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c_8d2cqab2s2j7 | In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury, says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix. Alternative names for this formula are the matrix inversion lemma, Sher... | Matrix inversion lemma |
c_n1i7j5r1rmnc | While the identity is primarily used on matrices, it holds in a general ring or in an Ab-category. The Woodbury matrix identity allows cheap computation of inverses and solutions to linear equations. However, little is known about the numerical stability of the formula. There are no published results concerning its err... | Matrix inversion lemma |
c_i3vorkraa8wv | In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A {\displaystyle A} acting on an inner product space is called positive-semidefinite (or non-negative) if, for every x ∈ Dom ( A ) {\displaystyle x\in \mathop {\text{Dom}} (A)} , ⟨ A x , x ⟩ ∈ ... | Positive operator |
c_0ntyx9jg0xsw | In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in R 2 {\displaystyle \mathbb {R} ^{2}} (the real-number plane) are called double integral... | Multiple integrals |
c_glk15gp7969k | In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid; that is, a braid in which the corresponding ends of the strings are connected in pairs. The theorem is named after James Waddell Alexander II, who published a proof in 1923.Braids were first considered as a tool of k... | Alexander's theorem |
c_kve32bft1by9 | A good construction example is found in Colin Adams's book.However, the correspondence between knots and braids is clearly not one-to-one: a knot may have many braid representations. For example, conjugate braids yield equivalent knots. This leads to a second fundamental question: Which closed braids represent the same... | Alexander's theorem |
c_1v7hf8u3qif4 | In mathematics Antoine's necklace is a topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected. It also serves as a counterexample to the claim that all Cantor spaces are ambiently homeomorphic to each other. It was discovered by Louis Antoine (1921). | Antoine's horned sphere |
c_8gxvpef9uvx1 | In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group G over a field K, and for any linear representation ρ of G on a K-vector space V, given v ≠ 0 in V that is fixed by the action of G, there is a G-invariant polynomial F on V, without const... | Haboush's theorem |
c_zcvkk2im2kr8 | In mathematics Nef polygons and Nef polyhedra are the sets of polygons and polyhedra which can be obtained from a finite set of halfplanes (halfspaces) by Boolean operations of set intersection and set complement. The objects are named after the Swiss mathematician Walter Nef (1919–2013), who introduced them in his 197... | Nef polygon |
c_ya8bxn487yfc | In mathematics a Cauchy–Euler operator is a differential operator of the form p ( x ) ⋅ d d x {\displaystyle p(x)\cdot {d \over dx}} for a polynomial p. It is named after Augustin-Louis Cauchy and Leonhard Euler. The simplest example is that in which p(x) = x, which has eigenvalues n = 0, 1, 2, 3, ... and corresponding... | Cauchy–Euler operator |
c_j39c7aykjwjz | In mathematics a Dirac structure is a geometric construction generalizing both symplectic structures and Poisson structures, and having several applications to mechanics. It is based on the notion of constraint introduced by Paul Dirac and was first introduced by Ted Courant and Alan Weinstein. In more detail, let V be... | Dirac structure |
c_rztmrarnaro0 | (Similar definitions can be made for vector spaces over other fields.) An alternative (equivalent) definition often used is that D {\displaystyle D} satisfies D = D ⊥ {\displaystyle D=D^{\perp }} , where orthogonality is with respect to the symmetric bilinear form on V × V ∗ {\displaystyle V\times V^{*}} given by ⟨ ( u... | Dirac structure |
c_7jnktpfn5nhi | In mathematics a Lie coalgebra is the dual structure to a Lie algebra. In finite dimensions, these are dual objects: the dual vector space to a Lie algebra naturally has the structure of a Lie coalgebra, and conversely. | Lie coalgebra |
c_8q5pwmymvhda | In mathematics a P-recursive equation can be solved for polynomial solutions. Sergei A. Abramov in 1989 and Marko Petkovšek in 1992 described an algorithm which finds all polynomial solutions of those recurrence equations with polynomial coefficients. The algorithm computes a degree bound for the solution in a first st... | Polynomial solutions of P-recursive equations |
c_easc89x6jrbo | In mathematics a P-recursive equation is a linear equation of sequences where the coefficient sequences can be represented as polynomials. P-recursive equations are linear recurrence equations (or linear recurrence relations or linear difference equations) with polynomial coefficients. These equations play an important... | P-recursive equation |
c_4pxhbutoafus | The sequences which are solutions of these equations are called holonomic, P-recursive or D-finite. From the late 1980s, the first algorithms were developed to find solutions for these equations. Sergei A. Abramov, Marko Petkovšek and Mark van Hoeij described algorithms to find polynomial, rational, hypergeometric and ... | P-recursive equation |
c_n4ogpkjvl6ol | In mathematics a Steinberg symbol is a pairing function which generalises the Hilbert symbol and plays a role in the algebraic K-theory of fields. It is named after mathematician Robert Steinberg. For a field F we define a Steinberg symbol (or simply a symbol) to be a function ( ⋅ , ⋅ ): F ∗ × F ∗ → G {\displaystyle (\... | Steinberg symbol |
c_4qtqr0apmeht | In mathematics a Yetter–Drinfeld category is a special type of braided monoidal category. It consists of modules over a Hopf algebra which satisfy some additional axioms. | Yetter–Drinfeld category |
c_yh05d0pfkm6z | In mathematics a cocycle is a closed cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likewise used in group cohomology. In autonomous dynamical systems, cocycles are used to describe particular kinds of map, as... | Cocycle |
c_ynw9a2b8h3ts | In mathematics a group is a set together with a binary operation on the set called multiplication that obeys the group axioms. The axiom of choice is an axiom of ZFC set theory which in one form states that every set can be wellordered. In ZF set theory, i.e. ZFC without the axiom of choice, the following statements ar... | Group structure and the axiom of choice |
c_vogz43mby46o | In mathematics a linear inequality is an inequality which involves a linear function. A linear inequality contains one of the symbols of inequality: < less than > greater than ≤ less than or equal to ≥ greater than or equal to ≠ not equal toA linear inequality looks exactly like a linear equation, with the inequality s... | System of linear inequalities |
c_qgxq9tmx6e9t | In mathematics a p-group G {\displaystyle G} is called power closed if for every section H {\displaystyle H} of G {\displaystyle G} the product of p k {\displaystyle p^{k}} powers is again a p k {\displaystyle p^{k}} th power. Regular p-groups are an example of power closed groups. On the other hand, powerful p-groups,... | Power closed |
c_zkrolcthxlal | In mathematics a partial differential algebraic equation (PDAE) set is an incomplete system of partial differential equations that is closed with a set of algebraic equations. | Partial differential algebraic equation |
c_rai5ej77trdb | In mathematics a polydivisible number (or magic number) is a number in a given number base with digits abcde... that has the following properties: Its first digit a is not 0. The number formed by its first two digits ab is a multiple of 2. The number formed by its first three digits abc is a multiple of 3. The number f... | Polydivisible number |
c_c79vb6yogv2d | In mathematics a primitive abundant number is an abundant number whose proper divisors are all deficient numbers.For example, 20 is a primitive abundant number because: The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number. The sums of the proper divisors of 1, 2, 4, 5 and 10 are 0, 1, ... | Primitive abundant number |
c_d2wgvnvg8lfi | In mathematics a radial basis function (RBF) is a real-valued function φ {\textstyle \varphi } whose value depends only on the distance between the input and some fixed point, either the origin, so that φ ( x ) = φ ^ ( ‖ x ‖ ) {\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)} , or some ... | Radial basis function |
c_fdciznnhhqmg | They are often used as a collection { φ k } k {\displaystyle \{\varphi _{k}\}_{k}} which forms a basis for some function space of interest, hence the name. Sums of radial basis functions are typically used to approximate given functions. This approximation process can also be interpreted as a simple kind of neural netw... | Radial basis function |
c_x6bol0ynw945 | In mathematics a regular Hadamard matrix is a Hadamard matrix whose row and column sums are all equal. While the order of a Hadamard matrix must be 1, 2, or a multiple of 4, regular Hadamard matrices carry the further restriction that the order be a square number. The excess, denoted E(H), of a Hadamard matrix H of ord... | Regular Hadamard matrices |
c_7yl0g4skvbtk | In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist. Descent theory is concerned with generalisations of s... | Stack (mathematics) |
c_yrvpkcnyzynn | The intuitive meaning of a stack is that it is a fibred category such that "all possible gluings work". The specification of gluings requires a definition of coverings with regard to which the gluings can be considered. It turns out that the general language for describing these coverings is that of a Grothendieck topo... | Stack (mathematics) |
c_fm6atqkvm2hm | In mathematics a topological space is called countably compact if every countable open cover has a finite subcover. | Countably compact space |
c_lr2dqkuh75rv | In mathematics a translation surface is a surface obtained from identifying the sides of a polygon in the Euclidean plane by translations. An equivalent definition is a Riemann surface together with a holomorphic 1-form. These surfaces arise in dynamical systems where they can be used to model billiards, and in Teichmü... | Translation surface |
c_3jbxe17o6g29 | In mathematics an Eberlein compactum, studied by William Frederick Eberlein, is a compact topological space homeomorphic to a subset of a Banach space with the weak topology. Every compact metric space, more generally every one-point compactification of a locally compact metric space, is Eberlein compact. The converse ... | Eberlein compactum |
c_28x3ztqyij94 | In mathematics an even integer, that is, a number that is divisible by 2, is called evenly even or doubly even if it is a multiple of 4, and oddly even or singly even if it is not. The former names are traditional ones, derived from ancient Greek mathematics; the latter have become common in recent decades. These names... | Singly and doubly even |
c_50gs3g8zrf2l | In mathematics an expression represents a single value. A function maps one or more values to one unique value. Inverses of functions are not always well defined as functions. Sometimes extra conditions are required to make an inverse of a function fit the definition of a function. | Narrowing of algebraic value sets |
c_sz5rq3g09bne | Some Boolean operations, in particular do not have inverses that may be defined as functions. In particular the disjunction "or" has inverses that allow two values. In natural language "or" represents alternate possibilities. | Narrowing of algebraic value sets |
c_ht29ty9yqrxy | Narrowing is based on value sets that allow multiple values to be packaged and considered as a single value. This allows the inverses of functions to always be considered as functions. To achieve this value sets must record the context to which a value belongs. | Narrowing of algebraic value sets |
c_wcg8aiiikppq | A variable may only take on a single value in each possible world. The value sets tag each value in the value set with the world to which it belongs. Possible worlds belong to world sets. | Narrowing of algebraic value sets |
c_6x4l5nwzj3mu | A world set is a set of all mutually exclusive worlds. Combining values from different possible worlds is impossible, because that would mean combining mutually exclusive possible worlds. The application of functions to value sets creates combinations of value sets from different worlds. | Narrowing of algebraic value sets |
c_8fm5qlsu2gqi | Narrowing reduces those worlds by eliminating combinations of different worlds from the same world set. Narrowing rules also detect situations where some combinations of worlds are shown to be impossible. No back tracking is required in the use of narrowing. By packaging the possible values in a value set all combinati... | Narrowing of algebraic value sets |
c_xdmpu5dz99u0 | In mathematics an orthogonal trajectory is a curve, which intersects any curve of a given pencil of (planar) curves orthogonally. For example, the orthogonal trajectories of a pencil of concentric circles are the lines through their common center (see diagram). Suitable methods for the determination of orthogonal traje... | Isogonal trajectory |
c_yzuhnzuf2gf2 | Both steps may be difficult or even impossible. In such cases one has to apply numerical methods. Orthogonal trajectories are used in mathematics for example as curved coordinate systems (i.e. elliptic coordinates) or appear in physics as electric fields and their equipotential curves. If the trajectory intersects the ... | Isogonal trajectory |
c_2hhqak2vz7nb | In mathematics and Fourier analysis, a rectangular mask short-time Fourier transform (rec-STFT) has the simple form of short-time Fourier transform. Other types of the STFT may require more computation time than the rec-STFT. The rectangular mask function can be defined for some bound (B) over time (t) as w ( t ) = { 1... | Rectangular mask short-time Fourier transform |
c_5li34uvimk4l | In mathematics and abstract algebra, a Bol loop is an algebraic structure generalizing the notion of group. Bol loops are named for the Dutch mathematician Gerrit Bol who introduced them in (Bol 1937). A loop, L, is said to be a left Bol loop if it satisfies the identity a ( b ( a c ) ) = ( a ( b a ) ) c {\displaystyle... | Bol loop |
c_bq3u946ickz2 | In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true. In logic, mathematics and theoretical computer science, a Boolean domain is usually written as {0, 1}, or B . {\displaystyle \mathbb {B} .} The algebraic structure that natural... | Boolean domain |
c_hhdla871r1qn | The initial object in the category of bounded lattices is a Boolean domain. In computer science, a Boolean variable is a variable that takes values in some Boolean domain. Some programming languages feature reserved words or symbols for the elements of the Boolean domain, for example false and true. However, many progr... | Boolean domain |
c_9y5qvrykqnci | In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the algebra 2 X 2 of all binary relations on a set X, that is, subsets of the cartesian square X2, with R•S interpreted ... | Relation algebra |
c_5rnh9th9bd1d | In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recu... | List of group theory topics |
c_wd7h2hrl5ju3 | Various physical systems, such as crystals and the hydrogen atom, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography. | List of group theory topics |
c_wuf2cp7j7gzt | In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose underlying set (or universe or carrier) B is the Boolean domain. The elements of the Boolean domain are 1 and 0 by convention, so that B = {0, 1}. Paul Halmos's name for this algebra "2" has some following in the literatur... | Two-element Boolean algebra |
c_o48cyf8dv0eu | In mathematics and analytic number theory, Vaughan's identity is an identity found by R. C. Vaughan (1977) that can be used to simplify Vinogradov's work on trigonometric sums. It can be used to estimate summatory functions of the form ∑ n ≤ N f ( n ) Λ ( n ) {\displaystyle \sum _{n\leq N}f(n)\Lambda (n)} where f is so... | Vaughan's identity |
c_421z05kjclq1 | In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In per... | Perturbation theory |
c_5ewq2i42yyzw | Successive terms in the series at higher powers of ε {\displaystyle \varepsilon } usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, usually by keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction. | Perturbation theory |
c_4iz0ddq1ui48 | Perturbation theory is used in a wide range of fields, and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. The field in general remains actively and heavily researched across multiple disciplines. | Perturbation theory |
c_c7nby6xhxnsp | In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of classical mechanics. A closely related concept also appears in quantum mechani... | Generalized momentum |
c_uabr7bt8uun7 | In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transfo... | Poisson commutativity |
c_yc254oy1qxcw | For instance, it is often possible to choose the Hamiltonian itself H = H ( q , p , t ) {\displaystyle H=H(q,p,t)} as one of the new canonical momentum coordinates. In a more general sense, the Poisson bracket is used to define a Poisson algebra, of which the algebra of functions on a Poisson manifold is a special case... | Poisson commutativity |
c_9w1eucwjxlhs | In mathematics and combinatorics, a centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. The following figures illustrate this arrangement for the first four centered hexagonal num... | Centered hexagonal number |
c_hc4z51i8j0l0 | In mathematics and computability theory, an elementary cellular automaton is a one-dimensional cellular automaton where there are two possible states (labeled 0 and 1) and the rule to determine the state of a cell in the next generation depends only on the current state of the cell and its two immediate neighbors. Ther... | Elementary cellular automaton |
c_z1jx4p8g0t5z | In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). Delaunay triangulations maximize the minimum of all ... | Delaunay triangulation |
c_cg3ppbsvfbkd | By considering circumscribed spheres, the notion of Delaunay triangulation extends to three and higher dimensions. Generalizations are possible to metrics other than Euclidean distance. However, in these cases a Delaunay triangulation is not guaranteed to exist or be unique. | Delaunay triangulation |
c_od7izchw5q2n | In mathematics and computational geometry, the Gabriel graph of a set S {\displaystyle S} of points in the Euclidean plane expresses one notion of proximity or nearness of those points. Formally, it is the graph G {\displaystyle G} with vertex set S {\displaystyle S} in which any two distinct points p ∈ S {\displaystyl... | Gabriel graph |
c_9z9xgi6sli7u | In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is t... | Euler's method |
c_5wl0td3zfp0b | In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to... | Distinct degree factorization |
c_oykv3bpyti3c | In mathematics and computer algebra, automatic differentiation (auto-differentiation, autodiff, or AD), also called algorithmic differentiation, computational differentiation, is a set of techniques to evaluate the partial derivative of a function specified by a computer program. Automatic differentiation exploits the ... | Computational differentiation |
c_3l4ztpq8dc8m | In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of computer... | Polynomial factorization |
c_3ucrcecwd0ge | But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra systems: When the long-known finite step algorithms were first put on computers, they turned out to be highly inefficient. The fact that almost any uni- or multivariate polynomial of degree up to 100 and with coefficient... | Polynomial factorization |
c_34d7ho1sulih | In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some variants are commonly referred to as square-and-multiply algorithms or bin... | Binary exponentiation |
c_47qkox9220b3 | In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to the elements of a list, a vector, or a matrix, depending on whether one is wr... | Indicial notation |
c_lm0ar0t37jod | In mathematics and computer programming, the order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression. These rules are formalized with a ranking of the operators. The rank of an operator is called its precedence, a... | Precedence rule |
c_hvqzm02w6ytg | For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation. Thus, in the expression 1 + 2 × 3, the multiplication is performed before addition, and the expression has the value 1 + (2 × 3) = 7, and not (1 + 2) × 3 = 9. When expo... | Precedence rule |
c_zhx5h5cmmqd1 | Thus 3 + 52 = 28 and 3 × 52 = 75. These conventions exist to avoid notational ambiguity while allowing notation to remain brief. Where it is desired to override the precedence conventions, or even simply to emphasize them, parentheses ( ) can be used. | Precedence rule |
c_gwnpeynvnv9m | For example, (2 + 3) × 4 = 20 forces addition to precede multiplication, while (3 + 5)2 = 64 forces addition to precede exponentiation. If multiple pairs of parentheses are required in a mathematical expression (such as in the case of nested parentheses), the parentheses may be replaced by brackets or braces to avoid c... | Precedence rule |
c_mg4r5nk01xnu | These rules are meaningful only when the usual notation (called infix notation) is used. When functional or Polish notation are used for all operations, the order of operations results from the notation itself. Internet memes sometimes present ambiguous infix expressions that cause disputes and increase web traffic. Mo... | Precedence rule |
c_vacpduzow6b1 | In mathematics and computer science in general, a fixed point of a function is a value that is mapped to itself by the function. In combinatory logic for computer science, a fixed-point combinator (or fixpoint combinator): page 26 is a higher-order function fix {\displaystyle {\textsf {fix}}} that returns some fixed po... | Fixed-point combinator |
c_dettoiq5u8y1 | In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and P... | Horner's method |
c_6nuvmwcosh8k | {\displaystyle {\begin{aligned}a_{0}&+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots +a_{n}x^{n}\\&=a_{0}+x{\bigg (}a_{1}+x{\Big (}a_{2}+x{\big (}a_{3}+\cdots +x(a_{n-1}+x\,a_{n})\cdots {\big )}{\Big )}{\bigg )}.\end{aligned}}} This allows the evaluation of a polynomial of degree n with only n {\displaystyle n} multiplications an... | Horner's method |
c_8rioiafinabd | In mathematics and computer science, Recamán's sequence is a well known sequence defined by a recurrence relation. Because its elements are related to the previous elements in a straightforward way, they are often defined using recursion. It takes its name after its inventor Bernardo Recamán Santos, a Colombian mathema... | Recamán's sequence |
c_5egl93ujryjq | In mathematics and computer science, Zeno machines (abbreviated ZM, and also called accelerated Turing machine, ATM) are a hypothetical computational model related to Turing machines that are capable of carrying out computations involving a countably infinite number of algorithmic steps. These machines are ruled out in... | Zeno machine |
c_gv581axajpf4 | In mathematics and computer science, a balanced boolean function is a boolean function whose output yields as many 0s as 1s over its input set. This means that for a uniformly random input string of bits, the probability of getting a 1 is 1/2. Examples of balanced boolean functions are the function that copies the firs... | Balanced boolean function |
c_lia68dlec6sm | In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an object and allows it to be identified in a unique way. The distinction between "... | Normal form (mathematics) |
c_fzxm7krzo4np | More generally, for a class of objects on which an equivalence relation is defined, a canonical form consists in the choice of a specific object in each class. For example: Jordan normal form is a canonical form for matrix similarity. The row echelon form is a canonical form, when one considers as equivalent a matrix a... | Normal form (mathematics) |
c_zf5iq2rieuos | In this context, a canonical form is a representation such that every object has a unique representation (with canonicalization being the process through which a representation is put into its canonical form). Thus, the equality of two objects can easily be tested by testing the equality of their canonical forms. Despi... | Normal form (mathematics) |
c_fri1lo7oj8uh | Therefore, in computer algebra, normal form is a weaker notion: A normal form is a representation such that zero is uniquely represented. This allows testing for equality by putting the difference of two objects in normal form. Canonical form can also mean a differential form that is defined in a natural (canonical) wa... | Normal form (mathematics) |
c_drszpyy14ext | In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: A simple base case (or cases) — a terminating scenario that does not use recursion to produce an answer A recursive step — a set of rules that reduces all successive cases toward the ... | Recursive step |
c_1zz25zg2kl17 | By this base case and recursive rule, one can generate the set of all natural numbers. Other recursively defined mathematical objects include factorials, functions (e.g., recurrence relations), sets (e.g., Cantor ternary set), and fractals. There are various more tongue-in-cheek definitions of recursion; see recursive ... | Recursive step |
c_eaxzzwinokl7 | In mathematics and computer science, a higher-order function (HOF) is a function that does at least one of the following: takes one or more functions as arguments (i.e. a procedural parameter, which is a parameter of a procedure that is itself a procedure), returns a function as its result.All other functions are first... | Functional form |
c_23dpn7ns4qrc | In mathematics and computer science, a history monoid is a way of representing the histories of concurrently running computer processes as a collection of strings, each string representing the individual history of a process. The history monoid provides a set of synchronization primitives (such as locks, mutexes or thr... | History monoid |
c_z5cnmdhhnmbe | History monoids were first presented by M.W. Shields.History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of dependency graphs. As such, they are free objects and are universal. The history monoid is a type of semi-abelian categorical product in the category of monoids. | History monoid |
c_1owrofo2ba1r | In mathematics and computer science, a matroid oracle is a subroutine through which an algorithm may access a matroid, an abstract combinatorial structure that can be used to describe the linear dependencies between vectors in a vector space or the spanning trees of a graph, among other applications. The most commonly ... | Circuit-finding oracle |
c_y9gcqewxegyk | In mathematics and computer science, a morphic word or substitutive word is an infinite sequence of symbols which is constructed from a particular class of endomorphism of a free monoid. Every automatic sequence is morphic. | Prolongable morphism |
c_um7f0u4o4pww | In mathematics and computer science, a pebble game is a type of mathematical game played by placing "pebbles" or "markers" on a directed acyclic graph according to certain rules: A given step of the game consists of either placing a pebble on an empty vertex or removing a pebble from a previously pebbled vertex. A vert... | Pebble game |
c_74ggv3mxt3sd | In mathematics and computer science, a primality certificate or primality proof is a succinct, formal proof that a number is prime. Primality certificates allow the primality of a number to be rapidly checked without having to run an expensive or unreliable primality test. "Succinct" usually means that the proof should... | Primality certificate |
c_83c56z838xdl | These problems already trivially lie in co-NP. This was the first strong evidence that these problems are not NP-complete, since if they were, it would imply that NP is subset of co-NP, a result widely believed to be false; in fact, this was the first demonstration of a problem in NP intersect co-NP not known, at the t... | Primality certificate |
c_01ibon2slol1 | In mathematics and computer science, a random tree is a tree or arborescence that is formed by a stochastic process. Types of random trees include: Uniform spanning tree, a spanning tree of a given graph in which each different tree is equally likely to be selected Random minimal spanning tree, spanning trees of a grap... | Random tree |
c_z4gn381q2roh | In mathematics and computer science, a rational series is a generalisation of the concept of formal power series over a ring to the case when the basic algebraic structure is no longer a ring but a semiring, and the indeterminates adjoined are not assumed to commute. They can be regarded as algebraic expressions of a f... | Rational series |
c_yvpdfh77ja2p | In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set (Aczel 1977:740ff). Some examples of recursively-definable objects include factorials, natural numbers, Fibonacci numbers, and the Cantor ternary set. A rec... | Recursive definition |
c_ow3vf4452g9c | is defined by the rules 0 ! = 1. | Recursive definition |
c_jy4vjh0kby2w | ( n + 1 ) ! = ( n + 1 ) ⋅ n ! . | Recursive definition |
c_4y0csrlrn20u | {\displaystyle {\begin{aligned}&0!=1.\\&(n+1)!=(n+1)\cdot n!.\end{aligned}}} This definition is valid for each natural number n, because the recursion eventually reaches the base case of 0. The definition may also be thought of as giving a procedure for computing the value of the function n!, starting from n = 0 and pr... | Recursive definition |
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