The rookie Dylan McIlrath, 23, will enter the lineup for Girardi, who was on the ice for three of the Penguins' four non-empty-net goals Wednesday.
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And even with Rinne ducking out of the way of one bad-angle shot Nick Ritchie fired and positioning himself way off his angle on a similarly harmless shot by Sami Vatanen, the Ducks were only able to squeak out a win by one non-empty-net goal.
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A function between topological spaces and is quasi-open if, for any non-empty open set , the interior of in is non-empty.
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Further simple completeness conditions arise from the consideration of all non-empty finite sets. An order in which all non-empty finite sets have both a supremum and an infimum is called a lattice. It suffices to require that all suprema and infima of two elements exist to obtain all non-empty finite ones. A straightforward induction shows that every finite non-empty supremum/infimum can be decomposed into a finite number of binary suprema/infima.
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Every Cauchy sequence in converges in (that is, to some point of ). 3. The expansion constant of is ≤ 2. 4. Every decreasing sequence of non-empty closed subsets of , with diameters tending to 0, has a non-empty intersection: if is closed and non-empty, for every , and , then there is a point common to all sets .
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This construction is used to define in terms of prefilter convergence. ;Products of prefilters Throughout, will be a non-empty family of non-empty sets and will be a family of non-empty sets where each . For every , let : denote the canonical projection. Recall that the product of the sets (which is defined above) is denoted by .
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Replaces switchprog S pm:n, provided it is a non- empty substring of S p.
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The property of being non-empty (that is, having at least one edge) is monotone, because adding another edge to a non-empty graph produces another non-empty graph. There is a simple algorithm for testing whether a graph is non-empty: loop through all of the pairs of vertices, testing whether each pair is connected by an edge. If an edge is ever found in this way, break out of the loop, and report that the graph is non-empty, and if the loop completes without finding any edges, then report that the graph is empty. On some graphs (for instance the complete graphs) this algorithm will terminate quickly, without testing every pair of vertices, but on the empty graph it tests all possible pairs before terminating.
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If T is empty then we use the convention that and . Any other value x is stored in the subtree where . The auxiliary tree keeps track of which children are non-empty, so contains the value j if and only if is non-empty.
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This leaves people to be placed into at most non-empty holes, so that the principle applies.
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By invariance of domain, a non-empty n-manifold cannot be an m-manifold for n ≠ m. The dimension of a non-empty n-manifold is n. Being an n-manifold is a topological property, meaning that any topological space homeomorphic to an n-manifold is also an n-manifold.
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A holomorphic functional calculus can be defined in a similar fashion for unbounded closed operators with non-empty resolvent set.
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Throughout we assume the following: 1. is any non-empty set and is a non-empty collection of subsets of directed by subset inclusion (i.e. for any there exists some such that ). 2. is a topological vector space (not necessarily Hausdorff or locally convex) and is a basis of neighborhoods of 0 in . 3.
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In the category of sets, the inverse limit of any inverse system of non-empty finite sets is non-empty. This may be seen as a generalization of Kőnig's lemma and can be proved with Tychonoff's theorem, viewing the finite sets as compact discrete spaces, and then using the finite intersection property characterization of compactness.
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A serial relation can be equivalently characterized as every element having a non-empty successor neighborhood. Similarly an inverse serial relation is a relation in which every element has non-empty "predecessor neighborhood". More commonly, an inverse serial relation is called a surjective relation, and is specified by a serial converse relation. Definition 5.8, page 57.
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A non- empty connected topological space X is a cut-point space if every point in X is a cut point of X.
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A groupoid is an algebraic structure (G,\ast) consisting of a non-empty set G and a binary partial function '\ast' defined on G.
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The condition that A,B are both non-empty is clearly necessary. This condition is not part of the multiplicative versions of BM stated below.
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In a game with finitely many actions, this process always terminates and leaves a non-empty set of actions for each player. These are the rationalizable actions.
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For a topological space X the following conditions are equivalent: #X is connected, that is, it cannot be divided into two disjoint non-empty open sets. #X cannot be divided into two disjoint non-empty closed sets. #The only subsets of X which are both open and closed (clopen sets) are X and the empty set. #The only subsets of X with empty boundary are X and the empty set.
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Zorn's lemma is sometimesFor example, , , and . stated as follows: Although this formulation appears to be formally weaker (since it places on P the additional condition of being non-empty, but obtains the same conclusion about P), in fact the two formulations are equivalent. To verify this, suppose first that P satisfies the condition that every chain in P has an upper bound in P. Then the empty subset of P is a chain, as it satisfies the definition vacuously; so the hypothesis implies that this subset must have an upper bound in P, and this upper bound shows that P is in fact non-empty. Conversely, if P is assumed to be non-empty and satisfies the hypothesis that every non-empty chain has an upper bound in P, then P also satisfies the condition that every chain has an upper bound, as an arbitrary element of P serves as an upper bound for the empty chain (that is, the empty subset viewed as a chain).
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The Bondareva–Shapley theorem, in game theory, describes a necessary and sufficient condition for the non-emptiness of the core of a cooperative game in characteristic function form. Specifically, the game's core is non-empty if and only if the game is balanced. The Bondareva–Shapley theorem implies that market games and convex games have non-empty cores. The theorem was formulated independently by Olga Bondareva and Lloyd Shapley in the 1960s.
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The maximal connected subsets (ordered by inclusion) of a non-empty topological space are called the connected components of the space. The components of any topological space X form a partition of X: they are disjoint, non-empty, and their union is the whole space. Every component is a closed subset of the original space. It follows that, in the case where their number is finite, each component is also an open subset.
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For Minkowski addition, the zero set containing only the zero vector has special importance: For every non-empty subset S of a vector space :; in algebraic terminology, is the identity element of Minkowski addition (on the collection of non-empty sets).The empty set is important in Minkowski addition, because the empty set annihilates every other subset: For every subset of a vector space, its sum with the empty set is empty: .
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Cantor's intersection theorem refers to two closely related theorems in general topology and real analysis, named after Georg Cantor, about intersections of decreasing nested sequences of non-empty compact sets.
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Zorn's lemma can be stated as: Variants of this formulation are sometimes used, such as requiring that the set P and the chains be non-empty. See the discussion below.
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The Algebraic Theory of Semigroups Vol. I (Second Edition). American Mathematical Society. pp. 2–3 However not all authors insist on the underlying set of a semigroup being non-empty.
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Theorem. Let X be a non-empty compact Hausdorff space that satisfies the property that no one-point set is open. Then X is uncountable. Proof. We will show that if U ⊆ X is non-empty and open, and if x is a point of X, then there is a neighbourhood V ⊂ U whose closure does not contain x (x may or may not be in U). Choose y in U different from x (if x is in U, then there must exist such a y for otherwise U would be an open one point set; if x is not in U, this is possible since U is non-empty). Then by the Hausdorff condition, choose disjoint neighbourhoods W and K of x and y respectively.
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Formally, let A be an alphabet and A∗ be the free monoid of finite strings over A. Every non-empty word w in A+ is a sesquipower of order 1\. If u is a sesquipower of order n then any word w = uvu is a sesquipower of order n + 1.Lothaire (2011) p. 135 The degree of a non-empty word w is the largest integer d such that w is a sesquipower of order d.
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The intersection of an arbitrary family of saturated families is a saturated family. Since the power set of X is saturated, any given non-empty family 𝒢 of subsets of X containing at least one non-empty set, the saturated hull of 𝒢 is well-defined. Note that a saturated family of subsets of X that covers X is a bornology on X. The set of all bounded subsets of a topological vector space is a saturated family.
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Let be any non-empty open subset of (e.g. could be a non-empty bounded open interval in ) and let denote the subspace topology on that inherits from (so ). Then the topology generated by on is equal to the union (see this footnote for an explanation),Since is a topology on and is an open subset of , it is easy to verify that is a topology on . Since isn't a topology on , is clearly the smallest topology on containing ).
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An important consequence of the domain invariance theorem is that cannot be homeomorphic to if . Indeed, no non-empty open subset of can be homeomorphic to any open subset of in this case.
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If X is space that can be written as the union of two open simply connected sets U and V with U ∩ V non-empty and path-connected, then X is simply connected.
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As with ideals, for every non-empty subset of , there exists a smallest band containing that subset, called the band generated by . A band generated by a singleton is called a principal band.
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In voting systems, the Schwartz set is the union of all Schwartz set components. A Schwartz set component is any non-empty set S of candidates such that # Every candidate inside the set S is pairwise unbeaten by every candidate outside S; and # No non-empty proper subset of S fulfills the first property. A set of candidates that meets the first requirement is also known as an undominated set. The Schwartz set provides one standard of optimal choice for an election outcome.
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In the namua phase, each player begins his or her move by introducing one of the seeds he or she has in hand into the board. The seed must be placed in a non- empty pit in the player's inner row. A "marker" pit is a pit of the inner row that faces a non-empty opponent's pit. If the first seed is placed in a marker pit, a capture occurs, and the player's turn will be called a mtaji turn.
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Then there is a bijection between the factors of for which and the subsets of the set of all prime factors of . The asserted result follows from the fact that every non-empty finite set has an equal number of odd- and even-cardinality subsets. This last fact can be shown easily by induction on the cardinality of a non-empty finite set . First, if , there is exactly one odd-cardinality subset of , namely itself, and exactly one even-cardinality subset, namely .
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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.
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In mathematics, specifically functional analysis, the barrier cone is a cone associated to any non-empty subset of a Banach space. It is closely related to the notions of support functions and polar sets.
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In an arbitrary graph that is not cop-win, the robber can win by removing all dominated vertices and playing within the remaining subgraph, which must be non-empty else the graph would be dismantlable.
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If X takes the values (non-empty, compact sets) K and L with probabilities 1 − λ and λ respectively, then Vitale's random Brunn–Minkowski inequality is simply the original Brunn–Minkowski inequality for compact sets.
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If C is a non-empty convex cone in X, then the linear span of C is equal to C - C and the largest vector subspace of X contained in C is equal to C ∩ (-C).
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In mathematical logic, the Scott–Curry theorem is a result in lambda calculus stating that if two non-empty sets of lambda terms A and B are closed under beta-convertibility then they are recursively inseparable.
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Let the non-empty set S be a counter-example to the axiom of regularity; that is, every element of S has a non-empty intersection with S. We define a binary relation R on S by aRb :\Leftrightarrow b \in S \cap a, which is entire by assumption. Thus, by the axiom of dependent choice, there is some sequence (an) in S satisfying anRan+1 for all n in N. As this is an infinite descending chain, we arrive at a contradiction and so, no such S exists.
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Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection. The compactness theorem is one of the two key properties, along with the downward Löwenheim–Skolem theorem, that is used in Lindström's theorem to characterize first-order logic. Although, there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them.
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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.
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The DRR scans all non empty queues in sequence. When a non empty queue i is selected, its deficit counter is incremented by its quantum value. Then, the value of the deficit counter is a maximal amount of bytes that can be sent at this turn: if the deficit counter is greater than the packet's size at the head of the queue (HoQ), this packet can be sent and the value of the counter is decremented by the packet size. Then, the size of the next packet is compared to the counter value, etc.
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In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a well-ordered set. In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering. Every non-empty well-ordered set has a least element.
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A subset of a set is said to be cofinite in is its complement in (i.e. the set ) is finite. The Fréchet filter on , denoted by , is the set of all non-empty cofinite subsets of . That is: ::.
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Let X be non-empty, F ⊆ 2X, F having the finite intersection property. Then there exists an U ultrafilter (in 2X) such that F ⊆ U. See details and proof in .. This result is known as the ultrafilter lemma.
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A clock constraint defines a set of valuations. Two kinds of such sets are considered in the literature. A zone is a non-empty set of valuations satisfying a clock constraint. Zones and clock constraints are implemented using difference bound matrix.
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We prove the finite version, using Radon's theorem as in the proof by . The infinite version then follows by the finite intersection property characterization of compactness: a collection of closed subsets of a compact space has a non-empty intersection if and only if every finite subcollection has a non-empty intersection (once you fix a single set, the intersection of all others with it are closed subsets of a fixed compact space). The proof is by induction: Base case: Let . By our assumptions, for every there is a point that is in the common intersection of all with the possible exception of .
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Let A be a non-empty set, X a subset of A, F a set of functions in A, and X_+ the inductive closure of X under F. Let be B any non-empty set and let G be the set of functions on B, such that there is a function d:F\to G in G that maps with each function f of arity n in F the following function d(f):B^n\to B in G (G cannot be a bijection). From this lemma we can now build the concept of unique homomorphic extension.
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One special form of a path still exists: the empty path `MAIL FROM:<>`, used for many auto replies and especially all bounces. In a strict sense, bounces sent with a non-empty `Return-Path` are incorrect. RFC 3834 offers some heuristics to identify incorrect bounces based on the local part (left hand side before the "@") of the address in a non-empty `Return- Path`, and it even defines a mail header field, `Auto-Submitted`, to identify auto replies. But the mail header is a part of the mail data (SMTP command `DATA`), and MTAs typically don't look into the mail.
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Transformation semigroups are of essential importance for the structure theory of finite state machines in automata theory. In particular, a semiautomaton is a triple (Σ,X,T), where Σ is a non-empty set called the input alphabet, X is a non-empty set called the set of states and T is a function :T\colon \Sigma\times X \to X called the transition function. Semiautomata arise from deterministic automata by ignoring the initial state and the set of accept states. Given a semiautomaton, let Ta: X → X, for a ∈ Σ, denote the transformation of X defined by Ta(x) = T(a,x).
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A graph with edges colored to illustrate path H-A-B (green), closed path or walk with a repeated vertex B-D-E-F-D-C-B (blue) and a cycle with no repeated edge or vertex H-D-G-H (red). In graph theory, a cycle in a graph is a non- empty trail in which the only repeated vertices are the first and last vertices. A directed cycle in a directed graph is a non-empty directed trail in which the only repeated vertices are the first and last vertices. A graph without cycles is called an acyclic graph.
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The master constraint for LQG was established as a genuine positive self-adjoint operator and the physical Hilbert space of LQG was shown to be non-empty, an obvious consistency test LQG must pass to be a viable theory of quantum General relativity.
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Every LP game v is a totally balanced game. So every subgame of v has a non-empty core. One imputation can be computed by solving the dual problem of P(N). Let \alpha be the optimal dual solution of P(N).
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Now we will show by contradiction that at least one of and is empty. Assume the contrary, i.e. that and are both non-empty and let the largest member of be and the largest member of be . Because and contain no common elements, .
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In the study of metric spaces in mathematics, there are various notions of two metrics on the same underlying space being "the same", or equivalent. In the following, X will denote a non-empty set and d_1 and d_2 will denote two metrics on X.
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If is a finite set then the Fréchet filter on consists of all non-empty subsets of . On the set of natural numbers, the set of infinite intervals } is a Fréchet filter base, i.e., the Fréchet filter on consists of all supersets of elements of .
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The set of subsets of of cardinality less than or equal to is sometimes denoted by or , and the set of subsets with cardinality strictly less than is sometimes denoted or . Similarly, the set of non-empty subsets of might be denoted by or .
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The non-emptiness problem (is the language of an input AFA non-empty?), the universality problem (is the complement of the language of an input AFA empty?), and the equivalence problem (do two input AFAs recognize the same language) are PSPACE-complete for AFAs.
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Then and mesh and generates a filter on that is strictly finer than . ;When prefilters mesh Given non-empty families and , let : where these sets always satisfy and . If is proper (resp. a prefilter, a filter subbase) then this is also true of both and .
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Every topological space is a dense subset of itself. For a set X equipped with the discrete topology, the whole space is the only dense subset. Every non- empty subset of a set X equipped with the trivial topology is dense, and every topology for which every non-empty subset is dense must be trivial. Denseness is transitive: Given three subsets A, B and C of a topological space X with such that A is dense in B and B is dense in C (in the respective subspace topology) then A is also dense in C. The image of a dense subset under a surjective continuous function is again dense.
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We work with first-order predicate calculus. Our languages allow constant, function and relation symbols. Structures consist of (non-empty) domains and interpretations of the relevant symbols as constant members, functions or relations over that domain. We assume classical logic (as opposed to intuitionistic logic for example).
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Assuming that someNode is some node in a non-empty list, this code traverses through that list starting with someNode (any node will do): Forwards node := someNode do do something with node.value node := node.next while node ≠ someNode Backwards node := someNode do do something with node.value node := node.
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Given a model M, it uses a finite number of constants in its clock constraints. Let K be the greatest constant used. A region is a non-empty zone in which no constraint greater than K are used, and furthermore, such that it is minimal for the inclusion.
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An important special case of an ideal is constituted by those ideals whose set-theoretic complements are filters, i.e. ideals in the inverse order. Such ideals are called prime ideals. Also note that, since we require ideals and filters to be non-empty, every prime ideal is necessarily proper.
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In mathematics, specifically in class theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets of sets. Informally it states that one can simultaneously choose an element from every non-empty set.
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If (A, ∧) is a meet-semilattice, then the meet may be extended to a well-defined meet of any non-empty finite set, by the technique described in iterated binary operations. Alternatively, if the meet defines or is defined by a partial order, some subsets of A indeed have infima with respect to this, and it is reasonable to consider such an infimum as the meet of the subset. For non-empty finite subsets, the two approaches yield the same result, and so either may be taken as a definition of meet. In the case where each subset of A has a meet, in fact (A, ≤) is a complete lattice; for details, see completeness (order theory).
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The colorful Helly theorem is an extension of Helly's theorem in which, instead of one collection, there are d+1 collections of convex subsets of . If, for every choice of a transversal – one set from every collection – there is a point in common to all the chosen sets, then for at least one of the collections, there is a point in common to all sets in the collection. Figuratively, one can consider the d+1 collections to be of d+1 different colors. Then the theorem says that, if every choice of one-set-per-color has a non-empty intersection, then there exists a color such that all sets of that color have a non-empty intersection.
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In general any skew diagram whose set of non-empty rows (or of non-empty columns) is not contiguous or does not contain the first row (respectively column) will be associated to more than one skew shape. therefore the shape of a skew diagram cannot always be determined from the set of filled squares only. Although many properties of skew tableaux only depend on the filled squares, some operations defined on them do require explicit knowledge of and , so it is important that skew tableaux do record this information: two distinct skew tableaux may differ only in their shape, while they occupy the same set of squares, each filled with the same entries.
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Syntax is centered around a line- oriented design, similar to that of Python. The structure of a file is defined using whitespace and other control characters. `#` is used as the line-comment character, and can be placed anywhere in a file. Instructions are any non- empty and non-comment line.
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We use induction on . If is empty, then the theorem is vacuously true and the base case for induction is verified. Assume is non-empty, let be an element of and write If is any -linear transformation on , by the induction hypothesis there exists such that for all in . Write .
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Remove a directory (delete a directory); by default the directories must be empty of files for the command to succeed. The command is available in MS-DOS versions 2 and later. The deltree command in some versions of MS-DOS and all versions of Windows 9x removes non-empty directories.
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A bounding volume for a set of objects is also a bounding volume for the single object consisting of their union, and the other way around. Therefore, it is possible to confine the description to the case of a single object, which is assumed to be non-empty and bounded (finite).
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In mathematics, the Ellis–Numakura lemma states that if S is a non-empty semigroup with a topology such that S is compact and the product is semi- continuous, then S has an idempotent element p, (that is, with pp = p). The lemma is named after Robert Ellis and Katsui Numakura.
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Another approach is to use Sturges' rule: use a bin so large that there are about 1+\log_2n non-empty bins (Scott, 2009). This works well for n under 200, but was found to be inaccurate for large n. For a discussion and an alternative approach, see Birgé and Rozenholc.
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Starting with only the weak topology, we may obtain a range of locally convex topologies by using polar sets. Such topologies are called polar topologies. The weak topology is the weakest topology of this range. Throughout, will be a pairing over and 𝒢 will be a non-empty collection of -bounded subsets of .
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Let be a set in a real or complex vector space. is star convex (star- shaped) if there exists an in such that the line segment from to any point in is contained in . Hence a non-empty convex set is always star-convex but a star-convex set is not always convex.
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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.
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Throughout this section we will assume that and are topological vector spaces. will be a non-empty collection of subsets of directed by inclusion. :Notation: will denote the vector space of all continuous linear maps from into . If is given the -topology inherited from then this space with this topology is denoted by .
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An event driven architecture may be built on four logical layers, starting with the sensing of an event (i.e., a significant temporal state or fact), proceeding to the creation of its technical representation in the form of an event structure and ending with a non-empty set of reactions to that event.
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Every non-empty totally ordered set is directed. Therefore, every function on such a set is a net. In particular, the natural numbers with the usual order form such a set, and a sequence is a function on the natural numbers, so every sequence is a net. Another important example is as follows.
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A quotient space is defined as follows: if X is a topological space and Y is a set, and if f : X→ Y is a surjective function, then the quotient topology on Y is the collection of subsets of Y that have open inverse images under f. In other words, the quotient topology is the finest topology on Y for which f is continuous. A common example of a quotient topology is when an equivalence relation is defined on the topological space X. The map f is then the natural projection onto the set of equivalence classes. The Vietoris topology on the set of all non-empty subsets of a topological space X, named for Leopold Vietoris, is generated by the following basis: for every n-tuple U1, ..., Un of open sets in X, we construct a basis set consisting of all subsets of the union of the Ui that have non-empty intersections with each Ui. The Fell topology on the set of all non-empty closed subsets of a locally compact Polish space X is a variant of the Vietoris topology, and is named after mathematician James Fell.
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In this case, the committee must be an independent transversal, where the underlying graph describes the "dislike" relations. Another generalization of the concept of a transversal would be a set that just has a non-empty intersection with each member of C. An example of the latter would be a Bernstein set, which is defined as a set that has a non-empty intersection with each set of C, but contains no set of C, where C is the collection of all perfect sets of a topological Polish space. As another example, let C consist of all the lines of a projective plane, then a blocking set in this plane is a set of points which intersects each line but contains no line.
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A property of an irreducible algebraic variety X is said to be true generically if it holds except on a proper Zariski-closed subset of X, in other words, if it holds on a non-empty Zariski-open subset. This definition agrees with the topological one above, because for irreducible algebraic varieties any non-empty open set is dense. For example, by the Jacobian criterion for regularity, a generic point of a variety over a field of characteristic zero is smooth. (This statement is known as generic smoothness.) This is true because the Jacobian criterion can be used to find equations for the points which are not smooth: They are exactly the points where the Jacobian matrix of a point of X does not have full rank.
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A graph H is the clique graph K(G) of another graph if and only if there exists a collection C of cliques in H whose union covers all the edges of H, such that C forms a Helly family. This means that, if S is a subset of C with the property that every two members of S have a non-empty intersection, then S itself should also have a non-empty intersection. However, the cliques in C do not necessarily have to be maximal cliques. When H =K(G), a family C of this type may be constructed in which each clique in C corresponds to a vertex v in G, and consists of the cliques in G that contain v.
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In combinatorics, a Helly family of order k is a family of sets such that any minimal subfamily with an empty intersection has k or fewer sets in it. Equivalently, every finite subfamily such that every k-fold intersection is non-empty has non-empty total intersection.. The k-Helly property is the property of being a Helly family of order k.. See in particular Section 2.5, "Helly Property", pp. 393–394. The number k is frequently omitted from these names in the case that k = 2\. Thus, a set-family has the Helly property if for every n sets s_1,\ldots,s_n in the family, if \forall i,j\in[n]: s_i \cap s_j eq\emptyset , then s_1 \cap \cdots \cap s_n eq\emptyset .
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Thus the central operations of lattices are binary suprema \vee and infima It is in this context that the terms meet for \wedge and join for \vee are most common. A poset in which only non-empty finite suprema are known to exist is therefore called a join-semilattice. The dual notion is meet-semilattice.
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Also note that the empty set usually has upper bounds (if the poset is non- empty) and thus a bounded-complete poset has a least element. One may also consider the subsets of a poset which are totally ordered, i.e. the chains. If all chains have a supremum, the order is called chain complete.
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The computational complexity depends on the algorithm used to sort each bucket, the number of buckets to use, and whether the input is uniformly distributed. Bucket sort works as follows: # Set up an array of initially empty "buckets". # Scatter: Go over the original array, putting each object in its bucket. # Sort each non-empty bucket.
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A partition of a set X is any collection of mutually disjoint non-empty sets whose union is X., p. 28. Every partition can equivalently be described by an equivalence relation, a binary relation that describes whether two elements belong to the same set in the partition. Disjoint-set data structures. and partition refinement.
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In adding a phrase to the dictionary there is a difference to the character-based version. The phrase from the next step will be called S1. If and S1 are both non-empty syllables, then we add a new phrase to the dictionary. The new phrase is created by the concatenation of S1 with the first syllable of .
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Thus, an n-ary group that is not reducible does not contain such elements. There exist n-ary groups with more than one neutral element. If the set of all neutral elements of an n-ary group is non-empty it forms an n-ary subgroup.Wiesław A. Dudek, Remarks to Głazek's results on n-ary groups, Discussiones Mathematicae.
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The difficulty appears when there is no natural choice of elements from each set. If we cannot make explicit choices, how do we know that our set exists? For example, suppose that X is the set of all non-empty subsets of the real numbers. First we might try to proceed as if X were finite.
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It then allow to influence the portion of capacity received by each queue/task. In computer networks, a service opportunity is the emission of one packet, if the selected queue is non empty. If all packets have the same size, WRR is the simplest approximation of generalized processor sharing (GPS). There exists several variations of WRR.
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In mathematics, a semigroup with no elements (the empty semigroup) is a semigroup in which the underlying set is the empty set. Many authors do not admit the existence of such a semigroup. For them a semigroup is by definition a non-empty set together with an associative binary operation.A H Clifford, G B Preston (1964).
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In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. The concept is essentially analogous to the concept of "almost everywhere" in measure theory.
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A homeomorphism of Rn is called stable if it is a product of homeomorphisms each of which is the identity on some non-empty open set. The stable homeomorphism conjecture states that every orientation-preserving homeomorphism of Rn is stable. previously showed that the stable homeomorphism conjecture is equivalent to the annulus conjecture, so it is true.
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For example, if X = (0, 1) and for each positive integer i, Xi is the set of elements of X having a decimal expansion with digit 0 in the ith decimal place, then any finite intersection is non- empty (just take 0 in those finitely many places and 1 in the rest), but the intersection of all Xi for i ≥ 1 is empty, since no element of (0, 1) has all zero digits. The finite intersection property is useful in formulating an alternative definition of compactness: a space is compact if and only if every collection of closed sets having the finite intersection property has non- empty intersection. This formulation of compactness is used in some proofs of Tychonoff's theorem and the uncountability of the real numbers (see next section).
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Each set in the countable sequence of sets (Si) = S1, S2, S3, ... contains a nonzero, and possibly infinite (or even uncountably infinite), number of elements. The axiom of countable choice allows us to arbitrarily select a single element from each set, forming a corresponding sequence of elements (xi) = x1, x2, x3, ... The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. I.e., given a function A with domain N (where N denotes the set of natural numbers) such that A(n) is a non-empty set for every n ∈ N, then there exists a function f with domain N such that f(n) ∈ A(n) for every n ∈ N.
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In automata theory, the class of unrestricted grammars (also called semi-Thue, type-0 or phrase structure grammars) is the most general class of grammars in the Chomsky hierarchy. No restrictions are made on the productions of an unrestricted grammar, other than each of their left-hand sides being non- empty. This grammar class can generate arbitrary recursively enumerable languages.
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Theorem: A real Banach space is reflexive if and only if every pair of non- empty disjoint closed convex subsets, one of which is bounded, can be strictly separated by a hyperplane. James' theorem: A Banach space B is reflexive if and only if every continuous linear functional on B attains its supremum on the closed unit ball in B.
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However, this is not necessary: see "Properties" below. An equivalent definition of support is as the largest C ∈ B(T) (with respect to inclusion) such that every open set which has non-empty intersection with C has positive measure, i.e. the largest C such that: :(\forall U \in T)(U \cap C eq \varnothing \implies \mu (U \cap C) > 0).
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A open neighbourhood of a subset of is the any set such that . A neighbourhood of in is subset such that contains open neighborhood of . Explicitly, this means that is a neighbourhood of in if and only if there is some open set such that . The neighbourhood system for any non-empty set is a filter called the neighbourhood filter for .
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Quantum mechanics can be used to describe spacetime as being non-empty at extremely small scales, fluctuating and generating particle pairs that appear and disappear incredibly quickly. It has been suggested by some such as Paul DiracDirac, Paul: "Is there an Aether?", Nature 168 (1951), p. 906. that this quantum vacuum may be the equivalent in modern physics of a particulate aether.
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Python 2.2 and earlier does not have an explicit boolean type. In all versions of Python, boolean operators treat zero values or empty values such as `""`, `0`, `None`, `0.0`, `[]`, and `{}` as false, while in general treating non-empty, non-zero values as true. In Python 2.2.1 the boolean constants `True` and `False` were added to the language (subclassed from 1 and 0).
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Now take two non- empty sets of distinct non-consecutive Fibonacci numbers and which have the same sum. Consider sets and which are equal to and from which the common elements have been removed (i.e. and ). Since and had equal sum, and we have removed exactly the elements from \cap from both sets, and must have the same sum as well.
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Suppose has more than one point, is a constant map, and then will consist of all non-empty subsets of . Although doesn't preserve properties of filters very well, if is downward closed (resp. closed under finite unions, an ideal) then this will also be true for . Using the duality between ideals and dual ideals allows for a construction of the following filter.
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A matroid on a set Q is specified by a class of non-empty subsets M of Q, called circuits, such that no element of M contains another, and if X and Y are in M, a is in X and Y, b is in X but not in Y, then there is some Z in M containing b but not a and contained in X∪Y. The subsets of Q that are unions of circuits are called flats. The elements of M are called 0-flats, the minimal non-empty flats that are not 0-flats are called 1-flats, the minimal nonempty flats that are not 0-flats or 1-flats are called 2-flats, and so on. A path is a finite sequence of 0-flats such that any two consecutive elements of the path lie in some 1-flat.
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Therefore, Ord is a proper class. So von Neumann's axiom implies that there is a function F that maps Ord onto V. To define a well-ordering of V, let G be the subclass of F consisting of the ordered pairs (α, x) where α is the least β such that (β, x) ∈ F; that is, G = {(α, x) ∈ F: ∀β((β, x) ∈ F ⇒ α ≤ β)}. The function G is a one-to-one correspondence between a subset of Ord and V. Therefore, x < y if G−1(x) < G−1(y) defines a well-ordering of V. This well-ordering defines a global choice function: Let Inf(x) be the least element of a non-empty set x. Since Inf(x) ∈ x, this function chooses an element of x for every non-empty set x.
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We consider several bipartite graphs with Y = {1, 2} and X = {A, B; a, b, c}. The Meshulam condition trivially holds for the empty set. It holds for subsets of size 1 iff the neighbor-graph of each vertex in Y is non-empty (so it requires at least one explosion to destroy), which is easy to check. It remains to check the subset Y itself.
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To prove that Tychonoff's theorem in its general version implies the axiom of choice, we establish that every infinite cartesian product of non-empty sets is nonempty. The trickiest part of the proof is introducing the right topology. The right topology, as it turns out, is the cofinite topology with a small twist. It turns out that every set given this topology automatically becomes a compact space.
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The running time is O(1) (amortized). The extract-min operation removes an element from bucket b[0] and returns it. If the bucket b[0] is not yet empty, the operation is terminated. If, however, it is empty, the next larger non-empty bucket is searched, its smallest element k tracked and u[0] is set to k (monotonicity is required for this).
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An empty graph is a graph that has an empty set of vertices (and thus an empty set of edges). The order of a graph is its number of vertices . The size of a graph is its number of edges . However, in some contexts, such as for expressing the computational complexity of algorithms, the size is (otherwise, a non-empty graph could have a size 0).
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In mathematics, the Bourbaki–Witt theorem in order theory, named after Nicolas Bourbaki and Ernst Witt, is a basic fixed point theorem for partially ordered sets. It states that if X is a non-empty chain complete poset, and : f : X \to X such that : f (x) \geq x for all x, then f has a fixed point. Such a function f is called inflationary or progressive.
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Zorn's lemma can be used to show that every nontrivial ring R with unity contains a maximal ideal. In the terminology above, the set P consists of all (two-sided) ideals in R except R itself. Since R is non- trivial, the set P contains the trivial ideal {0}, and therefore P is non- empty. This set P is partially ordered by set inclusion.
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A path w.r.t. H is a sequence of edges such that the intersection of each one with the next one is non-empty and not contained in the nodes of H. A set of edges is connected w.r.t. H if, for each pair of its edges, there is a path w.r.t. H of which the two nodes are the first and the last edge.
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It follows that E = 1 + E+: in words, "a set is either empty or non-empty". Equations like this can be read as referring to a single structure, as well as to the entire collection of structures. The original definition of the species inspired three directions of investigation. \- On the categorical side, ones needs a larger frame to content both the product and coproduct.
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The virtual start time is the maximum between the previous virtual finish time of the same queue and the current instant. With a virtual finishing time of all candidate packets (i.e., the packets at the head of all non-empty flow queues) computed, fair queuing compares the virtual finishing time and selects the minimum one. The packet with the minimum virtual finishing time is transmitted.
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In general relativity, the Vaidya metric describes the non-empty external spacetime of a spherically symmetric and nonrotating star which is either emitting or absorbing null dusts. It is named after the Indian physicist Prahalad Chunnilal Vaidya and constitutes the simplest non-static generalization of the non-radiative Schwarzschild solution to Einstein's field equation, and therefore is also called the "radiating(shining) Schwarzschild metric".
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It was proved by John C. Oxtoby that a non-empty topological space X is a Baire space if and only if Player I has no winning strategy. A nonempty topological space X in which Player II has a winning strategy is called a Choquet space. (Note that it is possible that neither player has a winning strategy.) Thus every Choquet space is Baire.
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There are some variations of the definition of a Kleinian group: sometimes Kleinian groups are allowed to be subgroups of PSL(2, C).2 (PSL(2, C) extended by complex conjugations), in other words to have orientation reversing elements, and sometimes they are assumed to be finitely generated, and sometimes they are required to act properly discontinuously on a non-empty open subset of the Riemann sphere.
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A graph is said to be prime (with respect to splits) if it has no nontrivial splits. Two splits are said to cross if each side of one split has a non-empty intersection with each side of the other split. A split is called strong when it is not crossed by any other split. As a special case, every trivial split is strong.
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One of many ways to express the axiom of choice is to say that it is equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. The proof that this is equivalent to the statement of the axiom in terms of choice functions is immediate: one needs only to pick an element from each set to find a representative in the product. Conversely, a representative of the product is a set which contains exactly one element from each component. The axiom of choice occurs again in the study of (topological) product spaces; for example, Tychonoff's theorem on compact sets is a more complex and subtle example of a statement that is equivalent to the axiom of choice, and shows why the product topology may be considered the more useful topology to put on a Cartesian product.
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Its domain is the power set of A (with the empty set removed), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets. With this alternate notion of choice function, the axiom of choice can be compactly stated as :Every set has a choice function.. which is equivalent to :For any set A there is a function f such that for any non-empty subset B of A, f(B) lies in B. The negation of the axiom can thus be expressed as: :There is a set A such that for all functions f (on the set of non-empty subsets of A), there is a B such that f(B) does not lie in B.
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In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i.e. certain suprema or infima. Roughly speaking, these functions map the supremum/infimum of a set to the supremum/infimum of the image of the set. Depending on the type of sets for which a function satisfies this property, it may preserve finite, directed, non-empty, or just arbitrary suprema or infima.
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A topological space X is said to be disconnected if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice.
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This is the case c>s=0, the case without saddles. Theorem:. Let M^n be a closed oriented connected manifold of dimension n\ge 2. Assume that M^n admits a C^1-transversely oriented codimension one foliation F with a non empty set of singularities all of them centers. Then the singular set of F consists of two points and M^n is homeomorphic to the sphere S^n.
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An infinite subset of the natural numbers cannot be bounded from above. An infinite subset of the integers may be bounded from below or bounded from above, but not both. An infinite subset of the rational numbers may or may not be bounded from below, and may or may not be bounded from above. Every finite subset of a non-empty totally ordered set has both upper and lower bounds.
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A graph is trivially a quotient graph of itself (each block of the partition is a single vertex), and the graph consisting of a single point is the quotient graph of any non-empty graph (the partition consisting of a single block of all vertices). The simplest non-trivial quotient graph is one obtained by identifying two vertices (vertex identification); if the vertices are connected, this is called edge contraction.
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One of the widely used convex optimization algorithms is projections onto convex sets (POCS). This algorithm is employed to recover/synthesize a signal satisfying simultaneously several convex constraints. Let f_i be the indicator function of non-empty closed convex set C_i modeling a constraint. This reduces to convex feasibility problem, which require us to find a solution such that it lies in the intersection of all convex sets C_i.
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One can strengthen the results by putting additional restrictions on the manifold. For example, the -sphere always embeds in – which is the best possible (closed -manifolds cannot embed in ). Any compact orientable surface and any compact surface with non-empty boundary embeds in , though any closed non-orientable surface needs . If is a compact orientable -dimensional manifold, then embeds in (for not a power of 2 the orientability condition is superfluous).
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A vector subspace of a Riesz space is called an ideal if it is solid, meaning if for and , we have: implies that . The intersection of an arbitrary collection of ideals is again an ideal, which allows for the definition of a smallest ideal containing some non-empty subset of , and is called the ideal generated by . An Ideal generated by a singleton is called a principal ideal.
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Each kind is detailed in its respective article, this one serving as a description of relations between them. A common property of all three kinds is that they describe coefficients relating three different sequences of polynomials that frequently arise in combinatorics. Moreover, all three can be defined as the number of partitions of n elements into k non- empty subsets, with different ways of counting orderings within each subset.
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In general, the Hausdorff distance d_H(S,T) can be infinite. Two sets are close to each other in the Hausdorff distance if every element of either set is close to some element of the other set. The Hausdorff distance d_H turns the set K(M) of all non-empty compact subsets of M into a metric space. One can show that K(M) is complete if M is complete.
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In general topology, a branch of mathematics, a collection A of subsets of a set X is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of A is non-empty. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of A is infinite. A centered system of sets is a collection of sets with the finite intersection property.
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For this reason, one considers instead a smaller collection of privileged subsets of X. These subsets will be called the measurable sets. They are closed under operations that one would expect for measurable sets; that is, the complement of a measurable set is a measurable set and the countable union of measurable sets is a measurable set. Non-empty collections of sets with these properties are called σ-algebras.
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Crawlers usually perform some type of URL normalization in order to avoid crawling the same resource more than once. The term URL normalization, also called URL canonicalization, refers to the process of modifying and standardizing a URL in a consistent manner. There are several types of normalization that may be performed including conversion of URLs to lowercase, removal of "." and ".." segments, and adding trailing slashes to the non-empty path component.
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Ordered trees can be naturally encoded by finite sequences of natural numbers. Denote ω⁎ the set of all finite sequences of natural numbers. Then any non-empty subset of ω⁎ that is closed under taking prefixes gives rise to an ordered tree: take the prefix order for and the lexicographical order for . Conversely, for an ordered tree assign each node the sequence w(x) of sibling indices, i.e.
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Additionally, while a collection of less than two sets is trivially disjoint, as there are no pairs to compare, the intersection of a collection of one set is equal to that set, which may be non-empty. For instance, the three sets have an empty intersection but are not disjoint. In fact, there are no two disjoint sets in this collection. Also the empty family of sets is pairwise disjoint.
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In formal language theory, in particular in algorithmic learning theory, a class C of languages has finite thickness if every string is contained in at most finitely many languages in C. This condition was introduced by Dana Angluin as a sufficient condition for C being identifiable in the limit. (citeseer.ist.psu.edu); here: Condition 3, p.123 mid. Angluin's original requirement (every non-empty string set be contained in at most finitely many languages) is equivalent.
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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 .
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This Whitehead/Russell theorem is described in more modern language by . # Every surjective function from P(P(S)) onto itself is one-to-one. # (Alfred Tarski) Every non-empty family of subsets of S has a minimal element with respect to inclusion., demonstrated that his definition (which is also known as I-finite) is equivalent to Kuratowski's set-theoretical definition, which he then noted is equivalent to the standard numerical definition via the proof by .
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There are various proofs of this theorem, by either analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one real root. Because of this fact, theorems that hold for any algebraically closed field apply to C. For example, any non-empty complex square matrix has at least one (complex) eigenvalue.
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In the setting with empty domains allowed, the drinker paradox must be formulated as follows: A set P satisfies :\exists x\in P.\ [D(x) \rightarrow \forall y\in P.\ D(y)] \, if and only if it is non-empty. Or in words: :If and only if there is someone in the pub, there is someone in the pub such that, if he is drinking, then everyone in the pub is drinking.
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A Discrete Global Grid (DGG) is a mosaic which covers the entire Earth's surface. Mathematically it is a space partitioning: it consists of a set of non-empty regions that form a partition of the Earth's surface. In a usual grid-modeling strategy, to simplify position calculations, each region is represented by a point, abstracting the grid as a set of region-points. Each region or region-point in the grid is called a cell.
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Then the algorithm shrinks the edge between s and t to search for non s-t cuts. The minimum cut found in all phases will be the minimum weighted cut of the graph. A cut is a partition of the vertices of a graph into two non- empty, disjoint subsets. A minimum cut is a cut for which the size or weight of the cut is not larger than the size of any other cut.
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Let S be a semigroup, I and Λ non-empty sets and P a matrix indexed by I and Λ with entries pi,λ taken from S. Then the Rees matrix semigroup M(S; I, Λ; P) is the set I×S×Λ together with the multiplication :(i, s, λ)(j, t, μ) = (i, spλ,j t, μ). Rees matrix semigroups are an important technique for building new semigroups out of old ones.
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The paradox for the square can be strengthened as follows: : Any two bounded subsets of the Euclidean plane with non-empty interiors are equidecomposable with respect to the area- preserving affine maps. This has consequences concerning the problem of measure. As von Neumann notes, :"Infolgedessen gibt es bereits in der Ebene kein nichtnegatives additives Maß (wo das Einheitsquadrat das Maß 1 hat), dass [sic] gegenüber allen Abbildungen von A2 invariant wäre."On p.
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In this case, "select the smallest number" is a choice function. Even if infinitely many sets were collected from the natural numbers, it will always be possible to choose the smallest element from each set to produce a set. That is, the choice function provides the set of chosen elements. However, no choice function is known for the collection of all non-empty subsets of the real numbers (if there are non- constructible reals).
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Some examples of non-paracompact manifolds in higher dimensions include the Prüfer manifold, products of any non-paracompact manifold with any non-empty manifold, the ball of long radius, and so on. The bagpipe theorem shows that there are 2ℵ1 isomorphism classes of non-paracompact surfaces. There are no complex analogues of the long line as every Riemann surface is paracompact, but gave an example of a non-paracompact complex manifold of complex dimension 2.
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The buffer works as a queuing system, where the data packets are stored temporarily until they are transmitted. With a link data-rate of R, at any given time the N active data flows (the ones with non-empty queues) are serviced each with an average data rate of R/N. In a short time interval the data rate may fluctuate around this value since the packets are delivered sequentially in turn.
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In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line. Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and complete, i.e., which "lacks gaps" in the sense that every non-empty subset with an upper bound has a least upper bound.
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This axiomatization is due to Felix Hausdorff. Let X be a set; the elements of X are usually called points, though they can be any mathematical object. We allow X to be empty. Let N be a function assigning to each x (point) in X a non-empty collection N(x) of subsets of X. The elements of N(x) will be called neighbourhoods of x with respect to N (or, simply, neighbourhoods of x).
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A complication is that this multivariate mutual information (as well as the interaction information) can be positive, negative, or zero, which makes this quantity difficult to interpret intuitively. In fact, for n random variables, there are 2^n-1 degrees of freedom for how they might be correlated in an information- theoretic sense, corresponding to each non-empty subset of these variables. These degrees of freedom are bounded by the various inequalities in information theory.
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At the same time, , so the Zeckendorf representation of does not contain . As a result, can be represented as the sum of and the Zeckendorf representation of . The second part of Zeckendorf's theorem (uniqueness) requires the following lemma: :Lemma: The sum of any non-empty set of distinct, non-consecutive Fibonacci numbers whose largest member is is strictly less than the next larger Fibonacci number . The lemma can be proven by induction on .
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If is a map then and . If then while if and are equivalent then . If and are principal then they are equivalent if and only if . ;Classifying families of sets by their kernels Family of examples: For any non-empty , the family is free but it is a filter subbase if and only if no finite union of the form covers , in which case the filter that it generates will also be free.
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In mathematical analysis, the word region usually refers to a subset of \R^n or \Complex^n that is open (in the standard Euclidean topology), simply connected and non-empty. A closed region is sometimes defined to be the closure of a region. Regions and closed regions are often used as domains of functions or differential equations. According to Kreyszig, :A region is a set consisting of a domain plus, perhaps, some or all of its boundary points.
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This gives rise to a number of useful categorical dualities between the categories of all complete semilattices with morphisms preserving all meets or joins, respectively. Another usage of "complete meet-semilattice" refers to a bounded complete cpo. A complete meet-semilattice in this sense is arguably the "most complete" meet-semilattice that is not necessarily a complete lattice. Indeed, a complete meet-semilattice has all non-empty meets (which is equivalent to being bounded complete) and all directed joins.
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A class of languages has finite thickness if every non- empty set of strings is contained in at most finitely many languages of the class. This is exactly Condition 3 in Angluin's paper.p.123 mid Angluin showed that if a class of recursive languages has finite thickness, then it is learnable in the limit.p.123 bot, Corollary 2 A class with finite thickness certainly satisfies MEF-condition and MFF-condition; in other words, finite thickness implies M-finite thickness.
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For any word in a dictionary that is formed by a non-empty prefix x and a suffix y, a GADDAG contains a direct, deterministic path for any string REV(x)+y, where + is a concatenation operator. For example, for the word "explain," a GADDAG will contain direct paths to the strings e+xplain xe+plain pxe+lain lpxe+ain alpxe+in ialpxe+n nialpxe This setup enables searching for a word given any letter that occurs in it.
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For example, lattice homomorphisms are those functions that preserve non-empty finite suprema and infima, i.e. the image of a supremum/infimum of two elements is just the supremum/infimum of their images. In domain theory, one often deals with so-called Scott-continuous functions that preserve all directed suprema. The background for the definitions and terminology given below is to be found in category theory, where limits (and co-limits) in a more general sense are considered.
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After identifying the required form, the original problem is reformulated into a master program and n subprograms. This reformulation relies on the fact that every point of a non-empty, bounded convex polyhedron can be represented as a convex combination of its extreme points. Each column in the new master program represents a solution to one of the subproblems. The master program enforces that the coupling constraints are satisfied given the set of subproblem solutions that are currently available.
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More generally, a partially ordered set has the least-upper-bound property if every non-empty subset of with an upper bound has a least upper bound (supremum) in . The least-upper-bound property is one form of the completeness axiom for the real numbers, and is sometimes referred to as Dedekind completeness.Willard says that an ordered space "X is Dedekind complete if every subset of X having an upper bound has a least upper bound." (pp.
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A subset C of a vector space X is called a cone if for all real r > 0, rC ⊆ C. A cone is called pointed if it contains the origin. A cone C is convex if and only if C + C ⊆ C. The intersection of any non-empty family of cones (resp. convex cones) is again a cone (resp. convex cone); the same is true of the union of an increasing (under set inclusion) family of cones (resp.
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If S is a stationary set and C is a club set, then their intersection S \cap C is also stationary. This is because if D is any club set, then C \cap D is a club set, thus (S \cap C) \cap D = S \cap (C \cap D) is non empty. Therefore, (S \cap C) must be stationary. See also: Fodor's lemma The restriction to uncountable cofinality is in order to avoid trivialities: Suppose \kappa has countable cofinality.
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By commutativity and associativity one can think of join and meet as operations on non-empty finite sets, rather than on pairs of elements. In a bounded lattice the join and meet of the empty set can also be defined (as 0 and 1, respectively). This makes bounded lattices somewhat more natural than general lattices, and many authors require all lattices to be bounded. The algebraic interpretation of lattices plays an essential role in universal algebra.
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As non-empty sets contain members while urelements do not, the unary relation is only needed to distinguish the empty set from urelements. Note that in this case, the axiom of extensionality must be formulated to apply only to objects that are not urelements. This situation is analogous to the treatments of theories of sets and classes. Indeed, urelements are in some sense dual to proper classes: urelements cannot have members whereas proper classes cannot be members.
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This definition makes \lor and \land binary operations. The first axiom says that L is a join-semilattice; the second says that L is a meet-semilattice. Both operations are monotone with respect to the order: a1 ≤ a2 and b1 ≤ b2 implies that a1 \lor b1 ≤ a2 \lor b2 and a1 \land b1 ≤ a2 \land b2. It follows by an induction argument that every non-empty finite subset of a lattice has a join (supremum) and a meet (infimum).
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The complete graph K4 has the ten matchings shown, so its Hosoya index is ten, the maximum for any four-vertex graph. The Hosoya index, also known as the Z index, of a graph is the total number of matchings in it. The Hosoya index is always at least one, because the empty set of edges is counted as a matching for this purpose. Equivalently, the Hosoya index is the number of non-empty matchings plus one.
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The Hahn–Mazurkiewicz theorem is the following characterization of spaces that are the continuous image of curves: :A non- empty Hausdorff topological space is a continuous image of the unit interval if and only if it is a compact, connected, locally connected, second-countable space. Spaces that are the continuous image of a unit interval are sometimes called Peano spaces. In many formulations of the Hahn–Mazurkiewicz theorem, second-countable is replaced by metrizable. These two formulations are equivalent.
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Here is the variant using only one condition variable and notifyAll: global volatile RingBuffer queue; // A thread-unsafe ring-buffer of tasks. global Lock queueLock; // A mutex for the ring-buffer of tasks. (Not a spin-lock.) global CV queueFullOrEmptyCV; // A single condition variable for when the queue is not ready for any thread // -- i.e., for producer threads waiting for the queue to become non-full // and consumer threads waiting for the queue to become non-empty.
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Some sources, including Kakutani's original paper, use the concept of upper hemicontinuity while stating the theorem: :Let S be a non-empty, compact and convex subset of some Euclidean space Rn. Let φ: S→2S be an upper hemicontinuous set-valued function on S with the property that φ(x) is non-empty, closed, and convex for all x ∈ S. Then φ has a fixed point. This statement of Kakutani's theorem is completely equivalent to the statement given at the beginning of this article. We can show this by using the closed graph theorem for set-valued functions, which says that for a compact Hausdorff range space Y, a set-valued function φ: X→2Y has a closed graph if and only if it is upper hemicontinuous and φ(x) is a closed set for all x. Since all Euclidean spaces are Hausdorff (being metric spaces) and φ is required to be closed-valued in the alternative statement of the Kakutani theorem, the Closed Graph Theorem implies that the two statements are equivalent.
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Formally, this reduction is executed via a log-space transducer. Such a machine has polynomially-many configurations, so log-space reductions are also polynomial-time reductions. However, log-space reductions are probably weaker than polynomial-time reductions; while any non- empty, non-full language in P is polynomial-time reducible to any other non- empty, non-full language in P, a log-space reduction from an NL-complete language to a language in L, both of which would be languages in P, would imply the unlikely L = NL. It is an open question if the NP-complete problems are different with respect to log-space and polynomial-time reductions. Log- space reductions are normally used on languages in P, in which case it usually does not matter whether many-one reductions or Turing reductions are used, since it has been verified that L, SL, NL, and P are all closed under Turing reductions, meaning that Turing reductions can be used to show a problem is in any of these classes.
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Suppose S is a compact connected oriented Riemann surface and M is a non-empty finite set of points in S that contains at least one point from each boundary component of S (the boundary of S is not assumed to be either empty or non- empty). The pair (S, M) is often referred to as a bordered surface with marked points. It has been shown by Fomin-Shapiro-Thurston that if S is not a closed surface, or if M has more than one point, then the (tagged) arcs on (S, M) parameterize the set of cluster variables of certain cluster algebra A(S, M), which depends only on (S, M) and the choice of some coefficient system, in such a way that the set of (tagged) triangulations of (S, M) is in one-to-one correspondence with the set of clusters of A(S, M), two (tagged) triangulations being related by a flip if and only if the clusters they correspond to are related by cluster mutation.
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Illustration of the axiom of choice, with each Si and xi represented as a jar and a colored marble, respectively family of sets indexed over the real numbers R; that is, there is a set Si for each real number i, with a small sample shown above. Each set contains at least one, and possibly infinitely many, elements. The axiom of choice allows us to arbitrarily select a single element from each set, forming a corresponding family of elements (xi) also indexed over the real numbers, with xi drawn from Si. In general, the collections may be indexed over any set I, not just R. In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite.
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In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right. It is named after Felix Hausdorff and Dimitrie Pompeiu. Informally, two sets are close in the Hausdorff distance if every point of either set is close to some point of the other set.
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To motivate the (conjectural) motivic Galois group, fix a field k and consider the functor :finite separable extensions K of k → non- empty finite sets with a (continuous) transitive action of the absolute Galois group of k which maps K to the (finite) set of embeddings of K into an algebraic closure of k. In Galois theory this functor is shown to be an equivalence of categories. Notice that fields are 0-dimensional. Motives of this kind are called Artin motives.
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Darboux functions are a quite general class of functions. It turns out that any real-valued function ƒ on the real line can be written as the sum of two Darboux functions.Bruckner, Andrew M: Differentiation of real functions, 2 ed, page 6, American Mathematical Society, 1994 This implies in particular that the class of Darboux functions is not closed under addition. A strongly Darboux function is one for which the image of every (non-empty) open interval is the whole real line.
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In geometry, the Chebyshev center of a bounded set Q having non-empty interior is the center of the minimal-radius ball enclosing the entire set Q, or alternatively (and non-equivalently) the center of largest inscribed ball of Q. In the field of parameter estimation, the Chebyshev center approach tries to find an estimator \hat x for x given the feasibility set Q , such that \hat x minimizes the worst possible estimation error for x (e.g. best worst case).
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For a graph G on n vertices, let e_k denote the number of colorings using exactly k colors up to renaming colors (so colorings that can be obtained from one another by permuting colors are counted as one; colorings obtained by automorphisms of G are still counted separately). In other words, e_k counts the number of partitions of the vertex set into k (non-empty) independent sets. Then k! \cdot e_k counts the number of colorings using exactly k colors (with distinguishable colors).
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On the other hand, the property of being empty is non-trivial, because the empty graph possesses this property, but non-empty graphs do not. A graph property is said to be monotone if the addition of edges does not destroy the property. Alternately, if a graph possesses a monotone property, then every supergraph of this graph on the same vertex set also possesses it. For instance, the property of being nonplanar is monotone: a supergraph of a nonplanar graph is itself nonplanar.
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The difference may seem subtle, but in many proofs that invoke Zorn's lemma one takes unions of some sort to produce an upper bound, and so the case of the empty chain may be overlooked; that is, the verification that all chains have upper bounds may have to deal with empty and non-empty chains separately. So many authors prefer to verify the non-emptiness of the set P rather than deal with the empty chain in the general argument.
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In this section, juxtaposed variables such as indicate the product , and the standard order of operations is assumed. A total order on the natural numbers is defined by letting if and only if there exists another natural number where . This order is compatible with the arithmetical operations in the following sense: if , and are natural numbers and , then and . An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element.
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However, in functional analysis we almost always want to be real-valued (i.e. to never take on the value of ), which happens if and only if the set } is non-empty for every . In order for to be real-valued, it suffices for the origin of to belong to the algebraic interior (or core) of in . If is absorbing in , where recall that this implies that , then the origin belongs to the algebraic interior of in and thus is real-valued.
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The following proof of the division theorem relies on the fact that a decreasing sequence of non-negative integers stops eventually. It is separated into two parts: one for existence and another for uniqueness of q and r. Other proofs use the well-ordering principle (i.e., the assertion that every non-empty set of non-negative integers has a smallest element) to make the reasoning simpler, but have the disadvantage of not providing directly an algorithm for solving the division (see for more).
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It isn't too hard to see that a manifold is an (n-1)-handlebody if and only if it has non- empty boundary. Any handlebody decomposition of a manifold defines a CW complex decomposition of the manifold, since attaching an r-handle is the same, up to homotopy equivalence, as attaching an r-cell. However, a handlebody decomposition gives more information than just the homotopy type of the manifold. For instance, a handlebody decomposition completely describes the manifold up to homeomorphism.
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Having introduced the usage of condition variables, let's use it to revisit and solve the classic bounded producer/consumer problem. The classic solution is to use two monitors, comprising two condition variables sharing one lock on the queue: global volatile RingBuffer queue; // A thread-unsafe ring-buffer of tasks. global Lock queueLock; // A mutex for the ring-buffer of tasks. (Not a spin- lock.) global CV queueEmptyCV; // A condition variable for consumer threads waiting for the queue to // become non-empty.
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By convention, the term shift is understood to refer to the full n-shift. A subshift is then any subspace of the full shift that is shift-invariant (that is, a subspace that is invariant under the action of the shift operator), non-empty, and closed for the product topology defined below. Some subshifts can be characterized by a transition matrix, as above; such subshifts are then called subshifts of finite type. Often, subshifts of finite type are called simply shifts of finite type.
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A sublattice M of a complete lattice L is called a complete sublattice of L if for every subset A of M the elements \bigwedge A and \bigvee A, as defined in L, are actually in M.Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. (A monograph available free online). If the above requirement is lessened to require only non-empty meet and joins to be in L, the sublattice M is called a closed sublattice of M.
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Thus, if one enlarges the group to allow arbitrary bijections of , then all sets with non- empty interior become congruent. Likewise, one ball can be made into a larger or smaller ball by stretching, or in other words, by applying similarity transformations. Hence, if the group is large enough, -equidecomposable sets may be found whose "size"s vary. Moreover, since a countable set can be made into two copies of itself, one might expect that using countably many pieces could somehow do the trick.
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In many specialized areas of order theory, one restricts to classes of partially ordered sets that are complete with respect to certain limit constructions. For example, in lattice theory, one is interested in orders where all finite non-empty sets have both a least upper bound and a greatest lower bound. In domain theory, on the other hand, one focuses on partially ordered sets in which every directed subset has a supremum. Complete lattices and orders with a least element (the "empty supremum") provide further examples.
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This results from the axiom of foundation – or the axiom of regularity – which enacts such a prohibition (cf. p. 190 in Being and Event). (This axiom states that every non-empty set A contains an element y that is disjoint from A.) Badiou's philosophy draws two major implications from this prohibition. Firstly, it secures the inexistence of the 'one': there cannot be a grand overarching set, and thus it is fallacious to conceive of a grand cosmos, a whole Nature, or a Being of God.
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In particular: # If the common intersection of all sets is not empty ( \bigcap X_i eq \emptyset), then obviously they cannot be partitioned to collections with disjoint unions. Hence the union of connected sets with non-empty intersection is connected. # If the intersection of each pair of sets is not empty (\forall i,j: X_i \cap X_j eq \emptyset) then again they cannot be partitioned to collections with disjoint unions, so their union must be connected. # If the sets can be ordered as a "linked chain", i.e.
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A semistable elliptic curve may be described more concretely as an elliptic curve that has bad reduction only of multiplicative type.Husemöller (1987) pp.116-117 Suppose E is an elliptic curve defined over the rational number field Q. It is known that there is a finite, non-empty set S of prime numbers p for which E has bad reduction modulo p. The latter means that the curve Ep obtained by reduction of E to the prime field with p elements has a singular point.
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Addition of species is defined by the disjoint union of sets, and corresponds to a choice between structures. For species F and G, define (F + G)[A] to be the disjoint union (also written "+") of F[A] and G[A]. It follows that (F + G)(x) = F(x) + G(x). As a demonstration, take E+ to be the species of non-empty sets, whose generating function is E+(x) = ex − 1, and 1 the species of the empty set, whose generating function is 1(x) = 1.
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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.
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Every contractible space is simply connected. ;Coproduct topology: If {Xi} is a collection of spaces and X is the (set-theoretic) disjoint union of {Xi}, then the coproduct topology (or disjoint union topology, topological sum of the Xi) on X is the finest topology for which all the injection maps are continuous. ;Cosmic space: A continuous image of some separable metric space. ;Countable chain condition: A space X satisfies the countable chain condition if every family of non-empty, pairswise disjoint open sets is countable.
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With additional assumptions, further conclusions may be possible; see Completeness (order theory) for more discussion of this subject. A bounded lattice has a greatest (or maximum) and least (or minimum) element, denoted 1 and 0 by convention (also called top and bottom). Any lattice can be converted into a bounded lattice by adding a greatest and least element, and every non- empty finite lattice is bounded, by taking the join (resp., meet) of all elements, denoted by \bigvee A=a_1\lor\cdots\lor a_n (resp.
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So M = M(\infty, \dots, \infty). We equip the space H of finite volume hyperbolic 3-manifolds with the geometric topology. Thurston's hyperbolic Dehn surgery theorem states: M(u_1, u_2, \dots, u_n) is hyperbolic as long as a finite set of exceptional slopes E_i is avoided for the i-th cusp for each i. In addition, M(u_1, u_2, \dots, u_n) converges to M in H as all p_i^2+q_i^2 \rightarrow \infty for all p_i/q_i corresponding to non-empty Dehn fillings u_i.
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The read permissions are needed to list the contents of the directory in order to delete them. This sometimes leads to an odd situation where a non-empty directory cannot be deleted because one doesn't have write permission to it and so cannot delete its contents; but if the same directory were empty, one would be able to delete it. If a file resides in a directory with the sticky bit set, then deleting the file requires one to be the owner of the file.
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In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A; that is, the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A -- for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, a subset A of a topological space X is dense in X if for any point x in X, any neighborhood of x contains at least one point from A (i.e., A has non-empty intersection with every non-empty open subset of X). Equivalently, A is dense in X if and only if the smallest closed subset of X containing A is X itself. This can also be expressed by saying that the closure of A is X, or that the interior of the complement of A is empty.
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Alternatively, a filter can be viewed as a "locating scheme": When trying to locate something (a point or a subset) in the space X, call a filter the collection of subsets of X that might contain "what is looked for". Then this "filter" should possess the following natural structure: #A locating scheme must be non-empty in order to be of any use at all. #If two subsets, E and F, both might contain "what is looked for", then so might their intersection. Thus the filter should be closed with respect to finite intersection.
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Meet and join are binary operations on S. A simple induction argument shows that the existence of all possible pairwise suprema (infima), as per the definition, implies the existence of all non- empty finite suprema (infima). A join-semilattice is bounded if it has a least element, the join of the empty set. Dually, a meet-semilattice is bounded if it has a greatest element, the meet of the empty set. Other properties may be assumed; see the article on completeness in order theory for more discussion on this subject.
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At each iteration, a new particle is drawn from the previous (weighted) particle set with probability proportional to its weight. Instead of the resampling done in classic MCL, the KLD–sampling algorithm draws particles from the previous, weighted, particle set and applies the motion and sensor updates before placing the particle into its bin. The algorithm keeps track of the number of non-empty bins, k. If a particle is inserted in a previously empty bin, the value of M_x is recalculated, which increases mostly linear in k.
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In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X is a connected set if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is locally connected, which neither implies nor follows from connectedness.
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Thus when defining formal Laurent series, one requires Laurent series with only finitely many negative terms. Similarly, the sum of two convergent Laurent series need not converge, though it is always defined formally, but the sum of two bounded below Laurent series (or any Laurent series on a punctured disk) has a non- empty annulus of convergence. Also, for a field F, by the sum and multiplication defined above, formal Laurent series would form a field F((x)) which is also the field of fractions of the ring Fx of formal power series.
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In mathematical logic, an uninterpreted function or function symbol is one that has no other property than its name and n-ary form. Function symbols are used, together with constants and variables, to form terms. The theory of uninterpreted functions is also sometimes called the free theory, because it is freely generated, and thus a free object, or the empty theory, being the theory having an empty set of sentences (in analogy to an initial algebra). Theories with a non-empty set of equations are known as equational theories.
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In relational database theory, a tuple-generating dependency (TGD) is a certain kind of constraint on a relational database. It is a subclass of the class of embedded dependencies (EDs). A TGD is a sentence in first-order logic of the form: ∀x1 ... xn, P(x1, ..., xn) → ∃y1, ..., ym, Q(x1, ..., xn,y1, ..., ym), where P is a possibly empty and Q is a non-empty conjunction of relational atoms. A relational atom has the form R(w1, ..., wh) where each of the w, ..., wh, wi, wj, are variables or constants.
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The interval enclosure of a subset X\subseteq \R is also the convex hull of X. The intersection of any collection of intervals is always an interval. The union of two intervals is an interval if and only if they have a non-empty intersection or an open end- point of one interval is a closed end-point of the other (e.g., (a,b) \cup [b,c] = (a,c]). If \R is viewed as a metric space, its open balls are the open bounded sets , and its closed balls are the closed bounded sets .
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If the p_i have a common zero (sometimes called base point), every irreducible component of the non-empty algebraic set defined by the p_i is an irreducible component of the algebraic set defined by I. It follows that, in this case, the direct elimination of the t_i provides an empty set of polynomials. Therefore, if k>1, two Gröbner basis computations are needed to implicitize: # Saturate I by p_0 to get a Gröbner basis G # Eliminate the t_i from G to get a Gröbner basis of the ideal (of the implicit equations) of the variety.
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Thus the proof theory of first-order logic becomes more complicated when empty structures are permitted. However, the gain in allowing them is negligible, as both the intended interpretations and the interesting interpretations of the theories people study have non-empty domains. Empty relations do not cause any problem for first-order interpretations, because there is no similar notion of passing a relation symbol across a logical connective, enlarging its scope in the process. Thus it is acceptable for relation symbols to be interpreted as being identically false.
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The empty set cannot belong to any collection with the finite intersection property. The condition is trivially satisfied if the intersection over the entire collection is non-empty (in particular, if the collection itself is empty), and it is also trivially satisfied if the collection is nested, meaning that the collection is totally ordered by inclusion (equivalently, for any finite subcollection, a particular element of the subcollection is contained in all the other elements of the subcollection), e.g. the nested sequence of intervals (0, 1/n). However, these are not the only possibilities.
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Continue this process whereby choosing a neighbourhood Un+1 ⊂ Un whose closure does not contain xn+1. Then the collection {Ui : i ∈ N} satisfies the finite intersection property and hence the intersection of their closures is non-empty by the compactness of X. Therefore, there is a point x in this intersection. No xi can belong to this intersection because xi does not belong to the closure of Ui. This means that x is not equal to xi for all i and f is not surjective; a contradiction. Therefore, X is uncountable.
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Second, the graph theory term digraph (a portmanteau from directed graph) is defined as, "A graph in which each line has a direction associated with it; a finite, non-empty set of elements together with a set of ordered pairs of these elements." The two digraph terms were first recorded in 1788 and 1955, respectively. The OED2 defines two digraphic meanings, "Pertaining to or of the nature of a digraph" and "Written in two different characters or alphabets." It gives their earliest examples in 1873 and 1880 (which was used meaning "digraphia").
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Fisher's inequality is valid for more general classes of designs. A pairwise balanced design (or PBD) is a set together with a family of non-empty subsets of (which need not have the same size and may contain repeats) such that every pair of distinct elements of is contained in exactly (a positive integer) subsets. The set is allowed to be one of the subsets, and if all the subsets are copies of , the PBD is called "trivial". The size of is and the number of subsets in the family (counted with multiplicity) is .
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It is also equal to the closure of the convex hull of and to the intersection of all closed convex subsets that contain . It is straightforward to show that the convex hull of the extreme points forms a subset of , so the main burden of the proof is to show that there are enough extreme points so that their convex hull covers all of . As a corollary, it follows that every non-empty compact convex subset of a Hausdorff locally convex TVS has extreme points (i.e. the set of its extreme points is not empty).
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An example of an NP-hard problem is the decision subset sum problem: given a set of integers, does any non-empty subset of them add up to zero? That is a decision problem and happens to be NP-complete. Another example of an NP-hard problem is the optimization problem of finding the least-cost cyclic route through all nodes of a weighted graph. This is commonly known as the traveling salesman problem.. There are decision problems that are NP-hard but not NP-complete such as the halting problem.
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In this case, `Nil` only matches the literal object `Nil`, but `pivot :: tail` matches a non-empty list, and simultaneously destructures the list according to the pattern given. In this case, the associated code will have access to a local variable named `pivot` holding the head of the list, and another variable `tail` holding the tail of the list. Note that these variables are read-only, and are semantically very similar to variable bindings established using the `let` operator in Lisp and Scheme. Pattern matching also happens in local variable declarations.
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The intersection of any two doubly infinite arithmetic progressions is either empty or another arithmetic progression, which can be found using the Chinese remainder theorem. If each pair of progressions in a family of doubly infinite arithmetic progressions have a non-empty intersection, then there exists a number common to all of them; that is, infinite arithmetic progressions form a Helly family.. See in particular Section 2.5, "Helly Property", pp. 393–394. However, the intersection of infinitely many infinite arithmetic progressions might be a single number rather than itself being an infinite progression.
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In probability experiments on a finite sample space, there is often no difference between almost surely and surely (since having a probability of 1 often entails including all the sample points). However, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, and the continuity of the paths of Brownian motion. The terms almost certainly (a.
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Morton and Hilbert curves of level 6 (45=1024 cells in the recursive square partition) plotting each address as different color in the RGB standard, and using Geohash labels. The neighborhoods have similar colors, but each curve offers different pattern of grouping similars in smaller scales. If a curve is not injective, then one can find two intersecting subcurves of the curve, each obtained by considering the images of two disjoint segments from the curve's domain (the unit line segment). The two subcurves intersect if the intersection of the two images is non-empty.
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In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also ). Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem.
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That means that if one were to partition the rationals into two non-empty sets A and B where A contains all rationals less than some irrational number (π, say) and B all rationals greater than it, then A has no largest member and B has no smallest member. The field of real numbers, by contrast, is both infinitely divisible and gapless. Any linearly ordered set that is infinitely divisible and gapless, and has more than one member, is uncountably infinite. For a proof, see Cantor's first uncountability proof.
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The Seifert manifold is often allowed to have a boundary (also fibered by circles, so it is a union of tori). When studying non-orientable manifolds, it is sometimes useful to allow fibers to have neighborhoods that look like the surface bundle of a reflection (rather than a rotation) of a disk, so that some fibers have neighborhoods looking like fibered Klein bottles, in which case there may be one-parameter families of exceptional curves. In both of these cases, the base B of the fibration usually has a non-empty boundary.
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A sparse file: The empty bytes don't need to be saved, thus they can be represented by metadata. In computer science, a sparse file is a type of computer file that attempts to use file system space more efficiently when the file itself is partially empty. This is achieved by writing brief information (metadata) representing the empty blocks to disk instead of the actual "empty" space which makes up the block, using less disk space. The full block size is written to disk as the actual size only when the block contains "real" (non- empty) data.
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Sierpiński space, then Scott-continuous functions are characteristic functions, and thus Sierpiński space is the classifying topos for open sets. A subset O of a partially ordered set P is called Scott-open if it is an upper set and if it is inaccessible by directed joins, i.e. if all directed sets D with supremum in O have non-empty intersection with O. The Scott-open subsets of a partially ordered set P form a topology on P, the Scott topology. A function between partially ordered sets is Scott-continuous if and only if it is continuous with respect to the Scott topology.
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Based on the fact that some limits can be expressed in terms of others, one can derive connections between the preservation properties. For example, a function f preserves directed suprema if and only if it preserves the suprema of all ideals. Furthermore, a mapping f from a poset in which every non-empty finite supremum exists (a so-called sup-semilattice) preserves arbitrary suprema if and only if it preserves both directed and finite (possibly empty) suprema. However, it is not true that a function that preserves all suprema would also preserve all infima or vice versa.
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The geometric mean of a non-empty data set of (positive) numbers is always at most their arithmetic mean. Equality is only obtained when all numbers in the data set are equal; otherwise, the geometric mean is smaller. For example, the geometric mean of 242 and 288 equals 264, while their arithmetic mean is 265. In particular, this means that when a set of non-identical numbers is subjected to a mean-preserving spread — that is, the elements of the set are "spread apart" more from each other while leaving the arithmetic mean unchanged — their geometric mean decreases.
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Conversely, every injection f with non-empty domain has a left inverse g, which can be defined by fixing an element a in the domain of f so that g(x) equals the unique preimage of x under f if it exists and g(x) = a otherwise. The left inverse g is not necessarily an inverse of f, because the composition in the other order, , may differ from the identity on Y. In other words, an injective function can be "reversed" by a left inverse, but is not necessarily invertible, which requires that the function is bijective.
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The number of abstract simplicial complexes on up to n elements (that is on a set S of size n) is one less than the nth Dedekind number. These numbers grow very rapidly, and are known only for ; they are (starting with n = 0): :1, 2, 5, 19, 167, 7580, 7828353, 2414682040997, 56130437228687557907787 . This corresponds to the number of non-empty antichains of subsets of an set. The number of abstract simplicial complexes whose vertices are exactly n labeled elements is given by the sequence "1, 2, 9, 114, 6894, 7785062, 2414627396434, 56130437209370320359966" , starting at n = 1.
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The LALR(j) parsers are incomparable with LL(k) parsers: for any j and k both greater than 0, there are LALR(j) grammars that are not LL(k) grammars and conversely. In fact, it is undecidable whether a given LL(1) grammar is LALR(k) for any k > 0. Depending on the presence of empty derivations, a LL(1) grammar can be equal to a SLR(1) or a LALR(1) grammar. If the LL(1) grammar has no empty derivations it is SLR(1) and if all symbols with empty derivations have non-empty derivations it is LALR(1).
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Ideals of a ring R are the submodules of R, i.e., the modules contained in R. In more detail, an ideal I is a non-empty subset of R such that for all r in R, i and j in I, both ri and i + j are in I. For various applications, understanding the ideals of a ring is of particular importance, but often one proceeds by studying modules in general. Any ring has two ideals, namely the zero ideal {0} and R, the whole ring. These two ideals are the only ones precisely if R is a field.
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A k-core of a graph G is a maximal connected subgraph of G in which all vertices have degree at least k. Equivalently, it is one of the connected components of the subgraph of G formed by repeatedly deleting all vertices of degree less than k. If a non- empty k-core exists, then, clearly, G has degeneracy at least k, and the degeneracy of G is the largest k for which G has a k-core. A vertex u has coreness c if it belongs to a c-core but not to any (c+1)-core.
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Given a set Rn of n-ary relation symbols for each natural number n ≥ 1, an (unsorted first-order) atomic formula is obtained by applying an n-ary relation symbol to n terms. As for function symbols, a relation symbol set Rn is usually non- empty only for small n. In mathematical logic, more complex formulas are built from atomic formulas using logical connectives and quantifiers. For example, letting ℝ denote the set of real numbers, ∀x: x ∈ ℝ ⇒ (x+1)⋅(x+1) ≥ 0 is a mathematical formula evaluating to true in the algebra of complex numbers.
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Firstly, a prefix sum operation is performed that calculates the boundaries of the buckets. However, since only full blocks are moved in this phase, the boundaries are rounded up to a multiple of the block size and a single overflow buffer is allocated. Before starting the block permutation, some empty blocks might have to be moved to the end of its bucket. Thereafter, a write pointer w_i is set to the start of the bucket b_i subarray for each bucket and a read pointer r_i is set to the last non empty block in the bucket b_i subarray for each bucket.
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All the implementations of von Neumann's self-reproducing machine require considerable resources to run on computer. For example, in the Nobili- Pesavento 32-state implementation shown above, while the body of the machine is just 6,329 non-empty cells (within a rectangle of size 97x170), it requires a tape that is 145,315 cells long, and takes 63 billion timesteps to replicate. A simulator running at 1,000 timesteps per second would take over 2 years to make the first copy. In 1995, when the first implementation was published, the authors had not seen their own machine replicate.
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And this formula gives the same truth value regardless of whether it is evaluated in , , or (some other standard model of ZF with the same ordinals) and we will suppose that the formula is false if either or is not in . It is well known that the axiom of choice is equivalent to the ability to well-order every set. Being able to well-order the proper class (as we have done here with ) is equivalent to the axiom of global choice, which is more powerful than the ordinary axiom of choice because it also covers proper classes of non-empty sets.
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Cuckoo hashing is a form of open addressing in which each non-empty cell of a hash table contains a key or key–value pair. A hash function is used to determine the location for each key, and its presence in the table (or the value associated with it) can be found by examining that cell of the table. However, open addressing suffers from collisions, which happen when more than one key is mapped to the same cell. The basic idea of cuckoo hashing is to resolve collisions by using two hash functions instead of only one.
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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.
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A torus. The standard torus is homogeneous under its diffeomorphism and homeomorphism groups, and the flat torus is homogeneous under its diffeomorphism, homeomorphism, and isometry groups. In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are called the symmetries of X. A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group.
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161 The basic principle is that each step of the algorithm reduces f inexorably; hence, if can be reduced only a finite number of times, the algorithm must stop in a finite number of steps. This principle relies on the well-ordering property of the non-negative integers, which asserts that every non-empty set of non-negative integers has a smallest member. The fundamental theorem of arithmetic applies to any Euclidean domain: Any number from a Euclidean domain can be factored uniquely into irreducible elements. Any Euclidean domain is a unique factorization domain (UFD), although the converse is not true.
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Attempts to axiomatize the empirical sciences, Carnap said, use a descriptive interpretation to model reality.: the aim of these attempts is to construct a formal system for which reality is the only interpretation.The Concept and the Role of the Model in Mathematics and Natural and Social Sciences \- the world is an interpretation (or model) of these sciences, only insofar as these sciences are true. Any non-empty set may be chosen as the domain of a descriptive interpretation, and all n-ary relations among the elements of the domain are candidates for assignment to any predicate of degree n.
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Any two distinct points in [-1,1] are topologically distinguishable under the overlapping interval topology as one can always find an open set containing one but not the other point. However, every non-empty open set contains the point 0 which can therefore not be separated from any other point in [-1,1], making [-1,1] with the overlapping interval topology an example of a T0 space that is not a T1 space. The overlapping interval topology is second countable, with a countable basis being given by the intervals [-1,s), (r,s) and (r,1] with r < 0 < s and r and s rational.
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As a simple example, in a function using division, the programmer may prove that the divisor will never equal zero, preventing a division by zero error. Let's say, the divisor 'X' was computed as 5 times the length of list 'A'. One can prove, that in the case of a non-empty list, 'X' is non-zero, since 'X' is the product of two non-zero numbers (5 and the length of 'A'). A more practical example would be proving through reference counting that the retain count on an allocated block of memory is being counted correctly for each pointer.
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Some instances of the smallest bounding circle, the case of the bounding sphere in 2 dimensions. In mathematics, given a non-empty set of objects of finite extension in d-dimensional space, for example a set of points, a bounding sphere, enclosing sphere or enclosing ball for that set is an d-dimensional solid sphere containing all of these objects. Used in computer graphics and computational geometry, a bounding sphere is a special type of bounding volume. There are several fast and simple bounding sphere construction algorithms with a high practical value in real-time computer graphics applications.
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The problem can be solved in pseudo-polynomial time using dynamic programming. Suppose the sequence is :x_1,\ldots, x_N sorted in the increasing order and we wish to determine if there is a nonempty subset which sums to zero. Define the boolean-valued function Q(i, s) to be the value (true or false) of :"there is a nonempty subset of x_1,\ldots, x_i which sums to s." Thus, the solution to the problem "Given a set of integers, is there a non-empty subset whose sum is zero?" is the value of Q(N, 0).
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A variant of this solution could use a single condition variable for both producers and consumers, perhaps named "queueFullOrEmptyCV" or "queueSizeChangedCV". In this case, more than one condition is associated with the condition variable, such that the condition variable represents a weaker condition than the conditions being checked by individual threads. The condition variable represents threads that are waiting for the queue to be non-full and ones waiting for it to be non-empty. However, doing this would require using notifyAll in all the threads using the condition variable and cannot use a regular signal.
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In the uniform matroid U_{0,n}, every element is a loop (an element that does not belong to any independent set), and in the uniform matroid U_{n,n}, every element is a coloop (an element that belongs to all bases). The direct sum of matroids of these two types is a partition matroid in which every element is a loop or a coloop; it is called a discrete matroid. An equivalent definition of a discrete matroid is a matroid in which every proper, non-empty subset of the ground set E is a separator.
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In statistics and computational geometry, the notion of centerpoint is a generalization of the median to data in higher-dimensional Euclidean space. Given a set of points in d-dimensional space, a centerpoint of the set is a point such that any hyperplane that goes through that point divides the set of points in two roughly equal subsets: the smaller part should have at least a 1/(d + 1) fraction of the points. Like the median, a centerpoint need not be one of the data points. Every non-empty set of points (with no duplicates) has at least one centerpoint.
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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).
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Google Profiles existed before Buzz and could be set by the user to be public or not. After Buzz was released, the last name field was required to be non empty and profiles set not to be indexed became indexed for a profile search. A 2010 New York Times article characterized Google as being "known for releasing new products before they are fully ready and then improving them over time". Google twice tried to address privacy concerns: first by making the option to disable public sharing of contact lists more prominent and later by changing one of Buzz's features from "auto-follow" to "auto-suggest".
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Every amorphous set is Dedekind-finite, meaning that it has no bijection to a proper subset of itself. To see this, suppose that S is a set that does have a bijection f to a proper subset. For each i ≥ 0 define Si to be the set of elements that belong to the image of the i-fold composition of f with itself but not to the image of the (i + 1)-fold composition. Then each Si is non- empty, so the union of the sets Si with even indices would be an infinite set whose complement is also infinite, showing that S cannot be amorphous.
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However, this existence theorem is purely theoretical, as such a base has never been explicitly described. The well-ordering theorem implies that the real numbers can be well-ordered if the axiom of choice is assumed: there exists a total order on R with the property that every non-empty subset of R has a least element in this ordering. (The standard ordering ≤ of the real numbers is not a well-ordering since e.g. an open interval does not contain a least element in this ordering.) Again, the existence of such a well-ordering is purely theoretical, as it has not been explicitly described.
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There are many other equivalent statements of the axiom of choice. These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it. One variation avoids the use of choice functions by, in effect, replacing each choice function with its range. :Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.. According to , this was the formulation of the axiom of choice which was originally given by .
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Until the late 19th century, the axiom of choice was often used implicitly, although it had not yet been formally stated. For example, after having established that the set X contains only non-empty sets, a mathematician might have said "let F(s) be one of the members of s for all s in X" to define a function F. In general, it is impossible to prove that F exists without the axiom of choice, but this seems to have gone unnoticed until Zermelo. Not every situation requires the axiom of choice. For finite sets X, the axiom of choice follows from the other axioms of set theory.
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Thurston's hyperbolic Dehn surgery theorem states: M(u_1, u_2, \dots, u_n) is hyperbolic as long as a finite set of exceptional slopes E_i is avoided for the i-th cusp for each i. In addition, M(u_1, u_2, \dots, u_n) converges to M in H as all p_i^2+q_i^2 \rightarrow \infty for all p_i/q_i corresponding to non-empty Dehn fillings u_i. This theorem is due to William Thurston and fundamental to the theory of hyperbolic 3-manifolds. It shows that nontrivial limits exist in H. Troels Jorgensen's study of the geometric topology further shows that all nontrivial limits arise by Dehn filling as in the theorem.
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BN-pairs can be used to prove that many groups of Lie type are simple modulo their centers. More precisely, if G has a BN-pair such that B is a solvable group, the intersection of all conjugates of B is trivial, and the set of generators of W cannot be decomposed into two non-empty commuting sets, then G is simple whenever it is a perfect group. In practice all of these conditions except for G being perfect are easy to check. Checking that G is perfect needs some slightly messy calculations (and in fact there are a few small groups of Lie type which are not perfect).
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Like polynomials, rational expressions can also be generalized to n indeterminates X1,..., Xn, by taking the field of fractions of F[X1,..., Xn], which is denoted by F(X1,..., Xn). An extended version of the abstract idea of rational function is used in algebraic geometry. There the function field of an algebraic variety V is formed as the field of fractions of the coordinate ring of V (more accurately said, of a Zariski-dense affine open set in V). Its elements f are considered as regular functions in the sense of algebraic geometry on non-empty open sets U, and also may be seen as morphisms to the projective line.
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Diophantine definitions can be provided by simultaneous systems of equations as well as by individual equations because the system :p_1=0,\ldots,p_k=0 is equivalent to the single equation :p_1^2+\cdots+p_k^2=0. Sets of natural numbers, of pairs of natural numbers (or even of n-tuples of natural numbers) that have Diophantine definitions are called Diophantine sets. In these terms, Hilbert's tenth problem asks whether there is an algorithm to determine if a given Diophantine set is non-empty. The work on the problem has been in terms of solutions in natural numbers (understood as the non-negative integers) rather than arbitrary integers.
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Let G be a linear algebraic group over a global field k, and A the adele ring of k. If S is a non-empty finite set of places of k, then we write AS for the ring of S-adeles and AS for the product of the completions ks, for s in the finite set S. For any choice of S, G(k) embeds in G(AS) and G(AS). The question asked in weak approximation is whether the embedding of G(k) in G(AS) has dense image. If the group G is connected and k-rational, then it satisfies weak approximation with respect to any set S .
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For every proper convex function f on Rn there exist some b in Rn and β in R such that :f(x) \ge x \cdot b - \beta for every x. The sum of two proper convex functions is convex, but not necessarily proper. For instance if the sets A \subset X and B \subset X are non-empty convex sets in the vector space X, then the characteristic functions I_A and I_B are proper convex functions, but if A \cap B = \emptyset then I_A + I_B is identically equal to +\infty. The infimal convolution of two proper convex functions is convex but not necessarily proper convex..
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The basic goal of the meld (also called merge) operation is to take two heaps (by taking each heaps root nodes), Q1 and Q2, and merges them, returning a single heap node as a result. This heap node is the root node of a heap containing all elements from the two subtrees rooted at Q1 and Q2. A nice feature of this meld operation is that it can be defined recursively. If either heaps are null, then the merge is taking place with an empty set and the method simply returns the root node of the non-empty heap. If both Q1 and Q2 are not nil, check if Q1 > Q2.
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A modeller could add a process to the model, by adding states disconnected and connected of Thing Set. The purpose of the model thus includes the action of transforming a disconnected Thing Set to a connected Thing Set using the Link Set as an instrument of connection. ; OPM model of Thing OPM model of Thing OPM model of Thing, is a model for an OPM Thing, showing its specialization into Object and Process, as depicted in the image of model of thing below. A set of States characterize Object, which can be empty, in a Stateless Object, or non-empty in the case of a Stateful Object.
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A group that satisfies the ascending chain condition on subgroups is called a Noetherian group, and a group that satisfies the descending chain condition is called an Artinian group (not to be confused with Artin groups), by analogy with Noetherian rings and Artinian rings. The ACC is equivalent to the maximal condition: every non-empty collection of subgroups has a maximal member, and the DCC is equivalent to the analogous minimal condition. A group can be Noetherian but not Artinian, such as the infinite cyclic group, and unlike for rings, a group can be Artinian but not Noetherian, such as the Prüfer group. Every finite group is clearly Noetherian and Artinian.
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In voting systems, the Smith set, named after John H. Smith, but also known as the top cycle, or as Generalized Top-Choice Assumption (GETCHA), is the smallest non-empty set of candidates in a particular election such that each member defeats every candidate outside the set in a pairwise election. The Smith set provides one standard of optimal choice for an election outcome. Voting systems that always elect a candidate from the Smith set pass the Smith criterion and are said to be "Smith-efficient". A set of candidates where every member of the set pairwise defeats every member outside of the set is known as a dominating set.
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The first shown partition contains five single-element subsets; the last partition contains one subset having five elements. The traditional Japanese symbols for the 54 chapters of the Tale of Genji are based on the 52 ways of partitioning five elements (the two red symbols represent the same partition, and the green symbol is added for reaching 54). In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation.
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A guarded bisimulation between two τ-structures A and B is a non-empty set I of finite partial isomorphic f: X → Y from A to B such that the back and forth conditions are satisfied. Back: For every f: X → Y in I and for every guarded set Y` ⊆ B, there exists a partial isomorphic g: X` → Y` in I such that f^-1 and g^-1 agree on Y ∩ Y`. Forth For every f: X → Y in I and for every guarded set X` ⊆ A, there exists a partial isomorphic g: X` → Y` in I such that f and g agree on X ∩ X`.
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Depending on definition, a tree may be required to have a root node (in which case all trees are non-empty), or may be allowed to be empty, in which case it does not necessarily have a root node. Being the topmost node, the root node will not have a parent. It is the node at which algorithms on the tree begin, since as a data structure, one can only pass from parents to children. Note that some algorithms (such as post-order depth-first search) begin at the root, but first visit leaf nodes (access the value of leaf nodes), only visit the root last (i.e.
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There is also the requirement that no two "downward" references point to the same node. In practice, nodes in a tree commonly include other data as well, such as next/previous references, references to their parent nodes, or nearly anything. Due to the use of to trees in the linked tree data structure, trees are often discussed implicitly assuming that they are being represented by references to the root node, as this is often how they are actually implemented. For example, rather than an empty tree, one may have a null reference: a tree is always non-empty, but a reference to a tree may be null.
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The subbase consisting of all semi-infinite open intervals of the form alone, where is a real number, does not generate the usual topology. The resulting topology does not satisfy the T1 separation axiom, since all open sets have a non-empty intersection. The initial topology on defined by a family of functions , where each has a topology, is the coarsest topology on such that each is continuous. Because continuity can be defined in terms of the inverse images of open sets, this means that the initial topology on is given by taking all , where ranges over all open subsets of , as a subbasis.
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Usually, on most filesystems, deleting a file requires write permission on the parent directory (and execute permission, in order to enter the directory in the first place). (Note that, confusingly for beginners, permissions on the file itself are irrelevant. However, GNU `rm` asks for confirmation if a write-protected file is to be deleted, unless the -f option is used.) To delete a directory (with `rm -r`), one must delete all of its contents recursively. This requires that one must have read and write and execute permission to that directory (if it's not empty) and all non-empty subdirectories recursively (if there are any).
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In particular, on a finite set , there are no free filter subbases (or prefilters), all prefilters are principal, and all filters on are principal filters generated by their (non- empty) kernels. The trivial filter is always a finite filter on and if is infinite then it is the only finite filter because a non-trivial finite filter on a set is possible if and only if is finite. However, on any infinite set there are non-trivial filter subbases and prefilters that are finite (although they cannot be filters). If is a singleton set then the trivial filter is the only proper subset of .
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The group action is transitive if and only if it has exactly one orbit, that is, if there exists x in X with . This is the case if and only if for all x in X (given that X is non-empty). The set of all orbits of X under the action of G is written as X/G (or, less frequently: G\X), and is called the quotient of the action. In geometric situations it may be called the ', while in algebraic situations it may be called the space of ', and written XG, by contrast with the invariants (fixed points), denoted XG: the coinvariants are a quotient while the invariants are a subset.
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Magmas are not often studied as such; instead there are several different kinds of magma, depending on what axioms the operation is required to satisfy. Commonly studied types of magma include: ;Quasigroup: A magma where division is always possible ;Loop: A quasigroup with an identity element ;Semigroup: A magma where the operation is associative ;Inverse semigroup: A semigroup with inverse. ;Semilattice: A semigroup where the operation is commutative and idempotent ;Monoid: A semigroup with an identity element ;Group: A monoid with inverse elements, or equivalently, an associative loop, or a non-empty associative quasigroup ;Abelian group: A group where the operation is commutative Note that each of divisibility and invertibility imply the cancellation property.
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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.
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Let i1, ..., iN be a finite collection of indices in I. Then the finite product Ai1 × ... × AiN is non-empty (only finitely many choices here, so AC is not needed); it merely consists of N-tuples. Let a = (a1, ..., aN) be such an N-tuple. We extend a to the whole index set: take a to the function f defined by f(j) = ak if j = ik, and f(j) = j otherwise. This step is where the addition of the extra point to each space is crucial, for it allows us to define f for everything outside of the N-tuple in a precise way without choices (we can already choose, by construction, j from Xj ).
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In a computer file system, a fork is a set of data associated with a file- system object. File systems without forks only allow a single set of data for the contents, while file systems with forks allow multiple such contents. Every non-empty file must have at least one fork, often of default type, and depending on the file system, a file may have one or more other associated forks, which in turn may contain primary data integral to the file, or just metadata. Unlike extended attributes, a similar file system feature which is typically of fixed size, forks can be of variable size, possibly even larger than the file's primary data fork.
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The local identity axiom would therefore imply that any two sections of F over the empty set are equal, but this is not true. A similar presheaf G that satisfies the local identity axiom over the empty set is constructed as follows. Let , where 0 is a one-element set. On all non-empty sets, give G the value Z. For each inclusion of open sets, G returns either the unique map to 0, if the smaller set is empty, or the identity map on Z. Intermediate step for the constant sheaf Notice that as a consequence of the local identity axiom for the empty set, all the restriction maps involving the empty set are boring.
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The exponential Bell polynomial encodes the information related to the ways a set can be partitioned. For example, if we consider a set {A, B, C}, it can be partitioned into two non-empty, non- overlapping subsets, which is also referred to as parts or blocks, in 3 different ways: :{{A}, {B, C}} :{{B}, {A, C}} :{{C}, {B, A}} Thus, we can encode the information regarding these partitions as :B_{3,2}(x_1,x_2) = 3 x_1 x_2. Here, the subscripts of B3,2 tells us that we are considering the partitioning of set with 3 elements into 2 blocks. The subscript of each xi indicates the presence of block with i elements (or block of size i) in a given partition.
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This variant of the lexicographical order is sometimes called shortlex order. In lexicographical order, the word "Thomas" appears before "Thompson" because they first differ at the fifth letter ('a' and 'p'), and letter 'a' comes before the letter 'p' in the alphabet. Because it is the first difference, in this case the 5th letter is the "most significant difference" for alphabetical ordering. An important property of the lexicographical order is that for each , the set of words of length is well- ordered by the lexicographical order (provided the alphabet is finite); that is, every decreasing sequence of words of length is finite (or equivalently, every non-empty subset has a least element).
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Any priority queue that can handle non-monotone extraction operations can also handle monotone extractions, but some priority queues are specialized to work only for monotone extractions or work better when the extractions are monotone. For instance, the bucket queue is a simple priority queue data structure consisting of an array indexed by priority, where each array cell contains a bucket of items with that priority. An extract-min operation performs a sequential search for the first non-empty bucket and chooses an arbitrary item in that bucket. For non-monotone extractions, each extract-min operation takes time (in the worst case) proportional to the array length (the number of distinct priorities).
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In other words, for every measurable set A, the density of A is 0 or 1 almost everywhere in Rn. However, it is a curious fact that if μ(A) > 0 and , then there are always points of Rn where the density is neither 0 nor 1\. For example, given a square in the plane, the density at every point inside the square is 1, on the edges is 1/2, and at the corners is 1/4. The set of points in the plane at which the density is neither 0 nor 1 is non-empty (the square boundary), but it is negligible. The Lebesgue density theorem is a particular case of the Lebesgue differentiation theorem.
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Theorem 1: The subspace spanned by a non-empty subset S of a vector space V is the set of all linear combinations of vectors in S. This theorem is so well known that at times, it is referred to as the definition of span of a set. Theorem 2: Every spanning set S of a vector space V must contain at least as many elements as any linearly independent set of vectors from V. Theorem 3: Let V be a finite-dimensional vector space. Any set of vectors that spans V can be reduced to a basis for V, by discarding vectors if necessary (i.e. if there are linearly dependent vectors in the set).
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SLD resolution implicitly defines a search tree of alternative computations, in which the initial goal clause is associated with the root of the tree. For every node in the tree and for every definite clause in the program whose positive literal unifies with the selected literal in the goal clause associated with the node, there is a child node associated with the goal clause obtained by SLD resolution. A leaf node, which has no children, is a success node if its associated goal clause is the empty clause. It is a failure node if its associated goal clause is non-empty but its selected literal unifies with the positive literal of no input clause.
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In mathematics, an algebraic structure (R,T) consisting of a non-empty set R and a ternary mapping T \colon R^3 \to R \, may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary system used by to construct projective planes by means of coordinates. A planar ternary ring is not a ring in the traditional sense, but any field gives a planar ternary ring where the operation T is defined by T(a,b,c) = ab + c. Thus, we can think of a planar ternary ring as a generalization of a field where the ternary operation takes the place of both addition and multiplication.
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Another equivalent formulation of the union-closed sets conjecture uses graph theory. In an undirected graph, an independent set is a set of vertices no two of which are adjacent to each other; an independent set is maximal if it is not a subset of a larger independent set. In any graph, the "heavy" vertices that appear in more than half of the maximal independent sets must themselves form an independent set, so there always exists at least one non- heavy vertex, a vertex that appears in at most half of the maximal independent sets. The graph formulation of the union-closed sets conjecture states that every finite non-empty graph contains two adjacent non-heavy vertices.
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Model checking of finite state systems can often be translated into various operations on Büchi automata. In addition to the closure operations presented above, the following are some useful operations for the applications of Büchi automata. ; Determinization Since deterministic Büchi automata are strictly less expressive than non- deterministic automata, there can not be an algorithm for determinization of Büchi automata. But, McNaughton's Theorem and Safra's construction provide algorithms that can translate a Büchi automaton into a deterministic Muller automaton or deterministic Rabin automaton.. ; Emptiness checking The language recognized by a Büchi automaton is non-empty if and only if there is a final state that is both reachable from the initial state, and lies on a cycle.
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A quasigroup is a non-empty set Q with a binary operation ∗ (that is, a magma), obeying the Latin square property. This states that, for each a and b in Q, there exist unique elements x and y in Q such that both :a ∗ x = b, :y ∗ a = b hold. (In other words: Each element of the set occurs exactly once in each row and exactly once in each column of the quasigroup's multiplication table, or Cayley table. This property ensures that the Cayley table of a finite quasigroup, and, in particular, finite group, is a Latin square.) The uniqueness requirement can be replaced by the requirement that the magma be cancellative.
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Lexical meanings are construed as concepts in a psychological sense: an n-place concept (with n > 0) is the property of being a (human) perception or conception in whose content a certain non-empty set of n-place attributes of real-world entities occurs as a subset. The set of attributes is called the (n-place) intension of the concept, and the set of real-world entities that have all attributes is called its (n-place) extension. The extension but not the intension may be empty. In the case of a 1-place concept, the attributes in the intension are properties, and the extension is a set of individual real-world objects.
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In computer science, an ambiguous grammar is a context-free grammar for which there exists a string that can have more than one leftmost derivation or parse tree, while an unambiguous grammar is a context-free grammar for which every valid string has a unique leftmost derivation or parse tree. Many languages admit both ambiguous and unambiguous grammars, while some languages admit only ambiguous grammars. Any non-empty language admits an ambiguous grammar by taking an unambiguous grammar and introducing a duplicate rule or synonym (the only language without ambiguous grammars is the empty language). A language that only admits ambiguous grammars is called an inherently ambiguous language, and there are inherently ambiguous context-free languages.
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There are different important equilibrium concepts in economics, the proofs of the existence of which often require variations of the Bolzano–Weierstrass theorem. One example is the existence of a Pareto efficient allocation. An allocation is a matrix of consumption bundles for agents in an economy, and an allocation is Pareto efficient if no change can be made to it which makes no agent worse off and at least one agent better off (here rows of the allocation matrix must be rankable by a preference relation). The Bolzano–Weierstrass theorem allows one to prove that if the set of allocations is compact and non-empty, then the system has a Pareto- efficient allocation.
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The real numbers with the usual topology have the rational numbers as a countable dense subset which shows that the cardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. The irrational numbers are another dense subset which shows that a topological space may have several disjoint dense subsets (in particular, two dense subsets may be each other's complements), and they need not even be of the same cardinality. Perhaps even more surprisingly, both the rationals and the irrationals have empty interiors, showing that dense sets need not contain any non-empty open set. The intersection of two dense open subsets of a topological space is again dense and open.
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In a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependent choice, this is equivalent to saying that the set is totally ordered and there is no infinite decreasing sequence (the latter being easier to visualize). In practice, the importance of well-ordering is justified by the possibility of applying transfinite induction, which says, essentially, that any property that passes on from the predecessors of an element to that element itself must be true of all elements (of the given well-ordered set). If the states of a computation (computer program or game) can be well-ordered—in such a way that each step is followed by a "lower" step—then the computation will terminate.
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Dedekind cuts can be generalized from the rational numbers to any totally ordered set by defining a Dedekind cut as a partition of a totally ordered set into two non-empty parts A and B, such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards, and A contains no greatest element. See also completeness (order theory). It is straightforward to show that a Dedekind cut among the real numbers is uniquely defined by the corresponding cut among the rational numbers. Similarly, every cut of reals is identical to the cut produced by a specific real number (which can be identified as the smallest element of the B set).
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In computer science, a loop variant is a mathematical function defined on the state space of a computer program whose value is monotonically decreased with respect to a (strict) well-founded relation by the iteration of a while loop under some invariant conditions, thereby ensuring its termination. A loop variant whose range is restricted to the non-negative integers is also known as a bound function, because in this case it provides a trivial upper bound on the number of iterations of a loop before it terminates. However, a loop variant may be transfinite, and thus is not necessarily restricted to integer values. A well-founded relation is characterized by the existence of a minimal element of every non-empty subset of its domain.
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In algebraic geometry and commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules on a scheme is flat or free. They are due to Alexander Grothendieck. Generic flatness states that if Y is an integral locally noetherian scheme, is a finite type morphism of schemes, and F is a coherent OX-module, then there is a non-empty open subset U of Y such that the restriction of F to u−1(U) is flat over U.EGA IV2, Théorème 6.9.1 Because Y is integral, U is a dense open subset of Y. This can be applied to deduce a variant of generic flatness which is true when the base is not integral.
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In abstract algebra, a semiheap is an algebraic structure consisting of a non- empty set H with a ternary operation denoted [x,y,z]\in H that satisfies a modified associativity property: :\forall \ a,b,c,d,e \in H \ \ \ \ a,b,c],d,e] = [a,[d,c,b],e] = [a,b,[c,d,e. A biunitary element h of a semiheap satisfies [h,h,k] = k = [k,h,h] for every k in H. A heap is a semiheap in which every element is biunitary. The term heap is derived from груда, Russian for "heap", "pile", or "stack". Anton Sushkevich used the term in his Theory of Generalized Groups (1937) which influenced Viktor Wagner, promulgator of semiheaps, heaps, and generalized heaps.
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Therefore, the query complexity of this algorithm is n(n − 1)/2: in the worst case, the algorithm performs n(n − 1)/2 tests. The algorithm described above is not the only possible method of testing for non-emptiness, but the Aanderaa–Karp–Rosenberg conjecture implies that every deterministic algorithm for testing non-emptiness has the same query complexity, n(n − 1)/2. That is, the property of being non-empty is evasive. For this property, the result is easy to prove directly: if an algorithm does not perform n(n − 1)/2 tests, it cannot distinguish the empty graph from a graph that has one edge connecting one of the untested pairs of vertices, and must give an incorrect answer on one of these two graphs.
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In mathematics, the constant sheaf on a topological space X associated to a set A is a sheaf of sets on X whose stalks are all equal to A. It is denoted by or AX. The constant presheaf with value A is the presheaf that assigns to each non-empty open subset of X the value A, and all of whose restriction maps are the identity map . The constant sheaf associated to A is the sheafification of the constant presheaf associated to A. In certain cases, the set A may be replaced with an object A in some category C (e.g. when C is the category of abelian groups, or commutative rings). Constant sheaves of abelian groups appear in particular as coefficients in sheaf cohomology.
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A partition of a set S is a family of non-empty disjoint subsets of S that have S as their union. A partition, together with a total order on the sets of the partition, gives a structure called by Richard P. Stanley an ordered partition. and by Theodore Motzkin a list of sets.. An ordered partition of a finite set may be written as a finite sequence of the sets in the partition: for instance, the three ordered partitions of the set {a, b} are :{a}, {b}, :{b}, {a}, and :{a, b}. In a strict weak ordering, the equivalence classes of incomparability give a set partition, in which the sets inherit a total ordering from their elements, giving rise to an ordered partition.
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Von Neumann's paper left open the possibility of a paradoxical decomposition of the interior of the unit square with respect to the linear group SL(2,R) (Wagon, Question 7.4). In 2000, Miklós Laczkovich proved that such a decomposition exists. More precisely, let A be the family of all bounded subsets of the plane with non-empty interior and at a positive distance from the origin, and B the family of all planar sets with the property that a union of finitely many translates under some elements of SL(2,R) contains a punctured neighbourhood of the origin. Then all sets in the family A are SL(2,R)-equidecomposable, and likewise for the sets in B. It follows that both families consist of paradoxical sets.
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A permutation group G acting transitively on a non-empty finite set M is imprimitive if there is some nontrivial set partition of M that is preserved by the action of G, where "nontrivial" means that the partition isn't the partition into a singleton sets nor the partition with only one part. Otherwise, if G is transitive but does not preserve any nontrivial partition of M, the group G is primitive. For example, the group of symmetries of a square is primitive on the vertices: if they are numbered 1, 2, 3, 4 in cyclic order, then the partition {{1, 3}, {2, 4}} into opposite pairs is preserved by every group element. On the other hand, the full symmetric group on a set M is always imprimitive.
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Since the set of f for which it can be solved is non-empty, and the set of all f is connected, this shows that it can be solved for all f. The map from smooth functions to smooth functions taking φ to F defined by ::F=(\omega+dd'\phi)^m/\omega^m is neither injective nor surjective. It is not injective because adding a constant to φ does not change F, and it is not surjective because F must be positive and have average value 1. So we consider the map restricted to functions φ that are normalized to have average value 0, and ask if this map is an isomorphism onto the set of positive F=e^f with average value 1.
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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).
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This was proven by Frobenius, starting in 1878 for a commuting pair, as discussed at commuting matrices. As for a single matrix, over the complex numbers these can be triangularized by unitary matrices. The fact that commuting matrices have a common eigenvector can be interpreted as a result of Hilbert's Nullstellensatz: commuting matrices form a commutative algebra K[A_1,\ldots,A_k] over K[x_1,\ldots,x_k] which can be interpreted as a variety in k-dimensional affine space, and the existence of a (common) eigenvalue (and hence a common eigenvector) corresponds to this variety having a point (being non-empty), which is the content of the (weak) Nullstellensatz. In algebraic terms, these operators correspond to an algebra representation of the polynomial algebra in k variables.
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In the category of sets, every monomorphism (injective function) with a non-empty domain is a section, and every epimorphism (surjective function) is a retraction; the latter statement is equivalent to the axiom of choice. In the category of vector spaces over a field K, every monomorphism and every epimorphism splits; this follows from the fact that linear maps can be uniquely defined by specifying their values on a basis. In the category of abelian groups, the epimorphism Z → Z/2Z which sends every integer to its remainder modulo 2 does not split; in fact the only morphism Z/2Z → Z is the zero map. Similarly, the natural monomorphism Z/2Z → Z/4Z doesn't split even though there is a non-trivial morphism Z/4Z → Z/2Z.
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Two fake projective planes are defined to be in the same class if their fundamental groups are both contained in the same maximal arithmetic subgroup of automorphisms of the unit ball. , used the volume formula for arithmetic groups from to list 28 non-empty classes of fake projective planes and show that there can at most be five extra classes which are not expected to exist. (See the addendum of the paper where the classification was refined and some errors in the original paper was corrected.) verified that the five extra classes indeed did not exist and listed all possibilities within the twenty-eight classes. There are exactly 50 fake projective planes classified up to isometry and hence 100 distinct fake projective planes classified up to biholomorphism.
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The strong version of the paradox claims: : Any two bounded subsets of 3-dimensional Euclidean space with non- empty interiors are equidecomposable. While apparently more general, this statement is derived in a simple way from the doubling of a ball by using a generalization of the Bernstein–Schroeder theorem due to Banach that implies that if is equidecomposable with a subset of and is equidecomposable with a subset of , then and are equidecomposable. The Banach–Tarski paradox can be put in context by pointing out that for two sets in the strong form of the paradox, there is always a bijective function that can map the points in one shape into the other in a one-to-one fashion. In the language of Georg Cantor's set theory, these two sets have equal cardinality.
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However, states that the assertion symbol in Gentzen-system sequents, which he denotes as ' ⇒ ', is part of the object language, not the metalanguage., defines sequents to have the form U ⇒ V for (possibly non-empty) sets of formulas U and V. Then he writes: : "Intuitively, a sequent represents 'provable from' in the sense that the formulas in U are assumptions for the set of formulas V that are to be proved. The symbol ⇒ is similar to the symbol ⊢ in Hilbert systems, except that ⇒ is part of the object language of the deductive system being formalized, while ⊢ is a metalanguage notation used to reason about deductive systems." According to Prawitz (1965): "The calculi of sequents can be understood as meta-calculi for the deducibility relation in the corresponding systems of natural deduction.".
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The ReactOS `rmdir` command Normal usage is identical to Unix-like operating systems: rmdir name_of_directory The equivalent command in MS-DOS and earlier (non-NT-based) versions of Microsoft Windows for deleting non- empty directories is . In later version of Windows: rd /s directory_name Windows based on the NT kernel (XP, Vista, 7, 8, Server 2003/2008) are case insensitive, just like their earlier predecessors, unless two files of the same name and different case exist. Then case sensitivity applies when selecting which file to use, or if the case does not match either file, one may be chosen by Windows. Having two files named the same with different case sensitivity is allowed either when Windows Services for Unix is installed or when the Windows Registry settings are set to allow it.
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The interesting number paradox is a semi-humorous paradox which arises from the attempt to classify every natural number as either "interesting" or "uninteresting". The paradox states that every natural number is interesting. The "proof" is by contradiction: if there exists a non-empty set of uninteresting natural numbers, there would be a smallest uninteresting number - but the smallest uninteresting number is itself interesting because it is the smallest uninteresting number, thus producing a contradiction. In a discussion between the mathematicians G. H. Hardy and Srinivasa Ramanujan about interesting and uninteresting numbers, Hardy remarked that the number 1729 of the taxicab he had ridden seemed "rather a dull one", and Ramanujan immediately answered that it is interesting, being the smallest number that is the sum of two cubes in two different ways.
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An order unit of a preordered vector space is any element x such that the set [−x, x] is absorbing. The set of all linear functionals on a preordered vector space V that map every order interval into a bounded set is called the order bound dual of V and denoted by Vb If a space is ordered then its order bound dual is a vector subspace of its algebraic dual. A subset A of a vector lattice E is called order complete if for every non-empty subset B ⊆ A such that B is order bounded in A, both \sup B and \inf B exist and are elements of A. We say that a vector lattice E is order complete is E is an order complete subset of E.
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An order unit of a preordered vector space is any element x such that the set [−x, x] is absorbing. The set of all linear functionals on a preordered vector space X that map every order interval into a bounded set is called the order bound dual of X and denoted by Xb. If a space is ordered then its order bound dual is a vector subspace of its algebraic dual. A subset A of an ordered vector space X is called order complete if for every non-empty subset B ⊆ A such that B is order bounded in A, both \sup B and \inf B exist and are elements of A. We say that an ordered vector space X is order complete is X is an order complete subset of X.
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The pumping lemma for context-free languages (called just "the pumping lemma" for the rest of this article) describes a property that all context-free languages are guaranteed to have. The property is a property of all strings in the language that are of length at least p, where p is a constant—called the pumping length—that varies between context-free languages. Say s is a string of length at least p that is in the language. The pumping lemma states that s can be split into five substrings, s = uvwxy, where vx is non-empty and the length of vwx is at most p, such that repeating v and x the same number of times (n) in s produces a string that is still in the language.
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Given any instance I of problem \Pi and witness W, if there exists a verifier V so that given the ordered pair (I, W) as input, V returns "yes" in polynomial time if the witness proves that the answer is "yes" or "no" in polynomial time otherwise, then \Pi is in NP. The "no"-answer version of this problem is stated as: "given a finite set of integers, does every non-empty subset have a nonzero sum?". The verifier-based definition of NP does not require an efficient verifier for the "no"-answers. The class of problems with such verifiers for the "no"-answers is called co-NP. In fact, it is an open question whether all problems in NP also have verifiers for the "no"-answers and thus are in co-NP.
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Their isolation lemma chooses a random number of random hyperplanes, and has the property that, with non-negligible probability, the intersection of any fixed non-empty solution space with the chosen hyperplanes contains exactly one element. This suffices to show the Valiant–Vazirani theorem: there exists a randomized polynomial-time reduction from the satisfiability problem for Boolean formulas to the problem of detecting whether a Boolean formula has a unique solution. introduced an isolation lemma of a slightly different kind: Here every coordinate of the solution space gets assigned a random weight in a certain range of integers, and the property is that, with non-negligible probability, there is exactly one element in the solution space that has minimum weight. This can be used to obtain a randomized parallel algorithm for the maximum matching problem.
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In mathematics, the Chabauty topology is a certain topological structure introduced in 1950 by Claude Chabauty, on the set of all closed subgroups of a locally compact group G. The intuitive idea may be seen in the case of the set of all lattices in a Euclidean space E. There these are only certain of the closed subgroups: others can be found by in a sense taking limiting cases or degenerating a certain sequence of lattices. One can find linear subspaces or discrete groups that are lattices in a subspace, depending on how one takes a limit. This phenomenon suggests that the set of all closed subgroups carries a useful topology. This topology can be derived from the Vietoris topology construction, a topological structure on all non-empty subsets of a space.
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In economics, the Edgeworth conjecture is the idea, named after Francis Ysidro Edgeworth, that the core of an economy shrinks to the set of Walrasian equilibria as the number of agents increases to infinity. The core of an economy is a concept from cooperative game theory defined as the set of feasible allocations in an economy that cannot be improved upon by subset of the set of the economy's consumers (a coalition). For general equilibrium economies typically the core is non-empty (there is at least one feasible allocation) but also "large" in the sense that there may be a continuum of feasible allocations that satisfy the requirements. The conjecture basically states that if the number of agents is also "large" then the only allocations in the core are precisely what a competitive market would produce.
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A topological space X is reducible if it can be written as a union X = X_1 \cup X_2 of two closed proper subsets X_1, X_2 of X. A topological space is irreducible (or hyperconnected) if it is not reducible. Equivalently, all non empty open subsets of X are dense or any two nonempty open sets have nonempty intersection. A subset F of a topological space X is called irreducible or reducible, if F considered as a topological space via the subspace topology has the corresponding property in the above sense. That is, F is reducible if it can be written as a union F = (G_1\cap F)\cup(G_2\cap F), where G_1,G_2 are closed subsets of X, neither of which contains F. An irreducible component of a topological space is a maximal irreducible subset.
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The main difficulty here lies in the fact that the unit square is not invariant under the action of the linear group SL(2, R), hence one cannot simply transfer a paradoxical decomposition from the group to the square, as in the third step of the above proof of the Banach–Tarski paradox. Moreover, the fixed points of the group present difficulties (for example, the origin is fixed under all linear transformations). This is why von Neumann used the larger group SA2 including the translations, and he constructed a paradoxical decomposition of the unit square with respect to the enlarged group (in 1929). Applying the Banach–Tarski method, the paradox for the square can be strengthened as follows: : Any two bounded subsets of the Euclidean plane with non-empty interiors are equidecomposable with respect to the area-preserving affine maps.
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However, other subclasses of P such as NC may not be closed under Turing reductions, and so many-one reductions must be used. Just as polynomial-time reductions are useless within P and its subclasses, log- space reductions are useless to distinguish problems in L and its subclasses; in particular, every non-empty, non-full problem in L is trivially L-complete under log-space reductions. While even weaker reductions exist, they are not often used in practice, because complexity classes smaller than L (that is, strictly contained or thought to be strictly contained in L) receive relatively little attention. The tools available to designers of log-space reductions have been greatly expanded by the result that L = SL; see SL for a list of some SL-complete problems that can now be used as subroutines in log- space reductions.
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As mentioned before, domain theory deals with partially ordered sets to model a domain of computation. The goal is to interpret the elements of such an order as pieces of information or (partial) results of a computation, where elements that are higher in the order extend the information of the elements below them in a consistent way. From this simple intuition it is already clear that domains often do not have a greatest element, since this would mean that there is an element that contains the information of all other elements—a rather uninteresting situation. A concept that plays an important role in the theory is that of a directed subset of a domain; a directed subset is a non-empty subset of the order in which any two elements have an upper bound that is an element of this subset.
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A fan F=[x_1,x_2,x_3] of v (dashed edges are uncolored), (v,x_1),(v,x_2),(v,x_3) are the fan edges. F'=[x_1,x_2] is also a fan of v, but it is not maximal. A color x is said to be free of an edge (u,v) on u if c(u,z) ≠ x for all (u,z) \in E(G) : z≠v. A fan of a vertex u is a sequence of vertices F[1:k] that satisfies the following conditions: #F[1:k] is a non-empty sequence of distinct neighbors of u #(F[1],u) \in E(G) is uncolored #The color of (F[i+1],u) is free on F[i] for 1 ≤ i < k Examples of cdx paths: ac,cg,gd is a red-green_c path and bd,dg is a red-oranged path.
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Along with the Shapley value, stochastic games, the Bondareva–Shapley theorem (which implies that convex games have non-empty cores), the Shapley–Shubik power index (for weighted or block voting power), the Gale–Shapley algorithm for the stable marriage problem, the concept of a potential game (with Dov Monderer), the Aumann–Shapley pricing, the Harsanyi–Shapley solution, the Snow–Shapley theorem for matrix games, and the Shapley–Folkman lemma & theorem bear his name. According to The Economist, Shapley "may have thought of himself as a mathematician, but he cannot avoid being remembered for his huge contributions to economics". The American Economic Association noted that Shapley was "one of the giants of game theory and economic theory". Besides, his early work with R. N. Snow and Samuel Karlin on matrix games was so complete that little has been added since.
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The degeneracy of an undirected graph is the smallest number such that every non-empty subgraph of has at least one vertex of degree at most . If one repeatedly removes a minimum-degree vertex from until no vertices are left, then the largest of the degrees of the vertices at the time of their removal will be exactly , and this method of repeated removal can be used to compute the degeneracy of any graph in linear time. Greedy coloring the vertices in the reverse of this removal ordering will automatically produce a coloring with at most colors, and for some graphs (such as complete graphs and odd-length cycle graphs) this number of colors is optimal. For colorings with colors, it may not be possible to move from one coloring to another by changing the color of one vertex at a time.
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As with any non-empty family of sets, is contained in some filter if and only if it is a filter subbase (meaning it has the finite intersection property). If denotes the -system generated by , which is the set : then there exists a filter on containing every as a subset if and only if , in which case is a prefilter and is the smallest filter on containing every as a subset; this makes the filter the supremum and the least upper bound of in and is equal to the intersection of all filters on containing . The least upper bound of a family of filters may fail to be a filter. Indeed, if contains at least 2 distinct elements then there exist filters and on for which there does _not_ exist a filter on that contains both and .
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Once the dynamical properties of complex sets of circuits had been disentangled in logical terms, it was tempting to come back to a more usual and quantitative description in terms of differential equations, taking advantage of the knowledge gained regarding qualitative behaviour. Two articles by Thomas and Marcelline Kaufman compare the logical and differential predictions of the number and nature of the steady states. Following articles by Thomas and Marcelle Kaufman, and by Thomas and Pascal Nardone showed that the phase space of the system can be partitioned into domains according to the signs and to the real or complex nature of the eigenvalues of the Jacobian matrix. Indeed, regulatory circuits can be defined formally as sets of non-empty elements of the Jacobian matrix (or of the interaction graph) of dynamical systems such that the line and column indices are in circular permutation.
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An embedding of a topological space X as a dense subset of a compact space is called a compactification of X. A linear operator between topological vector spaces X and Y is said to be densely defined if its domain is a dense subset of X and if its range is contained within Y. See also continuous linear extension. A topological space X is hyperconnected if and only if every nonempty open set is dense in X. A topological space is submaximal if and only if every dense subset is open. If \left(X, d_X\right) is a metric space, then a non-empty subset Y is said to be ε-dense if : (\forall x \in X)\, (\exists y \in Y)\, d_X(x, y) \leq \varepsilon. One can then show that D is dense in \left(X, d_X\right) if and only if it is ε-dense for every \varepsilon > 0.
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More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (M, d) and (N, d') are two metric spaces, then f:M \rightarrow N is a contractive mapping if there is a constant 0 \leq k < 1 such that :d'(f(x),f(y)) \leq k\,d(x,y) for all x and y in M. Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant k is no longer necessarily less than 1). A contraction mapping has at most one fixed point. Moreover, the Banach fixed-point theorem states that every contraction mapping on a non-empty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point.
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The next two segments that the main algorithm recurs on are (elements ≤ pivot) and (elements ≥ pivot) as opposed to and as in Lomuto's scheme. However, the partitioning algorithm guarantees which implies both resulting partitions are non-empty, hence there's no risk of infinite recursion. In pseudocode, algorithm quicksort(A, lo, hi) is if lo < hi then p := partition(A, lo, hi) quicksort(A, lo, p) quicksort(A, p + 1, hi) algorithm partition(A, lo, hi) is pivot := A[⌊(hi + lo) / 2⌋] i := lo - 1 j := hi + 1 loop forever do i := i + 1 while A[i] < pivot do j := j - 1 while A[j] > pivot if i ≥ j then return j swap A[i] with A[j] An important point in choosing the pivot item is to round the division result towards zero. This is the implicit behavior of integer division in some programming languages (e.g.
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The Axiom of Choice can be proven from the well-ordering theorem as follows. :To make a choice function for a collection of non-empty sets, E, take the union of the sets in E and call it X. There exists a well- ordering of X; let R be such an ordering. The function that to each set S of E associates the smallest element of S, as ordered by (the restriction to S of) R, is a choice function for the collection E. An essential point of this proof is that it involves only a single arbitrary choice, that of R; applying the well-ordering theorem to each member S of E separately would not work, since the theorem only asserts the existence of a well-ordering, and choosing for each S a well-ordering would not be easier than choosing an element.
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The Banach–Tarski paradox, proved by Stefan Banach and Alfred Tarski in 1924, states that it is possible to partition a three-dimensional unit ball into finitely many pieces and reassemble them into two unit balls, a single ball of larger or smaller area, or any other bounded set with a non- empty interior. Although it is a mathematical theorem, it is called a paradox because it is so counter-intuitive; in the preface to the book, Jan Mycielski calls it the most surprising result in mathematics. It is closely related to measure theory and the non-existence of a measure on all subsets of three- dimensional space, invariant under all congruences of space, and to the theory of paradoxical sets in free groups and the representation of these groups by three-dimensional rotations, used in the proof of the paradox. The topic of the book is the Banach–Tarski paradox, its proof, and the many related results that have since become known.
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Analytic continuation of natural logarithm (imaginary part) Suppose f is an analytic function defined on a non-empty open subset U of the complex plane \Complex. If V is a larger open subset of \Complex, containing U, and F is an analytic function defined on V such that :F(z) = f(z) \qquad \forall z \in U, then F is called an analytic continuation of f. In other words, the restriction of F to U is the function f we started with. Analytic continuations are unique in the following sense: if V is the connected domain of two analytic functions F1 and F2 such that U is contained in V and for all z in U :F_1(z) = F_2(z) = f(z), then :F_1=F_2 on all of V. This is because F1 − F2 is an analytic function which vanishes on the open, connected domain U of f and hence must vanish on its entire domain.
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Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets that are called well-ordered (so intimately linked, in fact, that some mathematicians make no distinction between the two concepts). A well-ordered set is a totally ordered set (given any two elements one defines a smaller and a larger one in a coherent way), in which every non- empty subset of the set has a least element. In particular, there is no infinite decreasing sequence. (However, there may be infinite increasing sequences.) Ordinals may be used to label the elements of any given well- ordered set (the smallest element being labelled 0, the one after that 1, the next one 2, "and so on"), and to measure the "length" of the whole set by the least ordinal that is not a label for an element of the set.
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The DE-9IM model is based on a 3×3 intersection matrix with the form: where is the dimension of the intersection (∩) of the interior (I), boundary (B), and exterior (E) of geometries a and b. The terms interior and boundary in this article are used in the sense used in algebraic topology and manifold theory, not in the sense used in general topology: for example, the interior of a line segment is the line segment without its endpoints, and its boundary is just the two endpoints (in general topology, the interior of a line segment in the plane is empty and the line segment is its own boundary). In the notation of topological space operators, the matrix elements can be expressed also as The dimension of empty sets (∅) are denoted as −1 or (false). The dimension of non-empty sets (¬∅) are denoted with the maximum number of dimensions of the intersection, specifically for points, for lines, for areas.
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In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads: : \forall x\,(x eq \varnothing \rightarrow \exists y \in x\,(y \cap x = \varnothing)). The axiom of regularity together with the axiom of pairing implies that no set is an element of itself, and that there is no infinite sequence (an) such that ai+1 is an element of ai for all i. With the axiom of dependent choice (which is a weakened form of the axiom of choice), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, in this context the axiom of regularity is equivalent to the sentence that there are no downward infinite membership chains.
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In mathematics, specifically in order theory and functional analysis, a subset A of an ordered vector space is said to be order complete in X if for every non-empty subset S of C that is order bounded in A (i.e. is contained in an interval [a, b] := { z ∈ X : a ≤ z and z ≤ b } for some a and b belonging to A), the supremum sup S and the infimum inf S both exist and are elements of A. An ordered vector space is called order complete, Dedekind complete, a complete vector lattice, or a complete Riesz space, if it is order complete as a subset of itself, in which case it is necessarily a vector lattice. An ordered vector space is said to be countably order complete if each countable subset that is bounded above has a supremum. Being an order complete vector space is an important property that is used frequently in the theory of topological vector lattices.
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There is a distinction between a tree as an abstract data type and as a concrete data structure, analogous to the distinction between a list and a linked list. As a data type, a tree has a value and children, and the children are themselves trees; the value and children of the tree are interpreted as the value of the root node and the subtrees of the children of the root node. To allow finite trees, one must either allow the list of children to be empty (in which case trees can be required to be non- empty, an "empty tree" instead being represented by a forest of zero trees), or allow trees to be empty, in which case the list of children can be of fixed size (branching factor, especially 2 or "binary"), if desired. As a data structure, a linked tree is a group of nodes, where each node has a value and a list of references to other nodes (its children).
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In computational geometry, the problem of computing the intersection of a polyhedron with a line has important applications in computer graphics, optimization, and even in some Monte Carlo methods. It can be viewed as a three-dimensional version of the line clipping problem.. If the polyhedron is given as the intersection of a finite number of halfspaces, then one may partition the halfspaces into three subsets: the ones that include only one infinite end of the line, the ones that include the other end, and the ones that include both ends. The halfspaces that include both ends must be parallel to the given line, and do not contribute to the solution. Each of the other two subsets (if it is non-empty) contributes a single endpoint to the intersection, which may be found by intersecting the line with each of the halfplane boundary planes and choosing the intersection point that is closest to the end of the line contained by the halfspaces in the subset.
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She echoed a quantitative stance towards narrative research by explaining > I can't review someone I feel sorry or hopeless about...I'm forced to feel > sorry because of the way they present themselves as: dissed blacks, abused > women, or disenfranchised homosexuals - as performers, in short, who make > victimhood victim art Croce illustrates what Tony E. Adams, Stacy Holman Jones, and Carolyn Ellis refer to as "illusory boundaries and borders between scholarship and criticism". These "borders" are seen to hide or take away from the idea that autoethnographic evaluation and criticism present another personal story about the experience of an experience. Or as Craig Gingrich-Philbrook wrote, "any evaluation of autoethnography...is simply another story from a highly situated, privileged, empowered subject about something he or she experienced". Prominent philosopher of science, Karl Popper, when claiming that falsifiability was a basic criteria of a scientific theory said: > A theory is falsifiable ... if there exists at least one non-empty class of > homotypic basic statements which are forbidden by it As autoethnography makes no claims that can be verified, it is no longer falsifiable.
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The coordinates (x_i,y_i,z_i) of the apexes of a solution to the tripod problem form a 2-comparable sets of triples, where two triples are defined as being 2-comparable if there are either at least two coordinates where one triple is smaller than the other, or at least two coordinates where one triple is larger than the other. This condition ensures that the tripods defined from these triples do not have intersecting rays. Another equivalent two-dimensional version of the question asks how many cells of an n\times n array of square cells (indexed from 1 to n) can be filled in by the numbers from 1 to n in such a way that the non-empty cells of each row and each column of the array form strictly increasing sequences of numbers, and the positions holding each value i form a monotone chain within the array. A collection of disjoint tripods with apexes (x_i,y_i,z_i) can be transformed into an array of this type by placing the number z_i in array cell (x_i,y_i) and vice versa.
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One of these algorithms, first described by , works by choosing vertices in the same order as the eventual topological sort. First, find a list of "start nodes" which have no incoming edges and insert them into a set S; at least one such node must exist in a non-empty acyclic graph. Then: L ← Empty list that will contain the sorted elements S ← Set of all nodes with no incoming edge while S is not empty do remove a node n from S add n to L for each node m with an edge e from n to m do remove edge e from the graph if m has no other incoming edges then insert m into S if graph has edges then return error (graph has at least one cycle) else return L (a topologically sorted order) If the graph is a DAG, a solution will be contained in the list L (the solution is not necessarily unique). Otherwise, the graph must have at least one cycle and therefore a topological sort is impossible.
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The nonnegative integer r is called the free rank or Betti number of the module M. The module is determined up to isomorphism by specifying its free rank , and for class of associated irreducible elements and each positive integer the number of times that occurs among the elementary divisors. The elementary divisors can be obtained from the list of invariant factors of the module by decomposing each of them as far as possible into pairwise relatively prime (non-unit) factors, which will be powers of irreducible elements. This decomposition corresponds to maximally decomposing each submodule corresponding to an invariant factor by using the Chinese remainder theorem for R. Conversely, knowing the multiset of elementary divisors, the invariant factors can be found, starting from the final one (which is a multiple of all others), as follows. For each irreducible element such that some power occurs in , take the highest such power, removing it from , and multiply these powers together for all (classes of associated) to give the final invariant factor; as long as is non-empty, repeat to find the invariant factors before it.
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A gerbe on a topological space X is a stack G of groupoids over X which is locally non- empty (each point in X has an open neighbourhood U over which the section category G(U) of the gerbe is not empty) and transitive (for any two objects a and b of G(U) for any open set U, there is an open covering {Vi}i of U such that the restrictions of a and b to each Vi are connected by at least one morphism). A canonical example is the gerbe of principal bundles with a fixed structure group H: the section category over an open set U is the category of principal H-bundles on U with isomorphism as morphisms (thus the category is a groupoid). As principal bundles glue together (satisfy the descent condition), these groupoids form a stack. The trivial bundle X x H over X shows that the local non-emptiness condition is satisfied, and finally as principal bundles are locally trivial, they become isomorphic when restricted to sufficiently small open sets; thus the transitivity condition is satisfied as well.
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The Ritt–Wu process, first devised by Ritt, subsequently modified by Wu, computes not a Ritt characteristic but an extended one, called Wu characteristic set or ascending chain. A non-empty subset T of the ideal generated by F is a Wu characteristic set of F if one of the following condition holds :(1) T = {a} with a being a nonzero constant, :(2) T is a triangular set and there exists a subset G of such that = and every polynomial in G is pseudo-reduced to zero with respect to T. Wu characteristic set is defined to the set F of polynomials, rather to the ideal generated by F. Also it can be shown that a Ritt characteristic set T of is a Wu characteristic set of F. Wu characteristic sets can be computed by Wu's algorithm CHRST-REM, which only requires pseudo-remainder computations and no factorizations are needed. Wu's characteristic set method has exponential complexity; improvements in computing efficiency by weak chains, regular chains, saturated chain were introducedChou S C, Gao X S; Ritt–Wu's decomposition algorithm and geometry theorem proving. Proc of CADE, 10 LNCS, #449, Berlin, Springer Verlag, 1990 207–220.
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