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c_6spg0i5sfq0l | Tiny games and up have certain curious relational characteristics. Specifically, though ⧾G is infinitesimal with respect to ↑ for all positive values of x, ⧾⧾⧾G is equal to up. Expansion of ⧾⧾⧾G into its canonical form yields {0|{{0|{{0|{0|-G}}|0}}|0}}. | Tiny and miny |
c_6qfs7atr2b5t | While the expression appears daunting, some careful and persistent expansion of the game tree of ⧾⧾⧾G + ↓ will show that it is a second player win, and that, consequently, ⧾⧾⧾G = ↑. Similarly curious, mathematician John Horton Conway noted, calling it "amusing," that "↑ is the unique solution of ⧾G = G." Conway's asser... | Tiny and miny |
c_8pauo8zqkm8t | In mathematics, to approximate a derivative to an arbitrary order of accuracy, it is possible to use the finite difference. A finite difference can be central, forward or backward. | Finite difference coefficients |
c_bu31kz1h86hj | In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as unknowns. A solution... | Solution (equation) |
c_docfakqd5sj0 | A solution of an equation is often called a root of the equation, particularly but not only for polynomial equations. The set of all solutions of an equation is its solution set. An equation may be solved either numerically or symbolically. | Solution (equation) |
c_ywlxmzmyndwn | Solving an equation numerically means that only numbers are admitted as solutions. Solving an equation symbolically means that expressions can be used for representing the solutions. For example, the equation x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the... | Solution (equation) |
c_41g1w4sbbqa4 | It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1. Or x and y can both be treated as unknowns, and then there are many solutions to the equation; a symbolic solution is (x, y) = (a + 1, a), where the variable a may take any value. Instantiating a symbolic solutio... | Solution (equation) |
c_51t6zm9xuy0k | The distinction between known variables and unknown variables is generally made in the statement of the problem, by phrases such as "an equation in x and y", or "solve for x and y", which indicate the unknowns, here x and y. However, it is common to reserve x, y, z, ... to denote the unknowns, and to use a, b, c, ... t... | Solution (equation) |
c_ujpajm3c3jsl | Depending on the context, solving an equation may consist to find either any solution (finding a single solution is enough), all solutions, or a solution that satisfies further properties, such as belonging to a given interval. When the task is to find the solution that is the best under some criterion, this is an opti... | Solution (equation) |
c_z2efq279g6wi | In mathematics, topological Galois theory is a mathematical theory which originated from a topological proof of Abel's impossibility theorem found by V. I. Arnold and concerns the applications of some topological concepts to some problems in the field of Galois theory. It connects many ideas from algebra to ideas in to... | Topological Galois theory |
c_rrxq816yok91 | In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological K-theory is due to Michael Atiyah and Friedrich Hirze... | Stably isomorphic |
c_7fgx1rjo4kwa | In mathematics, topological complexity of a topological space X (also denoted by TC(X)) is a topological invariant closely connected to the motion planning problem, introduced by Michael Farber in 2003. | Topological complexity |
c_dii92tj0z8qv | In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory. When one solution of an equation is easily found, degree theory can often be used to pro... | Topological degree theory |
c_zby7c9qmq2k3 | There are different types of degree for different types of maps: e.g. for maps between Banach spaces there is the Brouwer degree in Rn, the Leray-Schauder degree for compact mappings in normed spaces, the coincidence degree and various other types. There is also a degree for continuous maps between manifolds. Topologic... | Topological degree theory |
c_a01cgegw47tf | In mathematics, topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology. | Topological dynamical system |
c_c8wdrhk4cgd5 | In mathematics, topological graph theory is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces. It also studies immersions of graphs. | Graph topology |
c_erj40ehwwjaj | Embedding a graph in a surface means that we want to draw the graph on a surface, a sphere for example, without two edges intersecting. A basic embedding problem often presented as a mathematical puzzle is the three utilities problem. Other applications can be found in printing electronic circuits where the aim is to p... | Graph topology |
c_7ed0kv86tt4y | In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.Topological ... | Closed subgroup |
c_xc840bv3asbn | In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer n there is a topological space tmf n {\displaystyle \operatorname {tmf} ^{n}} , and these spaces are equipped with certain maps between them, so that for any topol... | Topological modular forms |
c_cdj9kkjuegl2 | The spectrum of topological modular forms is constructed as the global sections of a sheaf of E-infinity ring spectra on the moduli stack of (generalized) elliptic curves. This theory has relations to the theory of modular forms in number theory, the homotopy groups of spheres, and conjectural index theories on loop sp... | Topological modular forms |
c_2r4s6536118p | In mathematics, topological recursion is a recursive definition of invariants of spectral curves. It has applications in enumerative geometry, random matrix theory, mathematical physics, string theory, knot theory. | Topological recursion |
c_0p7mm4rpvave | In mathematics, topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing. A topological space is a set endowed with ... | List of topology topics |
c_8ndlxuzl7rep | A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle... | List of topology topics |
c_fq02dw8lho6l | The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 1... | List of topology topics |
c_933lwet0avbi | In mathematics, topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or p... | Topology |
c_wlgufgsyvi0o | A property that is invariant under such deformations is a topological property. The following are basic examples of topological properties: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distingui... | Topology |
c_zy6dsnwye047 | The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 1... | Topology |
c_14wk2xfluo1b | In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as (slightly modified) graphs in which some edges are labeled by matrices. The simplest trace diagrams represent the trace and determinant of a matrix. Several results in linear alg... | Trace diagram |
c_eacfxb5et3um | In mathematics, trailing zeros are a sequence of 0 in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow. Trailing zeros to the right of a decimal point, as in 12.340, don’t affect the value of a number and may be omitted if all that is of in... | Trailing zero |
c_7spki6exhaq8 | For example, in pharmacy, trailing zeros are omitted from dose values to prevent misreading. However, trailing zeros may be useful for indicating the number of significant figures, for example in a measurement. In such a context, "simplifying" a number by removing trailing zeros would be incorrect. | Trailing zero |
c_azsiguelf0gt | The number of trailing zeros in a non-zero base-b integer n equals the exponent of the highest power of b that divides n. For example, 14000 has three trailing zeros and is therefore divisible by 1000 = 103, but not by 104. This property is useful when looking for small factors in integer factorization. Some computer a... | Trailing zero |
c_iwmceiuocn2h | In mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used ... | Infinite number |
c_54hk555e2e3r | Nevertheless, the term transfinite also remains in use. Notable work on transfinite numbers was done by Wacław Sierpiński: Leçons sur les nombres transfinis (1928 book) much expanded into Cardinal and Ordinal Numbers (1958, 2nd ed. 1965). | Infinite number |
c_8yvyvq18s07d | In mathematics, transform theory is the study of transforms, which relate a function in one domain to another function in a second domain. The essence of transform theory is that by a suitable choice of basis for a vector space a problem may be simplified—or diagonalized as in spectral theory. | Transform theory |
c_87vpf08l4fuc | In mathematics, transformation geometry (or transformational geometry) is the name of a mathematical and pedagogic take on the study of geometry by focusing on groups of geometric transformations, and properties that are invariant under them. It is opposed to the classical synthetic geometry approach of Euclidean geome... | Transformation geometry |
c_3jb3njrniyj8 | For nearly a century this approach remained confined to mathematics research circles. In the 20th century efforts were made to exploit it for mathematical education. Andrei Kolmogorov included this approach (together with set theory) as part of a proposal for geometry teaching reform in Russia. These efforts culminated... | Transformation geometry |
c_iipkdvfqhwfg | In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the idea of a generic intersection in differential topology. It is defined by considering the linearizations of the intersecti... | Transversality (mathematics) |
c_ooht15j2293d | In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation group SO(8), arising because th... | Triality |
c_1iizod59pzn6 | The diagram has four nodes with one node located at the center, and the other three attached symmetrically. The symmetry group of the diagram is the symmetric group S3 which acts by permuting the three legs. This gives rise to an S3 group of outer automorphisms of Spin(8). | Triality |
c_4pptu5r7ufss | This automorphism group permutes the three 8-dimensional irreducible representations of Spin(8); these being the vector representation and two chiral spin representations. These automorphisms do not project to automorphisms of SO(8). The vector representation—the natural action of SO(8) (hence Spin(8)) on F8—consists o... | Triality |
c_aojnlcsmzof0 | No other connected Dynkin diagram has an automorphism group of order greater than 2; for other Dn (corresponding to other even Spin groups, Spin(2n)), there is still the automorphism corresponding to switching the two half-spin representations, but these are not isomorphic to the vector representation. Roughly speaking... | Triality |
c_svhbuuw579bs | In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangulable. Triangulation has various uses in different branches of mathematics, f... | Triangulable space |
c_6jrwovptlo9e | In mathematics, trigonometric integrals are a family of integrals involving trigonometric functions. | Trigonometric integral |
c_crz0ys3vyet4 | In mathematics, trigonometric interpolation is interpolation with trigonometric polynomials. Interpolation is the process of finding a function which goes through some given data points. For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of sines and cosines of given per... | Trigonometric interpolation |
c_ghxpzk0fmjgk | In mathematics, trigonometric substitution is the replacement of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities to simplify certain integrals containing radical expressions. Like other met... | Trigonometric substitution |
c_ibyuly5rcy5y | In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition: x ⊕ y = min { x , y } , {\displaystyle x\oplus y=\min\{x,y\},} x ⊗ y = x + y . {\displaystyle x\otimes y=x+y.} So for example, t... | Tropical geometry |
c_m3nam2whfcln | Such polynomials and their solutions have important applications in optimization problems, for example the problem of optimizing departure times for a network of trains. Tropical geometry is a variant of algebraic geometry in which polynomial graphs resemble piecewise linear meshes, and in which numbers belong to the t... | Tropical geometry |
c_fyq7r4uybjgx | In mathematics, twisted K-theory (also called K-theory with local coefficients) is a variation on K-theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory. More specifically, twisted K-theory with twist H is a particular variant of K-theory, in which the twist i... | Twisted K-theory |
c_1k6jxacox336 | This was provided in two steps; the first one was done in 1970 (Publ. Math. de l'IHÉS) by Peter Donovan and Max Karoubi; the second one in 1988 by Jonathan Rosenberg in Continuous-Trace Algebras from the Bundle Theoretic Point of View. | Twisted K-theory |
c_8lrmnxnvj153 | In physics, it has been conjectured to classify D-branes, Ramond-Ramond field strengths and in some cases even spinors in type II string theory. For more information on twisted K-theory in string theory, see K-theory (physics). In the broader context of K-theory, in each subject it has numerous isomorphic formulations ... | Twisted K-theory |
c_pbt8udrpjedc | In mathematics, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite abelian groups. They have been generalized by L. Ya. Kulikov. | Prüfer theorems |
c_xrsp9x5cnhuw | In mathematics, two elements x and y of a set P are said to be comparable with respect to a binary relation ≤ if at least one of x ≤ y or y ≤ x is true. They are called incomparable if they are not comparable. | Comparability |
c_b2eliht478ly | In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct § Topological equivalence of flows, are important in the study of iterated functions and more generally dynamical systems, sinc... | Topological conjugacy |
c_m11farq7q1ut | In mathematics, two functions have a contact of order k if, at a point P, they have the same value and k equal derivatives. This is an equivalence relation, whose equivalence classes are generally called jets. The point of osculation is also called the double cusp. | Contact (mathematics) |
c_s7j5oqc2s3hw | Contact is a geometric notion; it can be defined algebraically as a valuation. One speaks also of curves and geometric objects having k-th order contact at a point: this is also called osculation (i.e. kissing), generalising the property of being tangent. (Here the derivatives are considered with respect to arc length.... | Contact (mathematics) |
c_4fikopt0pcvw | In mathematics, two linear operators are called isospectral or cospectral if they have the same spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues, when those are counted with multiplicity. The theory of isospectral operators is markedly different depending on whether the space is finite... | Isospectral flow |
c_dx48vc1x3ego | In finite-dimensions, one essentially deals with square matrices. In infinite dimensions, the spectrum need not consist solely of isolated eigenvalues. However, the case of a compact operator on a Hilbert space (or Banach space) is still tractable, since the eigenvalues are at most countable with at most a single limit... | Isospectral flow |
c_di4lke8rvqtq | The most studied isospectral problem in infinite dimensions is that of the Laplace operator on a domain in R2. Two such domains are called isospectral if their Laplacians are isospectral. The problem of inferring the geometrical properties of a domain from the spectrum of its Laplacian is often known as hearing the sha... | Isospectral flow |
c_vxl4xameomdd | In mathematics, two links L 0 ⊂ S n {\displaystyle L_{0}\subset S^{n}} and L 1 ⊂ S n {\displaystyle L_{1}\subset S^{n}} are concordant if there exists an embedding f: L 0 × → S n × {\displaystyle f:L_{0}\times \to S^{n}\times } such that f ( L 0 × { 0 } ) = L 0 × { 0 } {\displaystyle f(L_{0}\times \{0\})=L_{0}\times ... | Link concordance |
c_j3m5lcdilerp | In mathematics, two major works were published in a single year, 1631. Thomas Harriot's Artis analyticae praxis, published ten years posthumously, and William Oughtred's Clavis mathematicae. Both contributed to the evolution of modern mathematical language; the former introduced the × {\displaystyle \times } sign for m... | Artisan Mannerism |
c_5v0zl4i3apvi | In mathematics, two metrics on the same underlying set are said to be equivalent if the resulting metric spaces share certain properties. Equivalence is a weaker notion than isometry; equivalent metrics do not have to be literally the same. Instead, it is one of several ways of generalizing equivalence of norms to gene... | Equivalence of metrics |
c_a7d4jbl6t9qw | In mathematics, two non-empty subsets A and B of a given metric space (X, d) are said to be positively separated if the infimum inf a ∈ A , b ∈ B d ( a , b ) > 0. {\displaystyle \inf _{a\in A,b\in B}d(a,b)>0.} (Some authors also specify that A and B should be disjoint sets; however, this adds nothing to the definition,... | Positively separated sets |
c_t1tzye4ypewr | In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio a/b is a rational number; otherwise a and b are called incommensurable. (Recall that a rational number is one that is equivalent to the ratio of two integers.) There is a more general notion of commensurability in group theory... | Commensurability (mathematics) |
c_inb6r7uixcgd | The numbers 3 {\displaystyle {\sqrt {3}}} and 2 3 {\displaystyle 2{\sqrt {3}}} are also commensurable because their ratio, 3 2 3 = 1 2 {\textstyle {\frac {\sqrt {3}}{2{\sqrt {3}}}}={\frac {1}{2}}} , is a rational number. However, the numbers 3 {\textstyle {\sqrt {3}}} and 2 are incommensurable because their ratio, 3 2 ... | Commensurability (mathematics) |
c_j36t5j2gr0q3 | In mathematics, two objects, especially systems of axioms or semantics for them, are called cryptomorphic if they are equivalent but not obviously equivalent. In particular, two definitions or axiomatizations of the same object are "cryptomorphic" if it is not obvious that they define the same object. Examples of crypt... | Cryptomorphism |
c_k1accs2035ux | In mathematics, two points of a sphere (or n-sphere, including a circle) are called antipodal or diametrically opposite if they are the intersections of the sphere with a diameter, a straight line passing through its center.Given any point on a sphere, its antipodal point is the unique point at greatest distance, wheth... | Antipodal point |
c_mmlhwlr1xc3e | In mathematics, two positive (or signed or complex) measures μ {\displaystyle \mu } and ν {\displaystyle \nu } defined on a measurable space ( Ω , Σ ) {\displaystyle (\Omega ,\Sigma )} are called singular if there exist two disjoint measurable sets A , B ∈ Σ {\displaystyle A,B\in \Sigma } whose union is Ω {\displaystyl... | Singular measures |
c_4ozehpohic1t | {\displaystyle \mu \perp \nu .} A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure. See below for examples. | Singular measures |
c_023md8y982a3 | In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a {\displaystyle a} and b {\displaystyle b} with a > b > 0 {\displaystyle a>b>0} , where the Greek letter phi ( φ {\displaystyle \varph... | The Golden Ratio |
c_729uw9js3xdz | A golden rectangle—that is, a rectangle with an aspect ratio of φ {\displaystyle \varphi } —may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on d... | The Golden Ratio |
c_14pizpu6gr5j | The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation. Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing it to be aesthetically pleasing. The... | The Golden Ratio |
c_oht5k3a0pi4j | In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational ... | Silver triangle |
c_bfjwrh6yky6r | Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed alg... | Silver triangle |
c_1q5dc1xnu3n2 | The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, ... | Silver triangle |
c_f3led1zalttk | In mathematics, two quantities are in the supergolden ratio if their quotient equals the unique real solution to the equation x 3 = x 2 + 1. {\displaystyle x^{3}=x^{2}+1.} This solution is commonly denoted ψ . {\displaystyle \psi .} | Supergolden ratio |
c_6y0dztk0tn4k | The name supergolden ratio results of a analogy with the golden ratio φ {\displaystyle \varphi } , which is the positive root of the equation x 2 = x + 1. {\displaystyle x^{2}=x+1.} Using formulas for the cubic equation, one can show that ψ = 1 3 ( 1 + 29 + 3 93 2 3 + 29 − 3 93 2 3 ) , {\displaystyle \psi ={\frac {1}{3... | Supergolden ratio |
c_emjpkumdxy8w | In mathematics, two real numbers p , q > 1 {\displaystyle p,q>1} are called conjugate indices (or Hölder conjugates) if 1 p + 1 q = 1. {\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1.} Formally, we also define q = ∞ {\displaystyle q=\infty } as conjugate to p = 1 {\displaystyle p=1} and vice versa. Conjugate indices are... | Hölder conjugates |
c_1cw1a1mtk21u | In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio. The ratio is called coefficient of proportionality (or proportionality constant) and its reciprocal is known as constant of normalization (or normalizing co... | Proportionality (mathematics) |
c_u2yoohr0nnyt | This meaning of variable is not the common meaning of the term in mathematics (see variable (mathematics)); these two different concepts share the same name for historical reasons. Two functions f ( x ) {\displaystyle f(x)} and g ( x ) {\displaystyle g(x)} are proportional if their ratio f ( x ) g ( x ) {\textstyle {\f... | Proportionality (mathematics) |
c_ctgdntoaruz9 | In mathematics, two sets are almost disjoint if their intersection is small in some sense; different definitions of "small" will result in different definitions of "almost disjoint". | Almost disjoint sets |
c_koekhtyhdlqf | In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two or more sets is called disjoi... | Pairwise disjoint sets |
c_ctztezwm1ygj | In mathematics, two sets or classes A and B are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from A to B such that for every element y of B, there is exactly one element x of A with f(x) = y. Equinumerous sets are said to have the same cardina... | Equinumerosity |
c_h2mmoido412t | The statement that two sets A and B are equinumerous is usually denoted A ≈ B {\displaystyle A\approx B\,} or A ∼ B {\displaystyle A\sim B} , or | A | = | B | . {\displaystyle |A|=|B|.} The definition of equinumerosity using bijections can be applied to both finite and infinite sets, and allows one to state whether two... | Equinumerosity |
c_i5is325n646e | Georg Cantor, the inventor of set theory, showed in 1874 that there is more than one kind of infinity, specifically that the collection of all natural numbers and the collection of all real numbers, while both infinite, are not equinumerous (see Cantor's first uncountability proof). In his controversial 1878 paper, Can... | Equinumerosity |
c_votcgv1xhhfa | This allows the definition of greater and greater infinite sets starting from a single infinite set. If the axiom of choice holds, then the cardinal number of a set may be regarded as the least ordinal number of that cardinality (see initial ordinal). Otherwise, it may be regarded (by Scott's trick) as the set of sets ... | Equinumerosity |
c_ebxor54msb8p | In mathematics, two square matrices A and B over a field are called congruent if there exists an invertible matrix P over the same field such that PTAP = Bwhere "T" denotes the matrix transpose. Matrix congruence is an equivalence relation. Matrix congruence arises when considering the effect of change of basis on the ... | Matrix congruence |
c_pliiuhbkauqs | In mathematics, two-center bipolar coordinates is a coordinate system based on two coordinates which give distances from two fixed centers c 1 {\displaystyle c_{1}} and c 2 {\displaystyle c_{2}} . This system is very useful in some scientific applications (e.g. calculating the electric field of a dipole on a plane). | Two-center bipolar coordinates |
c_qr69xnwsx7kq | In mathematics, umbral moonshine is a mysterious connection between Niemeier lattices and Ramanujan's mock theta functions. It is a generalization of the Mathieu moonshine phenomenon connecting representations of the Mathieu group M24 with K3 surfaces. | Umbral moonshine |
c_mcervqvr86ay | In mathematics, uncertainty is often characterized in terms of a probability distribution. From that perspective, epistemic uncertainty means not being certain what the relevant probability distribution is, and aleatoric uncertainty means not being certain what a random sample drawn from a probability distribution will... | Epistemic probability |
c_8ze89kuo4849 | In mathematics, uniform absolute-convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed. | Uniform absolute convergence |
c_e1p00lhl6twx | In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. | Uniform integrability |
c_6wwezld2u26q | In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936. | Uniformly convex Banach space |
c_69g3pdub99m5 | In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. | Unimodal distribution |
c_is3jlz5kryzo | In mathematics, unscented optimal control combines the notion of the unscented transform with deterministic optimal control to address a class of uncertain optimal control problems. It is a specific application of Riemmann-Stieltjes optimal control theory, a concept introduced by Ross and his coworkers. | Unscented optimal control |
c_mk6nelmw705q | In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as π or an integer such as 42. The value of a variable or a constant is any number or o... | Value (mathematics) |
c_kg8rxgf71k4k | The value of a mathematical expression is the result of the computation described by this expression when the variables and constants in it are assigned values. The value of a function, given the value(s) assigned to its argument(s), is the quantity assumed by the function for these argument values.For example, if the ... | Value (mathematics) |
c_lu53rovqlsrf | In mathematics, van der Corput's method generates estimates for exponential sums. The method applies two processes, the van der Corput processes A and B which relate the sums into simpler sums which are easier to estimate. The processes apply to exponential sums of the form ∑ n = a b e ( f ( n ) ) {\displaystyle \sum _... | Exponent pair |
c_c9zdne9fckgz | In mathematics, vanishing cycles are studied in singularity theory and other parts of algebraic geometry. They are those homology cycles of a smooth fiber in a family which vanish in the singular fiber. For example, in a map from a connected complex surface to the complex projective line, a generic fiber is a smooth Ri... | Vanishing cycle |
c_lvh1m9ve7byy | The loop in the smooth fibers gives an element of the first homology group of a surface, and the monodromy of the critical value is defined to be the monodromy of the first homology of the fibers as the loop is traversed, i.e. an invertible map of the first homology of a (real) surface of genus g. A classical result is... | Vanishing cycle |
c_zww030j5jo5l | There the definition uses derived categories, and looks very different. It involves a functor, the nearby cycle functor, with a definition by means of the higher direct image and pullbacks. The vanishing cycle functor then sits in a distinguished triangle with the nearby cycle functor and a more elementary functor. Thi... | Vanishing cycle |
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