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(countable, set theory, mathematical analysis) A collection of subsets of a given set, such that this collection contains the empty set, and the collection is closed under unions and complements (and thereby also under intersections and differences).
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(mathematics, computing) A structure B = (S,⊑₁ ,⊑₂) in which S is a non-empty set, and ⊑₁ and ⊑₂ are partial orderings each giving S the structure of a lattice, determining thus for each of the two lattices the corresponding operations of meet and join.
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(algebra) An algebraic structure (𝛴,∨,∧,∼,0,1) where ∨ and ∧ are idempotent binary operators, ∼ is a unary involutory operator (called "complement"), and 0 and 1 are nullary operators (i.e., constants), such that (𝛴,∨,0) is a commutative monoid, (𝛴,∧,1) is a commutative monoid, ∧ and ∨ distribute with respect to each other, and such that combining two complementary elements through one binary operator yields the identity of the other binary operator. (See Boolean algebra (structure)#Axiomatics.)
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(mathematics) A result on the combinatorics of block designs, stating that, if a (v, b, r, k, λ)-design exists with v = b (a symmetric block design), then: (i) if v is even, then k − λ is a square; (ii) if v is odd, then the following Diophantine equation has a nontrivial solution: x² − (k − λ)y² − (−1)^((v−1)/2) λ z² = 0.
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(mathematics, mathematical analysis) For a given m-order partial differential equation, the problem of finding a solution function u on ℝⁿ that satisfies the boundary conditions that, for a smooth manifold S⊂ℝⁿ, u(x)=f_0(x) and (∂ᵏu(x))/(∂nᵏ)=f_k(x), ∀x∈S, k=1…m-1, given specified functions f_k defined on, and vector n normal to, the manifold.
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(algebra) For a positive integer n, a polynomial whose roots are the primitive nᵗʰ roots of unity, so that its degree is Euler's totient function of n. That is, letting 𝜁ₙ=e^(i 2𝜋/n) be the first primitive nᵗʰ root of unity, then 𝛷ₙ(x)=∏_( stackrel )1
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Initialism of differential equation. [(calculus) an equation involving the derivatives of a function]
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(geometry) A theorem that states a lower bound on the number of lines determined by n points in a projective plane.
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(calculus) Of a function, another function, the value of which for any value of the independent variable is the instantaneous rate of change of the given function at that value of the independent variable.
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A proof, developed by Georg Cantor, to show that the set of real numbers is uncountably infinite.
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(mathematics) Any of a number of mathematical objects analogously derived from a given ordered set of objects.
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(algebra, field theory) A valuation (on some field) that takes integer values (including infinity).
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(algebra) The conjecture that any endomorphism of a Weyl algebra is an automorphism.
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(mathematics) A ring with no zero divisors; that is, in which no product of nonzero elements is zero.
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(mathematics) An algorithm for finding the roots of polynomial equations.
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(mathematics) An important structural theorem in the theory of Banach spaces, essentially stating that every sufficiently high-dimensional normed vector space will have low-dimensional subspaces that are approximately Euclidean.
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(algebra, field theory, algebraic geometry) Any pair of fields, denoted L/K, such that K is a subfield of L.
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(computing theory) An algorithm for approximating the number of distinct elements in a stream with a single pass and logarithmic space consumption.
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(mathematics) The conjecture that every domain of ℝᵈ (i.e. subset of ℝᵈ with positive finite Lebesgue measure) is a spectral set if and only if it tiles ℝᵈ by translation.
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An algorithm for finding a solution to the stable marriage problem in polynomial time.
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(complex analysis) A theorem that gives a geometric relation between the roots of a polynomial P and the roots of its derivative P′. It states that the roots of P′ all lie within the convex hull of the roots of P.
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(mathematics) A theorem that establishes the transcendence of a large class of numbers, stating that, if a and b are algebraic numbers with a ≠ 0, 1, and b irrational, then any value of aᵇ is a transcendental number.
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(mathematics) A theorem proving that it is always possible to tile a chessboard with dominoes if two squares of opposite colors are first removed from the board.
adj
Having to do with group theory.
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(mathematics) A theorem stating that, given n measurable "objects" in n-dimensional Euclidean space, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single (n−1)-dimensional hyperplane.
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(number theory) A method used to compute the residues about zero for the generating function of a series, as part of proving asymptotic behaviour of the series.
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(graph theory) A conjecture concerning the connection between graph coloring and the tensor product of graphs. A counterexample was found in 2019.
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(mathematics) A theorem stating that, if S, is an IP set and S=C_1∪C_2∪...∪C_n, then at least one C_i, contains an IP set.
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(mathematics, computing theory) An algorithm for polynomial evaluation that employs Horner's rule.
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(algebra) A subsemigroup with the property that if any semigroup element outside of it is added to any one of its members, the result must lie outside of it.
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(number theory, complex analysis, quaternion theory) An imaginary number (in the case of complex numbers, usually denoted i) that is defined as a solution to the equation i²=-1.
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(computing theory) A fast multiplication algorithm that reduces the multiplication of two n-digit numbers to at most n^(log ₂₃)≈n^(1.585) single-digit multiplications.
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(algebra) A De Morgan algebra which also satisfies the inequation x∧∼x⩽y,∨∼y for all x and y, where "∼" here denotes the De Morgan involution.
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(mathematics) A theorem stating that, if L is a complete lattice and f : L → L is an order-preserving function, then the set of fixed points of f in L is also a complete lattice. It has important applications in formal semantics of programming languages and abstract interpretation.
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(mathematics) Any of various theorems that saliently concern mean values.
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(mathematics) A theorem in complex analysis concerning the existence of meromorphic functions with prescribed poles.
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Synonym of Matiyasevich's theorem
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(mathematics) A theorem showing that all problems about algebraic number fields can be reduced to problems about their absolute Galois groups; one of the foundational results of anabelian geometry.
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(algebra, ring theory, of an element x of a semigroup or ring) Such that, for some positive integer n, xⁿ = 0.
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(physics) A theorem, proven by Emmy Noether in 1915 and published in 1918, stating that any differentiable symmetry of the action of a physical system has a corresponding conservation law.
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(mathematics) A theorem that establishes a fundamental relationship between geometry and algebra by relating algebraic sets to ideals in polynomial rings over algebraically closed fields.
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(algebra, field theory) algebraic number field
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(mathematics, of a function) Assuming each of the values in its codomain; having its range equal to its codomain.
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(algebra) An integral domain which has a subset whose elements are said to be "positive", such that this subset is closed under addition, closed under multiplication, and all elements of the integral domain satisfy a law of trichotomy; namely, that either that element is in the said subset, or it is the zero (additive identity), or its product with −1 (the additive inverse of the multiplicative identity) belongs to the said subset.
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(number theory) The ultrametric with prime number p as parameter defined as d_p(x,y)=|x-y|ₚ; i.e., such that the distance between two rational numbers is equal to the p-adic absolute value of the difference between those two numbers.
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(geometry) A result in plane geometry which cannot be derived from Euclid's postulates. It states that, given points a, b, c, and d on a line, if it is known that the points are ordered as (a, b, c) and (b, c, d), then it is also true that (a, b, d).
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(mathematics) An important theorem on the existence and uniqueness of solutions to first-order equations with given initial conditions.
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(mathematics) The theorem which states that any partition of a finite set of n elements into m (< n) subsets (allowing empty subsets) must include a subset with two or more elements; any of certain reformulations concerning the partition of infinite sets where the cardinality of the unpartitioned set exceeds that of the partition (so there is no one-to-one correspondence).
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(mathematics) A real algebraic integer greater than 1 all of whose Galois conjugates are less than 1 in absolute value.
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(complex analysis) For a meromorphic function f(z), any point a for which f(z)→∞ as z→a.
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(algebra, field theory) A polynomial over a given finite field whose roots are primitive elements; especially, the minimal polynomial of a primitive element of said finite field.
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(mathematics) A polynomial approximation of a power series, made up of monomials whose indices lie in the Newton diagram of the power series and which occur with the same coefficients as in the original power series.
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(mathematics) Any operation or a result thereof which generalises multiplication of numbers, like the multiplicative operation in a ring, product of types or a categorical product.
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(mathematics) A function where f(n+k) ≡ f(n) (mod k) for all integers n ≥ 0, k ≥ 1.
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(geometry) A theorem on convex sets, stating that any set of d + 2 points in Rᵈ can be partitioned into two sets whose convex hulls intersect.
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(algebra) The quotient ring of a commutative ring divided by one of its maximal ideals; by a certain theorem such a quotient ring must be a field.
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(mathematics, mathematical analysis, number theory) The conjecture that the zeros of the Riemann zeta function exist only at the negative even integers and certain complex numbers whose real part is ½.
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(countable) A usage of (a specified value of) the Riemann zeta function, such as in an equation.
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(mathematics) A lemma, of importance in harmonic analysis and asymptotic analysis, stating that the Fourier transform or Laplace transform of an L1 function vanishes at infinity.
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(algebra, logic) The form of a Boolean formula expressed using only the operators and constants of a Boolean ring (XOR, AND, 0, 1), some variables and possibly also coefficients, and without using any parentheses.
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(graph theory) A theorem stating that the undirected graphs, partially ordered by the graph-minor relationship, form a well-quasi-ordering.
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(computing theory) A theorem giving a relationship between deterministic and non-deterministic space complexity.
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(computing theory) An asymptotically fast recursive multiplication algorithm for large integers.
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(mathematics) A theorem about discrete dynamical systems. One of its implications is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.
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(mathematics) A combinatorial analog of the Brouwer fixed-point theorem, stating that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors.
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(mathematics) The positive number which, when squared, yields another number; the principal square root.
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(number theory) A theorem that states precisely which quadratic imaginary number fields admit unique factorization in their ring of integers.
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(mathematics) A theorem stating that a set in (n + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto n-dimensional space, and the resulting set is still definable in terms of polynomial identities and inequalities.
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(mathematics) A theorem that describes the properties of the support of the convolution of two functions.
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(algebra) The identity element, neutral element.
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(mathematics) (of an algebra) containing a multiplicative identity element (or unit), i.e. an element 1 with the property 1x = x1 = x for all elements x of the algebra.
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(mathematics, order theory) A subset of a poset which contains any ascending chain which starts at any element of itself.
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(mathematics, algebra) The additive identity element of a monoid or greater algebraic structure, particularly a group or ring.
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