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c_q71ykl3gofzh | In mathematical logic (a subtopic within the field of formal logic), two formulae are equisatisfiable if the first formula is satisfiable whenever the second is and vice versa; in other words, either both formulae are satisfiable or both are not. Equisatisfiable formulae may disagree, however, for a particular choice of variables. As a result, equisatisfiability is different from logical equivalence, as two equivalent formulae always have the same models. | Equisatisfiability | [
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c_vrkl2kut5feg | Whereas within equisatisfiable formulae, only the primitive proposition the formula imposes is valued. Equisatisfiability is generally used in the context of translating formulae, so that one can define a translation to be correct if the original and resulting formulae are equisatisfiable. Examples of translations involving this concept are Skolemization and some translations into conjunctive normal form. | Equisatisfiability | [
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c_fzrmhkzdb9nu | In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation complete theorem-proving technique for sentences in propositional logic and first-order logic. For propositional logic, systematically applying the resolution rule acts as a decision procedure for formula unsatisfiability, solving the (complement of the) Boolean satisfiability problem. For first-order logic, resolution can be used as the basis for a semi-algorithm for the unsatisfiability problem of first-order logic, providing a more practical method than one following from Gödel's completeness theorem. The resolution rule can be traced back to Davis and Putnam (1960); however, their algorithm required trying all ground instances of the given formula. This source of combinatorial explosion was eliminated in 1965 by John Alan Robinson's syntactical unification algorithm, which allowed one to instantiate the formula during the proof "on demand" just as far as needed to keep refutation completeness.The clause produced by a resolution rule is sometimes called a resolvent. | Resolution inference | [
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c_yyf8a91sb8zd | In mathematical finite group theory, the uniqueness case is one of the three possibilities for groups of characteristic 2 type given by the trichotomy theorem. The uniqueness case covers groups G of characteristic 2 type with e(G) ≥ 3 that have an almost strongly p-embedded maximal 2-local subgroup for all primes p whose 2-local p-rank is sufficiently large (usually at least 3). Aschbacher (1983a, 1983b) proved that there are no finite simple groups in the uniqueness case. | Uniqueness case | [
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c_kdboewthdmkt | In mathematical finance, the CEV or constant elasticity of variance model is a stochastic volatility model that attempts to capture stochastic volatility and the leverage effect. The model is widely used by practitioners in the financial industry, especially for modelling equities and commodities. It was developed by John Cox in 1975. | Constant elasticity of variance model | [
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c_elyf3hdf76bu | In mathematical logic and computer science, Gabbay's separation theorem, named after Dov Gabbay, states that any arbitrary temporal logic formula can be rewritten in a logically equivalent "past → future" form. I.e. the future becomes what must be satisfied. This form can be used as execution rules; a MetateM program is a set of such rules. == References == | Gabbay's separation theorem | [
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c_am4zgsvsscj7 | In mathematical finite group theory, a block, sometimes called Aschbacher block, is a subgroup giving an obstruction to Thompson factorization and pushing up. Blocks were introduced by Michael Aschbacher. | Aschbacher block | [
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c_a2ltk9yo1rrm | In mathematical invariant theory, a transvectant is an invariant formed from n invariants in n variables using Cayley's Ω process. | Transvectant | [
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c_pjehhdga13xl | In mathematical group theory, the automorphism group of a free group is a discrete group of automorphisms of a free group. The quotient by the inner automorphisms is the outer automorphism group of a free group, which is similar in some ways to the mapping class group of a surface. | Automorphism group of a free group | [
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c_pgwr14muxwi5 | In mathematical logic and computer science, some type theories and type systems include a top type that is commonly denoted with top or the symbol ⊤. The top type is sometimes called also universal type, or universal supertype as all other types in the type system of interest are subtypes of it, and in most cases, it contains every possible object of the type system. It is in contrast with the bottom type, or the universal subtype, which every other type is supertype of and it is often that the type contains no members at all. | Top type | [
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c_o8qptg0xez53 | In mathematical finite group theory, the Gorenstein–Harada theorem, proved by Gorenstein and Harada (1973, 1974) in a 464-page paper, classifies the simple finite groups of sectional 2-rank at most 4. It is part of the classification of finite simple groups.Finite simple groups of section 2 that rank at least 5, have Sylow 2-subgroups with a self-centralizing normal subgroup of rank at least 3, which implies that they have to be of either component type or of characteristic 2 type. Therefore, the Gorenstein–Harada theorem splits the problem of classifying finite simple groups into these two sub-cases. | Gorenstein–Harada theorem | [
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c_exnt22hj9qiu | In mathematical group theory, a normal p-complement of a finite group for a prime p is a normal subgroup of order coprime to p and index a power of p. In other words the group is a semidirect product of the normal p-complement and any Sylow p-subgroup. A group is called p-nilpotent if it has a normal p-complement. | Normal p-complement | [
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c_q789s4bz4c5i | In mathematical genetics, a genetic algebra is a (possibly non-associative) algebra used to model inheritance in genetics. Some variations of these algebras are called train algebras, special train algebras, gametic algebras, Bernstein algebras, copular algebras, zygotic algebras, and baric algebras (also called weighted algebra). The study of these algebras was started by Ivor Etherington (1939). In applications to genetics, these algebras often have a basis corresponding to the genetically different gametes, and the structure constant of the algebra encode the probabilities of producing offspring of various types. The laws of inheritance are then encoded as algebraic properties of the algebra. For surveys of genetic algebras see Bertrand (1966), Wörz-Busekros (1980) and Reed (1997). | Bernstein algebra | [
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] |
c_nkwvo4tkbocr | In mathematical heat conduction, the Green's function number is used to uniquely categorize certain fundamental solutions of the heat equation to make existing solutions easier to identify, store, and retrieve. | Green's function number | [
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c_yapyxqd3rsrv | In mathematical finite group theory, the L-balance theorem was proved by Gorenstein & Walter (1975). The letter L stands for the layer of a group, and "balance" refers to the property discussed below. | L-balance theorem | [
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c_ppjvpsbonm83 | In mathematical group theory, a subgroup of a group is termed a special abelian subgroup or SA-subgroup if the centralizer of any nonidentity element in the subgroup is precisely the subgroup(Curtis & Reiner 1981, p.354). Equivalently, an SA subgroup is a centrally closed abelian subgroup. Any SA subgroup is a maximal abelian subgroup, that is, it is not properly contained in another abelian subgroup. For a CA group, the SA subgroups are precisely the maximal abelian subgroups.SA subgroups are known for certain characters associated with them termed exceptional characters. | Special abelian subgroup | [
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c_u4haf00j4u93 | In mathematical finite group theory, an exceptional character of a group is a character related in a certain way to a character of a subgroup. They were introduced by Suzuki (1955, p. 663), based on ideas due to Brauer in (Brauer & Nesbitt 1941). | Exceptional character | [
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c_um4lrlodki7h | In mathematical finance, the Greeks are the quantities representing the sensitivity of the price of derivatives such as options to a change in underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the most common of these sensitivities are denoted by Greek letters (as are some other finance measures). Collectively these have also been called the risk sensitivities, risk measures: 742 or hedge parameters. | Option delta | [
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c_w8b47n4kr7nx | In mathematical logic and in particular in model theory, a potential isomorphism is a collection of finite partial isomorphisms between two models which satisfies certain closure conditions. Existence of a partial isomorphism entails elementary equivalence, however the converse is not generally true, but it holds for ω-saturated models. | Potential isomorphism | [
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c_0z5tr4y2ylud | In mathematical finance, the term is also used as an informal synonym for the principle of no-arbitrage. This principle states that a combination of securities that has the same cash-flows as another security must have the same net price in equilibrium. | No such thing as a free lunch | [
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c_8eji27x2uq6w | In mathematical field of combinatorial geometry, the Littlewood–Offord problem is the problem of determining the number of subsums of a set of vectors that fall in a given convex set. More formally, if V is a vector space of dimension d, the problem is to determine, given a finite subset of vectors S and a convex subset A, the number of subsets of S whose summation is in A. The first upper bound for this problem was proven (for d = 1 and d = 2) in 1938 by John Edensor Littlewood and A. Cyril Offord. This Littlewood–Offord lemma states that if S is a set of n real or complex numbers of absolute value at least one and A is any disc of radius one, then not more than ( c log n / n ) 2 n {\displaystyle {\Big (}c\,\log n/{\sqrt {n}}{\Big )}\,2^{n}} of the 2n possible subsums of S fall into the disc. In 1945 Paul Erdős improved the upper bound for d = 1 to ( n ⌊ n / 2 ⌋ ) ≈ 2 n 1 n {\displaystyle {n \choose \lfloor {n/2}\rfloor }\approx 2^{n}\,{\frac {1}{\sqrt {n}}}} using Sperner's theorem. | Littlewood–Offord problem | [
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c_3c4p52ybxjgh | This bound is sharp; equality is attained when all vectors in S are equal. In 1966, Kleitman showed that the same bound held for complex numbers. In 1970, he extended this to the setting when V is a normed space.Suppose S = {v1, …, vn}. | Littlewood–Offord problem | [
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c_kinaq5tehtgj | By subtracting 1 2 ∑ i = 1 n v i {\displaystyle {\frac {1}{2}}\sum _{i=1}^{n}v_{i}} from each possible subsum (that is, by changing the origin and then scaling by a factor of 2), the Littlewood–Offord problem is equivalent to the problem of determining the number of sums of the form ∑ i = 1 n ε i v i {\displaystyle \sum _{i=1}^{n}\varepsilon _{i}v_{i}} that fall in the target set A, where ε i {\displaystyle \varepsilon _{i}} takes the value 1 or −1. This makes the problem into a probabilistic one, in which the question is of the distribution of these random vectors, and what can be said knowing nothing more about the vi. == References == | Littlewood–Offord problem | [
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c_mq3j531oid1j | Elliott (1907) describes the early history of perpetuants and gives an annotated bibliography. MacMahon conjectured and Stroh proved that the dimension of the space of perpetuants of degree n>2 and weight w is the coefficient of xw of x 2 n − 1 − 1 ( 1 − x 2 ) ( 1 − x 3 ) ⋯ ( 1 − x n ) {\displaystyle {\frac {x^{2^{n-1}-1}}{(1-x^{2})(1-x^{3})\cdots (1-x^{n})}}} For n=1 there is just one perpetuant, of weight 0, and for n=2 the number is given by the coefficient of xw of x2/(1-x2). There are very few papers after about 1910 discussing perpetuants; (Littlewood 1944) is one of the few exceptions. (Kraft & Procesi 2020) exhibited an explicit base of the space of perpetuants. | Perpetuant | [
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c_gx60uskl35uh | In mathematical invariant theory, a perpetuant is informally an irreducible covariant of a form or infinite degree. More precisely, the dimension of the space of irreducible covariants of given degree and weight for a binary form stabilizes provided the degree of the form is larger than the weight of the covariant, and the elements of this space are called perpetuants. Perpetuants were introduced and named by Sylvester (1882, p.105). MacMahon (1884, 1885, 1894) and Stroh (1890) classified the perpetuants. | Perpetuant | [
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c_ws5ljpkjxpvp | In mathematical logic and graph theory, an implication graph is a skew-symmetric, directed graph G = (V, E) composed of vertex set V and directed edge set E. Each vertex in V represents the truth status of a Boolean literal, and each directed edge from vertex u to vertex v represents the material implication "If the literal u is true then the literal v is also true". Implication graphs were originally used for analyzing complex Boolean expressions. | Implication graph | [
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c_yrkc2q2o60to | In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers over both the set of natural numbers, N {\displaystyle \mathbb {N} } , and over functions from N {\displaystyle \mathbb {N} } to N {\displaystyle \mathbb {N} } . The analytical hierarchy of sets classifies sets by the formulas that can be used to define them; it is the lightface version of the projective hierarchy. | Analytical hierarchy | [
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] |
c_w383hudh7kf4 | , FN), and let Q {\displaystyle \mathbb {Q} } denote an equivalent martingale measure. Let U = (U0, U1, . . | Doob decomposition theorem | [
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c_hs02nnt30i63 | Let U = M + A denote the Doob decomposition with respect to Q {\displaystyle \mathbb {Q} } of the Snell envelope U into a martingale M = (M0, M1, . . . | Doob decomposition theorem | [
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c_yy6l52ugbn0e | , AN) with A0 = 0. Then the largest stopping time to exercise the American option in an optimal way is τ max := { N if A N = 0 , min { n ∈ { 0 , … , N − 1 } ∣ A n + 1 < 0 } if A N < 0. {\displaystyle \tau _{\text{max}}:={\begin{cases}N&{\text{if }}A_{N}=0,\\\min\{n\in \{0,\dots ,N-1\}\mid A_{n+1}<0\}&{\text{if }}A_{N}<0.\end{cases}}} Since A is predictable, the event {τmax = n} = {An = 0, An+1 < 0} is in Fn for every n ∈ {0, 1, . | Doob decomposition theorem | [
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c_53y99gt7o86j | For example, a co-dimension 2 link in 3-dimensional space is a subspace of 3-dimensional Euclidean space (or often the 3-sphere) whose connected components are homeomorphic to circles. The simplest nontrivial example of a link with more than one component is called the Hopf link, which consists of two circles (or unknots) linked together once. The circles in the Borromean rings are collectively linked despite the fact that no two of them are directly linked. The Borromean rings thus form a Brunnian link and in fact constitute the simplest such link. | Link (knot theory) | [
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c_l5bcz1hcs78g | In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a trivial reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link. | Link (knot theory) | [
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c_yw3mbukrfglv | However, not every total recursive function is a primitive recursive function—the most famous example is the Ackermann function. Other equivalent classes of functions are the functions of lambda calculus and the functions that can be computed by Markov algorithms. The subset of all total recursive functions with values in {0,1} is known in computational complexity theory as the complexity class R. | Mu-recursive function | [
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c_118525g6llis | In mathematical logic and computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "computable" in an intuitive sense – as well as in a formal one. If the function is total, it is also called a total recursive function (sometimes shortened to recursive function). In computability theory, it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines (this is one of the theorems that supports the Church–Turing thesis). The μ-recursive functions are closely related to primitive recursive functions, and their inductive definition (below) builds upon that of the primitive recursive functions. | Mu-recursive function | [
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c_uui5zih03bff | According to the Curry–Howard isomorphism, lambda calculus on its own can express theorems in intuitionistic logic only, and several classical logical theorems can't be written at all. However with these new operators one is able to write terms that have the type of, for example, Peirce's law. Semantically these operators correspond to continuations, found in some functional programming languages. | Lambda-mu calculus | [
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c_c7m22j7nswnm | In mathematical logic and computer science, the lambda-mu calculus is an extension of the lambda calculus introduced by M. Parigot. It introduces two new operators: the μ operator (which is completely different both from the μ operator found in computability theory and from the μ operator of modal μ-calculus) and the bracket operator. Proof-theoretically, it provides a well-behaved formulation of classical natural deduction. One of the main goals of this extended calculus is to be able to describe expressions corresponding to theorems in classical logic. | Lambda-mu calculus | [
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c_ak0r69vi6z84 | In mathematical folklore, the "no free lunch" (NFL) theorem (sometimes pluralized) of David Wolpert and William Macready, alludes to the saying "no such thing as a free lunch", that is, there are no easy shortcuts to success. It appeared in the 1997 "No Free Lunch Theorems for Optimization". Wolpert had previously derived no free lunch theorems for machine learning (statistical inference).In 2005, Wolpert and Macready themselves indicated that the first theorem in their paper "state that any two optimization algorithms are equivalent when their performance is averaged across all possible problems".The "no free lunch" (NFL) theorem is an easily stated and easily understood consequence of theorems Wolpert and Macready actually prove. It is weaker than the proven theorems, and thus does not encapsulate them. Various investigators have extended the work of Wolpert and Macready substantively. In terms of how the NFL theorem is used in the context of the research area, the no free lunch in search and optimization is a field that is dedicated for purposes of mathematically analyzing data for statistical identity, particularly search and optimization.While some scholars argue that NFL conveys important insight, others argue that NFL is of little relevance to machine learning research. | No free lunch theorem | [
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c_g9l46102t0bx | In mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty or with complicated features. The first application to option pricing was by Phelim Boyle in 1977 (for European options). In 1996, M. Broadie and P. Glasserman showed how to price Asian options by Monte Carlo. An important development was the introduction in 1996 by Carriere of Monte Carlo methods for options with early exercise features. | Monte Carlo option model | [
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c_n1nvd2wbi5o0 | In mathematical group theory, a 3-transposition group is a group generated by a conjugacy class of involutions, called the 3-transpositions, such that the product of any two involutions from the conjugacy class has order at most 3. They were first studied by Bernd Fischer (1964, 1970, 1971) who discovered the three Fischer groups as examples of 3-transposition groups. | 3-transposition group | [
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c_9iqr2fdvmuad | In mathematical finance for volatility smile modeling of options via local volatility, one has the problem of deriving a diffusion coefficient σ ( X t , t ) {\displaystyle {\sigma }(\mathbf {X} _{t},t)} consistent with a probability density obtained from market option quotes. The problem is therefore an inversion of the Fokker–Planck equation: Given the density f(x,t) of the option underlying X deduced from the option market, one aims at finding the local volatility σ ( X t , t ) {\displaystyle {\sigma }(\mathbf {X} _{t},t)} consistent with f. This is an inverse problem that has been solved in general by Dupire (1994, 1997) with a non-parametric solution. Brigo and Mercurio (2002, 2003) propose a solution in parametric form via a particular local volatility σ ( X t , t ) {\displaystyle {\sigma }(\mathbf {X} _{t},t)} consistent with a solution of the Fokker–Planck equation given by a mixture model. More information is available also in Fengler (2008), Gatheral (2008), and Musiela and Rutkowski (2008). | Smoluchowski equation | [
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c_f1n0z6ykaaxg | This is purely a geometrical fact, and it corresponds to the case of a complete three-dimensional, space-like, totally geodesic submanifold of a (3 + 1)-dimensional spacetime. Such a submanifold is often called a time-symmetric initial data set for a spacetime. The condition of (M, g) having nonnegative scalar curvature is equivalent to the spacetime obeying the dominant energy condition. | Penrose inequality | [
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] |
c_lsirgjald43t | This inequality was first proved by Gerhard Huisken and Tom Ilmanen in 1997 in the case where A is the area of the largest component of the outermost minimal surface. Their proof relied on the machinery of weakly defined inverse mean curvature flow, which they developed. In 1999, Hubert Bray gave the first complete proof of the above inequality using a conformal flow of metrics. Both of the papers were published in 2001. | Penrose inequality | [
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c_93zq8lrn6u68 | In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannian Penrose inequality is an important special case. Specifically, if (M, g) is an asymptotically flat Riemannian 3-manifold with nonnegative scalar curvature and ADM mass m, and A is the area of the outermost minimal surface (possibly with multiple connected components), then the Riemannian Penrose inequality asserts m ≥ A 16 π . {\displaystyle m\geq {\sqrt {\frac {A}{16\pi }}}.} | Penrose inequality | [
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c_gfo6ooudq94p | In mathematical group theory, the balance theorem states that if G is a group with no core then G either has disconnected Sylow 2-subgroups or it is of characteristic 2 type or it is of component type (Gorenstein 1983, p. 7). The significance of this theorem is that it splits the classification of finite simple groups into three major subcases. | Balance theorem | [
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c_96qtjxzbzzmi | In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantical validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true. | Completeness (logic) | [
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c_y4yzmru03fp6 | In mathematical knot theory, a Conway sphere, named after John Horton Conway, is a 2-sphere intersecting a given knot or link in a 3-manifold transversely in four points. In a knot diagram, a Conway sphere can be represented by a simple closed curve crossing four points of the knot, the cross-section of the sphere; such a curve does not always exist for an arbitrary knot diagram of a knot with a Conway sphere, but it is always possible to choose a diagram for the knot in which the sphere can be depicted in this way. A Conway sphere is essential if it is incompressible in the knot complement. Sometimes, this condition is included in the definition of Conway spheres. == References == | Conway sphere | [
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] |
c_kw6bfezbgqeq | In mathematical logic and computer science the symbol ⊢ ( ⊢ {\displaystyle \vdash } ) has taken the name turnstile because of its resemblance to a typical turnstile if viewed from above. It is also referred to as tee and is often read as "yields", "proves", "satisfies" or "entails". | ⊢ | [
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c_tjsfo7g8ztzm | In mathematical invariant theory, an invariant of a binary form is a polynomial in the coefficients of a binary form in two variables x and y that remains invariant under the special linear group acting on the variables x and y. | Invariant of a binary form | [
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c_vv046821bx6t | In mathematical finite group theory, a quadratic pair for the odd prime p, introduced by Thompson (1971), is a finite group G together with a quadratic module, a faithful representation M on a vector space over the finite field with p elements such that G is generated by elements with minimal polynomial (x − 1)2. Thompson classified the quadratic pairs for p ≥ 5. Chermak (2004) classified the quadratic pairs for p = 3. With a few exceptions, especially for p = 3, groups with a quadratic pair for the prime p tend to be more or less groups of Lie type in characteristic p. | Quadratic pair | [
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c_xyikywt681bu | In mathematical finance, the Cox–Ingersoll–Ross (CIR) model describes the evolution of interest rates. It is a type of "one factor model" (short-rate model) as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives. It was introduced in 1985 by John C. Cox, Jonathan E. Ingersoll and Stephen A. Ross as an extension of the Vasicek model. | Cox–Ingersoll–Ross model | [
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c_n7fzesqqb9ua | In mathematical finance, the stochastic volatility jump (SVJ) model is suggested by Bates. This model fits the observed implied volatility surface well. The model is a Heston process for stochastic volatility with an added Merton log-normal jump. It assumes the following correlated processes: d S = μ S d t + ν S d Z 1 + ( e α + δ ε − 1 ) S d q {\displaystyle dS=\mu S\,dt+{\sqrt {\nu }}S\,dZ_{1}+(e^{\alpha +\delta \varepsilon }-1)S\,dq} d ν = λ ( ν − ν ¯ ) d t + η ν d Z 2 {\displaystyle d\nu =\lambda (\nu -{\overline {\nu }})\,dt+\eta {\sqrt {\nu }}\,dZ_{2}} corr ( d Z 1 , d Z 2 ) = ρ {\displaystyle \operatorname {corr} (dZ_{1},dZ_{2})=\rho } prob ( d q = 1 ) = λ d t {\displaystyle \operatorname {prob} (dq=1)=\lambda dt} where S is the price of security, μ is the constant drift (i.e. expected return), t represents time, Z1 is a standard Brownian motion, q is a Poisson counter with density λ. == References == | Stochastic volatility jump | [
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c_wsi6kh0nb6hy | The philosophical implications of Skolem's paradox have received much study. One line of inquiry questions whether it is accurate to claim that any first-order sentence actually states "there are uncountable sets". This line of thought can be extended to question whether any set is uncountable in an absolute sense. More recently, the paper "Models and Reality" by Hilary Putnam, and responses to it, led to renewed interest in the philosophical aspects of Skolem's result. | Skolem paradox | [
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c_5a834ri7i3zu | Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countable sets, satisfies the first-order sentence that intuitively states "there are uncountable sets". A mathematical explanation of the paradox, showing that it is not a contradiction in mathematics, was given by Skolem (1922). Skolem's work was harshly received by Ernst Zermelo, who argued against the limitations of first-order logic, but the result quickly came to be accepted by the mathematical community. | Skolem paradox | [
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c_7uj5b2om3zdn | In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as non-absoluteness. Although it is not an actual antinomy like Russell's paradox, the result is typically called a paradox and was described as a "paradoxical state of affairs" by Skolem (1922: p. | Skolem paradox | [
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c_ernahugexvqf | In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema. Valuations are also called truth assignments. In propositional logic, there are no quantifiers, and formulas are built from propositional variables using logical connectives. | Valuation (logic) | [
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c_i2mz1cwtjw96 | Formulas are built out of atomic formulas using logical connectives and quantifiers. A structure consists of a set (domain of discourse) that determines the range of the quantifiers, along with interpretations of the constant, function, and relation symbols in the language. Corresponding to each structure is a unique truth assignment for all sentences (formulas with no free variables) in the language. | Valuation (logic) | [
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c_2vni3jgt2pdu | In this context, a valuation begins with an assignment of a truth value to each propositional variable. This assignment can be uniquely extended to an assignment of truth values to all propositional formulas. In first-order logic, a language consists of a collection of constant symbols, a collection of function symbols, and a collection of relation symbols. | Valuation (logic) | [
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c_r3p8ap0s6uyd | In mathematical finite group theory, a p-group of symplectic type is a p-group such that all characteristic abelian subgroups are cyclic. According to Thompson (1968, p.386), the p-groups of symplectic type were classified by P. Hall in unpublished lecture notes, who showed that they are all a central product of an extraspecial group with a group that is cyclic, dihedral, quasidihedral, or quaternion. Gorenstein (1980, 5.4.9) gives a proof of this result. The width n of a group G of symplectic type is the largest integer n such that the group contains an extraspecial subgroup H of order p1+2n such that G = H.CG(H), or 0 if G contains no such subgroup. Groups of symplectic type appear in centralizers of involutions of groups of GF(2)-type. | Group of symplectic type | [
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c_gbbl07o8q2oq | In mathematical finite group theory, Thompson's original uniqueness theorem (Feit & Thompson 1963, theorems 24.5 and 25.2) states that in a minimal simple finite group of odd order there is a unique maximal subgroup containing a given elementary abelian subgroup of rank 3. Bender (1970) gave a shorter proof of the uniqueness theorem. | Thompson uniqueness theorem | [
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c_t5rhdpv0dla3 | In older texts one occasionally finds (−)n, which means the same. When the standard presumption that the plus-or-minus signs all take on the same value of +1 or all −1 is not true, then the line of text that immediately follows the equation must contain a brief description of the actual connection, if any, most often of the form "where the ‘±’ signs are independent" or similar. If a brief, simple description is not possible, the equation must be re-written to provide clarity; e.g. by introducing variables such as s1, s2, ... and specifying a value of +1 or −1 separately for each, or some appropriate relation, like s 3 = s 1 ⋅ ( s 2 ) n , {\displaystyle s_{3}=s_{1}\cdot (s_{2})^{n}\,,} or similar. | Minus-plus sign | [
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c_y178lgcbs02q | In mathematical formulas, the ± symbol may be used to indicate a symbol that may be replaced by either the plus and minus signs, + or −, allowing the formula to represent two values or two equations.If x2 = 9, one may give the solution as x = ±3. This indicates that the equation has two solutions: x = +3 and x = −3. A common use of this notation is found in the quadratic formula x = − b ± b 2 − 4 a c 2 a , {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},} which describes the two solutions to the quadratic equation ax2+bx+c = 0. Similarly, the trigonometric identity sin ( A ± B ) = sin ( A ) cos ( B ) ± cos ( A ) sin ( B ) {\displaystyle \sin(A\pm B)=\sin(A)\cos(B)\pm \cos(A)\sin(B)} can be interpreted as a shorthand for two equations: one with + on both sides of the equation, and one with − on both sides. | Minus-plus sign | [
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] |
c_3jhwzv6p6i0q | The groups of each of these 8 types were classified by various authors. They consist mainly of groups of Lie type with all roots of the same length over the field with 2 elements, but also include many exceptional cases, including the majority of the sporadic simple groups. Smith (1980) gave a survey of this work. Smith (1979, p.279) gives a table of simple groups containing a large extraspecial 2-group. | Groups of GF(2) type | [
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c_193d4kj18vc1 | In mathematical finite group theory, a group of GF(2)-type is a group with an involution centralizer whose generalized Fitting subgroup is a group of symplectic type (Gorenstein 1982, definition 1.45). As the name suggests, many of the groups of Lie type over the field with 2 elements are groups of GF(2)-type. Also 16 of the 26 sporadic groups are of GF(2)-type, suggesting that in some sense sporadic groups are somehow related to special properties of the field with 2 elements. Timmesfeld (1978) showed roughly that groups of GF(2)-type can be subdivided into 8 types. | Groups of GF(2) type | [
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c_dh6da0mp1p1w | In mathematical finite group theory, the Thompson subgroup J ( P ) {\displaystyle J(P)} of a finite p-group P refers to one of several characteristic subgroups of P. John G. Thompson (1964) originally defined J ( P ) {\displaystyle J(P)} to be the subgroup generated by the abelian subgroups of P of maximal rank. More often the Thompson subgroup J ( P ) {\displaystyle J(P)} is defined to be the subgroup generated by the abelian subgroups of P of maximal order or the subgroup generated by the elementary abelian subgroups of P of maximal rank. In general these three subgroups can be different, though they are all called the Thompson subgroup and denoted by J ( P ) {\displaystyle J(P)} . | Thompson subgroup | [
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c_5su8uoydpfj7 | In mathematical finite group theory, the Dempwolff group is a finite group of order 319979520 = 215·32·5·7·31, that is the unique nonsplit extension 2 5 . G L 5 ( F 2 ) {\displaystyle 2^{5\,. }\mathrm {GL} _{5}(\mathbb {F} _{2})} of G L 5 ( F 2 ) {\displaystyle \mathrm {GL} _{5}(\mathbb {F} _{2})} by its natural module of order 2 5 {\displaystyle 2^{5}} . The uniqueness of such a nonsplit extension was shown by Dempwolff (1972), and the existence by Thompson (1976), who showed using some computer calculations of Smith (1976) that the Dempwolff group is contained in the compact Lie group E 8 {\displaystyle E_{8}} as the subgroup fixing a certain lattice in the Lie algebra of E 8 {\displaystyle E_{8}} , and is also contained in the Thompson sporadic group (the full automorphism group of this lattice) as a maximal subgroup. | Dempwolff group | [
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c_sthfrp1lzpns | G L 4 ( F 2 ) {\displaystyle 2^{4\,. }\mathrm {GL} _{4}(\mathbb {F} _{2})} is a maximal subgroup of the sporadic Conway group Co3. The nonsplit extension 2 5 . G L 5 ( F 2 ) {\displaystyle 2^{5\,. }\mathrm {GL} _{5}(\mathbb {F} _{2})} is a maximal subgroup of the Thompson sporadic group Th. | Dempwolff group | [
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c_wdxkphv4if9n | Huppert (1967, p.124) showed that any extension of G L n ( F q ) {\displaystyle \mathrm {GL} _{n}(\mathbb {F} _{q})} by its natural module F q n {\displaystyle \mathbb {F} _{q}^{n}} splits if q > 2 {\displaystyle q>2} , and Dempwolff (1973) showed that it also splits if n {\displaystyle n} is not 3, 4, or 5, and in each of these three cases there is just one non-split extension. These three nonsplit extensions can be constructed as follows: The nonsplit extension 2 3 . | Dempwolff group | [
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c_3af4p7rlowj4 | In mathematical finite group theory, the Thompson order formula, introduced by John Griggs Thompson (Held 1969, p.279), gives a formula for the order of a finite group in terms of the centralizers of involutions, extending the results of Brauer & Fowler (1955). | Thompson order formula | [
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c_2gw04ibdo00t | In mathematical finite group theory, an N-group is a group all of whose local subgroups (that is, the normalizers of nontrivial p-subgroups) are solvable groups. The non-solvable ones were classified by Thompson during his work on finding all the minimal finite simple groups. | N-group (finite group theory) | [
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c_c87kg8n4g9vi | In mathematical group theory, a Demushkin group (also written as Demuškin or Demuskin) is a pro-p group G having a certain properties relating to duality in group cohomology. More precisely, G must be such that the first cohomology group with coefficients in Fp = Z/p Z has finite rank, the second cohomology group has rank 1, and the cup product induces a non-degenerate pairing H1(G,Fp)× H1(G,Fp) → H2(G,Fp).Such groups were introduced by Demuškin (1959). Demushkin groups occur as the Galois groups of the maximal p-extensions of local number fields containing all p-th roots of unity. | Demushkin group | [
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c_rvclinxpsdcw | In mathematical group theory, the Hall–Higman theorem, due to Philip Hall and Graham Higman (1956, Theorem B), describes the possibilities for the minimal polynomial of an element of prime power order for a representation of a p-solvable group. | Hall–Higman theorem | [
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c_g7wdu4a6wexk | In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, the CoC and its variants have been the basis for Coq and other proof assistants. Some of its variants include the calculus of inductive constructions (which adds inductive types), the calculus of (co)inductive constructions (which adds coinduction), and the predicative calculus of inductive constructions (which removes some impredicativity). | Calculus of Constructions | [
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c_bb5u2511w2vx | In mathematical finite group theory, the Brauer–Fowler theorem, proved by Brauer & Fowler (1955), states that if a group G has even order g > 2 then it has a proper subgroup of order greater than g1/3. The technique of the proof is to count involutions (elements of order 2) in G. Perhaps more important is another result that the authors derive from the same count of involutions, namely that up to isomorphism there are only a finite number of finite simple groups with a given centralizer of an involution. This suggested that finite simple groups could be classified by studying their centralizers of involutions, and it led to the discovery of several sporadic groups. Later it motivated a part of the classification of finite simple groups. | Brauer–Fowler theorem | [
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c_qztcsv96vslb | In mathematical graph theory, the Higman–Sims graph is a 22-regular undirected graph with 100 vertices and 1100 edges. It is the unique strongly regular graph srg(100,22,0,6), where no neighboring pair of vertices share a common neighbor and each non-neighboring pair of vertices share six common neighbors. It was first constructed by Mesner (1956) and rediscovered in 1968 by Donald G. Higman and Charles C. Sims as a way to define the Higman–Sims group, a subgroup of index two in the group of automorphisms of the Higman–Sims graph. | Higman-Sims graph | [
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c_sha0219b34u9 | In mathematical finance, the SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for "stochastic alpha, beta, rho", referring to the parameters of the model. The SABR model is widely used by practitioners in the financial industry, especially in the interest rate derivative markets. It was developed by Patrick S. Hagan, Deep Kumar, Andrew Lesniewski, and Diana Woodward. | SABR volatility model | [
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c_gr5zveolqzuy | In mathematical finance, convexity refers to non-linearities in a financial model. In other words, if the price of an underlying variable changes, the price of an output does not change linearly, but depends on the second derivative (or, loosely speaking, higher-order terms) of the modeling function. Geometrically, the model is no longer flat but curved, and the degree of curvature is called the convexity. | Convexity (finance) | [
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c_7kdx9ojzebg7 | In mathematical invariant theory, an evectant is a contravariant constructed from an invariant by acting on it with a differential operator called an evector. Evectants and evectors were introduced by Sylvester (1854, p.95). | Evectant | [
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c_mw1d7w456sqx | In mathematical invariant theory, the catalecticant of a form of even degree is a polynomial in its coefficients that vanishes when the form is a sum of an unusually small number of powers of linear forms. It was introduced by Sylvester (1852); see Miller (2010). The word catalectic refers to an incomplete line of verse, lacking a syllable at the end or ending with an incomplete foot. | Catalecticant | [
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c_wiuqw5s7m1eg | In mathematical finite group theory, the vertex of a representation of a finite group is a subgroup associated to it, that has a special representation called a source. Vertices and sources were introduced by Green (1958–1959). | Vertex of a representation | [
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c_20pfm4qlhdy8 | A formal proof or derivation is a finite sequence of well-formed formulas (which may be interpreted as sentences, or propositions) each of which is an axiom or follows from the preceding formulas in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system. Formal proofs are useful because their theorems can be interpreted as true propositions. | Formal language theory | [
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c_le4clnew5ai0 | In mathematical logic, a formal theory is a set of sentences expressed in a formal language. A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules, which may be interpreted as valid rules of inference, or a set of axioms, or have both. | Formal language theory | [
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c_y8paf0szvldd | A formal system is used to derive one expression from one or more other expressions. Although a formal language can be identified with its formulas, a formal system cannot be likewise identified by its theorems. Two formal systems F S {\displaystyle {\mathcal {FS}}} and F S ′ {\displaystyle {\mathcal {FS'}}} may have all the same theorems and yet differ in some significant proof-theoretic way (a formula A may be a syntactic consequence of a formula B in one but not another for instance). | Formal language theory | [
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c_z0067ijyeq8i | New Foundations has a universal set, so it is a non-well-founded set theory. That is to say, it is an axiomatic set theory that allows infinite descending chains of membership, such as ... xn ∈ xn-1 ∈ ... ∈ x2 ∈ x1. It avoids Russell's paradox by permitting only stratifiable formulas to be defined using the axiom schema of comprehension. For instance, x ∈ y is a stratifiable formula, but x ∈ x is not. New Foundations is closely related to Russellian unramified typed set theory (TST), a streamlined version of the theory of types of Principia Mathematica with a linear hierarchy of types. | Typed set theory | [
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c_ldthcj2lr6ii | In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name. Much of this entry discusses NF with urelements (NFU), an important variant of NF due to Jensen and clarified by Holmes. In 1940 and in a revision in 1951, Quine introduced an extension of NF sometimes called "Mathematical Logic" or "ML", that included proper classes as well as sets. | Typed set theory | [
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c_syi6v2dp3b99 | In mathematical logic, Diaconescu's theorem, or the Goodman–Myhill theorem, states that the full axiom of choice is sufficient to derive the law of the excluded middle or restricted forms of it. The theorem was discovered in 1975 by Radu Diaconescu and later by Goodman and Myhill. Already in 1967, Errett Bishop posed the theorem as an exercise (Problem 2 on page 58 in Foundations of constructive analysis). | Diaconescu theorem | [
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c_v7clo0j0xmtr | In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was developed by Kurt Gödel for the proof of his incompleteness theorems. (Gödel 1931) A Gödel numbering can be interpreted as an encoding in which a number is assigned to each symbol of a mathematical notation, after which a sequence of natural numbers can then represent a sequence of symbols. These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic. Since the publishing of Gödel's paper in 1931, the term "Gödel numbering" or "Gödel code" has been used to refer to more general assignments of natural numbers to mathematical objects. | Gödel encoding | [
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c_yvcc2ejrqn1k | Frege's PC and standard PC share two common axioms: THEN-1 and THEN-2. Notice that axioms THEN-1 through THEN-3 only make use of (and define) the implication operator, whereas axioms FRG-1 through FRG-3 define the negation operator. | Frege's propositional calculus | [
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c_r6uc4nhqy2l3 | In mathematical logic, Frege's propositional calculus was the first axiomatization of propositional calculus. It was invented by Gottlob Frege, who also invented predicate calculus, in 1879 as part of his second-order predicate calculus (although Charles Peirce was the first to use the term "second-order" and developed his own version of the predicate calculus independently of Frege). It makes use of just two logical operators: implication and negation, and it is constituted by six axioms and one inference rule: modus ponens. Frege's propositional calculus is equivalent to any other classical propositional calculus, such as the "standard PC" with 11 axioms. | Frege's propositional calculus | [
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c_whljmdm08z2j | The following theorems will aim to find the remaining nine axioms of standard PC within the "theorem-space" of Frege's PC, showing that the theory of standard PC is contained within the theory of Frege's PC. (A theory, also called here, for figurative purposes, a "theorem-space", is a set of theorems that are a subset of a universal set of well-formed formulas. The theorems are linked to each other in a directed manner by inference rules, forming a sort of dendritic network. At the roots of the theorem-space are found the axioms, which "generate" the theorem-space much like a generating set generates a group.) | Frege's propositional calculus | [
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] |
c_k1ic5vx9d3hn | In mathematical logic the Löwenheim number of an abstract logic is the smallest cardinal number for which a weak downward Löwenheim–Skolem theorem holds. They are named after Leopold Löwenheim, who proved that these exist for a very broad class of logics. | Löwenheim number | [
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c_hq9x7cj3o5m1 | To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory. For first-order theories, interpretations are commonly called structures. Given a structure or interpretation, a sentence will have a fixed truth value. A theory is satisfiable when it is possible to present an interpretation in which all of its sentences are true. The study of algorithms to automatically discover interpretations of theories that render all sentences as being true is known as the satisfiability modulo theories problem. | Sentence (mathematical logic) | [
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c_05adoibsr7l3 | Sentences are then built up out of atomic formulas by applying connectives and quantifiers. A set of sentences is called a theory; thus, individual sentences may be called theorems. | Sentence (mathematical logic) | [
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c_jr05y5bko18c | In mathematical logic, a sentence (or closed formula) of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values: as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary. Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy to atomic formula. | Sentence (mathematical logic) | [
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c_vj4kepmn579a | Let T 1 {\displaystyle T_{1}} be a theory obtained from T {\displaystyle T} by extending its language with new constants a 1 , … , a m {\displaystyle a_{1},\ldots ,a_{m}} and adding a new axiom φ ( a 1 , … , a m ) {\displaystyle \varphi (a_{1},\ldots ,a_{m})} .Then T 1 {\displaystyle T_{1}} is a conservative extension of T {\displaystyle T} , which means that the theory T 1 {\displaystyle T_{1}} has the same set of theorems in the original language (i.e., without constants a i {\displaystyle a_{i}} ) as the theory T {\displaystyle T} . Such a theory can also be conservatively extended by introducing a new functional symbol:Suppose that a closed formula ∀ x → ∃ y φ ( y , x → ) {\displaystyle \forall {\vec {x}}\,\exists y\,\!\,\varphi (y,{\vec {x}})} is a theorem of a first-order theory T {\displaystyle T} , where we denote x → := ( x 1 , … , x n ) {\displaystyle {\vec {x}}:=(x_{1},\ldots ,x_{n})} . Let T 1 {\displaystyle T_{1}} be a theory obtained from T {\displaystyle T} by extending its language with a new functional symbol f {\displaystyle f} (of arity n {\displaystyle n} ) and adding a new axiom ∀ x → φ ( f ( x → ) , x → ) {\displaystyle \forall {\vec {x}}\,\varphi (f({\vec {x}}),{\vec {x}})} . | Extension by new constant and function names | [
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c_s9mxwt2dlnq5 | Then T 1 {\displaystyle T_{1}} is a conservative extension of T {\displaystyle T} , i.e. the theories T {\displaystyle T} and T 1 {\displaystyle T_{1}} prove the same theorems not involving the functional symbol f {\displaystyle f} ). Shoenfield states the theorem in the form for a new function name, and constants are the same as functions of zero arguments. In formal systems that admit ordered tuples, extension by multiple constants as shown here can be accomplished by addition of a new constant tuple and the new constant names having the values of elements of the tuple. | Extension by new constant and function names | [
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c_6a58hn5afjz4 | In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be done safely without uniqueness. Suppose that a closed formula ∃ x 1 … ∃ x m φ ( x 1 , … , x m ) {\displaystyle \exists x_{1}\ldots \exists x_{m}\,\varphi (x_{1},\ldots ,x_{m})} is a theorem of a first-order theory T {\displaystyle T} . | Extension by new constant and function names | [
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c_ksi9zdg0mk5x | In mathematical logic and theoretical computer science, an abstract rewriting system (also (abstract) reduction system or abstract rewrite system; abbreviated ARS) is a formalism that captures the quintessential notion and properties of rewriting systems. In its simplest form, an ARS is simply a set (of "objects") together with a binary relation, traditionally denoted with → {\displaystyle \rightarrow } ; this definition can be further refined if we index (label) subsets of the binary relation. Despite its simplicity, an ARS is sufficient to describe important properties of rewriting systems like normal forms, termination, and various notions of confluence. | Convergent term rewriting system | [
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c_rxcru5fd1jnz | Historically, there have been several formalizations of rewriting in an abstract setting, each with its idiosyncrasies. This is due in part to the fact that some notions are equivalent, see below in this article. The formalization that is most commonly encountered in monographs and textbooks, and which is generally followed here, is due to Gérard Huet (1980). | Convergent term rewriting system | [
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c_pzrucyli1d8x | Points in space and the real numbers under Euclidean geometry are another structure, with relations such as "the distance between point P1 and point P2 is real number R1"; equivalently, the "distance" relation includes the element (P1, P2, R1). Other structures include the Riemann space of general relativity and the Hilbert space of quantum mechanics. The entities in a mathematical structure do not have any independent existence outside their participation in relations. Two descriptions of a structure are considered equivalent, and to be describing the same underlying structure, if there is a correspondence between the descriptions that preserves all relations.Many proponents of structural realism formally or informally ascribe "properties" to the abstract objects; some argue that such properties, while they can perhaps be "shoehorned" into the formalism of relations, should instead be considered distinct from relations. | Epistemic structural realism | [
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Subsets and Splits