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c_0n461hwabmxr | In mathematics, a nonlocal operator is a mapping which maps functions on a topological space to functions, in such a way that the value of the output function at a given point cannot be determined solely from the values of the input function in any neighbourhood of any point. An example of a nonlocal operator is the Fo... | Nonlocal operator |
c_c3wbzweeem6v | In mathematics, a nonnegative matrix, written X ≥ 0 , {\displaystyle \mathbf {X} \geq 0,} is a matrix in which all the elements are equal to or greater than zero, that is, x i j ≥ 0 ∀ i , j . {\displaystyle x_{ij}\geq 0\qquad \forall {i,j}.} A positive matrix is a matrix in which all the elements are strictly greater t... | Nonnegative matrices |
c_rjvdf462f603 | The set of positive matrices is a subset of all non-negative matrices. While such matrices are commonly found, the term is only occasionally used due to the possible confusion with positive-definite matrices, which are different. A matrix which is both non-negative and is positive semidefinite is called a doubly non-ne... | Nonnegative matrices |
c_xbnqwrh2aqq3 | In mathematics, a nonrecursive filter only uses input values like x, unlike recursive filter where it uses previous output values like y. In signal processing, non-recursive digital filters are often known as Finite Impulse Response (FIR) filters, as a non-recursive digital filter has a finite number of coefficients in... | Nonrecursive filter |
c_5js83zv2pbfc | In mathematics, a nonstandard integer may refer to Hyperinteger, the integer part of a hyperreal number an integer in a non-standard model of arithmetic | Nonstandard integer |
c_omq9ggstw6mt | In mathematics, a norm form is a homogeneous form in n variables constructed from the field norm of a field extension L/K of degree n. That is, writing N for the norm mapping to K, and selecting a basis e1, ..., en for L as a vector space over K, the form is given by N(x1e1 + ... + xnen)in variables x1, ..., xn. In num... | Norm form |
c_0zgsakhyl394 | In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidea... | Vector length |
c_4lli4tcqu2t4 | A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space. The term pseudonorm has been used for several related meanings. | Vector length |
c_vjcvb84qj09a | It may be a synonym of "seminorm". A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality " ≤ {\displaystyle \,\leq \,} " in the homogeneity axiom. It can also refer to a norm that can take infinite values, or to certain functions parametrised by a directed set. | Vector length |
c_euq52mdamufp | In mathematics, a norm variety is a particular type of algebraic variety V over a field F, introduced for the purposes of algebraic K-theory by Voevodsky. The idea is to relate Milnor K-theory of F to geometric objects V, having function fields F(V) that 'split' given 'symbols' (elements of Milnor K-groups).The formula... | Norm variety |
c_ozrbw3yxr3k8 | The conditions on a norm variety V are that V is irreducible and a non-singular complete variety. Further it should have dimension d equal to p n − 1 − 1. {\displaystyle p^{n-1}-1.\ } The key condition is in terms of the d-th Newton polynomial sd, evaluated on the (algebraic) total Chern class of the tangent bundle of ... | Norm variety |
c_6o7vq18kflfa | In mathematics, a normal map is a concept in geometric topology due to William Browder which is of fundamental importance in surgery theory. Given a Poincaré complex X (more geometrically a Poincaré space), a normal map on X endows the space, roughly speaking, with some of the homotopy-theoretic global structure of a c... | Normal invariants |
c_65dzdj6udvs2 | If the dimension of X is ≥ {\displaystyle \geq } 5 there is then only the algebraic topology surgery obstruction due to C. T. C. Wall to X actually being homotopy equivalent to a closed manifold. Normal maps also apply to the study of the uniqueness of manifold structures within a homotopy type, which was pioneered by ... | Normal invariants |
c_uvf73oudqz8x | Depending on the category of manifolds (differentiable, piecewise-linear, or topological), there are similarly defined, but inequivalent, concepts of normal maps and normal invariants. It is possible to perform surgery on normal maps, meaning surgery on the domain manifold, and preserving the map. Surgery on normal map... | Normal invariants |
c_u0mfeu3uj2og | In mathematics, a normal surface is a surface inside a triangulated 3-manifold that intersects each tetrahedron so that each component of intersection is a triangle or a quad (see figure). A triangle cuts off a vertex of the tetrahedron while a quad separates pairs of vertices. A normal surface may have many components... | Normal surface |
c_eakbi8qc28ek | Dually, a normal surface can be considered to be a surface that intersects each handle of a given handle structure on the 3-manifold in a prescribed manner similar to the above. The concept of normal surface can be generalized to arbitrary polyhedra. There are also related notions of almost normal surface and spun norm... | Normal surface |
c_u3dcf7bf0aoj | The concept of normal surface is due to Hellmuth Kneser, who utilized it in his proof of the prime decomposition theorem for 3-manifolds. Later Wolfgang Haken extended and refined the notion to create normal surface theory, which is at the basis of many of the algorithms in 3-manifold theory. The notion of almost norma... | Normal surface |
c_yn5eu6blpj8v | In mathematics, a normed algebra A is an algebra over a field which has a sub-multiplicative norm: ∀ x , y ∈ A ‖ x y ‖ ≤ ‖ x ‖ ‖ y ‖ . {\displaystyle \forall x,y\in A\qquad \|xy\|\leq \|x\|\|y\|.} Some authors require it to have a multiplicative identity 1A such that ║1A║ = 1. | Normed algebra |
c_f2i4lnxvabz2 | In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. A norm is a generalization of the intuitive notion of "length" in the physical world. If V {\displaystyle V} is a vector space over K {\displaystyle K} , where K {\displaystyle K} is a fi... | Normed spaces |
c_x31uts5d58h7 | Absolute homogeneity: for every λ ∈ K {\displaystyle \lambda \in K} and x ∈ V {\displaystyle x\in V} , Triangle inequality: for every x ∈ V {\displaystyle x\in V} and y ∈ V {\displaystyle y\in V} ,If V {\displaystyle V} is a real or complex vector space as above, and ‖ ⋅ ‖ {\displaystyle \lVert \cdot \rVert } is a norm... | Normed spaces |
c_zma4pljy467j | If this metric space is complete then the normed space is a Banach space. Every normed vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed space but converse is not true. | Normed spaces |
c_bvqo6dvmfc2g | For example, the set of the finite sequences of real numbers can be normed with the Euclidean norm, but it is not complete for this norm. An inner product space is a normed vector space whose norm is the square root of the inner product of a vector and itself. The Euclidean norm of a Euclidean vector space is a special... | Normed spaces |
c_vfrk777wmn45 | In mathematics, a nowhere commutative semigroup is a semigroup S such that, for all a and b in S, if ab = ba then a = b. A semigroup S is nowhere commutative if and only if any two elements of S are inverses of each other. | Nowhere commutative semigroup |
c_y0jy1l3ztw8y | In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f {\displaystyle f} is a function from real numbers to real numbers, then f {\displaystyle f} is nowhere continuous if for each point x {\displaystyle x} t... | Nowhere continuous function |
c_v5ms8lc4pat7 | In mathematics, a null semigroup (also called a zero semigroup) is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero. If every element of a semigroup is a left zero then the semigroup is called a left zero semigroup; a right zero semigroup is defined analogously. Accor... | Null semigroup |
c_y975qootna5f | In mathematics, a nullform of a vector space acted on linearly by a group is a vector on which all invariants of the group vanish. Nullforms were introduced by Hilbert (1893). (Dieudonné & Carrell 1970, 1971, p.57). | Nullform |
c_cdzl7rhd1cq6 | In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians; the solutions to which are given by eigenvalues corresponding to their ... | Harmonic (mathematics) |
c_4ajcydfg1msl | In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations. The first result in the field was the Schauder fixed-point theorem, proved in 1930 by Ju... | Fixed-point theorems in infinite-dimensional spaces |
c_sktgcw6x7j3b | One way in which fixed-point theorems of this kind have had a larger influence on mathematics as a whole has been that one approach is to try to carry over methods of algebraic topology, first proved for finite simplicial complexes, to spaces of infinite dimension. For example, the research of Jean Leray who founded sh... | Fixed-point theorems in infinite-dimensional spaces |
c_84trweealrof | Tikhonov (Tychonoff) fixed-point theorem: Let V be a locally convex topological vector space. For any nonempty compact convex set X in V, any continuous function f: X → X has a fixed point. Browder fixed-point theorem: Let K be a nonempty closed bounded convex set in a uniformly convex Banach space. | Fixed-point theorems in infinite-dimensional spaces |
c_j25yb7czp6c3 | Then any non-expansive function f: K → K has a fixed point. (A function f {\displaystyle f} is called non-expansive if ‖ f ( x ) − f ( y ) ‖ ≤ ‖ x − y ‖ {\displaystyle \|f(x)-f(y)\|\leq \|x-y\|} for each x {\displaystyle x} and y {\displaystyle y} .) Other results include the Markov–Kakutani fixed-point theorem (1936-1... | Fixed-point theorems in infinite-dimensional spaces |
c_jmf87nownwcq | In mathematics, a numerical semigroup is a special kind of a semigroup. Its underlying set is the set of all nonnegative integers except a finite number and the binary operation is the operation of addition of integers. Also, the integer 0 must be an element of the semigroup. For example, while the set {0, 2, 3, 4, 5, ... | Numerical monoid |
c_usyfbn47lyhf | Numerical semigroups are commutative monoids and are also known as numerical monoids.The definition of numerical semigroup is intimately related to the problem of determining nonnegative integers that can be expressed in the form x1n1 + x2 n2 + ... + xr nr for a given set {n1, n2, ..., nr} of positive integers and for ... | Numerical monoid |
c_qjv733nw1e03 | In mathematics, a one-dimensional array corresponds to a vector, a two-dimensional array resembles a matrix; more generally, a tensor may be represented as an n-dimensional data cube. | Data cube |
c_91y9td74x3bb | In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism φ: R → G {\displaystyle \varphi :\mathbb {R} \rightarrow G} from the real line R {\displaystyle \mathbb {R} } (as an additive group) to some other topological group G {\displaystyle G} . If φ {\displaystyle \va... | 1-parameter group |
c_lcsj2afiaa6s | It is these infinitesimal transformations that generate a Lie algebra that is used to describe a Lie group of any dimension. The action of a one-parameter group on a set is known as a flow. A smooth vector field on a manifold, at a point, induces a local flow - a one parameter group of local diffeomorphisms, sending po... | 1-parameter group |
c_snpwqesqyuba | In mathematics, a p-adic distribution is an analogue of ordinary distributions (i.e. generalized functions) that takes values in a ring of p-adic numbers. | P-adic distribution |
c_nls765thp1l4 | In mathematics, a p-adic modular form is a p-adic analog of a modular form, with coefficients that are p-adic numbers rather than complex numbers. Serre (1973) introduced p-adic modular forms as limits of ordinary modular forms, and Katz (1973) shortly afterwards gave a geometric and more general definition. Katz's p-a... | P-adic modular form |
c_bx84zsxrkiux | In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose domain and target are p-adic (where p is a prime number). For example, the domain could be the p-adic integers Zp, a profinite p-group, or a p-adic ... | P-adic zeta function |
c_vp6uen9ky94h | For example, Kubota–Leopoldt used Kummer's congruences for Bernoulli numbers to construct a p-adic L-function, the p-adic Riemann zeta function ζp(s), whose values at negative odd integers are those of the Riemann zeta function at negative odd integers (up to an explicit correction factor). p-adic L-functions arising i... | P-adic zeta function |
c_jis57q5zhjae | A p-adic L-function arising in this way is typically called an arithmetic p-adic L-function as it encodes arithmetic data of the Galois module involved. The main conjecture of Iwasawa theory (now a theorem due to Barry Mazur and Andrew Wiles) is the statement that the Kubota–Leopoldt p-adic L-function and an arithmetic... | P-adic zeta function |
c_2wgn9rz9jtmx | In mathematics, a p-adically closed field is a field that enjoys a closure property that is a close analogue for p-adic fields to what real closure is to the real field. They were introduced by James Ax and Simon B. Kochen in 1965. | P-adically closed field |
c_b3doaa5zpwk4 | In mathematics, a p-constrained group is a finite group resembling the centralizer of an element of prime order p in a group of Lie type over a finite field of characteristic p. They were introduced by Gorenstein and Walter (1964, p.169) in order to extend some of Thompson's results about odd groups to groups with dihe... | P-constrained group |
c_kt6pmqhixdr5 | In mathematics, a packing in a hypergraph is a partition of the set of the hypergraph's edges into a number of disjoint subsets such that no pair of edges in each subset share any vertex. There are two famous algorithms to achieve asymptotically optimal packing in k-uniform hypergraphs. One of them is a random greedy a... | Packing in a hypergraph |
c_n99pxq94k18f | In mathematics, a pair of pants is a surface which is homeomorphic to the three-holed sphere. The name comes from considering one of the removed disks as the waist and the two others as the cuffs of a pair of pants. Pairs of pants are used as building blocks for compact surfaces in various theories. Two important appli... | Pair of pants (mathematics) |
c_tfs68xdv0d1h | In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number.Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers. | Cantor's pairing function |
c_hrl77iwmhdbk | In mathematics, a pairing is an R-bilinear map from the Cartesian product of two R-modules, where the underlying ring R is commutative. | Pairing |
c_unshrwg4xdym | In mathematics, a palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns. The first few decimal palindromic primes are: 2, 3, 5, 7, 11, 10... | Palindromic prime |
c_f9cqof1giz15 | In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1234567890 (one billion two hundred thirty four million five hundred sixty seven thousand eight hundred ninety) is a pandigital number in base 10. The first few... | Pandigital number |
c_vuz0yif45v3a | In a trivial sense, all positive integers are pandigital in unary (or tallying). In binary, all integers are pandigital except for 0 and numbers of the form 2 n − 1 {\displaystyle 2^{n}-1} (the Mersenne numbers). The larger the base, the rarer pandigital numbers become, though one can always find runs of b x {\displays... | Pandigital number |
c_qoekv6368wtf | Conversely, the smaller the base, the fewer pandigital numbers without redundant digits there are. 2 is the only such pandigital number in base 2, while there are more of these in base 10. Sometimes, the term is used to refer only to pandigital numbers with no redundant digits. | Pandigital number |
c_hodaoc9ff36u | In some cases, a number might be called pandigital even if it doesn't have a zero as a significant digit, for example, 923456781 (these are sometimes referred to as "zeroless pandigital numbers"). No base 10 pandigital number can be a prime number if it doesn't have redundant digits. The sum of the digits 0 to 9 is 45,... | Pandigital number |
c_l2eoj5y2mlix | The first base 10 pandigital prime is 10123457689; OEIS: A050288 lists more. For different reasons, redundant digits are also required for a pandigital number (in any base except unary) to also be a palindromic number in that base. The smallest pandigital palindromic number in base 10 is 1023456789876543201. | Pandigital number |
c_qg8ufe7x4b9t | The largest pandigital number without redundant digits to be also a square number is 9814072356 = 990662. Two of the zeroless pandigital Friedman numbers are: 123456789 = ((86 + 2 × 7)5 − 91) / 34, and 987654321 = (8 × (97 + 6/2)5 + 1) / 34. A pandigital Friedman number without redundant digits is the square: 217034856... | Pandigital number |
c_fn3ld2ny0f7f | While much of what has been said does not apply to Roman numerals, there are pandigital numbers: MCDXLIV, MCDXLVI, MCDLXIV, MCDLXVI, MDCXLIV, MDCXLVI, MDCLXIV, MDCLXVI. These, listed in OEIS: A105416, use each of the digits just once, while OEIS: A105417 has pandigital Roman numerals with repeats. Pandigital numbers ar... | Pandigital number |
c_yc6i851zkk1k | In mathematics, a pantachy or pantachie (from the Greek word πανταχη meaning everywhere) is a maximal totally ordered subset of a partially ordered set, especially a set of equivalence classes of sequences of real numbers. The term was introduced by du Bois-Reymond (1879, 1882) to mean a dense subset of an ordered set,... | Pantachy |
c_80m0d22851b5 | In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point (the focus) and a line (the directrix). The ... | Parabola |
c_chpqz6cuh94q | The parabola is the locus of points in that plane that are equidistant from the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface.The line per... | Parabola |
c_ysv5k0byipws | The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola that is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. | Parabola |
c_4kyozqr80uwq | Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar. Parabolas have the property that, if they are made of material that reflects light, then light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is ... | Parabola |
c_by80ig6ygq5m | The same effects occur with sound and other waves. This reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles. It is frequently used in p... | Parabola |
c_4qastil3fr4g | In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by Dieudonné (1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it ad... | Paracompact manifold |
c_a0vkhrqypqke | Sometimes paracompact spaces are defined so as to always be Hausdorff. Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets. | Paracompact manifold |
c_91vorkcdjd2x | A space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact. The notion of paracompact space is also studied in pointless topology, where it is more well-behaved. | Paracompact manifold |
c_1hvm6w28al3h | For example, the product of any number of paracompact locales is a paracompact locale, but the product of two paracompact spaces may not be paracompact. Compare this to Tychonoff's theorem, which states that the product of any collection of compact topological spaces is compact. However, the product of a paracompact sp... | Paracompact manifold |
c_vn1suokuz1t0 | In mathematics, a parallelization of a manifold M {\displaystyle M\,} of dimension n is a set of n global smooth linearly independent vector fields. | Parallelization (mathematics) |
c_u5unjc0d0xfj | In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, called parametric curve and parametric surf... | Parametric formula |
c_d1c264578iod | {\displaystyle (x,y)=(\cos t,\sin t).} Parametric representations are generally nonunique (see the "Examples in two dimensions" section below), so the same quantities may be expressed by a number of different parameterizations.In addition to curves and surfaces, parametric equations can describe manifolds and algebraic... | Parametric formula |
c_191uuij1uvni | Parametric equations are commonly used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a single parameter is often labeled t; however, parameters can represent other physical quantities (such as geometric variables) or can be ... | Parametric formula |
c_dg85w7ch2tj6 | In mathematics, a paramodular group is a special sort of arithmetic subgroup of the symplectic group. It is a generalization of the Siegel modular group, and has the same relation to polarized abelian varieties that the Siegel modular group has to principally polarized abelian varieties. It is the group of automorphism... | Paramodular group |
c_59x8yimejql2 | The corresponding group over the reals is called the parasymplectic group and is conjugate to a (real) symplectic group. A paramodular form is a Siegel modular form for a paramodular group. Paramodular groups were introduced by Conforto (1952) and named by Shimura (1958, section 8). | Paramodular group |
c_5zct9nf5kwig | In mathematics, a paraproduct is a non-commutative bilinear operator acting on functions that in some sense is like the product of the two functions it acts on. According to Svante Janson and Jaak Peetre, in an article from 1988, "the name 'paraproduct' denotes an idea rather than a unique definition; several versions ... | Paraproduct |
c_q4vbutuy6o5p | {\displaystyle fg=\Lambda (f,g)+\Lambda (g,f).} For any appropriate functions f {\displaystyle f} and h {\displaystyle h} with h ( 0 ) = 0 {\displaystyle h(0)=0} , it is the case that h ( f ) = Λ ( f , h ′ ( f ) ) {\displaystyle h(f)=\Lambda (f,h'(f))} . It should satisfy some form of the Leibniz rule.A paraproduct may... | Paraproduct |
c_5by9ki3e3rq4 | In mathematics, a paratopological group is a topological semigroup that is algebraically a group. In other words, it is a group G with a topology such that the group's product operation is a continuous function from G × G to G. This differs from the definition of a topological group in that the group inverse is not req... | Paratopological group |
c_39dlcagnjamz | In mathematics, a parent function is the core representation of a function type without manipulations such as translation and dilation. For example, for the family of quadratic functions having the general form y = a x 2 + b x + c , {\displaystyle y=ax^{2}+bx+c\,,} the simplest function is y = x 2 {\displaystyle y=x^{2... | Parent function |
c_9zjaslymx3p1 | For example, the graph of y = x2 − 4x + 7 can be obtained from the graph of y = x2 by translating +2 units along the X axis and +3 units along Y axis. This is because the equation can also be written as y − 3 = (x − 2)2. For many trigonometric functions, the parent function is usually a basic sin(x), cos(x), or tan(x). | Parent function |
c_7f8e2yuissr7 | For example, the graph of y = A sin(x) + B cos(x) can be obtained from the graph of y = sin(x) by translating it through an angle α along the positive X axis (where tan(α) = A⁄B), then stretching it parallel to the Y axis using a stretch factor R, where R2 = A2 + B2. This is because A sin(x) + B cos(x) can be written a... | Parent function |
c_x2tp0cium4jb | In mathematics, a partial cyclic order is a ternary relation that generalizes a cyclic order in the same way that a partial order generalizes a linear order. | Partial cyclic order |
c_u8hbk7fttl9w | In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The... | Partial derivatives |
c_i6asudb5d0rv | {\displaystyle {\tfrac {\partial z}{\partial x}}.} Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in: The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol... | Partial derivatives |
c_kx1z7fyicjmo | In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation... | Linear partial differential equation |
c_r35xjzg19lf0 | Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the m... | Linear partial differential equation |
c_t5gydb0zbv2m | For instance, they are foundational in the modern scientific understanding of sound, heat, diffusion, electrostatics, electrodynamics, thermodynamics, fluid dynamics, elasticity, general relativity, and quantum mechanics (Schrödinger equation, Pauli equation, etc.). They also arise from many purely mathematical conside... | Linear partial differential equation |
c_ctrxrwwj9yef | As such, it is usually acknowledged that there is no "general theory" of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields.Ordinary differential equations form a subclass of partial differential equations, corresponding to functions of a sing... | Linear partial differential equation |
c_gwytewry8y4x | In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) is a homogeneous binary relation that is symmetric and transitive. If the relation is also reflexive, then the relation is an equivalence relation. | Partial equivalence relation |
c_pc2cvzold8jj | In mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly the whole X itself) to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition or natural domain of f. If S equals X, that is, if f is defined on every element in X, then ... | Partial functions |
c_slcxc3cgw0qa | This is the case in calculus, where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator. For this reason, in calculus, and more generally in mathematical analysis, a partial function is generally called simply a function. | Partial functions |
c_m4z3lwbh7gyk | In computability theory, a general recursive function is a partial function from the integers to the integers; no algorithm can exist for deciding whether an arbitrary such function is in fact total. When arrow notation is used for functions, a partial function f {\displaystyle f} from X {\displaystyle X} to Y {\displa... | Partial functions |
c_mn7fs6bae8q0 | In mathematics, a partial group algebra is an associative algebra related to the partial representations of a group. | Partial group algebra |
c_he8mutoyglmw | In mathematics, a partial order or total order < on a set X {\displaystyle X} is said to be dense if, for all x {\displaystyle x} and y {\displaystyle y} in X {\displaystyle X} for which x < y {\displaystyle x | Dense relation |
c_7mfa3ta14ksh | In mathematics, a partially ordered set P is said to have Knaster's condition upwards (sometimes property (K)) if any uncountable subset A of P has an upwards-linked uncountable subset. An analogous definition applies to Knaster's condition downwards. The property is named after Polish mathematician Bronisław Knaster. | Knaster's condition |
c_rr74zo0y2eve | Knaster's condition implies the countable chain condition (ccc), and it is sometimes used in conjunction with a weaker form of Martin's axiom, where the ccc requirement is replaced with Knaster's condition. Not unlike ccc, Knaster's condition is also sometimes used as a property of a topological space, in which case it... | Knaster's condition |
c_97h87b51u0f6 | In mathematics, a partially ordered space (or pospace) is a topological space X {\displaystyle X} equipped with a closed partial order ≤ {\displaystyle \leq } , i.e. a partial order whose graph { ( x , y ) ∈ X 2 ∣ x ≤ y } {\displaystyle \{(x,y)\in X^{2}\mid x\leq y\}} is a closed subset of X 2 {\displaystyle X^{2}} . F... | Partially ordered space |
c_g0a81uhwygk3 | In mathematics, a partition matroid or partitional matroid is a matroid that is a direct sum of uniform matroids. It is defined over a base set in which the elements are partitioned into different categories. For each category, there is a capacity constraint - a maximum number of allowed elements from this category. Th... | Partition matroid |
c_2ivdjeyu67ft | In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence rela... | Set partition |
c_avx1b4326t2r | In mathematics, a partition of an interval on the real line is a finite sequence x0, x1, x2, …, xn of real numbers such that a = x0 < x1 < x2 < … < xn = b.In other terms, a partition of a compact interval I is a strictly increasing sequence of numbers (belonging to the interval I itself) starting from the initial poin... | Partition of an interval |
c_fyhi8gih92kv | In mathematics, a partition of unity of a topological space X {\displaystyle X} is a set R {\displaystyle R} of continuous functions from X {\displaystyle X} to the unit interval such that for every point x ∈ X {\displaystyle x\in X}: there is a neighbourhood of x {\displaystyle x} where all but a finite number of the... | Partition of unity |
c_dcns799j6cbr | In mathematics, a path in a topological space X {\displaystyle X} is a continuous function from the closed unit interval {\displaystyle } into X . {\displaystyle X.} Paths play an important role in the fields of topology and mathematical analysis. For example, a topological space for which there exists a path connecti... | Arc (topology) |
c_q5ibkdvtst5q | Any space may be broken up into path-connected components. The set of path-connected components of a space X {\displaystyle X} is often denoted π 0 ( X ) . {\displaystyle \pi _{0}(X).} | Arc (topology) |
c_076xhs9jwexx | One can also define paths and loops in pointed spaces, which are important in homotopy theory. If X {\displaystyle X} is a topological space with basepoint x 0 , {\displaystyle x_{0},} then a path in X {\displaystyle X} is one whose initial point is x 0 {\displaystyle x_{0}} . Likewise, a loop in X {\displaystyle X} is... | Arc (topology) |
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