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He then notes that for irreducible boundary-incompressible 3-manifolds this gives the algebraic definition. Jean-Pierre Otal (2001) uses the algebraic definition without additional restrictions. Bennett Chow (2007) uses the geometric definition, restricted to irreducible manifolds. | https://en.wikipedia.org/wiki/Atoroidal |
Michael Kapovich (2009) requires the algebraic variant of atoroidal manifolds (which he calls simply atoroidal) to avoid being one of three kinds of fiber bundle. He makes the same restriction on geometrically atoroidal manifolds (which he calls topologically atoroidal) and in addition requires them to avoid incompress... | https://en.wikipedia.org/wiki/Atoroidal |
In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. These automata represent the Cayley graph of the group. That is, they can tell if a given word representation of a group element is in a "canonical form" and can tell if two elements given in canonical words di... | https://en.wikipedia.org/wiki/Automatic_group |
In mathematics, an automatic semigroup is a finitely generated semigroup equipped with several regular languages over an alphabet representing a generating set. One of these languages determines "canonical forms" for the elements of the semigroup, the other languages determine if two canonical forms represent elements ... | https://en.wikipedia.org/wiki/Automatic_semigroup |
In mathematics, an automorphic L-function is a function L(s,π,r) of a complex variable s, associated to an automorphic representation π of a reductive group G over a global field and a finite-dimensional complex representation r of the Langlands dual group LG of G, generalizing the Dirichlet L-series of a Dirichlet cha... | https://en.wikipedia.org/wiki/Automorphic_L-function |
In mathematics, an automorphic factor is a certain type of analytic function, defined on subgroups of SL(2,R), appearing in the theory of modular forms. The general case, for general groups, is reviewed in the article 'factor of automorphy'. | https://en.wikipedia.org/wiki/Automorphic_factor |
In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group. | https://en.wikipedia.org/wiki/Automorphic_function |
In mathematics, an automorphic number (sometimes referred to as a circular number) is a natural number in a given number base b {\displaystyle b} whose square "ends" in the same digits as the number itself. | https://en.wikipedia.org/wiki/Automorphic_number |
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely ... | https://en.wikipedia.org/wiki/Trivial_automorphism |
In mathematics, an autonomous category is a monoidal category where dual objects exist. | https://en.wikipedia.org/wiki/Autonomous_category |
In mathematics, an autonomous convergence theorem is one of a family of related theorems which specify conditions guaranteeing global asymptotic stability of a continuous autonomous dynamical system. | https://en.wikipedia.org/wiki/Autonomous_convergence_theorem |
In mathematics, an autonomous system is a dynamic equation on a smooth manifold. A non-autonomous system is a dynamic equation on a smooth fiber bundle Q → R {\displaystyle Q\to \mathbb {R} } over R {\displaystyle \mathbb {R} } . For instance, this is the case of non-autonomous mechanics. An r-order differential equati... | https://en.wikipedia.org/wiki/Non-autonomous_system_(mathematics) |
A dynamic equation on Q → R {\displaystyle Q\to \mathbb {R} } is a differential equation which is algebraically solved for a higher-order derivatives. In particular, a first-order dynamic equation on a fiber bundle Q → R {\displaystyle Q\to \mathbb {R} } is a kernel of the covariant differential of some connection Γ {\... | https://en.wikipedia.org/wiki/Non-autonomous_system_(mathematics) |
{\displaystyle q_{t}^{i}=\Gamma (t,q^{i}).} For instance, this is the case of Hamiltonian non-autonomous mechanics. A second-order dynamic equation q t t i = ξ i ( t , q j , q t j ) {\displaystyle q_{tt}^{i}=\xi ^{i}(t,q^{j},q_{t}^{j})} on Q → R {\displaystyle Q\to \mathbb {R} } is defined as a holonomic connection ξ {... | https://en.wikipedia.org/wiki/Non-autonomous_system_(mathematics) |
This equation also is represented by a connection on an affine jet bundle J 1 Q → Q {\displaystyle J^{1}Q\to Q} . Due to the canonical embedding J 1 Q → T Q {\displaystyle J^{1}Q\to TQ} , it is equivalent to a geodesic equation on the tangent bundle T Q {\displaystyle TQ} of Q {\displaystyle Q} . A free motion equation... | https://en.wikipedia.org/wiki/Non-autonomous_system_(mathematics) |
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems. Many laws in physics, where the independent variable is usually as... | https://en.wikipedia.org/wiki/Autonomous_system_(mathematics) |
In mathematics, an axiom of countability is a property of certain mathematical objects that asserts the existence of a countable set with certain properties. Without such an axiom, such a set might not provably exist. | https://en.wikipedia.org/wiki/Countability_axiom |
In mathematics, an edge cycle cover (sometimes called simply cycle cover) of a graph is a family of cycles which are subgraphs of G and contain all edges of G. If the cycles of the cover have no vertices in common, the cover is called vertex-disjoint or sometimes simply disjoint cycle cover. In this case, the set of th... | https://en.wikipedia.org/wiki/Edge_cycle_cover |
In mathematics, an effaceable functor is an additive functor F between abelian categories C and D for which, for each object A in C, there exists a monomorphism u: A → M {\displaystyle u:A\to M} , for some M, such that F ( u ) = 0 {\displaystyle F(u)=0} . Similarly, a coeffaceable functor is one for which, for each A, ... | https://en.wikipedia.org/wiki/Effaceable_functor |
In mathematics, an eigenform (meaning simultaneous Hecke eigenform with modular group SL(2,Z)) is a modular form which is an eigenvector for all Hecke operators Tm, m = 1, 2, 3, .... Eigenforms fall into the realm of number theory, but can be found in other areas of math and science such as analysis, combinatorics, and... | https://en.wikipedia.org/wiki/Eigenform |
In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f {\displaystyle f} in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as for some scalar eigenvalue λ . {\... | https://en.wikipedia.org/wiki/Eigenfunction |
In mathematics, an eigenoperator, A, of a matrix H is a linear operator such that = λ A {\displaystyle =\lambda A\,} where λ {\displaystyle \lambda } is a corresponding scalar called an eigenvalue. == References == | https://en.wikipedia.org/wiki/Eigenoperator |
In mathematics, an eigenplane is a two-dimensional invariant subspace in a given vector space. By analogy with the term eigenvector for a vector which, when operated on by a linear operator is another vector which is a scalar multiple of itself, the term eigenplane can be used to describe a two-dimensional plane (a 2-p... | https://en.wikipedia.org/wiki/Eigenplane |
In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system A x = λ x {\displaystyle Ax=\lambda x} that is perturbed from one with known eigenvectors and eigenvalues A 0 x 0 = λ 0 x 0 {\displaystyle A_{0}x_{0}=\lambda _{0}x_{0}} . This is useful for studying how se... | https://en.wikipedia.org/wiki/Eigenvalue_perturbation |
In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. | https://en.wikipedia.org/wiki/Membership_(set_theory) |
In mathematics, an element a of a commutative ring A is called (relatively) prime to an ideal Q if whenever ab is an element of Q then b is also an element of Q. A proper ideal Q of a commutative ring A is said to be primal if the elements that are not prime to it form an ideal. | https://en.wikipedia.org/wiki/Primal_ideal |
In mathematics, an element p of a partial order (P, ≤) is a meet prime element when p is the principal element of a principal prime ideal. Equivalently, if P is a lattice, p ≠ top, and for all a, b in P, a∧b ≤ p implies a ≤ p or b ≤ p. | https://en.wikipedia.org/wiki/Prime_(order_theory) |
In mathematics, an element x of a *-algebra is normal if it satisfies x x ∗ = x ∗ x . {\displaystyle xx^{*}=x^{*}x.} This definition stems from the definition of a normal linear operator in functional analysis, where a linear operator A from a Hilbert space into itself is called unitary if A A ∗ = A ∗ A , {\displaystyl... | https://en.wikipedia.org/wiki/Normal_element |
In mathematics, an element x of a *-algebra is unitary if it satisfies x ∗ = x − 1 . {\displaystyle x^{*}=x^{-1}.} In functional analysis, a linear operator A from a Hilbert space into itself is called unitary if it is invertible and its inverse is equal to its own adjoint A∗ and that the domain of A is the same as tha... | https://en.wikipedia.org/wiki/Unitary_element |
In mathematics, an element x of a Lie group or a Lie algebra is called an n-Engel element, named after Friedrich Engel, if it satisfies the n-Engel condition that the repeated commutator ,y], ..., y] with n copies of y is trivial (where means xyx−1y−1 or the Lie bracket). It is called an Engel element if it satisfies ... | https://en.wikipedia.org/wiki/Engel_group |
Every nilpotent group or Lie algebra is Engel. Engel's theorem states that every finite-dimensional Engel algebra is nilpotent. (Cohn 1955) gave examples of non-nilpotent Engel groups and algebras. | https://en.wikipedia.org/wiki/Engel_group |
In mathematics, an element x {\displaystyle x} of a ring R {\displaystyle R} is called nilpotent if there exists some positive integer n {\displaystyle n} , called the index (or sometimes the degree), such that x n = 0 {\displaystyle x^{n}=0} . The term, along with its sister idempotent, was introduced by Benjamin Peir... | https://en.wikipedia.org/wiki/Nilpotent |
In mathematics, an element z {\displaystyle z} of a Banach algebra A {\displaystyle A} is called a topological divisor of zero if there exists a sequence x 1 , x 2 , x 3 , . . . {\displaystyle x_{1},x_{2},x_{3},...} of elements of A {\displaystyle A} such that The sequence z x n {\displaystyle zx_{n}} converges to the ... | https://en.wikipedia.org/wiki/Topological_divisor_of_zero |
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions (e.g., arcsin, lo... | https://en.wikipedia.org/wiki/Elementary_functions |
In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GLn(F) when F is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while... | https://en.wikipedia.org/wiki/Elementary_row_operation |
In mathematics, an elementary proof is a mathematical proof that only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. Historically, it was once thought that certain theorems, like the prime number theorem, could only be proved by invok... | https://en.wikipedia.org/wiki/Elementary_proof |
While there is generally no consensus as to what counts as elementary, the term is nevertheless a common part of the mathematical jargon. An elementary proof is not necessarily simple, in the sense of being easy to understand or trivial. In fact, some elementary proofs can be quite complicated — and this is especially ... | https://en.wikipedia.org/wiki/Elementary_proof |
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measu... | https://en.wikipedia.org/wiki/Gardener's_ellipse |
Analytically, the equation of a standard ellipse centered at the origin with width 2 a {\displaystyle 2a} and height 2 b {\displaystyle 2b} is: Assuming a ≥ b {\displaystyle a\geq b} , the foci are ( ± c , 0 ) {\displaystyle (\pm c,0)} for c = a 2 − b 2 {\textstyle c={\sqrt {a^{2}-b^{2}}}} . The standard parametric equ... | https://en.wikipedia.org/wiki/Gardener's_ellipse |
An angled cross section of a cylinder is also an ellipse. An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix: for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant. This constant ratio is the... | https://en.wikipedia.org/wiki/Gardener's_ellipse |
For example, the orbit of each planet in the Solar System is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the barycenter of the Sun–planet pair). The same is true for moons orbiting planets and all other systems of two astronomical bodies. The shapes of planets and stars are of... | https://en.wikipedia.org/wiki/Gardener's_ellipse |
A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under parallel or perspective projection. The ellipse is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency: a similar effect leads to elliptical p... | https://en.wikipedia.org/wiki/Gardener's_ellipse |
In mathematics, an elliptic Gauss sum is an analog of a Gauss sum depending on an elliptic curve with complex multiplication. The quadratic residue symbol in a Gauss sum is replaced by a higher residue symbol such as a cubic or quartic residue symbol, and the exponential function in a Gauss sum is replaced by an ellipt... | https://en.wikipedia.org/wiki/Elliptic_Gauss_sum |
In mathematics, an elliptic boundary value problem is a special kind of boundary value problem which can be thought of as the stable state of an evolution problem. For example, the Dirichlet problem for the Laplacian gives the eventual distribution of heat in a room several hours after the heating is turned on. Differe... | https://en.wikipedia.org/wiki/Elliptic_boundary_value_problem |
This is necessary because each category must be analyzed using different techniques. The present article deals with the category of boundary value problems known as linear elliptic problems. Boundary value problems and partial differential equations specify relations between two or more quantities. | https://en.wikipedia.org/wiki/Elliptic_boundary_value_problem |
For instance, in the heat equation, the rate of change of temperature at a point is related to the difference of temperature between that point and the nearby points so that, over time, the heat flows from hotter points to cooler points. Boundary value problems can involve space, time and other quantities such as tempe... | https://en.wikipedia.org/wiki/Elliptic_boundary_value_problem |
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K2, the Cartesian product of K with itself. If the field's characteristic is different from 2 and 3, then the curve can be d... | https://en.wikipedia.org/wiki/Elliptic_curves |
Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic curves; see § Elliptic curves over a general field below.) An elliptic curve is an abelian... | https://en.wikipedia.org/wiki/Elliptic_curves |
If y2 = P(x), where P is any polynomial of degree three in x with no repeated roots, the solution set is a nonsingular plane curve of genus one, an elliptic curve. If P has degree four and is square-free this equation again describes a plane curve of genus one; however, it has no natural choice of identity element. Mor... | https://en.wikipedia.org/wiki/Elliptic_curves |
Using the theory of elliptic functions, it can be shown that elliptic curves defined over the complex numbers correspond to embeddings of the torus into the complex projective plane. The torus is also an abelian group, and this correspondence is also a group isomorphism. Elliptic curves are especially important in numb... | https://en.wikipedia.org/wiki/Elliptic_curves |
They also find applications in elliptic curve cryptography (ECC) and integer factorization. An elliptic curve is not an ellipse in the sense of a projective conic, which has genus zero: see elliptic integral for the origin of the term. However, there is a natural representation of real elliptic curves with shape invari... | https://en.wikipedia.org/wiki/Elliptic_curves |
Specifically, the intersections of the Minkowski hyperboloid with quadric surfaces characterized by a certain constant-angle property produce the Steiner ellipses in H 2 {\displaystyle \mathbb {H} ^{2}} (generated by orientation-preserving collineations). Further, the orthogonal trajectories of these ellipses comprise ... | https://en.wikipedia.org/wiki/Elliptic_curves |
In mathematics, an elliptic divisibility sequence (EDS) is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined, and their arithmetic properties studied, by Morgan Ward in the 1940s. They attracted only sporadic attention until aro... | https://en.wikipedia.org/wiki/Elliptic_divisibility_sequence |
In mathematics, an elliptic hypergeometric series is a series Σcn such that the ratio cn/cn−1 is an elliptic function of n, analogous to generalized hypergeometric series where the ratio is a rational function of n, and basic hypergeometric series where the ratio is a periodic function of the complex number n. They wer... | https://en.wikipedia.org/wiki/Elliptic_hypergeometric_series |
In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed field such as the complex numbers, these fibers are elliptic curves, perha... | https://en.wikipedia.org/wiki/Quasi-elliptic_surface |
The surface and the base curve are assumed to be non-singular (complex manifolds or regular schemes, depending on the context). The fibers that are not elliptic curves are called the singular fibers and were classified by Kunihiko Kodaira. Both elliptic and singular fibers are important in string theory, especially in ... | https://en.wikipedia.org/wiki/Quasi-elliptic_surface |
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X {\displaystyle X} is said to be embedded in another object Y {\displaystyle Y} , the embedding is given by some injective and structure-pre... | https://en.wikipedia.org/wiki/Isometric_imbedding |
The fact that a map f: X → Y {\displaystyle f:X\rightarrow Y} is an embedding is often indicated by the use of a "hooked arrow" (U+21AA ↪ RIGHTWARDS ARROW WITH HOOK); thus: f: X ↪ Y . {\displaystyle f:X\hookrightarrow Y.} (On the other hand, this notation is sometimes reserved for inclusion maps.) | https://en.wikipedia.org/wiki/Isometric_imbedding |
Given X {\displaystyle X} and Y {\displaystyle Y} , several different embeddings of X {\displaystyle X} in Y {\displaystyle Y} may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational ... | https://en.wikipedia.org/wiki/Isometric_imbedding |
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention ... | https://en.wikipedia.org/wiki/Empty_product |
In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero. The natural way to extend non-empty sums is to let the empty sum be the additive identity. Let a 1 {\displaystyle a_{1}} , a 2 {\displaystyle a_{2}} , a 3 {\displaystyle a_{3}} , ... be a sequence of numbers, and let s m = ∑... | https://en.wikipedia.org/wiki/Empty_sum |
In other words, a "sum" s 1 {\displaystyle s_{1}} with only one term evaluates to that one term, while a "sum" s 0 {\displaystyle s_{0}} with no terms evaluates to 0. Allowing a "sum" with only 1 or 0 terms reduces the number of cases to be considered in many mathematical formulas. | https://en.wikipedia.org/wiki/Empty_sum |
Such "sums" are natural starting points in induction proofs, as well as in algorithms. For these reasons, the "empty sum is zero" extension is standard practice in mathematics and computer programming (assuming the domain has a zero element). For the same reason, the empty product is taken to be the multiplicative iden... | https://en.wikipedia.org/wiki/Empty_sum |
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space V is a linear map f: V → V, and an endomorphism of a group G is a group homomorphism f: G → G. In general, we can talk about ... | https://en.wikipedia.org/wiki/Endomorphism |
In mathematics, an equaliser is a set of arguments where two or more functions have equal values. An equaliser is the solution set of an equation. In certain contexts, a difference kernel is the equaliser of exactly two functions. | https://en.wikipedia.org/wiki/Equaliser_(mathematics) |
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, wh... | https://en.wikipedia.org/wiki/Mathematical_equations |
There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables.The "=" symbol, which appears in every equation, was invented in 1557 by Robert Recorde, who considered that nothing cou... | https://en.wikipedia.org/wiki/Mathematical_equations |
In mathematics, an equidistant set (also called a midset, or a bisector) is a set whose elements have the same distance (measured using some appropriate distance function) from two or more sets. The equidistant set of two singleton sets in the Euclidean plane is the perpendicular bisector of the segment joining the two... | https://en.wikipedia.org/wiki/Equidistant_set |
The concept of equidistant set is used to define frontiers in territorial domain controversies. For instance, the United Nations Convention on the Law of the Sea (Article 15) establishes that, in absence of any previous agreement, the delimitation of the territorial sea between countries occurs exactly on the median li... | https://en.wikipedia.org/wiki/Equidistant_set |
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number a is equal to itself (reflexive). | https://en.wikipedia.org/wiki/Equivalence_relation |
If a = b, then b = a (symmetric). If a = b and b = c, then a = c (transitive). Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. | https://en.wikipedia.org/wiki/Equivalence_relation |
In mathematics, an ergodic sequence is a certain type of integer sequence, having certain equidistribution properties. | https://en.wikipedia.org/wiki/Ergodic_sequence |
In mathematics, an essentially finite vector bundle is a particular type of vector bundle defined by Madhav V. Nori, as the main tool in the construction of the fundamental group scheme. Even if the definition is not intuitive there is a nice characterization that makes essentially finite vector bundles quite natural o... | https://en.wikipedia.org/wiki/Essentially_finite_vector_bundle |
In mathematics, an evasive Boolean function ƒ (of n variables) is a Boolean function for which every decision tree algorithm has running time of exactly n. Consequently, every decision tree algorithm that represents the function has, at worst case, a running time of n. | https://en.wikipedia.org/wiki/Evasive_Boolean_function |
In mathematics, an event that occurs with high probability (often shortened to w.h.p. or WHP) is one whose probability depends on a certain number n and goes to 1 as n goes to infinity, i.e. the probability of the event occurring can be made as close to 1 as desired by making n big enough. | https://en.wikipedia.org/wiki/With_high_probability |
In mathematics, an exact category is a concept of category theory due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition of such a sequence. | https://en.wikipedia.org/wiki/Exact_category |
In mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in Physics and engineering. | https://en.wikipedia.org/wiki/Exact_differential_equation |
In mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five of them: g 2 , f 4 , e 6 , e 7 , e 8 {\displaystyle {\mathfrak {g}}_{2},{\mathfrak {f}}_{4},{\mathfrak {e}}_{6},{\mathfrak {e}}_{7},{\mathfrak {e}}_{8}} ; their r... | https://en.wikipedia.org/wiki/Exceptional_Lie_algebra |
In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj of two families, usually infinite, of mathematical objects, which is incidental, in that it is not an instance of a general pattern of such isomorphisms. These coincidences are at times conside... | https://en.wikipedia.org/wiki/Accidental_isomorphism |
In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase "there exist(s)", or it might be a universal statement whose last quantifier is existential (e.g., "for all x, y, ... there exist(s) ..."). In the formal terms of symbo... | https://en.wikipedia.org/wiki/Purely_existential_proof |
For example, the statement that the sine function is continuous everywhere, or any theorem written in big O notation, can be considered as theorems which are existential by nature—since the quantification can be found in the definitions of the concepts used. A controversy that goes back to the early twentieth century c... | https://en.wikipedia.org/wiki/Purely_existential_proof |
In mathematics, an existence theorem is purely theoretical if the proof given for it does not indicate a construction of the object whose existence is asserted. Such a proof is non-constructive, since the whole approach may not lend itself to construction. In terms of algorithms, purely theoretical existence theorems b... | https://en.wikipedia.org/wiki/Existence_theorem |
Despite that, the purely theoretical existence results are nevertheless ubiquitous in contemporary mathematics. For example, John Nash's original proof of the existence of a Nash equilibrium in 1951 was such an existence theorem. An approach which is constructive was also later found in 1962. | https://en.wikipedia.org/wiki/Existence_theorem |
In mathematics, an exp algebra is a Hopf algebra Exp(G) constructed from an abelian group G, and is the universal ring R such that there is an exponential map from G to the group of the power series in R] with constant term 1. In other words the functor Exp from abelian groups to commutative rings is adjoint to the fun... | https://en.wikipedia.org/wiki/Exp_ring |
In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, b... | https://en.wikipedia.org/wiki/Polynomial_expansion |
It is customary to reintroduce powers in the final result when terms involve products of identical symbols. Simple examples of polynomial expansions are the well known rules ( x + y ) 2 = x 2 + 2 x y + y 2 {\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}} ( x + y ) ( x − y ) = x 2 − y 2 {\displaystyle (x+y)(x-y)=x^{2}-y^{2}} w... | https://en.wikipedia.org/wiki/Polynomial_expansion |
In mathematics, an explicit reciprocity law is a formula for the Hilbert symbol of a local field. The name "explicit reciprocity law" refers to the fact that the Hilbert symbols of local fields appear in Hilbert's reciprocity law for the power residue symbol. The definitions of the Hilbert symbol are usually rather rou... | https://en.wikipedia.org/wiki/Explicit_reciprocity_law |
In mathematics, an exponential field is a field that has an extra operation on its elements which extends the usual idea of exponentiation. | https://en.wikipedia.org/wiki/Exponential_ring |
In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function e ( x ) = exp ( 2 π i x ) . {\displaystyle e(x)=\exp(2\pi ix).\,} Therefore, a typical exponential sum may take the... | https://en.wikipedia.org/wiki/Exponential_sum |
In mathematics, an exposed point of a convex set C {\displaystyle C} is a point x ∈ C {\displaystyle x\in C} at which some continuous linear functional attains its strict maximum over C {\displaystyle C} . Such a functional is then said to expose x {\displaystyle x} . There can be many exposing functionals for x {\disp... | https://en.wikipedia.org/wiki/Exposed_point |
A stronger notion is that of strongly exposed point of C {\displaystyle C} which is an exposed point x ∈ C {\displaystyle x\in C} such that some exposing functional f {\displaystyle f} of x {\displaystyle x} attains its strong maximum over C {\displaystyle C} at x {\displaystyle x} , i.e. for each sequence ( x n ) ⊂ C ... | https://en.wikipedia.org/wiki/Exposed_point |
In mathematics, an expression is in closed form if it is formed with constants, variables and a finite set of basic functions connected by arithmetic operations (+, −, ×, ÷, and integer powers) and function composition. Commonly, the allowed functions are nth root, exponential function, logarithm, and trigonometric fun... | https://en.wikipedia.org/wiki/Solution_in_closed_form |
In mathematics, an expression must represent a single value. For example consider the equation, x 2 = 4 {\displaystyle x^{2}=4} which implies, x = 2 ∨ x = − 2 {\displaystyle x=2\lor x=-2} But this is a bit long winded, and it does not allow us to work with multiple values at the same time. If further conditions or cons... | https://en.wikipedia.org/wiki/Narrowing_of_algebraic_value_sets |
Each x must represent a single value in the expression. Either x is 2 or x = −2. This can be resolved by keeping track of the two values so that we make sure that the values are used consistently, and this is what a value set does. | https://en.wikipedia.org/wiki/Narrowing_of_algebraic_value_sets |
In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operatio... | https://en.wikipedia.org/wiki/Mathematical_expression |
For example, 8 x − 5 {\displaystyle 8x-5} is an expression, while 8 x − 5 ≥ 5 x − 8 {\displaystyle 8x-5\geq 5x-8} is a formula. However, in modern mathematics, and in particular in computer algebra, formulas are viewed as expressions that can be evaluated to true or false, depending on the values that are given to the ... | https://en.wikipedia.org/wiki/Mathematical_expression |
In mathematics, an extensive category is a category C with finite coproducts that are disjoint and well-behaved with respect to pullbacks. Equivalently, C is extensive if the coproduct functor from the product of the slice categories C/X × C/Y to the slice category C/(X + Y) is an equivalence of categories for all obje... | https://en.wikipedia.org/wiki/Extensive_category |
In mathematics, an extraneous solution (or spurious solution) is a solution, such as that to an equation, that emerges from the process of solving the problem but is not a valid solution to the problem. A missing solution is a solution that is a valid solution to the problem, but disappeared during the process of solvi... | https://en.wikipedia.org/wiki/Extraneous_solution |
In mathematics, an extremally disconnected space is a topological space in which the closure of every open set is open. (The term "extremally disconnected" is correct, even though the word "extremally" does not appear in most dictionaries, and is sometimes mistaken by spellcheckers for the homophone extremely disconnec... | https://en.wikipedia.org/wiki/Extremally_disconnected_space |
Every Stonean space is a Stone space, but not vice versa. In the duality between Stone spaces and Boolean algebras, the Stonean spaces correspond to the complete Boolean algebras. An extremally disconnected first-countable collectionwise Hausdorff space must be discrete. In particular, for metric spaces, the property o... | https://en.wikipedia.org/wiki/Extremally_disconnected_space |
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