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Often, the term linear equation refers implicitly to this particular case, in which the variable is sensibly called the unknown. In the case of two variables, each solution may be interpreted as the Cartesian coordinates of a point of the Euclidean plane. The solutions of a linear equation form a line in the Euclidean ... | https://en.wikipedia.org/wiki/First_degree_equation |
This is the origin of the term linear for describing this type of equations. More generally, the solutions of a linear equation in n variables form a hyperplane (a subspace of dimension n − 1) in the Euclidean space of dimension n. Linear equations occur frequently in all mathematics and their applications in physics a... | https://en.wikipedia.org/wiki/First_degree_equation |
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If V is a vector space over a field k, the set of all linear functionals from V to k is itself a vector space over k ... | https://en.wikipedia.org/wiki/Dual_vector |
It is often denoted Hom(V, k), or, when the field k is understood, V ∗ {\displaystyle V^{*}} ; other notations are also used, such as V ′ {\displaystyle V'} , V # {\displaystyle V^{\#}} or V ∨ . {\displaystyle V^{\vee }.} When vectors are represented by column vectors (as is common when a basis is fixed), then linear f... | https://en.wikipedia.org/wiki/Dual_vector |
In mathematics, a linear fractional transformation is, roughly speaking, an invertible transformation of the form z ↦ a z + b c z + d . {\displaystyle z\mapsto {\frac {az+b}{cz+d}}.} The precise definition depends on the nature of a, b, c, d, and z. In other words, a linear fractional transformation is a transformation... | https://en.wikipedia.org/wiki/Linear_fractional_transformations |
The invertibility condition is then ad – bc ≠ 0. Over a field, a linear fractional transformation is the restriction to the field of a projective transformation or homography of the projective line. When a, b, c, d are integer (or, more generally, belong to an integral domain), z is supposed to be a rational number (or... | https://en.wikipedia.org/wiki/Linear_fractional_transformations |
In this case, the invertibility condition is that ad – bc must be a unit of the domain (that is 1 or −1 in the case of integers).In the most general setting, the a, b, c, d and z are elements of a ring, such as square matrices. An example of such linear fractional transformation is the Cayley transform, which was origi... | https://en.wikipedia.org/wiki/Linear_fractional_transformations |
In mathematics, a linear map (or linear function) f ( x ) {\displaystyle f(x)} is one which satisfies both of the following properties: Additivity or superposition principle: f ( x + y ) = f ( x ) + f ( y ) ; {\displaystyle \textstyle f(x+y)=f(x)+f(y);} Homogeneity: f ( α x ) = α f ( x ) . {\displaystyle \textstyle f(\... | https://en.wikipedia.org/wiki/Nonlinear_science |
The conditions of additivity and homogeneity are often combined in the superposition principle f ( α x + β y ) = α f ( x ) + β f ( y ) {\displaystyle f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)} An equation written as f ( x ) = C {\displaystyle f(x)=C} is called linear if f ( x ) {\displaystyle f(x)} is a linear map (as... | https://en.wikipedia.org/wiki/Nonlinear_science |
In mathematics, a linear map or linear function f(x) is a function that satisfies the two properties: Additivity: f(x + y) = f(x) + f(y). Homogeneity of degree 1: f(αx) = α f(x) for all α.These properties are known as the superposition principle. In this definition, x is not necessarily a real number, but can in genera... | https://en.wikipedia.org/wiki/Linearity |
Additivity alone implies homogeneity for rational α, since f ( x + x ) = f ( x ) + f ( x ) {\displaystyle f(x+x)=f(x)+f(x)} implies f ( n x ) = n f ( x ) {\displaystyle f(nx)=nf(x)} for any natural number n by mathematical induction, and then n f ( x ) = f ( n x ) = f ( m n m x ) = m f ( n m x ) {\displaystyle nf(x)=f(... | https://en.wikipedia.org/wiki/Linearity |
In mathematics, a linear operator T on a vector space is semisimple if every T-invariant subspace has a complementary T-invariant subspace; in other words, the vector space is a semisimple representation of the operator T. Equivalently, a linear operator is semisimple if the minimal polynomial of it is a product of dis... | https://en.wikipedia.org/wiki/Semi-simple_operator |
In mathematics, a linear operator f: V → V {\displaystyle f:V\to V} is called locally finite if the space V {\displaystyle V} is the union of a family of finite-dimensional f {\displaystyle f} -invariant subspaces. : 40 In other words, there exists a family { V i | i ∈ I } {\displaystyle \{V_{i}\vert i\in I\}} of linea... | https://en.wikipedia.org/wiki/Locally_finite_operator |
In mathematics, a linearised polynomial (or q-polynomial) is a polynomial for which the exponents of all the constituent monomials are powers of q and the coefficients come from some extension field of the finite field of order q. We write a typical example as where each a i {\displaystyle a_{i}} is in F q m ( = GF (... | https://en.wikipedia.org/wiki/Linearized_polynomial |
In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property. | https://en.wikipedia.org/wiki/Linked_field |
In mathematics, a local language is a formal language for which membership of a word in the language can be determined by looking at the first and last symbol and each two-symbol substring of the word. Equivalently, it is a language recognised by a local automaton, a particular kind of deterministic finite automaton.Fo... | https://en.wikipedia.org/wiki/Local_language_(formal_language) |
In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local martingale that is bounded from below is a supermartingale, and every local ... | https://en.wikipedia.org/wiki/Local_martingale |
In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were in... | https://en.wikipedia.org/wiki/Local_coefficients |
In mathematics, a locally catenative sequence is a sequence of words in which each word can be constructed as the concatenation of previous words in the sequence.Formally, an infinite sequence of words w(n) is locally catenative if, for some positive integers k and i1,...ik: w ( n ) = w ( n − i 1 ) w ( n − i 2 ) … w ( ... | https://en.wikipedia.org/wiki/Locally_catenative_sequence |
In mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure. ... | https://en.wikipedia.org/wiki/Locally_compact_group |
For compact groups, modifications of these proofs yields similar results by averaging with respect to the normalized Haar integral. In the general locally compact setting, such techniques need not hold. The resulting theory is a central part of harmonic analysis. The representation theory for locally compact abelian gr... | https://en.wikipedia.org/wiki/Locally_compact_group |
In mathematics, a locally compact topological group G has property (T) if the trivial representation is an isolated point in its unitary dual equipped with the Fell topology. Informally, this means that if G acts unitarily on a Hilbert space and has "almost invariant vectors", then it has a nonzero invariant vector. Th... | https://en.wikipedia.org/wiki/Kazhdan's_property_(T) |
In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function. | https://en.wikipedia.org/wiki/Locally_constant_function |
In mathematics, a locally cyclic group is a group (G, *) in which every finitely generated subgroup is cyclic. | https://en.wikipedia.org/wiki/Locally_cyclic_group |
In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure. | https://en.wikipedia.org/wiki/Locally_finite_measure |
In mathematics, a locally finite poset is a partially ordered set P such that for all x, y ∈ P, the interval consists of finitely many elements. Given a locally finite poset P we can define its incidence algebra. Elements of the incidence algebra are functions ƒ that assign to each interval of P a real number ƒ(x, y)... | https://en.wikipedia.org/wiki/Locally_finite_partially_ordered_set |
{\displaystyle (f*g)(x,y):=\sum _{x\leq z\leq y}f(x,z)g(z,y).} There is also a definition of incidence coalgebra. In theoretical physics a locally finite poset is also called a causal set and has been used as a model for spacetime. | https://en.wikipedia.org/wiki/Locally_finite_partially_ordered_set |
In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to Lp spaces, but its ... | https://en.wikipedia.org/wiki/Locally_integrable_function |
In mathematics, a locally profinite group is a Hausdorff topological group in which every neighborhood of the identity element contains a compact open subgroup. Equivalently, a locally profinite group is a topological group that is Hausdorff, locally compact, and totally disconnected. Moreover, a locally profinite grou... | https://en.wikipedia.org/wiki/Locally_profinite_group |
In mathematics, a locally simply connected space is a topological space that admits a basis of simply connected sets. Every locally simply connected space is also locally path-connected and locally connected. The circle is an example of a locally simply connected space which is not simply connected. The Hawaiian earrin... | https://en.wikipedia.org/wiki/Locally_simply_connected_space |
The cone on the Hawaiian earring is contractible and therefore simply connected, but still not locally simply connected. All topological manifolds and CW complexes are locally simply connected. In fact, these satisfy the much stronger property of being locally contractible. | https://en.wikipedia.org/wiki/Locally_simply_connected_space |
A strictly weaker condition is that of being semi-locally simply connected. Both locally simply connected spaces and simply connected spaces are semi-locally simply connected, but neither converse holds. == References == | https://en.wikipedia.org/wiki/Locally_simply_connected_space |
In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices that do ha... | https://en.wikipedia.org/wiki/Logarithm_of_a_matrix |
In mathematics, a logical matrix may be described as d-disjunct and/or d-separable. These concepts play a pivotal role in the mathematical area of non-adaptive group testing. In the mathematical literature, d-disjunct matrices may also be called super-imposed codes or d-cover-free families.According to Chen and Hwang (... | https://en.wikipedia.org/wiki/Disjunct_matrix |
A matrix is said to be d-disjunct if no set of d columns has a boolean sum which is a superset of any other single column.The following relationships are "well-known": Every d + 1 ¯ {\displaystyle {\overline {d+1}}} -separable matrix is also d {\displaystyle d} -disjunct. Every d {\displaystyle d} -disjunct matrix is a... | https://en.wikipedia.org/wiki/Disjunct_matrix |
In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise. | https://en.wikipedia.org/wiki/Loop_group |
In mathematics, a loop in a topological space X is a continuous function f from the unit interval I = to X such that f(0) = f(1). In other words, it is a path whose initial point is equal to its terminal point.A loop may also be seen as a continuous map f from the pointed unit circle S1 into X, because S1 may be regar... | https://en.wikipedia.org/wiki/Loop_(topology) |
In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x1, ..., xN has a low discrepancy. Roughly speaking, the discrepancy of a sequence is low if the proportion of points in the sequence falling into an arbitrary set B is close to proportional to the measu... | https://en.wikipedia.org/wiki/Quasi-random_sequence |
In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a collection of integers arranged in an n × n × n pattern such that the sums of the numbers on each row, on each column, on each pillar and on each of the four main space diagonals are equal, the so-called magic constant of the cub... | https://en.wikipedia.org/wiki/Magic_cube |
In mathematics, a magic hypercube is the k-dimensional generalization of magic squares and magic cubes, that is, an n × n × n × ... × n array of integers such that the sums of the numbers on each pillar (along any axis) as well as on the main space diagonals are all the same. The common sum is called the magic constant... | https://en.wikipedia.org/wiki/Nasik_magic_hypercube |
Four-, five-, six-, seven- and eight-dimensional magic hypercubes of order three have been constructed by J. R. Hendricks. Marian Trenkler proved the following theorem: A p-dimensional magic hypercube of order n exists if and only if p > 1 and n is different from 2 or p = 1. A construction of a magic hypercube follows ... | https://en.wikipedia.org/wiki/Nasik_magic_hypercube |
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n {\displaystyle n} -dimensional manifold, or n {\displaystyle n} -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open ... | https://en.wikipedia.org/wiki/Manifold_theory |
Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simple... | https://en.wikipedia.org/wiki/Manifold_theory |
The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g. CT scans). Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure allows calculus to be done. | https://en.wikipedia.org/wiki/Manifold_theory |
A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. The study of manifolds requires working knowledge of calcul... | https://en.wikipedia.org/wiki/Manifold_theory |
In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper.The term map may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map i... | https://en.wikipedia.org/wiki/Map_(mathematics) |
In category theory, a map may refer to a morphism. The term transformation can be used interchangeably, but transformation often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory. | https://en.wikipedia.org/wiki/Map_(mathematics) |
In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some sufficiently small or arbitrarily small neighborhoods of points). | https://en.wikipedia.org/wiki/P-local_subgroup |
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by... | https://en.wikipedia.org/wiki/Principal_submatrix |
Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents the composition of linear maps. Not all matrices are related to linear algebra. | https://en.wikipedia.org/wiki/Principal_submatrix |
This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. This article focuses on matrices related to linear algebra, and, unless otherwise specified, all matrices represent linear maps or may be viewed as such. Square matrices, matrices with the same number of rows and columns, p... | https://en.wikipedia.org/wiki/Principal_submatrix |
Square matrices of a given dimension form a noncommutative ring, which is one of the most common examples of a noncommutative ring. The determinant of a square matrix is a number associated to the matrix, which is fundamental for the study of a square matrix; for example, a square matrix is invertible if and only if it... | https://en.wikipedia.org/wiki/Principal_submatrix |
In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimension. Matrices are used in most areas of mathematics and most scientific fields, either directly, or through their use in geometry and numerical analysis. ... | https://en.wikipedia.org/wiki/Principal_submatrix |
In mathematics, a matrix coefficient (or matrix element) is a function on a group of a special form, which depends on a linear representation of the group and additional data. Precisely, it is a function on a compact topological group G obtained by composing a representation of G on a vector space V with a linear map f... | https://en.wikipedia.org/wiki/Matrix_coefficient |
They arise naturally from finite-dimensional representations of G as the matrix-entry functions of the corresponding matrix representations. The Peter–Weyl theorem says that the matrix coefficients on G are dense in the Hilbert space of square-integrable functions on G. Matrix coefficients of representations of Lie gro... | https://en.wikipedia.org/wiki/Matrix_coefficient |
In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that every multivariate real-valued polynomial p without linear terms can be written as a AB = pI, where A and B are square matrices and I is the identity matrix. Given the po... | https://en.wikipedia.org/wiki/Matrix_factorization_of_a_polynomial |
In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a faithful, finite-dimensional representation over K). Any finite group is linear, becau... | https://en.wikipedia.org/wiki/Matrix_group |
In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes the behavior of a set of matrices within the framework of quantum mech... | https://en.wikipedia.org/wiki/M-theory |
In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional supergravity. These calculations led them to propose that the BFSS matrix model is exactly equivalent to M-theory. The BFSS matrix model can therefore be used as a prototy... | https://en.wikipedia.org/wiki/M-theory |
In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes the behavior of a set of matrices within the framework of quantum mech... | https://en.wikipedia.org/wiki/Gauge–gravity_duality |
In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional supergravity. These calculations led them to propose that the BFSS matrix model is exactly equivalent to M-theory. The BFSS matrix model can therefore be used as a prototy... | https://en.wikipedia.org/wiki/Gauge–gravity_duality |
This subject is a generalization of ordinary geometry in which mathematicians define new geometric notions using tools from noncommutative algebra. In a paper from 1998, Alain Connes, Michael R. Douglas, and Albert Schwarz showed that some aspects of matrix models and M-theory are described by a noncommutative quantum ... | https://en.wikipedia.org/wiki/Gauge–gravity_duality |
In mathematics, a matrix is conformable if its dimensions are suitable for defining some operation (e.g. addition, multiplication, etc.). | https://en.wikipedia.org/wiki/Conformable_matrix |
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). | https://en.wikipedia.org/wiki/Spectral_norm |
In mathematics, a matrix of ones or all-ones matrix is a matrix where every entry is equal to one. Examples of standard notation are given below: J 2 = ( 1 1 1 1 ) ; J 3 = ( 1 1 1 1 1 1 1 1 1 ) ; J 2 , 5 = ( 1 1 1 1 1 1 1 1 1 1 ) ; J 1 , 2 = ( 1 1 ) . {\displaystyle J_{2}={\begin{pmatrix}1&1\\1&1\end{pmatrix}};\quad J_... | https://en.wikipedia.org/wiki/Matrix_of_ones |
In mathematics, a matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial P ( x ) = ∑ i = 0 n a i x i = a 0 + a 1 x + a 2 x 2 + ⋯ + a n x n , {\displaystyle P(x)=\sum _{i=0}^{n}{a_{i}x^{i}}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},} this polynomial evaluated at... | https://en.wikipedia.org/wiki/Matrix_geometrical_series |
In mathematics, a matroid polytope, also called a matroid basis polytope (or basis matroid polytope) to distinguish it from other polytopes derived from a matroid, is a polytope constructed via the bases of a matroid. Given a matroid M {\displaystyle M} , the matroid polytope P M {\displaystyle P_{M}} is the convex hul... | https://en.wikipedia.org/wiki/Matroid_polytope |
In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the classification of Lie groups and especially semi-simple Lie groups. Maximal compact subgrou... | https://en.wikipedia.org/wiki/Maximal_compact_subgroup |
In mathematics, a meander or closed meander is a self-avoiding closed curve which crosses a given line a number of times, meaning that it intersects the line while passing from one side to the other. Intuitively, a meander can be viewed as a meandering river with a straight road crossing the river over a number of brid... | https://en.wikipedia.org/wiki/Meander_(mathematics) |
In mathematics, a measurable acting group is a special group that acts on some space in a way that is compatible with structures of measure theory. Measurable acting groups are found in the intersection of measure theory and group theory, two sub-disciplines of mathematics. Measurable acting groups are the basis for th... | https://en.wikipedia.org/wiki/Measurable_group_action |
In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal κ, or more generally on any set. For a cardinal κ, it can be described as a subdivision of all of its subsets into large and small sets such that κ itself i... | https://en.wikipedia.org/wiki/Measurable_cardinal |
In mathematics, a measurable group is a special type of group in the intersection between group theory and measure theory. Measurable groups are used to study measures is an abstract setting and are often closely related to topological groups. | https://en.wikipedia.org/wiki/Measurable_group |
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. | https://en.wikipedia.org/wiki/Measurable_space |
In mathematics, a measure algebra is a Boolean algebra with a countably additive positive measure. A probability measure on a measure space gives a measure algebra on the Boolean algebra of measurable sets modulo null sets. | https://en.wikipedia.org/wiki/Measure_algebra |
In mathematics, a measure is said to be saturated if every locally measurable set is also measurable. A set E {\displaystyle E} , not necessarily measurable, is said to be a locally measurable set if for every measurable set A {\displaystyle A} of finite measure, E ∩ A {\displaystyle E\cap A} is measurable. σ {\display... | https://en.wikipedia.org/wiki/Locally_measurable_set |
In mathematics, a measure on a real vector space is said to be transverse to a given set if it assigns measure zero to every translate of that set, while assigning finite and positive (i.e. non-zero) measure to some compact set. | https://en.wikipedia.org/wiki/Transverse_measure |
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special case of conservative systems. They provide the formal, mathematical basis for... | https://en.wikipedia.org/wiki/Measure_preserving_dynamical_system |
In mathematics, a metabelian group is a group whose commutator subgroup is abelian. Equivalently, a group G is metabelian if and only if there is an abelian normal subgroup A such that the quotient group G/A is abelian. Subgroups of metabelian groups are metabelian, as are images of metabelian groups over group homomor... | https://en.wikipedia.org/wiki/Metabelian_group |
In mathematics, a metasymplectic space, introduced by Freudenthal (1959) and Tits (1974, 10.13), is a Tits building of type F4 (a specific generalized incidence structure). The four types of vertices are called points, lines, planes, and symplecta. | https://en.wikipedia.org/wiki/Metasymplectic_space |
In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. This is equivalent to: A connection for which the covariant derivativ... | https://en.wikipedia.org/wiki/Riemannian_connection |
In this case, the bundle E is the tangent bundle TM of a manifold, and the metric on E is induced by a Riemannian metric on M. Another special case of a metric connection is a Yang–Mills connection, which satisfies the Yang–Mills equations of motion. Most of the machinery of defining a connection and its curvature can ... | https://en.wikipedia.org/wiki/Riemannian_connection |
In mathematics, a metric outer measure is an outer measure μ defined on the subsets of a given metric space (X, d) such that μ ( A ∪ B ) = μ ( A ) + μ ( B ) {\displaystyle \mu (A\cup B)=\mu (A)+\mu (B)} for every pair of positively separated subsets A and B of X. | https://en.wikipedia.org/wiki/Metric_outer_measure |
In mathematics, a metric space X with metric d is said to be doubling if there is some doubling constant M > 0 such that for any x ∈ X and r > 0, it is possible to cover the ball B(x, r) = {y | d(x, y) < r} with the union of at most M balls of radius r/2. The base-2 logarithm of M is called the doubling dimension of X.... | https://en.wikipedia.org/wiki/Doubling_dimension |
In mathematics, a metric space aimed at its subspace is a categorical construction that has a direct geometric meaning. It is also a useful step toward the construction of the metric envelope, or tight span, which are basic (injective) objects of the category of metric spaces. Following (Holsztyński 1966), a notion of ... | https://en.wikipedia.org/wiki/Metric_space_aimed_at_its_subspace |
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The mos... | https://en.wikipedia.org/wiki/Homogeneous_metric |
Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of character... | https://en.wikipedia.org/wiki/Homogeneous_metric |
Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and therefore admit the structure of a metric space, including Riemannian manifolds, normed vector spaces, and graphs. In abstract algebra, the p-adi... | https://en.wikipedia.org/wiki/Homogeneous_metric |
In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined. Much of analysis happens in some metric space; the most commonly used are the real line, the complex plane, Euclidean space, other vector spaces, and the integers. Examples of analysis without a ... | https://en.wikipedia.org/wiki/Hard_analysis |
In mathematics, a microbundle is a generalization of the concept of vector bundle, introduced by the American mathematician John Milnor in 1964. It allows the creation of bundle-like objects in situations where they would not ordinarily be thought to exist. For example, the tangent bundle is defined for a smooth manifo... | https://en.wikipedia.org/wiki/Microbundle |
In mathematics, a minimal K-type is a representation of a maximal compact subgroup K of a semisimple Lie group G that is in some sense the smallest representation of K occurring in a Harish-Chandra module of G. Minimal K-types were introduced by Vogan (1979) as part of an algebraic description of the Langlands classifi... | https://en.wikipedia.org/wiki/Minimal_K-type |
In mathematics, a minimal counterexample is the smallest example which falsifies a claim, and a proof by minimal counterexample is a method of proof which combines the use of a minimal counterexample with the ideas of proof by induction and proof by contradiction. More specifically, in trying to prove a proposition P, ... | https://en.wikipedia.org/wiki/Minimal_counterexample |
In which case, there may be multiple and more complex ways to structure the argument of the proof. The assumption that if there is a counterexample, there is a minimal counterexample, is based on a well-ordering of some kind. The usual ordering on the natural numbers is clearly possible, by the most usual formulation o... | https://en.wikipedia.org/wiki/Minimal_counterexample |
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models o... | https://en.wikipedia.org/wiki/Minimal_surfaces |
In mathematics, a minimal surface of revolution or minimum surface of revolution is a surface of revolution defined from two points in a half-plane, whose boundary is the axis of revolution of the surface. It is generated by a curve that lies in the half-plane and connects the two points; among all the surfaces that ca... | https://en.wikipedia.org/wiki/Minimal_surfaces_of_revolution |
In mathematics, a minimum bottleneck spanning tree (MBST) in an undirected graph is a spanning tree in which the most expensive edge is as cheap as possible. A bottleneck edge is the highest weighted edge in a spanning tree. A spanning tree is a minimum bottleneck spanning tree if the graph does not contain a spanning ... | https://en.wikipedia.org/wiki/Minimum_bottleneck_spanning_tree |
In mathematics, a mixed boundary condition for a partial differential equation defines a boundary value problem in which the solution of the given equation is required to satisfy different boundary conditions on disjoint parts of the boundary of the domain where the condition is stated. Precisely, in a mixed boundary v... | https://en.wikipedia.org/wiki/Mixed_boundary_condition |
In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight 1/2. The first examples of mock theta functions were described by Srinivasa Ramanujan in his last 1920 letter to G. H. Hardy and in his lost notebook. Sander ... | https://en.wikipedia.org/wiki/Mock_theta_functions |
In mathematics, a modular Lie algebra is a Lie algebra over a field of positive characteristic. The theory of modular Lie algebras is significantly different from the theory of real and complex Lie algebras. This difference can be traced to the properties of Frobenius automorphism and to the failure of the exponential ... | https://en.wikipedia.org/wiki/Modular_Lie_algebra |
In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problems. That is, given a number of functions on a moduli space, a modular equation is an equation holding between them, or in other words an identity for moduli. The most frequent use of the term modular equation i... | https://en.wikipedia.org/wiki/Modular_equation |
That implies that any two rational functions F and G, in the function field of the modular curve, will satisfy a modular equation P(F,G) = 0 with P a non-zero polynomial of two variables over the complex numbers. For suitable non-degenerate choice of F and G, the equation P(X,Y) = 0 will actually define the modular cur... | https://en.wikipedia.org/wiki/Modular_equation |
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