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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}}.
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https://en.wikipedia.org/wiki/Tiny_and_miny
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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 assertion is also easily verifiable with canonical forms and game trees.
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https://en.wikipedia.org/wiki/Tiny_and_miny
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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.
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https://en.wikipedia.org/wiki/Finite_difference_coefficients
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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 is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values (one for each unknown) such that, when substituted for the unknowns, the equation becomes an equality.
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https://en.wikipedia.org/wiki/Solution_(equation)
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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.
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https://en.wikipedia.org/wiki/Solution_(equation)
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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 equation results in (y + 1) + y = 2(y + 1) – 1, a true statement.
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https://en.wikipedia.org/wiki/Solution_(equation)
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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 solution with specific numbers gives a numerical solution; for example, a = 0 gives (x, y) = (1, 0) (that is, x = 1, y = 0), and a = 1 gives (x, y) = (2, 1).
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https://en.wikipedia.org/wiki/Solution_(equation)
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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, ... to denote the known variables, which are often called parameters. This is typically the case when considering polynomial equations, such as quadratic equations. However, for some problems, all variables may assume either role.
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https://en.wikipedia.org/wiki/Solution_(equation)
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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 optimization problem. Solving an optimization problem is generally not referred to as "equation solving", as, generally, solving methods start from a particular solution for finding a better solution, and repeating the process until finding eventually the best solution.
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https://en.wikipedia.org/wiki/Solution_(equation)
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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 topology. As described in Khovanskii's book: "According to this theory, the way the Riemann surface of an analytic function covers the plane of complex numbers can obstruct the representability of this function by explicit formulas. The strongest known results on the unexpressibility of functions by explicit formulas have been obtained in this way."
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https://en.wikipedia.org/wiki/Topological_Galois_theory
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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 Hirzebruch.
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https://en.wikipedia.org/wiki/Stably_isomorphic
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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.
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https://en.wikipedia.org/wiki/Topological_complexity
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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 prove existence of a second, nontrivial, solution.
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https://en.wikipedia.org/wiki/Topological_degree_theory
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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. Topological degree theory has applications in complementarity problems, differential equations, differential inclusions and dynamical systems.
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https://en.wikipedia.org/wiki/Topological_degree_theory
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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.
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https://en.wikipedia.org/wiki/Topological_dynamical_system
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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.
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https://en.wikipedia.org/wiki/Graph_topology
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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 print (embed) a circuit (the graph) on a circuit board (the surface) without two connections crossing each other and resulting in a short circuit.
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https://en.wikipedia.org/wiki/Graph_topology
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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 groups have been studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a very wide class of topological groups.Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional analysis.
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https://en.wikipedia.org/wiki/Closed_subgroup
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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 topological space X, one obtains an abelian group structure on the set tmf n ( X ) {\displaystyle \operatorname {tmf} ^{n}(X)} of homotopy classes of continuous maps from X to tmf n {\displaystyle \operatorname {tmf} ^{n}} . One feature that distinguishes tmf is the fact that its coefficient ring, tmf 0 {\displaystyle \operatorname {tmf} ^{0}} (point), is almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting the primes 2 and 3, but this inversion erases a lot of torsion information in the coefficient ring.
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https://en.wikipedia.org/wiki/Topological_modular_forms
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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 spaces of manifolds. tmf was first constructed by Michael Hopkins and Haynes Miller; many of the computations can be found in preprints and articles by Paul Goerss, Hopkins, Mark Mahowald, Miller, Charles Rezk, and Tilman Bauer.
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https://en.wikipedia.org/wiki/Topological_modular_forms
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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.
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https://en.wikipedia.org/wiki/Topological_recursion
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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 a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies.
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https://en.wikipedia.org/wiki/List_of_topology_topics
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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 from two non-intersecting circles.
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https://en.wikipedia.org/wiki/List_of_topology_topics
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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 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. This is a list of topology topics. See also: Topology glossary List of topologies List of general topology topics List of geometric topology topics List of algebraic topology topics List of topological invariants (topological properties) Publications in topology
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https://en.wikipedia.org/wiki/List_of_topology_topics
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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 passing through itself. A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies.
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https://en.wikipedia.org/wiki/Topology
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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 distinguishing a circle from two non-intersecting circles.
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https://en.wikipedia.org/wiki/Topology
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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 19th century; although, it was not until the first decades of the 20th century that the idea of a topological space was developed.
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https://en.wikipedia.org/wiki/Topology
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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 algebra, such as Cramer's Rule and the Cayley–Hamilton theorem, have simple diagrammatic proofs. They are closely related to Penrose's graphical notation.
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https://en.wikipedia.org/wiki/Trace_diagram
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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 interest is its numerical value. This is true even if the zeros recur infinitely.
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https://en.wikipedia.org/wiki/Trailing_zero
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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.
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https://en.wikipedia.org/wiki/Trailing_zero
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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 architectures have a count trailing zeros operation in their instruction set for efficiently determining the number of trailing zero bits in a machine word.
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https://en.wikipedia.org/wiki/Trailing_zero
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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 to provide an ordering of infinite sets. The term transfinite was coined in 1895 by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were, nevertheless, not finite. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as infinite numbers.
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https://en.wikipedia.org/wiki/Infinite_number
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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).
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https://en.wikipedia.org/wiki/Infinite_number
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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.
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https://en.wikipedia.org/wiki/Transform_theory
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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 geometry, that focuses on proving theorems. For example, within transformation geometry, the properties of an isosceles triangle are deduced from the fact that it is mapped to itself by a reflection about a certain line. This contrasts with the classical proofs by the criteria for congruence of triangles.The first systematic effort to use transformations as the foundation of geometry was made by Felix Klein in the 19th century, under the name Erlangen programme.
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https://en.wikipedia.org/wiki/Transformation_geometry
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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 in the 1960s with the general reform of mathematics teaching known as the New Math movement.
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https://en.wikipedia.org/wiki/Transformation_geometry
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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 intersecting spaces at the points of intersection.
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https://en.wikipedia.org/wiki/Transversality_(mathematics)
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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 the group has an outer automorphism of order three. There is a geometrical version of triality, analogous to duality in projective geometry. Of all simple Lie groups, Spin(8) has the most symmetrical Dynkin diagram, D4.
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https://en.wikipedia.org/wiki/Triality
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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).
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https://en.wikipedia.org/wiki/Triality
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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 over the real numbers of Euclidean 8-vectors and is generally known as the "defining module", while the chiral spin representations are also known as "half-spin representations", and all three of these are fundamental representations.
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https://en.wikipedia.org/wiki/Triality
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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, symmetries of the Dynkin diagram lead to automorphisms of the Tits building associated with the group. For special linear groups, one obtains projective duality. For Spin(8), one finds a curious phenomenon involving 1-, 2-, and 4-dimensional subspaces of 8-dimensional space, historically known as "geometric triality". The exceptional 3-fold symmetry of the D4 diagram also gives rise to the Steinberg group 3D4.
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https://en.wikipedia.org/wiki/Triality
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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, for instance in algebraic topology, in complex analysis or in modeling.
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https://en.wikipedia.org/wiki/Triangulable_space
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In mathematics, trigonometric integrals are a family of integrals involving trigonometric functions.
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https://en.wikipedia.org/wiki/Trigonometric_integral
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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 periods. This form is especially suited for interpolation of periodic functions. An important special case is when the given data points are equally spaced, in which case the solution is given by the discrete Fourier transform.
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https://en.wikipedia.org/wiki/Trigonometric_interpolation
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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 methods of integration by substitution, when evaluating a definite integral, it may be simpler to completely deduce the antiderivative before applying the boundaries of integration.
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https://en.wikipedia.org/wiki/Trigonometric_substitution
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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, the classical polynomial x 3 + 2 x y + y 4 {\displaystyle x^{3}+2xy+y^{4}} would become min { x + x + x , 2 + x + y , y + y + y + y } {\displaystyle \min\{x+x+x,\;2+x+y,\;y+y+y+y\}} .
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https://en.wikipedia.org/wiki/Tropical_geometry
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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 tropical semiring instead of a field. Because classical and tropical geometry are closely related, results and methods can be converted between them. Algebraic varieties can be mapped to a tropical counterpart and, since this process still retains some geometric information about the original variety, it can be used to help prove and generalize classical results from algebraic geometry, such as the Brill–Noether theorem, using the tools of tropical geometry.
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https://en.wikipedia.org/wiki/Tropical_geometry
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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 is given by an integral 3-dimensional cohomology class. It is special among the various twists that K-theory admits for two reasons. First, it admits a geometric formulation.
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https://en.wikipedia.org/wiki/Twisted_K-theory
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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.
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https://en.wikipedia.org/wiki/Twisted_K-theory
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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 and, in many cases, isomorphisms relating definitions in various subjects have been proven. It also has numerous deformations, for example, in abstract algebra K-theory may be twisted by any integral cohomology class.
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https://en.wikipedia.org/wiki/Twisted_K-theory
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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.
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https://en.wikipedia.org/wiki/Prüfer_theorems
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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.
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https://en.wikipedia.org/wiki/Comparability
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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, since, if the dynamics of one iterative function can be determined, then that for a topologically conjugate function follows trivially.To illustrate this directly: suppose that f {\displaystyle f} and g {\displaystyle g} are iterated functions, and there exists a homeomorphism h {\displaystyle h} such that g = h − 1 ∘ f ∘ h , {\displaystyle g=h^{-1}\circ f\circ h,} so that f {\displaystyle f} and g {\displaystyle g} are topologically conjugate. Then one must have g n = h − 1 ∘ f n ∘ h , {\displaystyle g^{n}=h^{-1}\circ f^{n}\circ h,} and so the iterated systems are topologically conjugate as well. Here, ∘ {\displaystyle \circ } denotes function composition.
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https://en.wikipedia.org/wiki/Topological_conjugacy
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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.
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https://en.wikipedia.org/wiki/Contact_(mathematics)
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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.) An osculating curve from a given family of curves is a curve that has the highest possible order of contact with a given curve at a given point; for instance a tangent line is an osculating curve from the family of lines, and has first-order contact with the given curve; an osculating circle is an osculating curve from the family of circles, and has second-order contact (same tangent angle and curvature), etc.
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https://en.wikipedia.org/wiki/Contact_(mathematics)
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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 or infinite dimensional.
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https://en.wikipedia.org/wiki/Isospectral_flow
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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 point λ = 0.
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https://en.wikipedia.org/wiki/Isospectral_flow
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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 shape of a drum.
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https://en.wikipedia.org/wiki/Isospectral_flow
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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 \{0\}} and f ( L 0 × { 1 } ) = L 1 × { 1 } {\displaystyle f(L_{0}\times \{1\})=L_{1}\times \{1\}} . By its nature, link concordance is an equivalence relation. It is weaker than isotopy, and stronger than homotopy: isotopy implies concordance implies homotopy. A link is a slice link if it is concordant to the unlink.
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https://en.wikipedia.org/wiki/Link_concordance
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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 multiplication and (::) sign for proportion. In philosophy, Thomas Hobbes (1588–1679) was already writing some of his works and evolving his key concepts, though they were not in print until after the end of the Caroline era.
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https://en.wikipedia.org/wiki/Artisan_Mannerism
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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 general metric spaces. Throughout the article, X {\displaystyle X} will denote a non-empty set and d 1 {\displaystyle d_{1}} and d 2 {\displaystyle d_{2}} will denote two metrics on X {\displaystyle X} .
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https://en.wikipedia.org/wiki/Equivalence_of_metrics
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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, since if A and B have some common point p, then d(p, p) = 0, and so the infimum above is clearly 0 in that case.) For example, on the real line with the usual distance, the open intervals (0, 2) and (3, 4) are positively separated, while (3, 4) and (4, 5) are not. In two dimensions, the graph of y = 1/x for x > 0 and the x-axis are not positively separated.
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https://en.wikipedia.org/wiki/Positively_separated_sets
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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. For example, the numbers 3 and 2 are commensurable because their ratio, 3/2, is a rational number.
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https://en.wikipedia.org/wiki/Commensurability_(mathematics)
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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 {\textstyle {\frac {\sqrt {3}}{2}}} , is an irrational number. More generally, it is immediate from the definition that if a and b are any two non-zero rational numbers, then a and b are commensurable; it is also immediate that if a is any irrational number and b is any non-zero rational number, then a and b are incommensurable. On the other hand, if both a and b are irrational numbers, then a and b may or may not be commensurable.
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https://en.wikipedia.org/wiki/Commensurability_(mathematics)
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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 cryptomorphic definitions abound in matroid theory and others can be found elsewhere, e.g., in group theory the definition of a group by a single operation of division, which is not obviously equivalent to the usual three "operations" of identity element, inverse, and multiplication. This word is a play on the many morphisms in mathematics, but "cryptomorphism" is only very distantly related to "isomorphism", "homomorphism", or "morphisms". The equivalence may in a cryptomorphism, if it is not actual identity, be informal, or may be formalized in terms of a bijection or equivalence of categories between the mathematical objects defined by the two cryptomorphic axiom systems.
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https://en.wikipedia.org/wiki/Cryptomorphism
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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, whether measured intrinsically (great-circle distance on the surface of the sphere) or extrinsically (chordal distance through the sphere's interior). Every great circle on a sphere passing through a point also passes through its antipodal point, and there are infinitely many great circles passing through a pair of antipodal points (unlike the situation for any non-antipodal pair of points, which have a unique great circle passing through both). Many results in spherical geometry depend on choosing non-antipodal points, and degenerate if antipodal points are allowed; for example, a spherical triangle degenerates to an underspecified lune if two of the vertices are antipodal. The point antipodal to a given point is called its antipodes, from the Greek ἀντίποδες (antípodes) meaning "opposite feet"; see Antipodes § Etymology. Sometimes the s is dropped, and this is rendered antipode, a back-formation.
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https://en.wikipedia.org/wiki/Antipodal_point
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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 {\displaystyle \Omega } such that μ {\displaystyle \mu } is zero on all measurable subsets of B {\displaystyle B} while ν {\displaystyle \nu } is zero on all measurable subsets of A . {\displaystyle A.} This is denoted by μ ⊥ ν .
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https://en.wikipedia.org/wiki/Singular_measures
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{\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.
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https://en.wikipedia.org/wiki/Singular_measures
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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 \varphi } or ϕ {\displaystyle \phi } ) denotes the golden ratio. The constant φ {\displaystyle \varphi } satisfies the quadratic equation φ 2 = φ + 1 {\displaystyle \varphi ^{2}=\varphi +1} and is an irrational number with a value of The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli, and also goes by several other names.Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron.
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https://en.wikipedia.org/wiki/The_Golden_Ratio
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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 dubious fits to data.
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https://en.wikipedia.org/wiki/The_Golden_Ratio
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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. These uses often appear in the form of a golden rectangle.
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https://en.wikipedia.org/wiki/The_Golden_Ratio
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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 mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is denoted by δS.
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https://en.wikipedia.org/wiki/Silver_triangle
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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 algebraically: 2 a + b a = a b ≡ δ S {\displaystyle {\frac {2a+b}{a}}={\frac {a}{b}}\equiv \delta _{S}} or equivalently, 2 + b a = a b ≡ δ S {\displaystyle 2+{\frac {b}{a}}={\frac {a}{b}}\equiv \delta _{S}} The silver ratio can also be defined by the simple continued fraction: 2 + 1 2 + 1 2 + 1 2 + ⋱ = δ S {\displaystyle 2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots }}}}}}=\delta _{S}} The convergents of this continued fraction (2/1, 5/2, 12/5, 29/12, 70/29, ...) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers.
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https://en.wikipedia.org/wiki/Silver_triangle
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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, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt2.
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https://en.wikipedia.org/wiki/Silver_triangle
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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 .}
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https://en.wikipedia.org/wiki/Supergolden_ratio
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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}}\left(1+{\sqrt{\frac {29+3{\sqrt {93}}}{2}}}+{\sqrt{\frac {29-3{\sqrt {93}}}{2}}}\right),} or, using the hyperbolic cosine, ψ = 2 3 cosh ( 1 3 cosh − 1 ( 29 2 ) ) + 1 3 . {\displaystyle \psi ={\frac {2}{3}}\cosh {\left({\frac {1}{3}}\cosh ^{-1}\left({\frac {29}{2}}\right)\right)}+{\frac {1}{3}}.} The decimal expansion of this number begins as 1.465571231876768026656731... ((sequence A092526 in the OEIS)).
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https://en.wikipedia.org/wiki/Supergolden_ratio
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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 used in Hölder's inequality, as well as Young's inequality for products; the latter can be used to prove the former. If p , q > 1 {\displaystyle p,q>1} are conjugate indices, the spaces Lp and Lq are dual to each other (see Lp space).
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https://en.wikipedia.org/wiki/Hölder_conjugates
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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 constant). Two sequences are inversely proportional if corresponding elements have a constant product, also called the coefficient of proportionality. This definition is commonly extended to related varying quantities, which are often called variables.
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https://en.wikipedia.org/wiki/Proportionality_(mathematics)
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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 {\frac {f(x)}{g(x)}}} is a constant function. If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a proportion, e.g., a/b = x/y = ⋯ = k (for details see Ratio). Proportionality is closely related to linearity.
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https://en.wikipedia.org/wiki/Proportionality_(mathematics)
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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".
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https://en.wikipedia.org/wiki/Almost_disjoint_sets
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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 disjoint if any two distinct sets of the collection are disjoint.
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https://en.wikipedia.org/wiki/Pairwise_disjoint_sets
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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 cardinality (number of elements). The study of cardinality is often called equinumerosity (equalness-of-number). The terms equipollence (equalness-of-strength) and equipotence (equalness-of-power) are sometimes used instead. Equinumerosity has the characteristic properties of an equivalence relation.
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https://en.wikipedia.org/wiki/Equinumerosity
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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 sets have the same size even if they are infinite.
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https://en.wikipedia.org/wiki/Equinumerosity
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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, Cantor explicitly defined the notion of "power" of sets and used it to prove that the set of all natural numbers and the set of all rational numbers are equinumerous (an example where a proper subset of an infinite set is equinumerous to the original set), and that the Cartesian product of even a countably infinite number of copies of the real numbers is equinumerous to a single copy of the real numbers. Cantor's theorem from 1891 implies that no set is equinumerous to its own power set (the set of all its subsets).
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https://en.wikipedia.org/wiki/Equinumerosity
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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 of minimal rank having that cardinality.The statement that any two sets are either equinumerous or one has a smaller cardinality than the other is equivalent to the axiom of choice.
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https://en.wikipedia.org/wiki/Equinumerosity
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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 Gram matrix attached to a bilinear form or quadratic form on a finite-dimensional vector space: two matrices are congruent if and only if they represent the same bilinear form with respect to different bases. Note that Halmos defines congruence in terms of conjugate transpose (with respect to a complex inner product space) rather than transpose, but this definition has not been adopted by most other authors.
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https://en.wikipedia.org/wiki/Matrix_congruence
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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).
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https://en.wikipedia.org/wiki/Two-center_bipolar_coordinates
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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.
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https://en.wikipedia.org/wiki/Umbral_moonshine
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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 be.
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https://en.wikipedia.org/wiki/Epistemic_probability
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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.
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https://en.wikipedia.org/wiki/Uniform_absolute_convergence
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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.
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https://en.wikipedia.org/wiki/Uniform_integrability
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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.
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https://en.wikipedia.org/wiki/Uniformly_convex_Banach_space
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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.
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https://en.wikipedia.org/wiki/Unimodal_distribution
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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.
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https://en.wikipedia.org/wiki/Unscented_optimal_control
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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 other mathematical object assigned to it.
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https://en.wikipedia.org/wiki/Value_(mathematics)
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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 function f is defined by f(x) = 2x2 – 3x + 1, then assigning the value 3 to its argument x yields the function value 10, since f(3) = 2·32 – 3·3 + 1 = 10. If the variable, expression or function only assumes real values, it is called real-valued. Likewise, a complex-valued variable, expression or function only assumes complex values.
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https://en.wikipedia.org/wiki/Value_(mathematics)
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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 _{n=a}^{b}e(f(n))\ } where f is a sufficiently smooth function and e(x) denotes exp(2πix).
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https://en.wikipedia.org/wiki/Exponent_pair
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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 Riemann surface of some fixed genus g and, generically, there will be isolated points in the target whose preimages are nodal curves. If one considers an isolated critical value and a small loop around it, in each fiber, one can find a smooth loop such that the singular fiber can be obtained by pinching that loop to a point.
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https://en.wikipedia.org/wiki/Vanishing_cycle
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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 the Picard–Lefschetz formula, detailing how the monodromy round the singular fiber acts on the vanishing cycles, by a shear mapping. The classical, geometric theory of Solomon Lefschetz was recast in purely algebraic terms, in SGA7. This was for the requirements of its application in the context of l-adic cohomology; and eventual application to the Weil conjectures.
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https://en.wikipedia.org/wiki/Vanishing_cycle
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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. This formulation has been of continuing influence, in particular in D-module theory.
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https://en.wikipedia.org/wiki/Vanishing_cycle
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