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wiki_5000_chunk_0 | Indeterminate (variable) | It is not an unknown that could be solved for. It is not a variable designating a function argument, or a variable being summed or integrated over. It is not any type of bound variable. It is just a symbol used in an entirely formal way.When used as placeholders, a common operation is to substitute mathematical express... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5001_chunk_0 | Convex series | In mathematics, particularly in functional analysis and convex analysis, a convex series is a series of the form ∑ i = 1 ∞ r i x i {\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}} where x 1 , x 2 , … {\displaystyle x_{1},x_{2},\ldots } are all elements of a topological vector space X {\displaystyle X} , and all r 1 , r ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5002_chunk_0 | Ursescu theorem | In mathematics, particularly in functional analysis and convex analysis, the Ursescu theorem is a theorem that generalizes the closed graph theorem, the open mapping theorem, and the uniform boundedness principle. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5003_chunk_0 | Approximation of the identity | In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5004_chunk_0 | Closed linear operator | In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function f: X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph.This property is studied because there are many theore... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5005_chunk_0 | Closed graph theorem (functional analysis) | In mathematics, particularly in functional analysis and topology, the closed graph theorem is a result connecting the continuity of certain kinds of functions to a topological property of their graph. In its most elementary form, the closed graph theorem states that a linear function between two Banach spaces is contin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5006_chunk_0 | Mackey space | In mathematics, particularly in functional analysis, a Mackey space is a locally convex topological vector space X such that the topology of X coincides with the Mackey topology τ(X,X′), the finest topology which still preserves the continuous dual. They are named after George Mackey. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5007_chunk_0 | Projective measurement | In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are formally similar to real-valued measures, except that their values are sel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5008_chunk_0 | Projective measurement | The Borel functional calculus for self-adjoint operators is constructed using integrals with respect to PVMs. In quantum mechanics, PVMs are the mathematical description of projective measurements. They are generalized by positive operator valued measures (POVMs) in the same sense that a mixed state or density matrix g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5009_chunk_0 | Seminormed space | In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm. A top... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5010_chunk_0 | Webbed space | In mathematics, particularly in functional analysis, a webbed space is a topological vector space designed with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps whose codomains are webbed spaces. A space is called webbed if there exists a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5011_chunk_0 | Krein–Smulian theorem | In mathematics, particularly in functional analysis, the Krein-Smulian theorem can refer to two theorems relating the closed convex hull and compactness in the weak topology. They are named after Mark Krein and Vitold Shmulyan, who published them in 1940. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5012_chunk_0 | Spectrum (functional analysis) | In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number λ {\displaystyle \lambda } is said to be in the spectrum of a bounded linear operato... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5013_chunk_0 | Spectrum (functional analysis) | The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues. For example, consider the right shift operator R on the Hilbert space ℓ2, ( x 1 , x 2 , … ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5014_chunk_0 | Spectrum (functional analysis) | If T is closed (which includes the case when T is bounded), boundedness of ( T − λ I ) − 1 {\displaystyle (T-\lambda I)^{-1}} follows automatically from its existence. The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5015_chunk_0 | Eilenberg–Ganea theorem | In mathematics, particularly in homological algebra and algebraic topology, the Eilenberg–Ganea theorem states for every finitely generated group G with certain conditions on its cohomological dimension (namely 3 ≤ cd ( G ) ≤ n {\displaystyle 3\leq \operatorname {cd} (G)\leq n} ), one can construct an aspherical CW c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5016_chunk_0 | Pointed model category | In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes (derived ca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5017_chunk_0 | Localization theorem | In mathematics, particularly in integral calculus, the localization theorem allows, under certain conditions, to infer the nullity of a function given only information about its continuity and the value of its integral. Let F(x) be a real-valued function defined on some open interval Ω of the real line that is continuo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5018_chunk_0 | Matrix analysis | In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5019_chunk_0 | Flag (linear algebra) | In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector space V. Here "increasing" means each is a proper subspace of the next (see filtration): { 0 } = V 0 ⊂ V 1 ⊂ V 2 ⊂ ⋯ ⊂ V k = V . {\displaystyle \{0\}=V_{0}\subset V_{1}\subset V_{2}\subset \cdots... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5020_chunk_0 | Antisymmetric matrices | In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition: p. 38 In terms of the entries of the matrix, if a i j {\textstyle a_{ij}} denotes the entry in the i {\textstyle i} -th ro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5021_chunk_0 | Matrix multiplication | In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5022_chunk_0 | Matrix multiplication | The product of matrices A and B is denoted as AB.Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, to represent the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5023_chunk_0 | Schur product theorem | In mathematics, particularly in linear algebra, the Schur product theorem states that the Hadamard product of two positive definite matrices is also a positive definite matrix. The result is named after Issai Schur (Schur 1911, p. 14, Theorem VII) (note that Schur signed as J. Schur in Journal für die reine und angewan... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5024_chunk_0 | Bounded (set theory) | In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The name club is a contraction of "closed and unbounded". | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5025_chunk_0 | Permutation matrices | In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say P, represents a permutation of m elements and, when used to multiply another matrix, say A, results in permuting the rows (whe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5026_chunk_0 | Furstenberg's proof of the infinitude of primes | In mathematics, particularly in number theory, Hillel Furstenberg's proof of the infinitude of primes is a topological proof that the integers contain infinitely many prime numbers. When examined closely, the proof is less a statement about topology than a statement about certain properties of arithmetic sequences. Unl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5027_chunk_0 | Continuous functional calculus | In mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5028_chunk_0 | Wold–von Neumann decomposition | In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5029_chunk_0 | Pseudocomplement | Every pseudocomplemented lattice is necessarily bounded, i.e. it has a 1 as well. Since the pseudocomplement is unique by definition (if it exists), a pseudocomplemented lattice can be endowed with a unary operation * mapping every element to its pseudocomplement; this structure is sometimes called a p-algebra. However... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5030_chunk_0 | Fodor's lemma | In mathematics, particularly in set theory, Fodor's lemma states the following: If κ {\displaystyle \kappa } is a regular, uncountable cardinal, S {\displaystyle S} is a stationary subset of κ {\displaystyle \kappa } , and f: S → κ {\displaystyle f:S\rightarrow \kappa } is regressive (that is, f ( α ) < α {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5031_chunk_0 | Club filter | In mathematics, particularly in set theory, if κ {\displaystyle \kappa } is a regular uncountable cardinal then club ( κ ) , {\displaystyle \operatorname {club} (\kappa ),} the filter of all sets containing a club subset of κ , {\displaystyle \kappa ,} is a κ {\displaystyle \kappa } -complete filter closed under diag... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5032_chunk_0 | Club filter | Obviously C = ⋂ C i {\displaystyle C=\bigcap C_{i}} is closed, since any sequence which appears in C {\displaystyle C} appears in every C i , {\displaystyle C_{i},} and therefore its limit is also in every C i . {\displaystyle C_{i}.} To show that it is unbounded, take some β < κ . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5033_chunk_0 | Club filter | Such a sequence can be constructed, since every C i {\displaystyle C_{i}} is unbounded. Since α < κ {\displaystyle \alpha <\kappa } and κ {\displaystyle \kappa } is regular, the limit of this sequence is less than κ . {\displaystyle \kappa .} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5034_chunk_0 | Club filter | We call it β 2 , {\displaystyle \beta _{2},} and define a new sequence ⟨ β 2 , i ⟩ {\displaystyle \langle \beta _{2,i}\rangle } similar to the previous sequence. We can repeat this process, getting a sequence of sequences ⟨ β j , i ⟩ {\displaystyle \langle \beta _{j,i}\rangle } where each element of a sequence is great... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5035_chunk_0 | Club filter | This limit is then contained in every C i , {\displaystyle C_{i},} and therefore C , {\displaystyle C,} and is greater than β . {\displaystyle \beta .} To see that club ( κ ) {\displaystyle \operatorname {club} (\kappa )} is closed under diagonal intersection, let ⟨ C i ⟩ , {\displaystyle \langle C_{i}\rangle ,} i < ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5036_chunk_0 | Aleph One | In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph ( ℵ {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5037_chunk_0 | Beth number | In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written ℶ 0 , ℶ 1 , ℶ 2 , ℶ 3 , … {\displaystyle \beth _{0},\ \beth _{1},\ \beth _{2},\ \beth _{3},\ \dots } , where ℶ {\displaystyle \beth } is the second... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5038_chunk_0 | Fully characteristic subgroup | In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, every characteristic subgroup is normal; though the converse is not guar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5039_chunk_0 | Discrete inverse | In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. In the standard notation of modular arithmetic this congruence is written as a x ≡ 1 ( mod m ) , {\displaystyle ax\equiv 1{\p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5040_chunk_0 | Discrete inverse | As with the analogous operation on the real numbers, a fundamental use of this operation is in solving, when possible, linear congruences of the form a x ≡ b ( mod m ) . {\displaystyle ax\equiv b{\pmod {m}}.} Finding modular multiplicative inverses also has practical applications in the field of cryptography, e.g. publ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5041_chunk_0 | Vague topology | In mathematics, particularly in the area of functional analysis and topological vector spaces, the vague topology is an example of the weak-* topology which arises in the study of measures on locally compact Hausdorff spaces. Let X {\displaystyle X} be a locally compact Hausdorff space. Let M ( X ) {\displaystyle M(X)}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5042_chunk_0 | Vague topology | In particular, the topology defined by duality with C c ( X ) {\displaystyle C_{c}(X)} can be metrizable whereas the topology defined by duality with C 0 ( X ) {\displaystyle C_{0}(X)} is not. One application of this is to probability theory: for example, the central limit theorem is essentially a statement that if μ n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5043_chunk_0 | Chow quotient | In mathematics, particularly in the field of algebraic geometry, a Chow variety is an algebraic variety whose points correspond to effective algebraic cycles of fixed dimension and degree on a given projective space. More precisely, the Chow variety Gr ( k , d , n ) {\displaystyle \operatorname {Gr} (k,d,n)} is the f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5044_chunk_0 | Preimage theorem | In mathematics, particularly in the field of differential topology, the preimage theorem is a variation of the implicit function theorem concerning the preimage of particular points in a manifold under the action of a smooth map. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5045_chunk_0 | Rank of a partition | In mathematics, particularly in the fields of number theory and combinatorics, the rank of a partition of a positive integer is a certain integer associated with the partition. In fact at least two different definitions of rank appear in the literature. The first definition, with which most of this article is concerned... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5046_chunk_0 | Ushiki's Theorem | In mathematics, particularly in the study of functions of several complex variables, Ushiki's theorem, named after S. Ushiki, states that certain well-behaved functions cannot have certain kinds of well-behaved invariant manifolds. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5047_chunk_0 | Semianalytic set | In mathematics, particularly in the subfield of real analytic geometry, a subanalytic set is a set of points (for example in Euclidean space) defined in a way broader than for semianalytic sets (roughly speaking, those satisfying conditions requiring certain real power series to be positive there). Subanalytic sets sti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5048_chunk_0 | Saturated set | In mathematics, particularly in the subfields of set theory and topology, a set C {\displaystyle C} is said to be saturated with respect to a function f: X → Y {\displaystyle f:X\to Y} if C {\displaystyle C} is a subset of f {\displaystyle f} 's domain X {\displaystyle X} and if whenever f {\displaystyle f} sends two p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5049_chunk_0 | Uniformly hyperfinite algebra | In mathematics, particularly in the theory of C*-algebras, a uniformly hyperfinite, or UHF, algebra is a C*-algebra that can be written as the closure, in the norm topology, of an increasing union of finite-dimensional full matrix algebras. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5050_chunk_0 | Weyl-Brauer matrices | In mathematics, particularly in the theory of spinors, the Weyl–Brauer matrices are an explicit realization of a Clifford algebra as a matrix algebra of 2⌊n/2⌋ × 2⌊n/2⌋ matrices. They generalize the Pauli matrices to n dimensions, and are a specific construction of higher-dimensional gamma matrices. They are named for ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5051_chunk_0 | Spectral theorem | In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5052_chunk_0 | Spectral theorem | In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5053_chunk_0 | Spectral theorem | Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. The spectral theorem also provides a canonical decomposition, called the spectral decomposition, of the underlying vector space on which the operator acts. Augustin-Louis Cauchy p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5054_chunk_0 | Spectral theorem | In addition, Cauchy was the first to be systematic about determinants. The spectral theorem as generalized by John von Neumann is today perhaps the most important result of operator theory. This article mainly focuses on the simplest kind of spectral theorem, that for a self-adjoint operator on a Hilbert space. However... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5055_chunk_0 | Gram-Schmidt theorem | In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn equipped with the standard inner product. The Gram–Schmidt process takes a finite, line... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5056_chunk_0 | Zero tensor | In mathematics, particularly linear algebra, a zero matrix is a matrix with all its entries being zero. It is alternately denoted by the symbol O {\displaystyle O} . Some examples of zero matrices are 0 1 , 1 = , 0 2 , 2 = , 0 2 , 3 = , {\displaystyle 0_{1,1}={\begin{bmatrix}0\end{bmatrix}},\ 0_{2,2}={\begin{bmatrix... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5057_chunk_0 | Zero tensor | 0 K m , n = {\displaystyle 0_{K_{m,n}}={\begin{bmatrix}0_{K}&0_{K}&\cdots &0_{K}\\0_{K}&0_{K}&\cdots &0_{K}\\\vdots &\vdots &&\vdots \\0_{K}&0_{K}&\cdots &0_{K}\end{bmatrix}}} The zero matrix is the additive identity in K m , n {\displaystyle K_{m,n}} . That is, for all A ∈ K m , n {\displaystyle A\in K_{m,n}}: 0 K m ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5058_chunk_0 | Zero tensor | In general, the zero element of a ring is unique, and typically denoted as 0 without any subscript to indicate the parent ring. Hence the examples above represent zero matrices over any ring. The zero matrix also represents the linear transformation which sends all vectors to the zero vector. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5059_chunk_0 | Zero matrix | In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of m × n {\displaystyle m\times n} matrices, and is denoted by the symbol O {\displaystyle O} or 0 {\displaystyle 0} followed by subscripts c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5060_chunk_0 | Orthogonal basis | In mathematics, particularly linear algebra, an orthogonal basis for an inner product space V {\displaystyle V} is a basis for V {\displaystyle V} whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5061_chunk_0 | Complete orthonormal basis | In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V {\displaystyle V} whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space R n {\displaystyle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5062_chunk_0 | Complete orthonormal basis | Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process. In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces. Given a pre-Hilbert space H , {\di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5063_chunk_0 | Complete orthonormal basis | For instance, any square-integrable function on the interval {\displaystyle } can be expressed (almost everywhere) as an infinite sum of Legendre polynomials (an orthonormal basis), but not necessarily as an infinite sum of the monomials x n . {\displaystyle x^{n}.} A different generalisation is to pseudo-inner produc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5064_chunk_0 | Schur–Horn theorem | In mathematics, particularly linear algebra, the Schur–Horn theorem, named after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues. It has inspired investigations and substantial generalizations in the setting of symplectic geometry. A few important generalizations are... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5065_chunk_0 | Pascal matrix | In mathematics, particularly matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an upper-triangular ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5066_chunk_0 | Stieltjes matrix | In mathematics, particularly matrix theory, a Stieltjes matrix, named after Thomas Joannes Stieltjes, is a real symmetric positive definite matrix with nonpositive off-diagonal entries. A Stieltjes matrix is necessarily an M-matrix. Every n×n Stieltjes matrix is invertible to a nonsingular symmetric nonnegative matrix,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5067_chunk_0 | Bandwidth (linear algebra) | In mathematics, particularly matrix theory, a band matrix or banded matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5068_chunk_0 | Lehmer matrix | In mathematics, particularly matrix theory, the n×n Lehmer matrix (named after Derrick Henry Lehmer) is the constant symmetric matrix defined by A i j = { i / j , j ≥ i j / i , j < i . {\displaystyle A_{ij}={\begin{cases}i/j,&j\geq i\\j/i,&j | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5069_chunk_0 | Sigma-ideal | In mathematics, particularly measure theory, a 𝜎-ideal, or sigma ideal, of a sigma-algebra (𝜎, read "sigma," means countable in this context) is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is in probability theory.Let ( X , Σ ) {\displaystyle (X,\Si... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5070_chunk_0 | Sigma-ideal | {\displaystyle X.} That is, when no 𝜎-algebra is specified, then one simply takes the full power set of the underlying set. For example, the meager subsets of a topological space are those in the 𝜎-ideal generated by the collection of closed subsets with empty interior. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5071_chunk_0 | Surface integral | In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate a scalar field (that is, a function of position which returns a sc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5072_chunk_0 | Bramble–Hilbert lemma | In mathematics, particularly numerical analysis, the Bramble–Hilbert lemma, named after James H. Bramble and Stephen Hilbert, bounds the error of an approximation of a function u {\displaystyle \textstyle u} by a polynomial of order at most m − 1 {\displaystyle \textstyle m-1} in terms of derivatives of u {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5073_chunk_0 | Bramble–Hilbert lemma | Lipschitz domains are reasonable enough, which includes convex domains and domains with continuously differentiable boundary. The main use of the Bramble–Hilbert lemma is to prove bounds on the error of interpolation of function u {\displaystyle \textstyle u} by an operator that preserves polynomials of order up to m −... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5074_chunk_0 | P-adic exponential function | In mathematics, particularly p-adic analysis, the p-adic exponential function is a p-adic analogue of the usual exponential function on the complex numbers. As in the complex case, it has an inverse function, named the p-adic logarithm. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5075_chunk_0 | Ramanujan theta function | In mathematics, particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5076_chunk_0 | Tarski finiteness | In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, { 2 , 4 , 6 , 8 , 10 } {\displaystyle \{2,4,6,8,10\}} is a finite set with five elements. The number of elements of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5077_chunk_0 | Tarski finiteness | {\displaystyle \{1,2,3,\ldots \}.} Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5078_chunk_0 | Nonrecursive ordinals | In mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using recursive ordinal notations. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5079_chunk_0 | Fourier cosine series | In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5080_chunk_0 | Dunkl operator | In mathematics, particularly the study of Lie groups, a Dunkl operator is a certain kind of mathematical operator, involving differential operators but also reflections in an underlying space. Formally, let G be a Coxeter group with reduced root system R and kv an arbitrary "multiplicity" function on R (so ku = kv when... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5081_chunk_0 | Coordinate map | In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an atlas has its more common meaning. In general, the notion of atlas un... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5082_chunk_0 | Locally discrete collection | In mathematics, particularly topology, collections of subsets are said to be locally discrete if they look like they have precisely one element from a local point of view. The study of locally discrete collections is worthwhile as Bing's metrization theorem shows. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5083_chunk_0 | K-topology | In mathematics, particularly topology, the K-topology is a topology that one can impose on the set of all real numbers which has some interesting properties. Relative to the set of all real numbers carrying the standard topology, the set K = {1/n | n is a positive integer} is not closed since it doesn't contain its (on... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5084_chunk_0 | Homeomorphism group | In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. Homeomorphism groups are very important in the theory of topological spaces and in general are examples of automo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5085_chunk_0 | Tube lemma | In mathematics, particularly topology, the tube lemma, also called Wallace's theorem, is a useful tool in order to prove that the finite product of compact spaces is compact. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5086_chunk_0 | Milman's reverse Brunn–Minkowski inequality | In mathematics, particularly, in asymptotic convex geometry, Milman's reverse Brunn–Minkowski inequality is a result due to Vitali Milman that provides a reverse inequality to the famous Brunn–Minkowski inequality for convex bodies in n-dimensional Euclidean space Rn. Namely, it bounds the volume of the Minkowski sum o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5087_chunk_0 | Perfectoid space | In mathematics, perfectoid spaces are adic spaces of special kind, which occur in the study of problems of "mixed characteristic", such as local fields of characteristic zero which have residue fields of characteristic prime p. A perfectoid field is a complete topological field K whose topology is induced by a nondiscr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5088_chunk_0 | Persymmetric matrix | In mathematics, persymmetric matrix may refer to: a square matrix which is symmetric with respect to the northeast-to-southwest diagonal; or a square matrix such that the values on each line perpendicular to the main diagonal are the same for a given line.The first definition is the most common in the recent literature... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5089_chunk_0 | Perturbation problem beyond all orders | In mathematics, perturbation theory works typically by expanding unknown quantity in a power series in a small parameter. However, in a perturbation problem beyond all orders, all coefficients of the perturbation expansion vanish and the difference between the function and the constant function 0 cannot be detected by ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5090_chunk_0 | Perturbation problem beyond all orders | This is because the function e − 1 / z {\displaystyle e^{-1/z}} possesses an essential singularity at z = 0 {\displaystyle z=0} in the complex z {\displaystyle z} -plane, and therefore the function is most appropriately modeled by a Laurent series -- a Taylor series has a zero radius of convergence. Thus, if a physical... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5091_chunk_0 | Sinc function | In mathematics, physics and engineering, the sinc function, denoted by sinc(x), has two forms, normalized and unnormalized. In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by Alternatively, the unnormalized sinc function is often called the sampling function, indicated as Sa(x).In digital... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5092_chunk_0 | Sinc function | It is used in the concept of reconstructing a continuous bandlimited signal from uniformly spaced samples of that signal. The only difference between the two definitions is in the scaling of the independent variable (the x axis) by a factor of π. In both cases, the value of the function at the removable singularity at ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5093_chunk_0 | Sinc function | The sinc function is then analytic everywhere and hence an entire function. The function has also been called the cardinal sine or sine cardinal function. The term sinc was introduced by Philip M. Woodward in his 1952 article "Information theory and inverse probability in telecommunication", in which he said that the f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5094_chunk_0 | Moiré effect | In television and digital photography, a pattern on an object being photographed can interfere with the shape of the light sensors to generate unwanted artifacts. They are also sometimes created deliberately – in micrometers they are used to amplify the effects of very small movements. In physics, its manifestation is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5095_chunk_0 | Introduction to the mathematics of general relativity | In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or – as here – simply a vector) is a geometric object that has both a magnitude (or length) and direction. A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "one who ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5096_chunk_0 | Vector direction | In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5097_chunk_0 | Vector direction | It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplicat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5098_chunk_0 | Vector direction | Vectors play an important role in physics: the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5099_chunk_0 | Spatial frequency | In image-processing applications, spatial frequency is often expressed in units of cycles per millimeter (mm) or equivalently line pairs per mm. In wave propagation, the spatial frequency is also known as wavenumber. Ordinary wavenumber is defined as the reciprocal of wavelength λ {\displaystyle \lambda } and is common... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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