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There is a correspondence between certain systems of partial differential equations (linear and having very special properties for their solutions) and possible monodromies of their solutions. Such a result was proved for algebraic connections with regular singularities by Pierre Deligne (1970, generalizing existing wo... | https://en.wikipedia.org/wiki/Riemann-Hilbert_correspondence |
In mathematics, the term adjoint applies in several situations. Several of these share a similar formalism: if A is adjoint to B, then there is typically some formula of the type (Ax, y) = (x, By).Specifically, adjoint or adjunction may mean: Adjoint of a linear map, also called its transpose Hermitian adjoint (adjoint... | https://en.wikipedia.org/wiki/Adjoint |
In mathematics, the term chaos game originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it. The fractal is created by iteratively creating a sequence of points, starting with the initial random point, in which each point in the sequence is a given fracti... | https://en.wikipedia.org/wiki/Chaos_game |
As the number of points is increased to a number N, the arrangement forms a corresponding (N-1)-dimensional Sierpinski Simplex. The term has been generalized to refer to a method of generating the attractor, or the fixed point, of any iterated function system (IFS). Starting with any point x0, successive iterations are... | https://en.wikipedia.org/wiki/Chaos_game |
The iterations converge to the fixed point of the IFS. Whenever x0 belongs to the attractor of the IFS, all iterations xk stay inside the attractor and, with probability 1, form a dense set in the latter. The "chaos game" method plots points in random order all over the attractor. | https://en.wikipedia.org/wiki/Chaos_game |
This is in contrast to other methods of drawing fractals, which test each pixel on the screen to see whether it belongs to the fractal. The general shape of a fractal can be plotted quickly with the "chaos game" method, but it may be difficult to plot some areas of the fractal in detail. With the aid of the "chaos game... | https://en.wikipedia.org/wiki/Chaos_game |
In mathematics, the term combinatorial proof is often used to mean either of two types of mathematical proof: A proof by double counting. A combinatorial identity is proven by counting the number of elements of some carefully chosen set in two different ways to obtain the different expressions in the identity. Since th... | https://en.wikipedia.org/wiki/Combinatorial_proof |
A bijective proof. Two sets are shown to have the same number of members by exhibiting a bijection, i.e. a one-to-one correspondence, between them.The term "combinatorial proof" may also be used more broadly to refer to any kind of elementary proof in combinatorics. However, as Glass (2003) writes in his review of Benj... | https://en.wikipedia.org/wiki/Combinatorial_proof |
In mathematics, the term cosocle (socle meaning pedestal in French) has several related meanings. In group theory, a cosocle of a group G, denoted by Cosoc(G), is the intersection of all maximal normal subgroups of G. If G is a quasisimple group, then Cosoc(G) = Z(G).In the context of Lie algebras, a cosocle of a symme... | https://en.wikipedia.org/wiki/Cosocle |
In mathematics, the term essentially unique is used to describe a weaker form of uniqueness, where an object satisfying a property is "unique" only in the sense that all objects satisfying the property are equivalent to each other. The notion of essential uniqueness presupposes some form of "sameness", which is often f... | https://en.wikipedia.org/wiki/Essentially_unique |
In mathematics, the term fiber (US English) or fibre (British English) can have two meanings, depending on the context: In naive set theory, the fiber of the element y {\displaystyle y} in the set Y {\displaystyle Y} under a map f: X → Y {\displaystyle f:X\to Y} is the inverse image of the singleton { y } {\displaystyl... | https://en.wikipedia.org/wiki/Fiber_(mathematics) |
In mathematics, the term linear function refers to two distinct but related notions: In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For distinguishing such a linear function from the other concept, the term affine func... | https://en.wikipedia.org/wiki/Linear_growth |
In mathematics, the term linear is used in two distinct senses for two different properties: linearity of a function (or mapping ); linearity of a polynomial.An example of a linear function is the function defined by f ( x ) = ( a x , b x ) {\displaystyle f(x)=(ax,bx)} that maps the real line to a line in the Euclidean... | https://en.wikipedia.org/wiki/Linearity |
Examples in physics include the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships, such as between velocity and kinetic energy, are nonlinear. Generalized for functions in more than one dimension, linea... | https://en.wikipedia.org/wiki/Linearity |
Linearity of a polynomial means that its degree is less than two. The use of the term for polynomials stems from the fact that the graph of a polynomial in one variable is a straight line. In the term "linear equation", the word refers to the linearity of the polynomials involved. | https://en.wikipedia.org/wiki/Linearity |
Because a function such as f ( x ) = a x + b {\displaystyle f(x)=ax+b} is defined by a linear polynomial in its argument, it is sometimes also referred to as being a "linear function", and the relationship between the argument and the function value may be referred to as a "linear relationship". This is potentially con... | https://en.wikipedia.org/wiki/Linearity |
In mathematics, the term local analysis has at least two meanings, both derived from the idea of looking at a problem relative to each prime number p first, and then later trying to integrate the information gained at each prime into a 'global' picture. These are forms of the localization approach. | https://en.wikipedia.org/wiki/Local_analysis |
In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra. In group theory, a maximal subgroup H of a group G is a proper subgroup, such that no proper subgroup K contains H strictly. In other words, H is a maximal element of the partially ordered set of subgroups... | https://en.wikipedia.org/wiki/Maximal_subgroup |
In mathematics, the term modulo ("with respect to a modulus of", the Latin ablative of modulus which itself means "a small measure") is often used to assert that two distinct mathematical objects can be regarded as equivalent—if their difference is accounted for by an additional factor. It was initially introduced into... | https://en.wikipedia.org/wiki/Modulo_(mathematics) |
In mathematics, the term permutation representation of a (typically finite) group G {\displaystyle G} can refer to either of two closely related notions: a representation of G {\displaystyle G} as a group of permutations, or as a group of permutation matrices. The term also refers to the combination of the two. | https://en.wikipedia.org/wiki/Permutation_representation |
In mathematics, the term simple is used to describe an algebraic structure which in some sense cannot be divided by a smaller structure of the same type. Put another way, an algebraic structure is simple if the kernel of every homomorphism is either the whole structure or a single element. Some examples are: A group is... | https://en.wikipedia.org/wiki/Simple_(abstract_algebra) |
A module is called a simple module if it does not contain a nontrivial submodule. An algebra is called a simple algebra if it does not contain a nontrivial two sided ideal.The general pattern is that the structure admits no non-trivial congruence relations. | https://en.wikipedia.org/wiki/Simple_(abstract_algebra) |
The term is used differently in semigroup theory. A semigroup is said to be simple if it has no nontrivial ideals, or equivalently, if Green's relation J is the universal relation. Not every congruence on a semigroup is associated with an ideal, so a simple semigroup may have nontrivial congruences. A semigroup with no... | https://en.wikipedia.org/wiki/Simple_(abstract_algebra) |
In mathematics, the term socle has several related meanings. | https://en.wikipedia.org/wiki/Socle_of_a_module |
In mathematics, the term standard L-function refers to a particular type of automorphic L-function described by Robert P. Langlands. Here, standard refers to the finite-dimensional representation r being the standard representation of the L-group as a matrix group. | https://en.wikipedia.org/wiki/Standard_L-function |
In mathematics, the term undefined is often used to refer to an expression which is not assigned an interpretation or a value (such as an indeterminate form, which has the possibility of assuming different values). The term can take on several different meanings depending on the context. For example: In various branche... | https://en.wikipedia.org/wiki/Undefined_(mathematics) |
As these terms are not defined in terms of other concepts, they may be referred to as "undefined terms". A function is said to be "undefined" at points outside of its domain – for example, the real-valued function f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} is undefined for negative x {\displaystyle x} (i.e., it assig... | https://en.wikipedia.org/wiki/Undefined_(mathematics) |
In mathematics, the term variational analysis usually denotes the combination and extension of methods from convex optimization and the classical calculus of variations to a more general theory. This includes the more general problems of optimization theory, including topics in set-valued analysis, e.g. generalized der... | https://en.wikipedia.org/wiki/Variational_analysis |
In mathematics, the term weak inverse is used with several meanings. | https://en.wikipedia.org/wiki/Weak_inverse |
In mathematics, the terms continuity, continuous, and continuum are used in a variety of related ways. | https://en.wikipedia.org/wiki/List_of_continuity-related_mathematical_topics |
In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broadest of the "Bertini theorems" applying to a linear system of divisors; sim... | https://en.wikipedia.org/wiki/Bertini's_theorem |
In mathematics, the theorem of the cube is a condition for a line bundle over a product of three complete varieties to be trivial. It was a principle discovered, in the context of linear equivalence, by the Italian school of algebraic geometry. The final version of the theorem of the cube was first published by Lang (1... | https://en.wikipedia.org/wiki/Theorem_of_the_cube |
In mathematics, the theory of Latin squares is an active research area with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Problems posed here appeared in, for instance, the Loops (Prague) conferences and the Milehigh (Denver) conferen... | https://en.wikipedia.org/wiki/Problems_in_Latin_squares |
In mathematics, the theory of fiber bundles with a structure group G {\displaystyle G} (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F 1 {\displaystyle F_{1}} to F 2 {\displaystyle F_{2}} , which are both topological spaces with a group a... | https://en.wikipedia.org/wiki/Associated_bundle |
In mathematics, the theory of finite sphere packing concerns the question of how a finite number of equally-sized spheres can be most efficiently packed. The question of packing finitely many spheres has only been investigated in detail in recent decades, with much of the groundwork being laid by László Fejes Tóth. The... | https://en.wikipedia.org/wiki/Finite_sphere_packing |
Atoms in crystal structures can be simplistically viewed as closely-packed spheres and treated as infinite sphere packings thanks to their large number. Sphere packing problems are distinguished between packings in given containers and free packings. This article primarily discusses free packings. | https://en.wikipedia.org/wiki/Finite_sphere_packing |
In mathematics, the theory of optimal stopping or early stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related... | https://en.wikipedia.org/wiki/Optimal_Stopping |
In mathematics, the theta correspondence or Howe correspondence is a mathematical relation between representations of two groups of a reductive dual pair. The local theta correspondence relates irreducible admissible representations over a local field, while the global theta correspondence relates irreducible automorph... | https://en.wikipedia.org/wiki/Theta_correspondence |
In mathematics, the theta divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers (and principally polarized) by the zero locus of the associated Riemann theta-function. It is therefore an algebraic subvariety of A of dimension dim A − 1. | https://en.wikipedia.org/wiki/Riemann–Kempf_singularity_theorem |
In mathematics, the theta function of a lattice is a function whose coefficients give the number of vectors of a given norm. | https://en.wikipedia.org/wiki/Theta_function_of_a_lattice |
In mathematics, the theta operator is a differential operator defined by θ = z d d z . {\displaystyle \theta =z{d \over dz}.} This is sometimes also called the homogeneity operator, because its eigenfunctions are the monomials in z: θ ( z k ) = k z k , k = 0 , 1 , 2 , … {\displaystyle \theta (z^{k})=kz^{k},\quad k=0,1,... | https://en.wikipedia.org/wiki/Theta_operator |
{\displaystyle \theta =\sum _{k=1}^{n}x_{k}{\frac {\partial }{\partial x_{k}}}.} As in one variable, the eigenspaces of θ are the spaces of homogeneous functions. (Euler's homogeneous function theorem) | https://en.wikipedia.org/wiki/Theta_operator |
In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics. It gains its name from the fact that the Jacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg group. The representation was popularized by David Mumford. | https://en.wikipedia.org/wiki/Theta_representation |
In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music. | https://en.wikipedia.org/wiki/Pythagorean_mean |
In mathematics, the three spheres inequality bounds the L 2 {\displaystyle L^{2}} norm of a harmonic function on a given sphere in terms of the L 2 {\displaystyle L^{2}} norm of this function on two spheres, one with bigger radius and one with smaller radius. | https://en.wikipedia.org/wiki/Three_spheres_inequality |
In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places n points on a circle, at angles of θ, 2θ, 3θ, ... from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. When there are ... | https://en.wikipedia.org/wiki/Three-gap_theorem |
In mathematics, the tilde operator (which can be represented by a tilde or the dedicated character U+223C ∼ TILDE OPERATOR), sometimes called "twiddle", is often used to denote an equivalence relation between two objects. Thus "x ~ y" means "x is equivalent to y". It is a weaker statement than stating that x equals y. ... | https://en.wikipedia.org/wiki/~ |
It can be used to denote the asymptotic equality of two functions. For example, f (x) ~ g(x) means that lim x → ∞ f ( x ) g ( x ) = 1 {\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=1} .A tilde is also used to indicate "approximately equal to" (e.g. 1.902 ~= 2). This usage probably developed as a typed alternat... | https://en.wikipedia.org/wiki/~ |
The symbol "≈" is also used for this purpose. In physics and astronomy, a tilde can be used between two expressions (e.g. h ~ 10−34 J s) to state that the two are of the same order of magnitude.In statistics and probability theory, the tilde means "is distributed as"; see random variable(e.g. X ~ B(n,p) for a binomial ... | https://en.wikipedia.org/wiki/~ |
A triple tilde (≋) is often used to show congruence, an equivalence relation in geometry. In graph theory, the tilde can be used to represent adjacency between vertices. The edge ( x , y ) {\displaystyle (x,y)} connects vertices x {\displaystyle x} and y {\displaystyle y} which can be said to be adjacent, and this adja... | https://en.wikipedia.org/wiki/~ |
In mathematics, the tombstone, halmos, end-of-proof, or Q.E.D. symbol "∎" (or "□") is a symbol used to denote the end of a proof, in place of the traditional abbreviation "Q.E.D." for the Latin phrase "quod erat demonstrandum". It is inspired by the typographic practice of end marks, an element that marks the end of an... | https://en.wikipedia.org/wiki/Halmos_box |
Its graphic form varies, as it may be a hollow or filled rectangle or square. In AMS-LaTeX, the symbol is automatically appended at the end of a proof environment \begin{proof} ... \end{proof}. It can also be obtained from the commands \qedsymbol, \qedhere or \qed (the latter causes the symbol to be right aligned).It i... | https://en.wikipedia.org/wiki/Halmos_box |
He got the idea of using it from seeing end marks in magazines, that is, typographic signs that indicate the end of an article. In his memoir I Want to Be a Mathematician, he wrote the following: The symbol is definitely not my invention — it appeared in popular magazines (not mathematical ones) before I adopted it, bu... | https://en.wikipedia.org/wiki/Halmos_box |
In mathematics, the topological entropy of a topological dynamical system is a nonnegative extended real number that is a measure of the complexity of the system. Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew. Their definition was modelled after the definition of the Kolmogorov–Sinai, ... | https://en.wikipedia.org/wiki/Topological_entropy |
Later, Dinaburg and Rufus Bowen gave a different, weaker definition reminiscent of the Hausdorff dimension. The second definition clarified the meaning of the topological entropy: for a system given by an iterated function, the topological entropy represents the exponential growth rate of the number of distinguishable ... | https://en.wikipedia.org/wiki/Topological_entropy |
In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this... | https://en.wikipedia.org/wiki/Total_derivative |
In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function f, defined on an interval ⊂ R, its total variation on the interval of definition is a measure of the one-dimen... | https://en.wikipedia.org/wiki/Total_variation_norm |
In mathematics, the trace field of a linear group is the field generated by the traces of its elements. It is mostly studied for Kleinian and Fuchsian groups, though related objects are used in the theory of lattices in Lie groups, often under the name field of definition. | https://en.wikipedia.org/wiki/Trace_field |
In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial differential equations with prescribed boundary conditions (boundary value problems), where weak sol... | https://en.wikipedia.org/wiki/Trace_operator |
In mathematics, the transcendental law of homogeneity (TLH) is a heuristic principle enunciated by Gottfried Wilhelm Leibniz most clearly in a 1710 text entitled Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potentiarum et differentiarum, et de lege homogeneorum transcendentali. Henk J. ... | https://en.wikipedia.org/wiki/Transcendental_law_of_homogeneity |
In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. In all usual cases, the largest eigenvalue is 1, and the corresponding eigenvector is the invariant measure of the system.... | https://en.wikipedia.org/wiki/Bernoulli_operator |
In mathematics, the transitive closure R+ of a homogeneous binary relation R on a set X is the smallest relation on X that contains R and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite sets R+ is the unique minimal transitive superset of R. F... | https://en.wikipedia.org/wiki/Transitive_closure_logic |
337). We have R+ = R if, and only if, R itself is transitive. Conversely, transitive reduction adduces a minimal relation S from a given relation R such that they have the same closure, that is, S+ = R+; however, many different S with this property may exist. Both transitive closure and transitive reduction are also us... | https://en.wikipedia.org/wiki/Transitive_closure_logic |
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclud... | https://en.wikipedia.org/wiki/Segment_addition_postulate |
In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proved without these theorems. The inequality can be viewed intuitively in either R2 or R3. | https://en.wikipedia.org/wiki/Segment_addition_postulate |
The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shor... | https://en.wikipedia.org/wiki/Segment_addition_postulate |
In mathematics, the tricorn, sometimes called the Mandelbar set, is a fractal defined in a similar way to the Mandelbrot set, but using the mapping z ↦ z ¯ 2 + c {\displaystyle z\mapsto {\bar {z}}^{2}+c} instead of z ↦ z 2 + c {\displaystyle z\mapsto z^{2}+c} used for the Mandelbrot set. It was introduced by W. D. Crow... | https://en.wikipedia.org/wiki/Tricorn_(mathematics) |
In mathematics, the trigamma function, denoted ψ1(z) or ψ(1)(z), is the second of the polygamma functions, and is defined by ψ 1 ( z ) = d 2 d z 2 ln Γ ( z ) {\displaystyle \psi _{1}(z)={\frac {d^{2}}{dz^{2}}}\ln \Gamma (z)} .It follows from this definition that ψ 1 ( z ) = d d z ψ ( z ) {\displaystyle \psi _{1}(z)={... | https://en.wikipedia.org/wiki/Trigamma_function |
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics... | https://en.wikipedia.org/wiki/Cotangent_(trigonometric_function) |
Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions. | https://en.wikipedia.org/wiki/Cotangent_(trigonometric_function) |
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine and cosine functions to functions whose domain is the whole real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; the... | https://en.wikipedia.org/wiki/Cotangent_(trigonometric_function) |
In mathematics, the trigonometric moment problem is formulated as follows: given a finite sequence {α0, ... αn }, does there exist a positive Borel measure μ on the interval such that α k = 1 2 π ∫ 0 2 π e − i k t d μ ( t ) . {\displaystyle \alpha _{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }e^{-ikt}\,d\mu (t).} In other w... | https://en.wikipedia.org/wiki/Trigonometric_moment_problem |
In mathematics, the truncated power function with exponent n {\displaystyle n} is defined as x + n = { x n: x > 0 0: x ≤ 0. {\displaystyle x_{+}^{n}={\begin{cases}x^{n}&:\ x>0\\0&:\ x\leq 0.\end{cases}}} In particular, x + = { x: x > 0 0: x ≤ 0. {\displaystyle x_{+}={\begin{cases}x&:\ x>0\\0&:\ x\leq 0.\end{cases}}} an... | https://en.wikipedia.org/wiki/Truncated_power_function |
In mathematics, the tunnel number of a knot, as first defined by Bradd Clark, is a knot invariant, given by the minimal number of arcs (called tunnels) that must be added to the knot so that the complement becomes a handlebody. The tunnel number can equally be defined for links. The boundary of a regular neighbourhood ... | https://en.wikipedia.org/wiki/Tunnel_number |
In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local information, by means of a local coefficient system. | https://en.wikipedia.org/wiki/Twisted_Poincaré_duality |
In mathematics, the two families cλn(x;k) and Bλn(x;k) of sieved ultraspherical polynomials, introduced by Waleed Al-Salam, W.R. Allaway and Richard Askey in 1984, are the archetypal examples of sieved orthogonal polynomials. Their recurrence relations are a modified (or "sieved") version of the recurrence relations fo... | https://en.wikipedia.org/wiki/Sieved_ultraspherical_polynomials |
In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin transform, the Z-transform and the ordinary or one-sided Laplace transform... | https://en.wikipedia.org/wiki/Bilateral_Laplace_transform |
The integral is most commonly understood as an improper integral, which converges if and only if both integrals ∫ 0 ∞ e − s t f ( t ) d t , ∫ − ∞ 0 e − s t f ( t ) d t {\displaystyle \int _{0}^{\infty }e^{-st}f(t)\,dt,\quad \int _{-\infty }^{0}e^{-st}f(t)\,dt} exist. There seems to be no generally accepted notation for... | https://en.wikipedia.org/wiki/Bilateral_Laplace_transform |
{\displaystyle {\mathcal {T}}\{f\}(s)=s{\mathcal {B}}\{f\}(s)=sF(s)=s\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.} In pure mathematics the argument t can be any variable, and Laplace transforms are used to study how differential operators transform the function. In science and engineering applications, the argument t oft... | https://en.wikipedia.org/wiki/Bilateral_Laplace_transform |
In these cases, the signals are transformed by filters, that work like a mathematical operator, but with a restriction. They have to be causal, which means that the output in a given time t cannot depend on an output which is a higher value of t. In population ecology, the argument t often represents spatial displaceme... | https://en.wikipedia.org/wiki/Bilateral_Laplace_transform |
In mathematics, the uncertainty exponent is a method of measuring the fractal dimension of a basin boundary. In a chaotic scattering system, the invariant set of the system is usually not directly accessible because it is non-attracting and typically of measure zero. Therefore, the only way to infer the presence of mem... | https://en.wikipedia.org/wiki/Uncertainty_exponent |
Suppose we start with a random trajectory and perturb it by a small amount, ϵ {\displaystyle \epsilon } , in a random direction. If the new trajectory ends up in a different basin from the old one, then it is called epsilon uncertain. If we take a large number of such trajectories, then the fraction of them that are ep... | https://en.wikipedia.org/wiki/Uncertainty_exponent |
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous... | https://en.wikipedia.org/wiki/Banach–Steinhaus_theorem |
In mathematics, the uniform limit theorem states that the uniform limit of any sequence of continuous functions is continuous. | https://en.wikipedia.org/wiki/Uniform_limit_theorem |
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere. In particular it admits a Riemannian metric of constant curvature. This classifies Riemannian surfaces as ellip... | https://en.wikipedia.org/wiki/Low_dimensional_topology |
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the p... | https://en.wikipedia.org/wiki/Uniformization_theorem |
It further follows that every Riemann surface admits a Riemannian metric of constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case. The uniformization theorem also yields a similar classification of closed orientable Riemannian 2-manifolds into ell... | https://en.wikipedia.org/wiki/Uniformization_theorem |
In mathematics, the unit doublet is the derivative of the Dirac delta function. It can be used to differentiate signals in electrical engineering: If u1 is the unit doublet, then ( x ∗ u 1 ) ( t ) = d x ( t ) d t {\displaystyle (x*u_{1})(t)={\frac {dx(t)}{dt}}} where ∗ {\displaystyle *} is the convolution operator.The ... | https://en.wikipedia.org/wiki/Unit_doublet |
The function can be thought of as the limiting case of two rectangles, one in the second quadrant, and the other in the fourth. The length of each rectangle is k, whereas their breadth is 1/k2, where k tends to zero. == References == | https://en.wikipedia.org/wiki/Unit_doublet |
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted I (capital letter I). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology... | https://en.wikipedia.org/wiki/Closed_unit_interval |
In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For... | https://en.wikipedia.org/wiki/Unitary_symmetry |
In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1, under multiplication. All the unitary groups contain copies of this group. The unitary group U(n) is a real Lie group of dimension n2. The Lie algebra of U(n) consists of n × n skew-Hermiti... | https://en.wikipedia.org/wiki/Unitary_symmetry |
In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group G, is a specific bundle over a classifying space BG, such that every bundle with the given structure group G over M is a pullback by means of a continuous map M → BG. | https://en.wikipedia.org/wiki/Universal_bundle |
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be... | https://en.wikipedia.org/wiki/Universal_enveloping_algebra |
Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent ... | https://en.wikipedia.org/wiki/Universal_enveloping_algebra |
In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfand–Naimark theorem, to contain the C* algebra of the corresponding Lie group. This relationship generalizes to the idea of Tannaka–Krein duality between co... | https://en.wikipedia.org/wiki/Universal_enveloping_algebra |
In mathematics, the universal invariant or u-invariant of a field describes the structure of quadratic forms over the field. The universal invariant u(F) of a field F is the largest dimension of an anisotropic quadratic space over F, or ∞ if this does not exist. Since formally real fields have anisotropic quadratic for... | https://en.wikipedia.org/wiki/U-invariant |
In mathematics, the universality of zeta functions is the remarkable ability of the Riemann zeta function and other similar functions (such as the Dirichlet L-functions) to approximate arbitrary non-vanishing holomorphic functions arbitrarily well. The universality of the Riemann zeta function was first proven by Serge... | https://en.wikipedia.org/wiki/Zeta_function_universality |
In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram. There are several types of unknotting algorithms. A major unresolved challenge is to determine if the problem admits a polynomial time algorithm; that is, whether t... | https://en.wikipedia.org/wiki/Unknotting_problem |
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which are defined similarly to the gamma function but with different or "incompl... | https://en.wikipedia.org/wiki/Upper_incomplete_gamma_function |
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