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Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the Artin–Wedderburn theorem classifying semi-simple rings. Results on direct integrals can be viewed as generalizations of results about finite-dimensional C*-algebras of matrices; in this case the resu... | https://en.wikipedia.org/wiki/Decomposable_operator |
In mathematics and game theory, Bulgarian solitaire is a card game that was introduced by Martin Gardner. In the game, a pack of N {\displaystyle N} cards is divided into several piles. Then for each pile, remove one card; collect the removed cards together to form a new pile (piles of zero size are ignored). If N {\di... | https://en.wikipedia.org/wiki/Bulgarian_solitaire |
In mathematics and geographic information science, a shortest-path graph is an undirected graph defined from a set of points in the Euclidean plane. The shortest-path graph is proposed with the idea of inferring edges between a point set such that the shortest path taken over the inferred edges will roughly align with ... | https://en.wikipedia.org/wiki/Shortest-path_graph |
In mathematics and geometry, a discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type o... | https://en.wikipedia.org/wiki/Discrete_symmetry |
a topological group with a discrete topology whose elements form a finite or a countable set. One of the most prominent discrete symmetries in physics is parity symmetry. It manifests itself in various elementary physical quantum systems, such as quantum harmonic oscillator, electron orbitals of Hydrogen-like atoms by ... | https://en.wikipedia.org/wiki/Discrete_symmetry |
In mathematics and group theory, a block system for the action of a group G on a set X is a partition of X that is G-invariant. In terms of the associated equivalence relation on X, G-invariance means that x ~ y implies gx ~ gyfor all g ∈ G and all x, y ∈ X. The action of G on X induces a natural action of G on any blo... | https://en.wikipedia.org/wiki/Block_(permutation_group_theory) |
The partition into singleton sets is a block system and if X is non-empty then the partition into one set X itself is a block system as well (if X is a singleton set then these two partitions are identical). A transitive (and thus non-empty) G-set X is said to be primitive if it has no other block systems. For a non-em... | https://en.wikipedia.org/wiki/Block_(permutation_group_theory) |
In mathematics and group theory, the term multiplicative group refers to one of the following concepts: the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred to as multiplication. In the case of a field F, the group is (F ∖ {0}, •), wh... | https://en.wikipedia.org/wiki/Multiplicative_group |
In mathematics and in computer programming, a variadic function is a function of indefinite arity, i.e., one which accepts a variable number of arguments. Support for variadic functions differs widely among programming languages. The term variadic is a neologism, dating back to 1936–1937. The term was not widely used u... | https://en.wikipedia.org/wiki/Variadic_functions |
In mathematics and in particular in combinatorics, the Lehmer code is a particular way to encode each possible permutation of a sequence of n numbers. It is an instance of a scheme for numbering permutations and is an example of an inversion table. The Lehmer code is named in reference to Derrick Henry Lehmer, but the ... | https://en.wikipedia.org/wiki/Lehmer_code |
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topologic... | https://en.wikipedia.org/wiki/Measurable_function |
In mathematics and in particular the field of complex analysis, Hurwitz's theorem is a theorem associating the zeroes of a sequence of holomorphic, compact locally uniformly convergent functions with that of their corresponding limit. The theorem is named after Adolf Hurwitz. | https://en.wikipedia.org/wiki/Hurwitz's_theorem_(complex_analysis) |
In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. It is named after Marshall Stone and John von Neumann. | https://en.wikipedia.org/wiki/Stone–von_Neumann_theorem |
In mathematics and information theory of probability, a sigma-martingale is a semimartingale with an integral representation. Sigma-martingales were introduced by C.S. Chou and M. Emery in 1977 and 1978. In financial mathematics, sigma-martingales appear in the fundamental theorem of asset pricing as an equivalent cond... | https://en.wikipedia.org/wiki/Sigma-martingale |
In mathematics and information theory, Sanov's theorem gives a bound on the probability of observing an atypical sequence of samples from a given probability distribution. In the language of large deviations theory, Sanov's theorem identifies the rate function for large deviations of the empirical measure of a sequence... | https://en.wikipedia.org/wiki/Sanov's_theorem |
Let A be a set of probability distributions over an alphabet X, and let q be an arbitrary distribution over X (where q may or may not be in A). Suppose we draw n i.i.d. samples from q, represented by the vector x n = x 1 , x 2 , … , x n {\displaystyle x^{n}=x_{1},x_{2},\ldots ,x_{n}} . | https://en.wikipedia.org/wiki/Sanov's_theorem |
Then, we have the following bound on the probability that the empirical measure p ^ x n {\displaystyle {\hat {p}}_{x^{n}}} of the samples falls within the set A: q n ( p ^ x n ∈ A ) ≤ ( n + 1 ) | X | 2 − n D K L ( p ∗ | | q ) {\displaystyle q^{n}({\hat {p}}_{x^{n}}\in A)\leq (n+1)^{|X|}2^{-nD_{\mathrm {KL} }(p^{*}||q)}... | https://en.wikipedia.org/wiki/Sanov's_theorem |
In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form: for given functions p ( x ) {\displaystyle p(x)} , q ( x ) {\displaystyle q(x)} and w ( x ) {\displaystyle w(x)} , together with some boundary conditions at extreme values of x {\displayst... | https://en.wikipedia.org/wiki/Sturm–Liouville_eigenproblem |
Such functions y {\displaystyle y} are called the eigenfunctions associated to each λ.Sturm–Liouville theory is the general study of Sturm–Liouville problems. In particular, for a "regular" Sturm–Liouville problem, it can be shown that there are an infinite number of eigenvalues each with a unique eigenfunction, and th... | https://en.wikipedia.org/wiki/Sturm–Liouville_eigenproblem |
This theory is important in applied mathematics, where Sturm–Liouville problems occur very frequently, particularly when dealing with separable linear partial differential equations. For example, in quantum mechanics, the one-dimensional time-independent Schrödinger equation is a Sturm–Liouville problem. Sturm–Liouvill... | https://en.wikipedia.org/wiki/Sturm–Liouville_eigenproblem |
In mathematics and its applications, a parametric family or a parameterized family is a family of objects (a set of related objects) whose differences depend only on the chosen values for a set of parameters.Common examples are parametrized (families of) functions, probability distributions, curves, shapes, etc. | https://en.wikipedia.org/wiki/Parametric_family_of_functions |
In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem is a particular kind of boundary value problem for a system of partial differential equations (PDE), in which the boundary between the phases can move with time. The classical Stefan problem aims to describe the evolution... | https://en.wikipedia.org/wiki/Stefan_problem |
To close the mathematical system a further equation, the Stefan condition, is required. This is an energy balance which defines the position of the moving interface. Note that this evolving boundary is an unknown (hyper-)surface; hence, Stefan problems are examples of free boundary problems. Analogous problems occur, f... | https://en.wikipedia.org/wiki/Stefan_problem |
In mathematics and its applications, the mean square is normally defined as the arithmetic mean of the squares of a set of numbers or of a random variable.It may also be defined as the arithmetic mean of the squares of the deviations between a set of numbers and a reference value (e.g., may be a mean or an assumed mean... | https://en.wikipedia.org/wiki/Mean_square |
In mathematics and its applications, the root mean square of a set of numbers x i {\displaystyle x_{i}} (abbreviated as RMS, RMS or rms and denoted in formulas as either x R M S {\displaystyle x_{\mathrm {RMS} }} or R M S x {\displaystyle \mathrm {RMS} _{x}} ) is defined as the square root of the mean square (the arith... | https://en.wikipedia.org/wiki/Root_Mean_Square |
In mathematics and its applications, the signed distance function (or oriented distance function) is the orthogonal distance of a given point x to the boundary of a set Ω in a metric space, with the sign determined by whether or not x is in the interior of Ω. The function has positive values at points x inside Ω, it de... | https://en.wikipedia.org/wiki/Signed_distance_function |
In mathematics and logic, Ackermann set theory (AST) is an axiomatic set theory proposed by Wilhelm Ackermann in 1956.AST differs from Zermelo–Fraenkel set theory (ZF) in that it allows proper classes, that is, objects that are not sets, including a class of all sets. It replaces several of the standard ZF axioms for c... | https://en.wikipedia.org/wiki/Axiom_of_heredity |
In its use of classes, AST differs from other alternative set theories such as Morse–Kelley set theory and Von Neumann–Bernays–Gödel set theory in that a class may be an element of another class. William N. Reinhardt established in 1970 that AST is effectively equivalent in strength to ZF, putting it on equal foundatio... | https://en.wikipedia.org/wiki/Axiom_of_heredity |
In mathematics and logic, a corollary ( KORR-ə-lerr-ee, UK: korr-OL-ər-ee) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another proposition; it might also be used more ca... | https://en.wikipedia.org/wiki/Corollary |
In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually axioms, existing lemmas and theorems, without making any further assumptions. In order to directly prove a conditional statement of the form "If p, then... | https://en.wikipedia.org/wiki/Direct_proof |
Common proof rules used are modus ponens and universal instantiation.In contrast, an indirect proof may begin with certain hypothetical scenarios and then proceed to eliminate the uncertainties in each of these scenarios until an inescapable conclusion is forced. For example, instead of showing directly p ⇒ q, one prov... | https://en.wikipedia.org/wiki/Direct_proof |
In mathematics and logic, a higher-order logic (abbreviated HOL) is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expressive, but their model-theoretic properties are less we... | https://en.wikipedia.org/wiki/Ordered_logic |
Here "simple" indicates that the underlying type theory is the theory of simple types, also called the simple theory of types. Leon Chwistek and Frank P. Ramsey proposed this as a simplification of the complicated and clumsy ramified theory of types specified in the Principia Mathematica by Alfred North Whitehead and B... | https://en.wikipedia.org/wiki/Ordered_logic |
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. It is sometimes said that a statement is vacuously true because it does not really say anything. For example... | https://en.wikipedia.org/wiki/Vacuous_truth |
More formally, a relatively well-defined usage refers to a conditional statement (or a universal conditional statement) with a false antecedent. One example of such a statement is "if Tokyo is in France, then the Eiffel Tower is in Bolivia". Such statements are considered vacuous truths, because the fact that the antec... | https://en.wikipedia.org/wiki/Vacuous_truth |
In essence, a conditional statement, that is based on the material conditional, is true when the antecedent ("Tokyo is in France" in the example) is false regardless of whether the conclusion or consequent ("the Eiffel Tower is in Bolivia" in the example) is true or false because the material conditional is defined in ... | https://en.wikipedia.org/wiki/Vacuous_truth |
This notion has relevance in pure mathematics, as well as in any other field that uses classical logic. Outside of mathematics, statements which can be characterized informally as vacuously true can be misleading. Such statements make reasonable assertions about qualified objects which do not actually exist. | https://en.wikipedia.org/wiki/Vacuous_truth |
For example, a child might truthfully tell their parent "I ate every vegetable on my plate", when there were no vegetables on the child's plate to begin with. In this case, the parent can believe that the child has actually eaten some vegetables, even though that is not true. In addition, a vacuous truth is often used ... | https://en.wikipedia.org/wiki/Vacuous_truth |
In mathematics and logic, ambiguity can be considered to be an instance of the logical concept of underdetermination—for example, X = Y {\displaystyle X=Y} leaves open what the value of X is—while its opposite is a self-contradiction, also called inconsistency, paradoxicalness, or oxymoron, or in mathematics an inconsi... | https://en.wikipedia.org/wiki/Lexical_ambiguity |
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system tha... | https://en.wikipedia.org/wiki/Axiomatic_proof |
In mathematics and logic, an operation is finitary if it has finite arity, i.e. if it has a finite number of input values. Similarly, an infinitary operation is one with an infinite number of input values. In standard mathematics, an operation is finitary by definition. Therefore these terms are usually only used in th... | https://en.wikipedia.org/wiki/Infinitary_operation |
In mathematics and logic, effective is used to describe metalogical methods that fit the criteria of an effective procedure. In group theory, a group element acts effectively (or faithfully) on a point, if that point is not fixed by the action. In physics, an effective theory is, similar to a phenomenological theory, a... | https://en.wikipedia.org/wiki/Effectiveness |
In heat transfer, effectiveness is a measure of the performance of a heat exchanger when using the NTU method. In medicine, effectiveness relates to how well a treatment works in practice, especially as shown in pragmatic clinical trials, as opposed to efficacy, which measures how well it works in explanatory clinical ... | https://en.wikipedia.org/wiki/Effectiveness |
In mathematics and logic, plural quantification is the theory that an individual variable x may take on plural, as well as singular, values. As well as substituting individual objects such as Alice, the number 1, the tallest building in London etc. for x, we may substitute both Alice and Bob, or all the numbers between... | https://en.wikipedia.org/wiki/Plural_quantification |
In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!" or "∃=1". For example, the formal sta... | https://en.wikipedia.org/wiki/Uniqueness_quantification |
In mathematics and mathematical biology, the Mackey–Glass equations, named after Michael Mackey and Leon Glass, refer to a family of delay differential equations whose behaviour manages to mimic both healthy and pathological behaviour in certain biological contexts, controlled by the equation's parameters. Originally, ... | https://en.wikipedia.org/wiki/Mackey–Glass_equations |
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. | https://en.wikipedia.org/wiki/Boolean_value |
Second, Boolean algebra uses logical operators such as conjunction (and) denoted as ∧, disjunction (or) denoted as ∨, and the negation (not) denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction, and division. Boolean algebra is therefore a formal w... | https://en.wikipedia.org/wiki/Boolean_value |
Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847), and set forth more fully in his An Investigation of the Laws of Thought (1854). According to Huntington, the term "Boolean algebra" was first suggested by Henry M. Sheffer in 1913, although Charles Sanders Peirce... | https://en.wikipedia.org/wiki/Boolean_value |
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel). I... | https://en.wikipedia.org/wiki/Infimal_convolution |
In mathematics and mathematical physics, Slater integrals are certain integrals of products of three spherical harmonics. They occur naturally when applying an orthonormal basis of functions on the unit sphere that transform in a particular way under rotations in three dimensions. Such integrals are particularly useful... | https://en.wikipedia.org/wiki/Slater_integrals |
In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold M is a set of basis vector fields {e1, ..., en} defined at every point P of a region of the manifold as e α = lim δ x α → 0 δ s δ x α , {\displaystyle \mathbf {e} _{\alpha }=\lim _{\delta x^{\alpha }\to 0}{\frac... | https://en.wikipedia.org/wiki/Holonomic_basis |
Since we have that u = uαeα, the identification is often made between a coordinate basis vector eα and the partial derivative operator ∂/∂xα, under the interpretation of vectors as operators acting on functions.A local condition for a basis {e1, ..., en} to be holonomic is that all mutual Lie derivatives vanish: = L e... | https://en.wikipedia.org/wiki/Holonomic_basis |
In mathematics and mathematical physics, a factorization algebra is an algebraic structure first introduced by Beilinson and Drinfel'd in an algebro-geometric setting as a reformulation of chiral algebras, and also studied in a more general setting by Costello to study quantum field theory. | https://en.wikipedia.org/wiki/Factorization_algebra |
In mathematics and mathematical physics, complex spacetime extends the traditional notion of spacetime described by real-valued space and time coordinates to complex-valued space and time coordinates. The notion is entirely mathematical with no physics implied, but should be seen as a tool, for instance, as exemplified... | https://en.wikipedia.org/wiki/Complex_spacetime |
In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called t... | https://en.wikipedia.org/wiki/Probabilistic_potential_theory |
For example, a result about the singularities of harmonic functions would be said to belong to potential theory whilst a result on how the solution depends on the boundary data would be said to belong to the theory of the Laplace equation. This is not a hard and fast distinction, and in practice there is considerable o... | https://en.wikipedia.org/wiki/Probabilistic_potential_theory |
In the continuous case, this is closely related to analytic theory. In the finite state space case, this connection can be introduced by introducing an electrical network on the state space, with resistance between points inversely proportional to transition probabilities and densities proportional to potentials. Even ... | https://en.wikipedia.org/wiki/Probabilistic_potential_theory |
In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions. | https://en.wikipedia.org/wiki/Raising_and_lowering_indices |
In mathematics and mechanics, the Euler–Rodrigues formula describes the rotation of a vector in three dimensions. It is based on Rodrigues' rotation formula, but uses a different parametrization. The rotation is described by four Euler parameters due to Leonhard Euler. The Rodrigues formula (named after Olinde Rodrigue... | https://en.wikipedia.org/wiki/Euler–Rodrigues_formula |
In mathematics and more precisely in functional analysis, the Aluthge transformation is an operation defined on the set of bounded operators of a Hilbert space. It was introduced by Ariyadasa Aluthge to study p-hyponormal linear operators. | https://en.wikipedia.org/wiki/Aluthge_transform |
In mathematics and more precisely in group theory, the commuting probability (also called degree of commutativity or commutativity degree) of a finite group is the probability that two randomly chosen elements commute. It can be used to measure how close to abelian a finite group is. It can be generalized to infinite g... | https://en.wikipedia.org/wiki/Commuting_probability |
In mathematics and more specifically in field theory, a radical extension of a field K is an extension of K that is obtained by adjoining a sequence of nth roots of elements. | https://en.wikipedia.org/wiki/Simple_radical_extension |
In mathematics and multivariate statistics, the centering matrix is a symmetric and idempotent matrix, which when multiplied with a vector has the same effect as subtracting the mean of the components of the vector from every component of that vector. | https://en.wikipedia.org/wiki/Centering_matrix |
In mathematics and numerical analysis, an adaptive step size is used in some methods for the numerical solution of ordinary differential equations (including the special case of numerical integration) in order to control the errors of the method and to ensure stability properties such as A-stability. Using an adaptive ... | https://en.wikipedia.org/wiki/Adaptive_stepsize |
However things are more difficult if one wishes to model the motion of a spacecraft taking into account both the Earth and the Moon as in the Three-body problem. There, scenarios emerge where one can take large time steps when the spacecraft is far from the Earth and Moon, but if the spacecraft gets close to colliding ... | https://en.wikipedia.org/wiki/Adaptive_stepsize |
In mathematics and numerical analysis, the Ricker wavelet ψ ( t ) = 2 3 σ π 1 / 4 ( 1 − ( t σ ) 2 ) e − t 2 2 σ 2 {\displaystyle \psi (t)={\frac {2}{{\sqrt {3\sigma }}\pi ^{1/4}}}\left(1-\left({\frac {t}{\sigma }}\right)^{2}\right)e^{-{\frac {t^{2}}{2\sigma ^{2}}}}} is the negative normalized second derivative of a Gau... | https://en.wikipedia.org/wiki/Mexican_hat_wavelet |
It is also known as the Marr wavelet for David Marr. ψ ( x , y ) = 1 π σ 4 ( 1 − 1 2 ( x 2 + y 2 σ 2 ) ) e − x 2 + y 2 2 σ 2 {\displaystyle \psi (x,y)={\frac {1}{\pi \sigma ^{4}}}\left(1-{\frac {1}{2}}\left({\frac {x^{2}+y^{2}}{\sigma ^{2}}}\right)\right)e^{-{\frac {x^{2}+y^{2}}{2\sigma ^{2}}}}} The multidimensional ge... | https://en.wikipedia.org/wiki/Mexican_hat_wavelet |
The scale normalized Laplacian (in L 1 {\displaystyle L_{1}} -norm) is frequently used as a blob detector and for automatic scale selection in computer vision applications; see Laplacian of Gaussian and scale space. The relation between this Laplacian of the Gaussian operator and the difference-of-Gaussians operator is... | https://en.wikipedia.org/wiki/Mexican_hat_wavelet |
In mathematics and numerical analysis, the van Wijngaarden transformation is a variant on the Euler transform used to accelerate the convergence of an alternating series. One algorithm to compute Euler's transform runs as follows: Compute a row of partial sums and form rows of averages between neighbors The first colum... | https://en.wikipedia.org/wiki/Van_Wijngaarden_transformation |
If a 0 , a 1 , … , a 12 {\displaystyle a_{0},a_{1},\ldots ,a_{12}} are available, then s 8 , 4 {\displaystyle s_{8,4}} is almost always a better approximation to the sum than s 12 , 0 {\displaystyle s_{12,0}} . In many cases the diagonal terms do not converge in one cycle so process of averaging is to be repeated with ... | https://en.wikipedia.org/wiki/Van_Wijngaarden_transformation |
In mathematics and optimization, a pseudo-Boolean function is a function of the form f: B n → R , {\displaystyle f:\mathbf {B} ^{n}\to \mathbb {R} ,} where B = {0, 1} is a Boolean domain and n is a nonnegative integer called the arity of the function. A Boolean function is then a special case, where the values are also... | https://en.wikipedia.org/wiki/Pseudo-Boolean_function |
In mathematics and particularly in algebra, a system of equations (either linear or nonlinear) is called consistent if there is at least one set of values for the unknowns that satisfies each equation in the system—that is, when substituted into each of the equations, they make each equation hold true as an identity. I... | https://en.wikipedia.org/wiki/Consistent_and_inconsistent_equations |
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value,: pp. 160 is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). For a system of order k (the number of time lags in discrete time, or the order of t... | https://en.wikipedia.org/wiki/Initial_condition |
In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. A corresponding problem exists for discrete time situations. While a closed form solution is not always possible to obtain, future values of ... | https://en.wikipedia.org/wiki/Initial_condition |
In mathematics and particularly in elementary geometry, a circumgon is a geometric figure which circumscribes some circle, in the sense that it is the union of the outer edges of non-overlapping triangles each of which has a vertex at the center of the circle and opposite side on a line that is tangent to the circle. :... | https://en.wikipedia.org/wiki/Circumgon |
A circumgonal region is the union of those triangular regions. Every triangle is a circumgonal region because it circumscribes the circle known as the incircle of the triangle. Every square is a circumgonal region. | https://en.wikipedia.org/wiki/Circumgon |
In fact, every regular polygon is a circumgonal region, as is more generally every tangential polygon. But not every polygon is a circumgonal region: for example, a non-square rectangle is not. A circumgonal region need not even be a convex polygon: for example, it could consist of three triangular wedges meeting only ... | https://en.wikipedia.org/wiki/Circumgon |
All circumgons have common properties regarding area–perimeter ratios and centroids. It is these properties that make circumgons interesting objects of study in elementary geometry. The concept and the terminology of a circumgon were introduced and their properties investigated first by Tom M. Apostol and Mamikon A. Mn... | https://en.wikipedia.org/wiki/Circumgon |
In mathematics and particularly in topology, pairwise Stone space is a bitopological space ( X , τ 1 , τ 2 ) {\displaystyle \scriptstyle (X,\tau _{1},\tau _{2})} which is pairwise compact, pairwise Hausdorff, and pairwise zero-dimensional. Pairwise Stone spaces are a bitopological version of the Stone spaces. Pairwise ... | https://en.wikipedia.org/wiki/Pairwise_Stone_space |
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each fu... | https://en.wikipedia.org/wiki/Spherical_harmonic_function |
Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions,... | https://en.wikipedia.org/wiki/Spherical_harmonic_function |
Spherical harmonics originate from solving Laplace's equation in the spherical domains. Functions that are solutions to Laplace's equation are called harmonics. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree ℓ {\d... | https://en.wikipedia.org/wiki/Spherical_harmonic_function |
The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence r ℓ {\displaystyle r^{\ell }} from the above-mentioned polynomial of degree ℓ {\displaystyle \ell } ; the remaining factor can be regarded as a function of the spherical angular coordinates ... | https://en.wikipedia.org/wiki/Spherical_harmonic_function |
Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below). A specific set of spherical harmonics, denoted Y ℓ m ( θ , φ ) {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} or Y ℓ m ( r ) {\displaystyle Y_{\ell }^{m}({\mathbf {r} })} , ar... | https://en.wikipedia.org/wiki/Spherical_harmonic_function |
These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above. Spherical harmonics are important in many theoretical and practical applications, including the representation of multipole electrostatic and electromagnetic fields, electron configura... | https://en.wikipedia.org/wiki/Spherical_harmonic_function |
In mathematics and physics CCR algebras (after canonical commutation relations) and CAR algebras (after canonical anticommutation relations) arise from the quantum mechanical study of bosons and fermions respectively. They play a prominent role in quantum statistical mechanics and quantum field theory. | https://en.wikipedia.org/wiki/Canonical_commutation_relation_algebra |
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as or where Δ = ∇ ⋅ ∇ = ∇ 2 {\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}} is the Laplace operator, ∇ ⋅ {\displaystyle \nabla... | https://en.wikipedia.org/wiki/Laplace's_equation |
Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Laplace's equation is also a special case of the Helmholtz equation. The general theory of solutions to Laplace's equation is known as potential theory. | https://en.wikipedia.org/wiki/Laplace's_equation |
The twice continuously differentiable solutions of Laplace's equation are the harmonic functions, which are important in multiple branches of physics, notably electrostatics, gravitation, and fluid dynamics. In the study of heat conduction, the Laplace equation is the steady-state heat equation. In general, Laplace's e... | https://en.wikipedia.org/wiki/Laplace's_equation |
In mathematics and physics, Lieb–Thirring inequalities provide an upper bound on the sums of powers of the negative eigenvalues of a Schrödinger operator in terms of integrals of the potential. They are named after E. H. Lieb and W. E. Thirring. The inequalities are useful in studies of quantum mechanics and differenti... | https://en.wikipedia.org/wiki/Lieb–Thirring_conjecture |
In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971. A diagram in the notation consists of several shapes linked together by lines. The notation widely appears in modern quantu... | https://en.wikipedia.org/wiki/Tensor_diagram_notation |
In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics... | https://en.wikipedia.org/wiki/Hamiltonian_flow |
In mathematics and physics, a global mode of a system is one in which the system executes coherent oscillations in time. Suppose a quantity y ( x , t ) {\displaystyle y(x,t)} which depends on space x {\displaystyle x} and time t {\displaystyle t} is governed by some partial differential equation which does not have an ... | https://en.wikipedia.org/wiki/Global_mode |
If ω {\displaystyle \omega } is complex, then the imaginary part corresponds to the mode exhibiting exponential growth or exponential decay. The concept of a global mode can be compared to that of a normal mode; the PDE may be thought of as a dynamical system of infinitely many equations coupled together. Global modes ... | https://en.wikipedia.org/wiki/Global_mode |
Philip Drazin introduced the concept of a global mode in his 1974 paper, and gave a technique for finding the normal modes of a linear PDE problem in which the coefficients or geometry vary slowly in x {\displaystyle x} . This technique is based on the WKBJ approximation, which is a special case of multiple-scale analy... | https://en.wikipedia.org/wiki/Global_mode |
In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function f ( x ) = e − 1 / x 2 , {\displaystyle f(x)=e^{-1/x^{2}},} which does not have a Taylor series at x = 0. Every coefficient of the Taylor expansion around x = 0 is exactly... | https://en.wikipedia.org/wiki/Non-perturbative |
In physics, such functions arise for phenomena which are impossible to understand by perturbation theory, at any finite order. In quantum field theory, 't Hooft–Polyakov monopoles, domain walls, flux tubes, and instantons are examples. A concrete, physical example is given by the Schwinger effect, whereby a strong elec... | https://en.wikipedia.org/wiki/Non-perturbative |
For not too strong fields, the rate per unit volume of this process is given by, Γ = ( e E ) 2 4 π 3 e − π m 2 e E {\displaystyle \Gamma ={\frac {(eE)^{2}}{4\pi ^{3}}}\mathrm {e} ^{-{\frac {\pi m^{2}}{eE}}}} which cannot be expanded in a Taylor series in the electric charge e {\displaystyle e} , or the electric field s... | https://en.wikipedia.org/wiki/Non-perturbative |
In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the Poincaré conjecture and the Calabi con... | https://en.wikipedia.org/wiki/Exact_solutions_of_nonlinear_partial_differential_equations |
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