It's all little apocalypses, little re-orderings, that happen and are then erased like they never happened.
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Now, the company wants to offer mobile orderings, previously available only to members of its loyalty program, to all customers.
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The probability of that happening depends on the total number of unique orderings — the number of permutations — those 14 phrases have.
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For Haruhi fans, Egan's construction gives explicit instructions for how to watch all possible orderings of season one in just 93,924,230,411 episodes.
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There are hierarchical groupings in which Ms. Mearns is central, dominant, with the others arrayed as if in attendance; but all such orderings here are impermanent.
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The strongest, such as "Flood," reveal even more: the mysterious, accumulating force of colors themselves, in charged, top-to-bottom orderings of sensations reminiscent of Hartley's soberly exultant paintings.
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The argument, which covered series with any number of episodes, showed that for the 14-episode first season of Haruhi, viewers would have to watch at least 13,884,313,611 episodes to see all possible orderings.
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Doing so, you can see, the total number of possible orderings of those 14 phrases that Trump has in common with Obama is 14*13*12*11*10*9*85003*7*6*5*4*3*2*1.
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Our ratings are merely rank orderings of credit risk, they are an assessment of 'will you get your money back in full and on time'," Colin Ellis, managing director for credit strategy at Moody's, told CNBC's "Squawk Box Europe.
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Chakraborty, D. and Ghosh, D. (2014) "Analytical fuzzy plane geometry II". Fuzzy Sets and Systems, 243, 84–109. fuzzy orderings,Zadeh L.A. (1971) "Similarity relations and fuzzy orderings". Inform. Sci., 3, 177–200.
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The triangular prism is the smallest graph for which one of its degeneracy orderings leads to a non-optimal coloring, and the square antiprism is the smallest graph that cannot be optimally colored using any of its degeneracy orderings.
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The set of all perfect elimination orderings of a chordal graph can be modeled as the basic words of an antimatroid; use this connection to antimatroids as part of an algorithm for efficiently listing all perfect elimination orderings of a given chordal graph.
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Kenneth Arrow (1963) generalizes the analysis. Along earlier lines, his version of a social welfare function, also called a 'constitution', maps a set of individual orderings (ordinal utility functions) for everyone in the society to a social ordering, a rule for ranking alternative social states (say passing an enforceable law or not, ceteris paribus). Arrow finds that nothing of behavioral significance is lost by dropping the requirement of social orderings that are real-valued (and thus cardinal) in favor of orderings, which are merely complete and transitive, such as a standard indifference curve map. The earlier analysis mapped any set of individual orderings to one social ordering, whatever it was.
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For instance, for n = 3, the permutohedron on three elements is just a regular hexagon. The face lattice of the hexagon (again, including the hexagon itself as a face, but not including the empty set) has thirteen elements: one hexagon, six edges, and six vertices, corresponding to the one completely tied weak ordering, six weak orderings with one tie, and six total orderings. The graph of moves on these 13 weak orderings is shown in the figure.
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Here the usual notation \partial_xz=z_x, \partial_yz=z_y,\ldots is used. If the number of functions is higher than one, these orderings have to be generalized appropriately, e.g. the orderings TOP or POT may be applied.W. Adams, P. Loustaunau, An introduction to Gröbner bases, American Mathematical Society, Providence, 1994.
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Though well-quasi-ordering is an appealing notion, many important infinitary operations do not preserve well-quasi- orderedness. An example due to Richard Rado illustrates this. In a 1965 paper Crispin Nash-Williams formulated the stronger notion of better-quasi-ordering in order to prove that the class of trees of height ω is well-quasi-ordered under the topological minor relation. Since then, many quasi-orderings have been proven to be well-quasi-orderings by proving them to be better-quasi- orderings.
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Many revision proposals involve orderings over models representing the relative plausibility of the possible alternatives. The problem of merging amounts to combine a set of orderings into a single one expressing the combined plausibility of the alternatives. This is similar with what is done in social choice theory, which is the study of how the preferences of a group of agents can be combined in a rational way. Belief revision and social choice theory are similar in that they combine a set of orderings into one.
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The 13 possible strict weak orderings on a set of three elements {a, b, c} In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the number of weak orderings on a set of n elements (orderings of the elements into a sequence allowing ties, such as might arise as the outcome of a horse race).. Because of this application, de Koninck calls these numbers "horse numbers", but this name does not appear to be in widespread use. Starting from n = 0, these numbers are :1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, ... . The ordered Bell numbers may be computed via a summation formula involving binomial coefficients, or by using a recurrence relation. Along with the weak orderings, they count several other types of combinatorial objects that have a bijective correspondence to the weak orderings, such as the ordered multiplicative partitions of a squarefree number or the faces of all dimensions of a permutohedron. (e.g.
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For a finite set of labeled items, every pair of weak orderings may be connected to each other by a sequence of moves that add or remove one dichotomy at a time to or from this set of dichotomies. Moreover, the undirected graph that has the weak orderings as its vertices, and these moves as its edges, forms a partial cube.. Geometrically, the total orderings of a given finite set may be represented as the vertices of a permutohedron, and the dichotomies on this same set as the facets of the permutohedron. In this geometric representation, the weak orderings on the set correspond to the faces of all different dimensions of the permutohedron (including the permutohedron itself, but not the empty set, as a face). The codimension of a face gives the number of equivalence classes in the corresponding weak ordering.. In this geometric representation the partial cube of moves on weak orderings is the graph describing the covering relation of the face lattice of the permutohedron.
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The order dimension of a partial order is the smallest number of total orderings whose intersection is the given partial order; such a set of orderings is called a realizer of the partial order. Schnyder's theorem states that a graph is planar if and only if the order dimension of is at most three.
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A merging operator can be expressed as a family of orderings over models, one for each possible multiset of knowledge bases to merge: the models of the result of merging a multiset of knowledge bases are the minimal models of the ordering associated to the multiset. A merging operator defined in this way satisfies the postulates for merging if and only if the family of orderings meets a given set of conditions. For the old definition of arbitration, the orderings are not on models but on pairs (or, in general, tuples) of models.
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The children of a Q node may be put in reverse order, but may not otherwise be reordered. A PQ tree represents all leaf node orderings that can be achieved by any sequence of these two operations. A PQ tree with many P and Q nodes can represent complicated subsets of the set of all possible orderings. However, not every set of orderings may be representable in this way; for instance, if an ordering is represented by a PQ tree, the reverse of the ordering must also be represented by the same tree.
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This social ordering selected the top-ranked feasible alternative from the economic environment as to resource constraints. Arrow proposed to examine mapping different sets of individual orderings to possibly different social orderings. Here the social ordering would depend on the set of individual orderings, rather than being imposed (invariant to them). Stunningly (relative to a course of theory from Adam Smith and Jeremy Bentham on), Arrow proved the general impossibility theorem which says that it is impossible to have a social welfare function that satisfies a certain set of "apparently reasonable" conditions.
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The ring homomorphisms from W(k) to Z correspond to the field orderings of k, by taking signature with respective to the ordering.
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9) uses individual preference orderings rather than real-valued measures of preferences. This excludes interpersonal comparisons of welfare in a precise sense (invariance of social choices to linear cardinalizations of individual preference orderings). Extended-sympathy interpersonal comparisons of welfare relax that constraint (Arrow, 1983, pp. 151–2). Such comparisons expand the informational base of welfare-theoretical decisions, as Amartya Sen (1982) has emphasized.
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Every ordered locally convex space is regularly ordered. The canonical orderings of subspaces, products, and direct sums of regularly ordered vector spaces are again regularly ordered.
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For a basic sentence with a single verb that takes two noun phrases as arguments, all orderings are possible, but having the verb final is less common.
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Tone rows, orderings used in the twelve-tone technique, are often considered this way due to the increase ease of comparing inverse intervals and forms (inversional equivalence).
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If, instead, different orderings of the sets are considered to be different partitions, then the number of these ordered partitions is given by the ordered Bell numbers.
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There are exactly 52 factorial (expressed in shorthand as 52!) possible orderings of the cards in a 52-card deck. In other words, there are 52 × 51 × 50 × 49 × ··· × 4 × 3 × 2 × 1 possible combinations of card sequence. This is approximately (80,658vigintillion) possible orderings, or specifically 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000. The magnitude of this number means that it is exceedingly improbable that two randomly selected, truly randomized decks will be the same.
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The values of -tuples of parameters are well-ordered by the product ordering. The formulas with parameters are well-ordered by the ordered sum (by Gödel numbers) of well-orderings. And is well-ordered by the ordered sum (indexed by ) of the orderings on . Notice that this well-ordering can be defined within itself by a formula of set theory with no parameters, only the free-variables and .
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A de Bruijn graph. With the vertex ordering shown, the partition of the edges into two subsets looping around the left and right sides of the drawing is a 2-queue layout of this graph. In mathematics, the queue number of a graph is a graph invariant defined analogously to stack number (book thickness) using first-in first-out (queue) orderings in place of last-in first-out (stack) orderings.
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The Banach spaces L^p\left( \mu \right) (1 \leq p \leq \infty) are Banach lattices under their canonical orderings. These spaces are order complete for p < \infty.
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Every fundamental cycle basis is weakly fundamental (for all linear orderings) but not necessarily vice versa. There exist graphs, and cycle bases for those graphs, that are not weakly fundamental..
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Two well-orderings W_1 and W_2 are similar and write W_1 \sim W_2 just in case there is a bijection f from the field of W_1 to the field of W_2 such that x W_1 y \leftrightarrow f(x)W_2f(y) for all x and y. Similarity is shown to be an equivalence relation in much the same way that equinumerousness was shown to be an equivalence relation above. In New Foundations (NFU), the order type of a well-ordering W is the set of all well-orderings which are similar to W. The set of ordinal numbers is the set of all order types of well-orderings. This does not work in ZFC, because the equivalence classes are too large.
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There are two equivalent common definitions of an ordered field. The definition of total order appeared first historically and is a first-order axiomatization of the ordering ≤ as a binary predicate. Artin and Schreier gave the definition in terms of positive cone in 1926, which axiomatizes the subcollection of nonnegative elements. Although the latter is higher-order, viewing positive cones as maximal prepositive cones provides a larger context in which field orderings are extremal partial orderings.
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This can be done by considering equivalence classes of smooth curves with the same direction. A directed smooth curve can then be defined as an ordered set of points in the complex plane that is the image of some smooth curve in their natural order (according to the parametrization). Note that not all orderings of the points are the natural ordering of a smooth curve. In fact, a given smooth curve has only two such orderings.
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In the field of computer science, a pre-topological order or pre-topological ordering of a directed graph is a linear ordering of its vertices such that if there is a directed path from vertex u to vertex v and v comes before u in the ordering, then there is also a directed path from vertex v to vertex u. If the graph is a directed acyclic graph (DAG), topological orderings are pre- topological orderings and vice versa. In other cases, any pre-topological ordering gives a partial order.
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Arrow originally rejected cardinal utility as a meaningful tool for expressing social welfare,"Modern economic theory has insisted on the ordinal concept of utility; that is, only orderings can be observed, and therefore no measurement of utility independent of these orderings has any significance. In the field of consumer's demand theory the ordinalist position turned out to create no problems; cardinal utility had no explanatory power above and beyond ordinal. Leibniz' Principle of the identity of indiscernibles demanded then the excision of cardinal utility from our thought patterns." Arrow (1967), as quoted on p.
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Kenneth Arrow's monograph Social Choice and Individual Values (1951, 2nd ed., 1963) and a theorem within it created modern social choice theory, a rigorous melding of social ethics and voting theory with an economic flavor. Somewhat formally, the "social choice" in the title refers to Arrow's representation of how social values from the set of individual orderings would be implemented under the constitution. Less formally, each social choice corresponds to the feasible set of laws passed by a "vote" (the set of orderings) under the constitution even if not every individual voted in favor of all the laws.
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They differ on how these orderings are interpreted: preferences in social choice theory; plausibility in belief revision. Another difference is that the alternatives are explicitly enumerated in social choice theory, while they are the propositional models over a given alphabet in belief revision.
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The Bell numbers, named after Eric Temple Bell, count the number of partitions of a set, and the weak orderings that are counted by the ordered Bell numbers may be interpreted as a partition together with a total order on the sets in the partition..
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In many cases another representation called a preferential arrangement based on a utility function is also possible. Weak orderings are counted by the ordered Bell numbers. They are used in computer science as part of partition refinement algorithms, and in the C++ Standard Library..
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A partially ordered set (poset) consists of a set of elements together with a binary relation on pairs of elements that is reflexive ( for every x), transitive (if and then ), and antisymmetric (if both and hold, then ). The usual numeric orderings on the integers or real numbers satisfy these properties; however, unlike the orderings on the numbers, a partial order may have two elements that are incomparable: neither nor holds. Another familiar example of a partial ordering is the inclusion ordering ⊆ on pairs of sets. If is a partially ordered set, a completion of means a complete lattice with an order-embedding of into .
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A graph is well-colored if and only if does not have two vertex orderings for which the greedy coloring algorithm produces different numbers of colors. Therefore, recognizing non-well-colored graphs can be performed within the complexity class NP. On the other hand, a graph G has Grundy number k or more if and only if the graph obtained from G by adding a (k-1)-vertex clique is well-colored. Therefore, by a reduction from the Grundy number problem, it is NP-complete to test whether these two orderings exist. It follows that it is co-NP-complete to test whether a given graph is well-colored.
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Explicitly, it is used to define the degree of a polynomial and the notion of homogeneous polynomial, as well as for graded monomial orderings used in formulating and computing Gröbner bases. Implicitly, it is used in grouping the terms of a Taylor series in several variables.
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Metaphysical debate about temporal orderings reaches back to the ancient Greek philosophers Heraclitus and Parmenides. Parmenides thought that reality is timeless and unchanging. Heraclitus, in contrast, believed that the world is a process of ceaseless change, flux and decay. Reality for Heraclitus is dynamic and ephemeral.
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Search in systems with faceted classification can enable a user to navigate information along multiple paths corresponding to different orderings of the facets. This contrasts with traditional taxonomies in which the hierarchy of categories is fixed and unchanging.Star, S. L. (1998, Fall). "Grounded classification: grounded theory and faceted classification".
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In 1978, Falmagne solved a well-known problem, posed in 1960 by the economists H.D. Block and Jacob Marschak in their article "Random Orderings and Stochastic Theories of Responses", concerning the representation of choice probabilities by random variables and published his findings in the Journal of Mathematical Psychology.
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The permutohedron on four elements, a three-dimensional convex polyhedron. It has 24 vertices, 36 edges, and 14 two-dimensional faces, which all together with the whole three-dimensional polyhedron correspond to the 75 weak orderings on four elements. Unlike for partial orders, the family of weak orderings on a given finite set is not in general connected by moves that add or remove a single order relation to or from a given ordering. For instance, for three elements, the ordering in which all three elements are tied differs by at least two pairs from any other weak ordering on the same set, in either the strict weak ordering or total preorder axiomatizations.
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They generalize strict weak orderings, in which items with equal scores may be tied but there is no margin of error. They are a special case of partial orders and of interval orders, and can be characterized among the partial orders by additional axioms, or by two forbidden four-item suborders.
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A closer look at a magnetization curve generally reveals a series of small, random jumps in magnetization called Barkhausen jumps. This effect is due to crystallographic defects such as dislocations. Magnetic hysteresis loops are not exclusive to materials with ferromagnetic ordering. Other magnetic orderings, such as spin glass ordering, also exhibit this phenomenon.
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Chapter 1 of Poizat's model theory textA Course in Model Theory, Bruno Poizat, tr. Moses Klein, New York: Springer- Verlag, 2000. contains an introduction to the Ehrenfeucht–Fraïssé game, and so do Chapters 6, 7, and 13 of Rosenstein's book on linear orders.Linear Orderings, Joseph G. Rosenstein, New York: Academic Press, 1982.
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In this article, a monomial is assumed to not include a coefficient. The defining property of monomial orderings implies that the order of the terms is kept when multiplying a polynomial by a monomial. Also, the leading term of a product of polynomials is the product of the leading terms of the factors.
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If k is a real Pythagorean field then the zero-divisors of W are the elements for which some signature is zero; otherwise, the zero- divisors are exactly the fundamental ideal.Lam (2005) p. 282 If k is an ordered field with positive cone P then Sylvester's law of inertia holds for quadratic forms over k and the signature defines a ring homomorphism from W(k) to Z, with kernel a prime ideal KP. These prime ideals are in bijection with the orderings Xk of k and constitute the minimal prime ideal spectrum MinSpec W(k) of W(k). The bijection is a homeomorphism between MinSpec W(k) with the Zariski topology and the set of orderings Xk with the Harrison topology.
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The majority of the theorems mentioned in the sections Galois theory, Constructing fields and Elementary notions can be found in Steinitz's work. linked the notion of orderings in a field, and thus the area of analysis, to purely algebraic properties. Emil Artin redeveloped Galois theory from 1928 through 1942, eliminating the dependency on the primitive element theorem.
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Maximal lotteries refers to a probabilistic voting system first considered by the French mathematician and social scientist Germain KrewerasG. Kreweras. Aggregation of preference orderings. In Mathematics and Social Sciences I: Proceedings of the seminars of Menthon-Saint-Bernard, France (1–27 July 1960) and of Gösing, Austria (3–27 July 1962), pages 73–79, 1965. in 1965.
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Rewriting s to t by a rule l::=r. If l and r are related by a rewrite relation, so are s and t. A simplifcation ordering always relates l and s, and similarly r and t. In theoretical computer science, in particular in automated reasoning about formal equations, reduction orderings are used to prevent endless loops.
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Many of Katchadourian's pieces involve bringing an incisive and playful order to the world. The "Sorted Books" series, for instance, ranges from ephemeral and impromptu arrangements of volumes on the shelves of friends, to commissioned photographed orderings of books in museum and library collections. The body of work is available as a book published by Chronicle Books.
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In statistics, Goodman and Kruskal's gamma is a measure of rank correlation, i.e., the similarity of the orderings of the data when ranked by each of the quantities. It measures the strength of association of the cross tabulated data when both variables are measured at the ordinal level. It makes no adjustment for either table size or ties.
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Richards, Marc D. (2008). "Desymmetrization: Parametric variation at the PF-Interface". The Canadian Journal of Linguistics 53 (2-3), p. 283. In this approach the relative positions of head and complement that are actually attested at this surface level, which are found to show variation both between and within languages (see above), must be treated as the "true" orderings.
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In fact, for any infinite cardinal κ, every κ-Suslin tree is a κ-Aronszajn tree (the converse does not hold). The Suslin conjecture was originally stated as a question about certain total orderings but it is equivalent to the statement: Every tree of height ω1 has an antichain of cardinality ω1 or a branch of length ω1.
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This introduces indeterminacy in the arrival order. Since the arrival orderings are indeterminate, they cannot be deduced from prior information by mathematical logic alone. Therefore, mathematical logic cannot implement concurrent computation in open systems. The authors claim that although mathematical logic cannot, in their view, implement general concurrency it can implement some special cases of concurrent computation, e.g.
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The partial order formed by three elements a, b, and c with a single comparability relationship, has three linear extensions, and In all three of these extensions, a is earlier than b. However, a is earlier than c in only two of them, and later than c in the third. Therefore, the pair of a and c have the desired property, showing that this partial order obeys the 1/3–2/3 conjecture. This example shows that the constants 1/3 and 2/3 in the conjecture are tight; if q is any fraction strictly between 1/3 and 2/3, then there would not exist a pair x, y in which x is earlier than y in a number of partial orderings that is between q and times the total number of partial orderings.
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A group is called finite if it has a finite number of elements. The number of elements is called the order of the group. An important class is the symmetric groups SN, the groups of permutations of N letters. For example, the symmetric group on 3 letters S3 is the group consisting of all possible orderings of the three letters ABC, i.e.
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Higher orders of stochastic dominance have also been analyzed, as have generalizations of the dual relationship between stochastic dominance orderings and classes of preference functions. Arguably the most powerful dominance criterion relies on the accepted economic assumption of decreasing absolute risk aversion. This involves several analytical challenges and a research effort is on its way to address those. See, e.g.
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Otter is based on resolution and paramodulation, constrained by term orderings similar to those in the superposition calculus. The prover also supports positive and negative hyperresolution and a set-of-support strategy. Proof search is based on saturation using a version of the given-clause algorithm, and is controlled by several heuristics. There also are meta-heuristics determining search parameters automatically.
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In d dimensions, there are 2d−1 such orderings. One such variation is due to Peacock (see also Gosset for a 3D version) and another to Fasano and Franceschini (see Lopes et al. for a comparison and computational details). Critical values for the test statistic can be obtained by simulations, but depend on the dependence structure in the joint distribution.
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Representative sample for parallel coordinates. When used for statistical data visualisation there are three important considerations: the order, the rotation, and the scaling of the axes. The order of the axes is critical for finding features, and in typical data analysis many reorderings will need to be tried. Some authors have come up with ordering heuristics which may create illuminating orderings.
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Cited in Wilson, Paul (1992).Bárdos, Lajos cited in Kárpáti 1994, 171 The term "acoustic scale" is sometimes used to describe a particular mode of this seven-note collection (e.g. the specific ordering C–D–E–F–G–A–B) and is sometimes used to describe the collection as a whole (e.g. including orderings such as E–F–G–A–B–C–D).
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An example of an algorithm that runs in factorial time is bogosort, a notoriously inefficient sorting algorithm based on trial and error. Bogosort sorts a list of n items by repeatedly shuffling the list until it is found to be sorted. In the average case, each pass through the bogosort algorithm will examine one of the n! orderings of the n items.
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He introduced, together with Prom Panitchpakdi, the injective metric spaces under the name of "hyperconvex metric spaces". Together with Kennan T. Smith, Aronszajn offered proof of the Aronszajn–Smith theorem. Also, the existence of Aronszajn trees was proven by Aronszajn; Aronszajn lines, also named after him, are the lexicographic orderings of Aronszajn trees. He also made a contribution to the theory of reproducing kernel Hilbert space.
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The following tree is of the same sentence from Kafka's story. The glossing conventions are those established by Lehmann. One can easily see the extent to which Japanese is head-final: The Metamorphosis-Japanese A large majority of head-dependent orderings in Japanese are head-final. This fact is obvious in this tree, since structure is strongly ascending as speech and processing move from left to right.
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This ordering was used by ,. and then again by .. As cited by . Brouwer does not cite any references, but Moschovakis argues that he may either have seen , or have been influenced by earlier work of the same authors leading to this work. Much later, studied the same ordering, and credited it to Brouwer.. See in particular section 26, "A digression concerning recursive linear orderings", pp. 419–422.
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In computing, the SEX assembly language mnemonic has often been used for the "`Sign EXtend`" machine instruction found in the Motorola 6809. A computer's or CPU's "sex" can also mean the endianness of the computer architecture used.For hardware, the Jargon File also reports the less common expression byte sex . It is unclear whether this terminology is also used when more than two orderings are possible.
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The Society is now concerned with many activities around the theme of imprecise probabilities. Imprecise probability is understood in a very wide sense. It is used as a generic term to cover all mathematical models which measure chance or uncertainty without sharp numerical probabilities. It includes both qualitative (comparative probability, partial preference orderings,...) and quantitative models (interval probabilities, belief functions, upper and lower previsions,...).
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The collection of von Neumann ordinals, like the collection in the Russell paradox, cannot be a set in any set theory with classical logic. But the collection of order types in New Foundations (defined as equivalence classes of well-orderings under similarity) is actually a set, and the paradox is avoided because the order type of the ordinals less than \Omega turns out not to be \Omega.
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Word order in Indonesian is generally subject- verb-object (SVO), similar to that of most modern European languages, such as English. However considerable flexibility in word ordering exists, in contrast with languages such as Japanese or Korean, for instance, which always end clauses with verbs. Indonesian, while allowing for relatively flexible word orderings, does not mark for grammatical case, nor does it make use of grammatical gender.
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The Bruhat order on the Schubert varieties of a flag manifold or a Grassmannian was first studied by , and the analogue for more general semisimple algebraic groups was studied by . started the combinatorial study of the Bruhat order on the Weyl group, and introduced the name "Bruhat order" because of the relation to the Bruhat decomposition introduced by François Bruhat. The left and right weak Bruhat orderings were studied by .
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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 this section, the central concepts and definitions of domain theory will be introduced. The above intuition of domains being information orderings will be emphasized to motivate the mathematical formalization of the theory. The precise formal definitions are to be found in the dedicated articles for each concept. A list of general order-theoretic definitions, which include domain theoretic notions as well can be found in the order theory glossary.
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Potency is defined as those immanent orderings of possibility not yet eliminated by present actuality. Potency draws its form from those collections of possibility that exemplify constellations whose status relative to actuality is “might-be.” Auxier argues that possibility is immediately experienced by human beings, and that its form is inferred by means of contrast with the experience of the “egress” of possibility. Actuality is cognized only indirectly.
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Reflecting the aristocratic atmosphere, the poetry was elegant and sophisticated and expressed emotions in a rhetorical style. Editing the resulting anthologies of poetry soon became a national pastime. The poem, now one of two standard orderings for the Japanese syllabary, was also developed during the early Heian period. The Tale of Genji (), written in the early 11th century by a woman named , is considered the pre-eminent novel of Heian fiction.
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In ZFC, the order type of a well- ordering W is then defined as the unique von Neumann ordinal which is equinumerous with the field of W and membership on which is isomorphic to the strict well-ordering associated with W. (the equinumerousness condition distinguishes between well-orderings with fields of size 0 and 1, whose associated strict well-orderings are indistinguishable). In ZFC there cannot be a set of all ordinals. In fact, the von Neumann ordinals are an inconsistent totality in any set theory: it can be shown with modest set theoretical assumptions that every element of a von Neumann ordinal is a von Neumann ordinal and the von Neumann ordinals are strictly well-ordered by membership. It follows that the class of von Neumann ordinals would be a von Neumann ordinal if it were a set: but it would then be an element of itself, which contradicts the fact that membership is a strict well-ordering of the von Neumann ordinals.
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A widely used method for drawing (sampling) a random vector x from the N-dimensional multivariate normal distribution with mean vector μ and covariance matrix Σ works as follows: # Find any real matrix A such that . When Σ is positive-definite, the Cholesky decomposition is typically used, and the extended form of this decomposition can always be used (as the covariance matrix may be only positive semi-definite) in both cases a suitable matrix A is obtained. An alternative is to use the matrix A = UΛ½ obtained from a spectral decomposition Σ = UΛU−1 of Σ. The former approach is more computationally straightforward but the matrices A change for different orderings of the elements of the random vector, while the latter approach gives matrices that are related by simple re-orderings. In theory both approaches give equally good ways of determining a suitable matrix A, but there are differences in computation time.
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A graph that has a topological ordering cannot have any cycles, because the edge into the earliest vertex of a cycle would have to be oriented the wrong way. Therefore, every graph with a topological ordering is acyclic. Conversely, every directed acyclic graph has at least one topological ordering. Therefore, this property can be used as an alternative definition of the directed acyclic graphs: they are exactly the graphs that have topological orderings.
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These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of science.
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Sarvani Vakkalanka, Ganesh Gopalakrishnan, and Robert M. Kirby, Dynamic Verification of MPI Programs with Reductions in Presence of Split Operations and Relaxed Orderings, Computer Aided Verification (CAV 2008), pp. 66-79, LNCS 5123. ISP has been used to successfully verify up to 14,000 lines of MPI/C code for deadlocks and assertion violations. It currently supports over 60 MPI 2.1 functions, and has been tested with MPICH2, OpenMPI,and Microsoft MPI libraries.
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The category is generated by coface and codegeneracy maps, which amount to inserting or deleting elements of the orderings. (See simplicial set for relations of these maps.) A simplicial object is a presheaf on \Delta, that is a contravariant functor from \Delta to another category. For instance, simplicial sets are contravariant with the codomain category being the category of sets. A cosimplicial object is defined similarly as a covariant functor originating from \Delta.
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Non-dictatorship is one of the necessary conditions in Arrow's impossibility theorem.Game Theory Second Edition Guillermo Owen Ch 6 pp124-5 Axiom 5 Academic Press, 1982 In Social Choice and Individual Values, Kenneth Arrow defines non-dictatorship as: :There is no voter i in {1, ..., n} such that for every set of orderings in the domain of the constitution and every pair of social states x and y, x P_i y implies x P y.
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Andreas Raphael Blass (born October 27, 1947) is a mathematician, currently a professor at the University of Michigan. He works in mathematical logic, particularly set theory, and theoretical computer science. Blass graduated from the University of Detroit, where he was a Putnam Fellow, in 1966 with a B.S. in physics. He received his Ph.D. in 1970 from Harvard University, with a thesis on Orderings of Ultrafilters written under the supervision of Frank Wattenberg.
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His research team made 35,000 microfilm photographs of the manuscripts, which he dated to early part of the 8th century. Puin has not published the entirety of his work, but noted unconventional verse orderings, minor textual variations, and rare styles of orthography. He also suggested that some of the parchments were palimpsests which had been reused. Puin believed that this implied a text that changed over time as opposed to one that remained the same.
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Support for multi-dimensional arrays may also be provided by external libraries, which may even support arbitrary orderings, where each dimension has a stride value, and row-major or column-major are just two possible resulting interpretations. Row-major order is the default in NumPy (for Python). Column-major order is the default in Eigen and Armadillo(both for C++). A special case would be OpenGL (and OpenGL ES) for graphics processing.
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Three well- orderings on the set of natural numbers with distinct order types (top to bottom): \omega, \omega+5, and \omega+\omega. Every well-ordered set is order- equivalent to exactly one ordinal number. The ordinal numbers are taken to be the canonical representatives of their classes, and so the order type of a well-ordered set is usually identified with the corresponding ordinal. For example, the order type of the natural numbers is .
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Chordal bipartite graphs have various characterizations in terms of perfect elimination orderings, hypergraphs and matrices. They are closely related to strongly chordal graphs. By definition, chordal bipartite graphs have a forbidden subgraph characterization as the graphs that do not contain any induced cycle of length 3 or of length at least 5 (so-called holes) as an induced subgraph. Thus, a graph G is chordal bipartite if and only if G is triangle-free and hole-free.
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The familiar notion of vector addition for velocities in the Euclidean plane can be done in a triangular formation, or since vector addition is commutative, the vectors in both orderings geometrically form a parallelogram (see "parallelogram law"). This does not hold for relativistic velocity addition; instead a hyperbolic triangle arises whose edges are related to the rapidities of the boosts. Changing the order of the boost velocities, one does not find the resultant boost velocities to coincide.
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Chapter nine discusses ways to weaken Ramsey's theorem, and the final chapter discusses stronger theorems in combinatorics including the Dushnik–Miller theorem on self-embedding of infinite linear orderings, Kruskal's tree theorem, Laver's theorem on order embedding of countable linear orders, and Hindman's theorem on IP sets. An appendix provides a proof of a theorem of Jiayi Liu, part of the collection of results showing that the graph Ramsey theorem does not fall into the big five subsystems.
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The oldest known copy of the Quran so far belongs to this collection: it dates to the end of the 7th–8th centuries. The German scholar Gerd R. Puin has been investigating these Quran fragments for years. His research team made 35,000 microfilm photographs of the manuscripts, which he dated to early part of the 8th century. Puin has not published the entirety of his work, but noted unconventional verse orderings, minor textual variations, and rare styles of orthography.
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This encapsulation was later accomplished by the serializer construct ([Hewitt and Atkinson 1977, 1979] and [Atkinson 1980]). The first models of computation (e.g., Turing machines, Post productions, the lambda calculus, etc.) were based on mathematics and made use of a global state to represent a computational step (later generalized in [McCarthy and Hayes 1969] and [Dijkstra 1976] see Event orderings versus global state). Each computational step was from one global state of the computation to the next global state.
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Each proper cone C in a real vector space induces an order on the vector space by defining x ≤ y if and only if y − x ∈ C, and furthermore, the positive cone of this ordered vector space will be C. Therefore, there exists a one-to-one correspondence between the proper convex cones of X and the vector partial orders on X. By a total vector ordering on X we mean a total order on X that is compatible with the vector space structure of X. The family of total vector orderings on a vector space X is in one-to-one correspondence with the family of all proper cones that are maximal under set inclusion. A total vector ordering cannot be Archimedean if its dimension, when considered as a vector space over the reals, is greater than 1. If R and S are two orderings of a vector space with positive cones P and Q, respectively, then we say that R is finer than S if P ⊆ Q.
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Cohen has been an important participant in the discussion surrounding intellectual property and copyright. There has been an ongoing debate about the use of technology instead of, or in addition to, copyright to protect intellectual property in digital form. She has expressed concern about the potential legal impact of these technologies, as well as mass market contracts as they threaten individual privacy and autonomy.Friedman, David D., "In Defense of Private Orderings: Comments on Julie Cohen's "Copyright and the Jurisprudence of Self-Help.
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It may be clearer to apply Von Neumann cardinal assignment to finite cases and to use Scott's trick for sets which are infinite or do not admit well orderings. Note that cardinal and ordinal arithmetic agree for finite numbers. The α-th infinite initial ordinal is written \omega_\alpha, it is always a limit ordinal. Its cardinality is written \aleph_\alpha. For example, the cardinality of ω0 = ω is \aleph_0, which is also the cardinality of ω2 or ε0 (all are countable ordinals).
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Present research focuses on the ongoing re-figuration of spaces. Löw’s inquiries are based on the theory that nowadays the constitution of space is increasingly polycontexturally structured, which implies that multiple spatial orderings are more and more frequently effective at the same time in our actions and interactions. Along with Jörg Stollmann she is currently heading the research project on Smart Cities: Everyday life in digitized environments as part of the Collaborative Research Centre SFB 1265 “The Re-Figuration of Spaces”.
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He was a student at the University of Chicago earning a bachelor of science in mathematics in the year of 1948, and a master of science in mathematics in the following year 1949. After his time at the University of Chicago Kruskal attended Princeton University, where he completed his Ph.D. in 1954, nominally under Albert W. Tucker and Roger Lyndon, but de facto under Paul Erdős with whom he had two very short conversations. Kruskal worked on well-quasi- orderings www.cs.tau.ac.il www.cs.tau.ac.
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PQ trees are used to solve problems where the goal is to find an ordering that satisfies various constraints. In these problems, constraints on the ordering are included one at a time, by modifying the PQ tree structure in such a way that it represents only orderings satisfying the constraint. Applications of PQ trees include creating a contig map from DNA fragments, testing a matrix for the consecutive ones property, recognizing interval graphs and determining whether a graph is planar.
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Thus, a PC tree can only represent sets of orderings in which any circular permutation or reversal of an ordering in the set is also in the set. However, a PQ tree on n elements may be simulated by a PC tree on n + 1 elements, where the extra element serves to root the PC tree. The data structure operations required to perform a planarity testing algorithm on PC trees are somewhat simpler than the corresponding operations on PQ trees.
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Laver received his PhD at the University of California, Berkeley in 1969, under the supervision of Ralph McKenzie,Ralph McKenzie has been a doctoral student of James Donald Monk, who has been a doctoral student of Alfred Tarski. with a thesis on Order Types and Well-Quasi-Orderings. The largest part of his career he spent as Professor and later Emeritus Professor at the University of Colorado at Boulder. Richard Laver died in Boulder, CO, on September 19, 2012 after a long illness.
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In the mathematical area of graph theory, an undirected graph G is dually chordal if the hypergraph of its maximal cliques is a hypertree. The name comes from the fact that a graph is chordal if and only if the hypergraph of its maximal cliques is the dual of a hypertree. Originally, these graphs were defined by maximum neighborhood orderings and have a variety of different characterizations.; ; ; ; Unlike for chordal graphs, the property of being dually chordal is not hereditary, i.e.
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His notes are greatly valued by architectural historians, as they frequently provide a brief but informed record of the buildings as they were before Victorian restorations and re-orderings. Glynne often revisited the churches on two or three occasions at several years remove, and so the notes also provide a record of changes over time. Lawrence Butler considers that "in some ways he was the precursor of the Royal Commission on Historical Monuments in terms of ordering his descriptions".Butler 2011, p. 8.
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The consecutive open-notes of all-fifths tuning are spaced seven semi-tones apart on the chromatic circle. While the notes of the open- notes of the all-fifths and all-fourths tunings agree, their orderings are reversed. New standard tuning substitutes a G for the high B of all-fifths tuning. :C2-G2-D3-A3-E4-B4 All-fifths tuning is a tuning in intervals of perfect fifths like that of a mandolin, cello or violin; other names include "perfect fifths" and "fifths".
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Special algorithms have been developed for factorizing large sparse matrices. These algorithms attempt to find sparse factors L and U. Ideally, the cost of computation is determined by the number of nonzero entries, rather than by the size of the matrix. These algorithms use the freedom to exchange rows and columns to minimize fill-in (entries that change from an initial zero to a non-zero value during the execution of an algorithm). General treatment of orderings that minimize fill-in can be addressed using graph theory.
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Comparative sociology involves comparison of the social processes between nation states, or across different types of society (for example capitalist and socialist). There are two main approaches to comparative sociology: some seek similarity across different countries and cultures whereas others seek variance. For example, structural Marxists have attempted to use comparative methods to discover the general processes that underlie apparently different social orderings in different societies. The danger of this approach is that the different social contexts are overlooked in the search for supposed universal structures.
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Let F be a field. There is a bijection between the field orderings of F and the positive cones of F. Given a field ordering ≤ as in the first definition, the set of elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in the second definition, one can associate a total ordering ≤P on F by setting x ≤P y to mean y − x ∈ P. This total ordering ≤P satisfies the properties of the first definition.
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A partial order of dimension 4 (shown as a Hasse diagram) and four total orderings that form a realizer for this partial order. In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order. This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order. first studied order dimension; for a more detailed treatment of this subject than provided here, see .
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An asymptotically equivalent permutation test can be created when there are too many possible orderings of the data to allow complete enumeration in a convenient manner. This is done by generating the reference distribution by Monte Carlo sampling, which takes a small (relative to the total number of permutations) random sample of the possible replicates. The realization that this could be applied to any permutation test on any dataset was an important breakthrough in the area of applied statistics. The earliest known reference to this approach is Dwass (1957).
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This can be modeled as a Markov chain whose states are orderings of the card deck and whose state-to-state transition probabilities are given by some mathematical model of random shuffling such as the Gilbert–Shannon–Reeds model. In this situation, rapid mixing of the Markov chain means that one does not have to perform a huge number of shuffles to reach a sufficiently randomized state. Beyond card games, similar considerations arise in the behavior of the physical systems studied in statistical mechanics, and of certain randomized algorithms.
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For any propositions H1, H2, ... Hn, and permutation σ(n) of the numbers 1 through n, it is the case that: :H1 \land H2 \land ... \land Hn is equivalent to :Hσ(1) \land Hσ(2) \land Hσ(n). For example, if H1 is :It is raining H2 is :Socrates is mortal and H3 is :2+2=4 then It is raining and Socrates is mortal and 2+2=4 is equivalent to Socrates is mortal and 2+2=4 and it is raining and the other orderings of the predicates.
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The equivalence rules of relational algebra are exploited, in other words, different query structures and orderings can be mathematically proven to yield the same result. For example, filtering on fields A and B, or cross joining R and S can be done in any order, but there can be a performance difference. Multiple operations may be combined, and operation orders may be altered. The result of query rewriting may not be at the same abstraction level or application programming interface (API) as the original set of queries (though often is).
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If an infinite set is a well-ordered set, then it must have a nonempty, nontrivial subset that has no greatest element. In ZF, a set is infinite if and only if the power set of its power set is a Dedekind-infinite set, having a proper subset equinumerous to itself.. See in particular pp. 32–33. If the axiom of choice is also true, then infinite sets are precisely the Dedekind-infinite sets. If an infinite set is a well-orderable set, then it has many well-orderings which are non-isomorphic.
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This stability persists even in varying situations. Identity fusion theory proposes that this is due to the strong emotions, beliefs, and intra-relationships experienced by the strongly fused individuals. That is, although the overall fusion of a group of persons may shift in response to powerful situational forces, the rank orderings of individuals within the group will remain stable. Researchers have tested the “once fused, always fused” hypothesis by comparing the temporal stability of fusion-with-country scores for highly fused individuals with those of moderately or weakly fused individuals.
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The existence of order types for all well-orderings is not a theorem of Zermelo set theory: it requires the Axiom of replacement. Even Scott's trick cannot be used in Zermelo set theory without an additional assumption (such as the assumption that every set belongs to a rank which is a set, which does not essentially strengthen Zermelo set theory but is not a theorem of that theory). In NFU, the collection of all ordinals is a set by stratified comprehension. The Burali-Forti paradox is evaded in an unexpected way.
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Thus e.g. a utility function defines a preference relation. In this context, weak orderings are also known as preferential arrangements.. If X is finite or countable, every weak order on X can be represented by a function in this way.. However, there exist strict weak orders that have no corresponding real function. For example, there is no such function for the lexicographic order on Rn. Thus, while in most preference relation models the relation defines a utility function up to order-preserving transformations, there is no such function for lexicographic preferences.
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However, a different kind of move is possible, in which the weak orderings on a set are more highly connected. Define a dichotomy to be a weak ordering with two equivalence classes, and define a dichotomy to be compatible with a given weak ordering if every two elements that are related in the ordering are either related in the same way or tied in the dichotomy. Alternatively, a dichotomy may be defined as a Dedekind cut for a weak ordering. Then a weak ordering may be characterized by its set of compatible dichotomies.
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Because optimal vertex orderings are hard to find, heuristics have been used that attempt to reduce the number of colors while not guaranteeing an optimal number of colors. A commonly used ordering for greedy coloring is to choose a vertex of minimum degree, order the subgraph with removed recursively, and then place last in the ordering. The largest degree of a removed vertex that this algorithm encounters is called the degeneracy of the graph, denoted . In the context of greedy coloring, the same ordering strategy is also called the smallest last ordering.
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Wansbrough's works were widely noted, but perhaps not widely read. In 1972 a cache of ancient Qur'ans in a mosque in Sana'a, Yemen was discovered – commonly known as the Sana'a manuscripts. The German scholar Gerd R. Puin and his research team, who investigated these Quran fragments for many years, made approximately 35,000 microfilm photographs of the manuscripts, which he dated to early part of the 8th century. Puin has not published the entirety of his work, but noted unconventional verse orderings, minor textual variations, and rare styles of orthography.
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The iroha poem, now one of two standard orderings for the Japanese syllabary, was also written during the early part of this period. The 10th-century Japanese narrative, The Tale of the Bamboo Cutter, can be considered an early example of proto-science fiction. The protagonist of the story, Kaguya-hime, is a princess from the Moon who is sent to Earth for safety during a celestial war, and is found and raised by a bamboo cutter in Japan. She is later taken back to the Moon by her real extraterrestrial family.
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They showed that when this phrase was heard only once, listeners perceived it as speech, but after several repetitions, they perceived it as song. This perceptual transformation required that the intervening repetitions be exact; it did not occur when they were transposed slightly, or presented with the syllables in jumbled orderings. In addition, when listeners were asked to repeat back the phrase after hearing it once, they repeated it back as speech. Yet when they were asked to repeat back the phrase after hearing it ten times, they repeated it back as song.
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The proof is in two parts (Arrow, 1963, pp. 97–100). The first part considers the hypothetical case of some one voter's ordering that prevails ('is decisive') as to the social choice for some pair of social states no matter what that voter's preference for the pair, despite all other voters opposing. It is shown that, for a constitution satisfying Unrestricted Domain, Pareto and Independence, that voter's ordering would prevail for every pair of social states, no matter what the orderings of others. So, the voter would be a Dictator.
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Equivalently, a DAG is a directed graph that has a topological ordering, a sequence of the vertices such that every edge is directed from earlier to later in the sequence. DAGs can model many different kinds of information. For example, a spreadsheet can be modeled as a DAG, with a vertex for each cell and an edge whenever the formula in one cell uses the value from another; a topological ordering of this DAG can be used to update all cell values when the spreadsheet is changed. Similarly, topological orderings of DAGs can be used to order the compilation operations in a makefile.
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It was initially a philosophical interest, which led him around 1897 to study Georg Cantor's work. Already, in the summer semester of 1901, Hausdorff gave a lecture on set theory. This was one of the first lectures on set theory at all; Ernst Zermelo's lectures in Göttingen College during the winter semester of 1900/1901 were a little earlier. That year, he published his first paper on order types in which he examined a generalization of well-orderings called graded order types, where a linear order is graded if no two of its segments share the same order type.
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In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called rigid circuit graphs.
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The title sequence included stock shots of Edmund riding his horse on location, interspersed with shots of him doing silly things (and, usually, a shot of King Richard IV to go with Brian Blessed's credit). The closing titles were a similar sequence of Edmund riding, eventually falling off his horse, and then chasing after it. All the credits of the first series featured eccentric orderings of the cast list (such as "order of precedence", "order of witchiness" and "order of disappearance") and included "with additional dialogue by William Shakespeare" and "made in glorious television".Credits at the Internet Movie Database.
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Even such a seemingly limited ordering makes it possible to fix systemic regularities of the sort shown by Feigenbaum numbers and strange attractors. (...) Different types of orderings in the chaos phase may be brought together under the notion of directing, for they point to a possible general direction of system development and even its extreme states. But even if a general path is known, enormous difficulties remain in linking algorithmically the present state with the final one and in operationalizing the algorithms. These objectives are realized in the next two large phases that I call predispositioning and programming.
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An important feature of the multipolar exchange Hamiltonian is its anisotropy. The value of coupling constant C_{K_{i}K_{j}}^{Q{i}Q_{j}} is usually very sensitive to the relative angle between two multipoles. Unlike conventional spin only exchange Hamiltonian where the coupling constants are isotropic in a homogeneous system, the highly anisotropic atomic orbitals (recall the shape of the s,p,d,f wave functions) coupling to the system's magnetic moments will inevitably introduce huge anisotropy even in a homogeneous system. This is one of the main reasons that most multipolar orderings tend to be non- colinear.
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Authorities disagree about the history of the letter's name. The Oxford English Dictionary says the original name of the letter was in Latin; this became in Vulgar Latin, passed into English via Old French , and by Middle English was pronounced . The American Heritage Dictionary of the English Language derives it from French hache from Latin haca or hic. Anatoly Liberman suggests a conflation of two obsolete orderings of the alphabet, one with H immediately followed by K and the other without any K: reciting the former's ..., H, K, L,... as when reinterpreted for the latter ..., H, L,... would imply a pronunciation for H.
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The natural sum of α and β is often denoted by α⊕β or α#β, and the natural product by α⊗β or α⨳β. The natural operations come up in the theory of well partial orders; given two well partial orders S and T, of types (maximum linearizations) o(S) and o(T), the type of the disjoint union is o(S)⊕o(T), while the type of the direct product is o(S)⊗o(T).D. H. J. De Jongh and R. Parikh, Well-partial orderings and hierarchies, Indag. Math. 39 (1977), 195–206.
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The larger one, Im, is the group of all fractional ideals relatively prime to m (which means these fractional ideals do not involve any prime ideal appearing in mf). The smaller one, Pm, is the group of principal fractional ideals (u/v) where u and v are nonzero elements of OK which are prime to mf, u ≡ v mod mf, and u/v > 0 in each of the orderings of m∞. (It is important here that in Pm, all we require is that some generator of the ideal has the indicated form. If one does, others might not.
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It is sometimes asserted that such methods may trivially fail the universality criterion. However, it is more appropriate to consider that such methods fail Arrow's definition of an aggregation rule (or that of a function whose domain consists of preference profiles), if preference orderings cannot uniquely translate into a ballot. Methods which don't, often called "rated" or "cardinal" (as opposed to "ranked", "ordinal", or "preferential") electoral system, can be viewed as using information that only cardinal utility can convey. In that case, it is not surprising if some of them satisfy all of Arrow's conditions that are reformulated.
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In the results of a poll, one candidate may be clearly ahead of another, or the two candidates may be statistically tied, meaning not that their poll results are equal but rather that they are within the margin of error of each other. However, if candidate x is statistically tied with y, and y is statistically tied with z, it might still be possible for x to be clearly better than z, so being tied is not in this case a transitive relation. Because of this possibility, rankings of this type are better modeled as semiorders than as weak orderings..
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Held at the Science Museum of London, England, Fleming’s exhibition Atomism and Animism is an amalgamation of scientific objects of the Cartesian Enlightenment period, distributed in juxtaposing displays across the museum. In a similar light to the collaborative installation La Musee de Sciences with Lapointe in 1984, the displays created a reflective and critical response to museum practices which reestablished new and multiple orderings of scientific tropes and meanings. Other museum exhibition and project collaborations include Split + Splice at the Medical Museion of the University of Copenhagen (2008-2009); and You are Here: The Design of Information, at the Design Museum in London.
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Kui thereupon made songs in imitation of the sounds of > the forests and valleys, he covered earthenware tubs with fresh hides and > beat on them, and he slapped stones and hit rocks to imitate the sounds of > the jade stone chimes of the Supreme Sovereign, with which he made the > hundred wild beasts dance. … [After Shun ascended] The Sovereign Shun than > ordered Kui to perform "Nine Summonings," "Six Orderings," and "Six > Flowers," through which he illuminated the Power of the Sovereign. Note that the Lüshi Chunqiu says Kui was music master for both Yao and Shun, instead of only Shun.
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For arbitrarily many points in one dimension, there are again only finitely many solutions, one for each of the linear orderings (up to reversal of the ordering) of the points on a line. For every set of point masses, and every dimension less than , there exists at least one central configuration of that dimension. For almost all -tuples of masses there are finitely many "Dziobek" configurations that span exactly dimensions. It is an unsolved problem, posed by and , whether there is always a bounded number of central configurations for five or more masses in two or more dimensions.
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The quantum counting algorithm can be used to speed up solution to problems which are NP-complete. An example of an NP-complete problem is the Hamiltonian cycle problem, which is the problem of determining whether a graph G=(V,E) has a Hamiltonian cycle. A simple solution to the Hamiltonian cycle problem is checking, for each ordering of the vertices of G, whether it is a Hamiltonian cycle or not. Searching through all the possible orderings of the graph's vertices can be done with quantum counting followed by Grover's algorithm, achieving a speedup of the square root, similar to Grover's algorithm.
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Directed acyclic graphs representations of partial orderings have many applications in scheduling for systems of tasks with ordering constraints., p. 469. An important class of problems of this type concern collections of objects that need to be updated, such as the cells of a spreadsheet after one of the cells has been changed, or the object files of a piece of computer software after its source code has been changed. In this context, a dependency graph is a graph that has a vertex for each object to be updated, and an edge connecting two objects whenever one of them needs to be updated earlier than the other.
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For example, they reflect the fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events, but always such that the speed of light is the same in all inertial reference frames. The invariance of light speed is one of the postulates of special relativity. Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism. The Lorentz transformation is in accordance with Albert Einstein's special relativity, but was derived first.
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Another application of perfect elimination orderings is finding a maximum clique of a chordal graph in polynomial-time, while the same problem for general graphs is NP-complete. More generally, a chordal graph can have only linearly many maximal cliques, while non-chordal graphs may have exponentially many. To list all maximal cliques of a chordal graph, simply find a perfect elimination ordering, form a clique for each vertex v together with the neighbors of v that are later than v in the perfect elimination ordering, and test whether each of the resulting cliques is maximal. The clique graphs of chordal graphs are the dually chordal graphs.
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It is also possible to define an order isomorphism to be a surjective order-embedding. The two assumptions that f cover all the elements of T and that it preserve orderings, are enough to ensure that f is also one-to-one, for if f(x)=f(y) then (by the assumption that f preserves the order) it would follow that x\le y and y\le x, implying by the definition of a partial order that x=y. Yet another characterization of order isomorphisms is that they are exactly the monotone bijections that have a monotone inverse.This is the definition used by and .
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The original motivation for introducing semiorders was to model human preferences without assuming (as strict weak orderings do) that incomparability is a transitive relation. For instance, if x, y, and z represent three quantities of the same material, and x and z differ by the smallest amount that is perceptible as a difference, while y is halfway between the two of them, then it is reasonable for a preference to exist between x and z but not between the other two pairs, violating transitivity., p. 179. Thus, suppose that X is a set of items, and u is a utility function that maps the members of X to real numbers.
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Massey has made many original contributions as a mathematician by developing a theory of “dynamical queueing systems”. Classical queueing models assumed that calling rates were constant so they could use the static, equilibrium analysis of time homogeneous Markov chains. However, real communication systems call for the large scale analysis of queueing models with time-varying rates. His thesis at Stanford University created a dynamic, asymptotic method for time inhomogeneous Markov chains called “uniform acceleration” to deal with such problems. Moreover, his research on queueing networks led to new methods of comparing multi- dimensional, Markov processes by viewing them as “stochastic orderings” on “partially ordered spaces”.
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It may not be obvious that it can be proven, without using AC, that there even exists a nonzero ordinal onto which there is no surjection from the reals (if there is such an ordinal, then there must be a least one because the ordinals are wellordered). However, suppose there were no such ordinal. Then to every ordinal α we could associate the set of all prewellorderings of the reals having length α. This would give an injection from the class of all ordinals into the set of all sets of orderings on the reals (which can to be seen to be a set via repeated application of the powerset axiom).
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Topological orderings are also closely related to the concept of a linear extension of a partial order in mathematics. In high-level terms, there is an adjunction between directed graphs and partial orders. A partially ordered set is just a set of objects together with a definition of the "≤" inequality relation, satisfying the axioms of reflexivity (x ≤ x), antisymmetry (if x ≤ y and y ≤ x then x = y) and transitivity (if x ≤ y and y ≤ z, then x ≤ z). A total order is a partial order in which, for every two objects x and y in the set, either x ≤ y or y ≤ x.
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The model R of real numbers with its usual order and the model Q of rational numbers with its usual order are elementarily equivalent, since they both interpret '<' as an unbounded dense linear ordering. This is sufficient to ensure elementary equivalence, because the theory of unbounded dense linear orderings is complete, as can be shown by the Łoś–Vaught test. More generally, any first-order theory with an infinite model has non- isomorphic, elementarily equivalent models, which can be obtained via the Löwenheim–Skolem theorem. Thus, for example, there are non-standard models of Peano arithmetic, which contain other objects than just the numbers 0, 1, 2, etc.
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For the purpose of dictionary orderingDifferent dictionaries use slightly different orderings; the system presented here is that used in the official Cambodian Dictionary, as described by Huffman (1970), p. 305. of words, main consonants, subscript consonants and dependent vowels are all significant; and when they appear in combination, they are considered in the order in which they would be spoken (main consonant, subscript, vowel). The order of the consonants and of the dependent vowels is the order in which they appear in the above tables. A syllable written without any dependent vowel is treated as if it contained a vowel character that precedes all the visible dependent vowels.
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A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (v0,...,vk), with the rule that two orderings define the same orientation if and only if they differ by an even permutation. Thus every simplex has exactly two orientations, and switching the order of two vertices changes an orientation to the opposite orientation. For example, choosing an orientation of a 1-simplex amounts to choosing one of the two possible directions, and choosing an orientation of a 2-simplex amounts to choosing what "counterclockwise" should mean.
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Different orderings of the vertices of a graph may cause the greedy coloring to use different numbers of colors, ranging from the optimal number of colors to, in some cases, a number of colors that is proportional to the number of vertices in the graph. For instance, a crown graph (a graph formed from two disjoint sets of vertices } and } by connecting to whenever ) can be a particularly bad case for greedy coloring. With the vertex ordering , a greedy coloring will use colors, one color for each pair . However, the optimal number of colors for this graph is two, one color for the vertices and another for the vertices .
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The Journal of Bisexuality was first published in 2000 by the Taylor & Francis Group under the Routledge imprint, and its editors-in-chief have included Fritz Klein, Jonathan Alexander, Brian Zamboni, James D. Weinrich, and M. Paz Galupo. In 2000, law scholar Kenji Yoshino published the influential article "The Epistemic Contract of Bisexual Erasure," which argues that "Straights and gays have an investment in stabilizing sexual orientation categories. The shared aspect of this investment is the security that all individuals draw from rigid social orderings." In 2001, Steven Angelides published A History of Bisexuality, in which he argues that bisexuality has operated historically as a structural other to sexual identity itself.
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The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual difference of two numbers, which is not given by the order). Other familiar examples of orderings are the alphabetical order of words in a dictionary and the genealogical property of lineal descent within a group of people. The notion of order is very general, extending beyond contexts that have an immediate, intuitive feel of sequence or relative quantity. In other contexts orders may capture notions of containment or specialization.
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In practice, corporate reputations are revealed by the relative rankings of companies created and propagated by information intermediaries. For example, business magazines and newspapers such as Fortune, Forbes, Business Week, Financial Times, and The Wall Street Journal regularly publish lists of the best places to work, the best business schools, or the most innovative companies. These rankings are explicit orderings of corporate reputations, and the relative positions of companies on these rankings are a reflection of the relative performance of companies on different cognitive attributes. Corporate reputations are found to influence the attractiveness of ranked companies' as suppliers of products, as prospective employers, and as investments.
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Therefore, it fits the requirements for the join operation of a lattice. Symmetrically, the operation P\wedge Q fits the requirements for the meet operation. Because they are defined using an element-wise minimum or element- wise maximum in the preference ordering, these two operations obey the same distributive laws obeyed by the minimum and maximum operations on linear orderings: for every three different matchings P, Q, and R, :P\wedge(Q\vee R)=(P\wedge Q)\vee (P\wedge R) and :P\vee(Q\wedge R)=(P\vee Q)\wedge (P\vee R) Therefore, the lattice of stable matchings is a distributive lattice.
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In a simple case, the intervals do not overlap and they can be inserted into a simple binary search tree and queried in O(\log n) time. However, with arbitrarily overlapping intervals, there is no way to compare two intervals for insertion into the tree since orderings sorted by the beginning points or the ending points may be different. A naive approach might be to build two parallel trees, one ordered by the beginning point, and one ordered by the ending point of each interval. This allows discarding half of each tree in O(\log n) time, but the results must be merged, requiring O(n) time.
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Abrahams challenges the dramatic orderings in her web performance of Huis Clos/No Exit – On Collaboration situated in CNES La Chartreuse in early 2010 for six performers whom became accustomed to reading material, illustrations that symbolized and depicted computer monitors. The purpose was to make the creation come to life through the performers. With the title pertaining to a theatrical terms as a claustrophobic piece of Satre, Abrahams integrates the virtual world and the performance device at a distance through the visual and audio recordings of a webcam clustered together to create a split screen. The objective of the piece was to test the levels of intimacy that can be gained from webcam sessions through a task based activity.
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The original definition of ordinal numbers, found for example in the Principia Mathematica, defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned in ZF and related systems of axiomatic set theory because these equivalence classes are too large to form a set. However, this definition still can be used in type theory and in Quine's axiomatic set theory New Foundations and related systems (where it affords a rather surprising alternative solution to the Burali-Forti paradox of the largest ordinal).
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A simple example of two cardinal utility functions u (first column) and v (second column) whose values in all circumstances are related by In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the value u(x_i) of one index u, occurring at any quantity x_i of the goods bundle being evaluated, the corresponding value v(x_i) of the other index v satisfies a relationship of the form :v(x_i) = au(x_i) + b\\!, for fixed constants a and b. Thus the utility functions themselves are related by :v(x) = au(x) + b.
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It assumes that the universe is dynamic and thus "the process of finding and creating order in the universe is viewed as fundamental to human life and inquiry." Dialectical thinking sets out to understand what is not known or understood about current ways of ordering and then works to create new orderings that are inclusive of what was unattended to previously. In this sense, dialectical thinkers show a tendency towards finding out what the best ordering of the universe could be in a current moment and are wary making claims about a presupposed order without critically evaluating many other perspectives and evidence. Styles of thought are relevant to making sense of the subjectivity involved in postformal thought.
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Orderings of the 3-subsets of {1, ..., 6}, represented as sets of red squares, increasing sequences (in blue), or by their indicator functions, converted in decimal notation (in grey). The grey numbers are also the rank of the subsets in all subsets of {1, ..., 6}, numbered in colexicographical order, and starting from 0. The lexicographical (lex) and colexicographical (colex) orders are on the top and the corresponding reverse orders (rev) on the bottom One passes from an order to its reverse order, either by reading bottom-up instead of up-bottom, or by exchanging red and white colors. In combinatorics, one has often to enumerate, and therefore to order the finite subsets of a given set .
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Then the sphere sequence (S^0, S^1,\dots) has the structure of a monoid and spectra are just modules over this monoid. If this monoid was commutative, then a monoidal structure on the category of modules over it would arise (as in algebra the modules over a commutative ring have a tensor product). But the monoid structure of the sphere sequence is not commutative due to different orderings of the coordinates. The idea is now that one can build the coordinate changes into the definition of a sequence: a symmetric sequence is a sequence of spaces (X_0, X_1, \dots) together with an action of the n-th symmetric group on X_n.
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As with the chordal graphs (and unlike the perfectly orderable graphs more generally) the graphs with width four are recognizable in polynomial time.; . A concept intermediate between the perfect elimination ordering of a chordal graph and a perfect ordering is a semiperfect elimination ordering: in an elimination ordering, there is no three-vertex induced path in which the middle vertex is the first of the three to be eliminated, and in a semiperfect elimination ordering, there is no four-vertex induced path in which one of the two middle vertices is the first to be eliminated. The reverse of this ordering therefore satisfies the requirements of a perfect ordering, so graphs with semiperfect elimination orderings are perfectly orderable.
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Recursion theory includes the study of generalized notions of this field such as arithmetic reducibility, hyperarithmetical reducibility and α-recursion theory, as described by Sacks (1990). These generalized notions include reducibilities that cannot be executed by Turing machines but are nevertheless natural generalizations of Turing reducibility. These studies include approaches to investigate the analytical hierarchy which differs from the arithmetical hierarchy by permitting quantification over sets of natural numbers in addition to quantification over individual numbers. These areas are linked to the theories of well-orderings and trees; for example the set of all indices of recursive (nonbinary) trees without infinite branches is complete for level \Pi^1_1 of the analytical hierarchy.
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In the 1960s a number of the fittings, including Pugin's screen, were removed and the interior repainted, to the detriment of the original design. The rood screen was rescued by an Anglican priest, who had it re-erected in the Anglican Holy Trinity Church, Reading. Other artefacts were removed to other churches, including the giant rood crucifix, which after its removal to the Church of the Sacred Heart & St Therese, in Coleshill, was reinstated in the cathedral within the Sanctuary on the instructions of Archbishop Maurice Couve de Murville. The Cathedral as it appears today is a result of post-Vatican II renovations and re-orderings, with only some of Pugin's work surviving.
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Chords are described here in terms of intervals relative to the root of the chord, arranged from smaller intervals to larger. This is a standard method used when describing jazz chords as it shows them hierarchically: Lower intervals (third, fifth and seventh) are more important in defining the function of the chord than the upper intervals or extensions (9th, 11th, 13th), which add color. Although it is possible to play the chords as described here literally, it is possible to use different orderings of the same notes, known as a voicings, or even by omitting certain notes.For instance, the dominant seventh 11 or Lydian dominant, C711, comprises the notes: :root (often omitted), 3, (5), 7, (9), 11, (13).
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Baumgartner's axiom A is an axiom for partially ordered sets introduced in . A partial order (P, ≤) is said to satisfy axiom A if there is a family ≤n of partial orderings on P for n = 0, 1, 2, ... such that # ≤0 is the same as ≤ #If p ≤n+1q then p ≤nq #If there is a sequence pn with pn+1 ≤n pn then there is a q with q ≤n pn for all n. #If I is a pairwise incompatible subset of P then for all p and for all natural numbers n there is a q such that q ≤n p and the number of elements of I compatible with q is countable.
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For, if the vertex separation number of a topological ordering is at most w, the minimum vertex separation among all orderings can be no larger, so the undirected graph formed by ignoring the orientations of the DAG described above must have pathwith at most w. It is possible to test whether this is the case, using the known fixed-parameter- tractable algorithms for pathwidth, and if so to find a path-decomposition for the undirected graph, in linear time given the assumption that w is a constant. Once a path decomposition has been found, a topological ordering of width w (if one exists) can be found using dynamic programming, again in linear time.
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The selected element is removed from all the lists where it appears as a head and appended to the output list. The process of selecting and removing a good head to extend the output list is repeated until all remaining lists are exhausted. If at some point no good head can be selected, because the heads of all remaining lists appear in any one tail of the lists, then the merge is impossible to compute due to inconsistent orderings of dependencies in the inheritance hierarchy and no linearization of the original class exists. A naive divide and conquer approach to computing the linearization of a class may invoke the algorithm recursively to find the linearizations of parent classes for the merge-subroutine.
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The Sophocles 2007 beta provides an interface for developing and maintaining a step outline in conjunction with the screenplay. Steps – the fundamental events of a plot – can be assigned to color-coded story threads, and scripts can contain multiple threads. Steps are maintained in two distinct orderings: presentation order and chronological order. Presentation order (or script order) is the order in which the steps will unfold on-screen; chronological order is the order in which the steps take place in the world of the story (the two would differ in the case of a flashback, for example). Steps can be flagged as implemented after they’ve been written into the script, or off-screen to indicate they won’t be implemented at all.
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This is not straightforward because the maximum difference between two joint cumulative distribution functions is not generally the same as the maximum difference of any of the complementary distribution functions. Thus the maximum difference will differ depending on which of \Pr(x < X \land y < Y) or \Pr(X < x \land Y > y) or any of the other two possible arrangements is used. One might require that the result of the test used should not depend on which choice is made. One approach to generalizing the Kolmogorov–Smirnov statistic to higher dimensions which meets the above concern is to compare the cdfs of the two samples with all possible orderings, and take the largest of the set of resulting K–S statistics.
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A natural number (which, in this context, includes the number 0) can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets, these two concepts coincide, and there is only one way to put a finite set into a linear sequence (up to isomorphism). When dealing with infinite sets, however, one has to distinguish between the notion of size, which leads to cardinal numbers, and the notion of position, which leads to the ordinal numbers described here. This is because while any set has only one size (its cardinality), there are many nonisomorphic well-orderings of any infinite set, as explained below.
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So ω can be identified with \aleph_0, except that the notation \aleph_0 is used when writing cardinals, and ω when writing ordinals (this is important since, for example, \aleph_0^2 = \aleph_0 whereas \omega^2 > \omega). Also, \omega_1 is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers: each such well-ordering defines a countable ordinal, and \omega_1 is the order type of that set), \omega_2 is the smallest ordinal whose cardinality is greater than \aleph_1, and so on, and \omega_\omega is the limit of the \omega_n for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the \omega_n).
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Sending Kenna to the flank of the enemy to block any retreats, Egerton opened his mountain battery guns orderings his troops to kneel or lay down. Upon the commencing of the fight the dervish rushed from cover to cover to through grass and brushes attacking the left. However they were unable to stand the intensity of the fire, the dervish again regrouped and tried to attack from the front and the right. Sargent Gibbs did an excellent job with the maxim gun, and the K.A.R and Sikh firing was remarkable, with in few minutes the dervish line collapsed and they retreated in full flight, pursued by the mounted Gedabursi Horses and Tribal Horses the rout of the dervish army was completed.
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Sending Kenna to the flank of the enemy to block any retreats, Egerton opened his mountain battery guns orderings his troops to kneel or lay down. Upon the commencing of the fight the dervish rushed from cover to cover to through grass and brushes attacking the left. However they were unable to stand the intensity of the fire, the dervish again regrouped and tried to attack from the front and the right. Sergeant Gibbs did an excellent job with the maxim gun, and the K.A.R and Sikh firing was remarkable, with in few minutes the dervish line collapsed and they retreated in full flight, pursued by the mounted Gedabursi Horses and Tribal Horses the rout of the dervish army was completed.
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In some cases, there are different ways to import the concepts into ZFC and NFU. For example, the usual definition of the first infinite ordinal \omega in ZFC is not suitable for NFU because the object (defined in purely set theoretical language as the set of all finite von Neumann ordinals) cannot be shown to exist in NFU. The usual definition of \omega in NFU is (in purely set theoretical language) the set of all infinite well-orderings all of whose proper initial segments are finite, an object which can be shown not to exist in ZFC. In the case of such imported objects, there may be different definitions, one for use in ZFC and related theories, and one for use in NFU and related theories.
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Here a modulus (or ray divisor) is a formal finite product of the valuations (also called primes or places) of K with positive integer exponents. The archimedean valuations that might appear in a modulus include only those whose completions are the real numbers (not the complex numbers); they may be identified with orderings on K and occur only to exponent one. The modulus m is a product of a non- archimedean (finite) part mf and an archimedean (infinite) part m∞. The non- archimedean part mf is a nonzero ideal in the ring of integers OK of K and the archimedean part m∞ is simply a set of real embeddings of K. Associated to such a modulus m are two groups of fractional ideals.
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33 by and so focused his theorem on preference rankings, but later stated that a cardinal score system with three or four classes "is probably the best". Arrow's framework assumes that individual and social preferences are "orderings" (i.e., satisfy completeness and transitivity) on the set of alternatives. This means that if the preferences are represented by a utility function, its value is an ordinal utility in the sense that it is meaningful so far as the greater value indicates the better alternative. For instance, having ordinal utilities of 4, 3, 2, 1 for alternatives a, b, c, d, respectively, is the same as having 1000, 100.01, 100, 0, which in turn is the same as having 99, 98, 1, .997.
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For something to be economically scarce it must necessarily have the exclusivity property—that use by one person excludes others from using it. These two justifications lead to different conclusions on what can be property. Intellectual property—incorporeal things like ideas, plans, orderings and arrangements (musical compositions, novels, computer programs)—are generally considered valid property to those who support an effort justification, but invalid to those who support a scarcity justification, since the things don't have the exclusivity property (however, those who support a scarcity justification may still support other "intellectual property" laws such as Copyright, as long as these are a subject of contract instead of government arbitration). Thus even ardent propertarians may disagree about IP. By either standard, one's body is one's property.
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One sociologist who employed comparative methods to understand variance was Max Weber, whose studies attempted to show how differences between cultures explained the different social orderings that had emerged (see for example The Protestant Ethic and the Spirit of Capitalism and Sociology of religion). There is some debate within sociology regarding whether the label of 'comparative' is suitable. Emile Durkheim argued in The Rules of Sociological Method (1895) that all sociological research was in fact comparative since social phenomenon are always held to be typical, representative or unique, all of which imply some sort of comparison. In this sense, all sociological analysis is comparative and it has been suggested that what is normally referred to as comparative research, may be more appropriately called cross-national research.
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Gerd R Puin photo of one of his Sana'a Quran parchments Gerd Puin was the head of a restoration project, commissioned by the Yemeni government, which spent a significant amount of time examining the ancient Quranic manuscripts discovered in Sana'a, Yemen, in 1972, in order to find criteria for systematically cataloging them. According to writer Toby Lester, his examination revealed "unconventional verse orderings, minor textual variations, and rare styles of orthography and artistic embellishment." The scriptures were written in the early Hijazi Arabic script, matching the pieces of the earliest Qurans known to exist. Some of the papyrus on which the text appears shows clear signs of earlier use, being that previous, washed-off writings are also visible on it.
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The New Order in Europe: German and other Axis conquests in Europe during World War II. The term Neuordnung originally had a different and more limited meaning than in its present usage. It is typically translated as New Order, but a more correct translation would actually be more akin to reorganization. When it was used in Germany during the Third Reich-era it referred specifically to the Nazis' desire to essentially redraw the contemporary state borders within Europe, thereby changing the then-existing geopolitical structures. In the same sense it has also been used now and in the past to denote similar re-orderings of the international political order such as the Peace of Westphalia in 1648, the Vienna Congress in 1815, and the Allied victory in 1945.
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Different choices of the sequence of vertices will typically produce different colorings of the given graph, so much of the study of greedy colorings has concerned how to find a good ordering. There always exists an ordering that produces an optimal coloring, but although such orderings can be found for many special classes of graphs, they are hard to find in general. Commonly used strategies for vertex ordering involve placing higher-degree vertices earlier than lower-degree vertices, or choosing vertices with fewer available colors in preference to vertices that are less constrained. Variations of greedy coloring choose the colors in an online manner, without any knowledge of the structure of the uncolored part of the graph, or choose other colors than the first available in order to reduce the total number of colors.
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The greedy coloring algorithm, when applied to a given ordering of the vertices of a graph G, considers the vertices of the graph in sequence and assigns each vertex its first available color, the minimum excluded value for the set of colors used by its neighbors. Different vertex orderings may lead this algorithm to use different numbers of colors. There is always an ordering that leads to an optimal coloring – this is true, for instance, of the ordering determined from an optimal coloring by sorting the vertices by their color – but it may be difficult to find. The perfectly orderable graphs are defined to be the graphs for which there is an ordering that is optimal for the greedy algorithm not just for the graph itself, but for all of its induced subgraphs.
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Relativity of simultaneity: Event B is simultaneous with A in the green reference frame, but it occurred before in the blue frame, and occurs later in the red frame. Einstein showed in his thought experiments that people travelling at different speeds, while agreeing on cause and effect, measure different time separations between events, and can even observe different chronological orderings between non-causally related events. Though these effects are typically minute in the human experience, the effect becomes much more pronounced for objects moving at speeds approaching the speed of light. Subatomic particles exist for a well known average fraction of a second in a lab relatively at rest, but when travelling close to the speed of light they are measured to travel farther and exist for much longer than when at rest.
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The great debate between defining notions of space and time as real objects themselves (absolute), or mere orderings upon actual objects (relational), began between physicists Isaac Newton (via his spokesman, Samuel Clarke) and Gottfried Leibniz in the papers of the Leibniz–Clarke correspondence. Arguing against the absolutist position, Leibniz offers a number of thought experiments with the purpose of showing that there is contradiction in assuming the existence of facts such as absolute location and velocity. These arguments trade heavily on two principles central to his philosophy: the principle of sufficient reason and the identity of indiscernibles. The principle of sufficient reason holds that for every fact, there is a reason that is sufficient to explain what and why it is the way it is and not otherwise.
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Schoenberg, inventor of twelve-tone technique The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve- note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law of the twelve tones" in 1919. In 1923, Arnold Schoenberg (1874–1951) developed his own, better-known version of 12-tone technique, which became associated with the "Second Viennese School" composers, who were the primary users of the technique in the first decades of its existence. The technique is a means of ensuring that all 12 notes of the chromatic scale are sounded as often as one another in a piece of music while preventing the emphasis of any one notePerle 1977, 2. through the use of tone rows, orderings of the 12 pitch classes.
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As an example, the collection of input triples :(2,1,3), (3,4,5), (1,4,5), (2,4,1), (5,2,3) is satisfied by the output ordering :3, 1, 4, 2, 5 but not by :3, 1, 2, 4, 5. In the first of these output orderings, for all five of the input triples, the middle item of the triple appears between the other two items However, for the second output ordering, item 4 is not between items 1 and 2, contradicting the requirement given by the triple (2,4,1). If an input contains two triples like (1,2,3) and (2,3,1) with the same three items but a different choice of the middle item, then there is no valid solution. However, there are more complicated ways of forming a set of triples with no valid solution, that do not contain such a pair of contradictory triples.
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Advantages of ion beam mixing as a means of synthesis over traditional modes of implantation include the process' ability to produce materials with high solute concentrations using lower amounts of irradiation, and better control of band gap variation and diffusion between layers. The cost of IM is also less prohibitive than that of other modes of film preparation on substrates, such as chemical vapor deposition (CVD) and molecular beam epitaxy (MBE). Disadvantages include the inability to completely direct and control lattice displacements initiated in the process, which can result in an undesirable degree of disorder in ion mixed samples, rendering them unsuitable for applications in which precise lattice orderings are paramount. Ion beams cannot be perfectly directed, nor the collision cascade controlled, once IM effects propagate, which can result in leaking, electron diffraction, radiation enhanced diffusion (RED), chemical migration and mismatch.
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If a topological sort has the property that all pairs of consecutive vertices in the sorted order are connected by edges, then these edges form a directed Hamiltonian path in the DAG. If a Hamiltonian path exists, the topological sort order is unique; no other order respects the edges of the path. Conversely, if a topological sort does not form a Hamiltonian path, the DAG will have two or more valid topological orderings, for in this case it is always possible to form a second valid ordering by swapping two consecutive vertices that are not connected by an edge to each other. Therefore, it is possible to test in linear time whether a unique ordering exists, and whether a Hamiltonian path exists, despite the NP- hardness of the Hamiltonian path problem for more general directed graphs.
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Some influential approaches within the sociology of law have challenged definitions of law in terms of official (state) law (see for example Eugen Ehrlich's concept of "living law" and Georges Gurvitch's "social law"). From this standpoint, law is understood broadly to include not only the legal system and formal (or official) legal institutions and processes, but also various informal (or unofficial) forms of nomativity and regulation which are generated within groups, associations and communities. The sociological studies of law are, thus, not limited to analysing how the rules or institutions of the legal system interact with social class, gender, race, religion, sexuality and other social categories. They also focus on how the internal normative orderings of various groups and "communities", such as the community of lawyers, businessmen, scientists, members of political parties, or members of the Mafia, interact with each other.
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Suppose that the Coxeter group W is the symmetric group Sn and P is the parabolic subgroup Sk×Sn–k. Then W/P can be identified with the k-element subsets of the n-element set {1,2,...,n} and the elements w of W correspond to the linear orderings of this set. A Coxeter matroid consists of k elements sets such that for each w there is a unique minimal element in the corresponding Bruhat ordering of k-element subsets. This is exactly the definition of a matroid of rank k on an n-element set in terms of bases: a matroid can be defined as some k-element subsets called bases of an n-element set such that for each linear ordering of the set there is a unique minimal base in the Gale ordering of k-element subsets.
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Semiorders generalize strict weak orderings, but do not assume transitivity of incomparability.. A strict weak order that is trichotomous is called a strict total order.. The total preorder which is the inverse of its complement is in this case a total order. For a strict weak order "<" another associated reflexive relation is its reflexive closure, a (non-strict) partial order "≤". The two associated reflexive relations differ with regard to different a and b for which neither a < b nor b < a: in the total preorder corresponding to a strict weak order we get a \lesssim b and b \lesssim a, while in the partial order given by the reflexive closure we get neither a ≤ b nor b ≤ a. For strict total orders these two associated reflexive relations are the same: the corresponding (non-strict) total order.
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According to Carnie, the sloppy identity problem can be explained using an LF-copying hypothesis. This hypothesis claims that before SPELLOUT, the elided VP has no structure and exists as an empty or null VP (as opposed to the PF-deletion hypothesis, which asserts that the elided VP has structure throughout the derivation). Only by copying structure from the VP antecedent does it have structure at LF. These copying processes occur covertly from SPELLOUT to LF. According to LF-copying, the ambiguity found in sloppy identity is due to different orderings of the copying rules for pronouns and verbs. Consider the following derivations of the sloppy reading and the strict reading for sentence (4), using the Covert VP-Copying Rule and the Anaphor-Copying Rule: 4) Calvin will strike himself and Otto will [vp Ø] too.
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Every directed acyclic graph has a topological ordering, an ordering of the vertices such that the starting endpoint of every edge occurs earlier in the ordering than the ending endpoint of the edge. The existence of such an ordering can be used to characterize DAGs: a directed graph is a DAG if and only if it has a topological ordering. In general, this ordering is not unique; a DAG has a unique topological ordering if and only if it has a directed path containing all the vertices, in which case the ordering is the same as the order in which the vertices appear in the path.. The family of topological orderings of a DAG is the same as the family of linear extensions of the reachability relation for the DAG,. so any two graphs representing the same partial order have the same set of topological orders.
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Martina Löw developed the idea of a "relational" model of space, which focuses on the “orderings”Löw, Martina (2008), »The Constitution of Space: The Structuration of Spaces Through the Simultaneity of Effects and Perception«, in: European Journal of Social Theory, 1, 11 of living entities and social goods, and examines how space is constituted in processes of perception, recall, or ideation to manifest itself as societal structure. From a social theory point of view, it follows on from the theory of structuration proposed by Anthony Giddens,Giddens, Anthony (1984), The Constitution of Society. Outline of the Theory of Structuration. Cambridge: Polity whose concept of the “duality of structure” Löw extends sociological terms into a “duality of space.” The basic idea is that individuals act as social agents (and constitute spaces in the process), but that their action depends on economic, legal, social, cultural, and, finally, spatial structures.
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However Napoleon defeated the Austrians and on 15 May 1796 entered in Milan, founding the Transpadane Republic and commandeering ecclesiastic properties. The population reacted with riots, particularly in Binasco and Pavia, and Visconti, to please the new ruler, tried to calm the turmoils and ordered prayers in all churches in favour of the army of the French First Republic. However the situation of the Milanese church got worse and worse due to the open anticlerical orderings of the just created Cisalpine Republic, such as the exclusion of the bishop from the appointments of the parish priests, the prohibition of processions in the streets, the covering of religious images on the wall of the houses, the disband of most chapters and of many religious orders. A brief break occurred between 1799 and 1800 when the Austrians returned in Milan for a few months: Visconti openly rejoiced for the change of rulers who revoked some of the anticlerical measures.
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After five introductory chapters on naive set theory and set-theoretic notation, and a sixth chapter on the axiom of choice, the book has four chapters on cardinal numbers, their arithmetic, and series and products of cardinal numbers, comprising approximately 50 pages. Following this, four longer chapters (totalling roughly 180 pages) cover orderings of sets, order types, well-orders, ordinal numbers, ordinal arithmetic, and the Burali-Forti paradox according to which the collection of all ordinal numbers cannot be a set. Three final chapters concern aleph numbers and the continuum hypothesis, statements equivalent to the axiom of choice, and consequences of the axiom of choice. The second edition makes only minor changes to the first except for adding footnotes concerning two later developments in the area: the proof by Paul Cohen of the independence of the continuum hypothesis, and the construction by Robert M. Solovay of the Solovay model in which all sets of real numbers are Lebesgue measurable.
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Among the other contributions in this book, Whitworth was the first to use ordered Bell numbers to count the number of weak orderings of a set, in the 1886 edition. These numbers had been studied earlier by Arthur Cayley, but for a different problem.. He was the first to publish Bertrand's ballot theorem, in 1878; the theorem is misnamed after Joseph Louis François Bertrand, who rediscovered the same result in 1887.. He is the inventor of the E[X] notation for the expected value of a random variable X, still commonly in use,. and he coined the name "subfactorial" for the number of derangements of n items.. Another of Whitworth's contributions, in geometry, concerns equable shapes, shapes whose area has the same numerical value (with a different set of units) as their perimeter. As Whitworth showed with D. Biddle in 1904, there are exactly five equable triangles with integer sides: the two right triangles with side lengths (5,12,13) and (6,8,10), and the three triangles with side lengths (6,25,29), (7,15,20), and (9,10,17)..
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The following conditions exploit the primal graph of the problem, which has a vertex for each variable and an edge between two nodes if the corresponding variables are in a constraint. The following are conditions on binary constraint satisfaction problems where enforcing local consistency is tractable and allows establishing satisfiability: # enforcing arc consistency, if the primal graph is acyclic; # enforcing directional arc consistency for an ordering of the variables that makes the ordered graph of constraint having width 1 (such an ordering exists if and only if the primal graph is a tree, but not all orderings of a tree generate width 1); # enforcing strong directional path consistency for an ordering of the variables that makes the primal graph having induced width 2. A condition that extends the last one holds for non-binary constraint satisfaction problems. Namely, for all problems for which there exists an ordering that makes the primal graph having induced width bounded by a constant i, enforcing strong directional i-consistency is tractable and allows establishing satisfiability.
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As mentioned above, weak orders have applications in utility theory. In linear programming and other types of combinatorial optimization problem, the prioritization of solutions or of bases is often given by a weak order, determined by a real- valued objective function; the phenomenon of ties in these orderings is called "degeneracy", and several types of tie-breaking rule have been used to refine this weak ordering into a total ordering in order to prevent problems caused by degeneracy.. Weak orders have also been used in computer science, in partition refinement based algorithms for lexicographic breadth-first search and lexicographic topological ordering. In these algorithms, a weak ordering on the vertices of a graph (represented as a family of sets that partition the vertices, together with a doubly linked list providing a total order on the sets) is gradually refined over the course of the algorithm, eventually producing a total ordering that is the output of the algorithm.. In the Standard Library for the C++ programming language, the set and multiset data types sort their input by a comparison function that is specified at the time of template instantiation, and that is assumed to implement a strict weak ordering.
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Badeldin Shogar, a Sudanese migrant, at the Jungle in October 2015 Migrants based in Calais were attempting to enter the United Kingdom via the Port of Calais or the Channel Tunnel by stowing away on lorries, ferries, cars, or trains. Some migrants were attempting to return to the United Kingdom having once lived there, whilst others were attempting to enter the British labour market to work illegally rather than claim asylum in France. Some migrants lived in the camp while seeking asylum in France, a choice they made because the French system did not provide for them while their claim was being processed, leaving them homeless for the duration.Francesca Ansaloni, 'Deterritorialising the Jungle: Understanding the Calais camp through its orderings' in Environment and Planning C: Politics and Space (25/02/20) doi.org/10.1177/2399654420908597 One migrant from Egypt, a politics graduate, told The Guardian that he had "paid $3,000 (£2,000) to leave Egypt, risked my life on a boat to Italy spending days at sea" and that in one month he had tried 20 times to reach England; another, an Eritrean woman with a one-year- old child, had paid €2,500 (£1,825) – and her husband the same – to sail to Italy, but her husband had drowned during the journey.
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Although developed in Douglas's earlier work, these two strands of her thought were first consciously woven together to form the fabric of a theory of risk perception in her and Wildavsky's 1982 book, Risk and Culture : An Essay on the Selection of Technical and Environmental Dangers. Focusing largely on political conflict over air pollution and nuclear power in the United States, Risk and Culture attributed political conflict over environmental and technological risks to a struggle between adherents of competing ways of life associated with the group–grid scheme: an egalitarian, collectivist (“low grid”, “high group”) one, which gravitates toward fear of environmental disaster as a justification for restricting commercial behavior productive of inequality; and individualistic ("low group") and hierarchical ("high grid") ones, which resist claims of environmental risk in order to shield private orderings from interference, and to defend established commercial and governmental elites from subversive rebuke. Later works in Cultural Theory systematized this argument. In these accounts, group–grid gives rise to either four or five discrete ways of life, each of which is associated with a view of nature (as robust, as fragile, as capricious, and so forth) that is congenial to its advancement in competition with the others.
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