the subset of natural numbers is a recursive set # the induced well-ordering on the subset of natural numbers is a recursive relation There are many such schemes of ordinal notations, including schemes by Wilhelm Ackermann, Heinz Bachmann, Wilfried Buchholz, Georg Cantor, Solomon Feferman, Gerhard Jäger, Isles, Pfeiffer, Wolfram Pohlers, Kurt Schütte, Gaisi Takeuti (called ordinal diagrams), Oswald Veblen.

In mathematics, infinity plus one has meaning for the hyperreals, and also as the number ω+1 (omega plus one) in the ordinal numbers and surreal numbers.

The cardinal and ordinal definitions are equivalent in the case of a convex consumption set with continuous preferences that are locally non-satiated in the first argument.

Voorhies, Barbara. (1992). Therefore, Ford's ordinal scales of measurement were not scientifically valid; "[He] simply does not know what the word 'measurement' denotes."Ford, James. A. (1954).

Divisions are commonly designated by combining an ordinal number and a type name (e.g.: "13th Infantry Division"). Nicknames are often assigned or adopted, although these often are not considered an official part of the unit's nomenclature, with divisions of the Italian Army being one of the exceptions. In some cases, divisional titles lack an ordinal number, often in the case of unique units or units serving as elite or special troops.

Further on, there will be ω3, then ω4, and so on, and ωω, then ωωω, then later ωωωω, and even later ε0 (epsilon nought) (to give a few examples of relatively small—countable—ordinals). This can be continued indefinitely (as every time one says "and so on" when enumerating ordinals, it defines a larger ordinal). The smallest uncountable ordinal is the set of all countable ordinals, expressed as ω1 or \Omega.

If α is any ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. This concept, a transfinite sequence (if α is infinite) or ordinal-indexed sequence, is a generalization of the concept of a sequence. An ordinary sequence corresponds to the case α = ω, while a finite α corresponds to a tuple (mathematics), a.k.a. string (computer science).

So F(0) is equal to 0 (the smallest ordinal of all). Now that F(0) is known, the definition applied to F(1) makes sense (it is the smallest ordinal not in the singleton set ), and so on (the and so on is exactly transfinite induction). It turns out that this example is not very exciting, since provably for all ordinals α, which can be shown, precisely, by transfinite induction.

It can also be applied to Ordinal data (ranked data): the MiniTab online documentation gives an example. However, this document notes: "When you have ordinal ratings, such as defect severity ratings on a scale of 1–5, Kendall's coefficients, which account for ordering, are usually more appropriate statistics to determine association than kappa alone." Keep in mind however, that Kendall rank coefficients are only appropriate for rank data.

However, existence of an -Erdős cardinal implies existence of zero sharp. If is the satisfaction relation for (using ordinal parameters), then existence of zero sharp is equivalent to there being an -Erdős ordinal with respect to . And this in turn, the zero sharp implies the falsity of axiom of constructibility, of Kurt Gödel. If κ is -Erdős, then it is -Erdős in every transitive model satisfying " is countable".

German grammar rules do not allow for leading zeros in dates at all, and there should always be a space after a dot. However, leading zeros were allowed according to machine writing standards if they helped aligning dates. The use of a dot as a separator matches the convention of pronouncing the day and the month as an ordinal number, because ordinal numbers are written in German followed by a dot.

There are several different measures for the degree of correlation in data, depending on the kind of data: principally whether the data is a measurement, ordinal, or categorical.

In ZFC, the order type of a well- ordering W is then defined as the unique von Neumann ordinal which is equinumerous with the field of W and membership on which is isomorphic to the strict well-ordering associated with W. (the equinumerousness condition distinguishes between well-orderings with fields of size 0 and 1, whose associated strict well-orderings are indistinguishable). In ZFC there cannot be a set of all ordinals. In fact, the von Neumann ordinals are an inconsistent totality in any set theory: it can be shown with modest set theoretical assumptions that every element of a von Neumann ordinal is a von Neumann ordinal and the von Neumann ordinals are strictly well-ordered by membership. It follows that the class of von Neumann ordinals would be a von Neumann ordinal if it were a set: but it would then be an element of itself, which contradicts the fact that membership is a strict well-ordering of the von Neumann ordinals.

Formally, if \kappa is a limit ordinal, then a set C\subseteq\kappa is closed in \kappa if and only if for every \alpha<\kappa, if \sup(C\cap \alpha)=\alpha e0, then \alpha\in C. Thus, if the limit of some sequence from C is less than \kappa, then the limit is also in C. If \kappa is a limit ordinal and C\subseteq\kappa then C is unbounded in \kappa if for any \alpha<\kappa, there is some \beta\in C such that \alpha<\beta. If a set is both closed and unbounded, then it is a club set. Closed proper classes are also of interest (every proper class of ordinals is unbounded in the class of all ordinals). For example, the set of all countable limit ordinals is a club set with respect to the first uncountable ordinal; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor unbounded.

There is a relation between computable ordinals and certain formal systems (containing arithmetic, that is, at least a reasonable fragment of Peano arithmetic). Certain computable ordinals are so large that while they can be given by a certain ordinal notation o, a given formal system might not be sufficiently powerful to show that o is, indeed, an ordinal notation: the system does not show transfinite induction for such large ordinals. For example, the usual first-order Peano axioms do not prove transfinite induction for (or beyond) ε0: while the ordinal ε0 can easily be arithmetically described (it is countable), the Peano axioms are not strong enough to show that it is indeed an ordinal; in fact, transfinite induction on ε0 proves the consistency of Peano's axioms (a theorem by Gentzen), so by Gödel's second incompleteness theorem, Peano's axioms cannot formalize that reasoning. (This is at the basis of the Kirby–Paris theorem on Goodstein sequences.) We say that ε0 measures the proof-theoretic strength of Peano's axioms.

The original definition of ordinal numbers, found for example in the Principia Mathematica, defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned in ZF and related systems of axiomatic set theory because these equivalence classes are too large to form a set. However, this definition still can be used in type theory and in Quine's axiomatic set theory New Foundations and related systems (where it affords a rather surprising alternative solution to the Burali-Forti paradox of the largest ordinal).

As discussed above, the Cantor Normal Form of ordinals below \varepsilon_0 can be expressed in an alphabet containing only the function symbols for addition, multiplication and exponentiation, as well as constant symbols for each natural number and for \omega. We can do away with the infinitely many numerals by using just the constant symbol 0 and the operation of successor, S (for example, the integer 4 may be expressed as S(S(S(S(0))))). This describes an ordinal notation: a system for naming ordinals over a finite alphabet. This particular system of ordinal notation is called the collection of arithmetical ordinal expressions, and can express all ordinals below \varepsilon_0, but cannot express \varepsilon_0.

Like Set, FinSet and FinOrd are topoi. As in Set, in FinSet the categorical product of two objects A and B is given by the cartesian product , the categorical sum is given by the disjoint union , and the exponential object BA is given by the set of all functions with domain A and codomain B. In FinOrd, the categorical product of two objects n and m is given by the ordinal product , the categorical sum is given by the ordinal sum , and the exponential object is given by the ordinal exponentiation nm. The subobject classifier in FinSet and FinOrd is the same as in Set. FinOrd is an example of a PRO.

Most definitions of ordinal collapsing functions found in the recent literature differ from the ones we have given in one technical but important way which makes them technically more convenient although intuitively less transparent. We now explain this. The following definition (by induction on \alpha) is completely equivalent to that of the function \psi above: :Let C(\alpha,\beta) be the set of ordinals generated starting from 0, 1, \omega, \Omega and all ordinals less than \beta by recursively applying the following functions: ordinal addition, multiplication and exponentiation, and the function \psi\upharpoonright_\alpha. Then \psi(\alpha) is defined as the smallest ordinal \rho such that C(\alpha,\rho) \cap \Omega = \rho.

Norton was first identified as the author of the 'Ordinal' in 1617 and has since become the widespread identification of the work's author. However, in 1932, two scholars, M. Nierenstein and P. F. Chapman, criticised this identification (which they named "the Maier-Ashmole hypothesis") under the grounds that, beyond the esoteric link drawn by Maier and Ashmole, very little contemporary evidence seemed to link Norton to the Ordinal. This criticism received little attention in its time and a 1957 article by historian J. Reidy roundly criticised the article, arguing conclusively in favour of the 'Maier-Ashmole hypothesis' by citing various contemporary pieces of evidence that imply that Norton was a significant alchemist and very likely wrote the Ordinal.

Other possible meanings of a "Julian date" of "36" include an astronomical Julian Day Number, or the year AD 36 in the Julian calendar, or a duration of 36 astronomical Julian years). This is why the terms "ordinal date" or "day-of-year" are preferred. In contexts where a "Julian date" means simply an ordinal date, calendars of a Gregorian year with formatting for ordinal dates are often called "Julian calendars", but this could also mean that the calendars are of years in the Julian calendar system. Historically, Julian dates were recorded relative to Greenwich Mean Time (GMT) (later, Ephemeris Time), but since 1997 the International Astronomical Union has recommended that Julian dates be specified in Terrestrial Time.

Mandinka -ri, Bambara -li process nouns), -ncè (ethnonymic, cf. Soninke -nke, Mandinka -nka), -anta (ordinal, cf. Soninke -ndi, Mandinka -njaŋ...), -anta (resultative participle, cf. Soninke -nte), -endi (causative, cf.

Ordinal numbers written in digits take a full stop (e.g. 3. sor '3rd line'). The full stop is retained even before the hyphen that connects suffixes (e.g. a 10.

Psychological research has shown that cardinal ratings (on a numerical or Likert scale, for instance) are more valid and convey more information than ordinal rankings in measuring human opinion.

But a certain exaggeration of emphasis may be pardoned in a writer seeking to attract the attention of an indifferent public. The Neoclassical Revolution, which would reshape economics, had been started. Jevons did not explicitly distinguish between the concepts of ordinal and cardinal utility. Cardinal utility allows the relative magnitude of utilities to be discussed, while ordinal utility only implies that goods can be compared and ranked according to which good provided the most utility.

Any ordinal number can be made into a topological space by endowing it with the order topology; this topology is discrete if and only if the ordinal is a countable cardinal, i.e. at most ω. A subset of ω + 1 is open in the order topology if and only if either it is cofinite or it does not contain ω as an element. See the Topology and ordinals section of the "Order topology" article.

In computer programming, an ordinal data type is a data type with the property that its values can be counted. That is, the values can be put in a one-to-one correspondence with the positive integers. For example, characters are ordinal because we can call 'A' the first character, 'B' the second, etc. The term is often used in programming for variables that can take one of a finite (often small) number of values.

A Kurepa tree can be "killed" by forcing the existence of a function whose value on any non-root node is an ordinal less than the rank of the node, such that whenever three nodes, one of which is a lower bound for the other two, are mapped to the same ordinal, then the three nodes are comparable. This can be done without collapsing ℵ1, and results in a tree with exactly ℵ1 branches.

Another example is the ordinals under the usual operations of ordinal arithmetic (here Clause 3 should be replaced with its symmetric form c · (a + b) = c · a + c · b. Strictly speaking, the class of all ordinals is not a set, so the above example should be more appropriately called a class near-semiring. We get a near-semiring in the standard sense if we restrict to those ordinals strictly less than some multiplicatively indecomposable ordinal.

All rating scales can be classified into one of these types: # Numeric Rating Scale (NRS) # Verbal Rating Scale (VRS) # Visual Analogue Scale (VAS) # Likert # Graphic rating scale # Descriptive graphic rating scale Some data are measured at the ordinal level. Numbers indicate the relative position of items, but not the magnitude of difference. Attitude and opinion scales are usually ordinal; one example is a Likert response scale: ; Statement: e.g. "I could not live without my computer".

In 1936, Gentzen published a proof that Peano Arithmetic is consistent. Gentzen's result shows that a consistency proof can be obtained in a system that is much weaker than set theory. Gentzen's proof proceeds by assigning to each proof in Peano arithmetic an ordinal number, based on the structure of the proof, with each of these ordinals less than ε0.Actually, the proof assigns a "notation" for an ordinal number to each proof.

Cardinal and Ordinal Numbers is a book on transfinite numbers, by Polish mathematician Wacław Sierpiński. It was published in 1958 by Państwowe Wydawnictwo Naukowe, as volume 34 of the series Monografie Matematyczne of the . Sierpiński wrote on the same topic earlier, in his 1928 book Leçons sur les nombres tranfinis, but his 1958 book on the topic was completely rewritten and significantly longer. A second edition of Cardinal and Ordinal Numbers was published in 1965.

Feferman introduced theta functions, described in as follows. The function for an ordinal α, θα is a function from ordinals to ordinals. Often θα(β) is written as θαβ. The set C(α,β) is defined by induction on α to be the set of ordinals that can be generated from 0, ω1, ω2, ..., ωω, together with the ordinals less than β by the operations of ordinal addition and the functions θξ for ξ<α.

Compare to expected value analysis, whose conclusion is of the form: "this strategy yields E(X)=n." Minimax thus can be used on ordinal data, and can be more transparent.

In mathematics, primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets or ordinals rather than natural numbers. They were introduced by .

The English translation is Cantor 1955. The first paper begins by defining set, subset, etc., in ways that would be largely acceptable now. The cardinal and ordinal arithmetic are reviewed.

The model cannot be consistently estimated using ordinary least squares; it is usually estimated using maximum likelihood. For details on how the equation is estimated, see the article Ordinal regression.

Spearman's coefficient is appropriate for both continuous and discrete ordinal variables.Scale types. Both Spearman's \rho and Kendall's \tau can be formulated as special cases of a more general correlation coefficient.

So ω can be identified with \aleph_0, except that the notation \aleph_0 is used when writing cardinals, and ω when writing ordinals (this is important since, for example, \aleph_0^2 = \aleph_0 whereas \omega^2 > \omega). Also, \omega_1 is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers: each such well-ordering defines a countable ordinal, and \omega_1 is the order type of that set), \omega_2 is the smallest ordinal whose cardinality is greater than \aleph_1, and so on, and \omega_\omega is the limit of the \omega_n for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the \omega_n).

Let S0 = S and let Sk+1 be the derived set of Sk. If there is a finite number n for which Sn is finite, then all the coefficients are zero. Later, Lebesgue proved that if there is a countably infinite ordinal α such that Sα is finite, then the coefficients of the series are all zero. Cantor's work on the uniqueness problem famously led him to invent transfinite ordinal numbers, which appeared as the subscripts α in Sα .

Items that compare equal receive the same ranking number, which is the mean of what they would have under ordinal rankings. Equivalently, the ranking number of 1 plus the number of items ranked above it plus half the number of items equal to it. This strategy has the property that the sum of the ranking numbers is the same as under ordinal ranking. For this reason, it is used in computing Borda counts and in statistical tests (see below).

According to Schultz, by 1931 the idea of ordinal utility was not yet embraced by American economists. The breakthrough occurred when a theory of ordinal utility was put together by John Hicks and Roy Allen in 1934. In fact pages 54–55 from this paper contain the first use ever of the term 'cardinal utility'. The first treatment of a class of utility functions preserved by affine transformations, though, was made in 1934 by Oskar Lange.

In German, Czech, Slovak, Slovenian, Serbian and some other languages, the role of the ordinal indicator is served by a simple dot, typographically identical to period or full stop. Writing out the endings for various cases, as sometimes happens in Czech and Slovak, is considered incorrect and uneducated. Should a period or full stop follow this dot, it is omitted. In Czech and Slovak, numerals with ordinal dot are mostly used only in tables, lists etc.

The competition was actually held from both 10 metre and 5 metre platforms. Divers performed a total of five dives: a standing dive and two running dives from the 10 metre platform, and a standing dive and a running dive from the 5 metre platform. Five judges scored each diver, giving two results. Each judge gave an ordinal placing for each diver in a group, with the five scores being summed to give a total ordinal points score.

Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets, the class of all ordinal numbers, and the class of all cardinal numbers. One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers. This method is used, for example, in the proof that there is no free complete lattice on three or more generators.

The idea that the collection of all ordinal numbers cannot logically exist seems paradoxical to many. This is related to Cesare Burali-Forti's "paradox" which states that there can be no greatest ordinal number. All of these problems can be traced back to the idea that, for every property that can be logically defined, there exists a set of all objects that have that property. However, as in Cantor's argument (above), this idea leads to difficulties.

In mathematical set theory, a set S is said to be ordinal definable if, informally, it can be defined in terms of a finite number of ordinals by a first-order formula. Ordinal definable sets were introduced by . A drawback to this informal definition is that requires quantification over all first-order formulas, which cannot be formalized in the language of set theory. However there is a different way of stating the definition that can be so formalized.

Ordinal numerals are formed by adding the thứ- ordinal prefix to cardinal numerals: thứ- + mười "ten" = thứ mười "tenth".Note that the affixal status of morphemes will be indicated with a hyphen in descriptions of the morphological structure of these words, but current Vietnamese orthographic practice does not use hyphens or write multisyllabic words without orthographic spaces. Other examples include: thứ nhất "first", thứ hai (or thứ nhì) "second", thứ ba "third", and thứ bốn (or thứ tư) "fourth".

In comparison to statistical classification and pattern recognition in a machine learning sense, two main distinguishing features of MCPs can be identified: # In MCPs the categories are defined in an ordinal way. This ordinal definition of the categories implicitly defines a preference structure. In contrast, machine learning is usually involved with nominal classification problems, where classes of observations are defined in a nominal way (i.e., collection of cases described by some common patterns), without any preferential implications.

Motion charts provide mechanisms for mapping ordinal, nominal and quantitative variables onto time, 2D coordinate axes, size, colors, glyphs and appearance characteristics, which facilitate the interactive display of multidimensional and temporal data.

The other paper gives, for the first time, a classification of the families in APG III which uses formal taxonomic ranks; previously only informal clade names were used above the ordinal level.

There are ten parts of speech, viz. Article, Substantive or Noun, Adjective, Numeral, Pronoun, Verb, Adverb, Preposition, Conjunction, and Interjection.", "NUMERALS. The numbers are divided into cardinal, ordinal, proportional, distributive, and collective.

Problems of ordinal optimization arise in many disciplines. Computer scientists study selection algorithms, which are simpler than sorting algorithms.Donald Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition.

One trillionth is the reciprocal or ordinal of a trillion, which means it has one of two definitions depending on whether the long scale or short scale numbering system is in use.

There are several different kinds of numeral words in Latin: the two most common are cardinal numerals and ordinal numerals. There are also several more rare numerals, e.g., distributive numerals and adverbial numerals.

Ordinal numbers (in reverse word order) are used for naming ruling members of a monarchy and the Popes. For example: Carol al II- lea, Papa Benedict al XVI-lea. See above for details.

It can be any number between n+1 and 2n-2. The smallest open case is n=4. The ordinal MMS condition can also be applied to asymmetric agents (agents with different entitlements).

This technique was proposed by L.Chen and F. Stentiford. A measurement of dissimilarity is made by combining the two aforementioned algorithms, Global temporal descriptors and Global ordinal measurement descriptors, in time and space.

In general, the counter words mentioned above are cardinal numbers, in that they indicate quantity. To transform a counter word into an ordinal number that denotes position in a sequence, 目 me is added to the end of the counter. Thus "one time" would be translated as 一回 ikkai, whereas "the first time" would be translated as 一回目 ikkaime. This rule is inconsistent, however, as counters without the me suffix are often used interchangeably with cardinal and ordinal meanings.

The ordinals less than \omega are finite. A finite sequence of finite ordinals always has a finite maximum, so \omega cannot be the limit of any sequence of type less than \omega whose elements are ordinals less than \omega, and is therefore a regular ordinal. \aleph_0 (aleph-null) is a regular cardinal because its initial ordinal, \omega, is regular. It can also be seen directly to be regular, as the cardinal sum of a finite number of finite cardinal numbers is itself finite.

Noun ellipsis (also N-ellipsis, N'-ellipsis, NP-ellipsis, NPE, ellipsis in the DP) occurs when the noun and potentially accompanying modifiers are omitted from a noun phrase.See Lobeck 2006 for an overview. Nominal ellipsis occurs with a limited set of determinatives in English (cardinal and ordinal numbers and possessive determiners), whereas it is much freer in other languages. The following examples illustrate nominal ellipsis with cardinal and ordinal numbers: ::Fred did three onerous tasks because Susan had done two onerous tasks.

If the dependent variable--the one whose value is determined to some extent by the other, independent variable-- is a categorical variable, such as the preferred brand of cereal, then probit or logit regression (or multinomial probit or multinomial logit) can be used. If both variables are ordinal, meaning they are ranked in a sequence as first, second, etc., then a rank correlation coefficient can be computed. If just the dependent variable is ordinal, ordered probit or ordered logit can be used.

There are two variants of the check digit algorithm to allow for the current NHI number format having a numeric check digit while the new format has an alphabetic check character. For the new format, each alphabetic character is given a numeric value equal to its ordinal position within a version of the alphabet that omits the letters I and O. The ordinal range is 1–24. This gives A=1 and Z=24, for example. Each numeric character is used with its face value 0–9 in the calculation. Each character’s equivalent numeric value is then multiplied by its reverse ordinal position within the NHI number. The first value is multiplied by 7, the second by 6, the third by 5, the fourth by 4, the fifth by 3 and the sixth by 2.

It exists without the assumption of the axiom of choice. Assume here for technical simplicity of proof that has no ordinal. Let denote multiplication in the group . For any there is an such that .

In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order preserving maps. It is used to define simplicial and cosimplicial objects.

They are declined like adjectives (paradigms pekný and cudzí). Note: Ordinal numerals are formed by adding adjective endings to the (slightly modified) cardinal numbers, for example: :5: päť – 5th: piaty, :20: dvadsať – 20th: dvadsiaty.

Numeric designations for Numbered Air Forces are written in full using ordinal words (e.g., Eighth Air Force), while cardinal numerals are used in abbreviations (e.g., 8 AF).AFH 33-337, The Tongue and Quill .

Moreover, the EF1 guarantee holds for any cardinal utilities consistent with the ordinal ranking, i.e., it is stochastic-dominance EF1 (sd-EF1). The algorithm uses as subroutines both the PS algorithm and the Birkhoff algorithm.

Essentially, an ordinal is intended to be defined as an isomorphism class of well-ordered sets: that is, as an equivalence class for the equivalence relation of "being order-isomorphic". There is a technical difficulty involved, however, in the fact that the equivalence class is too large to be a set in the usual Zermelo–Fraenkel (ZF) formalization of set theory. But this is not a serious difficulty. The ordinal can be said to be the order type of any set in the class.

The competition was actually held from both 10 metre and 5 metre platforms. Divers performed a standing plain dive and a running plain dive from the 10 metre platform, a running plain dive and a backward somersault from the 5 metre platform, and three dives of the competitor's choice from the 10 metre platform. Five judges scored each diver, giving two results. Each judge gave an ordinal placing for each diver in a group, with the five scores being summed to give a total ordinal points score.

Proving that the axiom of separation, axiom of replacement, and axiom of choice hold in requires (at least as shown above) the use of a reflection principle for . Here we describe such a principle. By induction on < , we can use ZF in to prove that for any ordinal , there is an ordinal > such that for any sentence (,...,) with ,..., in and containing fewer than symbols (counting a constant symbol for an element of as one symbol) we get that (,...,) holds in if and only if it holds in .

There are certain large cardinals that cannot exist in the constructible universe (L) of any model of set theory. Nevertheless, the constructible universe contains all the ordinal numbers that the original model of set theory contains. This "paradox" can be resolved by noting that the defining properties of some large cardinals are not absolute to submodels. One example of such a nonabsolute large cardinal axiom is for measurable cardinals; for an ordinal to be a measurable cardinal there must exist another set (the measure) satisfying certain properties.

In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy) is an ordinal-indexed family of rapidly increasing functions fα: N → N (where N is the set of natural numbers {0, 1, ...}, and α ranges up to some large countable ordinal). A primary example is the Wainer hierarchy, or Löb–Wainer hierarchy, which is an extension to all α < ε0. Such hierarchies provide a natural way to classify computable functions according to rate-of-growth and computational complexity.

Month-based ordinal dating ("Fourth Thursday in November", "Last Monday in May") will be obsolete. Two methods favored for perennializing the calendar are the introduction of so-called "blank days" and of a periodic "leap week".

Examples are SKU inventory codes and UPC bar codes. #Some data are measured at the ordinal level. Numbers indicate the relative position of items, but not the magnitude of difference. An example is a preference ranking.

Qualitative Measures Require Clearly Defined Rating Standards: Occasionally, quantitative measure is difficult or impossible to collect. An ordinal 'rating' standard is then used with clear definitions. Bottles and cans, and food matters trigger an automatic failure rating.

"A", "B", "AB" or "O", for blood type), ordinal (e.g. "large", "medium" or "small"), integer-valued (e.g. the number of occurrences of a particular word in an email) or real-valued (e.g. a measurement of blood pressure).

"A", "B", "AB" or "O", for blood type); ordinal (e.g. "large", "medium" or "small"); integer-valued (e.g. the number of occurrences of a particular word in an email); or real-valued (e.g. a measurement of blood pressure).

Centuries are named using ordinal numbers in reverse order: "14th century" is secolul al paisprezecelea (normally written secolul al XIV-lea). Cardinal numbers are often used although considered incorrect: secolul paisprezece. See above for details. Royal titles.

French uses the ordinal indicators (), in feminine (), (). French also uses the indicator for the variant ; in feminine this indicator becomes : . In plural, all these indicators take a S: (), (), (), (), (). These indicators use superscript formatting whenever it is available.

The Ophioparmaceae are a family of fungi in the Ascomycota, class Lecanoromycetes. Its relationship to other taxa in the Lecanoromycetes is not well understood, so it is considered to be incertae sedis with respect to ordinal placement.

The Fuscideaceae are a family of fungi in the Ascomycota, class Lecanoromycetes. Its relationship to other taxa in the Lecanoromycetes is not well understood, so it is considered to be incertae sedis with respect to ordinal placement.

Beon (), ho (), cha (), and hoe () are always used with Sino-Korean or Arabic ordinal numerals. For example, Yihoseon () is Line Number Two in a metropolitan subway system. Samsipchilbeongukdo () is highway number 37. They cannot be used interchangeably.

In statistics, ordered probit is a generalization of the widely used probit analysis to the case of more than two outcomes of an ordinal dependent variable (a dependent variable for which the potential values have a natural ordering, as in poor, fair, good, excellent). Similarly, the widely used logit method also has a counterpart ordered logit. Ordered probit, like ordered logit, is a particular method of ordinal regression. For example, in clinical research, the effect a drug may have on a patient may be modeled with ordered probit regression.

A solvable group is one whose derived series reaches the trivial subgroup at a finite stage. For an infinite group, the finite derived series may not stabilize, but the transfinite derived series always stabilizes. A group whose transfinite derived series reaches the trivial group is called a hypoabelian group, and every solvable group is a hypoabelian group. The first ordinal α such that G(α) = G(α+1) is called the (transfinite) derived length of the group G, and it has been shown that every ordinal is the derived length of some group .

In this case, it is assumed that each query-document pair in the training data has a numerical or ordinal score. Then the learning-to-rank problem can be approximated by a regression problem — given a single query-document pair, predict its score. A number of existing supervised machine learning algorithms can be readily used for this purpose. Ordinal regression and classification algorithms can also be used in pointwise approach when they are used to predict the score of a single query- document pair, and it takes a small, finite number of values.

It is when a Likert scale is symmetric and equidistant that it will behave more like an interval-level measurement. So while a Likert scale is indeed ordinal, if well presented it may nevertheless approximate an interval-level measurement. This can be beneficial since, if it was treated just as an ordinal scale, then some valuable information could be lost if the ‘distance’ between Likert items were not available for consideration. The important idea here is that the appropriate type of analysis is dependent on how the Likert scale has been presented.

The numero sign or numero symbol, №, (also represented as Nº, N _o_ , No./no.), is a typographic abbreviation of the word number(s) indicating ordinal numeration, especially in names and titles. For example, using the numero sign, the written long-form of the address is shortened to , yet both forms are spoken long. Typographically, the numero sign combines the uppercase Latin letter with a usually superscript lowercase letter , sometimes underlined, resembling the masculine ordinal indicator, as a single ligature. The ligature has a code point in Unicode as precomposed character, .

Arrow originally rejected cardinal utility as a meaningful tool for expressing social welfare,"Modern economic theory has insisted on the ordinal concept of utility; that is, only orderings can be observed, and therefore no measurement of utility independent of these orderings has any significance. In the field of consumer's demand theory the ordinalist position turned out to create no problems; cardinal utility had no explanatory power above and beyond ordinal. Leibniz' Principle of the identity of indiscernibles demanded then the excision of cardinal utility from our thought patterns." Arrow (1967), as quoted on p.

The term monotonic transformation (or monotone transformation) can also possibly cause some confusion because it refers to a transformation by a strictly increasing function. This is the case in economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform (see also monotone preferences).See the section on Cardinal Versus Ordinal Utility in . In this context, what we are calling a "monotonic transformation" is, more accurately, called a "positive monotonic transformation", in order to distinguish it from a “negative monotonic transformation,” which reverses the order of the numbers.

Using the facts above, we can define a (canonical) ordinal notation for every \gamma less than the Bachmann–Howard ordinal. We do this by induction on \gamma. If \gamma is less than \varepsilon_0, we use the iterated Cantor normal form of \gamma. Otherwise, there exists a largest \varepsilon-number \delta less or equal to \gamma (this is because the set of \varepsilon-numbers is closed): if \delta<\gamma then by induction we have defined a notation for \delta and the base \delta representation of \gamma gives one for \gamma, so we are finished.

Just as cardinal numbers form a (class) semiring, so do ordinal numbers form a near-ring, when the standard ordinal addition and multiplication are taken into account. However, the class of ordinals can be turned into a semiring by considering the so-called natural (or Hessenberg) operations instead. In category theory, a 2-rig is a category with functorial operations analogous to those of a rig. That the cardinal numbers form a rig can be categorified to say that the category of sets (or more generally, any topos) is a 2-rig.

The Church of England archbishops of Canterbury and York rejected the Pontiff's arguments in Saepius Officio in 1897.Temple, Frederick; Maclagan, William (1897). Answer of the Archbishops of England to the Apostolic Letter of Pope Leo XIII on English Ordinations. London: Longmans, Green, and Co. Retrieved 19 March 2018 This rebuttal was written to demonstrate the sufficiency of the form and intention used in the Anglican Ordinal: they archbishops wrote that in the preface to Ordinal the intention clearly is stated to continue the existing holy orders as received.

The competition was actually held from both 3 metre and 1 metre boards. Divers performed a running plain dive and a running forward somersault from the 1 metre board, a standing plain dive and a running plain dive from the 3 metre board, and three dives of the competitor's choice from the 3 metre board. Five judges scored each diver, giving two results. Each judge gave an ordinal placing for each diver in a group, with the five scores being summed to give a total ordinal points score.

In statistics, the Siegel–Tukey test, named after Sidney Siegel and John Tukey, is a non-parametric test which may be applied to data measured at least on an ordinal scale. It tests for differences in scale between two groups. The test is used to determine if one of two groups of data tends to have more widely dispersed values than the other. In other words, the test determines whether one of the two groups tends to move, sometimes to the right, sometimes to the left, but away from the center (of the ordinal scale).

In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relevance to proof theory still have computable ordinal notations. However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not (for reasons somewhat analogous to the unsolvability of the halting problem); various more-concrete ways of defining ordinals that definitely have notations are available.

First, if we take the powerset of any infinite set x, then that powerset will contain elements which are subsets of x of every finite cardinality (among other subsets of x). Proving the existence of those finite subsets may require either the axiom of separation or the axioms of pairing and union. Then we can apply the axiom of replacement to replace each element of that powerset of x by the initial ordinal number of the same cardinality (or zero, if there is no such ordinal). The result will be an infinite set of ordinals.

The slow-growing hierarchy grows much more slowly than the fast-growing hierarchy. Even gε0 is only equivalent to f3 and gα only attains the growth of fε0 (the first function that Peano arithmetic cannot prove total in the hierarchy) when α is the Bachmann–Howard ordinal. However, Girard proved that the slow-growing hierarchy eventually catches up with the fast-growing one. Specifically, that there exists an ordinal α such that for all integers n :gα(n) < fα(n) < gα(n + 1) where fα are the functions in the fast-growing hierarchy.

In particular, the rank of the empty set is zero, and every ordinal has a rank equal to itself. The sets in V are divided into the transfinite hierarchy Vα, called the cumulative hierarchy, based on their rank.

Going by different names in the sciences,Perreault J. Categories and Relators (International Classification, Vol 21, No 4, Frankfurt 1994) pp. 189ff where Perreault talks of ordinal relations (e.g. dimensions), determinative relations (e.g. activities) and subsumptive relations (e.g.

3,14 '3.14'). Numbers are usually written in Arabic numerals. Roman numerals are only used in some special traditional cases, only to express ordinal numbers (e.g. to express the numbering of monarchs, popes, districts of a city, congresses, etc.).

A hundredth is written as a decimal fraction as 0.01, and as a vulgar fraction as 1/100. “Hundredth” is also the ordinal number that follows “ninety-ninth” and precedes “hundred and first.” It is written as 100th.

DEF file which defines the ordinal position and name of each exported function. This allows the user to create a standard Windows DLL using Visual Basic (Version 6 or lower) which can be referenced through a "Declare" statement.

Categorical univariate data consist non-numerical observations that may be placed in categories. It includes labels or names used to identify an attribute of each element. Categorical univariate data usually use either nominal or ordinal scale of measurement.

The ordinal numbers are regular adjectives in Slovene. They have only definite forms, so the masculine nominative singular ends in -i. In writing, ordinals may be written in digit form followed by a period, as in German: 1., 2.

Independent nouns are created using na, which is added to the back of a noun to either indicate some kind of relationship or to change cardinal numbers to ordinal ones (see Numerals table at the bottom of the page).

Each stick corresponds to an ordinal of the form ω·m+n where m and n are natural numbers. Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, … After all natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on. (Exactly what addition means will be defined later on: just consider them as names.) After all of these come ω·2 (which is ω+ω), ω·2+1, ω·2+2, and so on, then ω·3, and then later on ω·4. Now the set of ordinals formed in this way (the ω·m+n, where m and n are natural numbers) must itself have an ordinal associated with it: and that is ω2.

The classes of successor ordinals and limit ordinals (of various cofinalities) as well as zero exhaust the entire class of ordinals, so these cases are often used in proofs by transfinite induction or definitions by transfinite recursion. Limit ordinals represent a sort of "turning point" in such procedures, in which one must use limiting operations such as taking the union over all preceding ordinals. In principle, one could do anything at limit ordinals, but taking the union is continuous in the order topology and this is usually desirable. If we use the Von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal (and this is a fitting observation, as cardinal derives from the Latin cardo meaning hinge or turning point): the proof of this fact is done by simply showing that every infinite successor ordinal is equinumerous to a limit ordinal via the Hotel Infinity argument.

While the concept of cardinality has fallen out of favor in neoclassical economics, the differences between cardinal utility and ordinal utility are minor for most applications. Cournot, Walras and Francis Ysidro Edgeworth are considered the precursors to modern mathematical economics.

"Methods for ordinal peer grading." Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, 2014. (2) algorithms to aggregate pairwise evaluation to more robustly estimate the global ranking of submissions,Chen, Xi, et al.

Phrase completion scales are a type of psychometric scale used in questionnaires. Developed in response to the problems associated with Likert scales, Phrase completions are concise, unidimensional measures that tap ordinal level data in a manner that approximates interval level data.

Cars of the 20x class would be smaller than cars of the 30x or 40x class, and cars of the same class would be usually replaced by the next ordinal number (e.g. the 201 was replaced with the Peugeot 202).

Tripus Aureus or The Golden Tripod is an alchemical book by Michael Maier published in 1618 by Lucas Jennis. It contains three alchemical texts: The "twelve keys" of Basil Valentine, Thomas Norton's Ordinal of Alchemy (1477), and The Testament of Cremer.

A user nicknamed "God God" created eight groups simply named after their ordinal numeral (hence the name "Nth Room"), and uploaded sexually exploitative pornography. Another user nicknamed "Watchman" advertised the link to these groups in another Telegram group named "Gotham room".

In mathematical optimization, ordinal optimization is the maximization of functions taking values in a partially ordered set ("poset").Dietrich, B. L.; Hoffman, A. J. On greedy algorithms, partially ordered sets, and submodular functions. IBM J. Res. Dev. 47 (2003), no.

Since the 1960s, the field of ordinal optimization has expanded in theory and in applications. In particular, antimatroids and the "max-plus algebra" have found application in network analysis and queuing theory, particularly in queuing networks and discrete-event systems.

Marina can run, jump, and boost (via jetpack) in the eight cardinal and ordinal directions. She can also slide, hover, and roll. The game has five worlds with roughly twelve levels apiece. Some levels are action-only while others include puzzles.

Ivy I-Ming Liu is a Taiwanese and New Zealander statistician specializing in categorical and ordinal data. She works as an associate professor and as head of the School of Mathematics and Statistics at Victoria University of Wellington in New Zealand.

In set theory, AD+ is an extension, proposed by W. Hugh Woodin, to the axiom of determinacy. The axiom, which is to be understood in the context of ZF plus DCR (the axiom of dependent choice for real numbers), states two things: # Every set of reals is ∞-Borel. # For any ordinal λ less than Θ, any subset A of ωω, and any continuous function π:λω→ωω, the preimage π−1[A] is determined. (Here λω is to be given the product topology, starting with the discrete topology on λ.) The second clause by itself is referred to as ordinal determinacy.

For example, if we write the names in Cyrillic and consider the Cyrillic ordering of letters, we might get a different result of evaluating "Smith < Johnson" than if we write the names in the standard Latin alphabet; and if we write the names in Chinese characters, we cannot meaningfully evaluate "Smith < Johnson" at all, because no consistent ordering is defined for such characters. However, if we do consider the names as written, e.g., in the Latin alphabet, and define an ordering corresponding to standard alphabetical order, then we have effectively converted them into ordinal variables defined on an ordinal scale.

On the other hand, we still have \tilde\psi(\Omega) = \zeta_0 (because \Omega \in C(\alpha,\rho) for all \rho so the extra condition does not come in play). Note in particular that \tilde\psi, unlike \psi, is not monotonic, nor is it continuous. Despite these changes, the \tilde\psi function also defines a system of ordinal notations up to the Bachmann–Howard ordinal: the notations, and the conditions for canonicalness, are slightly different (for example, \psi(\Omega+1+\alpha) = \tilde\psi(\tilde\psi(\Omega)+\alpha) for all \alpha less than the common value \psi(\Omega2) = \tilde\psi(\Omega+1)).

In statistics, a rank correlation is any of several statistics that measure an ordinal association—the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment of the ordering labels "first", "second", "third", etc. to different observations of a particular variable. A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess the significance of the relation between them. For example, two common nonparametric methods of significance that use rank correlation are the Mann–Whitney U test and the Wilcoxon signed-rank test.

An ordinal scale arises from the ordering or ranking objects, so that A is greater than B, B is greater than C, and so on. Many psychological experiments yield numbers of this sort; for example, a participant might be able to rank odors such that A is more pleasant than B, and B is more pleasant than C, but these rankings ("1, 2, 3 ...") would not tell by how much each odor differed from another. Some statistics can be computed from ordinal measures – for example, median, percentile, and order correlation – but others, such as standard deviation, cannot properly be used.

The bull took note of the fact that in 1662 the form introduced in the Edwardine ordinal of 1552 had added to it the words: "for the office and work of a priest". But it observed that this shows that the Anglicans themselves perceived that the first form was defective and inadequate. Rome felt that even if this addition could give the form its due signification, it was introduced too late. A century had already elapsed since the adoption of the Edwardine ordinal and as the hierarchy had become extinct there remained no power of ordaining.

Just as in the case of nimber addition, there is a means of computing the nimber product of finite ordinals. This is determined by the rules that # The nimber product of a Fermat 2-power (numbers of the form ) with a smaller number is equal to their ordinary product; # The nimber square of a Fermat 2-power is equal to as evaluated under the ordinary multiplication of natural numbers. The smallest algebraically closed field of nimbers is the set of nimbers less than the ordinal , where is the smallest infinite ordinal. It follows that as a nimber, is transcendental over the field.

More generally, Γα enumerates the ordinals that cannot be obtained from smaller ordinals using addition and the Veblen functions. It is, of course, possible to describe ordinals beyond the Feferman–Schütte ordinal. One could continue to seek fixed points in more and more complicated manner: enumerate the fixed points of \alpha\mapsto\Gamma_\alpha, then enumerate the fixed points of that, and so on, and then look for the first ordinal α such that α is obtained in α steps of this process, and continue diagonalizing in this ad hoc manner. This leads to the definition of the “small” and “large” Veblen ordinals.

However, only a rather large cardinal number can be both and thus weakly inaccessible. An ordinal is a weakly inaccessible cardinal if and only if it is a regular ordinal and it is a limit of regular ordinals. (Zero, one, and \aleph_0 are regular ordinals, but not limits of regular ordinals.) A cardinal which is weakly inaccessible and also a strong limit cardinal is strongly inaccessible. The assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a Grothendieck universe, the two ideas being intimately connected.

Other authors use the definition that for any ordinal α, a cardinal κ is α-hyper-inaccessible if and only if κ is hyper-inaccessible and for every ordinal β < α, the set of β-hyper-inaccessibles less than κ is unbounded in κ. Hyper-hyper-inaccessible cardinals and so on can be defined in similar ways, and as usual this term is ambiguous. Using "weakly inaccessible" instead of "inaccessible", similar definitions can be made for "weakly α-inaccessible", "weakly hyper-inaccessible", and "weakly α-hyper- inaccessible". Mahlo cardinals are inaccessible, hyper-inaccessible, hyper- hyper-inaccessible, ... and so on.

He further showed that the first α this holds for is the ordinal of the theory ID<ω of arbitrary finite iterations of an inductive definition. However, for the assignment of fundamental sequences found in the first match up occurs at the level ε0. For Buchholz style tree ordinals it could be shown that the first match up even occurs at \omega^2. Extensions of the result proved to considerably larger ordinals show that there are very few ordinals below the ordinal of transfinitely iterated \Pi^1_1-comprehension where the slow- and fast-growing hierarchy match up.

In lieu of testing differences in means with t-tests, differences in distributions of ordinal data from two independent samples can be tested with Mann-Whitney, runs, Smirnov, and signed-ranks tests. Test for two related or matched samples include the sign test and the Wilcoxon signed ranks test. Analysis of variance with ranks and the Jonckheere test for ordered alternatives can be conducted with ordinal data in place of independent samples ANOVA. Tests for more than two related samples include the Friedman two-way analysis of variance by ranks and the Page test for ordered alternatives.

For example, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limit ordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the terminology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of \varepsilon_\cdot ordinals, or the class of cardinals, are all closed unbounded; the set of regular cardinals, however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded. A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closed unbounded classes are stationary, and stationary classes are unbounded, but there are stationary classes that are not closed and stationary classes that have no closed unbounded subclass (such as the class of all limit ordinals with countable cofinality).

In ordinal ranking, all items receive distinct ordinal numbers, including items that compare equal. The assignment of distinct ordinal numbers to items that compare equal can be done at random, or arbitrarily, but it is generally preferable to use a system that is arbitrary but consistent, as this gives stable results if the ranking is done multiple times. An example of an arbitrary but consistent system would be to incorporate other attributes into the ranking order (such as alphabetical ordering of the competitor's name) to ensure that no two items exactly match. With this strategy, if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first") and D gets ranking number 4 ("fourth"), and either B gets ranking number 2 ("second") and C gets ranking number 3 ("third") or C gets ranking number 2 ("second") and B gets ranking number 3 ("third").

After five introductory chapters on naive set theory and set-theoretic notation, and a sixth chapter on the axiom of choice, the book has four chapters on cardinal numbers, their arithmetic, and series and products of cardinal numbers, comprising approximately 50 pages. Following this, four longer chapters (totalling roughly 180 pages) cover orderings of sets, order types, well-orders, ordinal numbers, ordinal arithmetic, and the Burali-Forti paradox according to which the collection of all ordinal numbers cannot be a set. Three final chapters concern aleph numbers and the continuum hypothesis, statements equivalent to the axiom of choice, and consequences of the axiom of choice. The second edition makes only minor changes to the first except for adding footnotes concerning two later developments in the area: the proof by Paul Cohen of the independence of the continuum hypothesis, and the construction by Robert M. Solovay of the Solovay model in which all sets of real numbers are Lebesgue measurable.

Transfinite numbers: Numbers that are greater than any natural number. Ordinal numbers: Finite and infinite numbers used to describe the order type of well-ordered sets. Cardinal numbers: Finite and infinite numbers used to describe the cardinalities of sets. Infinitesimals: Nilpotent numbers.

Then there is a -set which is a choice set for R , that is: # . # . A proof runs as follows: suppose for contradiction is a minimal counterexample, and fix , , and a good universal set for the -subsets of . Easily, must be a limit ordinal.

J. Zool. (London) A211: 747-770 At about the same time in 1987 Goryachev elevated the superfamily to ordinal status Seguenziiformes in the superorder Littorinimorpha, based on the taenioglossal radula.Goryachev, 1987. Ob'em i polozhenie semeistva Seguenziidae (Mollusca, Gastropoda, Seguenziidae) v klasse bryukhonogikh mollyuskov.

Schröder also made original contributions to algebra, set theory, lattice theory,"The Algebra of Logic Tradition". Stanford Encyclopedia of Philosophy. ordered sets and ordinal numbers. Along with Georg Cantor, he codiscovered the Cantor–Bernstein–Schröder theorem, although Schröder's proof (1898) is flawed.

"Exponential polynomials" in 0 and ω gives a system of ordinal notation for ordinals less than epsilon zero. There are many equivalent ways to write these; instead of exponential polynomials, one can use rooted trees, or nested parentheses, or the system described above.

Think of a rule as a "local-model" of the solution space. Rules can be represented in many different ways to handle different data types (e.g. binary, discrete-valued, ordinal, continuous- valued). Given binary data LCS traditionally applies a ternary rule representation (i.e.

In computability theory, computational complexity theory and proof theory, the slow-growing hierarchy is an ordinal-indexed family of slowly increasing functions gα: N → N (where N is the set of natural numbers, {0, 1, ...}). It contrasts with the fast-growing hierarchy.

Apart from the Caucasian Albanian palimpsests kept at Mt. Sinai, the most famous samples of Caucasian Albanian inscriptions were found in 1949 during excavations in Mingachevir region, Azerbaijan. Among the known Caucasian Albanian words are zow (I), own (and) and avel-om (much, ordinal form).

Numbers in Hlai language, including cardinal numbers, ordinal numbers, and numbers of approximation, usually act as subjects, predicate, or objects in a sentence. When numbers are used with classifiers, together they become a phrase that can be an attribute to modify the noun phrase.

In 1883, Cantor divided the infinite into the transfinite and the absolute.; English translation: Ewald 1996, pp. 916-917. The transfinite is increasable in magnitude, while the absolute is unincreasable. For example, an ordinal α is transfinite because it can be increased to α + 1\.

However, this cannot form the basis of a universal ordinal notation due to such self-referential representations as ε0 = ωε0. The so-called "natural" arithmetical operations retain commutativity at the expense of continuity. Interpreted as nimbers, ordinals are also subject to nimber arithmetic operations.

If there is no such noun (e.g. telephone numbers), the feminine form is used. For ordinal numbers greater than ten the cardinal is used and numbers above the value 20 have no gender. Jewish Town Hall building in Prague, with Hebrew numerals in counterclockwise order.

In the mathematical field of category theory, FinSet is the category whose objects are all finite sets and whose morphisms are all functions between them. FinOrd is the category whose objects are all finite ordinal numbers and whose morphisms are all functions between them.

The Durbin test is based on the following assumptions: #The b blocks are mutually independent. That means the results within one block do not affect the results within other blocks. #The data can be meaningfully ranked (i.e., the data have at least an ordinal scale).

The District Courts of Appeal originally consisted of three appellate districts, headquartered in San Francisco, Los Angeles, and Sacramento, with three justices each. These first nine justices were appointed by the Governor. Each district was assigned an ordinal number (i.e., first, second, and third).

Joannicius III (, ), (c. 1700 – 1793) was Archbishop of Peć and Serbian Patriarch from 1739 to 1746 and Archbishop of Constantinople and Ecumenical Patriarch from 1761 to 1763. The ordinal number of his title is III both for his office as Serbian Patriarch and of Constantinople.

Born in 1966 or 1967, Jordan grew up in London in a residence near Battersea Bridge. When she was 16, Jordan departed from school after receiving a low score on the GCE Ordinal Level. She studied at the secretarial school Lucie Clayton Charm Academy.

Classification methods have also been developed for ordinal data. The data are divided into different categories such that each observations are similar to each other. Dispersion is measured and minimized in each group to maximize classification results. The dispersion function is used in information theory.

However, if the data is numerical in nature (ordinal or interval/ratio) then the mode, median, or mean can all be used to describe the data. Using more than one of these measures provides a more accurate descriptive summary of central tendency for the univariate.

Macedonian numerals are words that are used in the Macedonian language for expressing quantity. The Macedonian numerals have three grammatical genders (masculine, feminine and neutral) and they can have articles. There are several types of numerals: cardinal numerals, ordinal numerals, collective numerals and multiplicative numerals.

An initial natural number is given, and players alternate choosing positive divisors of the initial number, but may not choose 1 or a multiple of a previously chosen divisor. This game models n-dimensional Chomp, where the initial natural number has n prime factors and the dimensions of the Chomp board are given by the exponents of the primes in its prime factorization. Ordinal Chomp is played on an infinite board with some of its dimensions ordinal numbers: for example a 2 × (ω + 4) bar. A move is to pick any block and remove all blocks with both indices greater than or equal the corresponding indices of the chosen block.

In March 1550, a new ordinal was published that was based on Martin Bucer's own treatise on the form of ordination. While Bucer had provided for only one service for all three orders of clergy, the English ordinal was more conservative and had separate services for deacons, priests and bishops. During his consecration as bishop of Gloucester, John Hooper objected to the mention of "all saints and the holy Evangelist" in the Oath of Supremacy and to the requirement that he wear a black chimere over a white rochet. Hooper was excused from invoking the saints in his oath, but he would ultimately be convinced to wear the offensive consecration garb.

It then follows by transfinite induction that there is one and only one function satisfying the recursion formula up to and including α. Here is an example of definition by transfinite recursion on the ordinals (more will be given later): define function F by letting F(α) be the smallest ordinal not in the set , that is, the set consisting of all F(β) for . This definition assumes the F(β) known in the very process of defining F; this apparent vicious circle is exactly what definition by transfinite recursion permits. In fact, F(0) makes sense since there is no ordinal , and the set is empty.

Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B and C each get ranking number 2.5 (average of "joint second/third") and D gets ranking number 4 ("fourth"). Here is an example: Suppose you have the data set 1.0, 1.0, 2.0, 3.0, 3.0, 4.0, 5.0, 5.0, 5.0. The ordinal ranks are 1, 2, 3, 4, 5, 6, 7, 8, 9. For v = 1.0, the fractional rank is the average of the ordinal ranks: (1 + 2) / 2 = 1.5. In a similar manner, for v = 5.0, the fractional rank is (7 + 8 + 9) / 3 = 8.0.

In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by . The Erdős cardinal is defined to be the least cardinal such that for every function there is a set of order type that is homogeneous for (if such a cardinal exists). In the notation of the partition calculus, the Erdős cardinal is the smallest cardinal such that : Existence of zero sharp implies that the constructible universe satisfies "for every countable ordinal , there is an -Erdős cardinal". In fact, for every indiscernible satisfies "for every ordinal , there is an -Erdős cardinal in (the Levy collapse to make countable)".

It is well known that the most subtle and delicate stage in a variant of the aggregated indices method is the stage of weights estimation because of usual shortage of information about exact values of weight-coefficients. As a rule, we have only non-numerical (ordinal) information, which can be represented by a system of equalities and inequalities for weights, and/or non-exact (interval) information, which can be represented by a system of inequalities, which determine only intervals for the weight-coefficients possible values. Usually ordinal and/or interval information is incomplete (i.e., this information is not enough for one-valued estimation of all weight-coefficients).

The pope went on to state that the Anglican ordinal had included what he felt were the errors of the English Reformation. It could not be used to confer valid orders, nor could it later be purged of this original defect, chiefly because he felt the words used in it had a meaning entirely different from what would be required to confer the sacrament. The pope felt that not only was the proper form for the sacrament lacking in the Anglican ordinal, but the intention was also lacking. He concluded by explaining how carefully and how prudently this matter has been examined by the Holy See.

John Scory and Miles Coverdale, the other two consecrators, were consecrated with the English Ordinal of 1550 on the same day in 1551 by Cranmer, Hodgkins and Ridley who were consecrated with the Latin Rite in 1532, 1537 and 1547 respectively.Project Canterbury, Supplementary Appendix A, Notes on the Consecration of Archbishop Parker, by Rev. Henry Barker, 2000; and the Register of the Diocese of Rochester on Ridley This ordinal was considered defective in form and intention. All four of Parker's consecrators were consecrated by bishops who themselves had been consecrated with the Roman Pontifical in the Church of England which at the time was in schism from Rome.

Records that absolutely must be managed by a record locking process are those which are processor shared. In TPF, most record accesses are done by using record type and ordinal. So if you had defined a record type in the TPF system of 'FRED' and gave it 100 records or ordinals, then in a processor shared scheme, record type 'FRED' ordinal '5' would resolve to exactly the same file address on DASD — clearly necessitating the use of a record locking mechanism. All processor shared records on a TPF system will be accessed via exactly the same file address which will resolve to exactly the same location.

Counting in the everyday sense typically starts from one, so it assigns to each object the size of the initial segment with that object as last element. Note that these numbers are one more than the formal ordinal numbers according to the isomorphic order, because these are equal to the number of earlier objects (which corresponds to counting from zero). Thus for finite n, the expression "n-th element" of a well-ordered set requires context to know whether this counts from zero or one. In a notation "β-th element" where β can also be an infinite ordinal, it will typically count from zero.

Different indices give different values of variation, and may be used for different purposes: several are used and critiqued in the sociology literature especially. If one wishes to simply make ordinal comparisons between samples (is one sample more or less varied than another), the choice of IQV is relatively less important, as they will often give the same ordering. Where the data is ordinal a method that may be of use in comparing samples is ORDANOVA. In some cases it is useful to not standardize an index to run from 0 to 1, regardless of number of categories or samples , but one generally so standardizes it.

Journal Geology 92: 583–597. , using a computer simulation on hypothetical data sets, and by Rubel and Pak Rubel, M. and Pak, D.N. (1984) Theory of stratigraphic correlation by means of ordinal scales. Computers Geosciences 10:43–57.in terms of the formal logic and stochastic theory.

In order-theoretic mathematics, the deviation of a poset is an ordinal number measuring the complexity of a partially ordered set. The deviation of a poset is used to define the Krull dimension of a module over a ring as the deviation of its poset of submodules.

Soviet and Russian staffed drifting ice stations are research stations built on the ice of the high latitudes of the Arctic Ocean. They are important contributors to exploration of the Arctic. The stations are named North Pole (NP; , ), followed by an ordinal number: North Pole-1, etc.

However, there exist examples of sequentially compact, first-countable spaces which are not compact (these are necessarily non-metric spaces). One such space is the ordinal space [0,ω1). Every first-countable space is compactly generated. Every subspace of a first-countable space is first-countable.

The Vezdaeaceae are a family of fungi in the Ascomycota, class Lecanoromycetes. Its relationship to other taxa in the Lecanoromycetes is not well understood, so it is considered to be incertae sedis with respect to ordinal placement. The family is monotypic, and contains the single genus Vezdaea.

The Elixiaceae are a family of fungi in the Ascomycota, class Lecanoromycetes. Its relationship to other taxa in the Lecanoromycetes is not well understood, so it is considered to be incertae sedis with respect to ordinal placement. The family is monotypic, and contains the single genus Elixia.

One motivation for the ZFC axioms is the cumulative hierarchy of sets introduced by John von Neumann., section 2. In this viewpoint, the universe of set theory is built up in stages, with one stage for each ordinal number. At stage 0 there are no sets yet.

Certainly this book formulates an ordinal hierarchy of values, which later appeared in Maslow's contribution to motivation theory. Aristotle's Nicomachean Ethics, particularly book V.v, has been called the most economically provocative analytic writing in ancient Greece.Lowry (2003), p. 20. Therein, Aristotle discusses justice in distribution and exchange.

Using the example: (pürev), (Khoyor myanga doloon) (ony) (Naiman) () (Арван). Ordinal Mongolian is a colloquial term used to express the day of the month instead of cardinal Mongolian. It is rarely used in formal writing. Using the example: (pürev), (Khoyor myanga doloon) (ony) (Naiman) () (Arvan dakhi).

Ordinal Mongolian is more often used when the month is understood from the context, i.e.: 10 дахь for the 10th. Weeks are most often identified by the last day of the week, either the Friday in business (e.g., "") or the Sunday in most other use (e.g.

The evaluation of decision alternatives is carried out by utility functions, which are represented in the form of decision rules. All attributes (function arguments and outcomes) are assumed to be discrete. Additionally, they can be preferentially ordered, so that a higher ordinal value represents a better preference.

The total element score and the program components score are added to give the total score for a competition segment (TSS). A skater's final placement is determined by the total of their scores in all segments of a competition. No ordinal rankings are used to determine the final results.

This typical setup can be modified in various ways. For example, instead of being a subset of X, each move might consist of a pair (I, p) where I \subset X and p \in x. Alternatively, the sequence of moves might have length some ordinal number other than ω1.

Generally speaking, treatment levels may be finite or infinite as well as ordinal or cardinal, which leads to a large collection of possible treatment effects to be studied in applications.Cattaneo, M. D. (2010): Multi-valued Treatment Effects. Encyclopedia of Research Design, ed. By N. J. Salkind, Sage Publications.

Although methodologically akin to principal components analysis, the MAP technique has been shown to perform quite well in determining the number of factors to retain in multiple simulation studies.Garrido, L. E., & Abad, F. J., & Ponsoda, V. (2012). A new look at Horn's parallel analysis with ordinal variables. Psychological Methods.

The choice of the ordinal collapsing function given as example below imitates greatly the system introduced by BuchholzBuchholz, 1986 (Ann. Pure Appl. Logic) but is limited to collapsing one cardinal for clarity of exposition. More on the relation between this example and Buchholz's system will be said below.

The two divers who scored the smallest number of points in each group of the first round plus the four best-scoring non-qualified divers of all groups advanced to the final. Ordinal placings were used to rank divers within the group, but were not used to determine qualification.

The two divers who scored the smallest number of points in each group of the first round plus the two best scoring non-qualified divers of all groups advanced to the final. Ordinal placings were used to rank divers within the group, but were not used to determine qualification.

The divers who scored the smallest number of points in each group of the first round plus the four best scoring non-qualified divers of all groups advanced to the final. Ordinal placings were used to rank divers within the group, but were not used to determine qualification.

Different people may have different preferences. Under certain conditions, a person's preferences can be represented by a numeric function. The article ordinal utility describes some properties of such functions and some ways by which they can be calculated. Another consideration that might complicate the decision problem is uncertainty.

Let Zi = Yi - Xi for i = 1, ... , n. # The differences Zi are assumed to be independent. # Each Zi comes from the same continuous population. # The values Xi and Yi represent are ordered (at least the ordinal scale), so the comparisons "greater than", "less than", and "equal to" are meaningful.

The two divers who scored the smallest number of points in each group of the first round plus the two best scoring non-qualified divers of all groups advanced to the final. Ordinal placings were used to rank divers within the group, but were not used to determine qualification.

Developed by Sherman and Marian McClellan in 1969, the oscillator is computed using the Exponential Moving Average (EMA) of the daily ordinal difference of advancing issues (stocks which gained in value) from declining issues (stocks which fell in value) over 39 trading day and 19 trading day periods.

Annual Progress in Child Psychiatry and Child Development 1997. Retrieved on 2008-03-18. A child's ordinal position in their family has also been shown to affect intelligence. A number of studies have indicated that as birth order increases IQ decreases with first borns having especially superior intelligence.

The proof starts by proving by contradiction that Ord, the class of all ordinals, is a proper class. Assume that Ord is a set. Since it is transitive set that is well-ordered by ∈, it is an ordinal. So Ord ∈ Ord, which contradicts Ord being well-ordered by ∈.

In statistics, the ordered logit model (also ordered logistic regression or proportional odds model) is an ordinal regression model—that is, a regression model for ordinal dependent variables—first considered by Peter McCullagh. For example, if one question on a survey is to be answered by a choice among "poor", "fair", "good", and "excellent", and the purpose of the analysis is to see how well that response can be predicted by the responses to other questions, some of which may be quantitative, then ordered logistic regression may be used. It can be thought of as an extension of the logistic regression model that applies to dichotomous dependent variables, allowing for more than two (ordered) response categories.

A more general class of epsilon numbers has been identified by John Horton Conway and Donald Knuth in the surreal number system, consisting of all surreals that are fixed points of the base ω exponential map x → ωx. defined gamma numbers (see additively indecomposable ordinal) to be numbers γ>0 such that α+γ=γ whenever α<γ, and delta numbers (see additively indecomposable ordinal#Multiplicatively indecomposable) to be numbers δ>1 such that αδ=δ whenever 0<α<δ, and epsilon numbers to be numbers ε>2 such that αε=ε whenever 1<α<ε. His gamma numbers are those of the form ωβ, and his delta numbers are those of the form ωωβ.

Scholars rate voting methods using mathematically-derived voting method criteria, which describe desirable features of a method. No ranked-preference method can meet all of the criteria, because some of them are mutually exclusive, as shown by statements such as Arrow's impossibility theorem and the Gibbard–Satterthwaite theorem. Many of the mathematical criteria by which voting methods are compared were formulated for voters with ordinal preferences. If voters vote according to the same ordinal preferences in both rounds, criteria can be applied to two-round systems of runoffs, and in that case, each of the criteria failed by IRV is also failed by the two-round system as they relate to automatic elimination of trailing candidates.

He placed fifth in the short program and first in the free skate, placing first overall. He was the first male skater since Terry Kubicka to win back-to-back Novice and Junior Men's titles in the United States. The win on the junior level was unusual in that Lysacek moved from third to first overall while sitting backstage, because he won through a tiebreak in the 6.0 ordinal system. Lysacek was tied with Parker Pennington in second place ordinals and had one more first place ordinal, giving him the win in the free skate in the Total Ordinals of Majority tiebreaker, which pushed him ahead in overall factored placements, allowing him to win the title overall.

Let A and B be countable abelian p-groups such that for every ordinal σ their Ulm factors are isomorphic, Uσ(A) ≅ Uσ(B) and the p-divisible parts of A and B are isomorphic, U∞(A) ≅ U∞(B). Then A and B are isomorphic. There is a complement to this theorem, first stated by Leo Zippin (1935) and proved in Kurosh (1960), which addresses the existence of an abelian p-group with given Ulm factors. : Let τ be an ordinal and {Aσ} be a family of countable abelian p-groups indexed by the ordinals σ < τ such that the p-heights of elements of each Aσ are finite and, except possibly for the last one, are unbounded.

Stephen Cole Kleene has a system of notations, called Kleene's O, which includes ordinal notations but it is not as well behaved as the other systems described here. Usually one proceeds by defining several functions from ordinals to ordinals and representing each such function by a symbol. In many systems, such as Veblen's well known system, the functions are normal functions, that is, they are strictly increasing and continuous in at least one of their arguments, and increasing in other arguments. Another desirable property for such functions is that the value of the function is greater than each of its arguments, so that an ordinal is always being described in terms of smaller ordinals.

Ordinal-linguistic personification (OLP, or personification for short) is a form of synesthesia in which ordered sequences, such as ordinal numbers, week-day names, months and alphabetical letters are associated with personalities or genders (). For example, the number 2 might be a young boy with a short temper, or the letter G might be a busy mother with a kind face. Although this form of synesthesia was documented as early as the 1890s (; ) researchers have, until recently, paid little attention to this form (see History of synesthesia research). This form of synesthesia was named as OLP in the contemporary literature by Julia Simner and colleagues although it is now also widely recognised by the term "sequence-personality" synesthesia.

The ordinal number given to the early Courtenay Earls of Devon depends on whether the earldom is deemed a new creation by the letters patent granted 22 February 1334/5 or whether it is deemed a restitution of the old dignity of the de Redvers family. Authorities differ in their opinions,Watson, in Cokayne, The Complete Peerage, new edition, IV, p.324 & footnote (c): "This would appear more like a restitution of the old dignity than the creation of a new earldom"; Debrett's Peerage however gives the ordinal numbers as if a new earldom had been created. (Montague-Smith, P.W. (ed.), Debrett's Peerage, Baronetage, Knightage and Companionage, Kelly's Directories Ltd, Kingston-upon-Thames, 1968, p.

Philip (29 August 1116 - 13 October 1131) was a king of France from 1129 to 1131, co-ruling with his father, Louis VI. As he predeceased his father and never reigned as sole king, he is not known by an ordinal or included in the traditional lists of French monarchs.

This order is typically induced by giving a numerical or ordinal score or a binary judgment (e.g. "relevant" or "not relevant") for each item. The ranking model purposes to rank, i.e. producing a permutation of items in new, unseen lists in a similar way to rankings in the training data.

Chen Chung Chang (Chinese: 张晨钟) was a mathematician who worked in model theory. He obtained his PhD from Berkeley in 1955 on "Cardinal and Ordinal Factorization of Relation Types" under Alfred Tarski. He wrote the standard text on model theory. Chang's conjecture and Chang's model are named after him.

He also proved the ordinal partition theorem (expressed in the arrow notation for Ramsey theory) ωω→(ωω,3)2, originally a problem of Erdős and Hajnal. He also introduced MV-algebras as models for Łukasiewicz logic. Chang was a professor at the mathematics department of the University of California, Los Angeles.

Revealed preference theory addresses the problem of how to observe ordinal preference relations in the real world. The challenge of revealed preference theory lies in part in determining what goods bundles were foregone, on the basis of them being less liked, when individuals are observed choosing particular bundles of goods.

Buchholz's psi-functions are a hierarchy of single-argument ordinal functions \psi_ u(\alpha) introduced by German mathematician Wilfried Buchholz in 1986. These functions are a simplified version of the \theta-functions, but nevertheless have the same strength as those. Later on this approach was extended by Jaiger and Schütte.

So instead of saying, for example, "three days", Hopi would say the equivalent of "on the third day", using ordinal numbers. Whorf argues that the Hopi do not consider the process of time passing to produce another new day, but merely as bringing back the daylight aspect of the world.

Often, categorical and ordinal data are grouped together; likewise for integer-valued and real-valued data. Furthermore, many algorithms work only in terms of categorical data and require that real-valued or integer-valued data be discretized into groups (e.g., less than 5, between 5 and 10, or greater than 10).

ORD: ordinal numeral INTR: interrogative MED: medial demonstrative SE: sentence ender CE: canonical ending REP : reportive NPST: nonpast Jeju is typologically similar to Korean, both being head-final agglutinative languages. However, the two languages show significant differences in the verbal paradigm, such as Jeju's use of a dedicated conditional suffix.

In the British Army and Royal Marines, the rank above second lieutenant is simply lieutenant (pronounced lef-tenant), with no ordinal attached. Before 1871, when the whole British Army switched to using the current rank of "lieutenant", the Royal Artillery, Royal Engineers and fusilier regiments used "first lieutenant" and "second lieutenant".

This type of construction is known as the "absolute accusative" (cf. absolute ablative in Latin grammar). Adverbs can be formed from adjectives, ordinal numerals: ' "frequently, a lot, often", ' "rarely", ' "firstly" or from nouns: ' "usually", ' "very". The second method to form adverbs is to use a preposition and a noun, e.g.

A nominal group only has members and non- members. That is, nothing more can be said about the members of the group other than they are part of the group. Nominal categories cannot be numerically organized or ranked. The members of a nominal group cannot be placed in ordinal (sequential) or ratio form.

The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs. Cardinality is studied for its own sake as part of set theory.

Many of the older streets are named for ordinal numbers (First Street through to Twentyfifth Street). There is no obvious pattern to the layout of streets or the order in which they are named. Some streets on the eastern side align to the contours where the land is too steep to build.

New Toronto Plant. In 1890, new streets for the Town of New Toronto were laid out in several series, essentially without names by simply using ordinal numbers (First, Second, Third, etc.). When the streets were laid out along Lake Shore Road (now Lake Shore Blvd. West), they had a single new starting point.

The Brigantiaeaceae are a family of fungi in the Ascomycota, class Lecanoromycetes. Its relationship to other taxa in the Lecanoromycetes is not well understood, so it is considered to be incertae sedis with respect to ordinal placement. Species in this family are lichenized with green algae, and are usually found growing on bark.

On a compact Hausdorff space the support of a non-zero measure is always non-empty, but may have measure 0. An example of this is given by adding the first uncountable ordinal Ω to the previous example: the support of the measure is the single point Ω, which has measure 0.

101 (one hundred [and] one) is the natural number following 100 and preceding 102. It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one". As an ordinal number, 101st (one hundred [and] first), rather than 101th, is the correct form.

Instead, a hyphen optionally replaces the missing letters: D-ro or Dro for Doktoro (Dr). With ordinal numerals, the adjectival a and accusative n may be superscripted: 13a or 13a (13th). The abbreviation k is used without a period for kaj (and); the ampersand (&) is not found. Roman numerals are also avoided.

By deed of 10 January 1348,Note: the deed records the date as 10 January, 23 Edward III. Regnal years are ordinal, so Edward III's first regnal year (i.e., 1 Edward III) ran from his coronation on 1 February 1327. 10 January, coming before 1 February, puts the calendar year at 1348.

This means that criteria and preference information can be uncertain, inaccurate or partially missing. Incomplete information is represented in SMAA using suitable probability distributions. The method is based on stochastic simulation by drawing random values for criteria measurements and weights from their corresponding distributions. SMAA can handle mixed cardinal and ordinal information.

For example, the set of 3-tuples of elements from a 2-element set has cardinality . In cardinal arithmetic, κ0 is always 1 (even if κ is an infinite cardinal or zero). Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process involving transfinite induction.

Line 744 in Thomas Norton's The Ordinal of Alchemy by John Rediry. The Early English Text Society no. 272. The stone was frequently praised and referred to in such terms. It needs to be noted that philosophorum does not mean "of the philosopher" or "the philosopher's" in the sense of a single philosopher.

The following is a timeline of the Tenrikyo religion, highlighting significant events since the birth of Tenrikyo's foundress Miki Nakayama. Specific dates are provided in parentheses; the lunar calendar is indicated with ordinal numbers (e.g. 18th day of 4th month) while the Gregorian calendar is indicated with name and number (e.g. August 15).

There have been several World Camps held by the Girl Guides and Girl Scouts, first held in 1924. Organized by the World Association of Girl Guides and Girl Scouts, unlike World Scout Jamborees, World Camps are not named with an ordinal number, nor is there an attempt to hold them at regular intervals.

For two agents, a simpler algorithm exists. 3\. Suppose the agents have ordinal rankings on items, without indifferences. Then, the problem of deciding whether a necessarily-proportional allocation exists can be solved in polynomial time. It is not known whether the same is true when the agents are allowed to express indifferences.

While all these notions are incompatible with Zermelo–Fraenkel set theory (ZFC), their \Pi^V_2 consequences do not appear to be false. There is no known inconsistency with ZFC in asserting that, for example: For every ordinal λ, there is a transitive model of ZF + Berkeley cardinal that is closed under λ sequences.

In mathematics, ordinal logic is a logic associated with an ordinal number by recursively adding elements to a sequence of previous logics.Solomon Feferman, Turing in the Land of O(z) in "The universal Turing machine: a half-century survey" by Rolf Herken 1995 page 111Concise Routledge encyclopedia of philosophy 2000 page 647 The concept was introduced in 1938 by Alan Turing in his PhD dissertation at Princeton in view of Gödel's incompleteness theorems.Alan Turing, Systems of Logic Based on Ordinals Proceedings London Mathematical Society Volumes 2–45, Issue 1, pp. 161–228. While Gödel showed that every system of logic suffers from some form of incompleteness, Turing focused on a method so that from a given system of logic a more complete system may be constructed.

33 by and so focused his theorem on preference rankings, but later stated that a cardinal score system with three or four classes "is probably the best". Arrow's framework assumes that individual and social preferences are "orderings" (i.e., satisfy completeness and transitivity) on the set of alternatives. This means that if the preferences are represented by a utility function, its value is an ordinal utility in the sense that it is meaningful so far as the greater value indicates the better alternative. For instance, having ordinal utilities of 4, 3, 2, 1 for alternatives a, b, c, d, respectively, is the same as having 1000, 100.01, 100, 0, which in turn is the same as having 99, 98, 1, .997.

Grade 5: full active range of motion & Normal muscle resistance Grade 4: full active range of motion & Reduced muscle resistance Grade 3: full active range of motion & No muscle resistance Grade 2: Reduced active range of motion & No muscle resistance Grade 1: No active range of motion & Palpable muscle contraction only Grade 0: No active range of motion & No palpable muscle contraction Manual muscle testing, however, has a number of limitations. One limitation is that the MRC scale is an ordinal scale with disproportional distances between grades. Another limitation of the MRC scale is that the scoring depends on the judgment of the examiner. Finally, with the 6-point ordinal MRC scale, it is difficult to identify relatively small but clinically relevant changes in muscle strength.

With Theodore Slaman, Groszek showed that (if they exist at all) non-constructible real numbers must be widespread, in the sense that every perfect set contains one of them, and they asked analogous questions of the non-computable real numbers. With Slaman, she has also shown that the existence of a maximally independent set of Turing degrees, of cardinality less than the cardinality of the continuum, is independent of ZFC. In the theory of ordinal definable sets, an unordered pair of sets is said to be a Groszek–Laver pair if the pair is ordinal definable but neither of its two elements is; this concept is named for Groszek and Richard Laver, who observed the existence of such pairs in certain models of set theory.

In the United Kingdom, the suffixes "Snr." and "Jnr." are rare, and not usually considered part of a person's name as such. Ordinal suffixes such as "III" are generally reserved for monarchs; however, the General Register Office has stated that, whereas it would normally reject a string of symbols or letters that "has no intrinsic sense of being a name" when registering a child, a suffix such as "III" would be accepted. Those who inherit a title of nobility do not use ordinal suffixes, but are distinguished from any ancestors with the same name by their position in the order of succession; for example Arthur Wellesley, 2nd Duke of Wellington is thus distinguished from his father, Arthur Wellesley, 1st Duke of Wellington.

Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values, and permit any order-preserving transformation. Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case with longitude and temperature measurements in degree Celsius or degree Fahrenheit), and permit any linear transformation. Ratio measurements have both a meaningful zero value and the distances between different measurements defined, and permit any rescaling transformation. Because variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, sometimes they are grouped together as categorical variables, whereas ratio and interval measurements are grouped together as quantitative variables, which can be either discrete or continuous, due to their numerical nature.

In zoology, the Linnaean orders were used more consistently. That is, the orders in the zoology part of the Systema Naturae refer to natural groups. Some of his ordinal names are still in use (e.g. Lepidoptera for the order of moths and butterflies, or Diptera for the order of flies, mosquitoes, midges, and gnats).

The Hebrew language has names for common numbers that range from zero to one million. Letters of the Hebrew alphabet are used to represent numbers in a few traditional contexts, for example in calendars. In other situations Arabic numerals are used. Cardinal and ordinal numbers must agree in gender with the noun they are describing.

The anti-tank battalion now included assault guns, tank destroyers, and towed anti-tank guns. Generally, the mechanization of these divisions increased compared to their previous organization. Since the Heer and the SS used their own ordinal systems, there were duplicate numbers (i.e. there was both a 9th Panzerdivision and a 9th SS-Panzerdivision).

The classification of the Phasmatodea is complex and the relationships between its members are poorly understood. Furthermore, there is much confusion over the ordinal name. Phasmida is preferred by many authors, though it is incorrectly formed; Phasmatodea is correctly formed, and is widely accepted. The order is divided into two, or sometimes three, suborders.

By repeating the process a sequence L1, L2, … of logics is obtained, each more complete than the previous one. A logic L can then be constructed in which the provable theorems are the totality of theorems provable with the help of the L1, L2, … etc. Thus Turing showed how one can associate a logic with any constructive ordinal.

Writing of ordinal numerals is similar to most European languages. The Czech language uses a decimal comma instead of a decimal point. When writing a long number, spaces between every three digits, including those in decimal places, may be used for better orientation in handwritten texts. The number 1,234,567.89101 may be written as 1234567,89101 or 1 234 567,891 01.

Signature of George Washington on the United States Constitution with superior letters, reading Go. Washington—Presidt. and deputy from Virginia. In English, superior letters are reserved for use with ordinal numerals, though this use is not mandatory and not always preferred: 1st, 2nd, 3rd, etc. Previously, in English-speaking countries, abbreviations of given names were used for recordkeeping.

OpenMx consists of an R library of functions and optimizers supporting the rapid and flexible implementation and estimation of SEM models. Models can be estimated based on either raw data (with FIML modelling) or on correlation or covariance matrices. Models can handle mixtures of continuous and ordinal data. The current version is OpenMx 2, and is available on CRAN.

In statistics, polychoric correlation is a technique for estimating the correlation between two theorised normally distributed continuous latent variables, from two observed ordinal variables. Tetrachoric correlation is a special case of the polychoric correlation applicable when both observed variables are dichotomous. These names derive from the polychoric and tetrachoric series which are used for estimation of these correlations.

In mathematics, the Veblen functions are a hierarchy of normal functions (continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If φ0 is any normal function, then for any non-zero ordinal α, φα is the function enumerating the common fixed points of φβ for β<α. These functions are all normal.

Charles VIIArticle Karl in Nordisk familjebok or Carl (Swedish: Karl Sverkersson; c. 1130 – 12 April 1167) was ruler of Götaland, and then King of Sweden from c. 1161 to 1167, when he was assassinated. He is the first historically known king of Sweden by the name of Charles, but use of the ordinal VII is widespread.

The duplicator's task is to always pick an element that is "similar" to the one the spoiler chose. The duplicator wins if and only if there exists an isomorphism between the eventual substructures chosen in the two different structures. The game lasts for a fixed number of steps (\gamma) (an ordinal, but usually a finite number or \omega).

WINMIRA 2001 is a program for analyses with the Rasch model for dichotomous and polytomous ordinal responses, with the latent class analysis, and with the Mixture Distribution Rasch model for dichotomous and polytomous item responses.Rost, J. (1990). Rasch models in latent classes: An integration of two approaches to item analysis. Applied Psychological Measurement, 14, 271-282.

Every spring, Siyuan will select excellent freshmen from all departments. After the writing test and interview, the list of new Siyuaners will be open to the public. Students graduated in different years are named after ordinal numbers. The first Siyuaners are the 2001 ones, who are called Siyuan 1st, so the most recent ones are called Siyuan 7th.

In the beginning of 1941, XXI Army Corps, already effectively an army-level unit since its designation as Gruppe XXI, was fully replaced and had its organizational structure transferred to the newly formed Armee Norwegen. With the ordinal number 21 freed up for German army corps, a new corps with that number, XXI Mountain Corps, was created in 1943.

In the American adaptation, the Mach 5 stems from the number 5 on the door. Although, in Japanese, is the word for the number 5, the Kanji character which is used in the car name actually means "item number" (i.e. it is an ordinal suffix). In addition, gogogo, is used as a general Japanese sound effect for rumble.

He described many common insects and suggested an ordinal classification of Insects. He described many Scarabaeidae as well as illustrating them for the first time. The study included 39 Scarabaeus species, 17 Copris species, seven Trox species, four Cetonia and four Trichius. Familiar beetles such as Canthon viridis, Macrodactylus angustatus and Osmoderma scabra were first described by him.

In statistics, Goodman and Kruskal's gamma is a measure of rank correlation, i.e., the similarity of the orderings of the data when ranked by each of the quantities. It measures the strength of association of the cross tabulated data when both variables are measured at the ordinal level. It makes no adjustment for either table size or ties.

They produce United States, and European summary rankings based on all five and a global summary ranking using the Wall Street Journal, Economist and Financial Times. The summary is based on underlying polls in which a school placed in the top ten using an average of the ordinal placements. The summary excludes the U.S. News & World Report results.

Ordinal numbers have grammatically no differences with adjectives. While forming them, upper three orders of numerals are agglutinated to nearest dividing power of 1000, which results in constructing some of the longest natural Russian words, e.g. (153,000-th), while the next is (153,001-st). In the latter example, only the last word is declined with noun.

Portrait by Henry Bryan Hall, 1839. The first result of co-operation and consultation between Cranmer and Bucer was the Ordinal, the liturgy for the ordination of priests. This was missing in the first Prayer Book and was not published until 1550. Cranmer adopted Bucer's draft and created three services for commissioning a deacon, a priest, and a bishop.

Rats have demonstrated time-place learning, and can also learn to infer correct timing for a specific task by following an order of events, suggesting that they might be able to use an ordinal timing mechanism. Like pigeons, rats are thought to have the ability to use a circadian timing mechanism for discriminating time of day.

It is the first such analysis of political systems from the 18th century to the present day. MaxRange1 provides nominal and ordinal rankings on a 1-100 scale, MaxRange2 on a 1-1,000 scale. The attributes underlying the categorization are also available. In addition to monthly rankings, yearly rankings are available from 1600 to the present.

Rota of Pope Alexander III, AD 1175 The rota is one of the symbols used by the Pope to authenticate documents such as papal bulls. It is a cross inscribed in two concentric circles. Pope Leo IX was the first pope to use it. The four inner quadrants contain: "Petrus", "Paulus", the Pope's name, and the Pope's ordinal number.

A key feature of minimax decision making is being non-probabilistic: in contrast to decisions using expected value or expected utility, it makes no assumptions about the probabilities of various outcomes, just scenario analysis of what the possible outcomes are. It is thus robust to changes in the assumptions, as these other decision techniques are not. Various extensions of this non-probabilistic approach exist, notably minimax regret and Info-gap decision theory. Further, minimax only requires ordinal measurement (that outcomes be compared and ranked), not interval measurements (that outcomes include "how much better or worse"), and returns ordinal data, using only the modeled outcomes: the conclusion of a minimax analysis is: "this strategy is minimax, as the worst case is (outcome), which is less bad than any other strategy".

An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size". Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted.

That every Dedekind-infinite set is infinite can be easily proven in ZF: every finite set has by definition a bijection with some finite ordinal n, and one can prove by induction on n that this is not Dedekind-infinite. By using the axiom of countable choice (denotation: axiom CC) one can prove the converse, namely that every infinite set X is Dedekind-infinite, as follows: First, define a function over the natural numbers (that is, over the finite ordinals) , so that for every natural number n, f(n) is the set of finite subsets of X of size n (i.e. that have a bijection with the finite ordinal n). f(n) is never empty, or otherwise X would be finite (as can be proven by induction on n).

Hugh de Courtenay, 4th/12th Earl of Devon (1389 – 16 June 1422) was an English nobleman, son of the 3rd/11th Earl of Devon, and father of the 5th/13th Earl. The ordinal number given to the early Courtenay Earls of Devon depends on whether the earldom is deemed a new creation by the letters patent granted 22 February 1334/5 or whether it is deemed a restitution of the old dignity of the de Redvers family. Authorities differ in their opinions,Watson, in Cokayne, The Complete Peerage, new edition, IV, p.324 & footnote (c): "This would appear more like a restitution of the old dignity than the creation of a new earldom"; Debrett's Peerage however gives the ordinal numbers as if a new earldom had been created.

Zero-based numbering is a way of numbering in which the initial element of a sequence is assigned the index 0, rather than the index 1 as is typical in everyday non-mathematical or non-programming circumstances. Under zero-based numbering, the initial element is sometimes termed the zeroth element, rather than the first element; zeroth is a coined ordinal number corresponding to the number zero. In some cases, an object or value that does not (originally) belong to a given sequence, but which could be naturally placed before its initial element, may be termed the zeroth element. There is not wide agreement regarding the correctness of using zero as an ordinal (nor regarding the use of the term zeroth) as it creates ambiguity for all subsequent elements of the sequence when lacking context.

After the incorporation, some of its streets were given new names. In 1931, north-south street names were standardized by continuing the ordinal numbers of New Toronto's streets, picking-up at Twenty-Third Street in the east through to Forty-Third Street in the west. For example; Lansdowne Avenue became Thirty- Third Street and Lake View Avenue became Thirty-Fifth Street.

Shungtangenoceras is a conical plectronoceratoid cephalopod from the Upper Cambrian of north-eastern China, described by Sun (1937) as a primitive endoceroid. Because of its apparently poor preservation its ordinal and familial position is uncertain. It could be included in either the Plectronocerida, family Plectronoceratidae, or the Ellesmerocerida, family Ellesmeroceratidae (Flower, 1954). On the other hand, Teichert (1964) included Shungtangendoceras in the Ellesmoerocatidae.

For the millions and above, -ti is suffixed and the vowels are not changed: milijónti/a/o (millionth), milijárdti/a/o (billionth). In ordinals from 100th and above, if the number is formed by multiple words, only the last word is changed into an ordinal. The others remain the same as the cardinal. So 200th is dvéstoti, but 201st is dvésto pŕvi.

The usual ordinal form is "twelfth" but "dozenth" or "duodecimal" (from the Latin word) is also used in some contexts, particularly base-12 numeration. Similarly, a group of twelve things is usually a "dozen" but may also be referred to as a "dodecad" or "duodecad". The adjective referring to a group of twelve is "duodecuple". As with eleven,Oxford English Dictionary, 1st ed.

The Stowe Breviary (British Library, Stowe MS 12) is an early-fourteenth- century illuminated manuscript Breviary from England, providing the divine office according to the Sarum ordinal and calendar (with Norwich additions). It is thought to be by the same scribe as the Macclesfield Psalter and the Douai Psalter. The manuscript forms part of the Stowe manuscripts in the British Library.

The “I Opt” model uses a 24-statement survey to assess preferences. The survey is designed in a way that creates ratio measurement (exact, like a ruler). This contrasts more typical ordinal measurement (e.g., rank ordered – big-bigger-biggest or none-some-lots) used by most other tools. Exact measurement allows “I Opt” to derive formulas that can be used by a computer.

There are arithmetic operations on ordinals by virtue of the one-to-one correspondence between ordinals and nimbers. Three common operations on nimbers are nimber addition, nimber multiplication, and minimum excludance (mex). Nimber addition is a generalization of the bitwise exclusive or operation on natural numbers. The of a set of ordinals is the smallest ordinal not present in the set.

On the other hand, because the ranked outcome is of utmost interest in many situations (e.g. allocating research grants to proposals or assigning letter grades to students), ways to systematically aggregate peer-wise assessment to recover the ranked order of submissions has many practical implications. To tackle this, some researchers studied (1) evaluation schemes (e.g. ordinal grading,Raman, Karthik, and Thorsten Joachims.

Indicator value is a term that has been used in ecology for two different indices. The older usage of the term refers to Ellenberg's indicator values, which are based on a simple ordinal classification of plants according to the position of their realized ecological niche along an environmental gradient.Ellenberg H. Zeigerwerte der Gefässpflanzen Mitteleuropas / H. Ellenberg // Scripta geobotanica. Göttingen, 1974.

He created "Crystal Memories" as the theme song for their 2017 smartphone video game Ordinal Strata. The following year, Toshi made his voice acting debut in the game voicing a version of himself. He also collaborated with hip-hop artist AK-69 on the rapper's May 14, 2018 digital single "Brave". On September 3, he released a food book titled .

It is not always possible to assign rankings uniquely. For example, in a race or competition two (or more) entrants might tie for a place in the ranking. When computing an ordinal measurement, two (or more) of the quantities being ranked might measure equal. In these cases, one of the strategies shown below for assigning the rankings may be adopted.

An additive preference relation can be represented by many different additive utility functions. However, all these functions are similar: they are not only increasing monotone transformations of each other (as are all utility functions representing the same relation); they are increasing linear transformations of each other. In short, ::An additive ordinal utility function is unique up to increasing linear transformation.

However, j might be different from the ultrapower(s) arising from such filter(s). If N and M are the same and j is the identity function on N, then j is called "trivial". If transitive class N is an inner model of ZFC and j has no critical point, i.e. every ordinal maps to itself, then j is trivial.

On typewriters and computers that do not support this symbol, it is acceptable and commonplace to replace it with the trigraph "No." (letter "N", letter "o", and a period (full stop)). On typewriters and computers that support the degree symbol a digraph "N°" may be used. If the masculine ordinal indicator is available, the better digraph "Nº" may be used.

The lodges are managed by a Worshipful Master, who will be assisted by one or more Deputy Masters (these are given Latin ordinal names: primary, secondary, and so forth). There are also the Primary and Secondary Wardens, a Master of Ceremonies, a Secretary, a Treasurer, an Orator, and a Director of Music. The lodge offices do not rotate as in e.g. American freemasonry.

The first value is multiplied by 7, the second by 6 and so on. The first 3 numeric characters are multiplied by the inverse of their ordinal position also. The sum of these multiplications modulus 11 subtracted from 11 is taken as the check digit (a result of 10 is translated to 0). This scheme is similar to the ISBN check digit scheme.

They all represent the ordering in which a is preferred to b to c to d. The assumption of ordinal preferences, which precludes interpersonal comparisons of utility, is an integral part of Arrow's theorem. For various reasons, an approach based on cardinal utility, where the utility has a meaning beyond just giving a ranking of alternatives, is not common in contemporary economics.

The legion received the ordinal "Fifth" because in Pannonia there were already four legions. The purpose of the legion, having her permanent camp in Bononia and an advanced castellum in Onagrinum, was to protect the imperial residence of Diocletian in Sirmium (Illyricum). The Notitia Dignitatum locates the legion still in Illyricum at the beginning of the 5th century.Notitia Dignitatum, occ XXXII v.

Philadelphia's 10th Street written in English and Chinese A numbered street is a street whose name is an ordinal number, as in Second Street or Tenth Avenue. Such forms are among the most common street names in North America, but also exist in other parts of the world, especially in the Middle East. Numbered streets were first used in PhiladelphiaRybczynski, Witold. "City Life".

Since the English Reformation in the 16th century, there have been more than fifty English-language translations and paraphrases of Veni Creator Spiritus.Charles S. Nutter; Wilbur F. Tillett. The Hymns and Hymn Writers of The Church (Smith & Lamar, 1911), p. 108. The version attributed to Archbishop Cranmer, his sole venture into English verse, first appeared in the Prayer Book Ordinal of 1550.

The memorial is of Portland stone with bronze sculpture. A statue of Queen Victoria stands on a tall pedestal facing South, "looking a little pensively over the square," according to Nikolaus Pevsner. The pedestal sits on a tall square plinth with rounded corners accompanied by four bronze lions at the ordinal points. Around the plinth is an unbroken bas relief frieze of bronze.

Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal numbers and in contexts similar to those in which Roman numerals are still used elsewhere in the West. For ordinary cardinal numbers, however, Greece uses Arabic numerals.

Thus by the previous results, nontrivial limits in H are taken to nontrivial limits in the set of volumes. In fact, one can further conclude, as did Thurston, that the set of volumes of finite volume hyperbolic 3-manifolds has ordinal type \omega^\omega. This result is known as the Thurston-Jørgensen theorem. Further work characterizing this set was done by Gromov.

The network with the minimum number of links is obtained when q = n − 1 and r = ∞, i.e., PFnet(n − 1, ∞). With ordinal-scale data (see level of measurement), the r-parameter should be infinity because the same PFnet would result from any positive monotonic transformation of the proximity data. Other values of r require data measured on a ratio scale.

In 1967, Bradshaw entered Westcott House, Cambridge, an Anglican theological college in the Liberal Anglo-Catholic tradition. While training for ordination at Westcott House, he also undertook postgraduate research at King's College London. He completed his Doctor of Philosophy (PhD) degree in 1971. His doctoral thesis was titled "The Anglican Ordinal: its history and development from the Reformation to the present day".

Development of the ascocarp in Glonium stellatum. Amer J Bot 40: 626–633. Initially, Luttrell (1953) was unsure whether the Hysteriaceae justified ordinal status, stating that the elongated hysteriaceous locule alone may not appear to be sufficient for the recognition of a separate order and the dothideaceous nature of the centrum at the earliest stages was not observed in his study.

However, the use of the adjective alone is fairly common in the case of superlatives such as biggest, ordinal numbers such as first, second, etc., and other related words such as next and last. Many adjectives, though, have undergone conversion so that they can be used regularly as countable nouns; examples include Catholic, Protestant, red (with various meanings), green, etc.

A processor unique record is one that is defined such that each processor expected to be in the loosely coupled complex has a record type of 'FRED' and perhaps 100 ordinals. However, if a user on any 2 or more processors examines the file address that record type 'FRED', ordinal '5' resolves to, they will note a different physical address is used.

Every sentence must contain formal words to designate what Husserl calls "formal categories". There are two kinds of categories: meaning categories and formal-ontological categories. Meaning categories relate judgments; they include forms of conjunction, disjunction, forms of plural, among others. Formal-ontological categories relate objects and include notions such as set, cardinal number, ordinal number, part and whole, relation, and so on.

Class I adjectives for which the last syllable in the masculine direct singular form is ور /‑wár/, ګر /‑gár/, جن /‑ján/, or م ن /‑mán/, as well as ordinal numbers ending in م /‑ám/, undergo a different vowel alternation: the vowel /á/ of the final syllable centralizes to /ə́/ in feminine non-direct singulars and in all plural forms, irrespective of gender.

The Loewy length and Loewy series were introduced by If M is a module, then define the Loewy series Mα for ordinals α by M0 = 0, Mα+1/Mα = socle M/Mα, Mα = ∪λ<α Mλ if α is a limit ordinal. The Loewy length of M is defined to be the smallest α with M = Mα, if it exists.

Perhaps doing this to compensate for being partly cut out of his father's will. Norton begun the Ordinal in 1477. Norton was rewarded with land confiscated from the rebels upon Edward IV's return from exile in 1471. In March 1479, Norton accused the incumbent mayor of Bristol of high treason, surrounding an argument concerning the legacy of his father as the mayor.

Each judge would then arrange the skaters in order of total score by that judge; these ordinal rankings were used to provide final placement for the skaters, using a "majority rule"--if a majority of the judges ranked a pair first, the pair won. If there was no majority, the total ordinals controlled. Ties were broken by total points.Official Report, pp. 558–65.

Multicriteria classification (sorting) is one of the problems considered within MCDA and can be stated as follows: given a set of objects evaluated by a set of criteria (attributes with preference-order domains), assign these objects to some pre-defined and preference-ordered decision classes, such that each object is assigned to exactly one class. Due to the preference ordering, improvement of evaluations of an object on the criteria should not worsen its class assignment. The sorting problem is very similar to the problem of classification, however, in the latter, the objects are evaluated by regular attributes and the decision classes are not necessarily preference ordered. The problem of multicriteria classification is also referred to as ordinal classification problem with monotonicity constraints and often appears in real-life application when ordinal and monotone properties follow from the domain knowledge about the problem.

The Mann–Whitney U test tests a null hypothesis of that the probability that a randomly drawn observation from one group is larger than a randomly drawn observation from the other is equal to 0.5 against an alternative that this probability is not 0.5 (see Mann–Whitney U test#Assumptions and formal statement of hypotheses). In contrast, a t-test tests a null hypothesis of equal means in two groups against an alternative of unequal means. Hence, except in special cases, the Mann–Whitney U test and the t-test do not test the same hypotheses and should be compared with this in mind. ;Ordinal data: The Mann–Whitney U test is preferable to the t-test when the data are ordinal but not interval scaled, in which case the spacing between adjacent values of the scale cannot be assumed to be constant.

Therefore, a researcher needs to develop nominal, ordinal, or interval coding scales to measure the particular types of variation he or she is interested in measuring. For example, the subject category "Techniques of Socialization" (OCM 861) will find passages that deal with cultural ideas about childtraining or general methods of discipline, but coding schemes need to be developed to measure dimensions of variation, such as "degree to which corporal punishment is employed," "degree to which threatening is employed," or "degree to which children are praised." It is not difficult, after a little practice, to develop ordinal scales that can allow for the coding of words into quantitative measures, and once that is done it is easy to use available software to test hypotheses, and compare, combine, and model the results. The indexed texts in HRAF are also amenable to qualitative cross-cultural comparisons.

However, circa the eleventh century there arose a tendency toward greater elaboration and precision in rubrical directions for the services, and at the same time the beginning of a more or less strongly marked division of these directions into two classes arose, which in the case of the Sarum Use were conveniently distinguished as the Customary and the Ordinal. Generally, the former of these rubrical books contained the principles and the latter their application; the former determined those matters that were constant and primarily the duties of persons, the latter dealt with the arrangements that varied from day to day and year to year. It is out of the Ordinal, often denominated the Ordinarium and Liber Ordinarius, that the Directorium or Pye, and later the Ordo recitandi evolved. These distinctions are not clear because the process was gradual.

But in the English and Continental Ordinals 2 different stages can be distinguished: first, the type of book in common use from the twelfth to fifteenth century, and represented by the Sarum Ordinal edited by W. H. Frere and the Ordinaria of Laon edited by Chevalier. In them was much miscellaneous information respecting feasts, the Divine Office and Mass to be prayed thereon according to the changes necessitated by the occurrence of Easter and the shifting of the Sundays, as well as the "Incipits" of the details of the liturgy, e. g. of the lessons to be read and the commemorations to be made. The second stage took the form of an adaptation of the Ordinal for ready use, an adaptation with which, in the case of Sarum, the name of Clement Maydeston is prominent connected.

So the nim-sum is written in binary as 1001, or in decimal as 9. This property of addition follows from the fact that both mex and XOR yield a winning strategy for Nim and there can be only one such strategy; or it can be shown directly by induction: Let and be two finite ordinals, and assume that the nim-sum of all pairs with one of them reduced is already defined. The only number whose XOR with is is , and vice versa; thus is excluded. On the other hand, for any ordinal , XORing with all of , and must lead to a reduction for one of them (since the leading 1 in must be present in at least one of the three); since , we must have or ; thus is included as or as , and hence is the minimum excluded ordinal.

Cranmer, however, assigned Nicholas Ridley, Bishop of London, to perform the consecration, and Ridley refused to do anything but follow the form of the ordinal as it had been prescribed by the Parliament of England. A reformist himself, and not always a strict follower of the ordinal, Ridley, it seems likely, had some particular objection to Hooper. It has been suggested that Henrician exiles like Hooper, who had experienced some of the more radically reformed churches on the continent, were at odds with English clergy who had accepted and never left the established church. John Henry Primus also notes that on July 24, 1550, the day after receiving instructions for Hooper's unique consecration, the church of the Austin Friars in London had been granted for use as a Stranger church with the freedom to employ their own rites and ceremonies.

"Honorable Mention: What Public Library National Ratings Say" (Nov/Dec 2008)Public Libraries p.36-41. They point out that, among other factors, imprecision in library statistics make ratings scores quite approximate, a fact rarely acknowledged by libraries receiving high ratings. The authors also note that HAPLR calculations perform invalid mathematical operations using ordinal rankings, making comparisons of scores between libraries and between years meaningless.

This is how this king is still referred to in Thai history books. His descendant Vajiravudh (Rama VI) who had studied in England, realised that most Siamese kings' names were difficult to reproduce and remember for Westerners. He therefore disposed to use for all kings of the Chakri dynasty the name Rama together with the respective ordinal number. So this king is Rama I in Western literature.

Welfare measurement. In 1990 Jorgenson presented econometric methods for welfare measurement in his Presidential Address to the Econometric Society. These methods have generated a new approach to cost of living measurement and new measures of the standard of living, inequality, and poverty. This has required dispensing with ordinal measures of individual welfare that are not comparable among individuals, as persuasively argued by Amartya Sen in 1977.

Lavielle and Morton was the first architecture firm west of the Mississippi River above New Orleans. As street commissioner in 1823–26, Joseph Laveille devised the city's street name grid, with ordinal numbers for north-south streets and arboreal names for east-west streets.Laveille and Morton - stlcin.missouri.org - Retrieved January 21, 2008 Missouri became a state in 1821, and the St. Louis population tripled in 10 years.

This process can be extended for all natural numbers n, and these are called n-categories. There is even a notion of ω-category corresponding to the ordinal number ω. Higher-dimensional categories are part of the broader mathematical field of higher-dimensional algebra, a concept introduced by Ronald Brown. For a conversational introduction to these ideas, see John Baez, 'A Tale of n-categories' (1996).

The theory of indifference curves was developed by Francis Ysidro Edgeworth, who explained in his 1881 book the mathematics needed for their drawing; later on, Vilfredo Pareto was the first author to actually draw these curves, in his 1906 book. The theory can be derived from William Stanley Jevons' ordinal utility theory, which posits that individuals can always rank any consumption bundles by order of preference.

An ordinal for ordination services was added in 1550. There was also a calendar and lectionary, which meant a Bible and a Psalter were the only other books required by a priest. It represented a "major theological shift" toward Protestantism. Cranmer's doctrinal concerns can be seen in the systematic amendment of source material to remove any idea that human merit contributed to an individual's salvation.

Two-Micron Sky Survey, or IRC, or Caltech infrared catalog is the astronomical catalogue of the infrared sources published in the 1969 by Neugebauer and Leighton. Catalogue index consists of two numbers - declination rounded to multiplier of 10 degrees, with sign, and star ordinal number within declination band. Catalog contains about 5000 objects between declinations +15 and -15 degrees. Most of the sources are M-type stars.

FinSet is a full subcategory of Set, the category whose objects are all sets and whose morphisms are all functions. Like Set, FinSet is a large category. FinOrd is a full subcategory of FinSet as by the standard definition, suggested by John von Neumann, each ordinal is the well-ordered set of all smaller ordinals. Unlike Set and FinSet, FinOrd is a small category.

The cardinal numbers in Frater: 1 - uni 2 - bi 3 - tri 4 - kuadri 5 - kuinti 6 - ses 7 - sep 8 - okta 9 - nona 10 - deka 11 - dekauni 12 - dekabi 13 - dekatri 20 - bideka 24 - bidekakuadri 30 - trideka 40 - kuadrideka 85 - oktadekakuinti 100 - senti 367 - trisenti-sesdeka-sep 600 - sessenti 1000 - mil 1000000 - milion Ordinal numbers are formed by placing the cardinal number after the noun.

Fiske proposed that the four discrete types of relationships correspond to Stevens's four levels of measurement. CS relationships resemble the categorical (nominal) scales of measurement in that all members of the relationship are equivalent. AR resembles an ordinal scale given that members of the relationship are placed in a linear ordering. EM relationships resemble interval measurement given that they are kept in balance by addition and subtraction.

Nouns are split into independent nouns and verbal nouns. Na is the only independent noun that is used in the Lau language. It is only added to nouns when one is expressing relationships, or it is added to cardinal numbers to form an ordinal number. Pronouns are words that replace a noun in a sentence and can function by themselves as a noun phrase.

Well-foundedness fails specifically for rank-into-rank extenders; but Itay Neeman showed in 2004 that it holds for all weaker types of extender. The Mitchell rank of a measure is the ordertype of its predecessors under ◅; since ◅ is well-founded this is always an ordinal. A cardinal which has measures of Mitchell rank α for each α < β is said to be β-measurable.

However, on the other hand, Cliff also suggested that there are viable and robust ordinal alternatives to mean comparisons. He introduced a measure of proportional difference (or dominance) between two sets of data often referred to as Cliff's delta. He has been president of the Psychometric Society and of the Society for Multivariate Experimental Psychology. Now an Emeritus Professor, he lives in New Mexico.

Cervini received all votes except of his own, which he gave to Gian Pietro Carafa. He retained his baptismal name, adding to it only an ordinal number (Marcellus II). On that same day, he was consecrated bishop of Rome by Cardinal Gian Pietro Carafa, bishop of Ostia e Velletri and Dean of the College of Cardinals, and crowned by Cardinal Francesco Pisani, Protodeacon of S. Marco.

As is standard in set theory, we denote by \omega the least infinite ordinal, which has cardinality \aleph_0; it may be identified with the set of all natural numbers. A number of cardinal characteristics naturally arise as cardinal invariants for ideals which are closely connected with the structure of the reals, such as the ideal of Lebesgue null sets and the ideal of meagre sets.

There is a subtle cardinal ≤ κ if and only if every transitive set S of cardinality κ contains x and y such that x is a proper subset of y and x ≠ Ø and x ≠ {Ø}. An infinite ordinal κ is subtle if and only if for every λ < κ, every transitive set S of cardinality κ includes a chain (under inclusion) of order type λ.

After his death, it would be 423 years before another pope would choose a name with an ordinal number less than IV (John Paul I). Cervini was the maternal uncle of Robert Bellarmine. Cervini's father and Pope Clement VII were personal friends. Cervini served in the household of Cardinal Alessandro Farnese. When Farnese became Pope, Cervini served as his secretary and was employed on some diplomatic missions.

Then S can be partitioned into \kappa many disjoint stationary sets. This result is due to Solovay. If \kappa is a successor cardinal, this result is due to Ulam and is easily shown by means of what is called an Ulam matrix. H. Friedman has shown that for every countable successor ordinal \beta, every stationary subset of \omega_1 contains a closed subset of order type \beta.

In Armenian, the stress falls on the last syllable unless the last syllable contains the definite article or , and the possessive articles and , in which case it falls on the penultimate one. For instance, , , but and . Exceptions to this rule are some words with the final letter ( in the reformed orthography) () and sometimes the ordinal numerals (, etc.), as well as , and a small number of other words.

These developments were accompanied by the introduction of new tools, such as indifference curves and the theory of ordinal utility. The level of mathematical sophistication of neoclassical economics increased. Paul Samuelson's Foundations of Economic Analysis (1947) contributed to this increase in mathematical modelling. The interwar period in American economics has been argued to have been pluralistic, with neoclassical economics and institutionalism competing for allegiance.

The Abbreviated Injury Scale (AIS) is an anatomically based consensus-derived global severity scoring system that classifies each injury in every body region according to its relative severity on a six-point ordinal scale: # Minor; # Moderate # Serious # Severe # Critical # Maximal (currently untreatable). There are nine AIS chapters corresponding to nine body regions: #Head #Face #Neck #Thorax #Abdomen #Spine #Upper Extremity #Lower Extremity #External and other.

Unlike choice voting where the numbers represent the order of a voter's ranking of candidates (i.e. they are ordinal numbers), in cumulative votes the numbers represent quantities (i.e. they are cardinal numbers). While giving voters more points may appear to give them a greater ability to graduate their support for individual candidates, it is not obvious that it changes the democratic structure of the method.

In such cases, the marginal utility of a good or service might actually be increasing. Without the presumption that utility is quantified, the diminishing of utility should not be taken to be itself an arithmetic subtraction. It is the movement from use of higher to lower priority, and may be no more than a purely ordinal change.Theodore-Angwenyi, Nicholas; "Utility", International Encyclopedia of the Social Sciences (1968).

Feferman was editor-in-chief of the five-volume Collected Works of Kurt Gödel, published by Oxford University Press between 2001 and 2013. In 2004, together with his wife Anita Burdman Feferman, he published a biography of Alfred Tarski: Alfred Tarski: Life and Logic. He worked on predicative mathematics, in particular introducing the Feferman–Schütte ordinal as a measure of the strength of certain predicative systems.

In computability theory, computational complexity theory and proof theory, the Hardy hierarchy, named after G. H. Hardy, is an ordinal-indexed family of functions hα: N → N (where N is the set of natural numbers, {0, 1, ...}). It is related to the fast-growing hierarchy and slow-growing hierarchy. The hierarchy was first described in Hardy's 1904 paper, "A theorem concerning the infinite cardinal numbers".

If λ is any ordinal, κ is λ-strong means that κ is a cardinal number and there exists an elementary embedding j from the universe V into a transitive inner model M with critical point κ and :V_\lambda\subseteq M That is, M agrees with V on an initial segment. Then κ is strong means that it is λ-strong for all ordinals λ.

In 1963, he received a D.Sc. in the same year he received his habilitation at the Faculty of Mathematics and Physics of Charles University, in 1966 he was entitled professor at this faculty. In 1973, he was awarded the Klement Gottwald State Prize for his work on the asymptotic theory of ordinal tests. He died at the age of 48 after a kidney transplant.

Age is an attribute that can be operationalized in many ways. It can be dichotomized so that only two values - "old" and "young" - are allowed for further data processing. In this case the attribute "age" is operationalized as a binary variable. If more than two values are possible and they can be ordered, the attribute is represented by ordinal variable, such as "young", "middle age", and "old".

Until 10, he grew up in the city of Ecbatana and lived as an ordinal citizen as his mother is a free citizen. ; : : :A young boy who serves Narsus. His parents were slaves freed by Narsus, hence his loyalty towards his master. Elam also grows into a friendship with Arslan and becomes a friend to him protecting him when his life is ever in danger.

In the United States, dates are traditionally written in the "month-day-year" order, with neither increasing nor decreasing order of significance. This order is used in both the traditional all-numeric date (e.g., "1/21/16" or "01/21/2016") and the expanded form (e.g., "January 21, 2016"—usually spoken with the year as a cardinal number and the day as an ordinal number, e.g.

Thus the modern umlaut ü was written as uͤ. Both vowels and consonants were used in this way, as in ſheͨzze and boͮsen. In modern typefaces, these letters are usually smaller than other superscripts, and their baseline is slightly above the base font's midline, making them extend no higher than a typical ordinal indicator. Superscripts are used for the standard abbreviations for service mark ℠ and trademark ™.

She went on to complete a PhD in theoretical computer science. Her supervisor was Harold Simmons who held a joint position in the School of Mathematics and the Department of Computer Science, with whom she wrote two papers Point-sensitive and point-free patch constructions. and An ordinal indexed hierarchy of separation properties. The first of these papers has eight citations — good for such esoteric mathematics.

RFC 3339 defines a profile of ISO 8601 for use in Internet protocols and standards. It explicitly excludes durations and dates before the common era. The more complex formats such as week numbers and ordinal days are not permitted.RFC 3339, section 5.6 RFC 3339 deviates from ISO 8601 in allowing a zero time zone offset to be specified as "-00:00", which ISO 8601 forbids.

The ordinal based selection methods include the tournament and ranking selection. Tournament selection involves the random selection of individuals of a population and the subsequent comparison of their fitness levels. The winners of these “tournaments” are the ones with the highest values and will be put into the mating pool as parents. In ranking selection all the individuals are sorted based on their fitness values.

Since the cardinal numbers are well-ordered by indexing with the ordinal numbers (see Cardinal number, formal definition), this also establishes that there is no greatest ordinal number; conversely, the latter statement implies Cantor's paradox. By applying this indexing to the Burali-Forti paradox we obtain another proof that the cardinal numbers are a proper class rather than a set, and (at least in ZFC or in von Neumann–Bernays–Gödel set theory) it follows from this that there is a bijection between the class of cardinals and the class of all sets. Since every set is a subset of this latter class, and every cardinality is the cardinality of a set (by definition!) this intuitively means that the "cardinality" of the collection of cardinals is greater than the cardinality of any set: it is more infinite than any true infinity. This is the paradoxical nature of Cantor's "paradox".

If only one monarch has used a particular name, no ordinal is used; for example, Queen Victoria is not known as "Victoria I", and ordinals are not used for English monarchs who reigned before the Norman conquest of England. The question of whether numbering for British monarchs is based on previous English or Scottish monarchs was raised in 1953 when Scottish nationalists challenged the Queen's use of "Elizabeth II", on the grounds that there had never been an "Elizabeth I" in Scotland. In MacCormick v Lord Advocate, the Scottish Court of Session ruled against the plaintiffs, finding that the Queen's title was a matter of her own choice and prerogative. The Home Secretary told the House of Commons that monarchs since the Acts of Union had consistently used the higher of the English and Scottish ordinals, which in the applicable four cases has been the English ordinal.

Lecture Notes in Computer Science 4702 (2007) 164-175. Having stated the probabilistic model for ordinal classification problems with monotonicity constraints, the concepts of lower approximations are extended to the stochastic case. The method is based on estimating the conditional probabilities using the nonparametric maximum likelihood method which leads to the problem of isotonic regression. Stochastic dominance-based rough sets can also be regarded as a sort of variable-consistency model.

Skepticism about the axiom of choice was reinforced by recently discovered paradoxes in naive set theory. Cesare Burali-Forti (1897) was the first to state a paradox: the Burali-Forti paradox shows that the collection of all ordinal numbers cannot form a set. Very soon thereafter, Bertrand Russell discovered Russell's paradox in 1901, and Jules Richard (1905) discovered Richard's paradox. Zermelo (1908b) provided the first set of axioms for set theory.

Hoffmann studied medicine and became an orthopaedic specialist. He served on the managing board of the Deutsche Eislauf-Union and has appeared as a figure skating judge. He judged the ladies' event at the 1994 Winter Olympics and was one of five judges who placed Oksana Baiul ahead of Nancy Kerrigan. Hoffman also judged the ladies competition at the 1998 Winter Olympics and gave his first-place ordinal to Michelle Kwan.

Fulk Bertrand IHis name appears as Fulco or Fulcho and Bertrannus in contemporary documents. It is Foulques in modern French. His ordinal is a reference to a second Bertrand, his son, who later reigned in Provence. (died 27 April 1051) was the joint Count of Provence with his elder brother William IV from 1018 and with his younger brother Geoffrey I from at least 1032 if not earlier.

Five items (five bipolar pairs of adjectives) have been proven to yield reliable findings, which highly correlate with alternative Likert numerical measures of the same attitude.Osgood, Suci and Tannebaum (1957). One problem with this scale is that its psychometric properties and level of measurement are disputed. The most general approach is to treat it as an ordinal scale, but it can be argued that the neutral response (i.e.

Nominal categories of data are often compared to ordinal and ratio data, to see if nominal categories play a role in determining these other factors. For example, the effect of race (nominal) on income (ratio) could be investigated by regressing the level of income upon one or more dummy variables that specify race. When nominal variables are to be explained, logistic regression or probit regression is commonly used.

A field army is composed of 100,000 to 300,000 troops. Specific field armies are usually named or numbered to distinguish them from "army" in the sense of an entire national land military force. In English, the typical orthographic style for writing out the names field armies is word numbers, such as "First Army"; whereas corps are usually distinguished by Roman numerals (e.g. I Corps) and subordinate formations with ordinal numbers (e.g.

The pair skating competition of the 1960 Winter Olympics was held at the Blyth Arena in Squaw Valley, California, United States. The event took place on Friday 19 February 1960. Each judge ranked the skaters by Ordinal Placement from first to last place. If a skater was ranked first by a majority of the judges, that skater was placed first overall; this process was repeated for each place.

Ordination is the process by which individuals are consecrated, that is, set apart as clergy to perform various religious rites and ceremonies such as celebrating the sacraments. The process and ceremonies of ordination varies by denomination. One who is in preparation for, or who is undergoing the process of ordination is sometimes called an ordinand. The liturgy used at an ordination is sometimes referred to as an ordinal.

The Berry paradox as formulated above arises because of systematic ambiguity in the word "definable". In other formulations of the Berry paradox, such as one that instead reads: "...not nameable in less..." the term "nameable" is also one that has this systematic ambiguity. Terms of this kind give rise to vicious circle fallacies. Other terms with this type of ambiguity are: satisfiable, true, false, function, property, class, relation, cardinal, and ordinal.

Milner's interest in set theory was sparked by visits of Paul Erdős to Singapore and by meeting András Hajnal while on sabbatical in Reading. He generalized Chen Chung Chang's ordinal partition theorem (expressed in the arrow notation for Ramsey theory) ωω→(ωω,3)2 to ωω→(ωω,k)2 for arbitrary finite k. He is also known for the Milner–Rado paradox. He has 15 joint papers with Paul Erdős.

The Barthel scale is an ordinal scale used to measure performance in activities of daily living (ADL). Each performance item is rated on this scale with a given number of points assigned to each level or ranking. It uses ten variables describing ADL and mobility. A higher number is associated with a greater likelihood of being able to live at home with a degree of independence following discharge from hospital.

The five response categories are often believed to represent an interval level of measurement. But this can only be the case if the intervals between the scale points correspond to empirical observations in a metric sense. Reips and Funke (2008) show that this criterion is much better met by a visual analogue scale. In fact, there may also appear phenomena which even question the ordinal scale level in Likert scales.

Norman Cliff (born September 1, 1930) is an American psychologist. He received his Ph.D. from Princeton in psychometrics in 1957. After research positions in the US Public Health Service and at Educational Testing Service he joined the University of Southern California in 1962. He has had a number of research interests, including quantification of cognitive processes, scaling and measurement theory, computer-interactive psychological measurement, multivariate statistics, and ordinal methods.

Nicholas V, Duke of Krnov (also known as Nicholas II of Opava-Ratibor;The ordinal II comes from only considering the Opava-Ratibor branch. In this view, Nicholas IV, Duke of Ratibor-Bruntál would be called Nicholas I of Krnov ; –1452) was a member of the Přemyslid dynasty. He was Duke of Racibórz, Krnov, Bruntál and Rybnik. All these duchies were situated in Silesia, then part of the Crown of Bohemia.

This modifier is usually attached to the last part of the noun phrase and gives a reference to the head noun in the noun phrase with regard to the rest of the sentence. However, noun phrases used as prepositional phrases do not have determiners. Quantifiers such as cardinal numerals can go either before or after the adjective in a sentence. Ordinal numerals on the other hand, such as first, second, etc.

On the topic of infinite chess, Hamkins, Brumleve and Schlicht proved that the mate-in-n problem of infinite chess is decidable. Hamkins and Evans investigated transfinite game values in infinite chess, proving that every countable ordinal arises as the game value of a position in infinite three-dimensional chess.C. D. A. Evans and J. D. Hamkins, "Transfinite game values in infinite chess," Integers, volume 14, Paper No. G2, 36, 2014.

The MOS is subject to certain mathematical properties and biases. In general, there is an ongoing debate on the usefulness of the MOS to quantify Quality of Experience in a single scalar value. When the MOS is acquired using a categorical rating scales, it is based on – similar to Likert scales – an ordinal scale. In this case, the ranking of the scale items is known, but their interval is not.

Another way to deal with multiple sincere votes is to augment the ordinal preference model with an approval or acceptance threshold. An approval threshold divides all of the candidates into two sets, those the voter approves of and those the voter does not approve of. A voter can approve of more than one candidate and still prefer one approved candidate to another approved candidate. Acceptance thresholds are similar.

In set theory, there are exponential operations for cardinal and ordinal numbers. If κ and λ are cardinal numbers, the expression κλ represents the cardinality of the set of functions from any set of cardinality λ to any set of cardinality κ.Nicolas Bourbaki, Elements of Mathematics, Theory of Sets, Springer-Verlag, 2004, III.§3.5. If κ and λ are finite, then this agrees with the ordinary arithmetic exponential operation.

The naming of the component conflicts within the Italian Wars has never been standardized and varies among historians of the period. Some wars may be split or combined differently, causing ordinal numbering systems to be inconsistent among different sources. The wars may be referred to by their dates or by the monarchs fighting them. Usually, the Italian Wars are grouped into three major phases: 1494-1516; 1521-1530; and 1535-1559.

In the old days, Saigon's roads were simply named by ordinal numbers. Starting from the Saigon River bank, Đồng Khởi was the Sixth Road. In 1865, the French Commander Admiral De La Grandiere renamed these roads and Sixth Road became Rue Catinat, a bustling place. Across the street from the future Continental site, the first foundations and floors for factories were built, the first one for Denis Frere.

Sico II (died 855) was the first prince of Salerno. Son and successor of Siconulf, he reigned from his father's death in 851 to his deposition in 853. He is given the ordinal "II" because he was the second Sico to rule in the south, the previous Sico being prince of Benevento. Sico was a minor when his father died and was put under the tutelage of Peter.

It is required that the ordinal structure of the top level nodes be "built up" as the direct limit of the ordinals in the branch to that node by the maps π, so the lower level nodes can be thought of as approximations to the (larger) top level node. A long list of further axioms is imposed to have this happen in a particularly "nice" way.K. Devlin. Constructibility. Springer, Berlin, 1984.

The first spelling of "Rhyncophthirina" by Ferris was a lapse, and in subsequent use of the term he spelled it "Rhynchophthirina" adding the second "h". Ordinal names are not covered by the International Code of Nomenclature and thus the name and spelling comes down to a matter of personal preference. The majority of phthirapterists spell the suborder as "Rhynchophthirina" as did Hopkins and Clay, 1952, and Price et al., 2003.

Standard map symbol for a numbered Army, the 'X's are not substituting the army's number A particular army can be named or numbered to distinguish it from military land forces in general. For example, the First United States Army and the Army of Northern Virginia. In the British Army it is normal to spell out the ordinal number of an army (e.g. First Army), whereas lower formations use figures (e.g.

GRNN has been implemented in many computer languages including MATLAB, R- programming language, Python (programming language) and Node.js. Neural networks (specifically Multi-layer Perceptron) can delineate non-linear patterns in data by combining with generalized linear models by considering distribution of outcomes (sightly different from original GRNN). There have been several successful developments, including Poisson regression, ordinal logistic regression, quantile regression and multinomial logistic regression that described by Fallah in 2009.

Each stage is assigned an ordinal number. The lowest stage, stage 0, consists of all entities having no members. We assume that the only entity at stage 0 is the empty set, although this stage would include any urelements we would choose to admit. Stage n, n>0, consists of all possible sets formed from elements to be found in any stage whose number is less than n.

The Han court was responsible for the major efforts of disaster relief when natural disasters such as earthquakes devastated the lives of commoners.Ebrey (1986), 621. To better prepare for calamities, Zhang Heng invented a seismometer in 132 CE, which provided instant alert to authorities in the capital Luoyang that an earthquake had occurred in a location indicated by a specific cardinal or ordinal direction.de Crespigny (2007), 1050; Morton & Lewis (2005), 70.

In axiomatic set theory, a function f : Ord → Ord is called normal (or a normal function) if and only if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions: # For every limit ordinal γ (i.e. γ is neither zero nor a successor), f(γ) = sup {f(ν) : ν < γ}. # For all ordinals α < β, f(α) < f(β).

Philologists have debated the origin and meaning of these names since classical antiquity. However, many of the meanings popularly assigned to various praenomina appear to have been no more than "folk etymology". The names derived from numbers are the most certain. The masculine names Quintus, Sextus, Septimus, Octavius and Decimus, and the feminine names Prima, Secunda, Tertia, Quarta, Quinta, Sexta, Septima, Octavia, Nona and Decima are all based on ordinal numbers.

The proof relies on basic properties of the Gromov norm. Jørgensen also showed that the volume function on this space is a continuous, proper function. Thus by the previous results, nontrivial limits in H are taken to nontrivial limits in the set of volumes. In fact, one can further conclude, as did Thurston, that the set of volumes of finite volume hyperbolic 3-manifolds has ordinal type \omega^\omega.

If a monarch reigns in more than one realm, he or she may carry different ordinals in each one, as some realms may have had different numbers of rulers of the same regnal name. For example, the same person was both King James I of England and King James VI of Scotland. The ordinal is not normally used for the first ruler of the name, but is used in historical references once the name is used again. Thus, Queen Elizabeth I of England was called simply "Elizabeth of England" until the accession of Queen Elizabeth II almost four centuries later in 1952; subsequent historical references to the earlier queen retroactively refer to her as Elizabeth I. However, Tsar Paul I of Russia, King Umberto I of Italy, King Juan Carlos I of Spain, Emperor Haile Selassie I of Ethiopia and Pope John Paul I all used the ordinal I (first) during their reigns, while Pope Francis does not.

Subtraction is a type of analysis called ordinal analysis > ...let space be now regarded as the field of progression which is to be > studied, and POINTS as states of that progression. ...I am led to regard the > word "Minus," or the mark −, in geometry, as the sign or characteristic of > analysis of one geometric position (in space), as compared with another > (such) position. The comparison of one mathematical point with another with > a view to the determination of what may be called their ordinal relation, or > their relative position in space... The first example of subtraction is to take the point A to represent the earth, and the point B to represent the sun, then an arrow drawn from A to B represents the act of moving or vection from A to B. ::B − A this represents the first example in Hamilton's lectures of a vector. In this case the act of traveling from the earth to the moon.

Upon receiving in "vision" a revelation of her distress and being commanded by the spirit of God to do so, Abel immediately took leave to attend to her, whereupon he "rebuked" for her sake the oppressive spirit so afflicting her, explaining, after the evil had fled, that he had been commanded to "rebuke the power that was destroying" her. The other ordinal action was on behalf of Eunice's husband. But for Mr. Kinney it was not an ordinance of blessing but one, rather, of biblical cursing — referred to as a "dusting of the feet." The same was revealed by Christ to His apostles (see Matt 10:14-15; Luke 10:10-12; Acts 13:51), and it was an ordinal measure that in July 1830 had been restored through modern revelation to Joseph Smith (D&C; 24:15; 84:92-95), and then directed and recommended, as prompted by the spirit of God, to ordained ministers of the gospel, of which Abel was one.

In mathematics, specifically in axiomatic set theory, a Hartogs number is a particular kind of ordinal number. In particular, if X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X can be well-ordered then the cardinal number of α is a minimal cardinal greater than that of X. If X cannot be well-ordered then there cannot be an injection from X to α. However, the cardinal number of α is still a minimal cardinal not less than or equal to the cardinality of X. (If we restrict to cardinal numbers of well-orderable sets then that of α is the smallest that is not not less than or equal to that of X.) The map taking X to α is sometimes called Hartogs's function. This mapping is used to construct the aleph numbers, which are all the cardinal numbers of infinite well- orderable sets.

Rathjen does not state how shrewd cardinals compare to unfoldable cardinals, however. λ-shrewdness is an improved version of λ-indescribability, as defined in Drake; this cardinal property differs in that the reflected substructure must be (Vα+λ, ∈, A ∩ Vα), making it impossible for a cardinal κ to be κ-indescribable. Also, the monotonicity property is lost: a λ-indescribable cardinal may fail to be α-indescribable for some ordinal α < λ.

Permutation tests exist for any test statistic, regardless of whether or not its distribution is known. Thus one is always free to choose the statistic which best discriminates between hypothesis and alternative and which minimizes losses. Permutation tests can be used for analyzing unbalanced designs and for combining dependent tests on mixtures of categorical, ordinal, and metric data (Pesarin, 2001) . They can also be used to analyze qualitative data that has been quantitized (i.e.

KXEG (1280 AM) is a radio station licensed to Phoenix, Arizona, United States, it serves the Phoenix area. The station is currently owned by Jacob Barker, through licensee Gabrielle Broadcasting Licensee Ordinal I FCC, LLC. First put on the air on October 23, 1956, the station has also gone by the call letters KHEP and KTKP, and it was said to be Arizona's oldest Christian radio station until it fell silent in February 2019.

Either Pearson's r, Kendall's τ, or Spearman's \rho can be used to measure pairwise correlation among raters using a scale that is ordered. Pearson assumes the rating scale is continuous; Kendall and Spearman statistics assume only that it is ordinal. If more than two raters are observed, an average level of agreement for the group can be calculated as the mean of the r, τ, or \rho values from each possible pair of raters.

Since the adoption of Arabic numerals, numbers have become written in Arabic numerals more and more often. Counters and ordinal numbers are typically written in Arabic numbers, such as 3人 (three people), 7月 (July, "seventh-month"), 20歳 (age 20), etc., although 三人、七月、and 二十歳 are also acceptable to write (albeit less common). However, numbers that are part of lexemes are typically written in kanji.

Implementations of indexed files vary between vendors, although common implementations, such as C‑ISAM and VSAM, are based on IBM's ISAM. Relative files, like indexed files, have a unique record key, but they do not have alternate keys. A relative record's key is its ordinal position; for example, the 10th record has a key of 10. This means that creating a record with a key of 5 may require the creation of (empty) preceding records.

Mokken scales can come in two forms: first as the Double Monotonicity model, where the items can differ in their difficulty. It is essentially an ordinal version of Rasch scale; and second, as the Monotone Homogeneity model, where items differ in their discrimination parameter, which means that there can be a weaker relationship between some items and the latent variable and other items and the latent variable. Double Monotonicity models are used most often.

A large number of tests were developed in the latter half of the 20th century (e.g., all the multivariate tests). Popular techniques (such as Hierarchical Linear Model, Arnold, 1992, Structural Equation Modeling, Byrne, 1996 and Independent Component Analysis, Hyvarinën, Karhunen and Oja, 2001) are relatively recent. In 1946, psychologist Stanley Smith Stevens organized levels of measurement into four scales, Nominal, Ordinal, Ratio, and Interval, in a paper that is still often cited.

Cook, Wade D. and Moshe Kress, "Ordinal Ranking with Intensity of Preference," Management Science (US), Vol. 31, No. 1 (Jan., 1985), pp. 26-32. In an analysis of voting, for example, the intensity of preference is a measure of an individual voter's (or group of voters') willingness to incur the costs or inconvenience of the act of officially registering a preferential choice at the time and place required, not the vote itself.

The general rule is that (for 1 and 2) or (for all other numbers, except , et cetera, but including and ) is appended to the numeral. The reason is that and respectively end the ordinal number words. The ordinals for 1 and 2 may however be given an form ( and instead of and ) when used about a male person (masculine natural gender), and if so they are written and . When indicating dates, suffixes are never used.

In the first taxonomy of Mollicutes, the classification was based on requiring or not requiring cholesterol for growth. The old order Mycoplasmatales consisted of two families: Mycoplasmataceae, which requires cholesterol, and the sterol-nonrequiring Acholeplasmataceae. In view of the many properties in which the acholeplasmas distinguish from species in Mycoplasmataceae and Spiroplasmataceae, Freundt et al. proposed in 1984 to elevate the family Acholeplasmataceae to the ordinal rank Acholeplasmatales, thus separating it from Mycoplasmatales.

Level of measurement or scale of measure is a classification that describes the nature of information within the values assigned to variables. Psychologist Stanley Smith Stevens developed the best-known classification with four levels, or scales, of measurement: nominal, ordinal, interval, and ratio. This framework of distinguishing levels of measurement originated in psychology and is widely criticized by scholars in other disciplines. Other classifications include those by Mosteller and Tukey, and by Chrisman.

Synesthesia is a neurological condition in which two or more bodily senses are coupled. For example, in a form of synesthesia known as Grapheme → color synesthesia, letters or numbers may be perceived as inherently colored. In another, called number → form synesthesia, numbers are automatically and consistently associated with locations in space. In yet another form of synesthesia, called ordinal linguistic personification, either numbers, days of the week, or months of the year evoke personalities.

The Lithuanian calendar is unusual among Western countries in that neither the names of the months nor the names of the weekdays are derived from Greek or Norse mythology. They were formalized after Lithuania regained independence in 1918, based on historic names, and celebrate natural phenomena; three months are named for birds, two for trees, and the remainder for seasonal activities and features. The days of the week are simply ordinal numbers.

As an aspect of color, value refers to how light or dark an object appears. Value effectively connotes "more" and "less," an ordinal measure; this makes it a very useful form of symbology in thematic maps, especially choropleth maps. Value contributes strongly to Visual hierarchy; elements that contrast most with the value of the background tend to stand out most (e.g., black on a white sheet of paper, white on a black computer screen).

However, the Western Schism was reinterpreted when Pope John XXIII (1958–1963) chose to reuse the ordinal XXIII, citing "twenty-two Johns of indisputable legitimacy." This is reflected in modern editions of the Annuario Pontificio, which extend Gregory XII's reign to 1415. The Pisan popes Alexander V and John XXIII are now considered to be antipopes. Gregory XII's resignation (in 1415) was the last time a pope resigned until Benedict XVI in 2013.

In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets, and is analogous to a set of non-zero measure in measure theory. There are at least three closely related notions of stationary set, depending on whether one is looking at subsets of an ordinal, or subsets of something of given cardinality, or a powerset.

The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the axiom of global choice, this need not be the case, and in such theories it is not necessarily true that the surreals are a universal ordered field.

Next to its upper course, on the two sides there were empty spaces where the remains of a market have been found. Being a shrine's approach, the avenue passes under three torii, or Shinto gates, called respectively Ichi no Torii (first gate), Ni no Torii (second gate) and San no Torii (third gate). The ordinal number decreases with the distance from the shrine, so the closest to Tsurugaoka Hachiman-gū is actually San no Torii.

The book built on ordinal utility and mainstreamed the now-standard distinction between the substitution effect and the income effect for an individual in demand theory for the 2-good case. It generalised the analysis to the case of one good and a composite good, that is, all other goods. It aggregated individuals and businesses through demand and supply across the economy. It anticipated the aggregation problem, most acutely for the stock of capital goods.

They asserted also that the Book of Common Prayer as a whole contained a strong sacrificial theology in the ordinal. They agreed that, at the time of the reunion of the churches under Queen Mary, many Edwardian priests were deprived for various reasons. They then demonstrated that not one priest was deprived on account of defect of order. Some were voluntarily reordained and others received anointing as a supplement to their previous ordination.

Instead they proposed recognizing five to eight families in a separate order, the Boraginales. In that system Boraginaceae is treated in a narrow sense (sensu stricto or s.s.). Subsequently the order Boraginales was added to the 2016 revision (APG IV) to include Boraginaceae at the ordinal level, in which it was the sole family. The consensus was to continue the broad usage rather than split it into separate families, based on its monophyletic composition.

PS satisfies an efficiency property called stochastic-dominace Pareto efficiency (sd-efficiency, also called: ordinal efficiency). Informally it means that, considering the resulting probability matrix, there is no other matrix that all agents weakly-sd-prefer and at least one agent strictly-sd- prefers. Here, the ex-ante notion of sd-efficiency is stronger than the ex- post notion: sd-efficiency implies that every allocation selected by the lottery is sd-Pareto-efficient.

When tested with displays containing items they had never seen before, they continued to respond to them in order. The authors conclude that monkeys can represent the numerosities 1 to 9 at least on an ordinal scale. Ants are able to use quantitative values and transmit this information. For instance, ants of several species are able to estimate quite precisely numbers of encounters with members of other colonies on their feeding territories.

In formal expressions, the ordinal number used before the word order refers to the highest order of derivative in the series expansion used in the approximation. The expressions: a zeroth-order approximation, a first-order approximation, a second-order approximation, and so forth are used as fixed phrases. The expression a zero order approximation is also common. Cardinal numerals are occasionally used in expressions like an order zero approximation, an order one approximation, etc.

Cladogram of the relations between the families of the Eodiscina, according to Jell, 1975 cited in The Eodiscina are mostly considered the more primitive suborder of the Agnostida, and the Agnostina the more advanced. Some scholars do not consider the Agnostina true trilobites, and consequently rejected the idea that they were related to the Eodiscina. Consequently, these scientists have proposed to elevate the group to ordinal level, which would thus be called Eodiscida Kobayashi, 1939.

Sierpiński authored 724 papers and 50 books, mostly in Polish. His book Cardinal and Ordinal Numbers was originally published in English in 1958. Two books, Introduction to General Topology (1934) and General Topology (1952) were translated into English by Canadian mathematician Cecilia Krieger. Another book, Pythagorean Triangles (1954), was translated into English by Indian mathematician Ambikeshwar Sharma, published in 1962, and republished by Dover Books in 2003; it also has a Russian translation.

Often there is a choice between Metric MDS (which deals with interval or ratio level data), and Nonmetric MDS (which deals with ordinal data). # Decide number of dimensions – The researcher must decide on the number of dimensions they want the computer to create. Interpretability of the MDS solution is often important, and lower dimensional solutions will typically be easier to interpret and visualize. However, dimension selection is also an issue of balancing underfitting and overfitting.

Lieutenant (; Lt) is a junior officer rank in the British Army and Royal Marines. It ranks above second lieutenant and below captain and has a NATO ranking code of OF-1 and it is the senior subaltern rank. Unlike some armed forces which use first lieutenant, the British rank is simply lieutenant, with no ordinal attached. The rank is equivalent to that of a flying officer in the Royal Air Force (RAF).

Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space, ω1 is often written as [0,ω1), to emphasize that it is the space consisting of all ordinals smaller than ω1. If the axiom of countable choice holds, every increasing ω-sequence of elements of [0,ω1) converges to a limit in [0,ω1). The reason is that the union (i.e.

In that case, Jensen's covering lemma holds: :For every uncountable set x of ordinals there is a constructible y such that x ⊂ y and y has the same cardinality as x. This deep result is due to Ronald Jensen. Using forcing it is easy to see that the condition that x is uncountable cannot be removed. For example, consider Namba forcing, that preserves \omega_1 and collapses \omega_2 to an ordinal of cofinality \omega.

In game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If one can change the identities of the players without changing the payoff to the strategies, then a game is symmetric. Symmetry can come in different varieties. Ordinally symmetric games are games that are symmetric with respect to the ordinal structure of the payoffs.

An alchemical master "sits at a table in the background surrounded by the symbols of alchemy: a silver crescent moon, a gold ball and a vessel used in experimentation". From the above-mentioned manuscript. The Ordinal of Alchemy is an alchemical poem composed of 3,102 lines of irregularly rhymed Middle English. In the beginning of the poem, Norton lambasts the naivete of others who have bastardized the ideas of the ancient masters of alchemy.

An n-ary partial operation ω from to Y is a partial function . An n-ary partial operation can also be viewed as an -ary relation that is unique on its output domain. The above describes what is usually called a finitary operation, referring to the finite number of operands (the value n). There are obvious extensions where the arity is taken to be an infinite ordinal or cardinal, or even an arbitrary set indexing the operands.

In proof theory, a branch of mathematical logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic, is the system of arithmetic with the usual elementary properties of 0, 1, +, ×, xy, together with induction for formulas with bounded quantifiers. EFA is a very weak logical system, whose proof theoretic ordinal is ω3, but still seems able to prove much of ordinary mathematics that can be stated in the language of first-order arithmetic.

A Relative Record Data Set (RRDS) is a type of data set organization used by the VSAM computer data storage system. Records are accessed based on their ordinal position in the file (relative record number, RRN). For example, the desired record to be accessed might be the 42nd record in the file out of 999 total. The concept of RRDS is similar to sequential access method, but it can access with data in random access and dynamic access.

Baumgartner, 2003, p. 142. Pope Gregory XIV (1590–1591) began the practice of creating cardinal-nephews whose formal appointment coincided de facto with their nomination, and was thus separate from the ordinal process for creating cardinals, and, when he fell ill, he authorized his cardinal-nephew, Paolo Emilio Sfondrato, to use the Fiat ut petitur, a power which was later diminished at the urging of the College.Tizon-Germe, Anne-Cécile, Levillain, ed., 2002, "Gregory XIV", p. 666.

Due to the frequent contacts made between the Li (黎族) and the Han (汉族) over a relatively lengthy stretch of time, the Hlai language has been influenced by the Chinese language and its grammar. As previously mentioned, the Hlai counting system for dates, ordinal numbers, and measurements have been influenced by Chinese. In this chapter, the Chinese influence in Hlai's word order of attribute phrases, verb-object-complement phrases, and interrogative sentences is discussed.

As part of her thesis work, in 1952, Morel found two different countable ordinal numbers whose squares are equal. After Wacław Sierpiński simplified her construction, they published it jointly. In 1955, Morel published a converse to the Knaster–Tarski theorem, according to which every incomplete lattice has an increasing function with no fixed point. Her 1965 paper with Thomas Frayne and Dana Scott, "Reduced direct products", provides the main definitions of reduced products in model theory.

This is an eponymous album as he used one of his stage names, Aleph-1. The concept of the album and its name, Aleph-1, derive from the theories of German mathematician Georg Cantor, who was a teacher in Halle, Saxony-Anhalt, Germany, a city, to which Alva Noto is deeply connected with through his family. In mathematical terms, \aleph_1 is the cardinality of the set of all countable ordinal numbers or a number of elements in endless successions.

The ability to recognize incorrect pitch (musical) is most often tested by using the Distorted Tunes Test (DTT). The DTT was originally developed in the 1940s and was used in large studies in the British population. The DTT measures musical pitch recognition ability on an ordinal scale, scored as the number of correctly classified tunes. More specifically the DTT is used to evaluate subjects on how well they judge whether simple popular melodies contain notes with incorrect pitch.

They again finished ahead of Cain-Gribble/LeDuc. Denney/Frazier struggled at the 2020 U.S. Championships, beginning in the short program where Denney fell on their throw and then popped their planned triple jump as well, resulting in them finishing sixth in that segment. Despite further side-by-side jump errors and another throw fall in the free, they rose one ordinal to fifth overall. On March 25, they announced that they were ending their partnership.

If the axiom of choice holds, then every cardinal κ has a successor, denoted κ+, where κ+ > κ and there are no cardinals between κ and its successor. (Without the axiom of choice, using Hartogs' theorem, it can be shown that for any cardinal number κ, there is a minimal cardinal κ+ such that \kappa^+ leq\kappa. ) For finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from the successor ordinal.

Rosser (1939) formally identified the three notions-as-definitions: Kleene proposes Church's Thesis: This left the overt expression of a "thesis" to Kleene. In his 1943 paper Recursive Predicates and Quantifiers Kleene proposed his "THESIS I": (22) references Church 1936; (23) references Turing 1936–7 Kleene goes on to note that: (24) references Post 1936 of Post and Church's Formal definitions in the theory of ordinal numbers, Fund. Math. vol 28 (1936) pp.11–21 (see ref.

The only doctrinal documents agreed upon in the Anglican Communion are the Apostles' Creed, the Nicene Creed of AD 325, and the Chicago-Lambeth Quadrilateral. Beside these documents, authorised liturgical formularies, such as Prayer Book and Ordinal, are normative. The several provincial editions of Prayer Books (and authorised alternative liturgies) are, however, not identical, although they share a greater or smaller amount of family resemblance. No specific edition of the Prayer Book is therefore binding for the entire Communion.

In set theory, Scott's trick is a method for giving a definition of equivalence classes for equivalence relations on a proper class (Jech 2003:65) by referring to levels of the cumulative hierarchy. The method relies on the axiom of regularity but not on the axiom of choice. It can be used to define representatives for ordinal numbers in ZF, Zermelo–Fraenkel set theory without the axiom of choice (Forster 2003:182). The method was introduced by .

"Ghusl tartibi" means an ordinal bath, performed in three stages. After washing away the najasat (e.g., semen or blood) from the body and after niyyat, the body has to be washed in three stages: head down to the neck; then the right side of the body from the shoulder down to the foot; then the left side of the body. Each part should be washed thoroughly in such a way that the water reaches the skin.

The species was first described by mycologists Jan Kohlmeyer, Brigitte Volkmann-Kohlmeyer, and Ove Eriksson in a 1996 Mycological Research publication. The generic name is derived from the Latin aquamarinus, meaning "a clear sea-green verging towards blue". The specific epithet speciosa is Latin for 'beautiful' or 'splendid', and refers to the "beautifully colored ascomata". The relationship of this taxon to other taxa within the Dothideomycetes is unknown (incertae sedis) with respect to ordinal and familial placement.

Baxter was consecrated on 9 March 2003 in Newmarket by the Rt. Rev'd Christopher Andrew Jukes of Calgary, Alberta, who at that time was a bishop in the Communion of Evangelical Episcopal Churches, using the traditional ordinal of the Book of Common Prayer (1962 Canada). He also established the Federation of Independent Anglican Churches of North America with himself as self-styled archbishop; this organisation was incorporated by Federal Canadian Letters Patent on 1 October 2003.

Recent simulation studies in the field of psychometrics suggest that the parallel analysis, minimum average partial, and comparative data techniques can be improved for different data situations. For example, in simulation studies, the performance of the minimum average partial test, when ordinal data is concerned, can be improved by utilizing polychoric correlations, as opposed to Pearson correlations. Courtney (2013) details how each of these three procedures can be optimized and carried out simultaneously from within the SPSS interface.

Non-parametric methods are widely used for studying populations that take on a ranked order (such as movie reviews receiving one to four stars). The use of non-parametric methods may be necessary when data have a ranking but no clear numerical interpretation, such as when assessing preferences. In terms of levels of measurement, non-parametric methods result in "ordinal" data. As non-parametric methods make fewer assumptions, their applicability is much wider than the corresponding parametric methods.

It was published by Erdős and Rado in 1956. Rado's theorem is another Ramsey-theoretic result concerning systems of linear equations, proved by Rado in his thesis. The Milner–Rado paradox, also in set theory, states the existence of a partition of an ordinal into subsets of small order-type; it was published by Rado and E. C. Milner in 1965. The Erdős–Ko–Rado theorem can be described either in terms of set systems or hypergraphs.

If the transfinite upper central series stabilizes at the whole group, then the group is called hypercentral. Hypercentral groups enjoy many properties of nilpotent groups, such as the normalizer condition (the normalizer of a proper subgroup properly contains the subgroup), elements of coprime order commute, and periodic hypercentral groups are the direct sum of their Sylow p-subgroups . For every ordinal λ there is a group G with Zλ(G) = G, but Zα(G) ≠ G for α < λ, and .

The only contemporary home version of Space Dungeon was for the Atari 5200 system in 1983. The game cartridge came prepackaged with a dual- controller holder, allowing players to snap two stock controllers in and play like in the arcade. The game differs from the arcade original in that most of the objects are approximately four times the size, spores can be launched by enemies only in the eight ordinal directions, and the enemies are less aggressive.

Since September 2010 Feld has been professor for Economic Policy and Ordinal Economics at the University of Freiburg and serves as lead executive at the Walter Eucken Institute. Since March 2011, Feld has been a member of the German Council of Economic Experts; in 2020, he became the body's chairman. Since 2013, he has also been serving on the advisory board of the Stability Council, a body devised as part of Germany’s national implementation of the European Fiscal Compact.

In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC. The rank of a well-founded set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set.; ; .

Adolf Lindenbaum (12 June 1904 – August 1941), was a Polish-Jewish logician and mathematician best known for Lindenbaum's lemma and Lindenbaum algebras. He was born and brought up in Warsaw. He earned a Ph.D. in 1928 under Wacław Sierpiński and habilitated at the University of Warsaw in 1934. He published works on mathematical logic, set theory, cardinal and ordinal arithmetic, the axiom of choice, the continuum hypothesis, theory of functions, measure theory, point-set topology, geometry and real analysis.

They intended to add multiple weapons, but chose to keep their first one—the bow and arrow—due to its feel. The arrow was designed to fire without charging and to bias towards targets so as to give the player "more leeway". Thorson also chose to limit the aim direction to the eight ordinal directions rather than affording complex 360 degree controls. They also added levels, items, a store, and a story based on ascending a tower.

For abstract large numbers the numeral suffix -zig (as in zwanzig = 20, vierzig = 40, sechzig = 60) is used like 'umpteen': Das habe ich schon zigmal gesagt! ('I already said so umpteen times'). An unknown ordinal number is was-weiß-ich-wievielte/r/s ('what do I know how many-th') or drölf (fictional integer whose name is a portmanteau of the words zwölf, 12, and dreizehn, 13). Exponents of 10 are also used as in English.

The Practice and Theory of Individual Psychology is a work on psychology by Alfred Adler, first published in 1924. In his work, Adler develops his personality theory, suggesting that the situation into which a person is born, such as family size, sex of siblings, and birth order, plays an important part in personality development.Dimond, Richard E., and David C. Munz. "Ordinal position of birth and self-disclosure in high school students." Psychological Reports 21, no. 3 (1967): 829-833.

If there is a set in that is a standard model of ZF, and the ordinal is the set of ordinals that occur in , then is the of . If there is a set that is a standard model of ZF, then the smallest such set is such a . This set is called the minimal model of ZFC. Using the downward Löwenheim–Skolem theorem, one can show that the minimal model (if it exists) is a countable set.

The paradoxes of naive set theory can be explained in terms of the inconsistent tacit assumption that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper (i.e., that they are not sets). For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper.

In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called the rank of the Borel set. The Borel hierarchy is of particular interest in descriptive set theory. One common use of the Borel hierarchy is to prove facts about the Borel sets using transfinite induction on rank.

Hyperarithmetical theory studies those sets that can be computed from a computable ordinal number of iterates of the Turing jump of the empty set. This is equivalent to sets defined by both a universal and existential formula in the language of second order arithmetic and to some models of Hypercomputation. Even more general recursion theories have been studied, such as E-recursion theory in which any set can be used as an argument to an E-recursive function.

It is generally also the tone most often repeated in the piece, and finally the range delimits the upper and lower tones for a given mode. The eight modes can be further divided into four categories based on their final (finalis). Medieval theorists called these pairs maneriae and labeled them according to the Greek ordinal numbers. Those modes that have d, e, f, and g as their final are put into the groups protus, deuterus, tritus, and tetrardus respectively.

Types of goods in economics. Goods' diversity allows for their classification into different categories based on distinctive characteristics, such as tangibility and (ordinal) relative elasticity. A tangible good like an apple differs from an intangible good like information due to the impossibility of a person to physically hold the latter, whereas the former occupies physical space. Intangible goods differ from services in that final (intangible) goods are transferable and can be traded, whereas a service cannot.

For instance: ` myColor = TRIANGLE ` can be forbidden, whilst ` myColor = RED ` is accepted, even if TRIANGLE and RED are both internally represented as 1. Conceptually, an enumerated type is similar to a list of nominals (numeric codes), since each possible value of the type is assigned a distinctive natural number. A given enumerated type is thus a concrete implementation of this notion. When order is meaningful and/or used for comparison, then an enumerated type becomes an ordinal type.

Formally, Southwark Cathedral is "the cathedral and collegiate church of St Saviour and St Mary Overie, Southwark" - a Church of England cathedral, whose origins lie in a convent established on the south bank of the River Thames in the year 606 AD.The APC Southwark Cathedral Ordinal booklet 19 October 2013 A Joint Service of Celebration, Ordination & Licensing of Recognised Ministers hosted by the Apostolic Pastoral Congress and Awards by the Order of St Hadrian of Canterbury (51 pages).

1: un; 2: du; 3: tri; 4: kwer; 5: pin; 6: ses; 7: sep; 8: oc; 9: nev; 10: des; 100: sunte; 1000: tilie. 357: trisunte pindes-sep. Ordinal numbers are formed by adding -i or -j (after a vowel): duj: "second"; trij: "third", kweri: "fourth", pini: "fifth"; the exception is pri: "first". Fractions are formed by adding -t to numbers: u trit: "a third", u kwert: "a fourth, a quarter"; the exception is mij: "half".

The 271st Infantry Division () was an infantry division of the German Heer during World War II. In total, three infantry formations used the ordinal number 271 within the Wehrmacht. The first 271st Infantry Division's deployment was aborted in May 1940, whereas the second iteration of the division saw its deployment completed in November 1943 and was destroyed in August 1944. Subsequently, a division designated 271st Volksgrenadier Division () was deployed in August 1944 and remained in combat until 1945.

This covers the definition and basic properties of cardinals. A cardinal is defined to be an equivalence class of similar classes (as opposed to ZFC, where a cardinal is a special sort of von Neumann ordinal). Each type has its own collection of cardinals associated with it, and there is a considerable amount of bookkeeping necessary for comparing cardinals of different types. PM define addition, multiplication and exponentiation of cardinals, and compare different definitions of finite and infinite cardinals.

383 – 390, 1967 Existence of a partition of the ordinal number \omega_2 into two colors with no monochromatic uncountable sequentially closed subset is independent of ZFC, ZFC + CH, and ZFC + ¬CH, assuming consistency of a Mahlo cardinal.Shelah, S., Proper and Improper Forcing, Springer 1992Schlindwein, Chaz, Shelah's work on non-semiproper iterations I, Archive for Mathematical Logic (47) 2008 pp. 579 – 606Schlindwein, Chaz, Shelah's work on non-semiproper iterations II, Journal of Symbolic Logic (66) 2001, pp.

Magnus Haakonsson (, ; 1 (or 3) May 1238 – 9 May 1280) was King of Norway (as Magnus VI) from 1263 to 1280 (junior king from 1257).Magnus 6 Håkonsson Lagabøte – utdypning (Store norske leksikon) One of his greatest achievements was the modernisation and nationalisation of the Norwegian law-code, after which he is known as Magnus the Law-mender (, ). He was the first Norwegian monarch known to have used an ordinal number, although originally counting himself as "IV".

That original plat was 28 blocks with a central town square. the town, with a population of 261, essentially still reflects that platting and its relation to the surrounding farmland. The town square is the central ordinal for directional prefixes (N,S,E,W) and numbering (beginning with hundreds) of street names and addresses. Town streets are uncurbed and typically wide and have at grade drains with metal grills flowing into an underground storm sewer system.

If α is a limit ordinal, an α-inaccessible is a fixed point of every ψβ for β < α (the value ψα(λ) is the λth such cardinal). This process of taking fixed points of functions generating successively larger cardinals is commonly encountered in the study of large cardinal numbers. The term hyper-inaccessible is ambiguous and has at least three incompatible meanings. Many authors use it to mean a regular limit of strongly inaccessible cardinals (1-inaccessible).

On August 2, a video detailing the game's Summer Games event contained a code in Base64 cipher that decrypted into a salted hash requiring a key for further decryption. Players also discovered clues in the video arranging nine playable heroes into cardinal and ordinal directions. Where previous puzzles were solved in hours, players struggled to make progress for days. On August 4, a comment from developer Jeffrey Kaplan prompted players to search the sky of the Dorado map.

Each skater had to complete four voluntary figures. Scores from 0 to 6 were given for each figure for both (a) content (difficulty and novelty) and (b) performance. The total possible score was therefore 48. Each judge would then arrange the skaters in order of total score by that judge; these ordinal rankings were used to provide final placement for the skaters, using a "majority rule"--if a majority of the judges ranked a pair first, the pair won.

Each skater performed a five-minute free skate. Scores from 0 to 6 were given for each figure for both (a) content (difficulty and variety) and (b) performance. The total possible score was therefore 12. Each judge would then arrange the pairs in order of total score by that judge; these ordinal rankings were used to provide final placement for the pairs, using a "majority rule"--if a majority of the judges ranked a pair first, the pair won.

When Queen Elizabeth came to the throne in 1558, a solution was thought to have been found. To minimise bloodshed over religion in her dominions, the religious settlement between the factions of Rome and Geneva was brought about. It was compellingly articulated in the development of the 1559 Book of Common Prayer, the Thirty- Nine Articles, the Ordinal, and the two Books of Homilies. These works, issued under Archbishop Matthew Parker, were to become the basis of all subsequent Anglican doctrine and identity.

The two won the 2017 CS Finlandia Trophy, their first international gold medal together. The Grand Prix was a disappointment, with Peng/Jin finishing fifth at both the 2017 Skate America and 2017 Internationaux de France. At the 2018 Chinese Championships, they finished second behind Yu/Zhang, and were named to China's team for the 2018 Winter Olympics. Peng/Jin competed in the pairs event in Pyeongchang, finishing seventeenth in the short program and thus missing the free skate by a single ordinal.

Unlike the Europeans all 3 teams in position to win gold simply by winning the free dance. In the free dance they received 3 1st place ordinals and 6 2nd place ordinals, but lost the gold to Grishuk & Platov who received 5 1st place ordinals, 1 2nd place ordinal, and 3 3rd place ordinals, losing the free dance and gold based on the majority rule, despite having no judges place them 3rd and a lower total of ordinals than Grishuk & Platov.

During his patriarchate he spared no effort to improve the relations both with the Holy See and within the Chaldean Church, after the eventful reign of his predecessor Joseph Audo. He died in Mosul at the age of 54 on June 27, 1894. The ordinal number of his title is sometime XIV, sometime XIII, while among scholars Eliya XII is often preferred. This is due to the uncertain list of the Patriarchal line of Alqosh in the 16th and 17th century.

Naruhito, Emperor of Japan since 2019 Monarchs and other royalty, for example Napoleon, have traditionally availed themselves of the privilege of using a mononym, modified when necessary by an ordinal or epithet (e.g., Queen Elizabeth II or Charles the Great). This is not always the case: King Carl XVI Gustaf of Sweden has two names. While many European royals have formally sported long chains of names, in practice they have tended to use only one or two and not to use surnames.

The two won the 2017 CS Finlandia Trophy, their first international gold medal together. The Grand Prix was a disappointment, with Peng/Jin finishing fifth at both the 2017 Skate America and 2017 Internationaux de France. At the 2018 Chinese Championships, they finished second behind Yu/Zhang, and were named to China's team for the 2018 Winter Olympics. Peng/Jin competed in the pairs event in Pyeongchang, finishing seventeenth in the short program and thus missing the free skate by a single ordinal.

The family of sets of uniqueness, considered as a set inside the space of compact sets (see Hausdorff distance), was located inside the analytical hierarchy. A crucial part in this research is played by the index of the set, which is an ordinal between 1 and ω1, first defined by Pyatetskii- Shapiro. Nowadays the research of sets of uniqueness is just as much a branch of descriptive set theory as it is of harmonic analysis. See the Kechris- Louveau book referenced below.

The two indices differ only with respect to scale and origin. Thus if one is concave, so is the other, in which case there is often said to be diminishing marginal utility. Thus the use of cardinal utility imposes the assumption that levels of absolute satisfaction exist, so that the magnitudes of increments to satisfaction can be compared across different situations. In consumer choice theory, ordinal utility with its weaker assumptions is preferred because results that are just as strong can be derived.

Somewhat unexpectedly, there was support for a clade comprising Opiliones, Ricinulei and Solifugae, a combination not found in most other studies. In early 2019, a molecular phylogenetic analysis placed the horseshoe crabs, Xiphosura, as the sister group to Ricinulei. It also grouped pseudoscorpions with mites and ticks, which the authors considered may be due to long branch attraction. Morphological analyses including fossils tend to recover the Tetrapulmonata, including the extinct group the Haptopoda, but recover other ordinal relationships with low support.

On string instruments, a string change is a change from playing on one string to another. This may also involve a simultaneous change in fingering and/or position (shift), all of which must be done skillfully to avoid noticeable string noise. String may be indicated through Roman numerals (I-IV) or simply the string's base note's letter (e.g. - A, E, G, etc.), fingering may be indicated through numbers for the fingers (1-4), and position may be indicated through ordinal numbers (e.g. 2nd).

"Ordinary" comes from the same root as our word "ordinal", and in this sense means "the counted weeks". In the Roman Catholic Church and in some Protestant traditions, these are the common weeks which do not belong to a proper season. In Latin, these seasons are called the weeks per annum, or "through the year". In the current form of the Roman Rite adopted following the Second Vatican Council, Ordinary Time consists of 33 or 34 Sundays and is divided into two sections.

As for the Wadge lemma, this holds for any pointclass Γ, assuming the axiom of determinacy. If we associate with each set A the collection of all sets strictly below A on the Wadge hierarchy, this forms a pointclass. Equivalently, for each ordinal α ≤ θ the collection Wα of sets that show up before stage α is a pointclass. Conversely, every pointclass is equal to some Wα. A pointclass is said to be self-dual if it is closed under complementation.

The Annuario Pontificio has historically recognized the decisions of the Council of Pisa (1409). Until the mid-20th century, the Annuario Pontificio listed Gregory XII's reign as 1406–1409, followed by Alexander V (1409–1410) and John XXIII (1410–1415). However, the Western Schism was reinterpreted when Pope John XXIII (1958–1963) chose to reuse the ordinal XXIII, citing "twenty-two Johns of indisputable legitimacy". This is reflected in modern editions of the Annuario Pontificio, which extend Gregory XII's reign to 1415.

The lakes in the chain have been given ordinal designations: First Lake, Second Lake, Third Lake, Fourth Lake, Fifth Lake, Sixth Lake, Seventh Lake, and Eighth Lake. The chain begins with a small lake not counted in the series called Old Forge Pond. In reality, First, Second and Third Lake are one long lake separated by narrow straits. At the east end of Fourth Lake, by Inlet in Hamilton County, a stream, or channel, allows access to the small Fifth Lake.

In computer programming, cohesion refers to the degree to which the elements inside a module belong together. In one sense, it is a measure of the strength of relationship between the methods and data of a class and some unifying purpose or concept served by that class. In another sense, it is a measure of the strength of relationship between the class's methods and data themselves. Cohesion is an ordinal type of measurement and is usually described as “high cohesion” or “low cohesion”.

If is any standard model of ZF sharing the same ordinals as , then the defined in is the same as the defined in . In particular, is the same in and , for any ordinal . And the same formulas and parameters in Def () produce the same constructible sets in . Furthermore, since is a subclass of and, similarly, is a subclass of , is the smallest class containing all the ordinals that is a standard model of ZF. Indeed, is the intersection of all such classes.

Figure skating was formerly judged on a 6.0 scale. This scale is sometimes called "the old scale", or "old system". Skaters were judged on "technical merit" (in the free skate), "required elements" (in the short program), and "presentation" (in both programs). The marks for each program ran from 0.0 to 6.0 and were used to determine a preference ranking, or "ordinal", separately for each judge; the judges' preferences were then combined to determine placements for each skater in each program.

13r-155v) and an antiphonary fragment (ff. 156r-162v) which has the Matins for Palm Sunday, St. Blasius and St. Hylarius in the conventional liturgical order, but with tonal rubrics.This form was also used in a contemporary Aquitanian abridged antiphonary or breviary (Paris, Bibliothèque Nationale, fonds lat., Ms. 1085), the only difference is that every chant is just represented by an incipit and that the tonal classification is a Latin ordinal according to the system of Hucbald (tonus I-VIII).

Osmund invented or introduced little himself, though the Sarum rite had some peculiarities distinct from that of other churches. He made selections of the practices he saw round him and arranged the offices and services. Intended primarily for his own diocese, the Ordinal of Osmund, regulating the Divine Office, Mass, and Calendar, was used, within a hundred years, almost throughout England, Wales, and Ireland, and was introduced into Scotland about 1250. The unifying influence of the Norman Conquest made its spread more easy.

So, one can say that there is only non-numerical (ordinal), non-exact (interval), and non- complete information (NNN-information) I about weight-coefficient. As information I about weights is incomplete, then weight-vector w=(w(1),…,w(m)) is ambiguously determined, i.e., this vector is determined with accuracy to within a set W(I) of all admissible (from the point of view of NNN-information I) weight-vectors. To model such uncertainty we shall address ourselves to the concept of Bayesian randomization.

The 66th Confluence scout group was formed in November 2012 following a BPSA Brownsea Training Camp that took place September 14–16, 2012 at Klondike Park in Augusta, Missouri. Several of the newly received Rover Squires from that training returned to St. Louis to begin the formation of the scout group, the ordinal "66" coming from the U.S. Route 66 highway that once crossed St. Louis, as well as the confluence of the Missouri and Mississippi Rivers, which is also in St. Louis.

All the members of the two extant genera, Aspidodiadema and Plesiodiadema, are found in tropical seas at bathyal and abyssal depths, often on the submarine slopes of islands. The genera Culozoma, Eosalenia, and Gymnotiara are known only from the fossil record. Study of the larval development of Aspidodiadema jacobyi suggests that the family should be elevated to ordinal status as a sister clade of the order Diadematoida, or possibly be regarded as a sister clade of the other families within that order.

The chiefs have their own territories, and apart from overseeing them, they have a function at the courts of their paramount chiefs as their ministers. Most of the functions are traditional, while some have been created recently: A chief arbitrates and decides political and economic questions in his area. When he is installed, he receives a stool name. Usually, all chiefs who belong to a reigning lineage have the same name – an ordinal being added to distinguish between all of them.

However the name was not applied to a new state; both England and Scotland continued to be governed independently. Its validity as a name of the Crown is also questioned, given that monarchs continued using separate ordinals (e.g., James VI/I, James VII/II) in England and Scotland. To avoid confusion historians generally avoid using the term King of Great Britain until 1707 and instead to match the ordinal usage call the monarchs kings or queens of England and New Zealand.

The women's single skating competition of the 1960 Winter Olympics was held at the Blyth Arena in Squaw Valley, California, United States. The compulsory figures section took place on Sunday 21 February 1960 with the free skating section concluding the event two days later. Each judge ranked each skater by Ordinal Placement from first to last place. If a skater was ranked first by a majority of the judges, that skater was placed first overall, this process was repeated for each place.

The men's single skating competition of the 1960 Winter Olympics was held at the Blyth Arena in Squaw Valley, California, United States. The compulsory figures section took place on Wednesday 24 February 1960 with the free skating section concluding the event two days later. Each judge ranked each skater by Ordinal Placement from first to last place. If a skater was ranked first by a majority of the judges, that skater was placed first overall, this process was repeated for each place.

The more recent shorter convention is that an act amending "Foo Act yyy1" will have short title "Foo (Amendment) Act yyy2". If a law is passed with the same title as another law passed in the same year, an ordinal number will be added to distinguish it from the others; this is particularly common for Finance Acts (Finance (No. 3) Act 2010) and commencement orders that bring parts of an Act into force (Environment Act 1995 (Commencement No.13) (Scotland) Order 1998).

Archaeolepis mane is the earliest known named Lepidopteran fossil. It dates from the Lower Jurassic (ca ) and according to Grimaldi & Engel (2005) a recent re-examination of the specimen has given additional support to its ordinal placement. The fossil consists of a pair of wings with scales that are characteristically similar to the wing venation pattern found in Trichoptera (caddisflies). The fossil was found in the Charmouth Mudstone Formation, Dorset, United Kingdom by J. F. Jackson (1894-1966), of Charmouth.

If λ ≤ κ is a limit ordinal and κ is α-inaccessible for all α < λ, then every β < λ is also less than α for some α < λ. So this case is trivial. In particular, κ is κ-inaccessible and thus hyper-inaccessible. To show that κ is a limit of hyper-inaccessibles and thus 1-hyper-inaccessible, we need to show that the diagonal set of cardinals μ < κ which are α-inaccessible for every α < μ is club in κ.

Linear trends are also used to find associations between ordinal data and other categorical variables, normally in a contingency tables. A correlation r is found between the variables where r lies between -1 and 1. To test the trend, a test statistic: : M^2 = (n-1)r^2 is used where n is the sample size. R can be found by letting u_1 \leq u_2 \leq ... \leq u_I be the row scores and v_1 \leq v_2 \leq ... \leq v_I be the column scores.

In the 2000s understanding of the relationships among eutherian mammals has experienced a virtual revolution. Molecular phylogenomics, new fossil finds and innovative morphological interpretations now group the more than 4600 extant species of eutherians into four major super-ordinal clades: Euarchontoglires (including Primates, Dermoptera, Scandentia, Rodentia, and Lagomorpha), Laurasiatheria (Cetartiodactyla, Perissodactyla, Carnivora, Chiroptera, Pholidota, and Eulipotyphla), Xenarthra, and Afrotheria (Proboscidea, Sirenia, Hyracoidea, Afrosoricida, Tubulidentata, and Macroscelidea). This tree is very useful in unifying the parts of a puzzle in comparative mammalian cytogenetics.

These superscripts typically share a baseline with numerator digits, the top of which are aligned with the top of the full-height numerals of the base font; lowercase ascenders may extend above. Ordinal indicators are sometimes written as superscripts (, , , , rather than 1st, 2nd, 3rd, 4th), although many English-language style guides recommend against this use. Other languages use a similar convention, such as 1er or 2e in French, or 4ª and 4º in Portuguese, Galician and Italian, or 4.ª and 4.

However, the statistical techniques employed by Michell (1990) in testing Thurstone's theory and multidimensional scaling did not take into consideration the ordinal constraints imposed by the cancellation axioms . , Kyngdon (2006), Michell (1994) and tested the cancellation axioms of upon the interstimulus midpoint orders obtained by the use of Coombs' (1964) theory of unidimensional unfolding. Coombs' theory in all three studies was applied to a set of six statements. These authors found that the axioms were satisfied, however, these were applications biased towards a positive result.

It produces a composite score in order to provide an ordinal ranking of countries on the impact of terrorism. The GTI is based on data from the Global Terrorism Database (GTD) which is collected and collated by the National Consortium for the Study of Terrorism and Responses to Terrorism (START) at the University of Maryland. The GTD has codified over 190,000 cases of terrorism, it covers 163 countries, consisting 99.7% of the world's population. The GTI was developed in consultation with the Global Peace Index expert panel.

For ordinal variables the median can be calculated as a measure of central tendency and the range (and variations of it) as a measure of dispersion. For interval level variables, the arithmetic mean (average) and standard deviation are added to the toolbox and, for ratio level variables, we add the geometric mean and harmonic mean as measures of central tendency and the coefficient of variation as a measure of dispersion. For interval and ratio level data, further descriptors include the variable's skewness and kurtosis.

Gentzen's consistency proof is a result of proof theory in mathematical logic, published by Gerhard Gentzen in 1936. It shows that the Peano axioms of first- order arithmetic do not contain a contradiction (i.e. are "consistent"), as long as a certain other system used in the proof does not contain any contradictions either. This other system, today called "primitive recursive arithmetic with the additional principle of quantifier-free transfinite induction up to the ordinal ε0", is neither weaker nor stronger than the system of Peano axioms.

The high point allows for panoramic views over five counties; landmarks such as Lincoln Cathedral and Bolsover Castle can be seen from the summit. The site also overlooks the Elizabethan Hardwick Hall, as well as the more modern M1 motorway. The flat area of the viewpoint was originally laid out with stone blocks, at the ordinal points of the compass. A bronze statue of a kneeling coal miner with a Davy lamp was a later addition, located on a rock plinth in the centre of the viewpoint.

The twenty arrondissements are arranged in the form of a clockwise spiral (often likened to a snail shell), starting from the middle of the city, with the first on the Right Bank (north bank) of the Seine. In French, notably on street signs, the number is often given in Roman numerals. For example, the Eiffel Tower belongs to the VIIe arrondissement while Gare de l'Est is in the Xe arrondissement. In daily speech, people use only the ordinal number corresponding to the arrondissement, e.g.

A regnal year is a year of the reign of a sovereign, from the Latin regnum meaning kingdom, rule. Regnal years considered the date as an ordinal, not a cardinal number. For example, a monarch could have a first year of rule, a second year of rule, a third year of rule, and so on, but not a zeroth year of rule. Applying this ancient epoch system to modern calculations of time, which include zero, is what led to the debate over when the third millennium began.

So apartment 8º-D (not 8D) means the 8th floor (hence the character "º" meaning ordinal number), apartment D (counting in clockwise direction, for those who are in the floor entrance). But a very common form for buildings with three apartments per floor is, Esq.-Frt./Fte. (Frente, en: Front - for the apartment located between left and right)-Dto. These universal rules simplify finding an apartment in a building, particularly for blind people, who do not need to ask where a given apartment is.

Hence, George's preferences can also be represented by the following function v: :v(A)=9, v(B)=2, v(C)=1 The functions u and v are ordinally equivalent – they represent George's preferences equally well. Ordinal utility contrasts with cardinal utility theory: the latter assumes that the differences between preferences are also important. In u the difference between A and B is much smaller than between B and C, while in v the opposite is true. Hence, u and v are not cardinally equivalent.

It also has first tone when used as an ordinal number (or part of one), and when it is immediately followed by any digit (including another ; hence both syllables of the word yīyī and its compounds have first tone). # When is used between two reduplicated words, it may become neutral in tone (e.g. kànyikàn ("to take a look of")). The numbers qī ("seven") and bā ("eight") sometimes display similar tonal behavior as yī, but for most modern speakers they are always pronounced with first tone.

The confusion between cardinality and measurability was not to be solved until the works of Armen Alchian, William Baumol, and John Chipman. The title of Baumol's paper, "The cardinal utility which is ordinal", expressed well the semantic mess of the literature at the time. It is helpful to consider the same problem as it appears in the construction of scales of measurement in the natural sciences. In the case of temperature there are two degrees of freedom for its measurement - the choice of unit and the zero.

Screenshot of gameplay In Puzzlejuice, the player turns falling tetrominos into letters, and those letters into words and points. The player taps and drags on the touchscreen to rotate and position multicolored tetrominos that fall from the top of the screen. When the player completes a solid row of tiles, or arranges the fallen blocks such that four or more like- colored tiles touch, the color tiles turn into letters. Players connect these letter tiles with their eight adjacent tiles (in ordinal directions) to make words.

In the Ruscio and Roche study (2012), the OC method was correct 74.03% of the time rivaling the PA technique (76.42%). The AF method was correct 45.91% of the time with a tendency toward under-estimation. Both the OC and AF methods, generated with the use of Pearson correlation coefficients, were reviewed in Ruscio and Roche's (2012) simulation study. Results suggested that both techniques performed quite well under ordinal response categories of two to seven (C = 2-7) and quasi- continuous (C = 10 or 20) data situations.

In organic chemistry, the carbon number of a compound is the number of carbon atoms in each molecule.Nava Dayan, Lambros Kromidas (ed.) Formulating, Packaging, and Marketing of Natural Cosmetic Products John Wiley & Sons, 2011; ; page 218 The properties of hydrocarbons can be correlated with the carbon number, although the carbon number alone does not give an indication of the saturation of the organic compound. When describing a particular molecule, the "carbon number" is also the ordinal position of a particular carbon atom in a chain.

Public transportation was installed, and industry began to develop in the area. As designers of University Park, Methodist Episcopal officials aligned streets along a northeastsouthwest axis and a northwestsoutheast axis in order to maximize exposure to sunlight. Church officials also chose street names in remembrance not only of institutions of higher learning but also persons and places important to Methodists of the time. As a result, University Park is a neighborhood without an ordinal relationship among streets; therefore, navigation requires knowledge of all streets.

The multivariate probit model is a standard method of estimating a joint relationship between several binary dependent variables and some independent variables. For categorical variables with more than two values there is the multinomial logit. For ordinal variables with more than two values, there are the ordered logit and ordered probit models. Censored regression models may be used when the dependent variable is only sometimes observed, and Heckman correction type models may be used when the sample is not randomly selected from the population of interest.

The son of Sir George Norton of Abbots Leigh in Somerset, he was great-grandson of Thomas Norton, author of the Ordinal of Alchemy. He studied for some time at St John's College, Cambridge, but records show no degree. On the death of his father, in 1584, he succeeded to the estates. Early in 1585 he was in the commission of the peace for the county, but apparently suffered removal; he was reappointed in October 1589, on the recommendation of Thomas Godwin, bishop of Bath and Wells .

The lay peers joined the bishops in their opposition and succeeded in amending the bill considerably. The Ordinal and Prayer Book provisions were removed and the Mass left unchanged, with the exception of allowing communion under both kinds. The Pope's authority was removed, but rather than granting the Queen the title of Supreme Head, it merely said she could adopt it herself. This bill would have returned the Church to its position at the death of Henry VIII rather than to that when Edward VI died.

For example, they can be constructed by taking powersets, or they can be separated as subsets of sets already "given". This, he says, eliminates contradictory ideas like "the set of all sets" or "the set of all ordinal numbers". He disposes of the Russell paradox by means of this Theorem: "Every set M possesses at least one subset M_0 that is not an element of M ". Let M_0 be the subset of M for which, by AXIOM III, is separated out by the notion "x otin x".

Dodgson's' method elects a Condorcet winner if there is one, and otherwise elects the candidate who can become the Condorcet winner after the fewest ordinal preference swaps on voters' ballots. Later-No-Help can be considered not applicable to Dodgson if the method is assumed to not accept truncated preference listings from the voter. On the other hand, Later- No-Help can be applied to Dodgson if the method is assumed to apportion possible rankings among unlisted candidates equally, as shown in the example below.

The Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek. KP is considerably weaker than Zermelo–Fraenkel set theory (ZFC), and can be thought of as roughly the predicative part of ZFC. The consistency strength of KP with an axiom of infinity is given by the Bachmann–Howard ordinal. Unlike ZFC, KP does not include the power set axiom, and KP includes only limited forms of the axiom of separation and axiom of replacement from ZFC.

Those lists also both designated ordinal teams as first team, second team, etc., while the McDonald's and Jordan Brand teams were selected without distinction among selectees. In basketball, some All-American teams are composed by position, while others were not. In 2006, 47 boys' high-school basketball players were selected to major honorary All-American teams. For the 2006 selections, McDonald's selected 24 players to its All-American team; the Jordan Brand included 21 players, and the USA Today All-USA prep basketball team had 15.

In fact the contested distinction between the seventh and eighth tones surrounds the very issue of tone sandhi (between glottal stop (-m) and low rising (-d) tones). High and high-falling tones (marked by -b and -j in the RPA orthography, respectively) trigger sandhi in subsequent words bearing particular tones. A frequent example can be found in the combination for numbering objects (ordinal number + classifier + noun): ib (one) + tus (classifier) + dev (dog) > ib tug dev (note tone change on the classifier from -s to -g).

Approval voting experts describe sincere votes as those "... that directly reflect the true preferences of a voter, i.e., that do not report preferences 'falsely.'" They also give a specific definition of a sincere approval vote in terms of the voter's ordinal preferences as being any vote that, if it votes for one candidate, it also votes for any more preferred candidate. This definition allows a sincere vote to treat strictly preferred candidates the same, ensuring that every voter has at least one sincere vote.

The mobilization model for the Wehrmacht's active and reserve forces in multiple waves was first issued in the annual mobilization plan of 8 December 1938. The system initially had four waves, the first of which would be the peacetime army and the other three raised in anticipation of the invasion of Poland. The first wave (the peacetime army) consisted of divisions with ordinal numbers of one to 50. The second wave, reservists who had completed their compulsory training, consisted of divisions numbered 51 to 100.

Thorne grouped most of his Magnolianae into two large orders, Magnoliales and Berberidales, although his Magnoliales was divided into suborders along lines similar to the ordinal groupings used by both Cronquist and Dahlgren. Thorne revised his system in 2000, restricting the name Magnoliidae to include only the Magnolianae, Nymphaeanae, and Rafflesianae, and removing the Berberidales and other previously included groups to his subclass Ranunculidae. This revised system diverges from the Cronquist system, but agrees more closely with the circumscription later published under APG II.

Every non-empty set x contains a member y such that x and y are disjoint sets. : \forall x [\exists a ( a \in x) \Rightarrow \exists y ( y \in x \land \lnot \exists z (z \in y \land z \in x))]. or in modern notation: \forall x\,(x eq \varnothing \Rightarrow \exists y \in x\,(y \cap x = \varnothing)). This (along with the Axiom of Pairing) implies, for example, that no set is an element of itself and that every set has an ordinal rank.

One millionth is equal to 0.000 001, or 1 x 10−6 in scientific notation. It is the reciprocal of a million, and can be also written as 1/1 000 000.. Units using this fraction can be indicated using the prefix "micro-" from Greek, meaning "small".. Numbers of this quantity are expressed in terms of μ (the Greek letter mu).. "Millionth" can also mean the ordinal number that comes after the nine hundred, ninety-nine thousand, nine hundred, ninety-ninth and before the million and first..

The Principia covered only set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism. It was also clear how lengthy such a development would be. A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third.

Except for some very fluent speakers (like news anchors), even in English-language media, dates are also often read with a cardinal instead of an ordinal number (e.g. "January one" rather than "January first" or "January the first") even if the written form is the same. This is mostly because educated Filipinos were taught to count English numbers cardinally, thus it carried over to their style of reading dates. In reading the day-month-year date notation used by some areas in the government (e.g.

Spatial memory recall is built upon a hierarchical structure. That is to say that people remember the general layout of a particular space and then "cue target locations" within that spatial set. This paradigm includes an ordinal scale of features that an individual must attend to in order to inform his or her cognitive map. Recollection of spatial details is a top-down procedure that requires an individual to recall the superordinate features of a cognitive map, followed by the ordinate and subordinate features.

In machine learning, alternatives to the latent-variable models of ordinal regression have been proposed. An early result was PRank, a variant of the perceptron algorithm that found multiple parallel hyperplanes separating the various ranks; its output is a weight vector and a sorted vector of thresholds , as in the ordered logit/probit models. The prediction rule for this model is to output the smallest rank such that . Other methods rely on the principle of large-margin learning that also underlies support vector machines.

1\. Suppose the agents have cardinal utility functions on items. Then, the problem of deciding whether a proportional allocation exists is NP-complete: it can be reduced from the partition problem. 2\. Suppose the agents have ordinal rankings on items, with or without indifferences. Then, the problem of deciding whether a necessarily-proportional allocation exists can be solved in polynomial time: it can be reduced to the problem of checking whether a bipartite graph admits a feasible b-matching (a matching when the edges have capacities).

Limitation of Size plus I being a set (hence the universe is nonempty) renders provable the sethood of the empty set; hence no need for an axiom of empty set. Such an axiom could be added, of course, and minor perturbations of the above axioms would necessitate this addition. The set I is not identified with the limit ordinal \omega, as I could be a set larger than \omega. In this case, the existence of \omega would follow from either form of Limitation of Size.

David White acted as an agent to receive payments from Menard on behalf of the Republic of Texas. White claimed that Menard made the payments, but it is not clear about the form of the payments and how much, if any, was forwarded to the Republic of Texas. John D. Groesbeck completed his orthogonal plan for Galveston in 1838. He named the eastwest streets according to letters from the alphabet, and used ordinal numbers for northsouth streets, though many of these streets were renamed.

Examples are documents dated 8 August 2013; 17 January 2014 ; 2 April 2014 Popes who have an ordinal numeral in their name traditionally place the abbreviation "PP." before the ordinal numeral, as in "Benedictus PP. XVI" (Pope Benedict XVI), except in bulls of canonization and decrees of ecumenical councils, which a pope signs with the formula, "Ego N. Episcopus Ecclesiae catholicae", without the numeral, as in "Ego Benedictus Episcopus Ecclesiae catholicae" (I, Benedict, Bishop of the Catholic Church). The pope's signature is followed, in bulls of canonization, by those of all the cardinals resident in Rome, and in decrees of ecumenical councils, by the signatures of the other bishops participating in the council, each signing as Bishop of a particular see. Papal bulls are headed N. Episcopus Servus Servorum Dei ("Name, Bishop, Servant of the Servants of God"). In general, they are not signed by the pope, but John Paul II introduced in the mid-1980s the custom by which the pope signs not only bulls of canonization but also, using his normal signature, such as "Benedictus PP. XVI", bulls of nomination of bishops.

On the basis of that list, Eric XIV and Charles IX chose to use high ordinals; previous monarchs with those names are traditionally numbered counting backward from Eric XIV and Charles IX. In contemporary Swedish usage, medieval kings are usually not given any ordinal at all. A list of Swedish monarchs, represented on the map of the Estates of the Swedish Crown, created by French engraver Jacques Chiquet (1673–1721) and published in Paris in 1719, starts with Canute I and shows Eric XIV and Charles IX as Eric IV and Charles II respectively, while the only Charles who holds his traditional ordinal in the list is Charles XII, being the highest enumerated. Sweden has been ruled by queens regnant on three occasions: by Margaret (1389–1412), Christina (1632–1654) and Ulrika Eleonora (1718–1720) respectively, and earlier, briefly, by a female regent Duchess Ingeborg (1318–1319). In addition to the list below, the Swedish throne was also claimed by the kings of the Polish–Lithuanian Commonwealth from 1599 to 1660. Following his abdication Sigismund continued to claim the throne from 1599 to his death in 1632.

Set theory is the study of sets, which are abstract collections of objects. Many of the basic notions, such as ordinal and cardinal numbers, were developed informally by Cantor before formal axiomatizations of set theory were developed. The first such axiomatization, due to Zermelo (1908b), was extended slightly to become Zermelo–Fraenkel set theory (ZF), which is now the most widely used foundational theory for mathematics. Other formalizations of set theory have been proposed, including von Neumann–Bernays–Gödel set theory (NBG), Morse–Kelley set theory (MK), and New Foundations (NF).

The functional morphology of the middle ear apparatus is reconsidered in this light, and it is proposed that adaptations towards low-frequency airborne hearing might have predisposed golden moles towards the evolution of seismic sensitivity through inertial bone conduction. The morphology of the middle ear apparatus sheds little light on the disputed ordinal position of the Chrysochloridae.” Not so long ago, there was a lot of uncertainty regarding how clades of living mammals were interrelated. Many mammalian systematists believed that golden moles (Chrysochloridae) were “insectivorans” along with shrews and hedgehogs.

Using Michell's schema, Ben Richards (Kyngdon & Richards, 2007) discovered that some instances of the triple cancellation axiom are "incoherent" as they contradict the single cancellation axiom. Moreover, he identified many instances of the triple cancellation which are trivially true if double cancellation is supported. The axioms of the theory of conjoint measurement are not stochastic; and given the ordinal constraints placed on data by the cancellation axioms, order restricted inference methodology must be used . George Karabatsos and his associates (Karabatsos, 2001; ) developed a Bayesian Markov chain Monte Carlo methodology for psychometric applications.

Skating was formerly judged for "technical merit" (in the free skate), "required elements" (in the short program), and "presentation" (in both programs). The marks for each program ran from 0.0 to 6.0, the latter being the highest. These marks were used to determine a preference ranking, or "ordinal", separately for each judge; the judges' preferences were then combined to determine placements for each skater in each program. The placements for the two programs were then combined, with the free skate placement weighted more heavily than the short program.

The family Rallidae was introduced (as Rallia) by the French polymath Constantine Samuel Rafinesque in 1815. The family has traditionally been grouped with two families of larger birds, the cranes and bustards, as well as several smaller families of usually "primitive" midsized amphibious birds, to make up the order Gruiformes. The alternative Sibley-Ahlquist taxonomy, which has been widely accepted in America, raises the family to ordinal level as the Ralliformes. Given uncertainty about gruiform monophyly, this may or may not be correct; it certainly seems more justified than most of the Sibley-Ahlquist proposals.

The three four-axle narrow gauge railcars of BDŽ class 05 01-03 which were procured in 1941 had proven very good in operation on the Rhodope Railway. In order to cope with the increase in traffic after the Second World War, the BDŽ procured four diesel railcars from Ganz Works Budapest in 1952 again. They were similar in construction and appearance to the vehicles of 1941, but more powerfully motorized with a power of . Originally, they were given the same series designation 05 with the ordinal numbers 04-07.

The sequence is infinite—and this statement requires some proof. The proof depends on the observation that the English names of all ordinal numbers, except those that end in 2, must contain at least one "t".. Aronson's sequence is closely related to autograms . There are many generalizations of Aronson's sequence and research into the topic is ongoing. write that Aronson's sequence is "a classic example of a self-referential sequence"; however, they criticize it for being ambiguously defined due to the variation in naming of numbers over one hundred in different dialects of English.

The Cantonese name for the Octopus card, Baat Daaht Tùng (), translates literally as "eight-arrived pass", where Baat Daaht may translate as "reaching everywhere". Less literally though the meaning is taken as the "go-everywhere pass". It was selected by the head of the MTR Corporation, the parent company of Octopus Cards Limited, in a naming competition held in 1996. The number eight refers to the cardinal and ordinal directions, and the four-character idiom sei tùng baat daaht (), a common expression loosely translated as "reachable in all directions".

A "Gassenhauer" usually denotes a (normally simple) tune that many people (in the Gassen) have taken up and sing or whistle for themselves, the tune as such having become rather independent from its compositional origins. A rare word in contemporary German, rough modern equivalents of the term include "hit" (success) or "schlager". Other composers who used this melody include Joseph von Eybler, Johann Nepomuk Hummel and Niccolò Paganini. Because of its unique scoring in Beethoven's output, there is some uncertainty as to whether to include it in the ordinal numbering of Beethoven's piano trios.

If the dependent variable is referred to as an "explained variable" then the term "predictor variable" is preferred by some authors for the independent variable. Variables may also be referred to by their form: continuous or categorical, which in turn may be binary/dichotomous, nominal categorical, and ordinal categorical, among others. An example is provided by the analysis of trend in sea level by . Here the dependent variable (and variable of most interest) was the annual mean sea level at a given location for which a series of yearly values were available.

The Annuario Pontificio has historically regarded the Roman line of popes as legitimate until 1409, followed by the Pisan popes. Until the mid-20th century, the Annuario Pontificio listed the last three popes of the schism as Gregory XII (1406–1409), Alexander V (1409–1410), and John XXIII (1410–1415). However, the Great Schism was reinterpreted when Pope John XXIII (1958–1963) chose to reuse the ordinal XXIII, citing "twenty-two Johns of indisputable legitimacy." This is reflected in modern editions of the Annuario Pontificio, which extend Gregory XII's reign to 1415.

The New Welfare Economics approach is based on the work of Pareto, Hicks, and Kaldor. It explicitly recognizes the differences between the efficiency aspect of the discipline and the distribution aspect and treats them differently. Questions of efficiency are assessed with criteria such as Pareto efficiency and the Kaldor–Hicks compensation tests, while questions of income distribution are covered in social welfare function specification. Further, efficiency dispenses with cardinal measures of utility, replacing it with ordinal utility, which merely ranks commodity bundles (with an indifference-curve map, for example).

In fact, Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.The biographical material in this article is mostly drawn from Dauben 1979. Grattan-Guinness 1971, and Purkert and Ilgauds 1985 are useful additional sources. Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive – even shocking – that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri PoincaréDauben 2004, p. 1.

Mertens, Jean-François, 1992. "The Small Worlds Axiom for Stable Equilibria," Games and Economic Behavior, 4: 553-564. emphasized also the importance of the small worlds principle that a solution concept should depend only on the ordinal properties of players' preferences, and should not depend on whether the game includes extraneous players whose actions have no effect on the original players' feasible strategies and payoffs. Kohlberg and Mertens demonstrated via examples that not all of these properties can be obtained from a solution concept that selects single Nash equilibria.

Felipe Massa during pre-season testing at Jerez in February 2011. In February 2011, Ford declared its intention to sue Ferrari over the use of the F150 name, to which Ford owns a trademark. In response, Ferrari began to refer to the car as the "F150th Italia", claiming the original F150 moniker was simply an abbreviation. On March 3, Ferrari announced that the name had again been tweaked to "Ferrari 150º Italia" (the 150º pronounced as the Italian ordinal centocinquantesimo), and that Ford had withdrawn its legal challenge.

As a result, research into this class of formal systems began to address both logical and computational aspects; this area of research came to be known as modern type theory. Advances were also made in ordinal analysis and the study of independence results in arithmetic such as the Paris–Harrington theorem. This was also a period, particularly in the 1950s and afterwards, when the ideas of mathematical logic begin to influence philosophical thinking. For example, tense logic is a formalised system for representing, and reasoning about, propositions qualified in terms of time.

Provided there exists an order isomorphism between two well-ordered sets, the order isomorphism is unique: this makes it quite justifiable to consider the two sets as essentially identical, and to seek a "canonical" representative of the isomorphism type (class). This is exactly what the ordinals provide, and it also provides a canonical labeling of the elements of any well-ordered set. Every well-ordered set (S,<) is order-isomorphic to the set of ordinals less than one specific ordinal number under their natural ordering. This canonical set is the order type of (S,<).

Transfinite induction holds in any well-ordered set, but it is so important in relation to ordinals that it is worth restating here. : Any property that passes from the set of ordinals smaller than a given ordinal α to α itself, is true of all ordinals. That is, if P(α) is true whenever P(β) is true for all , then P(α) is true for all α. Or, more practically: in order to prove a property P for all ordinals α, one can assume that it is already known for all smaller .

Transfinite induction can be used not only to prove things, but also to define them. Such a definition is normally said to be by transfinite recursion – the proof that the result is well-defined uses transfinite induction. Let F denote a (class) function F to be defined on the ordinals. The idea now is that, in defining F(α) for an unspecified ordinal α, one may assume that F(β) is already defined for all and thus give a formula for F(α) in terms of these F(β).

The Prioress's Tale, a painting by Edward Coley Burne-Jones The Prioress's Tale () follows The Shipman's Tale in Geoffrey Chaucer's The Canterbury Tales. Because of fragmentation of the manuscripts, it is impossible to tell where it comes in ordinal sequence, but it is second in group B2, followed by Chaucer's Tale of Sir Topas. The General Prologue names the prioress as Madame Eglantine, and describes her impeccable table manners and soft-hearted ways. Her portrait suggests she is likely in religious life as a means of social advancement, given her aristocratic manners and mispronounced French.

The 6.0 system was divided into "technical merit" (in the free skate), "required elements" (in the short program), and "presentation" (in both programs). The marks for each program ranged from 0.0 to 6.0, the latter being the highest. These marks were used to determine a preference ranking, or "ordinal", separately for each judge; the judges' preferences were then combined to determine placements for each skater in each program. The placements for the two programs were then combined, with the free skate placement weighted more heavily than the short program.

Ernst Specker, 1982 Ernst Paul Specker (11 February 1920, Zurich – 10 December 2011, Zurich) was a Swiss mathematician. Much of his most influential work was on Quine’s New Foundations, a set theory with a universal set, but he is most famous for the Kochen–Specker theorem in quantum mechanics, showing that certain types of hidden variable theories are impossible. He also proved the ordinal partition relation ω2 → (ω2,3)2, thereby solving a problem of Erdős. Specker received his Ph.D. in 1949 from ETH Zurich, where he remained throughout his professional career.

Until the 19th century, this commission made most of the economic decisions of Great Britain (England, before the Act of Union 1707). However, starting during the 19th century, these positions became sinecure positions, with the First Lord serving almost invariably as Prime Minister, the Second Lord invariably as Chancellor of the Exchequer, and the junior lords serving as whips in Parliament. As an office in commission, technically all Lords Commissioners of the Treasury are of equal rank, with their ordinal numbers connoting seniority rather than authority over their fellow Lords Commissioners.

This can be circumvented by the use of the Axiom of Choice to select a representative from each equivalence class to replace [x]_R, which will be at the same type as x, or by choosing a canonical representative if there is a way to do this without invoking Choice (the use of representatives is hardly unknown in ZFC, either). In NFU, the use of equivalence class constructions to abstract properties of general sets is more common, as for example in the definitions of cardinal and ordinal number below.

An anime series premiered in July 2012, and was followed by a for-TV movie Sword Art Online Extra Edition on December 31, 2013 a second anime series, Sword Art Online II, in July 2014, a theatrical film adaptation, Sword Art Online The Movie: Ordinal Scale, in February 2017, and the first of two seasons for the third anime series, Sword Art Online: Alicization, in October 2018. The Isolator was serialized online starting in 2004, and began publishing in print in June 2014. Five light novels and four manga have been written.

For example, in a set of items , , rated with a Likert scale circular relations like > , > and > can appear. This violates the axiom of transitivity for the ordinal scale. Research by Labovitz and Traylor provide evidence that, even with rather large distortions of perceived distances between scale points, Likert-type items perform closely to scales that are perceived as equal intervals. So these items and other equal-appearing scales in questionnaires are robust to violations of the equal distance assumption many researchers believe are required for parametric statistical procedures and tests.

In the late 19th century, Carl Menger and his followers from the Austrian school of economics undertook the first successful departure from measurable utility, in the clever form of a theory of ranked uses. Despite abandoning the thought of quantifiable utility (i.e. psychological satisfaction mapped into the set of real numbers) Menger managed to establish a body of hypothesis about decision-making, resting solely on a few axioms of ranked preferences over the possible uses of goods and services. His numerical examples are "illustrative of ordinal, not cardinal, relationships".

Another common example is in ordinal and cardinal numbers – "1" is read as one, while "1st" is read as fir-st. Note that word, morpheme (constituent part of word), and reading may be distinct: in "1", "one" is at once the word, the morpheme, and the reading, while in "1st", the word and the morpheme are "first", while the reading is fir, as the -st is written separately, and in "Xmas" the word is "Christmas" while the morphemes are Christ and -mas, and the reading "Christ" coincides with the first morpheme.

The letter i in the word zeci (both as a separate word and in compounds), although thought by native speakers to indicate an independent sound, is only pronounced as a palatalization of the previous consonant. It does not form a syllable by itself: patruzeci "forty" is pronounced . The same applies to the last i in cinci: , including compounds: 15 is pronounced and 50 is . However, in the case of ordinal numbers in the masculine form, before -lea the nonsylabic i becomes a full syllabic i in words like douăzecilea "20th" and in cincilea "5th" .

A programming bug confusing these two year numbers is probably the cause of some Android users of Twitter being unable to log in around midnight of 29 December 2014 UTC. The ISO week calendar relies on the Gregorian calendar, which it augments, to define the new year day (Monday of week 01). As a result, extra weeks are spread across the 400-year cycle in a complex, seemingly random pattern. There is no simple algorithm to determine whether a year has 53 weeks from its ordinal number alone.

They were calculated for "every tenth microturn", which referred to the ordinal tenths, not the fractional tenths. Those values, calculated on early digital computers, were made available to be used by Phototrack participants for making decimal calculations. Besides a "limited draft edition" of the computation handbook, published in August 1958, a later version was published by the National Academy of Sciences – National Research Council in January 1959 as Number 7 in its "IGY Satellite Report Series". The book of tables was also re-published by the Society of Photographic Scientists and Engineers in 1964.

The GAFCON statement contains the "Jerusalem Declaration", a doctrinal confession which was intended to form the basis of a new "Fellowship of Confessing Anglicans" (FCA). The declaration upholds the Holy Scriptures as containing "all things necessary for salvation", the first four Ecumenical councils and three Creeds as expressing the church's rule of faith, and the Thirty-Nine Articles as authoritative for Anglicans today. In addition, the 1662 Book of Common Prayer is called "a true and authoritative standard of worship and prayer" and the Anglican Ordinal is recognised as an authoritative standard.

The term Pentecost comes from the Greek () meaning "fiftieth". It refers to the festival celebrated on the fiftieth day after Passover, also known as the "Feast of Weeks" and the "Feast of 50 days" in rabbinic tradition. The Septuagint uses the term to refer to the "Feast of Pentecost" only twice, in the deuterocanonical Book of Tobit and 2 Maccabees. The Septuagint writers also used the word in two other senses: to signify the year of Jubilee (), an event which occurs every 50th year, and in several passages of chronology as an ordinal number.

Historically, some Eastern Orthodox bishops have assisted in the consecration of Anglican bishops; for example, in 1870, the Most Reverend Alexander Lycurgus, the Greek Orthodox Archbishop of Syra and Tinos, was one of the bishops who consecrated Henry MacKenzie as the Suffragan Bishop of Nottingham. Because of changes in the Ordinal (the rites of Holy Orders) under King Edward VI, the Roman Catholic Church does not fully recognize all Anglican Holy Orders as valid, but the latter are recognized (and participated in) by Old Catholics, whose Holy Orders are considered valid by Rome.

However, there are also models that include invertible infinitesimals. Other mathematical systems exist which include infinitesimals, including nonstandard analysis and the surreal numbers. Smooth infinitesimal analysis is like nonstandard analysis in that (1) it is meant to serve as a foundation for analysis, and (2) the infinitesimal quantities do not have concrete sizes (as opposed to the surreals, in which a typical infinitesimal is 1/ω, where ω is a von Neumann ordinal). However, smooth infinitesimal analysis differs from nonstandard analysis in its use of nonclassical logic, and in lacking the transfer principle.

It was used for the universe of sets in 1889 by Peano, the letter V signifying "Verum", which he used both as a logical symbol and to denote the class of all individuals.. See pages VIII and XI. Peano's notation V was adopted also by Whitehead and Russell for the class of all sets in 1910.. See page 229. The V notation (for the class of all sets) was not used by von Neumann in his 1920s papers about ordinal numbers and transfinite induction. Paul Cohen. See page 88.

The concepts of divine kingship and royal ceremonies are clear examples of the influence of Brahmanism. The Coronation of the Thai monarch are practiced more or less in its original form even up to the present reign. The Thai idea that the king is a reincarnation of the Hindu deity Vishnu was adopted from Indian tradition. (Though this belief no longer exists today, the tradition to call each Thai king of the present Chakri dynasty Rama (Rama is an incarnation of Vishnu) with an ordinal number, such as Rama I, Rama II etc.

Well ahead of modern phylogenetic analyses, she and Bryce Kendrick published an exploratory paper on numerical taxonomy, "Attempting Neo- Adansonian computer taxonomy at the ordinal level in Basidiomycetes", in 1966. Her studies on corticioid fungi led her to investigated sclerotium-producing Basidiomycetes, first among the corticioid fungi and then among other groups such as Typhula. She whimiscally and imaginatively named the fungus genus Minimedusa because of its "medusoid" (like the mythical decapitated Medusa’s head) tangled hyphae forming the bulbil. Her work involved major taxonomic revisions and studies on applied pathology problems.

In set theory, an extender is a system of ultrafilters which represents an elementary embedding witnessing large cardinal properties. A nonprincipal ultrafilter is the most basic case of an extender. A (κ, λ)-extender can be defined as an elementary embedding of some model M of ZFC− (ZFC minus the power set axiom) having critical point κ ε M, and which maps κ to an ordinal at least equal to λ. It can also be defined as a collection of ultrafilters, one for each n-tuple drawn from λ.

Musée de l'Aventure Peugeot. The exhibit label (2012) states: « Jusqu'à présente le modèles Peugeot se suivaient dans une numérotation plus ou moins logique qui d’ailleurs n’était que peu utilise par la publicité. Par exemple le 190 S était présenté dans les brochures sous le nom « Le 5 CV Peugeot ». Avec la 201 une ère novelle commence.» Cars after the Type 190 use names three digits long with a zero in the middle, beginning with the 201, abandoning Peugeot's original procedure of naming each new model with a successive ordinal number.

The majority of conformists were part of the Reformed consensus that included the Puritans; what divided the parties were disputes over church government. Whitgift's first move against the Puritans was a requirement that all clergy subscribe to three articles, the second of which stated that the Prayer Book and Ordinal contained "nothing ... contrary to the word of God". Whitgift's demands produced widespread turmoil, and around 400 ministers were suspended for refusal to subscribe. Under pressure from the Privy Council, Whitgift was forced to accept conditional subscriptions from defiant ministers.

Ny < nyugat 'west'). Units of measurement are used in accordance with the international standard, depending on whether the sign comes from a common name (m < méter) or a proper name (N < newton after Isaac Newton). Standard forms of abbreviations are not to be altered even in full- capitalized inscriptions (ÁRA: 100 Ft 'PRICE: 100 HUF').AkH. 277. Some abbreviations are written without a full stop, such as names of currencies, cardinal and ordinal directions, country codes of cars, codes of country names, chemical, physical, mathematicals symbols, symbols of units, etc.

At the corners, facing ordinal points, are four figurative sculptures, each depicting an allegory of Freedom (northeast), Truth (southeast), Wisdom (southwest) and Justice (northwest). On the four cardinal faces are near life size likenesses of fifty three prominent British figures from the Victorian era. Of the fifty three persons depicted upon the plinth of the Queen Victoria Monument only two are women: George Eliot and Florence Nightingale. Five of those depicted were born in Lancaster or the surrounding area: William Turner, Edward Frankland, Richard Owen, William Whewell, James Williamson, 1st Baron Ashton.

Portuguese (Portugal) keyboard layout Essentially, the Portuguese keyboard contains dead keys for five variants of diacritics; the letter Ç, the only application of the cedilla in Portuguese, has its own key, but there are also a dedicated key for the ordinal indicators and a dedicated key for quotation marks. The + combination for producing the euro sign € (Unicode 0x20AC) has become standard. On some QWERTY keyboards the key labels are translated, but the majority are labelled in English. During the 20th century, a different keyboard layout, HCESAR, was in widespread use in Portugal.

Least absolute deviations shares the ability to be relatively insensitive to large deviations in outlying observations, although even better methods of robust regression are available. The quantiles of a random variable are preserved under increasing transformations, in the sense that, for example, if is the median of a random variable , then is the median of , unless an arbitrary choice has been made from a range of values to specify a particular quantile. (See quantile estimation, above, for examples of such interpolation.) Quantiles can also be used in cases where only ordinal data are available.

The points of the compass are the vectors by which planet-based directions are conventionally defined. A compass rose is primarily composed of four cardinal directions—north, east, south, and west—each separated by 90 degrees, and secondarily divided by four ordinal (intercardinal) directions—northeast, southeast, southwest, and northwest—each located halfway between two cardinal directions. Some disciplines such as meteorology and navigation further divide the compass with additional vectors. Within European tradition, a fully defined compass has 32 'points' (and any finer subdivisions are described in fractions of points).

School 2 would be the champion despite receiving fewer first place scores because School 2's ordinal score is lower than School 1's. Thus, overall placement in the caption area for each judge is more important than the raw score awarded by the judge. In all other musical competitions (held in the spring semester), conference alignments are disregarded except for the rules regarding sight reading. Advancement in solo and small ensemble competition is from region to state, and at state the top two soloists and top ensemble are awarded medals.

A construction of the surreal numbers as a maximal binary pseudo-tree with simplicity (ancestor) and ordering relations is due to Philip Ehrlich, The difference from the usual definition of a tree is that the set of ancestors of a vertex is well-ordered, but may not have a maximal element (immediate predecessor); in other words the order type of that set is a general ordinal number, not just a natural number. This construction fulfills Alling's axioms as well and can easily be mapped to the sign-sequence representation.

Instead, in most instances in print, television, and radio, the race was referred to as the "2009 Indianapolis 500". Since the race was not held during the United States' participation in the two World Wars (1917–1918, 1942–1945), the advertised Centennial Era occurred during the 93rd to 95th runnings. To avoid confusion between the 100th anniversary, and the actual number of times the race has been run, references to the ordinal during the Centennial Era were curtailed. Six years later, in 2016, the race celebrated its 100th running with about 350,000 in attendance.

In applied mathematics and decision making, the aggregated indices randomization method (AIRM) is a modification of a well-known aggregated indices method, targeting complex objects subjected to multi-criteria estimation under uncertainty. AIRM was first developed by the Russian naval applied mathematician Aleksey Krylov around 1908. The main advantage of AIRM over other variants of aggregated indices methods is its ability to cope with poor-quality input information. It can use non-numeric (ordinal), non-exact (interval) and non-complete expert information to solve multi-criteria decision analysis (MCDM) problems.

At the 2018–19 Japanese Championships, they won the event after placing first in both segments. They placed ninth at the 2019 Four Continents Championships after placing ninth in both segments. Komatsubara/Koleto represented Japan at their first World Championships, held in Saitama, where they placed twenty-first in the rhythm dance, missing the free dance by one ordinal. To conclude the season, they participated in the 2019 World Team Trophy as part of Team Japan, which won the silver medal, though Komatsubara/Koleto placed sixth of sixth competitors in each of their segments.

"It is quite certain that the ape which most nearly approaches man is either the Chimpanzee, or the Gorilla..." (p86). "In the general proportions of the body and limbs there is a remarkable difference between the Gorilla and man (p87)... [but]... in whatever proportion the Gorilla differs from man, the other apes depart still more widely from the Gorilla and that, consequently, such differences of proportion can have no ordinal value" (p89). Put simply, Huxley rejects the idea that man should occupy an order separate from the apes. Therefore, they are primates.

His face is covered with an Aër, the liturgical veil with which the Holy Mysteries (chalice and paten) are covered during the Divine Liturgy. Also a Gospel Book is laid upon his breast (a similar practice was found in the West in the early Spanish Ordinal). When a bishop dies, he is vested by the clergy in his full episcopal vestments, including mitre. As each vestment is placed on him, a Protodeacon swings the censer and reads the vesting prayers, exactly as was done for him when he served the Divine Liturgy.

Hugh de Courtenay, 1st/9th Earl of Devon (14 September 1276 – 23 December 1340). of Tiverton Castle, Okehampton Castle, Plympton Castle and Colcombe Castle, all in Devon, feudal baron of Okehampton and feudal baron of Plympton, was an English nobleman. In 1335, forty-one years after the death of his second-cousin once removed Isabel de Redvers, suo jure 8th Countess of Devon (died 1293) he was officially declared Earl of Devon, although whether as a new creation or in succession to her is unknown, thus alternative ordinal numbers exist for this Courtenay earldom.

Mechanisms of entrapment include pinchouts and local changes of permeability – forms of stratigraphic traps – and structural traps such as oil-bearing units blockaded by unrelated, impermeable units put there by motion along faults. Three separate producing horizons, or vertical zones, are present in the Puente Formation, and are given ordinal numbers: First, Second, and Third zones. In addition to these zones, small pockets of oil have been found throughout the upper part of the Puente.DOGGR, 258–259 The average depth of the three zones from top to bottom is 900, 1,100, and 1,500 feet.

Unlike the full version of Spore, the main game is roughly an hour long, and divided into 18 separate sections, or 30 sections in the iPhone and iPod touch version, with the player attacking and eating other organisms while avoiding being eaten by superior ones. On some devices, movement is achieved by pressing the phone keys in ordinal directions. Other devices also support touching the screen to move the creature. Certain iPod devices use the click wheel as an input method, and users of the iPhone, iPod Touch, and iPod Nano may use the accelerometer.

The median of a sample is equivariant for a much larger group of transformations, the (strictly) monotonic functions of the real numbers. This analysis indicates that the median is more robust against certain kinds of changes to a data set, and that (unlike the mean) it is meaningful for ordinal data.. Revision of a chapter in Disseminations of the International Statistical Applications Institute (4th ed.), vol. 1, 1995, Wichita: ACG Press, pp. 61–66. The concepts of an invariant estimator and equivariant estimator have been used to formalize this style of analysis.

Combinatorial game theory provides alternative reals as well, with infinite Blue-Red Hackenbush as one particularly relevant example. In 1974, Elwyn Berlekamp described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of data compression. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = . However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the surreal number , where ω is the first infinite ordinal; the relevant game is LRRRR... or 0.000...2.

A simple counting argument will verify that any non-empty finite totally ordered set (and hence any non-empty subset thereof) has a least element. Thus every finite total order is in fact a well order. Either by direct proof or by observing that every well order is order isomorphic to an ordinal one may show that every finite total order is order isomorphic to an initial segment of the natural numbers ordered by <. In other words, a total order on a set with k elements induces a bijection with the first k natural numbers.

A new bill to amend the constitution is usually named with the ordinal number next after that of the last amendment passed. Multiple pending bills will often use the same number, and be distinguished by year of introduction and/or a parenthetical number or description. However, if the government introduces multiple bills, these are numbered consecutively. There are several gaps in the numbering of passed amendments, corresponding to government bills which did not pass: ;Twelfth: Amendments 12, 13, and 14, all relating to abortion, were put to referendums on the same day.

Minimum excluded values of subclasses of the ordinal numbers are used in combinatorial game theory to assign nim-values to impartial games. According to the Sprague–Grundy theorem, the nim-value of a game position is the minimum excluded value of the class of values of the positions that can be reached in a single move from the given position. Minimum excluded values are also used in graph theory, in greedy coloring algorithms. These algorithms typically choose an ordering of the vertices of a graph and choose a numbering of the available vertex colors.

For the precise definition, suppose that is a set of nodes. Using the reflexivity of partial orders, we can identify any tree on a subset of with its partial order - a subset of . The set of all relations that form a well-founded tree on a subset of is defined in stages , so that }. For each ordinal number , let belong to the -th stage if and only if is equal to : where is a subset of such that elements of are pairwise disjoint, and is a node that does not belong to .

A tree is a partially ordered set (poset) (T, <) such that for each t ∈ T, the set {s ∈ T : s < t} is well-ordered by the relation <. In particular, each well-ordered set (T, <) is a tree. For each t ∈ T, the order type of {s ∈ T : s < t} is called the height of t (denoted ht(t, T)). The height of T itself is the least ordinal greater than the height of each element of T. A root of a tree T is an element of height 0.

Others state that it "serves instead of a lamp at night", has "the appearance of a glowing fire", or of that "of a great flame of fire." Due to its luminescence, Marco Polo called it "The Red Palace Illuminator" (Ball 1938: 499). The English alchemist John Norton wrote a 1470 poem entitled "Ordinal, or a manual of the chemical art", in which he proposed erecting a gold bridge over the River Thames and illuminating it with carbuncles set on golden pinnacles, "A glorious thing for men to beholde" (Ashmole 1652: 27).

They won their first ice dance title at the 2018-19 Japan Championships in December 2018. They placed ninth at the 2019 Four Continents Championships after placing ninth in both segments. Komatsubara/Koleto represented Japan at their first World Championships, held in Saitama, where they placed twenty-first in the rhythm dance, missing the free dance by one ordinal. To conclude the season, they participated in the 2019 World Team Trophy as part of Team Japan, which won the silver medal, though Komatsubara/Koleto placed sixth of sixth competitors in each of their segments.

Some place names have become synonymous with battles, such as the Passchendaele, Pearl Harbor, the Alamo, Thermopylae and Waterloo. Military operations, many of which result in battle, are given codenames, which are not necessarily meaningful or indicative of the type or the location of the battle. Operation Market Garden and Operation Rolling Thunder are examples of battles known by their military codenames. When a battleground is the site of more than one battle in the same conflict, the instances are distinguished by ordinal number, such as the First and Second Battles of Bull Run.

The month name is written where enough space is provided for the date; the month is in genitive case (because of the meaning e.g., “first day of May”) and the ordinals are often incorrectly"Polish Language Dictionary" by PWN, rule 87.4: dot after ordinals (in Polish) followed by a full stop to indicate they are ordinal; the date is often preceded by the abbreviation "" (day) and followed by the abbreviation "" (year), as in "". The month name can be abbreviated to three initial letters where an actual date stamping device is used, e.g.

The "degree" terminology comes from a generic ordinal indicator used for classes in the early years of the Academy — for example, "2°" was read as "second class." In recent years, "degree" has been further shortened to "dig", as in "4 digs", "3 digs", etc. First-class cadets (seniors) are referred to as "firsties." In the military structure of the Cadet Wing, first class cadets hold the positions of cadet officers, second class cadets act as the cadet non-commissioned officers and third class cadets represent the cadet junior non-commissioned officers.

Until the mid-20th century, the Annuario Pontificio regarded the Roman line as legitimate until 1409, followed by the Pisan line until 1415. The last three popes of the schism were listed as Gregory XII (1406–1409), Alexander V (1409–1410), and John XXIII (1410–1415). However, the Western Schism was reinterpreted when Pope John XXIII (1958–1963) chose to reuse the ordinal XXIII, citing "twenty-two Johns [sic] of indisputable legitimacy." This is reflected in modern editions of the Annuario Pontificio, which extend Gregory XII's reign to 1415.

However, in even-numbered years schools in Conferences 2A, 4A, and 6A can advance from region to area and state, and in odd-numbered years schools from Conferences 3A and 5A can advance from region to area and state, in addition to schools in Conference 1A may advance from region to state (There are no Area contests for Conference 1A, only for 2A and up). In order for bands to advance from Region to Area, they need to get an overall Division 1 from at least two Region judges, whether it be a 1-2-1, 2-1-1, 1-1-2, etc... the bands then will advance to the Area Competition. In order to make State, the band needs to make it to the Area Final competition AND make it to the top 3-5 bands to get to State. The Area and State winner is the school with the lowest ordinal score when the rankings from each judge are tabulated. For example, School 1 receives a first place score from three judges, a second place score from the fourth judge, and a fourth place score from the fifth judge. The ordinal score for School 1 is 9 (1+1+1+2+4).

'Do you want to...?']) # The absence of ordinal numerals higher than 'sixth', so that 'seventh' is col che a fà set 'the one which makes seven'. # The existence of three affirmative interjections (that is, three ways to say yes): si, sè (from Latin sic est, as in Italian); é (from Latin est, as in Portuguese); òj (from Latin hoc est, as in Occitan, or maybe hoc illud, as in Franco-Provençal, French and Old Catalan and Occitan). # The absence of the voiceless postalveolar fricative (like the sh in English sheep), for which an alveolar S sound (as in English sun) is usually substituted.

The (IJAAS) used a straightforward system based on year of service and type, nearly identical to the Navy's long type and model number system. This system was used from 1927, replacing an earlier system in which a manufacturer type code from a Japanese Kanji ordinal from the Heavenly stems was assigned to the aircraft from each company, as well as a type number. With additional types being added, this system quickly became cumbersome. Assigned letters included 甲 (Ko) for Nieuport, 乙 (Otsu) for Salmson, 丙 (Hei) for SPAD, 丁 (Tei) for Farman, 戊 (Bo) for Caudron, and 己 (Ki) for Hanriot.

Turing's thesis considers iterating the process to infinity, creating a system with an infinite set of axioms. The thesis was completed at Princeton under Alonzo Church and was a classic work in mathematics which introduced the concept of ordinal logic.Solomon Feferman, Turing in the Land of O(z) in "The universal Turing machine: a half-century survey" by Rolf Herken 1995 page 111 Martin Davis states that although Turing's use of a computing oracle is not a major focus of the dissertation, it has proven to be highly influential in theoretical computer science, e.g. in the polynomial time hierarchy.

It generalizes several specialized agreement coefficients by accepting any number of observers, being applicable to nominal, ordinal, interval, and ratio levels of measurement, being able to handle missing data, and being corrected for small sample sizes. Alpha emerged in content analysis where textual units are categorized by trained coders and is used in counseling and survey research where experts code open-ended interview data into analyzable terms, in psychometrics where individual attributes are tested by multiple methods, in observational studies where unstructured happenings are recorded for subsequent analysis, and in computational linguistics where texts are annotated for various syntactic and semantic qualities.

Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of well-ordered sets and ordinal numbers. Cantor attempts to prove that if A and B are sets with A equivalent to a subset of B and B equivalent to a subset of A, then A and B are equivalent. Ernst Schröder had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the name Cantor–Bernstein–Schröder theorem.

Anglican Christians around the world are held together by common forms of worship, such as the Book of Common Prayer and its modern alternatives, which embody its doctrine. Other formularies, such as the Ordinal, the Thirty-Nine Articles and the First and Second Book of Homilies provide a shared theological tradition. Other instruments of unity in the Anglican Communion are, locally, its bishops and, internationally, the Archbishop of Canterbury, and, in more recent times, the Lambeth Conferences, the Anglican Communion Primates' Meeting, and the biennial Anglican Consultative Council. These last four instruments of unity have moral but not legislative authority over individual provinces.

In China, a person's official age is based on the Gregorian calendar; for traditional use, age is based on the Chinese sui calendar. At birth, a child is considered the first year of lifetime using ordinal numeral (instead of "zero" using cardinal numeral); after each Chinese New Year, one year is added to their traditional age. Because of the potential for confusion, infant ages are often given in months instead of years. After the Gregorian calendar's introduction in China, the Chinese traditional age was referred to as the "nominal age" () and the Gregorian age was known as the "real age" ().

One section of note is the import address table (IAT), which is used as a lookup table when the application is calling a function in a different module. It can be in the form of both import by ordinal and import by name. Because a compiled program cannot know the memory location of the libraries it depends upon, an indirect jump is required whenever an API call is made. As the dynamic linker loads modules and joins them together, it writes actual addresses into the IAT slots, so that they point to the memory locations of the corresponding library functions.

Gentzen's theorem is concerned with first-order arithmetic: the theory of the natural numbers, including their addition and multiplication, axiomatized by the first-order Peano axioms. This is a "first-order" theory: the quantifiers extend over natural numbers, but not over sets or functions of natural numbers. The theory is strong enough to describe recursively defined integer functions such as exponentiation, factorials or the Fibonacci sequence. Gentzen showed that the consistency of the first-order Peano axioms is provable over the base theory of primitive recursive arithmetic with the additional principle of quantifier- free transfinite induction up to the ordinal ε0.

The Roman Catholic Church judged Anglican orders invalid when Pope Leo XIII in 1896 wrote in Apostolicae curae that Anglican orders lack validity because the rite by which priests were ordained was not correctly worded from 1547 to 1553 and from 1559 to the time of Archbishop William Laud (Archbishop of Canterbury 1633–1645). The papacy claimed the form and matter was inadequate to make a Catholic bishop. The actual "mechanical" succession, prayer and laying on hands, was not disputed. Two of the four consecrators of Matthew Parker in 1559 had been consecrated using the English Ordinal and two using the Roman Pontifical.

One of the bank's shareholders became King Peter I Karađorđević. At the idea of King Peter I, in 1905, through the Board of Directors of Prometna Banka, "'Serbia', the first Serbian insurance company" was opened, ie the first insurance company in Serbia (except for the insurance department of the Belgrade Cooperative, which already existed). Shares with the ordinal number from 1 to 300 were also bought by King Peter I, which helped the idea of realizing such a society. From the very beginning, Savčić was in the management of that company and actively managed its affairs.

Intensity of preference, also known as intensity preference,Harvey, Charles M. "Aggregation of individuals' preference intensities into social preference intensity," Social Choice and Welfare, January 1999, Volume 16, Issue 1, pp 65-79; retrieved 2012-12-12. is a term popularized by the work of the economist Kenneth Arrow, who was a co-recipient of the 1972 Nobel Memorial Prize in Economics. This term is used in reference to models for aggregating ordinal rankings. This term is used in economics, politics, marketing, management science and other areas in which methods to derive the consensus ranking are developed.

Poetto is popularly divided into "fermate" (stops), which means the various stretches of beach are recognized by the ordinal number of bus stops or urban lines linking the city centre district. The most popular is the 1st stop, adjacent to the port of Marina Piccola, just below the Sella del Diavolo. Very famous and popular are the 2nd (D'Aquila), 3rd (Lido) 4th and the 6th, home of an old Hospital and of major events (Championships of Beach Volleyball, Beach soccer, Beach Football and concerts). Around the 10th stop the coastline belonging to the Comune of Quartu Sant'Elena begins.

On the new electoral register compiled for the 1945 general election, the constituency had 74,676 electors on the civilian residence register, 67 on the Business Premises register, and 5,166 on the service register."Return showing, with regard to each Parliamentary Constituency in England and Wales, the total number of Electors on the register now in force", HCP 107 of session 1944-45, p. 5. A new Boundary Commission review began in 1965 by which time Coventry's electorate had increased and the city was allocated four seats; they were named after the ordinal points of the compass.

The further a curve is from the origin, the greater is the level of utility. The slope of the curve (the negative of the marginal rate of substitution of X for Y) at any point shows the rate at which the individual is willing to trade off good X against good Y maintaining the same level of utility. The curve is convex to the origin as shown assuming the consumer has a diminishing marginal rate of substitution. It can be shown that consumer analysis with indifference curves (an ordinal approach) gives the same results as that based on cardinal utility theory — i.e.

Suppose a person has a bundle (x_0,y_0) and claims that he is indifferent between this bundle and the bundle (x_0-\lambda\cdot\delta,y_0+\delta). This means that he is willing to give \lambda\cdot\delta units of x to get \delta units of y. If this ratio is kept as \delta\to 0, we say that \lambda is the marginal rate of substitution (MRS) between x and y at the point (x_0,y_0). This definition of the MRS is based only on the ordinal preference relation – it does not depend on a numeric utility function.

For quantitative analysis, data is coded usually into measured and recorded as nominal or ordinal variables. Questionnaire data can be pre-coded (process of assigning codes to expected answers on designed questionnaire), field-coded (process of assigning codes as soon as data is available, usually during fieldwork), post-coded (coding of open questions on completed questionnaires) or office-coded (done after fieldwork). Note that some of the above are not mutually exclusive. In social sciences, spreadsheets such as Excel and more advanced software packages such as R, Matlab, PSPP/SPSS, DAP/SAS, MiniTab and Stata are often used.

In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself. p. 323 Suppose that j: N \to M is an elementary embedding where N and M are transitive classes and j is definable in N by a formula of set theory with parameters from N. Then j must take ordinals to ordinals and j must be strictly increasing. Also j(\omega) = \omega. If j(\alpha) = \alpha for all \alpha < \kappa and j(\kappa) > \kappa, then \kappa is said to be the critical point of j.

After the questionnaire is completed, each item may be analyzed separately or in some cases item responses may be summed to create a score for a group of items. Hence, Likert scales are often called summative scales. Whether individual Likert items can be considered as interval-level data, or whether they should be treated as ordered-categorical data is the subject of considerable disagreement in the literature, with strong convictions on what are the most applicable methods. This disagreement can be traced back, in many respects, to the extent to which Likert items are interpreted as being ordinal data.

Around the turn of the 19th century neoclassical economists started to embrace alternative ways to deal with the measurability issue. By 1900, Pareto was hesitant about accurately measuring pleasure or pain because he thought that such a self-reported subjective magnitude lacked scientific validity. He wanted to find an alternative way to treat utility that did not rely on erratic perceptions of the senses. Pareto's main contribution to ordinal utility was to assume that higher indifference curves have greater utility, but how much greater does not need to be specified to obtain the result of increasing marginal rates of substitution.

Dates. Calendar dates in Romanian are expressed using cardinal numbers, unlike English. For example, "the 21st of April" is 21 aprilie (read douăzeci și unu aprilie). For the first day of a month the ordinal number întâi is often used: 1 Decembrie (read Întâi Decembrie; upper case is used for names of national or international holidays). Normally the masculine form of the number is used everywhere, but when the units digit is 2, the feminine is also frequent: 2 ianuarie can be read both doi ianuarie and două ianuarie; the same applies for days 12 and 22. Centuries.

The fixed points of the "epsilon mapping" x \mapsto \varepsilon_x form a normal function, whose fixed points form a normal function, whose …; this is known as the Veblen hierarchy (the Veblen functions with base φ0(α) = ωα). In the notation of the Veblen hierarchy, the epsilon mapping is φ1, and its fixed points are enumerated by φ2. Continuing in this vein, one can define maps φα for progressively larger ordinals α (including, by this rarefied form of transfinite recursion, limit ordinals), with progressively larger least fixed points φα+1(0). The least ordinal not reachable from 0 by this procedure—i. e.

Pope Innocent III (Fresco at the cloister Sacro Speco, c. 1219) The fanon was mentioned in the oldest known Roman Ordinal, consequently its use in the eighth century can be proved. It was then called anabolagium (anagolagium), and was not yet at that period a vestment reserved for the use of the pope. This limitation of its use did not appear until the other ecclesiastics at Rome began to put the vestment on under the alb instead of over it, that is, when it became customary among the clergy to use the fanon as an ordinary amice.

Washington, D.C., is administratively divided into four geographical quadrants of unequal size, each delineated by their ordinal directions from the medallion located in the Crypt under the Rotunda of the Capitol. Street and number addressing, centered on the Capitol, radiates out into each of the quadrants, producing a number of intersections of identically named cross-streets in each quadrant. Originally, the District of Columbia was a near-perfect square. However, even then the Capitol was never located at the geographic center of the territory (the geographic center was located near the present-day intersection of 17th Street, NW and Constitution Ave.).

On return to Canada in 1919, the 87th Bn was demobilised; its name was perpetuated by the 1st Battalion, The Canadian Grenadier Guards (87th Bn CEF) in 1920. At the same time the 2nd Battalion, Canadian Grenadier Guards (245th Bn CEF) perpetuated the other Great War Battalion of the CEF. With this reorganisation, the regiment lost the ordinal title of "First Regiment", as numerals for all regiments were discarded.Annex A, The Canadian Grenadier Guards' Regimental Standing Orders The return to peace permitted steps to be taken to enhance the status of the regiment as a Regiment of Foot Guards.

In social sciences in general, psychology and psychiatry included, data about differences between individuals, like any data, can be collected and measured using different levels of measurement. Those levels include dichotomous (a person either has a personality trait or not) and non- dichotomous approaches. While the non-dichotomous approach allows for understanding that everyone lies somewhere on a particular personality dimension, the dichotomous (nominal categorical and ordinal) approaches only seek to confirm that a particular person either has or does not have a particular mental disorder. Expert witnesses particularly are trained to help courts in translating the data into the legal (e.g.

These findings, presented at the first Monocot Conference in 1993, with the addition of several studies that had become available in the interim, formed the basis of the 1998 consensus Angiosperm Phylogeny Group (APG) ordinal scheme. Among other things, the Alismatales were expanded and new orders such as Acorales (a placement for Acorus) and Pandanales (which now represented the stemonoids as well as new families) added. While not formally assigning any supraordinal ranks, the classification did recognize an informal grouping of monocot orders as the commelinoids. Otherwise the APG recognized only six monocot orders (Acorales, Alismatales, Asparagales, Dioscoreales, Liliales and Pandanales).

It is sometimes asserted that such methods may trivially fail the universality criterion. However, it is more appropriate to consider that such methods fail Arrow's definition of an aggregation rule (or that of a function whose domain consists of preference profiles), if preference orderings cannot uniquely translate into a ballot. Methods which don't, often called "rated" or "cardinal" (as opposed to "ranked", "ordinal", or "preferential") electoral system, can be viewed as using information that only cardinal utility can convey. In that case, it is not surprising if some of them satisfy all of Arrow's conditions that are reformulated.

At the end of the year, the best contestants (those who score thirty-nine points, or thirty-six points and also win the "Three-in-Ten") return for a "Champions League PopMaster", the structure of which is different. The contestants start with their original score from their first appearance, and then proceed to answer ten questions which are worth their ordinal values i.e. question 1 is worth one point, question 2 is worth two points and so on. The contestants still choose a bonus subject, but this is only worth its value in the order of the questions.

The vast majority of Paraguay's passports has the traces of handwriting of Konstanty Rokicki, but there are also several passports filled with a different character. The most probable version is that they are filled either by Juliusz Kühl or Stefan Ryniewicz, himself an experienced consul. Passports were issued for Jewish citizens of Poland, the Netherlands, Slovakia and Hungary as well as for Jews deprived of their Germany citizenship. The ordinal numbers of passports found in the Silberschein archives in Yad Vashem suggest that at least three series of these documents had been produced, tallying altogether to least 1056 pieces.

On April 23, 1997, after the decline of BBS popularity, Lang ceased development work on Renegade and passed it on to two Renegade BBS utility authors: Patrick Spence and Gary Hall. Spence and Hall maintained Renegade for three years, releasing three updates with their new, ordinal date version scheme. Jeff Herrings, another former third-party software developer, was handed the source by Spence in January 2000 after offering help when he found there was no Y2K-compliant version of the software. Herrings released a public alpha version of Renegade in March 2000 addressing Y2K-compliance problems.

Jech (2003) p.684 The proof uses Shelah's theories of semiproper forcing and iteration with revised countable supports. MM implies that the value of the continuum is \aleph_2Jech (2003) p.685 and that the ideal of nonstationary sets on ω1 is \aleph_2-saturated.Jech (2003) p.687 It further implies stationary reflection, i.e., if S is a stationary subset of some regular cardinal κ≥ω2 and every element of S has countable cofinality, then there is an ordinal α<κ such that S∩α is stationary in α. In fact, S contains a closed subset of order type ω1.

A compass rose showing the four cardinal directions, the four intercardinal directions, and eight more divisions. The four cardinal directions, or cardinal points, are the directions north, east, south, and west, commonly denoted by their initials N, E, S, and W. East and west are perpendicular (at right angles) to north and south, with east being in the clockwise direction of rotation from north and west being directly opposite east. Points between the cardinal directions form the points of the compass. The intercardinal (also called the intermediate directions and, historically, ordinal) directions are northeast (NE), southeast (SE), southwest (SW), and northwest (NW).

Issues ranged between 40 and 64 pages in length, printed mostly in black-and-white with a color cover but occasionally including sections printed in one or two colors, notably a series of stories by Al Columbia. Zero Zero's release schedule was irregular, fluctuating between bimonthly and quarterly intervals over the course of its run. A total of 27 issues were released. Early issues of Zero Zero were not numbered; however, the back cover of each issue featured a captioned illustration depicting an ordinal "Sign of the Impending Apocalypse" which also served as an ad hoc numbering system.

In 2009, the Indianapolis Motor Speedway began a three-year-long "Centennial Era" to celebrate the 100th anniversary of the opening of the track (1909), and the 100th anniversary of the first Indy 500 (1911). As a gesture to the nostalgic Centennial Era celebration (2009–2011), tickets for the 2009 race donned the moniker "93rd 500 Mile International Sweepstakes". It is the first time since 1980 that the "Sweepstakes" title has been used. During the month of May 2009, the ordinal (93rd) was used very sparingly, and for the first time since 1981, was not identified on the annual logo.

They also pointed out that none of the priests ordained with the English Ordinal were re-ordained as a requirement by Queen Mary - some did so voluntarily and some were re-anointed, a practice common at the time.Saepius Officio, VI., On the contrary, the Queen, unhappy about married clergy, ordered all of them, estimated at 15% of the total at the beginning of her reign in 1553, to put their wives away.Christopher Haigh, The Tudor Revolutions, Religion, Politics, Society under the Tudors, 1991 pp. 226-227. Parker was ordained in 1527 in the Latin Rite and before the break with Rome.

The Bucklin procedure is one way to ensure that the winning candidate will be among those with the highest median vote. When used with a cardinal voting scale instead of ordinal ranking, Bucklin's balloting method is the same as that of highest median rules like the Majority Judgment. However, Bucklin's selection algorithm starts with the highest rated votes and adds lower ones until a median winner is reached, whereas Majority Judgment starts with the median votes and removes them until all but one candidate is eliminated. Due to this difference, Bucklin passes some voting criteria that Majority Judgment fails, and vice versa.

The polyvagal theory is another way to describe the pathways in the autonomic nervous system that mediate HRV. The polyvagal theory highlights three main ordinal processes, inactive response to an environmental threat, the active response to an environmental threat, and the fluctuation between the connect and disconnect to an environmental threat. This theory decomposes heart rate variability based on frequency domain characteristics with an emphasis on respiratory sinus arrhythmia and its transmission by a neural pathway that is distinct from other components of HRV. There is anatomic and physiological evidence for a polyvagal control of the heart.

From 1985 to 1990 a new style of poetry arrived and was called "Transfinite," a word already used by the German mathematician, Georg CantorGeorg Cantor established a hierarchy among infinite sets, as well as transfinite ordinal numbers and calculations for working with them. (1845–1918). For the poet, "the Transfinite is the union of the finite and infinite in a transcendent synthesis : Its domain is Illumination.François Brousse, Conference, Paris, 18 February 1994." Poetry is a path, a type of ascension to the most ideal metamorphosis for night pilgrimsFrançois Brousse, Les Pèlerins de la nuit (The Night Pilgrims).

This incident was initiated by John Hooper, a follower of Heinrich Bullinger who had recently returned from Zürich. Hooper was unhappy with Cranmer's Prayer Book and Ordinal and he particularly objected to the use of ceremonies and vestments. When the Privy Council selected him to be the Bishop of Gloucester on 15 May 1550, he laid down conditions that he would not wear the required vestments. He found an ally among the Continental reformers in Jan Łaski who had become a leader of the Stranger church in London, a designated place of worship for Continental Protestant refugees.

Rankings are ordinal numbers that reflect only the athletes' relative positions, not their playing skill as measured by a standard yardstick. UTR, in contrast, rates each athlete on a single, standard metric. Therefore, tennis players' UTRs are largely independent of each other, aside from the algorithm's weighting of the strength of opponents who compete directly with the rated player. Nearly all tennis ranking systems use a "points per round" (PPR) method that assigns points depending on what round a player reaches in a given tournament, along with the rated "strength" of that tournament in terms of the players it accepts into the draw.

Buttonquail were traditionally placed in Gruiformes or Galliformes (the crane and pheasant orders). The Sibley-Ahlquist taxonomy elevated them to ordinal status as the Turniciformes and basal to other Neoaves either because their accelerated rate of molecular evolution exceeded the limits of sensitivity of DNA-DNA hybridization or because the authors did not perform the appropriate pairwise comparisons or both. Morphological, DNA-DNA hybridization and sequence data indicate that turnicids correctly belong to the shorebirds (Charadriiformes). They seem to be an ancient group among these, as indicated by the buttonquail-like Early Oligocene fossil Turnipax and the collected molecular data.

114 There is also a decimal counting system, which has become relatively widely used, though less so in giving the time, ages, and dates (it features no ordinal numbers). This system originated in Patagonian Welsh and was subsequently introduced to Wales in the 1940s. Whereas 39 in the vigesimal system is ("four on fifteen on twenty") or even ("two twenty minus one"), in the decimal system it is ("three tens nine"). Although there is only one word for "one" (), it triggers the soft mutation () of feminine nouns, where possible, other than those beginning with "ll" or "rh".

The distinction between strict and non-strict well orders is often ignored since they are easily interconvertible. Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can be well ordered. If a set is well ordered (or even if it merely admits a well-founded relation), the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set.

Either V contains no strong inaccessible or, taking κ to be the smallest strong inaccessible in V, Vκ is a standard model of ZFC which contains no strong inaccessibles. Thus, the consistency of ZFC implies consistency of ZFC+"there are no strong inaccessibles". Similarly, either V contains no weak inaccessible or, taking κ to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of V, then Lκ is a standard model of ZFC which contains no weak inaccessibles. So consistency of ZFC implies consistency of ZFC+"there are no weak inaccessibles".

Assuming ZFC, the inaccessible cardinal axiom is equivalent to the universe axiom of Grothendieck and Verdier: every set is contained in a Grothendieck universe. The axioms of ZFC along with the universe axiom (or equivalently the inaccessible cardinal axiom) are denoted ZFCU (which could be confused with ZFC with urelements). This axiomatic system is useful to prove for example that every category has an appropriate Yoneda embedding. This is a relatively weak large cardinal axiom since it amounts to saying that ∞ is 1-inaccessible in the language of the next section, where ∞ denotes the least ordinal not in V, i.e.

Rutherford and others had noted the disparity between the mass of an atom, computed in atomic mass units, and the approximate charge required on the nucleus for the Rutherford model to work. The required charge of the atomic nucleus was usually about half its atomic mass. Antonius van den Broek boldly hypothesized that the required charge, denoted by Z, was not half of the atomic weight for elements, but instead was exactly equal to the element's ordinal position in the periodic table. At that time, the positions of the elements in the periodic table were not known to have any physical significance.

Subject to that, we have to give as many people as possible their next-best (#2) item, and so on. In the special case in which each person should receive a single item (for example, when the "items" are tasks and each task has to be done by a single person), the problem is called rank-maximal matching or greedy matching. The idea is similar to that of utilitarian cake-cutting, where the goal is to maximize the sum of utilities of all participants. However, the utilitarian rule works with cardinal (numeric) utility functions, while the RM rule works with ordinal utilities (rankings).

The first syllables of the first four chapters, showing the name of Thomas Norton (collage). In 1617 Michael Maier in his Symbola Aureae Mensae, identifies Norton as such: Similarly, in 1652 Elias Ashmole in his Theatrum Chemicum Britannicum identifies Norton: Both of these men were drawing from the rather esoterically hidden message found in the Ordinal - where, by connecting the syllables from the first lines of the first 7 chapters of the book, a message is found. This message being: "To Mais Nor Ton Of Brise To", or as Maier and Ashmole interpreted it "Thomas Norton of Bristol".

A cardinal tree (or trie) of degree k, by analogy with cardinal numbers and by opposition with ordinal trees, is a rooted tree in which each node has k positions for an edge to a child. "Representing trees of higher degree" (2005) by David Benoit , Erik D. Demaine , J. Ian Munro , Rajeev Raman , Venkatesh Raman and S. Srinivasa Rao Each node has up to k children and each child of a given node is labeled by a unique integer from the set {1, 2, . . . , k}. For instance, a binary tree is a cardinal tree of degree 2.

The book is divided into five parts, the first titled The General Services, and consists of both a brief and full orders of worship, and the rituals of baptism, confirmation, the Lord's Supper (including a brief form), marriage, burial and the ordination services. Until this book was published the service of confirmation was referred to as The Reception of Members, with the term confirmation first appearing in Methodist ritual in this book. The ordination services were replaced in 1981 with a new Ordinal. The second section is Aids for the Ordering of Worship, which was divided into two parts.

In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning strategy. They motivated AD by its interesting consequences, and suggested that AD could be true in the least natural model L(R) of a set theory, which accepts only a weak form of the axiom of choice (AC) but contains all real and all ordinal numbers.

"In the jargon of psychological measurement theory, IQ is an ordinal scale, where we are simply rank-ordering people. ... It is not even appropriate to claim that the 10-point difference between IQ scores of 110 and 100 is the same as the 10-point difference between IQs of 160 and 150" While one standard deviation is 15 points, and two SDs are 30 points, and so on, this does not imply that mental ability is linearly related to IQ, such that IQ 50 means half the cognitive ability of IQ 100. In particular, IQ points are not percentage points.

In 1978, Albino Luciani became the first pope to use two names for his regnal name when he took the name John Paul I, including the "I". He took the "John Paul" name to honor both John XXIII and Paul VI. With the unexpected death of John Paul I a little over a month later, Karol Wojtyła took the name John Paul II to honor his immediate predecessor. Antipopes also have regnal names, and also use the ordinal to show their position in the line of previous pontiffs with their names. For example, David Bawden took the name Michael I when declared pope in 1990.

There was widespread public debate leading up to the celebrations of the year 2000 as to whether the beginning of that year should be understood (and celebrated) as the beginning of "the" new millennium. Historically, there has been debate around the turn of previous decades, centuries and millennia. The issue arises from the difference between the convention of using ordinal numbers to count years and millennia (as in "the third millennium"), or cardinally using "the two thousands". The first convention is common in English-speaking countries, but the latter is favoured in, for example, Sweden (tvåtusentalet, which translates literally as the two thousands period).

Some, such as economists in the tradition of the Austrian School, doubt whether a cardinal utility function, or cardinal social welfare function, is of any value. The reason given is that it is difficult to aggregate the utilities of various people that have differing marginal utility of money, such as the wealthy and the poor. Also, the economists of the Austrian School question the relevance of Pareto optimal allocation considering situations where the framework of means and ends is not perfectly known, since neoclassical theory always assumes that the ends-means framework is perfectly defined. The value of ordinal utility functions has been questioned.

'''' Measurement is the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events. The scope and application of measurement are dependent on the context and discipline. In the natural sciences and engineering, measurements do not apply to nominal properties of objects or events, which is consistent with the guidelines of the International vocabulary of metrology published by the International Bureau of Weights and Measures. However, in other fields such as statistics as well as the social and behavioural sciences, measurements can have multiple levels, which would include nominal, ordinal, interval and ratio scales.

The incident was referred to the FIA World Motor Sport Council for review, and no further action was taken. Alonso won further races at Monza, Singapore and the inaugural race in Korea as he finished the season second to Sebastian Vettel. Ferrari launched its 2011 car, the Ferrari 150º Italia in January 2011, with Ford declaring intentions to sue over the use of the F150 name – under which the car had been launched – Ferrari began referring to the car as the "F150th Italia". In March 2011, the car's name was changed again to "150º Italia", with the Italian language ordinal indicator º being used to replace the English language -th.

As there was no prospect of a settlement at home, Livingstone went over to Ireland in 1630 on the invitation of Lord Clandeboye, and soon afterwards became minister of Killinshie or Killinchy in the diocese of Down. Livingstone came after being urged by Robert Cunningham, minister of Holywood. He was ordained by some Scottish ministers under the presidency of Andrew Knox, bishop of Raphoe, who, to accommodate his countrymen, omitted those portions of the English ordinal to which they objected. In 1631 Livingstone was suspended for nonconformity by Robert Echlin, the Bishop of Down and Connor, but was restored on the intervention of Archbishop Ussher.

In set theory, a prewellordering is a binary relation \le that is transitive, connex, and wellfounded (more precisely, the relation x\le y\land y leq x is wellfounded). In other words, if \leq is a prewellordering on a set X, and if we define \sim by :x\sim y\iff x\leq y \land y\leq x then \sim is an equivalence relation on X, and \leq induces a wellordering on the quotient X/\sim. The order-type of this induced wellordering is an ordinal, referred to as the length of the prewellordering. A norm on a set X is a map from X into the ordinals.

In mathematical set theory, an ω-Jónsson function for a set x of ordinals is a function f:[x]^\omega\to x with the property that, for any subset y of x with the same cardinality as x, the restriction of f to [y]^\omega is surjective on x. Here [x]^\omega denotes the set of strictly increasing sequences of members of x, or equivalently the family of subsets of x with order type \omega, using a standard notation for the family of subsets with a given order type. Jónsson functions are named for Bjarni Jónsson. showed that for every ordinal λ there is an ω-Jónsson function for λ.

In his work with his coauthor (and Ph.D. student) Dipjyoti Majumdar, he weakens the notion of non-manipulability in the GS theorem to Ordinal Bayesian Incentive Compatibility, first studied in an important paper by Claude d'Aspremont and Gerard Varet. Arunava's work with Dipjyoti Majumdar shows that whether well-behaved voting rules exist with this weakening of non-manipulability depends on the beliefs of the voters on other voters' preferences. If beliefs are uniformly distributed, then many well- behaved voting rules exist and they provide a comprehensive description of such voting rules. However, if voters have generic beliefs (which are independent), a GS theorem type impossibility reappears.

This means that the conversion standard is set beforehand, and the distribution of percentiles can vary during the scoring of any particular LSAT. Adjusted scores lie in a bell curve, tapering off at the extremes and concentrating near the median. For example, there might be a 3–5 question difference between a score of 175 and a score of 180, but the difference between a 155 from a 160 could be 9 or more questions—this is because the LSAT uses an ordinal grading system. Although the exact percentile of a given score will vary slightly between examinations, there tends to be little variance.

Rather, media outlets generally engage in some degree of community journalism, as measured by the types of practices they follow and the intensity with which they follow them. A summated scale of multiple ordinal-level items would be an appropriate measure of community journalism. This is because community journalism is on a scale on which data is shown simply in order of magnitude since there is no standard of measurement of differences. In addition, numerous studies in this analysis suggest that any scale measure of community journalism should accommodate the impact of the community's power structure on news decisions and should address the need for inclusion of less powerful voices.

Some authors have commented on the misleading nature of the terms "cardinal utility" and "ordinal utility", as used in economic jargon: There remain economists who believe that utility, if it cannot be measured, at least can be approximated somewhat to provide some form of measurement, similar to how prices, which have no uniform unit to provide an actual price level, could still be indexed to provide an "inflation rate" (which is actually a level of change in the prices of weighted indexed products). These measures are not perfect but can act as a proxy for the utility. Lancaster's characteristics approach to consumer demand illustrates this point.

Under MPJ, each debate team ranks the judging pool according to their preferences and judges are selected such that both teams prefer the chosen judge equally (if possible). Under earlier systems, debaters ranked judges into categories; for the last decade, an ordinal system has been adopted in which each judge is ranked from most to least preferred. Algorithms have been developed that create judge assignments that maximize preference and mutuality, optimizing assignment first for teams with three losses, then teams with two losses, then teams with one loss, then undefeated teams, and finally teams with more than three losses (and who therefore are ineligible for elimination rounds).

However, the LCS stabilizes at the zeroth step if and only if it is perfect, while the UCS stabilizes at the zeroth step if and only if it is centerless, which are distinct concepts, and show that the lengths of the LCS and UCS (interpreted to mean the length before stabilization) need not agree in general. For a perfect group, the UCS always stabilizes by the first step, a fact called Grün's lemma. However, a centerless group may have a very long lower central series: a free group on two or more generators is centerless, but its lower central series does not stabilize until the first infinite ordinal.

In the sixteenth century a solid body of Anglican opinion emerged which saw the theological importance of the historic episcopate but refused to 'unchurch' those churches which did not retain it. The preface to the Ordinal limits itself to stating historical reasons why episcopal orders are to 'be continued and reverently used in the Church of England'. Before 1662 it was assumed that the foreign Reformed (Presbyterian) Churches were genuine ones with an authentic ministry of Word and Sacrament. The 1662 Act of Uniformity formally excluded from pastoral office in England any who lacked episcopal ordination but this was a reaction against the abolition of episcopacy in the Commonwealth period.

As defined by the Austrian School of economics the marginal use of a good or service is the specific use to which an agent would put a given increase, or the specific use of the good or service that would be abandoned in response to a given decrease.von Wieser, Friedrich; Über den Ursprung und die Hauptgesetze des wirtschaftlichen Wertes [The Nature and Essence of Theoretical Economics] (1884), p. 128. The usefulness of the marginal use thus corresponds to the marginal utility of the good or service.Mc Culloch, James Huston; “The Austrian Theory of the Marginal Use and of Ordinal Marginal Utility”, Zeitschrift für Nationalökonomie 37 (1977) #3&4 (September).

Stevens proposed his typology in a 1946 Science article titled "On the theory of scales of measurement". In that article, Stevens claimed that all measurement in science was conducted using four different types of scales that he called "nominal", "ordinal", "interval", and "ratio", unifying both "qualitative" (which are described by his "nominal" type) and "quantitative" (to a different degree, all the rest of his scales). The concept of scale types later received the mathematical rigour that it lacked at its inception with the work of mathematical psychologists Theodore Alper (1985, 1987), Louis Narens (1981a, b), and R. Duncan Luce (1986, 1987, 2001). As Luce (1997, p.

In mathematics, the height of an element g of an abelian group A is an invariant that captures its divisibility properties: it is the largest natural number N such that the equation Nx = g has a solution x ∈ A, or the symbol ∞ if there is no such N. The p-height considers only divisibility properties by the powers of a fixed prime number p. The notion of height admits a refinement so that the p-height becomes an ordinal number. Height plays an important role in Prüfer theorems and also in Ulm's theorem, which describes the classification of certain infinite abelian groups in terms of their Ulm factors or Ulm invariants.

In modern Greek, omicron represents the mid back rounded vowel , the same sound as omega. Letters that arose from omicron include Roman O and Cyrillic O. The upper-case letter of omicron (O) was originally used in mathematics as a symbol for Big O notation (representing a function's asymptotic growth rate), but has fallen out of favor because omicron is indistinguishable from the Latin letter O and easily confused with the digit zero (0). Omicron is used to designate the fifteenth star in a constellation group, its ordinal placement a function of both magnitude and position. Such stars include Omicron Andromedae, Omicron Ceti, and Omicron Persei.

Mathematical Games, September 1976 Scientific American Volume 235, Issue 3 The book is roughly divided into two sections: the first half (or Zeroth Part), on numbers, the second half (or First Part), on games. In the first section, Conway provides an axiomatic construction of numbers and ordinal arithmetic, namely, the integers, reals, the countable infinity, and entire towers of infinite ordinals, using a notation that is essentially an almost trite (but critically important) variation of the Dedekind cut. As such, the construction is rooted in axiomatic set theory, and is closely related to the Zermelo–Fraenkel axioms. The section also covers what Conway (adopting Knuth's nomenclature) termed the "surreal numbers".

An attempt in 2002 to combine analysis of RNA features of modern chelicerates and anatomical features of modern and fossil ones produced credible results for many lower-level groups, but its results for the high- level relationships between major sub-groups of chelicerates were unstable, in other words minor changes in the inputs caused significant changes in the outputs of the computer program used (POY). An analysis in 2007 using only anatomical features produced the cladogram on the right, but also noted that many uncertainties remain. In recent analyses the clade Tetrapulmonata is reliably recovered, but other ordinal relationships remain in flux. The position of scorpions is particularly controversial.

A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems. In Quine's set-theoretical writing, the phrase "ultimate class" is often used instead of the phrase "proper class" emphasising that in the systems he considers, certain classes cannot be members, and are thus the final term in any membership chain to which they belong. Outside set theory, the word "class" is sometimes used synonymously with "set".

As of the beginning of the 17th century, the running of the Treasury was frequently entrusted to a commission, rather than to a single individual. Since 1714, it has permanently been in commission. The commissioners have always since that date been referred to as Lords Commissioners of the Treasury, and adopted ordinal numbers to describe their seniority. Eventually in the middle of the same century, the First Lord of the Treasury came to be seen as the natural head of the overall ministry running the country, and, as of the time of Robert Walpole (Whig), began to be known, unofficially, as the Prime Minister.

In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after Cesare Burali-Forti, who, in 1897, published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Cantor. Bertrand Russell subsequently noticed the contradiction, and when he published it in his 1903 book Principles of Mathematics, he stated that it had been suggested to him by Burali-Forti's paper, with the result that it came to be known by Burali-Forti's name.

There are some variations of Reinhardt cardinals, forming a hierarchy of hypotheses asserting the existence of elementary embeddings V\to V. J3: There is a nontrivial elementary embedding j: V\to V J2: There is a nontrivial elementary embedding j: V\to V and DC\lambda holds, where \lambda is the least fixed-point above the critical point. J1: There is a cardinal \kappa such that for every ordinal \alpha, there is an elementary embedding j : V\to V with j(\kappa)>\alpha and having critical point \kappa. Each of J1 and J2 immediately imply J3. A cardinal \kappa as in J1 is known as a super Reinhardt cardinal.

By contrast, Edward's reign saw radical progress in the Reformation. In those six years, the Church transferred from an essentially Catholic liturgy and structure to one that is usually identified as Protestant. In particular, the introduction of the Book of Common Prayer, the Ordinal of 1550, and Cranmer's Forty-two Articles formed the basis for English Church practices that continue to this day.; Edward himself fully approved these changes, and though they were the work of reformers such as Thomas Cranmer, Hugh Latimer, and Nicholas Ridley, backed by Edward's determinedly evangelical Council, the fact of the king's religion was a catalyst in the acceleration of the Reformation during his reign.

In some cases, approval voting can sincerely elect any one of the candidates, including a Condorcet winner and a Condorcet loser, without the voter preferences changing. To the extent that electing a Condorcet winner and not electing a Condorcet loser is considered desirable outcomes for a voting system, approval voting can be considered vulnerable to sincere, strategic voting. In one sense, conditions where this can happen are robust and are not isolated cases. On the other hand, the variety of possible outcomes has also been portrayed as a virtue of approval voting, representing the flexibility and responsiveness of approval voting, not just to voter ordinal preferences, but cardinal utilities as well.

For the 1981 race, the name "65th Indianapolis 500-Mile Race" was officially adopted, with all references as the "International Sweepstakes" dropped. Since 1981, the race has been formally advertised in this fashion, complete with a unique annual logo with the ordinal almost always included. Around that same time, in the wake of the 1979 entry controversy, and the formation of CART, the race changed to an invitational event, rather than an Open, rendering the "sweepstakes" description inappropriate. For nearly a century, the race eschewed any sort of naming rights or title sponsor, a move, though uncommon in the modern sports world, that was well-received by fans.

Indexes are often referred to as scales, but in fact not all indexes are scales. Whereas indexes are usually created by aggregating scores assigned to individual attributes of various variables, scales are more nuanced and take into account differences in intensity among the attribute of the same variable in question. Indexes and scales should provide an ordinal ranking of cases on a given variable, through scales are usually more efficient at this. While indexes are based on a simple aggregation of indicators of a variable, scales are more advanced, and their calculations may be more complex, using for example scaling procedures such as semantic differential.

Frederick Temple, Archbishop of Canterbury, and William Maclagan, Archbishop of York, answered Pope Leo's charges in their written response, Saepius officio: Answer of the Archbishops of Canterbury and York to the bull Apostolicae Curae of H.H. Leo XIII. It was written to prove the sufficiency of the form and intention used in the Anglican ordinal rites since the time of the English Reformation. According to this view, the required references to the sacrificial priesthood never existed in many ancient Catholic ordination liturgies and also in certain current Eastern-rite ordination liturgies that the Roman Catholic Church considered valid. First, they asserted that the ordination ceremonies in question were biblically valid.

Hechler forcing (after Stephen Herman Hechler) is used to show that Martin's axiom implies that every family of less than c functions from ω to ω is eventually dominated by some such function. P is the set of pairs where s is a finite sequence of natural numbers (considered as functions from a finite ordinal to ω) and E is a finite subset of some fixed set G of functions from ω to ω. The element (s, E) is stronger than if t is contained in s, F is contained in E, and if k is in the domain of s but not of t then for all h in F.

While abbreviations typically exclude the initials of short function words (such as "and", "or", "of", or "to"), this is not always the case. Sometimes function words are included to make a pronounceable acronym, such as CORE (Congress of Racial Equality). Sometimes the letters representing these words are written in lower case, such as in the cases of "TfL" ("Transport for London") and LotR (Lord of the Rings); this usually occurs when the acronym represents a multi-word proper noun. Numbers (both cardinal and ordinal) in names are often represented by digits rather than initial letters, as in "4GL" ("fourth generation language") or "G77" ("Group of 77").

The symbol is known as the number sign, hash, or (in North American usage) pound sign. The symbol has historically been used for a wide range of purposes, including the designation of an ordinal number and as a ligatured abbreviation for pounds avoirdupois – having been derived from the now-rare . Since 2007, widespread usage of the symbol to introduce metadata tags on social media platforms has led to such tags being known as "hashtags", and from that, the symbol itself is sometimes called a hashtag. The symbol is distinguished from similar symbols by its combination of level horizontal strokes and right-tilting vertical strokes.

In this way, a particular count word may be used generally in a very open-ended manner and up to the construal or creativity of the speaker. There are two systems of numerals in Korean: native Korean and Sino-Korean. Native Korean numerals are used with most counter words, and usually count the number of an object, while Sino-Korean numerals are generally used for indicating a specific object in series, such as a specific lesson in a book, as well as monetary units and scientific measurements. Sometimes both types of numerals may be used, usually native Korean numerals indicating a quantity and Sino-Korean numerals indicating an ordinal.

It is of importance, however, in the study of non-ω-models. The system consisting of ACA0 plus induction for all formulas is sometimes called ACA with no subscript. The system ACA0 is a conservative extension of first-order arithmetic (or first-order Peano axioms), defined as the basic axioms, plus the first-order induction axiom scheme (for all formulas φ involving no class variables at all, bound or otherwise), in the language of first-order arithmetic (which does not permit class variables at all). In particular it has the same proof-theoretic ordinal ε0 as first-order arithmetic, owing to the limited induction schema.

Kenneth Arrow (1963) generalizes the analysis. Along earlier lines, his version of a social welfare function, also called a 'constitution', maps a set of individual orderings (ordinal utility functions) for everyone in the society to a social ordering, a rule for ranking alternative social states (say passing an enforceable law or not, ceteris paribus). Arrow finds that nothing of behavioral significance is lost by dropping the requirement of social orderings that are real-valued (and thus cardinal) in favor of orderings, which are merely complete and transitive, such as a standard indifference curve map. The earlier analysis mapped any set of individual orderings to one social ordering, whatever it was.

The characters Bo, Meng, Zhong, Shu and Ji are originally ordinals used in courtesy names to indicate a person's rank among his or her siblings of the same gender who survived to adulthood. The eldest brother's courtesy name would be prefixed with the word "Bo" (or "Meng" if he was born to a secondary wife), the second with "Zhong", the youngest with "Shu", and the rest with "Ji". For instance, Confucius' courtesy name was Zhongni. As the power of the Three Huan became hereditary, the descendants of Duke Zhuang's brothers used the ordinal numbers as family names to distinguish their branches of the House of Ji.

The term Julian date may also refer, outside of astronomy, to the day-of-year number (more properly, the ordinal date) in the Gregorian calendar, especially in computer programming, the military and the food industry,USDA c. 1963. or it may refer to dates in the Julian calendar. For example, if a given "Julian date" is "October 5, 1582", this means that date in the Julian calendar (which was October 15, 1582, in the Gregorian calendar—the date it was first established). Without an astronomical or historical context, a "Julian date" given as "36" most likely means the 36th day of a given Gregorian year, namely February 5.

If there were two or more aircraft with the same type and function, the latter was enhanced to further differentiate them. An example is the Type 2 single-seat fighter (the Nakajima Ki-44) and the Type 2 two-seat fighter (Kawasaki Ki-45). Major modifications (such as a different engine) were indicated with a subtype number, officially in kanji but often in Roman numerals. Small-scale modifications (such as armament) are indicated with a Japanese Kanji ordinal from the Heavenly stems:- ko (甲), otsu (乙), hei (丙), tei (丁), bo (戊), ki (己), which equate to:- a (first), b (second), c (third), d (fourth), e (fifth), but are NOT direct translations.

Systems of Logic Based on Ordinals was the PhD dissertation of the mathematician Alan Turing. Turing’s thesis is not about a new type of formal logic, nor was he interested in so-called ‘ranked logic’ systems derived from ordinal or relative numbering, in which comparisons can be made between truth- states on the basis of relative veracity. Instead, Turing investigated the possibility of resolving the Godelian incompleteness condition using Cantor’s method of infinites. This condition can be stated thus- in all systems with finite sets of axioms, an exclusive-or condition applies to expressive power and provability; ie one can have power and no proof, or proof and no power, but not both.

Haussmann's portrait of Bach depicts him holding the manuscript to BWV 1076, which is also the thirteenth canon in the Goldberg Canon cycle. When Bach's personal copy of the printed edition of the "Goldberg Variations" (see above) was discovered in 1974, it was found to include an appendix in the form of fourteen canons built on the first eight bass notes from the aria. It is speculated that the number 14 refers to the ordinal values of the letters in the composer's name: B(2) + A(1) + C(3) + H(8) = 14.See Chapter Seven of Richard Taruskin (2009) Music in the Seventeenth and Eighteenth Centuries: The Oxford History of Western Music.

In the United States Army, a battalion is a unit composed of a headquarters and two to six batteries, companies, or troops. They are normally identified by ordinal numbers (1st Battalion, 2nd Squadron, etc.) and normally have subordinate units that are identified by single letters (Battery A, Company A, Troop A, etc.). Battalions are tactical and administrative organizations with a limited capability to plan and conduct independent operations and are normally organic components of brigades, groups, or regiments. A U.S. Army battalion includes the battalion commander (lieutenant colonel), executive officer (major), command sergeant major (CSM), headquarters staff, and usually three to five companies, with a total of 300 to 1,000 (but typically 500 to 600) soldiers.

In 1974, the Worship and Doctrine Measure, passed by the new General Synod allowed the production of a new book which was to contain everything that would be required of priest and congregation: daily Morning and Evening Prayer, Holy Communion, initiation services (Baptism and Confirmation), marriage, funeral services, the Ordinal, Sunday readings, a lectionary and a psalter. Once again, after a gap of nearly fifteen years, parishes which did not want to use the Book of Common Prayer had in their hands all the words, including readings ordered according to themes and with a two-year cycle. Discussion in General Synod was lengthy. Hundreds of amendments to the initial proposals were debated on the floor of the chamber.

The survey, in the form of an encyclopedia, is important as a comprehensive, multivolume treatment of the vascular plants, with keys to and descriptions of all families and genera, mostly by specialists in those groups. The Kubitzki system served as the basis for classification in Mabberley's Plant-Book, a dictionary of the vascular plants. Mabberley states, in his Introduction on page xi of the 2008 edition, that the Kubitzki system "has remained the standard to which other literature is compared". In ordinal and family arrangements, the classification system in the initial angiosperm volumes closely resembles the Dahlgren system in Monocots and the Cronquist system in Dicots, but later volumes have been influenced by recent molecular phylogenetic studies.

However, they have been widely denounced by many higher education experts. Detractors argue that they ignore individual fit by comparing institutions with widely diverging missions on the same scale, imply a false precision by deriving an ordinal ranking from questionable data, encourage gamesmanship by institutions looking to improve their rank, and contribute to the admissions frenzy by unduly highlighting prestige. In addition to the rankings, U.S. News & World Report also publishes college guides in book form, and ranks American graduate schools and academic programs in a number of specific disciplines, including business, law, engineering, nursing, and medicine. In October 2014, the magazine began publishing a Best Global Universities ranking that focuses more on research and includes non-American schools.

In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member. The cofinality of a partially ordered set A can alternatively be defined as the least ordinal x such that there is a function from x to A with cofinal image. This second definition makes sense without the axiom of choice. If the axiom of choice is assumed, as will be the case in the rest of this article, then the two definitions are equivalent.

Buchler's keystone work, Metaphysics of Natural Complexes, builds upon two major principles: ontological parity,"This principle asserts the equal reality of whatever is: attributes are as real as substances, relations as real as entities, the impermanent as real as the fixed, the mental as real as the physical, human beings and the human order as real as God and the divine order, fictional entities as real as physical ones, and so on. Buchler's metaphysical orientation is, thus, nonreductionistic, nonhierarchical, and all−inclusive." Biography of Justus Buchler by Kathleen A. Wallace (Dictionary of Literary Biography, 2005−06). which asserts the equal reality of whatever is, and ordinal metaphysics, which asserts the indefinite complexity of whatever is.

To commemorate her fifth year as a solo artist, LiSA released her Letters To U EP as a limited edition LP on March 23, 2016. She released a mini-album titled Lucky Hi Five! on April 20, 2016. She released the single "Brave Freak Out", which was used as the first opening theme to the 2016 anime television series Qualidea Code, and the single also includes the song "AxxxiS", which was used as the second opening theme to Qualidea Code, on August 24, 2016. She released the single "Catch the Moment" on February 15, 2017; the title track was used as the theme song to the 2017 anime film Sword Art Online The Movie: Ordinal Scale.

In fact, every vertex order of a cograph is a perfect order which further implies that max clique finding and min colouring can be found in linear time with any greedy colouring and without the need for a cotree decomposition. Every cograph is a distance-hereditary graph, meaning that every induced path in a cograph is a shortest path. The cographs may be characterized among the distance-hereditary graphs as having diameter two in each connected component. Every cograph is also a comparability graph of a series-parallel partial order, obtained by replacing the disjoint union and join operations by which the cograph was constructed by disjoint union and ordinal sum operations on partial orders.

The Cochran–Armitage test for trend, named for William Cochran and Peter Armitage, is used in categorical data analysis when the aim is to assess for the presence of an association between a variable with two categories and an ordinal variable with k categories. It modifies the Pearson chi-squared test to incorporate a suspected ordering in the effects of the k categories of the second variable. For example, doses of a treatment can be ordered as 'low', 'medium', and 'high', and we may suspect that the treatment benefit cannot become smaller as the dose increases. The trend test is often used as a genotype-based test for case-control genetic association studies.

The ISO week date system is effectively a leap week calendar system that is part of the ISO 8601 date and time standard issued by the International Organization for Standardization (ISO) since 1988 (last revised in 2004) and, before that, it was defined in ISO (R) 2015 since 1971. It is used (mainly) in government and business for fiscal years, as well as in timekeeping. This was previously known as "Industrial date coding". The system specifies a week year atop the Gregorian calendar by defining a notation for ordinal weeks of the year. The Gregorian leap cycle, which has 97 leap days spread across 400 years, contains a whole number of weeks ().

Choose a collection of 2ℵ0 measure 0 subsets of R such that every measure 0 subset is contained in one of them. By the continuum hypothesis, it is possible to enumerate them as Sα for countable ordinals α. For each countable ordinal β choose a real number xβ that is not in any of the sets Sα for α < β, which is possible as the union of these sets has measure 0 so is not the whole of R. Then the uncountable set X of all these real numbers xβ has only a countable number of elements in each set Sα, so is a Sierpiński set. It is possible for a Sierpiński set to be a subgroup under addition.

This is a list of tables of the oldest people in the world in ordinal ranks. To avoid including false or unconfirmed claims of old age, names here are restricted to those people whose ages have been validated by an international body that specifically deals in longevity research, such as the Gerontology Research Group (GRG) or Guinness World Records (GWR), and others who have otherwise been reliably sourced. According to this criterion, the longest human lifespan is that of Jeanne Calment of France (1875–1997), who lived to age 122 years, 164 days. She supposedly met Vincent van Gogh when she was 12 or 13. She received news media attention in 1985, after turning 110.

In set theory, a branch of mathematics, the condensation lemma is a result about sets in the constructible universe. It states that if X is a transitive set and is an elementary submodel of some level of the constructible hierarchy Lα, that is, (X,\in)\prec (L_\alpha,\in), then in fact there is some ordinal \beta\leq\alpha such that X=L_\beta. More can be said: If X is not transitive, then its transitive collapse is equal to some L_\beta, and the hypothesis of elementarity can be weakened to elementarity only for formulas which are \Sigma_1 in the Lévy hierarchy. Also, the assumption that X be transitive automatically holds when \alpha=\omega_1.

A Forte number, "consists of two numbers separated by a hyphen....The first number is the cardinality of the set form...and the second number refers to the ordinal position..." Major and minor chords on C . In the 12-TET tuning system (or in any other system of tuning that splits the octave into twelve semitones), each pitch class may be denoted by an integer in the range from 0 to 11 (inclusive), and a pitch class set may be denoted by a set of these integers. The prime form of a pitch class set is the most compact (i.e., leftwards packed or smallest in lexicographic order) of either the normal form of a set or of its inversion.

Radar charts are primarily suited for strikingly showing outliers and commonality, or when one chart is greater in every variable than another, and primarily used for ordinal measurements – where each variable corresponds to "better" in some respect, and all variables on the same scale. Conversely, radar charts have been criticized as poorly suited for making trade-off decisions – when one chart is greater than another on some variables, but less on others.You are NOT spider man, so why do you use radar charts?, by Chandoo, September 18th, 2008 Further, it is hard to visually compare lengths of different spokes, because radial distances are hard to judge, though concentric circles help as grid lines.

The Pac-Man character appears in the film Pixels (2015), with Denis Akiyama playing series creator Toru Iwatani.Tarek Bazley: Pac-man at 35: the video game that changed the world Pac-Man is referenced and makes an appearance in the 2017 film Guardians of the Galaxy 2. The game, the character, and the ghosts all also appear in the film Wreck-It Ralph, as well as the sequel Ralph Breaks the Internet. In Sword Art Online The Movie: Ordinal Scale where Kirito and his friends beat a virtual reality game called PAC-Man 2024.. Al Jazeera English, May 25, 2015 Iwatani makes a cameo at the beginning of the film as an arcade technician.

Maya Usova & Alexander Zhulin skated a strong free dance that seemed to ensure the title, but had drawn first in the final flight, and received a wide spread of marks from the judges. Despite receiving four first place ordinals in the free dance, a strange ordinal situation caused them to place third in the free dance and drop from first to third in the end. In the 1991–92 season, Usova/Zhulin won silver at the 1992 European Championships in Lausanne, Switzerland and then captured their first Olympic medal, bronze, at the 1992 Winter Olympics in Albertville, France. Usova/Zhulin ended their season with silver at the 1992 World Championships in Oakland, California.

In two-competitor games where ties are rare or impossible, competitors are typically ranked by number of wins, with ties counting half; each competitor's listings are usually ordered wins–losses(–ties). Giving a half-point for a draw in chess was introduced in 1868 by the British Chess Association; previously, drawn games in chess tournaments were replayed. Where draws are more common, the award may be 2 points for a win and 1 for a draw, which is mathematically equivalent but avoids having half-points in the listings. These are usually ordered wins–draws–losses. If there are more than two competitors per match, points may be ordinal—for example, 3 for first, 2 for second, 1 for third.

His career has been devoted to different possibilities of expressing and living with uncertain beliefs, unsettled experiences, and inexhaustible realities. He has explored different forms of writing and varied terminologies for expressing what resists expression, based on the conviction that such a resistance requires constant vigilance and innovative writing, the constant production and transformation of forms of knowledge, especially including philosophy, whose relations to art, literature, science, and religion enrich it profoundly with novel and imaginative questions and answers. He began in American pragmatism, reading it to question itself fundamentally and to entail the inexhaustibility of nature and reason, the mysteriousness of things. He wrote several books on ordinality and an ordinal metaphysics, influenced by his mentor, Justus Buchler.

For example, in the early fifteenth century Clement Maydeston, probably following foreign precedents, titled his reorganized Sarum Ordinal the "Directorium Sacerdotum". In this way the words "Directorium Sacerdotum" came to be included in the beginning of many books, some of them among the earliest products of the printing press in England, that served to instruct clergy as to the form of Divine Office and Mass to be prayed each day of the year. The use of "directorium" was not peculiar to England. For example, and not as the earliest one, a very similar work was published at Augsburg in 1501 with the title Index sive Directorium Missarum Horarumque secundum ritum chori Constanciensis diocesis dicendarumn.

Indeed, there is no reason to stop at two levels: using \omega+1 new cardinals in this way, \Omega_1,\Omega_2,\ldots,\Omega_\omega, we get a system essentially equivalent to that introduced by Buchholz, the inessential difference being that since Buchholz uses \omega+1 ordinals from the start, he does not need to allow multiplication or exponentiation; also, Buchholz does not introduce the numbers 1 or \omega in the system as they will also be produced by the \psi functions: this makes the entire scheme much more elegant and more concise to define, albeit more difficult to understand. This system is also sensibly equivalent to the earlier (and much more difficult to grasp) “ordinal diagrams” of TakeutiTakeuti, 1967 (Ann.

This corresponded to the influence on the subject of mathematical methods used in the natural sciences. Neoclassical economics systematized supply and demand as joint determinants of price and quantity in market equilibrium, affecting both the allocation of output and the distribution of income. It dispensed with the labour theory of value inherited from classical economics in favour of a marginal utility theory of value on the demand side and a more general theory of costs on the supply side. In the 20th century, neoclassical theorists moved away from an earlier notion suggesting that total utility for a society could be measured in favour of ordinal utility, which hypothesizes merely behaviour-based relations across persons.

Ruins of Tiverton Castle, seat of the Earls of Devon Edward de Courtenay, 3rd/11th Earl of Devon (c.1357 - 5 December 1419), known by the epithet the "Blind Earl", was the son of Sir Edward de Courtenay and Emeline Dawnay, and in 1377 succeeded his grandfather, Hugh Courtenay, 10th Earl of Devon, as Earl of Devon. The ordinal number given to the early Courtenay Earls of Devon depends on whether the earldom is deemed a new creation by the letters patent granted 22 February 1334/5 or whether it is deemed a restitution of the old dignity of the de Redvers family. Authorities differ in their opinions,Watson, in Cokayne, The Complete Peerage, new edition, IV, p.

In modern American baseball, some batting positions have nicknames: "leadoff" for first, "cleanup" for fourth, and "last" for ninth. Others are known by the ordinal numbers or the term #-hole (3rd place hitter would be 3-hole). In similar fashion, the third, fourth, and fifth batters are often collectively referred to as the "heart" or "meat" of the batting order, while the seventh, eighth, and ninth batters are called the "bottom of the lineup," a designation generally referring both to their hitting position and to their typical lack of offensive prowess. At the start of each inning, the batting order resumes where it left off in the previous inning, rather than resetting to start with the #1 hitter again.

They then provided pages of quotations, detailing Roman and Orthodox liturgies that they considered guilty of the same alleged offenses. According to the archbishops, if the ordinations of the bishops and priests in the Anglican churches were invalid then, by the same measure, so must be the ordinations of clergy in the Roman Catholic and Orthodox churches. They pointed out that the preface to the ordinal explicitly stated that no new type of orders were to be conferred but were the continuation of the apostolic succession. On the charge of intent, the response argued that the readmission of the required phrases in 1662 were addressed more to the Presbyterian rather than the Roman controversy.

The ARIA Awards are given in four fields: ARIA Awards (for general and genre categories), Fine Arts, Artisan and Public Vote. With the exception of the Public Vote field, all award winners and nominees are determined by either a "voting academy" or a "judging school"; the nominees for the public voted categories are determined by ARIA with the public choosing the winner. In the following tables, all the categories are listed in order of the year they were first given; any box in the "last awarded" column that says "N/A" is a current award. The years are linked to their corresponding ceremony and the ordinal numbers beside the year correspond to the order they were presented.

In linguistics, a numeral (or number word) in the broadest sense is a word or phrase that describes a numerical quantity. Some theories of grammar use the word "numeral" to refer to cardinal numbers that act as a determiner to specify the quantity of a noun, for example the "two" in "two hats". Some theories of grammar do not include determiners as a part of speech and consider "two" in this example to be an adjective. Some theories consider "numeral" to be a synonym for "number" and assign all numbers (including ordinal numbers like the compound word "seventy-fifth") to a part of speech called "numerals"Charles Follen: A Practical Grammar of the German Language.

Here, the basic open sets are the half open intervals [a, b). This topology on R is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it. If Γ is an ordinal number, then the set Γ = [0, Γ) may be endowed with the order topology generated by the intervals (a, b), [0, b) and (a, Γ) where a and b are elements of Γ. Outer space of a free group Fn consists of the so-called "marked metric graph structures" of volume 1 on Fn.

In these competitions, it is common for more than one wine to receive any given medal. These competitions often also include a "Best in Class" award, producing a clear category winner among those vintages awarded any particular medal, as seen in the Los Angeles International Wine & Spirits Competition, the New York International Wine Competition, and The Decanter World Wine Awards.Decanter World Wine Awards 2016 Winners In still other competitions, instead of giving numerous awards, the wines in each wine category are ranked by number from high to low, a process known as ordinal ranking. In these competitions, there is only one first-place winner, one second place, one third place, and so on down to the lowest place.

Mohs hardness kit, containing one specimen of each mineral on the ten-point hardness scale The Mohs scale of mineral hardness () is a qualitative ordinal scale characterizing scratch resistance of various minerals through the ability of harder material to scratch softer material. Created in 1812 by German geologist and mineralogist Friedrich Mohs, it is one of several definitions of hardness in materials science, some of which are more quantitative."Mohs hardness" in Encyclopædia Britannica Online The method of comparing hardness by observing which minerals can scratch others is of great antiquity, having been mentioned by Theophrastus in his treatise On Stones, , followed by Pliny the Elder in his Naturalis Historia, .Theophrastus on Stones. Farlang.com.

The church preserved considerable independence in judicial matters, but gave up its old claim that the Norwegian kingdom was a fief under the ultimate authority of the Catholic Church.Sættargjerden i Tunsberg (Store norske leksikon) In cultural terms Magnus continued his father's policy of introducing European courtly culture to Norway. In 1277 he replaced the old Norse titles lendmann and skutilsvein with the European titles baron and riddar (knight), at the same time giving them certain extra privileges and the right to be addressed as lord (herra). Magnus is probably also the first Norwegian king to have named himself using an ordinal number - he called himself "Magnus IV" (he did not count Magnus Haraldsson (II) and Magnus Sigurdsson (IV)).

Leo XIII condemned the Anglican ordinals and deemed the Anglican orders "absolutely null and utterly void". Some Changes in the Anglican Ordinal since King Edward VI, and a fuller appreciation of the pre-Reformation ordinals suggest, according to some private theologians, that the correctness of the dismissal of Anglican orders may be questioned; however remains Roman Catholic definitive teaching and was reinforced by then-Cardinal Joseph Ratzinger, who later became Pope Benedict XVI. Since 1896 many Anglican bishops have been consecrated by bishops of the Old Catholic Church. Nevertheless, all Anglican clergymen who desire to enter the Catholic Church do so as laymen and must be ordained in the Catholic Church in order to serve as priests.

Woodin cardinals are important in descriptive set theory. By a resultA Proof of Projective Determinacy of Martin and Steel, existence of infinitely many Woodin cardinals implies projective determinacy, which in turn implies that every projective set is measurable, has the Baire property (differs from an open set by a meager set, that is, a set which is a countable union of nowhere dense sets), and the perfect set property (is either countable or contains a perfect subset). The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses. Working in ZF+AD+DC one can prove that \Theta _0 is Woodin in the class of hereditarily ordinal-definable sets.

Ridley played a major part in the vestments controversy. John Hooper, having been exiled during King Henry's reign, returned to England in 1548 from the churches in Zürich that had been reformed by Zwingli and Heinrich Bullinger in a highly iconoclastic fashion. When Hooper was invited to give a series of Lenten sermons before the king in February 1550, he spoke against Cranmer's 1549 ordinal whose oath mentioned "all saints" and required newly elected bishops and those attending the ordination ceremony to wear a cope and surplice. In Hooper's view, these requirements were vestiges of Judaism and Roman Catholicism, which had no biblical warrant for Christians since they were not used in the early Christian church.

Called before them on 15 May 1550, a compromise was reached. Vestments were to be considered a matter of adiaphora, or Res Indifferentes ("things indifferent", as opposed to an article of faith), and Hooper could be ordained without them at his discretion, but he must allow that others could wear them. Hooper passed confirmation of the new office again before the king and council on 20 July 1550 when the issue was raised again, and Cranmer was instructed that Hooper was not to be charged "with an oath burdensome to his conscience". Cranmer assigned Ridley to perform the consecration, and Ridley refused to do anything but follow the form of the ordinal as it had been prescribed by Parliament.

In the United Kingdom, however, the practice of terminating all bills upon prorogation has slightly altered; public bills may be re-introduced in the next legislative session, and fast-tracked directly to the stage they reached in the prorogued legislative session. A new session will often begin on the same day that the previous session ended. In most cases, when parliament reconvenes for a new legislative session, the head of state, or a representative thereof, will address the legislature in an opening ceremony. In both parliamentary and presidential systems, sessions are referred to by the name of the body and an ordinal numberfor example, the 2nd Session of the 39th Canadian Parliament or the 1st Session of the 109th United States Congress.

At the 1991 World Figure Skating Championships they were very close to winning. They led after both the compulsory dances and original dance (although finishing 2nd in the original dance portion), and in the free dance received 4 1st place ordinals from the 9 judges. Nonetheless a strange ordinal situation led to them finishing only 3rd in the free dance and dropping to 3rd overall behind the Duchensays and Klimova and Ponomarenko. In the 1991–92 season, Usova/Zhulin won silver at the 1992 European Championships in Lausanne, Switzerland and then captured their first Olympic medal, bronze, at the 1992 Winter Olympics in Albertville, France. Usova/Zhulin ended their season with silver at the 1992 World Championships in Oakland, California.

Morel graduated in 1941 from the University of California, Los Angeles. She began graduate study in mathematics in 1942 at the University of California, Berkeley, but left her studies to serve in the WAVES (the United States Naval Women's Reserve) during World War II. She returned to her studies in Berkeley in 1946, and completed her Ph.D. in 1953. Her dissertation, A Study in the Arithmetic of Order Types, was supervised by Alfred Tarski, and concerned ordinal arithmetic. After two years as an assistant professor at Berkeley, and positions at the University of California, Davis and the Institute for Advanced Study (1959–1960), she joined the mathematics faculty at the University of Washington in 1960, and became a tenured associate professor there in 1961.

A natural number (which, in this context, includes the number 0) can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets, these two concepts coincide, and there is only one way to put a finite set into a linear sequence (up to isomorphism). When dealing with infinite sets, however, one has to distinguish between the notion of size, which leads to cardinal numbers, and the notion of position, which leads to the ordinal numbers described here. This is because while any set has only one size (its cardinality), there are many nonisomorphic well-orderings of any infinite set, as explained below.

For a given proof, such a procedure produces a tree of proofs, with the given one serving as the root of the tree, and the other proofs being, in a sense, "simpler" than the given one. This increasing simplicity is formalized by attaching an ordinal < ε0 to every proof, and showing that, as one moves down the tree, these ordinals get smaller with every step. He then shows that if there were a proof of a contradiction, the reduction procedure would result in an infinite descending sequence of ordinals smaller than ε0 produced by a primitive recursive operation on proofs corresponding to a quantifier-free formula.See for a full presentation of Gentzen's proof and various comments on the historic and philosophical significance of the result.

Nonetheless, they believed that this caused a break of continuity in apostolic succession, making all further ordinations null and void. Eastern Orthodox bishops have, on occasion, granted "economy" when Anglican priests convert to Orthodoxy. Various Orthodox churches have also declared Anglican orders valid subject to a finding that the bishops in question did indeed maintain the true faith, the Orthodox concept of apostolic succession being one in which the faith must be properly adhered to and transmitted, not simply that the ceremony by which a man is made a bishop is conducted correctly. Changes in the Anglican Ordinal since King Edward VI, and a fuller appreciation of the pre-Reformation ordinals, suggest that the correctness of the enduring dismissal of Anglican orders is questionable.

Title page of the 1662 Prayer Book The Savoy Conference ended in disagreement late in July 1661, but the initiative in prayer book revision had already passed to the Convocations and from there to Parliament. The Convocations made some 600 changes, mostly of details, which were "far from partisan or extreme". However, Edwards states that more of the changes suggested by high Anglicans were implemented (though by no means all ) and Spurr comments that (except in the case of the Ordinal) the suggestions of the "Laudians" (Cosin and Matthew Wren) were not taken up possibly due to the influence of moderates such as Sanderson and Reynolds. For example, the inclusion in the intercessions of the Communion rite of prayer for the dead was proposed and rejected.

The title of the album is the first to break from the title style of previous Radwimps albums, featuring Radwimps followed by an ordinal number, and followed by a subtitle. This change was intended to show how the mood of the band had changed between this album and Radwimps 4. The title of the album, Altocolony no Teiri, started with the word , which Noda interprets as a key to a problem, as a theorem is in mathematics a statement that is true based on previous mathematical proofs. After listening to the completed album, Noda felt like the album was a theorem explaining himself, that it was a solution to all of his worries and the puzzles before him when he wrote the songs.

In some cases, there are different ways to import the concepts into ZFC and NFU. For example, the usual definition of the first infinite ordinal \omega in ZFC is not suitable for NFU because the object (defined in purely set theoretical language as the set of all finite von Neumann ordinals) cannot be shown to exist in NFU. The usual definition of \omega in NFU is (in purely set theoretical language) the set of all infinite well-orderings all of whose proper initial segments are finite, an object which can be shown not to exist in ZFC. In the case of such imported objects, there may be different definitions, one for use in ZFC and related theories, and one for use in NFU and related theories.

Two well-orderings W_1 and W_2 are similar and write W_1 \sim W_2 just in case there is a bijection f from the field of W_1 to the field of W_2 such that x W_1 y \leftrightarrow f(x)W_2f(y) for all x and y. Similarity is shown to be an equivalence relation in much the same way that equinumerousness was shown to be an equivalence relation above. In New Foundations (NFU), the order type of a well-ordering W is the set of all well-orderings which are similar to W. The set of ordinal numbers is the set of all order types of well-orderings. This does not work in ZFC, because the equivalence classes are too large.

The National Electoral Council (CNE) is composed of five persons; three of them nominated by civil society, one by the faculties of law and political science at national universities, and one by the Citizen Power. The three members nominated by civil society shall have six alternates in ordinal sequence, and each appointed by the universities and the Citizen Power has two alternates, respectively. Members of the National Electoral Council last seven years in office and be elected separately: the three nominated by civil society at the beginning of each period of the National Assembly, and the other two in the middle of it. Members of the National Electoral Council shall be appointed by the National Assembly with the vote of two thirds of its members.

Shapirovsky, "PSPACE-decidability of Japaridze's polymodal logic". Advances in Modal Logic 7 (2008), pp. 289-304. and the PSPACE-hardness of its variable-free fragment was proven by F.Pakhomov. Among the most notable applications of GLP has been its use in proof-theoretically analyzing Peano arithmetic, elaborating a canonical way for recovering ordinal notation system up to 0 from the corresponding algebra, and constructing simple combinatorial independent statements (Beklemishev L. Beklemishev, "Provability algebras and proof-theoretic ordinals, I". Annals of Pure and Applied Logic 128 (2004), pp. 103–123.). An extensive survey of GLP in the context of provability logics in general was given by George Boolos in his book “The Logic of Provability”.G. Boolos, “The Logic of Provability”.

Shakespeare and Cervantes seemingly died on exactly the same date (23 April 1616), but Cervantes predeceased Shakespeare by ten days in real time (as Spain used the Gregorian calendar, but Britain used the Julian calendar). This coincidence encouraged UNESCO to make 23 April the World Book and Copyright Day. Astronomers avoid this ambiguity by the use of the Julian day number. For dates before the year 1, unlike the proleptic Gregorian calendar used in the international standard ISO 8601, the traditional proleptic Gregorian calendar (like the Julian calendar) does not have a year 0 and instead uses the ordinal numbers 1, 2, ... both for years AD and BC. Thus the traditional time line is 2 BC, 1 BC, AD 1, and AD 2.

In dynastic China, the kè was a unit that represented of a day (it has since been refined to of a day, or 15 minutes). In France, a decimal time system in place from 1793 to 1805 divided the day into 10 hours, each divided into 100 minutes, in turn each divided into 100 seconds; the French Republican Calendar further extended this by assembling days into ten-day "weeks". Ordinal dates and Julian days, the latter of which has seen use in astronomy as it is not subject to leap year complications) allow for the expression of a decimal portion of the day.See, for instance, the current Mean Julian Day , precise to 6 decimal place, from the United States Naval Observatory.

George Mackey and Irving Kaplansky generalized Ulm's theorem to certain modules over a complete discrete valuation ring. They introduced invariants of abelian groups that lead to a direct statement of the classification of countable periodic abelian groups: given an abelian group A, a prime p, and an ordinal α, the corresponding αth Ulm invariant is the dimension of the quotient : pαA[p]/pα+1A[p], where B[p] denotes the p-torsion of an abelian group B, i.e. the subgroup of elements of order p, viewed as a vector space over the finite field with p elements. : A countable periodic reduced abelian group is determined uniquely up to isomorphism by its Ulm invariants for all prime numbers p and countable ordinals α.

Aarhus has a temperate oceanic climate (Köppen: Cfb), Note:The Köppen World Map is rather course-scaled, and not very useful or precise on scales the size of Denmark. and the weather is constantly influenced by major weather systems from all four ordinal directions, resulting in unstable conditions throughout the year. Temperature varies a great deal across the seasons with a mild spring in April and May, warmer summer months from June to August, frequently rainy and windy autumn months in October and September and cooler winter months, often with frost and occasional snow, from December to March. The city centre experiences the same climatic effects as other larger cities with higher wind speeds, more fog, less precipitation and higher temperatures than the surrounding, open land.

To this day, Turing machines are a central object of study in theory of computation. From September 1936 to July 1938, Turing spent most of his time studying under Church at Princeton University, in the second year as a Jane Eliza Procter Visiting Fellow. In addition to his purely mathematical work, he studied cryptology and also built three of four stages of an electro- mechanical binary multiplier. In June 1938, he obtained his PhD from the Department of Mathematics at Princeton; his dissertation, Systems of Logic Based on Ordinals, introduced the concept of ordinal logic and the notion of relative computing, in which Turing machines are augmented with so-called oracles, allowing the study of problems that cannot be solved by Turing machines.

Beta is often used to denote a variable in mathematics and physics, where it often has specific meanings for certain applications. In physics a stream of unbound energetic electrons is commonly referred to as beta radiation or beta rays. In regression analysis, symbolizes nonstandardized partial slope coefficients, whereas represents standardized (standard deviation-score form) coefficients; in both cases, the coefficients reflect the change in the criterion Y per one- unit change in the value of the associated predictor X. β is also used in biology, for instance in β-Carotene, a primary source of provitamin A, or the β cells in pancreatic islets, which produce insulin. β is sometimes used as a placeholder for an ordinal number if α is already used.

Although it is widely claimed that Gödel's theorem rules out the possibility of a finitistic consistency proof for Peano arithmetic, this depends on exactly what one means by a finitistic proof. Gödel himself pointed out the possibility of giving a finitistic consistency proof of Peano arithmetic or stronger systems by using finitistic methods that are not formalizable in Peano arithmetic, and in 1958, Gödel published a method for proving the consistency of arithmetic using type theory. In 1936, Gerhard Gentzen gave a proof of the consistency of Peano's axioms, using transfinite induction up to an ordinal called ε0. Gentzen explained: "The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles".

Formally, assuming the axiom of choice, the cardinality of a set X is the least ordinal α such that there is a bijection between X and α. This definition is known as the von Neumann cardinal assignment. If the axiom of choice is not assumed we need to do something different. The oldest definition of the cardinality of a set X (implicit in Cantor and explicit in Frege and Principia Mathematica) is as the set of all sets that are equinumerous with X: this does not work in ZFC or other related systems of axiomatic set theory because this collection is too large to be a set, but it does work in type theory and in New Foundations and related systems.

As a result, subsequent formal treatments of calculus tended to drop the infinitesimal viewpoint in favor of limits, which can be performed using the standard reals. Infinitesimals regained popularity in the 20th century with Abraham Robinson's development of nonstandard analysis and the hyperreal numbers, which showed that a formal treatment of infinitesimal calculus was possible, after a long controversy on this topic by centuries of mathematics. Following this was the development of the surreal numbers, a closely related formalization of infinite and infinitesimal numbers that includes both the hyperreal numbers and ordinal numbers, and which is the largest ordered field. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were infinitely small.

In the liturgy of the post-Vatican II Roman Rite, Ordinary Time is that part of the Christian liturgical year outside of Advent, Christmastide, Lent, the Easter Triduum, and Eastertide,Universal Norms on the Liturgical Year and the Calendar, 43 and is divided into two periods: that between Christmastide and Lent, and that between Eastertide and Advent. In this season the Church celebrates the public ministry of Jesus from his baptism to the time of his final suffering and death. The word "ordinary" as used here comes from the ordinal numerals by which the weeks are identified or counted, from the 1st week of Ordinary Time in January to the 34th week that begins toward the end of November; Ordinary Time is interrupted by Lent and Eastertide.

In fair division, a topic in economics, a preference relation is weakly additive if the following condition is met: : If A is preferred to B, and C is preferred to D (and the contents of A and C do not overlap) then A together with C is preferable to B together with D. Every additive utility function is weakly-additive. However, additivity is applicable only to cardinal utility functions, while weak additivity is applicable to ordinal utility functions. Weak additivity is often a realistic assumption when dividing up goods between claimants, and simplifies the mathematics of certain fair division problems considerably. Some procedures in fair division do not need the value of goods to be additive and only require weak additivity.

The nimber multiplicative inverse of the nonzero ordinal is given by , where is the smallest set of ordinals (nimbers) such that # 0 is an element of ; # if and is an element of , then is also an element of . For all natural numbers , the set of nimbers less than form the Galois field of order . In particular, this implies that the set of finite nimbers is isomorphic to the direct limit as of the fields . This subfield is not algebraically closed, since no other field (so with not a power of 2) is contained in any of those fields, and therefore not in their direct limit; for instance the polynomial , which has a root in , does not have a root in the set of finite nimbers.

Note that if α is a successor ordinal, then α is compact, in which case its one-point compactification α+1 is the disjoint union of α and a point. As topological spaces, all the ordinals are Hausdorff and even normal. They are also totally disconnected (connected components are points), scattered (every non-empty set has an isolated point; in this case, just take the smallest element), zero- dimensional (the topology has a clopen basis: here, write an open interval (β,γ) as the union of the clopen intervals (β,γ'+1)=[β+1,γ'] for γ'<γ). However, they are not extremally disconnected in general (there are open sets, for example the even numbers from ω, whose closure is not open).

In the analysis of multivariate observations designed to assess subjects with respect to an attribute, a Guttman Scale (named after Louis Guttman) is a single (unidimensional) ordinal scale for the assessment of the attribute, from which the original observations may be reproduced. The discovery of a Guttman Scale in data depends on their multivariate distribution's conforming to a particular structure (see below). Hence, a Guttman Scale is a hypothesis about the structure of the data, formulated with respect to a specified attribute and a specified population and cannot be constructed for any given set of observations. Contrary to a widespread belief, a Guttman Scale is not limited to dichotomous variables and does not necessarily determine an order among the variables.

Pareto was the first to realize that cardinal utility could be dispensed with and economic equilibrium thought of in terms of ordinal utility – that is, it was not necessary to know how much a person valued this or that, only that he preferred X of this to Y of that. Utility was a preference-ordering. With this, Pareto not only inaugurated modern microeconomics, but he also demolished the alliance of economics and utilitarian philosophy (which calls for the greatest good for the greatest number; Pareto said "good" cannot be measured). He replaced it with the notion of Pareto-optimality, the idea that a system is enjoying maximum economic satisfaction when no one can be made better off without making someone else worse off.

John Hooper, having been exiled during King Henry's reign, returned to England in 1548 from the churches in Zürich that had been reformed by Zwingli and Bullinger in a highly iconoclastic fashion. Hooper became a leading Protestant reformer in England under the patronage of Edward Seymour, 1st Duke of Somerset and subsequently John Dudley, 1st Duke of Northumberland. Hooper's fortunes were unchanged when power shifted from Somerset to Northumberland, since Northumberland also favoured Hooper's reformist agenda. When Hooper was invited to give a series of Lenten sermons before the king in February 1550, he spoke against the 1549 ordinal whose oath mentioned "all saints" and required newly elected bishops and those attending the ordination ceremony to wear a cope and surplice.

At the 1815 Battle of Waterloo 14 sergeants of the 40th (the 2nd Somersetshire) Regiment of Foot were killed or wounded while serving in the colour party and at the 1854 Battle of the Alma the colour party of the 21st (Royal North British Fusilier) Regiment of Foot lost 3 officers and 17 sergeants. Colours were first regulated by order of George II in 1747. The recent Jacobite rising of 1745 had prompted the king to set in place army reforms to standardise uniforms, drill and tactics. He was keen to ensure the soldiers' loyalty to the crown rather than the colonel of their regiment (regiments up to this time were known by their colonel's name rather than an ordinal number).

The more general conception of utility is that of use or usefulness, and this conception is at the heart of marginalism; the term "marginal utility" arose from translation of the German "Grenznutzen",von Wieser, Friedrich; Der natürliche Werth [Natural Value] (1889), Bk I Ch V "Marginal Utility" (HTML). which literally means border use, referring directly to the marginal use, and the more general formulations of marginal utility do not treat quantification as an essential feature.Mc Culloch, James Huston; "The Austrian Theory of the Marginal Use and of Ordinal Marginal Utility", Zeitschrift für Nationalökonomie 37 (1973) #3&4 (September). On the other hand, none of the early marginalists insisted that utility were not quantified,Stigler, George Joseph; "The Development of Utility Theory" Journal of Political Economy (1950).

In the Sprague–Grundy theory the minimum excluded ordinal is used to determine the nimber of a normal-play impartial game. In such a game, either player has the same moves in each position and the last player to move wins. The nimber is equal to 0 for a game that is lost immediately by the first player, and is equal to the mex of the nimbers of all possible next positions for any other game. For example, in a one-pile version of Nim, the game starts with a pile of stones, and the player to move may take any positive number of stones. If is zero stones, the nimber is 0 because the mex of the empty set of legal moves is the nimber 0.

Numbering sequences starting at 0 is quite common in mathematics notation, in particular in combinatorics, though programming languages for mathematics usually index from 1. In computer science, array indices usually start at 0 in modern programming languages, so computer programmers might use zeroth in situations where others might use first, and so forth. In some mathematical contexts, zero-based numbering can be used without confusion, when ordinal forms have well established meaning with an obvious candidate to come before first; for instance a zeroth derivative of a function is the function itself, obtained by differentiating zero times. Such usage corresponds to naming an element not properly belonging to the sequence but preceding it: the zeroth derivative is not really a derivative at all.

His work on auctions with Robert Weber introduced the concept of affiliation of random variables, to indicate systems of unknown quantities where learning that any one of them is higher than some given level would cause beliefs about others to be higher. His work with John Roberts and Chris Shannon advanced the use of supermodularity as a property of individuals' preferences that can yield general monotonicity results in economic analysis. The work of Milgrom and Shannon (1994) showed that comparative statics results could often be obtained through more relevant and intuitive ordinal conditions. Indeed, they show that their concept of quasi-supermodularity (a generalization of supermodular function) along with the single-crossing property, is necessary and sufficient for comparative statics to obtain on arbitrary choice sets.

Similarly, for all x and y in X and a in A, (a, x) > (a, y) is implied for every d in A such that (d, x) > (d, y). What this means is that if any two levels, a, b, are ordered, then this order holds irrespective of each and every level of X. The same holds for any two levels, x and y of X with respect to each and every level of A. Single cancellation is so-called because a single common factor of two levels of P cancel out to leave the same ordinal relationship holding on the remaining elements. For example, a cancels out of the inequality (a, x) > (a, y) as it is common to both sides, leaving x > y. Krantz, et al.

Cello first position fingerings Fingered music for guitar: the numbers 1 to 4 indicate the stopping fingers, 0 an open note, circled numbers strings, and dashed numbers slipping On string instruments fingers are numbered from 1 to 4, beginning with the index finger, the thumb not being counted because it does not normally play on a string, and 0 indicating an open string. In those cases on string instruments where the thumb is used (such as high notes on a cello in thumb position), it is represented by a symbol the shape of an O with a vertical stem below(somewhat similar to Ǫ or ϙ, for instance). Guitar music indicates thumb, occasionally used to finger bass notes on the low E string, with a 'T'. Position may be indicated through ordinal numbers (e.g.

In English-language outside North America (mostly in Anglophone Europe and some countries in Australasia), full dates are written as 7 December 1941 (or 7th December 1941) and spoken as "the seventh of December, nineteen forty-one" (exceedingly common usage of "the" and "of"), with the occasional usage of December 7, 1941 ("December the seventh, nineteen forty-one"). In common with most continental European usage, however, all- numeric dates are invariably ordered dd/mm/yyyy. In Canada and the United States, the usual written form is December 7, 1941, spoken as "December seventh, nineteen forty-one" or colloquially "December the seventh, nineteen forty-one". Ordinal numerals, however, are not always used when writing and pronouncing dates, and "December seven, nineteen forty-one" is also an accepted pronunciation of the date written December 7, 1941.

He could show that this proposition can neither be proved nor disproved within the formalism. This can mean only two things: either the reasoning by which a proof of consistency is given must contain some argument that has no formal counterpart within the system, i.e., we have not succeeded in completely formalizing the procedure of mathematical induction; or hope for a strictly "finitistic" proof of consistency must be given up altogether. When G. Gentzen finally succeeded in proving the consistency of arithmetic he trespassed those limits indeed by claiming as evident a type of reasoning that penetrates into Cantor's "second class of ordinal numbers." made the following comment in 1952 on the significance of Gentzen's result, particularly in the context of the formalist program which was initiated by Hilbert.

Large cardinals are understood in the context of the von Neumann universe V, which is built up by transfinitely iterating the powerset operation, which collects together all subsets of a given set. Typically, models in which large cardinal axioms fail can be seen in some natural way as submodels of those in which the axioms hold. For example, if there is an inaccessible cardinal, then "cutting the universe off" at the height of the first such cardinal yields a universe in which there is no inaccessible cardinal. Or if there is a measurable cardinal, then iterating the definable powerset operation rather than the full one yields Gödel's constructible universe, L, which does not satisfy the statement "there is a measurable cardinal" (even though it contains the measurable cardinal as an ordinal).

In 1958, Shepard took a job at Bell Labs, whose computer facilities made it possible for him to expand earlier work on generalization. He reports, "This led to the development of the methods now known as nonmetric multidimensional scalingfirst by me (Shepard, 1962a, 1962b) and then, with improvements, by my Bell Labs mathematical colleague Joseph Kruskal (1964a, 1964b)." According to the American Psychological Association, "nonmetric multidimensional scaling .. has provided the social sciences with a tool of enormous power for uncovering metric structures from ordinal data on similarities." Awarding Shepard its Rumelhart Prize in 2006, the Cognitive Science Society called nonmetric multidimensional scaling a "highly influential early contribution," explaining that: > This method provided a new means of recovering the internal structure of > mental representations from qualitative measures of similarity.

The Spanish era (), sometimes called the era of Caesar, was a calendar era (year numbering system) commonly used in the states of the Iberian Peninsula from the 5th century until the 15th, when it was phased out in favour of the Anno Domini (AD) system. The epoch (start date) of the Spanish era was 1 January 38 BC. To convert an Anno Domini date to the corresponding year in the Spanish era, add 38 to the Anno Domini year, such that Era 941 would be equivalent to AD 903. A date given in the Spanish era always uses the word era followed by a feminine ordinal number (when written out instead of given in Roman numerals). This contrasts with the AD system that uses the masculine anno (year): i.e.

Thomas Friedman's formula for CQ Friedman's claim is that Curiosity quotient plus Passion quotient is greater than Intelligence Quotient. There is no evidence that this inequality is true. Friedman may believe that curiosity and passion are 'greater' than intelligence, but there is no evidence to suggest that the sum of a person's curiosity and passion quotients will always exceed their IQ. Indeed, given the ordinal nature of psychometric quotients, it is not clear whether it makes sense to add the curiosity and passion quotients or even if they can have numerical values attributed to them. According to Friedman, curiosity and passion are key components for education in a world where information is readily available to everyone and where global markets reward those who have learned how to learn and are self-motivated to learn.

The concept of Bolshevism arose at the Second Congress of the Russian Social Democratic Labour Party (1903) as a result of the split of the party into two factions: supporters of Lenin and the rest.Despite the ordinal number adopted in Soviet historiography, the London Congress was actually a constituent one, since the Minsk Congress had no practical significance One of the main reasons for the split was the question of a party of a new type. In the course of work on the Charter of the Russian Social Democratic Labor Party, Vladimir Lenin and Yuliy Martov proposed two different wordings of the clause on party membership. Lenin – a party member is a citizen who recognizes the program and charter, pays membership fees and works in one of the party organizations.

In some other Anglican churches they can be deacons instead of priests; such archdeacons often work with the bishop to help with deacons' assignments to congregations and assist the bishop at ordinations and other diocesan liturgies. The Anglican ordinal presupposes (it is policy by default) that every Archdeacon helps to examine candidates for ordination and presents the most suitable candidate(s) to the ordaining bishop. In some parts of the Communion where women cannot be consecrated as bishops, the position is the most senior office a female cleric can hold: this being so, for instance, in the (Anglican) Diocese of Sydney. Very rarely, "lay archdeacons" have been arisen, most notably the former Anglican Communion Observer to the United Nations, Taimalelagi Fagamalama Tuatagoloa-Leota, who retained her title after having served as Archdeacon of Samoa.

Gentzen's proof is arguably finitistic, since the transfinite ordinal ε0 can be encoded in terms of finite objects (for example, as a Turing machine describing a suitable order on the integers, or more abstractly as consisting of the finite trees, suitably linearly ordered). Whether or not Gentzen's proof meets the requirements Hilbert envisioned is unclear: there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition. The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as Gentzen's proof. A small number of philosophers and mathematicians, some of whom also advocate ultrafinitism, reject Peano's axioms because accepting the axioms amounts to accepting the infinite collection of natural numbers.

To actually define the function b, we need to employ the axiom of choice. Using the function b, we are going to define elements a0 < a1 < a2 < a3 < ... in P. This sequence is really long: the indices are not just the natural numbers, but all ordinals. In fact, the sequence is too long for the set P; there are too many ordinals (a proper class), more than there are elements in any set, and the set P will be exhausted before long and then we will run into the desired contradiction. The ai are defined by transfinite recursion: we pick a0 in P arbitrary (this is possible, since P contains an upper bound for the empty set and is thus not empty) and for any other ordinal w we set aw = b({av : v < w}).

This definition of "infinite set" should be compared with the usual definition: a set A is infinite when it cannot be put in bijection with a finite ordinal, namely a set of the form } for some natural number n – an infinite set is one that is literally "not finite", in the sense of bijection. During the latter half of the 19th century, most mathematicians simply assumed that a set is infinite if and only if it is Dedekind-infinite. However, this equivalence cannot be proved with the axioms of Zermelo–Fraenkel set theory without the axiom of choice (AC) (usually denoted "ZF"). The full strength of AC is not needed to prove the equivalence; in fact, the equivalence of the two definitions is strictly weaker than the axiom of countable choice (CC).

In set theory, a branch of mathematics, the minimal model is the minimal standard model of ZFC. The minimal model was introduced by and rediscovered by . The existence of a minimal model cannot be proved in ZFC, even assuming that ZFC is consistent, but follows from the existence of a standard model as follows. If there is a set W in the von Neumann universe V that is a standard model of ZF, and the ordinal κ is the set of ordinals that occur in W, then Lκ is the class of constructible sets of W. If there is a set that is a standard model of ZF, then the smallest such set is such a Lκ. This set is called the minimal model of ZFC, and also satisfies the axiom of constructibility V=L.

Parsimony analysis, bootstrapping and the minimum evolution principle led to groups of species, further described by conidial and colony morphology. The species were re-inoculated into apples grown at an Iowa State University research station in Gilbert, Iowa, re-isolated, sequenced, and morphology compared. Thirty isolates fulfilled Koch’s postulates as new species, all Dothideomycetes, 27 were within Dothideales, one was within Pleosporales and two with undetermined ordinal level. Only 2 species (Peltaster fructicola and Zygophiala jamaicensis) had previously been associated with SBFS A 2008 publication of the same sample plus a 2005 sample of 30 more orchards in 10 eastern U.S. states, (39 US apple orchards in 14 states) speciated by DNA- and phylogenetic analyses reported 58 putative species belonging to the Dothideomycetes, 52 of which were Capnodiales, and 36 were part of the Mycosphaerellaceae.

In Pascal, an enumerated type can be implicitly declared by listing the values in a parenthesised list: var suit: (clubs, diamonds, hearts, spades); The declaration will often appear in a type synonym declaration, such that it can be used for multiple variables: type cardsuit = (clubs, diamonds, hearts, spades); card = record suit: cardsuit; value: 1 .. 13; end; var hand: array [ 1 .. 13 ] of card; trump: cardsuit; The order in which the enumeration values are given matters. An enumerated type is an ordinal type, and the `pred` and `succ` functions will give the prior or next value of the enumeration, and `ord` can convert enumeration values to their integer representation. Standard Pascal does not offer a conversion from arithmetic types to enumerations, however. Extended Pascal offers this functionality via an extended `succ` function.

Writing in May 1982 in the Roman Catholic magazine The Tablet, Timothy Dufort argued that "a way is open for the recognition of the Orders held in the Church of England today without the necessity of contradicting Pope Leo XIII". He argued that the present Book of Common Prayer wording introduced in the 1662 ordinal signifies the orders being bestowed in the clearest of terms and would meet Leo's requirements, while that of 1552 and 1559 did not. Furthermore the answer of the archbishops in his view has in itself removed another obstacle, as it shows an intention on the part of the archbishops that is clearly adequate by the tests of Trent and the Holy Office. The final obstacle, the gap between 1552 and 1662, to which Pope Leo refers, has also disappeared.

Nimber addition (also known as nim-addition) can be used to calculate the size of a single nim heap equivalent to a collection of nim heaps. It is defined recursively by :, where the minimum excludant of a set of ordinals is defined to be the smallest ordinal that is not an element of . For finite ordinals, the nim-sum is easily evaluated on a computer by taking the bitwise exclusive or (XOR, denoted by ) of the corresponding numbers. For example, the nim-sum of 7 and 14 can be found by writing 7 as 111 and 14 as 1110; the ones place adds to 1; the twos place adds to 2, which we replace with 0; the fours place adds to 2, which we replace with 0; the eights place adds to 1.

It is eventually in a subset Y of V if there exists an a in [0, c) such that for every x ≥ a, the point f(x) is in Y. We have limx → c f(x) = L if and only if for every neighborhood Y of L, f is eventually in Y. The net f is frequently in a subset Y of V if and only if for every a in [0, c) there exists some x in [a, c) such that f(x) is in Y. A point y in V is a cluster point of the net f if and only if for every neighborhood Y of y, the net is frequently in Y. The first example is a special case of this with c = ω. See also ordinal-indexed sequence.

Here x \succ_i^p y means that individual i prefers alternative x to y at profile p. A simple game with ordinal preferences is a pair (W, p) consisting of a simple game W and a profile p. Given (W, p), a dominance (social preference) relation \succ^p_W is defined on X by x \succ^p_W y if and only if there is a winning coalition S \in W satisfying x \succ_i^p y for all i \in S. The core C(W,p) of (W, p) is the set of alternatives undominated by \succ^p_W (the set of maximal elements of X with respect to \succ^p_W): :x \in C(W,p) if and only if there is no y\in X such that y \succ^p_W x.

It does not satisfy the majority criterion, but it satisfies a weakened form of it: a majority can force their choice to win, although they might not exercise that capability. To address this point, some proponents of score voting argue for the inclusion of an extra instant-runoff round in which a majority preference is established between the two top-rated candidates. As it satisfies the criteria of a deterministic voting method, with non-imposition, non-dictatorship, monotonicity, and independence of irrelevant alternatives, it may appear that it violates Arrow's impossibility theorem. The reason that score voting is not a counter-example to Arrow's theorem is that it is a cardinal voting method, while the "universality" criterion of Arrow's theorem effectively restricts that result to ordinal voting methods.

Henie won the first of an unprecedented ten consecutive World Figure Skating Championships in 1927 at the age of fourteen. The results of 1927 World Championships, where Henie won in 3–2 decision (or 7 vs. 8 ordinal points) over the defending Olympic and World Champion Herma Szabo of Austria, was controversial, as three of the five judges that gave Henie first-place ordinals were Norwegian (1 + 1 + 1 + 2 + 2 = 7 points) while Szabo received first-place ordinals from an Austrian and a German Judge (1 + 1 + 2 + 2 + 2 = 8 points). Henie went on to win first of her three Olympic gold medals the following year, became one of the youngest figure skating Olympic champions. She defended her Olympic titles in 1932 and in 1936, and her world titles annually until 1936.

The date on the plate was used to study time and calendar in the Maya world, and the plate remains one of the earliest examples of the usage of a cyclical calendar in the Central-American world. It is remarkable for being the oldest known usage of a Maya ordinal zero,André Cauty, Jean-Michel Hoppan, Et un, et deux zéros mayas, in Pour la science, Dossier mathématiques exotiques, April/June 2005. which symbol (graphically derived from the drawing of a sitting man, typically representing a king's crowning) appears two times, one to form the date "0 Yaxkin" from the first day of the seventh month of the festive year in Haab' calendar, and one to denote the Moon-Bird king accessing its throne on the other side of the plaque.

The Condorcet jury theorem has recently been used to conceptualize score integration when several physician readers (radiologists, endoscopists, etc.) independently evaluate images for disease activity. This task arises in central reading performed during clinical trials and has similarities to voting. According to the authors, the application of the theorem can translate individual reader scores into a final score in a fashion that is both mathematically sound (by avoiding averaging of ordinal data), mathematically tractable for further analysis, and in a manner that is consistent with the scoring task at hand (based on decisions about the presence or absence of features, a subjective classification task) The Condorcet jury theorem is also used in ensemble learning in the field of machine learning. An ensemble method combines the predictions of many individual classifiers by majority voting.

Even container types had short forms that programmers could use to save typing. Thus the code above is equivalent to the shorter form: put fld "typehere" into theValue Objects within a given context—the card or background, for instance—were also given a runtime number based on their z-order on the screen. To assist in using their position for navigation, HyperTalk also included a variety of ordinal and cardinal referencing systems to simplify the syntax further. Assuming the field "typehere" is the only field on the card, the code above could also be written: put the first card field into theValue or: put card field 1 into theValue The choice of addressing style was left to the programmer; often different styles were used in different statements in order to make the code more readable.

Even then there were some third-party applications that enhanced the interface or replaced the system shell. As a telephony device, the newer N-Gage no longer supported three GSM frequency bands 900/1800/1900; instead it came in two dual-band variants, one for the American market and another for the European and Asian markets. The only change made to the device's buttons was the replacement of the original five-way controller (four ordinal directions and a center "click" or confirm) with a simpler four- way directional controller and a separate "OK" button with a check logo. The QD was running the same software version as its predecessor, despite the newer Symbian 7.0s Series 60 2nd Edition having been shipped on several smartphones by the time the QD was announced.

For ordinal numbers, when the numerals are preceded by the prefix tē (第), the colloquial set is used with the exception of numeral 1 and 2; when the numerals are preceded by the prefix thâu (頭), there is no exception to use the colloquial set when the number is smaller than 10, but once the number is greater than 10, the exception of numeral 1 and 2 appears again. Note that the system with prefix thâu is usually added by counter words, and it means "the first few"; for example, thâu-gō͘ pái means "the first five times". Thâu-chhit (number seven) sometimes means thâu-chhit kang (first seven days). It means the first seven days after a person died, which is a Hokkien cultural noun that should usually be avoided.

Instead, they had to memorize the order of each photograph. Because the simultaneous training paradigm requires the subject to represent each item’s ordinal position, it provides an opportunity to study animal cognition. In the early 80s, Terrace helped organize an international conference on animal cognition at Columbia University that discussed the simultaneous training paradigm and other instances in which animals are able to represent stimuli. Since then, animal cognition has become a dominant area in comparative cognition. In 1985, Terrace began a primate cognition laboratory in which he studied how monkeys use representations in various serial learning tasks, for example, to respond in the correct order to ascending and descending series of numerically defined stimuli, to acquire serial expertise [the ability to become progressively better at learning arbitrary sequences] and to imitate another monkey’s sequential performance.

Richard Hooker (1554–1600), one of the most influential figures in shaping Anglican theology and self- identity Canterbury Cathedral houses the cathedra or episcopal chair of the Archbishop of Canterbury and is the cathedral of the Diocese of Canterbury and the mother church of the Church of England as well as a focus for the Anglican Communion The canon law of the Church of England identifies the Christian scriptures as the source of its doctrine. In addition, doctrine is also derived from the teachings of the Church Fathers and ecumenical councils (as well as the ecumenical creeds) in so far as these agree with scripture. This doctrine is expressed in the Thirty-Nine Articles of Religion, the Book of Common Prayer, and the Ordinal containing the rites for the ordination of deacons, priests, and the consecration of bishops.Canon A5.

Because compulsory figures were scored using a wider range of marks than the short program or free skating, this system allowed skaters to take a large lead in that segment of the competition, which made them effectively unreachable in later segments. The system of factored placements, in which ordinals were computed for each competition segment separately and factors applied to the relative placements rather than the raw marks, was proposed as early as 1971 by former Hungarian champion and World Referee Pál Jaross, and finally adopted for the 1980–1981 season. In 1998, the method by which placements within a segment were computed was changed from "best of majority"—ranking skaters by the highest ordinal for which they received a majority vote of the judges—to a system of "one-by-one" comparisons between the ordinals of all the skaters.

There is a natural order on the ordinals defined by \alpha\leq \beta if and only if some (and so any) W_1 \in \alpha is similar to an initial segment of some (and so any) W_2\in \beta. Further, it can be shown that this natural order is a well-ordering of the ordinals and so must have an order type \Omega. It would seem that the order type of the ordinals less than \Omega with the natural order would be \Omega, contradicting the fact that \Omega is the order type of the entire natural order on the ordinals (and so not of any of its proper initial segments). But this relies on one's intuition (correct in ZFC) that the order type of the natural order on the ordinals less than \alpha is \alpha for any ordinal \alpha.

ERP waveforms consist of a series of positive and negative voltage deflections, which are related to a set of underlying components. Though some ERP components are referred to with acronyms (e.g., contingent negative variation – CNV, error-related negativity – ERN), most components are referred to by a letter (N/P) indicating polarity (negative/positive), followed by a number indicating either the latency in milliseconds or the component's ordinal position in the waveform. For instance, a negative-going peak that is the first substantial peak in the waveform and often occurs about 100 milliseconds after a stimulus is presented is often called the N100 (indicating its latency is 100 ms after the stimulus and that it is negative) or N1 (indicating that it is the first peak and is negative); it is often followed by a positive peak, usually called the P200 or P2.

"While he argues that the rank originated with the Apostles, enjoyed divine approval, and flourished throughout Christendom, he rejects the view inherent in the Catholic position that the office is divinely commanded or is a result of divine law." The preface to the Ordinal limits itself to stating historical reasons why episcopal orders are to 'be continued and reverently used in the Church of England'. The "foreign Reformed [Presbyterian] Churches" were genuine ones despite the lack of apostolic succession because they had been abandoned by their bishops at the Reformation. This view was of the reformed churches was questioned during the earlier part of the seventeenth century and the 1662 Act of Uniformity formally excluded from pastoral office in England any who lacked episcopal ordination but this was a reaction against the abolition of episcopacy in the Commonwealth period.

The English Reformation advanced under pressure from two directions: from the traditionalists on the one hand and the zealots on the other, who led incidents of iconoclasm (image-smashing) and complained that reform did not go far enough. Reformed doctrines were made official, such as justification by faith alone and communion for laity as well as clergy in both kinds, of bread and wine. The Ordinal of 1550 replaced the divine ordination of priests with a government-run appointment system, authorising ministers to preach the gospel and administer the sacraments rather than, as before, "to offer sacrifice and celebrate mass both for the living and the dead".; ; Cranmer set himself the task of writing a uniform liturgy in English, detailing all weekly and daily services and religious festivals, to be made compulsory in the first Act of Uniformity of 1549.

Parker's consecration was, however, legally valid only by the plenitude of the Royal Supremacy approved by the Commons and reluctantly by a vote of the Lords 21–18; the Edwardine Ordinal, which was used, had been repealed by Mary Tudor and not re-enacted by the parliament of 1559. Parker mistrusted popular enthusiasm, and he wrote in horror of the idea that "the people" should be the reformers of the church. He was convinced that if ever Protestantism was to be firmly established in England at all, some definite ecclesiastical forms and methods must be sanctioned to secure the triumph of order over anarchy, and he vigorously set about the repression of what he thought a mutinous individualism incompatible with a catholic spirit. He was not an inspiring leader and no dogma or prayer book is associated with his name.

Frontispiece of the Breviary, depicting Eleanor of Viseu in prayer before a prie-dieu draped with her personal arms and device The Breviary of Eleanor of Portugal is an early 16th-century Flemish illuminated manuscript Breviary, providing the divine office according to the Roman ordinal and calendar. It contains the work of several leading miniaturists of the Ghent-Bruges school of Flemish illumination. The "Master of the First Prayerbook of Maximilian" seems to have led the team of artists that produced the codex, which included the Master of James IV of Scotland (who some scholars identify with Gerard Horenbout, court artist to Margaret of Austria), who painted many of the historiated borders, the calendar, as well as the small miniatures in the Ferial Psalter, and the Master of the Prayerbooks of c. 1500 or an artist in his circle.

Odex's subtitling has been criticized by the Singapore anime community for having font with lower quality and sometimes inaccurate translations, as compared to fansubs or imports (an example would be its release of "Sword Art Online: Ordinal Scale" which suffers from some glaring mistakes such as misnaming the character Eiji by his voice actor's name). Allegations were made by the online community that Odex had passed off fansubs as its own work. Sing admitted that this was partially true as Odex had hired anime fans to do subtitling in 2004 who had taken "the easy way out and copied word for word the subtitles on fansubs they downloaded". Sing explained that when Odex released its anime, the company did not realise what the anime fans had done, and it has been "paying for this mistake ever since".

While the 273rd Infantry Division itself initially did not see full redeployment, its subordinate regiments did. The three infantry regiments were deployed on 27 January 1942 at Milowitz military base, were redesignated Grenadier Regiments 544, 545, and 546 on 15 October 1942, and destroyed as part of the 389th Infantry Division at the Battle of Stalingrad between January and February 1943. They were subsequently reassembled under supervision of the 7th Army, sent back to the Eastern Front to fight in the Korsun–Cherkassy Pocket and were eventually trapped at Danzig in 1945. In November 1943, the ordinal number 273 was used for the 273rd Reserve Panzer Division, which was active until March 1944. In April 1945, a second 273rd Infantry Division was deployed as one of the last desperate formations during the final stages of the war.

Sample ballot of ranked voting using written numbers Ranked voting is any election voting system in which voters use a ranked (or preferential) ballot to rank choices in a sequence on the ordinal scale: 1st, 2nd, 3rd, etc. There are multiple ways in which the rankings can be counted to determine which candidate (or candidates) is (or are) elected (and different methods may choose different winners from the same set of ballots). The other major voting system is cardinal voting, (used in democracies such as that in the United States) where candidates are independently rated, rather than ranked. The similar term "Ranked Choice Voting" (RCV) is used by the US organization FairVote to refer to the use of ranked ballots with specific counting methods: either instant-runoff voting for single-winner elections or single transferable vote for multi-winner elections.

This means that, for every set S of cardinality \kappa, and every partition of the ordered pairs of elements of S into two subsets P_1 and P_1, there exists either a subset S_1\subset S of cardinality \kappa or a subset S_2\subset S of cardinality \alef_0, such that all pairs of elements of S_i belong to P_i. Here, P_1 can be interpreted as the edges of a graph having S as its vertex set, in which S_1 (if it exists) is a clique of cardinality \kappa, and S_2 (if it exists) is a countably infinite independent set. If S is taken to be the cardinal number \kappa itself, the theorem can be formulated in terms of ordinal numbers with the notation \kappa\rightarrow(\kappa,\omega)^2, meaning that S_2 (when it exists) has order type \omega.

By 1270 they had incorporated the principles of the Arab torquetum. A modern replica of Han dynasty polymath scientist Zhang Heng's seismometer of 132 CE Seismology: To better prepare for calamities, Zhang Heng invented a seismometer in 132 CE which provided instant alert to authorities in the capital Luoyang that an earthquake had occurred in a location indicated by a specific cardinal or ordinal direction.de Crespigny (2007), 1050; Morton & Lewis (2005), 70. Although no tremors could be felt in the capital when Zhang told the court that an earthquake had just occurred in the northwest, a message came soon afterwards that an earthquake had indeed struck 400 km (248 mi) to 500 km (310 mi) northwest of Luoyang (in what is now modern Gansu).Minford & Lau (2002), 307; Balchin (2003), 26–27; Needham (1986a), 627; Needham (1986c), 484; Krebs (2003), 31.

They found support for the cancellation axioms, however, their study was biased by the small size of the conjoint arrays (3 × 3 is size) and by statistical techniques that did not take into consideration the ordinal restrictions imposed by the cancellation axioms. Kyngdon (2011) used Karabatsos's (2001) order-restricted inference framework to test a conjoint matrix of reading item response proportions (P) where the examinee reading ability comprised the rows of the conjoint array (A) and the difficulty of the reading items formed the columns of the array (X). The levels of reading ability were identified via raw total test score and the levels of reading item difficulty were identified by the Lexile Framework for Reading . Kyngdon found that satisfaction of the cancellation axioms was obtained only through permutation of the matrix in a manner inconsistent with the putative Lexile measures of item difficulty.

Any well-ordered set is similar (order-isomorphic) to a unique ordinal number \alpha; in other words, its elements can be indexed in increasing fashion by the ordinals less than \alpha. This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some \alpha. The same holds, with a slight modification, for classes of ordinals (a collection of ordinals, possibly too large to form a set, defined by some property): any class of ordinals can be indexed by ordinals (and, when the class is unbounded in the class of all ordinals, this puts it in class-bijection with the class of all ordinals). So the \gamma-th element in the class (with the convention that the "0-th" is the smallest, the "1-st" is the next smallest, and so on) can be freely spoken of.

The input to this voting system consists of the agents' ordinal preferences over outcomes (not lotteries over outcomes), but a relation on the set of lotteries is constructed in the following way: if p and q are different lotteries over outcomes, p\succ q if the expected value of the margin of victory of an outcome selected with distribution p in a head-to-head vote against an outcome selected with distribution q is positive. While this relation is not necessarily transitive, it does always contain at least one maximal element. It is possible that several such maximal lotteries exist, but unicity can be proven in the case where the margins between any pair of alternatives is always an odd number.Gilbert Laffond, Jean-François Laslier and Michel Le Breton A theorem on two–player symmetric zero–sum games Journal of Economic Theory 72: 426–431, 1997.

Available here A useful application of this is when α and β are both subsets of some larger total order; then their union has order type at most α⊕β. If they are both subsets of some ordered abelian group, then their sum has order type at most α⊗β. We can also define the natural sum of α and β inductively (by simultaneous induction on α and β) as the smallest ordinal greater than the natural sum of α and γ for all γ < β and of γ and β for all γ < α. There is also an inductive definition of the natural product (by mutual induction), but it is somewhat tedious to write down and we shall not do so (see the article on surreal numbers for the definition in that context, which, however, uses surreal subtraction, something which obviously cannot be defined on ordinals).

While not strictly a regnal year, time in the United States of America can be derived from the Declaration of Independence (July 4, 1776). For example, the U.S. Constitution is dated as signed in "the Year of our Lord one thousand seven hundred and Eighty seven and of the Independence of the United States of America the Twelfth," and Presidential proclamations will often be ended "IN WITNESS WHEREOF, I have hereunto set my hand this [ordinal] day of [month], in the year of our Lord [year], and of the Independence of the United States of America the [year]." is the year of the Independence of the United States of America on and after July 4 of that year. Time is also sometimes reckoned in terms (and sessions, if necessary) of Congress; e.g. House of Representatives Bill 2 of the 112th Congress is dated "112th CONGRESS, 1st Session".

The project, which was launched over the course of March 2015, is > being developed purposely through a range of thirteenth sub-projects > represented graphically by a logo, a ring-shape dodecagon. The circle added > to the twelve-sided polygon symbolizes the thirteeth line. The graphic > choice, driven by the age of the band, is without a doubt well-thought since > a twelve-star polygon is culturally linked to the Earthly Branches, an > ancient means through which time is measured (duration, age). These sub- > projects, commonly titled ‘movement’ — the notion of time is a corollary of > the notion of movement — yet dissociated by ordinal numbers, are then > revealed in further details in dribs and drabs. Album ‘DOGMA’, being the > very first movement (located at the far north on the logo), is the core > piece of the puzzle which encompasses other audiovisual and text elements > created in association with 18 artists.

In static games of complete, perfect information, a normal-form representation of a game is a specification of players' strategy spaces and payoff functions. A strategy space for a player is the set of all strategies available to that player, whereas a strategy is a complete plan of action for every stage of the game, regardless of whether that stage actually arises in play. A payoff function for a player is a mapping from the cross-product of players' strategy spaces to that player's set of payoffs (normally the set of real numbers, where the number represents a cardinal or ordinal utility—often cardinal in the normal- form representation) of a player, i.e. the payoff function of a player takes as its input a strategy profile (that is a specification of strategies for every player) and yields a representation of payoff as its output.

In France, floors are usually marked the same way as in Spain; however, the letters for the ground floor are RC (rez-de-chaussée), seldom simplified to R. Where these exist, there are high ground RCH (rez-de-chaussée haut) and lower ground RCB (rez-de-chaussée bas), or garden ground RJ (rez-de-jardin) and former ground RC. In Portugal, the letters corresponding to the ground floor are R/C (rés-do-chão) or simply R. For example, in the Polish language there is a clear distinction: the word parter means ground floor and piętro means a floor above the parter, usually with an ordinal: 1st piętro, 2nd piętro etc. Therefore, a parter is the zeroth piętro. Older elevators in Poland have button marked P for the ground floor (parter) and S for basement (suterena). Elevators installed since 1990 have 0 for parter and -1, -2 etc.

By contrast, a breadth-first search will never reach the grandchildren, as it seeks to exhaust the children first. A more sophisticated analysis of running time can be given via infinite ordinal numbers; for example, the breadth-first search of the depth 2 tree above will take ω·2 steps: ω for the first level, and then another ω for the second level. Thus, simple depth-first or breadth-first searches do not traverse every infinite tree, and are not efficient on very large trees. However, hybrid methods can traverse any (countably) infinite tree, essentially via a diagonal argument ("diagonal"—a combination of vertical and horizontal—corresponds to a combination of depth and breadth). Concretely, given the infinitely branching tree of infinite depth, label the root (), the children of the root (1), (2), …, the grandchildren (1, 1), (1, 2), …, (2, 1), (2, 2), …, and so on.

The Book of Common Prayer (BCP) used in Canada was originally compiled in 1962, and is a national expression of a tradition of Christian worship stemming from the original Book of Common Prayer published by the Church of England in 1549. The original 1549 BCP was itself a revision of the medieval forms of worship in use within the English Church prior to the Reformation. The BCP simplified older forms, and made the Bible itself the standard of all Christian worship. The BCP contains in one volume what previously had been contained in many separate tomes: The Daily Offices (which are the Church's daily Morning Prayer and Evening Prayer), the Liturgy of the Holy Communion, the Ordinal (services for the ordinations of bishops, priests, and deacons), as well as many other services of the Church such as the Penitential Rite (used on Ash Wednesday), and the Baptism services.

One-to-one correspondence between an infinite set and its proper subset A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets. Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers).

A cardinal κ is called η-C(n)-extendible if there is an elementary embedding j witnessing that κ is η-extendible (that is, j is elementary from Vκ+η to some Vλ with critical point κ) such that furthermore, Vj(κ) is Σn-correct in V. That is, for every Σn formula φ, φ holds in Vj(κ) if and only if φ holds in V. A cardinal κ is said to be C(n)-extendible if it is η-C(n)-extendible for every ordinal η. Every extendible cardinal is C(1)-extendible, but for n≥1, the least C(n)-extendible cardinal is never C(n+1)-extendible (Bagaria 2011). Vopěnka's principle implies the existence of extendible cardinals; in fact, Vopěnka's principle (for definable classes) is equivalent to the existence of C(n)-extendible cardinals for all n (Bagaria 2011). All extendible cardinals are supercompact cardinals (Kanamori 2003).

A formal, distinct, and unique 6-part name is given to each term for test or observation identity. The database currently has over 71,000 observation terms that can be accessed and understood universally. Each database record includes six fields for the unique specification of each identified single test, observation, or measurement: # Component- what is measured, evaluated, or observed (example: urea,...) # Kind of property- characteristics of what is measured, such as length, mass, volume, time stamp and so on # Time aspect- interval of time over which the observation or measurement was made # System- context or specimen type within which the observation was made (example: blood, urine,...) # Type of scale- the scale of measure. The scale may be quantitative, ordinal, nominal or narrative # Type of method- procedure used to make the measurement or observation A unique code (format: nnnnn-n) is assigned to each entry upon registration.

Agama has a particular interest in the Christian Church of the first millennium in the British Isles and in the early monastics such as those at Whithorn (Candida Casa) (St Ninian) in Galloway, Scotland, at Iona, at Lindisfarne, and the north African Coptic (Coptic Church) (Berber) scholar-monk St Hadrian of Canterbury.Source: the APC's 2013 ordinal booklet Agama wrote on how insights from the first millennium can help people in their personal devotion and prayer life in the 21st century. Agama has said that recession is nothing new to the black community in the United Kingdom, and that the experience of the majority of the members of the black community over many years has been of hardship and exile. It has been said that, in the same way, Bob Marley and other black musicians and poets often sang and wrote of the experience of living in Babylon.

However, for laws that amend other laws, this ordinal numbering does not reset every year (For example, even though only two amendments were made to the Israeli Criminal Procedure Law in 2018, these amendments are numbered No.81 and No.82 in their titles.) In Ireland, the Thirty-First Amendment of the Constitution (Children) Act 2012 was enacted in 2015 rather than 2012. It was passed by both houses of the Oireachtas in 2012 but not signed into law by the President until 2015, after an intervening referendum and court challenge. Section 2(2) of the act, which assigns the short title, could not be amended between the houses' passing the bill and its being enacted (though it could still be amended by a subsequent act of the Oireachtas). This act's short title is longer than its long title ("An Act to Amend the Constitution", as required by the constitution).

Solovay suggested in his paper that the use of an inaccessible cardinal might not be necessary. Several authors proved weaker versions of Solovay's result without assuming the existence of an inaccessible cardinal. In particular showed there was a model of ZFC in which every ordinal-definable set of reals is measurable, Solovay showed there is a model of ZF + DC in which there is some translation-invariant extension of Lebesgue measure to all subsets of the reals, and showed that there is a model in which all sets of reals have the Baire property (so that the inaccessible cardinal is indeed unnecessary in this case). The case of the perfect set property was solved by , who showed (in ZF) that if every set of reals has the perfect set property and the first uncountable cardinal ℵ1 is regular then ℵ1 is inaccessible in the constructible universe.

The idea is that κ cannot be distinguished (looking from below) from smaller cardinals by any formula of n+1-th order logic with m-1 alternations of quantifiers even with the advantage of an extra unary predicate symbol (for A). This implies that it is large because it means that there must be many smaller cardinals with similar properties. The cardinal number κ is called totally indescribable if it is Π-indescribable for all positive integers m and n. If α is an ordinal, the cardinal number κ is called α-indescribable if for every formula φ and every subset U of Vκ such that φ(U) holds in Vκ+α there is a some λ<κ such that φ(U ∩ Vλ) holds in Vλ+α. If α is infinite then α-indescribable ordinals are totally indescribable, and if α is finite they are the same as Π-indescribable ordinals.

Suppose that X is the first uncountable ordinal, with the finite measure where the measurable sets are either countable (with measure 0) or the sets of countable complement (with measure 1). The (non-measurable) subset E of X×X given by pairs (x,y) with x The stronger versions of Fubini's theorem on a product of two unit intervals with Lebesgue measure, where the function is no longer assumed to be measurable but merely that the two iterated integrals are well defined and exist, are independent of the standard Zermelo–Fraenkel axioms of set theory. The continuum hypothesis and Martin's axiom both imply that there exists a function on the unit square whose iterated integrals are not equal, while showed that it is consistent with ZFC that a strong Fubini-type theorem for [0, 1] does hold, and whenever the two iterated integrals exist they are equal. See List of statements undecidable in ZFC.

The Tertiary Phase, Quandary Phase, Quintessential Phase and Hexagonal Phase are respectively the third, fourth, fifth and sixth series of The Hitchhiker's Guide to the Galaxy radio series. Produced in 2003, 2004 and 2018 by Above the Title Productions for BBC Radio 4, they are radio adaptations of the third, fourth, fifth and sixth books in Douglas Adams' The Hitchhiker's Guide to the Galaxy series: Life, the Universe and Everything; So Long, and Thanks For All the Fish; Mostly Harmless and And Another Thing.... These radio series consisted of a total of twenty episodes, following on from the twelve episodes from the original two series (the Primary and Secondary Phases) which originally aired in 1978 and 1980. The producers chose not to continue the ordinal sequence established by the Primary, Secondary and Tertiary phases. If they had done so, the fourth, fifth and sixth series would have been termed quaternary, quinary and senary.

The ACC is divided into four ecclesiastical provinces – British Columbia and the Yukon, Canada (encompassing the Atlantic provinces and Quebec), Ontario, and Rupert's Land (encompassing the prairie provinces, Nunavut, the Northwest Territories, and portions of Ontario). Within the provinces are 29 dioceses and one grouping of churches in British Columbia that functions equivalently to a diocese. Each province has its own archbishop, known as the Metropolitan, and each diocese has a bishop, although there are no metropolitical dioceses (or archdioceses) as such; a metropolitan is styled "Archbishop of [his or her own diocese], and Metropolitan of [the ecclesiastical province]." As with other churches in the Anglican tradition, each diocese is divided up into geographical regions called parishes, where certain authority resides in the rector or priest-in-charge (as laid out in the induction service, the ordinal, and the cleric's licence) and in the parish council (or vestry) as defined in diocesan canons.

In recent years, the Nebelhorn Trophy has also been used by the International Skating Union to experiment with new judging and scoring systems for figure skating. Specifically, the 1997 competition was used as the test event for the switch from the "best of majority" ordinal system to the "one-by-one" method; the 2002 event was used for an initial test of the ISU Judging System which was then under development, and the 2003 event was the first competition where that system was used to determine the official results; and the 2006 event was used for a trial of using separate panels of judges for technical elements and program components. The competition also serves as a testing ground for judges working towards international status. The 2009 competition was used as the final qualifying opportunity for the 2010 Winter Olympics and the 2013 event served the same purpose for the 2014 Olympics and the 2018 Olympics.

In 1606 he became dean of the Chapel Royal at Holyrood, on the revival of that office by King James. In 1606 the general assembly appointed him constant moderator of the presbytery of Kirkcudbright, and three years later he was sent up to court by the other titular bishops to confer with the king as to further measures which were in contemplation for the advancement of their order. The church having agreed in 1610 to the restoration of the ecclesiastical power of bishops, Hamilton, with John Spottiswoode, archbishop of Glasgow, and Andrew Lamb, bishop of Brechin, were called up to London by the king, and were consecrated 21 October of that year in the chapel of London House according to the English ordinal by the bishop of London, the bishop of Ely, bishop of Rochester, and the bishop of Worcester. They were not reordained, as the validity of ordination by presbyters was then recognised by the English church and state.