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c_jr3rib6bwnvh | Equivalently, it is the ratio of the infinitesimal change of the logarithm of a function with respect to the infinitesimal change of the logarithm of the argument. Generalisations to multi-input-multi-output cases also exist in the literature.The elasticity of a function is a constant α {\displaystyle \alpha } if and o... | Elasticity of a function |
c_1bwf23osriio | In mathematics, the elliptic gamma function is a generalization of the q-gamma function, which is itself the q-analog of the ordinary gamma function. It is closely related to a function studied by Jackson (1905), and can be expressed in terms of the triple gamma function. It is given by Γ ( z ; p , q ) = ∏ m = 0 ∞ ∏ n ... | Elliptic gamma function |
c_c351so0ucll0 | {\displaystyle \Gamma (z;p,q)=\prod _{m=0}^{\infty }\prod _{n=0}^{\infty }{\frac {1-p^{m+1}q^{n+1}/z}{1-p^{m}q^{n}z}}.} It obeys several identities: Γ ( z ; p , q ) = 1 Γ ( p q / z ; p , q ) {\displaystyle \Gamma (z;p,q)={\frac {1}{\Gamma (pq/z;p,q)}}\,} Γ ( p z ; p , q ) = θ ( z ; q ) Γ ( z ; p , q ) {\displaystyle \G... | Elliptic gamma function |
c_jcuq3d2jsn45 | In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are v... | Nonempty set |
c_aeatt6c6q2xv | Any set other than the empty set is called non-empty. In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). The empty set may also b... | Nonempty set |
c_rxfqkyx3ry2a | In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. | Endomorphism algebra |
c_4zc8fmswtjqb | Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map 0: x ↦ 0 {\textstyle 0:x\mapsto 0} as additive identity and the identity map 1: x ↦ x {\textstyle 1:x\mapsto x} as multiplicative identity.The functions involved are restricted to what is defined as a homomorph... | Endomorphism algebra |
c_zpxv7a9h2tgb | An abelian group is the same thing as a module over the ring of integers, which is the initial object in the category of rings. In a similar fashion, if R is any commutative ring, the endomorphisms of an R-module form an algebra over R by the same axioms and derivation. In particular, if R is a field, its modules M are... | Endomorphism algebra |
c_wn0ahce11v3b | In mathematics, the energy of a graph is the sum of the absolute values of the eigenvalues of the adjacency matrix of the graph. This quantity is studied in the context of spectral graph theory. More precisely, let G be a graph with n vertices. | Graph energy |
c_znuuvn8o5v0r | It is assumed that G is simple, that is, it does not contain loops or parallel edges. Let A be the adjacency matrix of G and let λ i {\displaystyle \lambda _{i}} , i = 1 , … , n {\displaystyle i=1,\ldots ,n} , be the eigenvalues of A. Then the energy of the graph is defined as: E ( G ) = ∑ i = 1 n | λ i | . {\displayst... | Graph energy |
c_l12xd1vnldlk | In mathematics, the entropy influence conjecture is a statement about Boolean functions originally conjectured by Ehud Friedgut and Gil Kalai in 1996. | Entropy influence conjecture |
c_fq0x3r2mv0au | In mathematics, the epigraph or supergraph of a function f: X → {\displaystyle f:X\to } valued in the extended real numbers = R ∪ { ± ∞ } {\displaystyle =\mathbb {R} \cup \{\pm \infty \}} is the set, denoted by epi f , {\displaystyle \operatorname {epi} f,} of all points in the Cartesian product X × R {\displaystyl... | Epigraph (mathematics) |
c_6xbqad4rtymx | If the function takes ± ∞ {\displaystyle \pm \infty } as a value then graph f {\displaystyle \operatorname {graph} f} will not be a subset of its epigraph epi f . {\displaystyle \operatorname {epi} f.} For example, if f ( x 0 ) = ∞ {\displaystyle f\left(x_{0}\right)=\infty } then the point ( x 0 , f ( x 0 ) ) = ( x... | Epigraph (mathematics) |
c_diniqzyuqo7i | {\displaystyle \operatorname {epi} f.} These two sets are nevertheless closely related because the graph can always be reconstructed from the epigraph, and vice versa. The study of continuous real-valued functions in real analysis has traditionally been closely associated with the study of their graphs, which are sets ... | Epigraph (mathematics) |
c_09sn7dero057 | Epigraphs serve this same purpose in the fields of convex analysis and variational analysis, in which the primary focus is on convex functions valued in {\displaystyle } instead of continuous functions valued in a vector space (such as R {\displaystyle \mathbb {R} } or R 2 {\displaystyle \mathbb {R} ^{2}} ). This is b... | Epigraph (mathematics) |
c_r662b26d6fln | In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplicati... | Epsilon number |
c_bljj0mwr7pmj | Larger ordinal fixed points of the exponential map are indexed by ordinal subscripts, resulting in ε 1 , ε 2 , … , ε ω , ε ω + 1 , … , ε ε 0 , … , ε ε 1 , … , ε ε ε ⋅ ⋅ ⋅ , … {\displaystyle \varepsilon _{1},\varepsilon _{2},\ldots ,\varepsilon _{\omega },\varepsilon _{\omega +1},\ldots ,\varepsilon _{\varepsilon _{0}},... | Epsilon number |
c_7ng1nnvgsqbr | The smallest epsilon number ε0 appears in many induction proofs, because for many purposes, transfinite induction is only required up to ε0 (as in Gentzen's consistency proof and the proof of Goodstein's theorem). Its use by Gentzen to prove the consistency of Peano arithmetic, along with Gödel's second incompleteness ... | Epsilon number |
c_f6fy2t548hco | A more general class of epsilon numbers has been identified by John Horton Conway and Donald Knuth in the surreal number system, consisting of all surreals that are fixed points of the base ω exponential map x → ωx. Hessenberg (1906) defined gamma numbers (see additively indecomposable ordinal) to be numbers γ>0 such t... | Epsilon number |
c_yhakhs76r4pg | In mathematics, the equal sign can be used as a simple statement of fact in a specific case ("x = 2"), or to create definitions ("let x = 2"), conditional statements ("if x = 2, then ..."), or to express a universal equivalence ("(x + 1)2 = x2 + 2x + 1"). The first important computer programming language to use the equ... | Not equal sign |
c_5kdwh35wap5t | A rival programming-language usage was pioneered by the original version of ALGOL, which was designed in 1958 and implemented in 1960. ALGOL included a relational operator that tested for equality, allowing constructions like if x = 2 with essentially the same meaning of = as the conditional usage in mathematics. The e... | Not equal sign |
c_8bo2ecc169vx | Both usages have remained common in different programming languages into the early 21st century. As well as Fortran, = is used for assignment in such languages as C, Perl, Python, awk, and their descendants. But = is used for equality and not assignment in the Pascal family, Ada, Eiffel, APL, and other languages. | Not equal sign |
c_9kb95ya4b6kb | A few languages, such as BASIC and PL/I, have used the equal sign to mean both assignment and equality, distinguished by context. However, in most languages where = has one of these meanings, a different character or, more often, a sequence of characters is used for the other meaning. Following ALGOL, most languages th... | Not equal sign |
c_kmfospblebrk | Fortran did not have an equality operator (it was only possible to compare an expression to zero, using the arithmetic IF statement) until FORTRAN IV was released in 1962, since when it has used the four characters .EQ. to test for equality. The language B introduced the use of == with this meaning, which has been copi... | Not equal sign |
c_zvuzhdrq9afc | In mathematics, the equations governing the isomonodromic deformation of meromorphic linear systems of ordinary differential equations are, in a fairly precise sense, the most fundamental exact nonlinear differential equations. As a result, their solutions and properties lie at the heart of the field of exact nonlinear... | Schlesinger equations |
c_rlxeypgnkjox | In mathematics, the equidistribution theorem is the statement that the sequence a, 2a, 3a, ... mod 1is uniformly distributed on the circle R / Z {\displaystyle \mathbb {R} /\mathbb {Z} } , when a is an irrational number. It is a special case of the ergodic theorem where one takes the normalized angle measure μ = d θ 2 ... | Equidistribution theorem |
c_lgaw1brgoxcn | In mathematics, the equilateral dimension of a metric space is the maximum size of any subset of the space whose points are all at equal distances to each other. Equilateral dimension has also been called "metric dimension", but the term "metric dimension" also has many other inequivalent usages. The equilateral dimens... | Equilateral dimension |
c_diw4l6gamr0p | In mathematics, the equioscillation theorem concerns the approximation of continuous functions using polynomials when the merit function is the maximum difference (uniform norm). Its discovery is attributed to Chebyshev. | Equioscillation theorem |
c_2p467qnz5mtx | In mathematics, the equivariant algebraic K-theory is an algebraic K-theory associated to the category Coh G ( X ) {\displaystyle \operatorname {Coh} ^{G}(X)} of equivariant coherent sheaves on an algebraic scheme X with action of a linear algebraic group G, via Quillen's Q-construction; thus, by definition, K i G ( ... | Equivariant algebraic K-theory |
c_tfenx6gtsxif | Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem. Equivalently, K i G ( X ) {\displaystyle K_{i}^{G}(X)} may be defined as the K i {\displaystyle K_{i}} of the category of coherent sheaves on the quotient stack {\displaystyle } . (Hence, the equivariant K-theory is a... | Equivariant algebraic K-theory |
c_5mryznih9mf8 | In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as: erf z = 2 π ∫ 0 z e − t 2 d t . {\displaystyle \operatorname {erf} z={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,\mathrm {d} t.} Some authors define erf {\disp... | Complementary error function |
c_vrrt4rdd9g4l | In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real. In statistics, for non-negative values of x, the error function has the following interpretation: for a random variable Y that is normally distributed with mean 0 and standard d... | Complementary error function |
c_dvnl8n5wtqn6 | In mathematics, the essence of counting a set and finding a result n, is that it establishes a one-to-one correspondence (or bijection) of the set with the subset of positive integers {1, 2, ..., n}. A fundamental fact, which can be proved by mathematical induction, is that no bijection can exist between {1, 2, ..., n}... | Inclusive counting |
c_w858st19jly4 | Many sets that arise in mathematics do not allow a bijection to be established with {1, 2, ..., n} for any natural number n; these are called infinite sets, while those sets for which such a bijection does exist (for some n) are called finite sets. Infinite sets cannot be counted in the usual sense; for one thing, the ... | Inclusive counting |
c_hr58y906rnk1 | The notion of counting may be extended to them in the sense of establishing (the existence of) a bijection with some well-understood set. For instance, if a set can be brought into bijection with the set of all natural numbers, then it is called "countably infinite." This kind of counting differs in a fundamental way f... | Inclusive counting |
c_ppbonfxhl6di | For instance, the set of all integers (including negative numbers) can be brought into bijection with the set of natural numbers, and even seemingly much larger sets like that of all finite sequences of rational numbers are still (only) countably infinite. Nevertheless, there are sets, such as the set of real numbers, ... | Inclusive counting |
c_gy4dzetf7bwq | Beyond the cardinalities given by each of the natural numbers, there is an infinite hierarchy of infinite cardinalities, although only very few such cardinalities occur in ordinary mathematics (that is, outside set theory that explicitly studies possible cardinalities). Counting, mostly of finite sets, has various appl... | Inclusive counting |
c_ydlza5tz72vo | One important principle is that if two sets X and Y have the same finite number of elements, and a function f: X → Y is known to be injective, then it is also surjective, and vice versa. A related fact is known as the pigeonhole principle, which states that if two sets X and Y have finite numbers of elements n and m wi... | Inclusive counting |
c_bnx91iqgs9fb | In mathematics, the essential spectrum of a bounded operator (or, more generally, of a densely defined closed linear operator) is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, "fails badly to be invertible". | Essential spectrum |
c_a8xz6unvyw0u | In mathematics, the eta invariant of a self-adjoint elliptic differential operator on a compact manifold is formally the number of positive eigenvalues minus the number of negative eigenvalues. In practice both numbers are often infinite so are defined using zeta function regularization. It was introduced by Atiyah, Pa... | Eta invariant |
c_lxb861yo52nl | In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any non-empty set and p ∈ X. The collection T = { S ⊆ X: p ∉ S } ∪ { X } {\displaystyle T=\{S\subseteq X:p\notin S\}\cup \{X\}} of subsets of X is then the excluded point topology on X. T... | Excluded point topology |
c_dlq0ejoa686a | In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic... | Riemann's explicit formula |
c_ef2nf247fxh2 | In mathematics, the exponential function can be characterized in many ways. The following characterizations (definitions) are most common. This article discusses why each characterization makes sense, and why the characterizations are independent of and equivalent to each other. As a special case of these consideration... | Characterizations of the exponential function |
c_3mmrzm3hpliu | In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument. | Well function |
c_cgcx86e8mst3 | In mathematics, the exponential response formula (ERF), also known as exponential response and complex replacement, is a method used to find a particular solution of a non-homogeneous linear ordinary differential equation of any order. The exponential response formula is applicable to non-homogeneous linear ordinary di... | Exponential response formula |
c_fm3eypz52707 | In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry. Let M be a complex manifold, and write OM for the sheaf of holomorphic functions on M. Let OM* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of abelia... | Exponential sequence |
c_g4kfz4qml1rk | Its kernel is the sheaf 2πiZ of locally constant functions on M taking the values 2πin, with n an integer. The exponential sheaf sequence is therefore 0 → 2 π i Z → O M → O M ∗ → 0. {\displaystyle 0\to 2\pi i\,\mathbb {Z} \to {\mathcal {O}}_{M}\to {\mathcal {O}}_{M}^{*}\to 0.} | Exponential sequence |
c_ctuvl31vqeux | The exponential mapping here is not always a surjective map on sections; this can be seen for example when M is a punctured disk in the complex plane. The exponential map is surjective on the stalks: Given a germ g of an holomorphic function at a point P such that g(P) ≠ 0, one can take the logarithm of g in a neighbor... | Exponential sequence |
c_zn63f722cb1k | In other words, there is a potential topological obstruction to taking a global logarithm of a non-vanishing holomorphic function, something that is always locally possible. A further consequence of the sequence is the exactness of ⋯ → H 1 ( O M ) → H 1 ( O M ∗ ) → H 2 ( 2 π i Z ) → ⋯ . {\displaystyle \cdots \to H^{1}(... | Exponential sequence |
c_t5kryqw90fjw | In mathematics, the extended natural numbers is a set which contains the values 0 , 1 , 2 , … {\displaystyle 0,1,2,\dots } and ∞ {\displaystyle \infty } (infinity). That is, it is the result of adding a maximum element ∞ {\displaystyle \infty } to the natural numbers. Addition and multiplication work as normal for fini... | Extended natural numbers |
c_i1qxhyg5t3gr | In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and thei... | Grassmann algebra |
c_2jhfis62gnck | More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning that u ∧ v = − ( v ∧ u ) {\displaystyle u\wedge v=-(v\wedge u)} for all vectors u {\displaystyle u} and v , ... | Grassmann algebra |
c_s8pd6g19hxrz | More generally, the exterior product of any number k {\displaystyle k} of vectors can be defined and is sometimes called a k {\displaystyle k} -blade (or decomposable, or simple, by some authors). It lives in a space known as the k {\displaystyle k} -th exterior power (generalizing exterior square and exterior cubic). ... | Grassmann algebra |
c_rhje6r0gstny | If Euclidean product is given for the vectors, the magnitude (that is, a scalar) of the resulting k {\displaystyle k} -blade is the oriented hypervolume of the k {\displaystyle k} -dimensional parallelotope whose edges are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensi... | Grassmann algebra |
c_2ljswaj68o9h | The exterior algebra contains objects that are not only k {\displaystyle k} -blades, but sums of k {\displaystyle k} -blades; such a sum is called a k-vector. Combining k {\displaystyle k} -blades in a linear structure by adding and scalar multiplication is the core of Projective Geometry, (see Plücker coordinates). k ... | Grassmann algebra |
c_ie9lmliz2nyo | For any k {\displaystyle k} -vector more associated objects exist: rank is defined to be the smallest number of simple elements of which it is a sum; support is defined as the minimal subspace the k {\displaystyle k} -vector lives in; the divisor space (some authors might have other names for it, like kernel or factor)... | Grassmann algebra |
c_jvhqhrcw8sun | {\displaystyle \alpha ,\beta ,\gamma .} . | Grassmann algebra |
c_b6zwpulz0c58 | As said, the k {\displaystyle k} -vectors are a lot like homogeneous polynomials of degree k {\displaystyle k} , such that when elements of different degrees are multiplied, the degrees add for the degree of the product. This means that the exterior algebra is a graded algebra. Exterior algebra emerged on two paths: as... | Grassmann algebra |
c_tqcni1scdaoe | This twofold approach exists in almost all cases, but an exception has to be singled out: when the vector spaces in the construction are over a field of characteristic 2. Hence, whenever antisymmetric/alternating tensors are considered in connection to exterior algebra, the basic field is supposed of 0 or odd character... | Grassmann algebra |
c_7jklj6jnab0b | On the first path, the abstract one, both ingredients are clearly given (pretty abstract, though) and this is its main power. On the second path, the vector space is clear and less abstract, but the exterior product can be defined in more (equivalent) ways, and much care is needed to avoid mistakes. The definition of t... | Grassmann algebra |
c_gmi4vx44snjl | Moreover, the field the vector spaces are based on may not be numeric, as real or complex numbers, but other (less usual) field (finite or not) with zero or positive characteristic. In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstr... | Grassmann algebra |
c_tcid59rkx5o8 | The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra... | Grassmann algebra |
c_xox1skvqzve1 | In mathematics, the factorial of a non-negative integer n {\displaystyle n} , denoted by n ! {\displaystyle n!} , is the product of all positive integers less than or equal to n {\displaystyle n} . The factorial of n {\displaystyle n} also equals the product of n {\displaystyle n} with the next smaller factorial: For e... | Factorial function |
c_nbd7s6wyy78h | is 1, according to the convention for an empty product.Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book Sefer Yetzirah. The factorial operation is encountered in many areas of mathematics, not... | Factorial function |
c_ijrv8pqt12qb | In mathematical analysis, factorials are used in power series for the exponential function and other functions, and they also have applications in algebra, number theory, probability theory, and computer science. Much of the mathematics of the factorial function was developed beginning in the late 18th and early 19th c... | Factorial function |
c_4bo6ql19q7fg | Legendre's formula describes the exponents of the prime numbers in a prime factorization of the factorials, and can be used to count the trailing zeros of the factorials. Daniel Bernoulli and Leonhard Euler interpolated the factorial function to a continuous function of complex numbers, except at the negative integers,... | Factorial function |
c_is50mozmqyia | In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, rising sequential product, or upper factorial) is def... | Falling factorial power |
c_6hfw8xllx1z0 | {\displaystyle {\tbinom {x}{n}}.} In this article, the symbol (x)n is used to represent the falling factorial, and the symbol x(n) is used for the rising factorial. These conventions are used in combinatorics, although Knuth's underline and overline notations x n _ {\displaystyle x^{\underline {n}}} and x n ¯ {\display... | Falling factorial power |
c_a6t6ibyx91mh | In mathematics, the family of Debye functions is defined by D n ( x ) = n x n ∫ 0 x t n e t − 1 d t . {\displaystyle D_{n}(x)={\frac {n}{x^{n}}}\int _{0}^{x}{\frac {t^{n}}{e^{t}-1}}\,dt.} The functions are named in honor of Peter Debye, who came across this function (with n = 3) in 1912 when he analytically computed th... | Debye function |
c_9d16cr581hln | In mathematics, the fibbinary numbers are the numbers whose binary representation does not contain two consecutive ones. That is, they are sums of distinct and non-consecutive powers of two. | Fibbinary number |
c_li6fgufpxb06 | In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundle from a given base space, fiber and a suitable set of transition functions. The theorem also gives conditions under which two such bundles are isomorphic. The theorem is important in the associated bundle construction wher... | Fibre bundle construction theorem |
c_t6t3au68iflj | In mathematics, the field T L E {\displaystyle \mathbb {T} ^{LE}} of logarithmic-exponential transseries is a non-Archimedean ordered differential field which extends comparability of asymptotic growth rates of elementary nontrigonometric functions to a much broader class of objects. Each log-exp transseries represents... | Transseries |
c_1gyfnd2uc2kw | The field T L E {\displaystyle \mathbb {T} ^{LE}} was introduced independently by Dahn-Göring and Ecalle in the respective contexts of model theory or exponential fields and of the study of analytic singularity and proof by Ecalle of the Dulac conjectures. It constitutes a formal object, extending the field of exp-log ... | Transseries |
c_ekzjf18cd5uq | In mathematics, the field of definition of an algebraic variety V is essentially the smallest field to which the coefficients of the polynomials defining V can belong. Given polynomials, with coefficients in a field K, it may not be obvious whether there is a smaller field k, and other polynomials defined over k, which... | Field of definition |
c_8qfpgq1i3yk9 | In mathematics, the field trace is a particular function defined with respect to a finite field extension L/K, which is a K-linear map from L onto K. | Field trace |
c_t4himboxo5ir | In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French–English pun, Fun. The name "field with one element" and the notation F1 are only suggestive, as ther... | Field with one element |
c_gx60zwmjxzij | Many theories of F1 have been proposed, but it is not clear which, if any, of them give F1 all the desired properties. While there is still no field with a single element in these theories, there is a field-like object whose characteristic is one. | Field with one element |
c_wst916r5l4ec | Most proposed theories of F1 replace abstract algebra entirely. Mathematical objects such as vector spaces and polynomial rings can be carried over into these new theories by mimicking their abstract properties. This allows the development of commutative algebra and algebraic geometry on new foundations. | Field with one element |
c_0gev9qjbkzou | One of the defining features of theories of F1 is that these new foundations allow more objects than classical abstract algebra does, one of which behaves like a field of characteristic one. The possibility of studying the mathematics of F1 was originally suggested in 1956 by Jacques Tits, published in Tits 1957, on th... | Field with one element |
c_t07gqjna0yyx | In mathematics, the finite lattice representation problem, or finite congruence lattice problem, asks whether every finite lattice is isomorphic to the congruence lattice of some finite algebra. | Finite lattice representation problem |
c_kr1txhzelwlp | In mathematics, the finite-dimensional representations of the complex classical Lie groups G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} , S L ( n , C ) {\displaystyle SL(n,\mathbb {C} )} , O ( n , C ) {\displaystyle O(n,\mathbb {C} )} , S O ( n , C ) {\displaystyle SO(n,\mathbb {C} )} , S p ( 2 n , C ) {\displaysty... | Representations of classical Lie groups |
c_gu5ygbls2uiy | In mathematics, the first Blakers–Massey theorem, named after Albert Blakers and William S. Massey, gave vanishing conditions for certain triad homotopy groups of spaces. | Blakers–Massey theorem |
c_13axkoyudav9 | In mathematics, the first uncountable ordinal, traditionally denoted by ω 1 {\displaystyle \omega _{1}} or sometimes by Ω {\displaystyle \Omega } , is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. When considered as a set, the el... | First uncountable ordinal |
c_mxfc9iqaebmm | Like any ordinal number (in von Neumann's approach), ω 1 {\displaystyle \omega _{1}} is a well-ordered set, with set membership serving as the order relation. ω 1 {\displaystyle \omega _{1}} is a limit ordinal, i.e. there is no ordinal α {\displaystyle \alpha } such that ω 1 = α + 1 {\displaystyle \omega _{1}=\alpha +1... | First uncountable ordinal |
c_8303xccyhiub | The ordinal ω 1 {\displaystyle \omega _{1}} is thus the initial ordinal of ℵ 1 {\displaystyle \aleph _{1}} . Under the continuum hypothesis, the cardinality of ω 1 {\displaystyle \omega _{1}} is ℶ 1 {\displaystyle \beth _{1}} , the same as that of R {\displaystyle \mathbb {R} } —the set of real numbers.In most construc... | First uncountable ordinal |
c_jwimx6bhg0xc | In mathematics, the fixed-point index is a concept in topological fixed-point theory, and in particular Nielsen theory. The fixed-point index can be thought of as a multiplicity measurement for fixed points. The index can be easily defined in the setting of complex analysis: Let f(z) be a holomorphic mapping on the com... | Fixed-point index |
c_dhueszgxff2j | In real Euclidean space, the fixed-point index is defined as follows: If x0 is an isolated fixed point of f, then let g be the function defined by g ( x ) = x − f ( x ) | | x − f ( x ) | | . {\displaystyle g(x)={\frac {x-f(x)}{||x-f(x)||}}.} Then g has an isolated singularity at x0, and maps the boundary of some delete... | Fixed-point index |
c_z1dw9v9tqy06 | In mathematics, the flat topology is a Grothendieck topology used in algebraic geometry. It is used to define the theory of flat cohomology; it also plays a fundamental role in the theory of descent (faithfully flat descent). The term flat here comes from flat modules. There are several slightly different flat topologi... | Flat cohomology |
c_gvrnuqekbc2r | fppf stands for fidèlement plate de présentation finie, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat and of finite presentation. fpqc stands for fidèlement plate et quasi-compacte, and in this topology, a morphism of affine schemes is a covering morphism if it is fa... | Flat cohomology |
c_n1jsitdyjy9i | In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover. These topologies are closely related to descent. The "pure" faithfully flat topology without any further finiteness conditions such as quasi compactness or finite presentation is not used much as is not subcanonical; in other words, repres... | Flat cohomology |
c_6qfabqjofynj | Unfortunately the terminology for flat topologies is not standardized. Some authors use the term "topology" for a pretopology, and there are several slightly different pretopologies sometimes called the fppf or fpqc (pre)topology, which sometimes give the same topology. Flat cohomology was introduced by Grothendieck in... | Flat cohomology |
c_xpgr39c7ieu2 | In mathematics, the flatness (symbol: ⏥) of a surface is the degree to which it approximates a mathematical plane. The term is often generalized for higher-dimensional manifolds to describe the degree to which they approximate the Euclidean space of the same dimensionality. (See curvature. )Flatness in homological alge... | Flatness (mathematics) |
c_rrtoay5lztaw | In mathematics, the folded spectrum method (FSM) is an iterative method for solving large eigenvalue problems. Here you always find a vector with an eigenvalue close to a search-value ε {\displaystyle \varepsilon } . This means you can get a vector Ψ {\displaystyle \Psi } in the middle of the spectrum without solving t... | Folded spectrum method |
c_zh8d8usr8dh2 | Ψ i + 1 = Ψ i − α ( H − ε 1 ) 2 Ψ i {\displaystyle \Psi _{i+1}=\Psi _{i}-\alpha (H-\varepsilon \mathbf {1} )^{2}\Psi _{i}} , with 0 < α < 1 {\displaystyle 0<\alpha ^{\,}<1} and 1 {\displaystyle \mathbf {1} } the Identity matrix. In contrast to the Conjugate gradient method, here the gradient calculates by twice multipl... | Folded spectrum method |
c_km9wh6f8ahse | In mathematics, the following matrix was given by Indian mathematician Brahmagupta: B ( x , y ) = . {\displaystyle B(x,y)={\begin{bmatrix}x&y\\\pm ty&\pm x\end{bmatrix}}.} It satisfies B ( x 1 , y 1 ) B ( x 2 , y 2 ) = B ( x 1 x 2 ± t y 1 y 2 , x 1 y 2 ± y 1 x 2 ) . {\displaystyle B(x_{1},y_{1})B(x_{2},y_{2})=B(x_{1}x... | Brahmagupta matrix |
c_f5fy0bdyffdk | {\displaystyle B^{n}={\begin{bmatrix}x&y\\ty&x\end{bmatrix}}^{n}={\begin{bmatrix}x_{n}&y_{n}\\ty_{n}&x_{n}\end{bmatrix}}\equiv B_{n}.} The x n {\displaystyle \ x_{n}} and y n {\displaystyle \ y_{n}} are called Brahmagupta polynomials. The Brahmagupta matrices can be extended to negative integers: B − n = − n = ≡ B − ... | Brahmagupta matrix |
c_a53rlw7c5ors | In mathematics, the formal derivative is an operation on elements of a polynomial ring or a ring of formal power series that mimics the form of the derivative from calculus. Though they appear similar, the algebraic advantage of a formal derivative is that it does not rely on the notion of a limit, which is in general ... | Formal derivative |
c_izt7jh7hgf5l | In mathematics, the four color theorem, or the four color map theorem, states that given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map—so that no two adjacent regions have the same color. Two regions are called a... | Conjecture |
c_nwe7t9n6og6o | Möbius mentioned the problem in his lectures as early as 1840. The conjecture was first proposed on October 23, 1852 when Francis Guthrie, while trying to color the map of counties of England, noticed that only four different colors were needed. The five color theorem, which has a short elementary proof, states that fi... | Conjecture |
c_a1gnyg5po15t | A number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852. The four color theorem was ultimately proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer. | Conjecture |
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