149 Sentences With "finiteness" | Random Sentence Generator

But with themed puzzles, I've enjoyed the pleasant finiteness to the four or five entries that undergird the grid.

Mr Bezos's ultimate justification for pursuing such megaprojects is his worry about the mismatch between the exponential process of population growth and the finiteness of Earth's resources.

The humble person has done the deep work of reflection and self-exploration and knows and embraces their finiteness and fragility as well as their potential and capacity.

The rigid, inflexible selfThe rigid self is a propped-up god, a custom made tyrant designed to keep us from encountering unpleasant realities like our own fragility or finiteness.

So maybe the thing that defines Montrose is that you're surrounded by other people who have to think about their finiteness too — and, for a little while, we just don't.

Unlike Mr. Lee's New York stories, which give their neighborhoods the finiteness and theatricality of stage sets, Mr. Singleton examines a more sprawling form of claustrophobia and a more adolescent angst.

No doubt the ending of this covenant of reconciliation, if that is what we are seeing, will bring back into world history those single-minded utopian forces that never accepted the finiteness of man's worldly condition nor the limits and complexity of human justice.

It's a beautiful and at-times funny meditation on our own cosmic search for a place to belong, and even if you find yourself bewildered throughout the film (and you probably will), the journey is worth taking—and the film's conclusion, an elegant slice of finiteness, is undeniably satisfying and soul-nourishing.

The exact density spectrum is given by the Frank–Tamm formula. In this case, the finiteness of the frequency range comes from the finiteness of the range over which a material can have a refractive index greater than unity. Cherenkov radiation also appears as a bright blue color, for these reasons.

In mathematics, finiteness properties of a group are a collection of properties that allow the use of various algebraic and topological tools, for example group cohomology, to study the group. It is mostly of interest for the study of infinite groups. Special cases of groups with finiteness properties are finitely generated and finitely presented groups.

Tate (1994), Proposition 4.3. For a regular integral scheme of dimension 2 which is flat and proper over the ring of integers of a number field, and which has a section, the finiteness of the Brauer group is equivalent to the finiteness of the Tate–Shafarevich group Ш for the Jacobian variety of the general fiber (a curve over a number field).Grothendieck (1968), Le groupe de Brauer III, Proposition 4.5. The finiteness of Ш is a central problem in the arithmetic of elliptic curves and more generally abelian varieties.

Isaacs, Corollary 13.16, p. 187 This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson.Jacobson, Nathan "Structure Theory of Simple Rings Without Finiteness Assumptions" This can be viewed as a kind of generalization of the Artin- Wedderburn theorem's conclusion about the structure of simple Artinian rings.

Early notable results of Bowditch include clarifying the classic notion of geometric finiteness for higher-dimensional Kleinian groups in constant and variable negative curvature. In a 1993 paper Bowditch proved that five standard characterisations of geometric finiteness for discrete groups of isometries of hyperbolic 3-space and hyperbolic plane, (including the definition in terms of having a finitely-sided fundamental polyhedron) remain equivalent for groups of isometries of hyperbolic n-space where n ≥ 4\. He showed, however, that in dimensions n ≥ 4 the condition of having a finitely-sided Dirichlet domain is no longer equivalent to the standard notions of geometric finiteness. In a subsequent paperB.

In mathematics, compact objects, also referred to as finitely presented objects, or objects of finite presentation, are objects in a category satisfying a certain finiteness condition.

Kim has made contributions to the application of arithmetic homotopy theory to the study of Diophantine problems, especially to finiteness theorems of the Faltings–Siegel type.

1, pp. 1–59 settling a long-standing open problem. In a 1997 paperBestvina, Mladen and Brady, Noel, Morse theory and finiteness properties of groups. Inventiones Mathematicae, vol.

Finiteness of the set of points is not essential in this definition, as long as each line can intersect the set in only a finite number of points.

The finite basis question for semigroups of order less than six. Semigroup Forum, 27(1983), 387-389.A.N. Trahtman. Finiteness of a basis of identities of 5-element semigroups.

Kneser-Haken finiteness says that for each 3-manifold, there is a constant C such that any collection of surfaces of cardinality greater than C must contain parallel elements.

In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.

For the derived category of constructible sheaves, see a section in ℓ-adic sheaf. The finiteness theorem in étale cohomology states that the higher direct images of a constructible sheaf are constructible.

On the other hand, for the Neumann Laplacian there is an essential spectrum unless cusps shrinks at infinity faster than the negative exponent (so the finiteness of the volume is not sufficient).

In the mathematical theory of Kleinian groups, the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleinian group. The theorem was proved by , apart from a gap that was filled by . The Ahlfors finiteness theorem states that if Γ is a finitely-generated Kleinian group with region of discontinuity Ω, then Ω/Γ has a finite number of components, each of which is a compact Riemann surface with a finite number of points removed.

Later Neoplatonic Philosophers and idealists like Plotinus treated the dyad as a second cause (demiurge), which was the divine mind (nous) that via a reflective nature (finiteness) causes matter to "appear" or become perceivable.

Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous finiteness theorem. Twenty years earlier, Paul Gordan had demonstrated the theorem of the finiteness of generators for binary forms using a complex computational approach. Attempts to generalize his method to functions with more than two variables failed because of the enormous difficulty of the calculations involved. To solve what had become known in some circles as Gordan's Problem, Hilbert realized that it was necessary to take a completely different path.

H. Bowditch, "Geometrical finiteness with variable negative curvature" Duke Mathematical Journal, vol. 77 (1995), no. 1, 229–274 Bowditch considered a similar problem for discrete groups of isometries of Hadamard manifold of pinched (but not necessarily constant) negative curvature and of arbitrary dimension n ≥ 2\. He proved that four out of five equivalent definitions of geometric finiteness considered in his previous paper remain equivalent in this general set-up, but the condition of having a finitely-sided fundamental polyhedron is no longer equivalent to them.

The tangent- chord process (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century. The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group E(Q)/2E(Q) which forms a major step in the proof. Certainly the finiteness of this group is a necessary condition for E(Q) to be finitely- generated; and it shows that the rank is finite. This turns out to be the essential difficulty.

Are we justified in telling the truth when another person finds the truth unbearable? We have to act in particular situations, "inventing an original solution" each time, but remembering that "man is man only through situations whose particularity is precisely a universal fact." The brief Conclusion sums up de Beauvoir's view of human freedom: "... we are absolutely free today if we choose to will our existence in its finiteness, a finiteness which is open on the infinite." She ends with a call for us to realize and act on this fundamental truth of our existence.

Synthesis of Yoga Part I ch. II-IIILetters on Yoga vol. II pp.585ff (3rd ed.) Guided by the evolving divine soul within, the sadhak moves away from ego, ignorance, finiteness, and the limitations of the outer being.

In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Hazewinkel, Gubareni & Kirichenko (2004), p.6, Prop. 1.1.4.Fraleigh & Katz (1967), p.

He is now a professor at the Steklov Institute in Moscow, where he is the head of the Department of Algebra, and he is also a professor at Moscow State University. Parshin proved in 1968 that the Mordell conjecture is a logical consequence of Shafarevich's finiteness conjecture concerning isomorphism classes of abelian varieties via what is known as Parshin's trick, which gives an embedding of an algebraic curve into the Siegel modular variety. In 1983 Gerd Faltings proved Shafarevich's finiteness conjecture (and thereby the Mordell conjecture). Shafarevich proved his conjecture for the case with genus g = 1\.

LQG offers a geometric explanation of the finiteness of the entropy and of the proportionality of the area of the horizon. These calculations have been generalized to rotating black holes. deficit angle or quantized amount of curvature. These deficit angles add up to 4 \pi.

The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the Lindelöf property of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.

If a group is of type FPn then its cohomology groups H^i(\Gamma) are finitely generated for 0 \le i \le n. If it is of type FP then it is of finite cohomological dimension. Thus finiteness properties play an important role in the cohomology theory of groups.

H. G. Wells advocated a finite God in his book God the Invisible King. Brightman suggested that Wells was the "first modern writer to devote an entire book to the concept of God's finiteness." Wells dissociated his God in any respect from the Biblical God.Wagar, W. Warren. (2004).

Another application of Minkowski's theorem is the result that every class in the ideal class group of a number field contains an integral ideal of norm not exceeding a certain bound, depending on , called Minkowski's bound: the finiteness of the class number of an algebraic number field follows immediately.

Often, an object is dualizable only when it satisfies some finiteness or compactness property. A category in which each object has a dual is called autonomous or rigid. The category of finite- dimensional vector spaces with the standard tensor product is rigid, while the category of all vector spaces is not.

General results of Dmitry Burago and Serge Ferleger on the uniform estimation on the number of collisions in non-degenerate semi-dispersing billiards allow to establish finiteness of its topological entropy and no more than exponential growth of periodic trajectories. In contrast, degenerate semi-dispersing billiards may have infinite topological entropy.

Finiteness is discussed at length in chpts. 9 and 16 of The Trouble with Physics, especially pp. 278-81. Some experts in the theoretical physics community disagree with these statements.Polchinski, Joseph (2007) "All Strung Out?" a review of The Trouble with Physics and Not Even Wrong, American Scientist 95(1):1.

It deals with the finiteness of life. It was based on two different texts in which they were working. It was the last song to be worked on in studio. Musically, the song features some elements of indigenous music which, according to Britto, were also present in Cabeça Dinossauro's title track.

It is used as reference-concept in all scientific hierarchy definitions, and many technical approaches, like the tree in computational data structures or nested set model of relational databases. Sometimes the concept is confused with a "set of sets" with a hereditary property (like the finiteness in a hereditarily finite set).

In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumptions of finiteness. Geometric lattices and matroid lattices, respectively, form the lattices of flats of finite and infinite matroids, and every geometric or matroid lattice comes from a matroid in this way.

Now standard proofs of the fact that the set of geodesic words in a word-hyperbolic group is a regular language also use finiteness of the number of cone types. Cannon's work also introduced an important notion of almost convexity for Cayley graphs of finitely generated groups,James W. Cannon. Almost convex groups. Geometriae Dedicata, vol.

An important but difficult question is on the finiteness of the integral closure of a finitely generated algebra. There are several known results. The integral closure of a Dedekind domain in a finite extension of the field of fractions is a Dedekind domain; in particular, a noetherian ring. This is a consequence of the Krull–Akizuki theorem.

He decided that it is possible for natural and real numbers to be definite sets, and that if one rejects the axiom of Euclidean finiteness (that states that actualities, singly and in aggregates, are necessarily finite), then one is not involved in any contradiction. The philosophical problem of actual infinity concerns whether the notion is coherent and epistemically sound.

Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ.Math.IHES 69(1989), 119-171; Addendum: ibid, 71(1990); with A.Borel. [7]. Values of isotropic quadratic forms at S-integral points, Compositio Mathematica, 83 (1992), 347-372; with A.Borel. [8]. Unrefined minimal K-types for p-adic groups, Inventiones Math. 116(1994), 393-408; with Allen Moy. [9].

Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers"). He also resolved a significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers.Reid 1996, p.

Bredon (1997), Theorem II.17.4; Borel (1984), V.3.17. For example, for a compact Hausdorff space X that is locally contractible (in the weak sense discussed above), the sheaf cohomology group Hj(X,Z) is finitely generated for every integer j. One case where the finiteness result applies is that of a constructible sheaf. Let X be a topologically stratified space.

1\. Locally discrete collections are always locally finite. See the page on local finiteness. 2\. If a collection of subsets of a topological space X is locally discrete, it must satisfy the property that each point of the space belongs to at most one element of the collection. This means that only collections of pairwise disjoint sets can be locally discrete. 3\.

Every affine or projective algebraic set is defined as the set of the zeros of an ideal in a polynomial ring. In this case, the irreducible components are the varieties associated to the minimal primes over the ideal. This is the identification that allows to prove the uniqueness and the finiteness of the decomposition. This decomposition is strongly related with the primary decomposition of the ideal.

Mathematician and cryptographer Neal Koblitz was one of Katz's students. Katz studied, with Sarnak among others, the connection of the eigenvalue distribution of large random matrices of classical groups to the distribution of the distances of the zeros of various L and zeta functions in algebraic geometry. He also studied trigonometric sums (Gauss sums) with algebro-geometric methods. He introduced the Katz–Lang finiteness theorem.

Isaacs, Corollary 13.16, p. 187 This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson. This can be viewed as a kind of generalization of the Artin- Wedderburn theorem's conclusion about the structure of simple Artinian rings. More formally, the theorem can be stated as follows: :The Jacobson Density Theorem.

If R is Noetherian, and U is finitely generated, then U is a Noetherian module over R, and the conclusion is satisfied. Somewhat remarkable is that the weaker assumption, namely that U is finitely generated as an R-module (and no finiteness assumption on R), is sufficient to guarantee the conclusion. This is essentially the statement of Nakayama's lemma. Precisely, one has the following.

The axioms of Zermelo–Fraenkel set theory without the axiom of choice (ZF) are not strong enough to prove that every infinite set is Dedekind-infinite, but the axioms of Zermelo–Fraenkel set theory with the axiom of countable choice () are strong enough. Other definitions of finiteness and infiniteness of sets than that given by Dedekind do not require the axiom of choice for this, see .

A collection in a space is countably locally finite (or σ-locally finite) if it is the union of a countable family of locally finite collections of subsets of X. Countable local finiteness is a key hypothesis in the Nagata–Smirnov metrization theorem, which states that a topological space is metrizable if and only if it is regular and has a countably locally finite basis.

List of Fellows of the American Mathematical Society, retrieved 2015-01-13. In 1978 he was an invited speaker at the International Congress of Mathematicians held in Helsinki (the theme was "Minimal Surfaces: Stability and Finiteness").. Do Carmo is also known for his textbooks. They were translated into many languages and used in courses from universities such as Harvard and Columbia. His students included Celso Costa, Marcos Dajczer and Keti Tenenblat.

George Sterman's research focuses on quantum field theory and its applications in quantum chromodynamics. With Steven Weinberg he proved the infrared finiteness of jet cross sections, thus proving that perturbation theory is a safe method in that regime. He also worked on reformulation and proof of factorization theorems with Stephen Libby, John C. Collins and Davison E. Soper. He authored a textbook entitled An Introduction to Quantum Field Theory in 1993.

Glaz was born in Bucharest, Romania, and earned a bachelor's degree in 1972 at Tel Aviv University, Israel. She came to the US for her graduate education in mathematics, completing a Ph.D. in 1977 at Rutgers University. Her dissertation, Finiteness and Differential Properties of Ideals, was supervised by Wolmer Vasconcelos. After postdoctoral research at Case Western Reserve University, Glaz became an assistant professor at Wesleyan University in 1980.

Its heated emotional subject of abduction and potential rape is not only depicted, but forcefully expressed through a personal visual language of pulsating and diaphanous color- forms.Martínez, Juan. Cuban Art and National Identity. Florida: University Press Florida, 1994: 120 Enríquez's paintings are about ecstasy when they are not about violence, for in both themes he identified one of the fundamental characteristics of his latitudes-the strident, orgasmic, experience of finiteness.

Naturally, if U is a Noetherian module, this holds. If R is Noetherian, and U is finitely generated, then U is a Noetherian module over R, and the conclusion is satisfied. Somewhat remarkable is that the weaker assumption, namely that U is finitely generated as an R-module (and no finiteness assumption on R), is sufficient to guarantee the conclusion. This is essentially the statement of Nakayama's lemma.

By finiteness, S must have a point P and a connecting line \ell that are a positive distance apart but are closer than all other point-line pairs. Kelly proved that \ell is ordinary, by contradiction. Assume that \ell is not ordinary. Then it goes through at least three points of S. At least two of these are on the same side of P', the perpendicular projection of P on \ell.

Artin conjectured that every proper scheme over the integers has finite Brauer group.Milne (1980), Question IV.2.19. This is far from known even in the special case of a smooth projective variety X over a finite field. Indeed, the finiteness of the Brauer group for surfaces in that case is equivalent to the Tate conjecture for divisors on X, one of the main problems in the theory of algebraic cycles.

Let m be a measure on the σ-algebra of Borel sets of a Hausdorff topological space X. The measure m is called inner regular or tight if, for any open set U, m(U) is the supremum of m(K) over all compact subsets K of U. The measure m is called outer regular if, for any Borel set B, m(B) is the infimum of m(U) over all open sets U containing B. The measure m is called locally finite if every point of X has a neighborhood U for which m(U) is finite. If m is locally finite, then it follows that m is finite on compact sets, and for locally compact Hausdorff spaces, the converse holds, too. Thus, in this case, local finiteness may be equivalently replaced by finiteness on compact subsets. The measure m is called a Radon measure if it is inner regular, outer regular and locally finite.

That is, every non-empty subset of S has both a least and a greatest element in the subset. # Every one-to-one function from P(P(S)) into itself is onto. That is, the powerset of the powerset of S is Dedekind-finite (see below).The equivalence of the standard numerical definition of finite sets to the Dedekind-finiteness of the power set of the power set was shown in 1912 by .

129 (1997), no. 3, pp. 445–470 Bestvina and Brady developed a version of discrete Morse theory for cubical complexes and applied it to study homological finiteness properties of subgroups of right-angled Artin groups. In particular, they constructed an example of a group which provides a counter-example to either the Whitehead asphericity conjecture or to the Eilenberg−Ganea conjecture, thus showing that at least one of these conjectures must be false.

After spending two years at the University of Toronto, Nelson returned to Hamilton to study at McMaster University. She received her B.Sc in mathematics from McMaster in 1965, followed by an M.Sc in mathematics from McMaster in 1967. She succeeded in having her thesis work published in the Canadian Journal of Mathematics, also in 1967; the article was entitled "Finiteness of semigroups of operators in universal algebra". Nelson completed her Ph.D in 1970.

Grammatical aspect is a formal property of a language, distinguished through overt inflection, derivational affixes, or independent words that serve as grammatically required markers of those aspects. For example, the K'iche' language spoken in Guatemala has the inflectional prefixes k- and x- to mark incompletive and completive aspect;Pye, Clifton (2001). "The Acquisition of Finiteness in K'iche' Maya". BUCLD 25: Proceedings of the 25th annual Boston University Conference on Language Development, pp. 645-656.

The templates include photographs, exhibition catalogues, newspaper clippings, magazines and illustrations or even cloth patterns. Many of the works show nocturnal urban scenes, fires, abstract shapes and patterns as well as short excerpts of stencilled texts. By focussing on the virtually infinite time span before his birth, van Eeden accepts his own existence as a merely insignificant part of a genuine time stream and discards the finiteness of his own existence.Galerie Zink (ed.): Sensational.

Every quasinormal subgroup of a finite group is a subnormal subgroup. This follows from the somewhat stronger statement that every conjugate permutable subgroup is subnormal, which in turn follows from the statement that every maximal conjugate permutable subgroup is normal. (The finiteness is used crucially in the proofs.) In summary, a subgroup H of a finite group G is permutable in G if and only if H is both modular and subnormal in G.

Stone adds finiteness of the process, and definiteness (having no ambiguity in the instructions) to this definition. produce, in a "reasonable" time,Knuth, loc. cit output-integer y at a specified place and in a specified format. The concept of algorithm is also used to define the notion of decidability—a notion that is central for explaining how formal systems come into being starting from a small set of axioms and rules.

If 1 is centrally primitive, we are done. If not, it is a sum of central orthogonal idempotents, which in turn are primitive or sums of more central idempotents, and so on. The problem that may occur is that this may continue without end, producing an infinite family of central orthogonal idempotents. The condition "R does not contain infinite sets of central orthogonal idempotents" is a type of finiteness condition on the ring.

Zariski's formulation of Hilbert's fourteenth problem asks whether, for a quasi-affine algebraic variety X over a field k, possibly assuming X normal or smooth, the ring of regular functions on X is finitely generated over k. Zariski's formulation was shown to be equivalent to the original problem, for X normal. (See also: Zariski's finiteness theorem.) Éfendiev F.F. (Fuad Efendi) provided symmetric algorithm generating basis of invariants of n-ary forms of degree r.

A ring is called Noetherian (in honor of Emmy Noether, who developed this concept) if every ascending chain of ideals :0 ⊆ I0 ⊆ I1 ... ⊆ In ⊆ In + 1 ⊆ ... becomes stationary, i.e. becomes constant beyond some index n. Equivalently, any ideal is generated by finitely many elements, or, yet equivalent, submodules of finitely generated modules are finitely generated. Being Noetherian is a highly important finiteness condition, and the condition is preserved under many operations that occur frequently in geometry.

He proved that planar polynomial vector fields have only finitely many limit cycles. Jean Écalle independently proved the same result, and an earlier attempted proof by Henri Dulac (in 1923) was shown to be defective by Ilyashenko in the 1970s. He was an Invited Speaker of the ICM in 1978 at Helsinki and in 1990 with talk Finiteness theorems for limit cycles at Kyoto. In 2017 he was elected a Fellow of the American Mathematical Society.

A class of groups is a set theoretical collection of groups satisfying the property that if G is in the collection then every group isomorphic to G is also in the collection. This concept arose from the necessity to work with a bunch of groups satisfying certain special property (for example finiteness or commutativity). Since set theory does not admit the "set of all groups", it is necessary to work with the more general concept of class.

The dimensionality of F∞ is countably infinite. A standard basis consists of the vectors ei which contain a 1 in the i-th slot and zeros elsewhere. This vector space is the coproduct (or direct sum) of countably many copies of the vector space F. Note the role of the finiteness condition here. One could consider arbitrary sequences of elements in F, which also constitute a vector space with the same operations, often denoted by FN \- see below.

In number theory, the Katz–Lang finiteness theorem, proved by , states that if X is a smooth geometrically connected scheme of finite type over a field K that is finitely generated over the prime field, and Ker(X/K) is the kernel of the maps between their abelianized fundamental groups, then Ker(X/K) is finite if K has characteristic 0, and the part of the kernel coprime to p is finite if K has characteristic p > 0\.

In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension. A collection of subsets of a topological space X is said to be locally finite, if each point in the space has a neighbourhood that intersects only finitely many of the sets in the collection. Note that the term locally finite has different meanings in other mathematical fields.

Suppose X is a normally distributed random variable with expectation μ and variance σ2. Further suppose g is a function for which the two expectations E(g(X) (X − μ)) and E(g ′(X)) both exist. (The existence of the expectation of any random variable is equivalent to the finiteness of the expectation of its absolute value.) Then :E\bigl(g(X)(X-\mu)\bigr)=\sigma^2 E\bigl(g'(X)\bigr). In general, suppose X and Y are jointly normally distributed.

Shortly afterwards the finiteness of covolume was proven in full generality by Borel and Harish-Chandra. Meanwhile, there was progress on the general theory of lattices in Lie groups by Atle Selberg, Grigori Margulis, David Kazhdan, M. S. Raghunathan and others. The state of the art after this period was essentially fixed in Raghunathan's treatise, published in 1972. In the seventies Margulis revolutionised the topic by proving that in "most" cases the arithmetic constructions account for all lattices in a given Lie group.

A property that is undecidable already for context-free languages or finite intersections of them, must be undecidable also for conjunctive grammars; these include: emptiness, finiteness, regularity, context-freeness,Given a conjunctive grammar, is its generated language empty / finite / regular / context-free? inclusion and equivalence.Given two conjunctive grammars, is the first's generated language a subset of / equal to the second's? The family of conjunctive languages is closed under union, intersection, concatenation and Kleene star, but not under string homomorphism, prefix, suffix, and substring.

In essence each of them is part of the Attribute Extension, which is active in each of them. But the finiteness of each of them is due to the fact that it is restrained or hedged in, so to say, by other finite modes. This limitation or determination is negation in the sense that each finite mode is not the whole attribute Extension; it is not the other finite modes. But each mode is positively real and ultimate as part of the Attribute.

The finiteness of the speed of light would mean that it would take a certain amount of time before the darkness from the sun's absence would reach the orbiting planet. Therefore, why would the planet instantaneously start traveling in a straight line before the arrival of information that the sun's disappearance has occurred? The cosmic catastrophe thought experiment led Einstein to the invention of the General Theory of Relativity and the creation of the concept of spacetime. Spacetime allowed Einstein to fix the deficiency in Newton's theory.

Einstein, H. A. Lorentz, H. Weyl, H. Minkowski, The Principle of Relativity, Dover Publications, Inc, 2000, softcover 216 pages, , See pp. 37–65 for an English translation of Einstein's original 1905 paper. postulated the constancy and finiteness of the speed of light for all observers. He showed that this postulate, together with a reasonable definition for what it means for two events to be simultaneous, requires that distances appear compressed and time intervals appear lengthened for events associated with objects in motion relative to an inertial observer.

25): Minsky states that although a machine may be finite, and finite automata "have a number of theoretical limitations": :...the magnitudes involved should lead one to suspect that theorems and arguments based chiefly on the mere finiteness [of] the state diagram may not carry a great deal of significance. (Minsky p. 25) It can also be decided automatically whether a nondeterministic machine with finite memory halts on none, some, or all of the possible sequences of nondeterministic decisions, by enumerating states after each possible decision.

For categories C with a well- behaved tensor product (more formally, C is required to be a monoidal category), there is another condition imposing some kind of finiteness, namely the condition that an object is dualizable. If the monoidal unit in C is compact, then any dualizable object is compact as well. For example, R is compact as an R-module, so this observation can be applied. Indeed, in the category of R-modules the dualizable objects are the finitely presented projective modules, which are in particular compact.

Siegel's work spans analytic number theory; and his theorem on the finiteness of the integer points of curves, for genus > 1, is historically important as a major general result on diophantine equations, when the field was essentially undeveloped. He worked on L-functions, discovering the (presumed illusory) Siegel zero phenomenon. His work, derived from the Hardy–Littlewood circle method on quadratic forms, appeared in the later, adele group theories encompassing the use of theta-functions. The Siegel modular varieties, which describe Siegel modular forms, are recognised as part of the moduli theory of abelian varieties.

David Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers"). He also resolved a significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers. He then had little more to publish on the subject; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area.

Ecopop (ECOlogie et POPulation, "Ecology and Population") is a Swiss voluntary association established in 1971 dedicated to the preservation of non-renewable resources and the reduction of overpopulation. Ecopop is a member of European Population Alliance (founded 2012). Founded under the mathematical trend of exponential growth of world population and the finiteness of natural resources, and inspired by the Club of Rome, the association is formed as part of the nascent ecology movement as Arbeitsgruppe Bevölkerungsentwicklung ("working group for population development"). It was renamed to its current name in 1987.

As a child, she went to a gymnasium in Basel and then studied at the University of Basel, graduating in 1982 with a diploma directed by Heinz Huber "On finiteness of the isometry group of a compact negatively curved Riemannian manifold". She received her PhD in 1988, from the same university, with a thesis entitled "On the volumes of hyperbolic polytopes in dimensions three and four". Her advisor was Hans-Christoph Im Hof. During the year 1983–84 she also studied at the University of Grenoble (Fourier Institute).

Roquette worked on number and function fields and especially local p-adic fields. He applied the methods of model theory (Nonstandard arithmetic) in number theory, joint with Abraham Robinson, with whom he worked on Mahler's theorem (on the finiteness of integral points on a curve of genus g> 0) using non-standard methods. He authored a number of works on the history of mathematics, in particular on the schools of Helmut Hasse and Emmy Noether. In 1975 Roquette was co-editor of the collected essays by Helmut Hasse.

This can be looked at another way, by considering the representation of G on the symmetric algebra of V, and then the whole subalgebra R of G-invariants. Then nd is the dimension of the homogeneous part of R of dimension d, when we look at it as graded ring. In this way a Molien series is also a kind of Hilbert series. Without further hypotheses not a great deal can be said, but assuming some conditions of finiteness it is then possible to show that the Molien series is a rational function.

In a sense one can understand them as affirming the existence of general bridges from thoughts to things. Both however can, like the postulates concerning specific constructions, be understood as "finiteness principles" affirming the existence of new arithmoi. Mayberry's “corrected” Euclid would thus underpin the sister disciplines of Geometry and Arithmetic with Common Notions, applicable to both, supplemented by two sets of Postulates, one for each discipline. Indeed, in so far as Geometry does rely on the notion of arithmos – it does so even in defining triangles, quadrilaterals, pentagons etc.

It is an elementary result in group theory that a finite cancellative semigroup is a group. Let S be a finite cancellative semigroup. Cancellativity and finiteness taken together imply that for all a in S. So given an element a in S, there is an element ea, depending on a, in S such that . Cancellativity now further implies that this ea is independent of a and that for all x in S. Thus ea is the identity element of S, which may from now on be denoted by e.

A break in research while he was involved in trying to meet 1960s student activism halfway caused him (by his own description) difficulties in picking up the threads afterwards. He wrote on modular forms and modular units, the idea of a 'distribution' on a profinite group, and value distribution theory. He made a number of conjectures in diophantine geometry: Mordell–Lang conjecture, Bombieri–Lang conjecture, Lang–Trotter conjecture, and the Lang conjecture on analytically hyperbolic varieties. He introduced the Lang map, the Katz–Lang finiteness theorem, and the Lang–Steinberg theorem (cf.

The covariant, or "spin foam", version of the dynamics was developed jointly over several decades by research groups in France, Canada, UK, Poland, and Germany. It was completed in 2008, leading to the definition of a family of transition amplitudes, which in the classical limit can be shown to be related to a family of truncations of general relativity. The finiteness of these amplitudes was proven in 2011. It requires the existence of a positive cosmological constant, which is consistent with observed acceleration in the expansion of the Universe.

Hilbert himself proved the finite generation of invariant rings in the case of the field of complex numbers for some classical semi-simple Lie groups (in particular the general linear group over the complex numbers) and specific linear actions on polynomial rings, i.e. actions coming from finite- dimensional representations of the Lie-group. This finiteness result was later extended by Hermann Weyl to the class of all semi-simple Lie-groups. A major ingredient in Hilbert's proof is the Hilbert basis theorem applied to the ideal inside the polynomial ring generated by the invariants.

In mathematics, a sofic group is a group whose Cayley graph is an initially subamenable graph, or equivalently a subgroup of an ultraproduct of finite- rank symmetric groups such that every two elements of the group have distance 1.Ceccherini-Silberstein & Coornaert (2010) p. 276 They were introduced by as a common generalization of amenable and residually finite groups. The name "sofic", from the Hebrew word meaning "finite", was later applied by , following Weiss's earlier use of the same word to indicate a generalization of finiteness in sofic subshifts.

These precursors may suggest that an underlying cause of these large moves—in the absence of significant change in valuation—may be due to the positioning of traders in advance of anticipated news. For example, suppose many traders are anticipating positive news and buy the stock. If the positive news does not materialize they are inclined to sell in large numbers, thereby suppressing the price significantly below the previous levels. This interpretation is inconsistent with EMH but is consistent with asset flow differential equations (AFDE) that incorporate behavioral concepts with the finiteness of assets.

For structures with N_{eq} < 10^4, which are common for quasibrittle materials, the Weibull theory does not apply. But the underlying weakest-link model, expressed by Eq. (1) for P_f, does, albeit with a finite N, which is a crucial point. The finiteness of the weakest-link chain model causes major deviations from the Weibull distribution. As the structure size, measured by N_{eq}, increases, the grafting point of the Weibullian left part moves to the right until, at about N_{eq} = 10^4, the entire distribution becomes Weibullian.

Caginalp and Balenovich in 1994 used their asset-flow differential equations model to show that the major patterns of technical analysis could be generated with some basic assumptions. Some of the patterns such as a triangle continuation or reversal pattern can be generated with the assumption of two distinct groups of investors with different assessments of valuation. The major assumptions of the models are that the finiteness of assets and the use of trend as well as valuation in decision making. Many of the patterns follow as mathematically logical consequences of these assumptions.

In the late 20th century, the possibilities of genetic engineering became practical for the first time, and a massive international effort began in 1990 to map out an entire human genome (the Human Genome Project). Earthrise, the Earth from above the Moon, Apollo 8. This 1968 NASA image by astronaut William Anders helped create awareness of the finiteness of Earth, and the limits of its natural resources. The discipline of ecology typically traces its origin to the synthesis of Darwinian evolution and Humboldtian biogeography, in the late 19th and early 20th centuries.

According to Hippolytus, the worldview was inspired by the Pythagoreans, who called the first thing that came into existence the "monad", which begat (bore) the dyad (from the Greek word for two), which begat the numbers, which begat the point, begetting lines or finiteness, etc.Diogenes Laërtius, Lives and Opinions of Eminent Philosophers. It meant divinity, the first being, or the totality of all beings, referring in cosmogony (creation theories) variously to source acting alone and/or an indivisible origin and equivalent comparators.Fairbanks, Arthur, Ed., "The First Philosophers of Greece".

Fedor Bogomolov (Head of the HSE Laboratory of Algebraic Geometry), one of the creators of the theory of hyperkahler manifolds. Boris Feigin (Tenured Professor), a well- known expert in representation theory. Yulij Ilyashenko (Head of the Academic Council, Master of Science program in Mathematics), rector of the Independent University of Moscow, the author of the finiteness result for the number of limit cycles of a polynomial vector field in the plane. Igor Krichever (Director of the HSE Master of Science program in Mathematics and Mathematical Physics, previously the dean of the mathematics department at Columbia University).

In Zermelo–Fraenkel set theory without the axiom of choice (ZF), the following conditions are all equivalent: # S is a finite set. That is, S can be placed into a one-to-one correspondence with the set of those natural numbers less than some specific natural number. # (Kazimierz Kuratowski) S has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time. (See below for the set-theoretical formulation of Kuratowski finiteness.) # (Paul Stäckel) S can be given a total ordering which is well-ordered both forwards and backwards.

Born in Loenen aan de Vecht, Griffioen studied at the Vrije Universiteit, where in 1971 he received his MA cum laude in Political economy, in 1975 another MA in Philosophy,"Dr. Griffioen nu lector aan de VU". Reformatorisch Dagblag 7 March 1979. and in 1976 his PhD cum laude with a thesis entitled "De roos en het kruis: De waardering van de eindigheid in het latere denken van Hegel" (The Rose and the Cross: The valuation of the finiteness in the later thought of Hegel).Sander Griffioen pages Griffioen had joined the Vrije Universiteit faculty in 1971.

A year later Dmitri Egorov published his independently proved results,In the note and the theorem became widely known under his name: however, it is not uncommon to find references to this theorem as the Severini–Egoroff theorem or Severini–Egorov Theorem. The first mathematicians to prove independently the theorem in the nowadays common abstract measure space setting were , and in :According to and . an earlier generalization is due to Nikolai Luzin, who succeeded in slightly relaxing the requirement of finiteness of measure of the domain of convergence of the pointwise converging functions in the ample paper .According to .

The third division deals with the doctrine of God, the divine attributes, and similar theological problems. Like Saadia Gaon and Bahya ibn Paquda, though more precisely and more systematically, Joseph proves the creation of the world (and consequently the existence of a Creator) from its finiteness. He criticizes the theory of the Motekallamin (as expounded in the Machkimat Peti of Joseph ha-Ro'eh), who assert that the world was produced by the created will of God. For him the will of God has existed from all eternity, and can not be separated from the essence of God.

The extended Weyl law fails in certain situations. In particular, the extended Weyl law "claims" that there is no essential spectrum if and only if the right-hand expression is finite for all E. If one considers domains with cusps (i.e. "shrinking exits to infinity") then the (extended) Weyl law claims that there is no essential spectrum if and only if the volume is finite. However for the Dirichlet Laplacian there is no essential spectrum even if the volume is infinite as long as cusps shrinks at infinity (so the finiteness of the volume is not necessary).

In 1968 Parshin proved a special case (for = the empty set) of the following theorem: If is a smooth complex curve and is a finite subset of then there exist only finitely many families (up to isomorphism) of smooth curves of fixed genus g ≥ 2 over . The general case (for non-empty ) of the preceding theorem was proved by Arakelov. At the same time, Parshin gave a new proof (without an application of the Shafarevich finiteness condition) of the Mordell conjecture in function fields (already proved by Yuri Manin in 1963 and by Hans Grauert in 1965).Parshin, Algebraic curves over function fields.

The same groups also appeared in analytic number theory as the study of classical modular forms and their generalisations developed. Of course the two topics were related, as can be seen for example in Langlands' computation of the volume of certain fundamental domains using analytic methods. This classical theory culminated with the work of Siegel, who showed the finiteness of the volume of a fundamental domain in many cases. For the modern theory to begin foundational work was needed, and was provided by the work of Armand Borel, André Weil, Jacques Tits and others on algebraic groups.

He introduced the Grothendieck–Ogg–Shafarevich formula and the Néron–Ogg–Shafarevich criterion. He also formulated the Shafarevich conjecture, which stated the finiteness of the set of Abelian varieties over a number field having fixed dimension and prescribed set of primes of bad reduction. The conjecture was proved by Gerd Faltings as a step in his proof of the Mordell conjecture. Shafarevich was a student of Boris Delaunay, and his students included Yuri Manin, Alexey Parshin, Igor Dolgachev, Evgeny Golod, Alexei Kostrikin, Igor A. Kostrikin, Suren Arakelov, G. V. Belyi, Victor Abrashkin, Andrey N. Tyurin, and Victor Kolyvagin.

The Bers area inequality is a quantitative refinement of the Ahlfors finiteness theorem proved by . It states that if Γ is a non-elementary finitely-generated Kleinian group with N generators and with region of discontinuity Ω, then :Area(Ω/Γ) ≤ 4π(N − 1) with equality only for Schottky groups. (The area is given by the Poincaré metric in each component.) Moreover, if Ω1 is an invariant component then :Area(Ω/Γ) ≤ 2Area(Ω1/Γ) with equality only for Fuchsian groups of the first kind (so in particular there can be at most two invariant components).

Logicians in the early 20th century aimed to solve the problem of foundations, such as, "What is the true base of mathematics?" The program was to be able to rewrite all mathematics using an entirely syntactical language without semantics. In the words of David Hilbert (referring to geometry), "it does not matter if we call the things chairs, tables and beer mugs or points, lines and planes." The stress on finiteness came from the idea that human mathematical thought is based on a finite number of principles and all the reasonings follow essentially one rule: the modus ponens.

Using this formula and certain number theoretic and Galois-cohomological estimates, Armand Borel and Gopal Prasad proved several finiteness theorems about arithmetic groups, [6]. The volume formula, together with number-theoretic and Bruhat-Tits theoretic considerations led to a classification, by Gopal Prasad and Sai-Kee Yeung, of fake projective planes (in the theory of smooth projective complex surfaces) into 28 non-empty classes [21] (see also [22] and [23]). This classification, together with computations by Donald Cartwright and Tim Steger, has led to a complete list of fake projective planes. This list consists of exactly 50 fake projective planes, up to isometry (distributed among the 28 classes).

Such scale breaking terms arise more naturally in a constraint formalism, the so-called BF scheme, in which the gauge curvature is denoted by F. In the case of gravity, it departs from the meta-linear group SL(5,R) in four dimensions, thus generalizing (Anti-)de Sitter gauge theories of gravity. After applying spontaneous symmetry breaking to the corresponding topological BF theory, again Einstein spaces emerge with a tiny cosmological constant related to the scale of symmetry breaking. Here the 'background' metric is induced via a Higgs-like mechanism. The finiteness of such a deformed topological scheme may convert into asymptotic safeness after quantization of the spontaneously broken model.

In papers in 1977 and 1978, Barry Mazur proved the torsion conjecture giving a complete list of the possible torsion subgroups of elliptic curves over the rational numbers. Mazur's first proof of this theorem depended upon a complete analysis of the rational points on certain modular curves. In 1996, the proof of the torsion conjecture was extended to all number fields by Loïc Merel. In 1983, Gerd Faltings proved the Mordell conjecture, demonstrating that a curve of genus greater than 1 has only finitely many rational points (where the Mordell–Weil theorem only demonstrates finite generation of the set of rational points as opposed to finiteness).

The Engelbert–Schmidt zero–one law is a theorem that gives a mathematical criterion for an event associated with a continuous, non-decreasing additive functional of Brownian motion to have probability either 0 or 1, without the possibility of an intermediate value. This zero-one law is used in the study of questions of finiteness and asymptotic behavior for stochastic differential equations. (A Wiener process is a mathematical formalization of Brownian motion used in the statement of the theorem.) This 0-1 law, published in 1981, is named after Hans-Jürgen Engelbert and the probabilist Wolfgang Schmidt (not to be confused with the number theorist Wolfgang M. Schmidt).

Fang assembled a group of young faculty members of USTC around him to conduct astrophysics research. At the time, conducting research on relativity theory and cosmology in China was very risky politically, because these theories were considered to be "idealistic" theories in contradiction with the dialectical materialism theory, which is the official philosophy of the Communist Party. According to the dialectical materialism philosophy, both time and space must be infinite, while the Big Bang theory allows the possibility of the finiteness of space and time. During the Cultural Revolution, campaigns were waged against Albert Einstein and the Theory of Relativity in Beijing and Shanghai.

One of the main points of Lossky's онтология or ontology is, the world is an organic whole as understood by human consciousness. Intuition, insight (noesis in Greek) is the direct contemplation of objects, and furthermore the assembling of the entire set of cognition from sensory perception into a complete and undivided organic whole, i.e. experience. This expression of consciousness as without thought, raw and uninterpreted by the rational faculty in the mind. Thus the mind's dianoia (rational or logical faculty) in its deficiency, finiteness or inconclusiveness (due to logic's incompleteness) causes the perceived conflict between the objectivism (materialism, external world) and idealism (spiritual, inner experience) forms of philosophy.

At the end of season one, Picard's human form expires on Coppelius due to a pre- existing medical condition, and he encounters Data's consciousness, still alive in a simulated matrix. Data pleads for Picard to finally let him die, as he considers the finiteness of life to be a defining human characteristic. Picard is brought back to life in a synthetic body, specifically configured to only give him the lifespan he would have lived if he didn't have his previous disease, and obliges Data's request, staying by him in the simulated matrix as Data dies. Picard then continues exploring space in his new body and crew.

Hebraic thought trends had much more of an influence on the important concepts of existentialism. Much of modern existentialism may be seen as more Jewish than Greek. Several core concepts found in the ancient Hebrew tradition that are often cited as the most important concepts explored by existentialism, for example, the "uneasiness" "deep within Biblical man", also his "sinfulness" and "feebleness and finiteness". While "the whole impulse of philosophy for Plato arises from an ardent search for escape from the evils of the world and the curse of time", Biblical Judaism recognizes the impossibility of trying to transcend the world entirely via intellectualism, lofty thoughts, and ideals.

There is a strong finiteness result on sheaf cohomology. Let X be a compact Hausdorff space, and let R be a principal ideal domain, for example a field or the ring Z of integers. Let E be a sheaf of R-modules on X, and assume that E has "locally finitely generated cohomology", meaning that for each point x in X, each integer j, and each open neighborhood U of x, there is an open neighborhood V ⊂ U of x such that the image of Hj(U,E) → Hj(V,E) is a finitely generated R-module. Then the cohomology groups Hj(X,E) are finitely generated R-modules.

The mean strength can be computed from this distribution and, as it turns out, its plot is identical with the plot of Eq. 5 seen in Fig. 2g. The point of deviation from the Weibull asymptote is determined by the location of the grafting point on the strength distribution of one RVE (Fig. 2g). Note that the finiteness of the chain in the weakest-link model captures the deterministic part of size effect. This theory has also been extended to the size effect on the Evans and Paris' laws of crack growth in quasibrittle materials, and to the size effect on the static and fatigue lifetimes.

In mathematics, an Euler system is a collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper and the work of . Euler systems are named after Leonhard Euler because the factors relating different elements of an Euler system resemble the Euler factors of an Euler product. Euler systems can be used to construct annihilators of ideal class groups or Selmer groups, thus giving bounds on their orders, which in turn has led to deep theorems such as the finiteness of some Tate-Shafarevich groups.

A major debate in understanding language acquisition is how these capacities are picked up by infants from the linguistic input. Input in the linguistic context is defined as "All words, contexts, and other forms of language to which a learner is exposed, relative to acquired proficiency in first or second languages". Nativists such as Chomsky have focused on the hugely complex nature of human grammars, the finiteness and ambiguity of the input that children receive, and the relatively limited cognitive abilities of an infant. From these characteristics, they conclude that the process of language acquisition in infants must be tightly constrained and guided by the biologically given characteristics of the human brain.

A projective plane satisfying Pappus's theorem universally is called a Pappian plane. Alternative, not necessarily associative, division algebras like the octonions correspond to Moufang planes. There is no known purely geometric proof of the purely geometric statement that Desargues' theorem implies Pappus' theorem in a finite projective plane (finite Desarguesian planes are Pappian). (The converse is true in any projective plane and is provable geometrically, but finiteness is essential in this statement as there are infinite Desarguesian planes which are not Pappian.) The most common proof uses coordinates in a division ring and Wedderburn's theorem that finite division rings must be commutative; give a proof that uses only more "elementary" algebraic facts about division rings.

FD-DEVS was originally named as ``Schedule-Controlable DEVS`` [Hwang05] and designed to support verification analysis of its networks which had been an open problem of DEVS formalism for 30 years. In addition, it was also designated to resolve the so-called ``OPNA`` problem of SP-DEVS. From the viewpoint of Classic DEVS, FD-DEVS has three restrictions # finiteness of event sets and state set, # the lifespan of a state can be scheduled by a rational number or infinity, and # the internal schedule can be either preserved or updated by an input event. The third restriction can be also seen as a relaxation from SP-DEVS where the schedule is always preserved by any input events.

SP-DEVS has been designed to support verification analysis of its networks by guaranteeing to obtain a finite-vertex reachability graph of the original networks, which had been an open problem of DEVS formalism for roughly 30 years. To get such a reachability graph of its networks, SP-DEVS has been imposed the three restrictions: # finiteness of event sets and state set, # the lifespan of a state can be scheduled by a rational number or infinity, and # preserving the internal schedule from any external events. Thus, SP-DEVS is a sub-class of both DEVS and FD-DEVS. These three restrictions lead that SP-DEVS class is closed under coupling even though the number of states are finite.

Kenkichi Iwasawa independently discovered essentially the same method (without an analog of the local theory in Tate's thesis) during the Second World War and announced it in his 1950 International Congress of Mathematicians paper and his letter to Jean Dieudonné written in 1952. Hence this theory is often called Iwasawa–Tate theory. Iwasawa in his letter to Dieudonné derived on several pages not only the meromorphic continuation and functional equation of the L-function, he also proved finiteness of the class number and Dirichlet's theorem on units as immediate byproducts of the main computation. The theory in positive characteristic was developed one decade earlier by Ernst Witt, Wilfried Schmid, and Oswald Teichmüller.

Two central configurations are considered to be equivalent if they are similar, that is, they can be transformed into each other by some combination of rotation, translation, and scaling. With this definition of equivalence, there is only one configuration of one or two points, and it is always central. In the case of three bodies, there are three one-dimensional central configurations, found by Leonhard Euler. The finiteness of the set of three- point central configurations was shown by Joseph-Louis Lagrange in his solution to the three-body problem; Lagrange showed that there is only one non-collinear central configuration, in which the three points form the vertices of an equilateral triangle.

He interpreted the Virasoro algebra discovered in consistency conditions as a geometrical symmetry of a world- sheet conformal field theory, formulating string theory in terms of two dimensional quantum field theory. He used the conformal invariance to calculate tree level string amplitudes on many worldsheet domains. Mandelstam was the first to explicitly construct the fermion scattering amplitudes in the Ramond and Neveu–Schwarz sectors of superstring theory, and later gave arguments for the finiteness of string perturbation theory. In quantum field theory, Mandelstam and independently Sidney Coleman extended work of Tony Skyrme to show that the two dimensional quantum Sine-Gordon model is equivalently described by a Thirring model whose fermions are the kinks.

A seeming paradox is that there are non- standard models of the theory of hereditarily finite sets which contain infinite sets, but these infinite sets look finite from within the model. (This can happen when the model lacks the sets or functions necessary to witness the infinitude of these sets.) On account of the incompleteness theorems, no first-order predicate, nor even any recursive scheme of first- order predicates, can characterize the standard part of all such models. So, at least from the point of view of first-order logic, one can only hope to describe finiteness approximately. More generally, informal notions like set, and particularly finite set, may receive interpretations across a range of formal systems varying in their axiomatics and logical apparatus.

In another direction, that of torsors, these were already implicit in the infinite descent arguments of Fermat for elliptic curves. Numerous direct calculations were done, and the proof of the Mordell–Weil theorem had to proceed by some surrogate of a finiteness proof for a particular H1 group. The 'twisted' nature of objects over fields that are not algebraically closed, which are not isomorphic but become so over the algebraic closure, was also known in many cases linked to other algebraic groups (such as quadratic forms, simple algebras, Severi–Brauer varieties), in the 1930s, before the general theory arrived. The needs of number theory were in particular expressed by the requirement to have control of a local-global principle for Galois cohomology.

To say that the domain has countable cardinality, use the sentence that says that there is a bijection between every two infinite subsets of the domain. It follows from the compactness theorem and the upward Löwenheim–Skolem theorem that it is not possible to characterize finiteness or countability, respectively, in first-order logic. Certain fragments of second order logic like ESO are also more expressive than first-order logic even though they are strictly less expressive than the full second-order logic. ESO also enjoys translation equivalence with some extensions of first-order logic which allow non-linear ordering of quantifier dependencies, like first-order logic extended with Henkin quantifiers, Hintikka and Sandu's independence- friendly logic, and Väänänen's dependence logic.

See page 21 of Lee Smolin, Recent Developments in Non-Perturbative Quantum Gravity, The direct proof of finiteness of canonical LQG in the presence of all forms of matter has been provided by Thiemann.Thomas Thiemann, Modern Canonical Quantum General Relativity, Cambridge University Press However, it has been suggested that loop quantum gravity violates background independence by introducing a preferred frame of reference ('spin foams'). Perturbative string theory (in addition to a number of non-perturbative formulations) is not 'obviously' background independent, because it depends on boundary conditions at infinity, similarly to how perturbative general relativity is not 'obviously' background dependent. However some sectors of string theory admit formulations in which background independence is manifest, including most notably the AdS/CFT.

In general relativity, the relativistic disk expression refers to a class of axi-symmetric self-consistent solutions to Einstein's field equations corresponding to the gravitational field generated by axi-symmetric isolated sources. To find such solutions, one has to pose correctly and solve together the ‘outer’ problem, a boundary value problem for vacuum Einstein's field equations whose solution determines the external field, and the ‘inner’ problem, whose solution determines the structure and the dynamics of the matter source in its own gravitational field. Physically reasonable solutions must satisfy some additional conditions such as finiteness and positiveness of mass, physically reasonable kind of matter and finite geometrical size. Exact solutions describing relativistic static thin disks as their sources were first studied by Bonnor and Sackfield and Morgan and Morgan.

A virtually polycyclic group is a group that has a polycyclic subgroup of finite index, an example of a virtual property. Such a group necessarily has a normal polycyclic subgroup of finite index, and therefore such groups are also called polycyclic-by-finite groups. Although polycyclic-by-finite groups need not be solvable, they still have many of the finiteness properties of polycyclic groups; for example, they satisfy the maximal condition, and they are finitely presented and residually finite. In the textbook and some papers, an M-group refers to what is now called a polycyclic-by-finite group, which by Hirsch's theorem can also be expressed as a group which has a finite length subnormal series with each factor a finite group or an infinite cyclic group.

Using the axioms of Zermelo–Fraenkel set theory with the originally highly controversial axiom of choice included (ZFC) one can show that a set is Dedekind-finite if and only if it is finite in the sense of having a finite number of elements. However, there exists a model of Zermelo–Fraenkel set theory without the axiom of choice (ZF) in which there exists an infinite, Dedekind-finite set, showing that the axioms of ZF are not strong enough to prove that every set that is Dedekind-finite has a finite number of elements. There are definitions of finiteness and infiniteness of sets besides the one given by Dedekind that do not depend on the axiom of choice. A vaguely related notion is that of a Dedekind-finite ring.

In higher dimensions, one unifying goal is the Bombieri–Lang conjecture that, for any variety X of general type over a number field k, the set of k-rational points of X is not Zariski dense in X. (That is, the k-rational points are contained in a finite union of lower-dimensional subvarieties of X.) In dimension 1, this is exactly Faltings's theorem, since a curve is of general type if and only if it has genus at least 2. Lang also made finer conjectures relating finiteness of rational points to Kobayashi hyperbolicity.Hindry & Silverman (2000), section F.5.2. For example, the Bombieri–Lang conjecture predicts that a smooth hypersurface of degree d in projective space Pn over a number field does not have Zariski dense rational points if d ≥ n + 2.

Wilkie was elected a Fellow of the Royal Society in 2001. To quote the citation :Wilkie has combined logical techniques and differential-geometric techniques to establish fundamental Finiteness Theorems for sets definable using the exponential function, and more general Pfaffian functions. The results, going far beyond those obtained by conventional methods, have already had striking applications to Lie groups.NOTICES, Bulletin of Symbolic Logic, Vol 7, No 3, p436, 2001 Wilkie received the Carol Karp Prize (the highest award made by the Association for Symbolic Logic, every five years) jointly with Ehud Hrushovski in 1993.NOTICES Carol Karp Prize, J. Symbolic logic, Volume 58, Number 2, June 1993 He was elected to the Council of the London Mathematical Society in 2007, vice-president of the Association for Symbolic Logic (2006) and president of the Association for Symbolic Logic in 2009.

To show that the tree contains every primitive Pythagorean triple, but no more than once, it suffices to show that for any such triple there is exactly one path back through the tree to the starting node (3, 4, 5). This can be seen by applying in turn each of the unimodular inverse matrices A−1, B−1, and C−1 to an arbitrary primitive Pythagorean triple (d, e, f), noting that by the above reasoning primitivity and the Pythagorean property are retained, and noting that for any triple larger than (3, 4, 5) exactly one of the inverse transition matrices yields a new triple with all positive entries (and a smaller hypotenuse). By induction, this new valid triple itself leads to exactly one smaller valid triple, and so forth. By the finiteness of the number of smaller and smaller potential hypotenuses, eventually (3, 4, 5) is reached.

In theoretical linguistics, she has worked extensively on the nature of finiteness, case, and agreement regarding clausal architecture in the world's languages, as well as the structure of reduced relative clauses and embedded clauses, specificity, scrambling, the copula. In applied linguistics and language education, she cultivated an approach to English grammar, a descriptive linguistics approach not only to assist language learners of all levels to develop and refine their English language use, but also to help those language users understand the deeper structures behind the language, empowering them to not depend on simple memorization. In her book English Grammar: a descriptive linguistics approach, she bridges the so-called contradiction between descriptive and prescriptive grammar by developing a new framework with which prescriptive grammar can be re-analyzed using the linguistic tools of descriptive linguistics. This approach has been adopted to teach English grammar to native speakers or non- native speakers of English.

The geometric lattices are cryptomorphic to (finite, simple) matroids, and the matroid lattices are cryptomorphic to simple matroids without the assumption of finiteness. Like geometric lattices, matroids are endowed with rank functions, but these functions map sets of elements to numbers rather than taking individual elements as arguments. The rank function of a matroid must be monotonic (adding an element to a set can never decrease its rank) and they must be submodular functions, meaning that they obey an inequality similar to the one for semimodular lattices: :r(X)+r(Y)\ge r(X\cap Y)+r(X\cup Y). \, The maximal sets of a given rank are called flats. The intersection of two flats is again a flat, defining a greatest lower bound operation on pairs of flats; one can also define a least upper bound of a pair of flats to be the (unique) maximal superset of their union that has the same rank as their union.

There are only several dozen of transitive verbs which take an accusative patient, all of which are monosyllabic and have distinct finite and nonfinite forms. It has been suggested that all transitive verbs which satisfy both conditions (monosyllabicity and a formal finiteness distinction), and only them, select for accusative patients, while all remaining transitive verbs take absolutive patients in Canela and all other Northern Jê languages. All subordinate clauses as well as recent past clauses (which are historically derived from subordinate clauses and are headed by a nonfinite verb) are ergatively organized: the agents of transitive verbs (A) are encoded by ergative postpositional phrases, whereas the patients of transitive verbs (P) and the sole arguments of all intransitive predicates (S) receive the absolutive case (also called internal case). Evaluative, progressive, continuous, completive, and negated clauses (which are historically derived from former biclausal constructions with an ergatively organized subordinate clause and a split-S matrix clause) in Canela have the cross-linguistically rare nominative-absolutive alignment pattern.

These dozens of paintings of flowers, alone or arranged in vivid grids, constitute peace offerings to Lebanon, full of hopeful and celebratory symbolism, and envision the possibility of a non- violent world. In 2016 and 2019, she also exhibited a collection of large- scale paintings, How Many How Many More, marked by systematic repetitive series of colored stripes within which she inserted archival photographs, buttons, beads, open medication capsules, and strips of newspaper, creating works at once cheerful and elegiac. The series started as a reaction to the ongoing wars in the Arab world, notably in Syria, and led to a personal reflection on the finiteness of human life, as well as a plea for peace. These works are simultaneously abstract paintings, collages, and landscapes that underscore the way memory is at once fragmentary, personal, and collective, as the repetition of stripes is meant to echo one counting the days of life.

The pronominal anaphor does not select a referent in its discourse, but it is bound by the preceding quantificational operator. If a pronominal anaphor does not have a quantificational antecedent, but instead it has a proper name or a definite description as in example (35), the non-quantificational antecedent is called rigid designator. In this environment, the pronominal anaphor refers back to its antecedent. (35) (i) [Beverly]1 believes that [she]1 is underpaid. (ii) [The female lawyer]1 believes that [she]1 is underpaid. (Déchaine and Wiltschko, 2014: 4(9)) In (35i), as indicated by the index, ‘she’ refers back to Beverly,’ while in (35ii), ‘she’ refers back to its preceding antecedent ‘the female lawyer.’ Condition Two states that BVA requires a co-varying anaphoric expression. Not only overt pronominal anaphors (classified in terms of finiteness) (36i), but also various types of covert anaphors (‘big PRO’) (36ii) should be considered in the discussion of 3rd person pronouns.

Knuth (1968, 1973) has given a list of five properties that are widely accepted as requirements for an algorithm: # Finiteness: "An algorithm must always terminate after a finite number of steps ... a very finite number, a reasonable number" # Definiteness: "Each step of an algorithm must be precisely defined; the actions to be carried out must be rigorously and unambiguously specified for each case" # Input: "...quantities which are given to it initially before the algorithm begins. These inputs are taken from specified sets of objects" # Output: "...quantities which have a specified relation to the inputs" # Effectiveness: "... all of the operations to be performed in the algorithm must be sufficiently basic that they can in principle be done exactly and in a finite length of time by a man using paper and pencil" Knuth offers as an example the Euclidean algorithm for determining the greatest common divisor of two natural numbers (cf. Knuth Vol. 1 p. 2).

The generalized Fermat equation generalizes the statement of Fermat's last theorem by considering positive integer solutions a, b, c, m, n, k satisfying In particular, the exponents m, n, k need not be equal, whereas Fermat's last theorem considers the case The Beal conjecture, also known as the Mauldin conjecture and the Tijdeman-Zagier conjecture, states that there are no solutions to the generalized Fermat equation in positive integers a, b, c, m, n, k with a, b, and c being pairwise coprime and all of m, n, k being greater than 2. The Fermat–Catalan conjecture generalizes Fermat's last theorem with the ideas of the Catalan conjecture. The conjecture states that the generalized Fermat equation has only finitely many solutions (a, b, c, m, n, k) with distinct triplets of values (am, bn, ck), where a, b, c are positive coprime integers and m, n, k are positive integers satisfying The statement is about the finiteness of the set of solutions because there are 10 known solutions.