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In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
-
$$
x(xy) = (xx)y
$$
-
$$
(yx)x = y(xx)
$$
for all x and y in the algebra.
Every associative algebra is obviously alternative, but so too are some strictly non-associati... | https://en.wikipedia.org/wiki/Alternative_algebra |
In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional logic. It can be thought of as the law of excluded middle written in a form that involves only one sort of connective, namely implication.
In propositional c... | https://en.wikipedia.org/wiki/Peirce%27s_law |
A functional differential equation is a differential equation with deviating argument. That is, a functional differential equation is an equation that contains a function and some of its derivatives evaluated at different argument values.
Functional differential equations find use in mathematical models that assume a s... | https://en.wikipedia.org/wiki/Functional_differential_equation |
In computer science, 2-satisfiability, 2-SAT or just 2SAT is a computational problem of assigning values to variables, each of which has two possible values, in order to satisfy a system of constraints on pairs of variables. It is a special case of the general Boolean satisfiability problem, which can involve constrain... | https://en.wikipedia.org/wiki/2-satisfiability |
In mathematics, the Yoneda lemma is a fundamental result in category theory. It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It also gen... | https://en.wikipedia.org/wiki/Yoneda_lemma |
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. A pyramid is a conic solid with a polygonal base. Many types of pyramids can be found by determining the shape of bases, either by based on a regular... | https://en.wikipedia.org/wiki/Pyramid_%28geometry%29 |
In mathematics, a codomain, counter-domain, or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the codomain or the image of a function.
A codomain is part of a ... | https://en.wikipedia.org/wiki/Codomain |
Simulation noise is a function that creates a divergence-free vector field. This signal can be used in artistic simulations for the purpose of increasing the perception of extra detail.
The function can be calculated in three dimensions by dividing the space into a regular lattice grid. With each edge is associated a... | https://en.wikipedia.org/wiki/Simulation_noise |
In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orde... | https://en.wikipedia.org/wiki/P-group |
In mathematical logic, New Foundations (NF) is a non-well-founded, finitely axiomatizable set theory conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica.
## Definition
The well-formed formulas of NF are the standard formulas of propositional calculus with two primiti... | https://en.wikipedia.org/wiki/New_Foundations |
In the mathematical subfield numerical analysis, tricubic interpolation is a method for obtaining values at arbitrary points in 3D space of a function defined on a regular grid. The approach involves approximating the function locally by an expression of the form
$$
f(x,y,z)=\sum_{i=0}^3 \sum_{j=0}^3 \sum_{k=0}^3 a_{... | https://en.wikipedia.org/wiki/Tricubic_interpolation |
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation a → b called implication such that (c ∧ a) ≤ b is equivalent to c ≤ (a → b). From a logical stand... | https://en.wikipedia.org/wiki/Heyting_algebra |
In number theory, a perfect digital invariant (PDI) is a number in a given number base (
$$
b
$$
) that is the sum of its own digits each raised to a given power (
$$
p
$$
). PDIs by Harvey Heinz
## Definition
Let
$$
n
$$
be a natural number. The perfect digital invariant function (also known as a happy function, fro... | https://en.wikipedia.org/wiki/Perfect_digital_invariant |
The following is a timeline of numerical analysis after 1945, and deals with developments after the invention of the modern electronic computer, which began during Second World War. For a fuller history of the subject before this period, see timeline and history of mathematics.
## 1940s
- Monte Carlo simulation (voted ... | https://en.wikipedia.org/wiki/Timeline_of_numerical_analysis_after_1945 |
In modular arithmetic, a number is a primitive root modulo if every number coprime to is congruent to a power of modulo . That is, is a primitive root modulo if for every integer coprime to , there is some integer for which ≡ (mod ). Such a value is called the index or discrete logarithm of to the base mo... | https://en.wikipedia.org/wiki/Primitive_root_modulo_n |
Process mining is a family of techniques for analyzing event data to understand and improve operational processes. Part of the fields of data science and process management, process mining is generally built on logs that contain case id, a unique identifier for a particular process instance; an activity, a description ... | https://en.wikipedia.org/wiki/Process_mining |
Weakened weak form (or W2 form) is used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings. These numerical methods are applicable to solid mechanics as well as fluid dynamics problems.
## Description
For simplicity we choose elasticity problems (2nd order ... | https://en.wikipedia.org/wiki/Weakened_weak_form |
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the latter half of the
### 19th century
after the commercialization of the elect... | https://en.wikipedia.org/wiki/Electrical_engineering |
In mathematics, a singular perturbation problem is a problem containing a small parameter that cannot be approximated by setting the parameter value to zero. More precisely, the solution cannot be uniformly approximated by an asymptotic expansion
$$
\varphi(x) \approx \sum_{n=0}^N \delta_n(\varepsilon) \psi_n(x) \,
$$
... | https://en.wikipedia.org/wiki/Singular_perturbation |
The Level-set method (LSM) is a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. LSM can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects. LSM makes it easier to perform computations on s... | https://en.wikipedia.org/wiki/Level-set_method |
In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of... | https://en.wikipedia.org/wiki/Galois_representation |
In computer graphics and computational geometry, a bounding volume (or bounding region) for a set of objects is a closed region that completely contains the union of the objects in the set. Bounding volumes are used to improve the efficiency of geometrical operations, such as by using simple regions, having simpler way... | https://en.wikipedia.org/wiki/Bounding_volume |
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to properties of the Lie algebra. It was later extended by to coefficients in an... | https://en.wikipedia.org/wiki/Lie_algebra_cohomology |
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
- Group with a partial function replacing the binary operation;
- Category in which every morphism is inver... | https://en.wikipedia.org/wiki/Groupoid |
In mathematics, particularly topology, a comb space is a particular subspace of
$$
\R^2
$$
that resembles a comb. The comb space has properties that serve as a number of counterexamples. The topologist's sine curve has similar properties to the comb space. The deleted comb space is a variation on the comb space.
## ... | https://en.wikipedia.org/wiki/Comb_space |
In probability theory, a real-valued stochastic process X is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the largest class of processes with respect to which the Itô integral and the Strat... | https://en.wikipedia.org/wiki/Semimartingale |
In category theory, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise distance) as its morphisms. This is a category because the composition of two metric maps is again a metric map. It was first considered by .
## Arrow... | https://en.wikipedia.org/wiki/Category_of_metric_spaces |
In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers, usually represented by the capital letter S, boldface or blackboard bold
$$
\mathbb S
$$
.
The sedenions are obtained by applying the Cayley–Dickson construction to the octonions, which can be mat... | https://en.wikipedia.org/wiki/Sedenion |
In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. Pure modules are complementary to flat modules and generalize Prüfer's notion of pure subgroups. While flat modules are those modu... | https://en.wikipedia.org/wiki/Pure_submodule |
In computer graphics, per-pixel lighting refers to any technique for lighting an image or scene that calculates illumination for each pixel on a rendered image. This is in contrast to other popular methods of lighting such as vertex lighting, which calculates illumination at each vertex of a 3D model and then interpola... | https://en.wikipedia.org/wiki/Per-pixel_lighting |
In oceanography, a tidal resonance occurs when the tide excites one of the resonant modes of the ocean.
The effect is most striking when a continental shelf is about a quarter wavelength wide. Then an incident tidal wave can be reinforced by reflections between the coast and the shelf edge, the result producing a mu... | https://en.wikipedia.org/wiki/Tidal_resonance |
In mathematics, an algebraic number field (or simply number field) is an extension field
$$
K
$$
of the field of rational numbers such that the field extension
$$
K / \mathbb{Q}
$$
has finite degree (and hence is an algebraic field extension).
Thus
$$
K
$$
is a field that contains
$$
\mathbb{Q}
$$
and has fini... | https://en.wikipedia.org/wiki/Algebraic_number_field |
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune
### Dirichlet
's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on a... | https://en.wikipedia.org/wiki/Analytic_number_theory |
In linear algebra, the order-r Krylov subspace generated by an n-by-n matrix A and a vector b of dimension n is the linear subspace spanned by the images of b under the first r powers of A (starting from
$$
A^0=I
$$
), that is,
$$
\mathcal{K}_r(A,b) = \operatorname{span} \, \{ b, Ab, A^2b, \ldots, A^{r-1}b \}.
$$
## B... | https://en.wikipedia.org/wiki/Krylov_subspace |
In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur ind... | https://en.wikipedia.org/wiki/Poisson_point_process |
In logic, a substructural logic is a logic lacking one of the usual structural rules (e.g. of classical and intuitionistic logic), such as weakening, contraction, exchange or associativity. Two of the more significant substructural logics are relevance logic and linear logic.
## Examples
In a sequent calculus, one wri... | https://en.wikipedia.org/wiki/Substructural_logic |
In complex analysis, the argument principle (or Cauchy's argument principle) is a theorem relating the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative.
## Formulation
If f is a meromorphic function inside and on some closed contou... | https://en.wikipedia.org/wiki/Argument_principle |
{{DISPLAYTITLE:T1 space}}
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is one in which this holds for every pair of topologically distinguishable points. The properti... | https://en.wikipedia.org/wiki/T1_space |
In mathematics,
## Darboux's theorem
is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the differentiation of another function has the intermediate value property: the image of an interval is also an interval.
When ƒ is continuously differentiable (ƒ in C1... | https://en.wikipedia.org/wiki/Darboux%27s_theorem_%28analysis%29 |
In mathematics, the special linear group
$$
\operatorname{SL}(n,R)
$$
of degree
$$
n
$$
over a commutative ring
$$
R
$$
is the set of
$$
n\times n
$$
matrices with determinant
$$
1
$$
, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general ... | https://en.wikipedia.org/wiki/Special_linear_group |
Luck is the phenomenon and belief that defines the experience of improbable events, especially improbably positive or negative ones. The naturalistic interpretation is that positive and negative events may happen at any time, both due to random and non-random natural and artificial processes, and that even improbable ... | https://en.wikipedia.org/wiki/Luck |
The space mapping methodology for modeling and design optimization of engineering systems was first discovered by John Bandler in 1993. It uses relevant existing knowledge to speed up model generation and design optimization of a system. The knowledge is updated with new validation information from the system when avai... | https://en.wikipedia.org/wiki/Space_mapping |
In 3D computer graphics, hidden-surface determination (also known as shown-surface determination, hidden-surface removal (HSR), occlusion culling (OC) or visible-surface determination (VSD)) is the process of identifying what surfaces and parts of surfaces can be seen from a particular viewing angle. A hidden-surface d... | https://en.wikipedia.org/wiki/Hidden-surface_determination |
In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups.
## Definition
A set N together with two binary operations + (called addition) and ⋅ (called multiplication) is called a (right) near-ring... | https://en.wikipedia.org/wiki/Near-ring |
In mathematics, a superparticular ratio, also called a superparticular number or epimoric ratio, is the ratio of two consecutive integer numbers.
More particularly, the ratio takes the form:
$$
\frac{n + 1}{n} = 1 + \frac{1}{n}
$$
where is a positive integer.
Thus:
Superparticular ratios were written about by Nicoma... | https://en.wikipedia.org/wiki/Superparticular_ratio |
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds,... | https://en.wikipedia.org/wiki/Cotangent_bundle |
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes (derived ca... | https://en.wikipedia.org/wiki/Model_category |
The Monty Hall problem is a brain teaser, in the form of a probability puzzle, based nominally on the American television game show Let's Make a Deal and named after its original host, Monty Hall. The problem was originally posed (and solved) in a letter by Steve Selvin to the American Statistician in 1975. It became f... | https://en.wikipedia.org/wiki/Monty_Hall_problem |
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number p/q is a "good" approx... | https://en.wikipedia.org/wiki/Diophantine_approximation |
A Bézier triangle is a special type of Bézier surface that is created by (linear, quadratic, cubic or higher degree) interpolation of control points.
## nth-order Bézier triangle
A general nth-order Bézier triangle has (n +1)(n + 2)/2 control points αiβjγk where i, j, k are non-negative integers such that i + j + k = n... | https://en.wikipedia.org/wiki/B%C3%A9zier_triangle |
A sum-of-squares optimization program is an optimization problem with a linear cost function and a particular type of constraint on the decision variables. These constraints are of the form that when the decision variables are used as coefficients in certain polynomials, those polynomials should have the polynomial SOS... | https://en.wikipedia.org/wiki/Sum-of-squares_optimization |
In computing, a Las Vegas algorithm is a randomized algorithm that always gives correct results; that is, it always produces the correct result or it informs about the failure. However, the runtime of a Las Vegas algorithm differs depending on the input. The usual definition of a Las Vegas algorithm includes the restri... | https://en.wikipedia.org/wiki/Las_Vegas_algorithm |
In mathematics, the arithmetic–geometric mean (AGM or agM) of two positive real numbers and is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The arithmetic–geometric mean is used in fast algorithms for exponential, trigonometric functions, and other special functions, as well a... | https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean |
In topology and related fields of mathematics, a topological space X is called a regular space if every closed subset C of X and a point p not contained in C have non-overlapping open neighborhoods. Thus p and C can be separated by neighborhoods. This condition is known as Axiom T3. The term "T3 space" usually means "a... | https://en.wikipedia.org/wiki/Regular_space |
In geometry, the Conway triangle notation, named after English mathematician John Horton Conway, allows trigonometric functions of a triangle to be managed algebraically. However, though the notation was promoted by Conway, a much earlier reference to the notation goes back to the Spanish nineteenth century mathematici... | https://en.wikipedia.org/wiki/Conway_triangle_notation |
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomoto... | https://en.wikipedia.org/wiki/Homotopy |
In computer graphics, a triangle strip is a subset of triangles in a triangle mesh with shared vertices, and is a more memory-efficient method of storing information about the mesh. They are more efficient than un-indexed lists of triangles, but usually equally fast or slower than indexed triangle lists. The primary re... | https://en.wikipedia.org/wiki/Triangle_strip |
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces.
More generally, one can also join manifolds toge... | https://en.wikipedia.org/wiki/Connected_sum |
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901.Russell, Bertrand. The Principles of Mathematics. 2d. ed. Reprint, New York: W. W. Norton & Company, 1996. (First published in 1903.) R... | https://en.wikipedia.org/wiki/Russell%27s_paradox |
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form
$$
\int_{a(x)}^{b(x)} f(x,t)\,dt,
$$
where
$$
-\infty < a(x), b(x) < \infty
$$
and the integrands are functions dependent on
$$
x,
$$
the derivative of t... | https://en.wikipedia.org/wiki/Leibniz_integral_rule |
A conceptual graph (CG) is a formalism for knowledge representation. In the first published paper on CGs, John F. Sowa used them to represent the conceptual schemas used in database systems. The first book on CGs applied them to a wide range of topics in artificial intelligence, computer science, and cognitive science.... | https://en.wikipedia.org/wiki/Conceptual_graph |
In mathematics, Birkhoff interpolation is an extension of polynomial interpolation. It refers to the problem of finding a polynomial
$$
P(x)
$$
of degree
$$
d
$$
such that only certain derivatives have specified values at specified points:
$$
P^{(n_i)}(x_i) = y_i \qquad\mbox{for } i=1,\ldots,d,
$$
where the data po... | https://en.wikipedia.org/wiki/Birkhoff_interpolation |
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, and their inverses (e.g., arcsin, log, or x1/n).
All element... | https://en.wikipedia.org/wiki/Elementary_function |
A maze is a path or collection of paths, typically from an entrance to a goal. The word is used to refer both to branching tour puzzles through which the solver must find a route, and to simpler non-branching ("unicursal") patterns that lead unambiguously through a convoluted layout to a goal. The term "labyrinth" is ... | https://en.wikipedia.org/wiki/Maze |
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total co... | https://en.wikipedia.org/wiki/Ackermann_function |
In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it. In other words, a complex number z0 is an isolated singularity of a function f if there exists an open disk D centered at z0 such that f is holomorphic on D \ {z0}, that is, on the set obtained from... | https://en.wikipedia.org/wiki/Isolated_singularity |
In mathematics, a well-posed problem is one for which the following properties hold:
1. The problem has a solution
1. The solution is unique
1. The solution's behavior changes continuously with the initial conditions
Examples of archetypal well-posed problems include the Dirichlet problem for Laplace's equation, and th... | https://en.wikipedia.org/wiki/Well-posed_problem |
In astronomy, the Roche lobe is the region around a star in a binary system within which orbiting material is gravitationally bound to that star. It is an approximately teardrop-shaped region bounded by a critical gravitational equipotential, with the apex of the teardrop pointing towards the other star (the apex is at... | https://en.wikipedia.org/wiki/Roche_lobe |
In probability theory, the martingale representation theorem states that a random variable with finite variance that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion.
The theorem only asserts the existence of the ... | https://en.wikipedia.org/wiki/Martingale_representation_theorem |
In computational complexity theory, the space hierarchy theorems are separation results that show that both deterministic and nondeterministic machines can solve more problems in (asymptotically) more space, subject to certain conditions. For example, a deterministic Turing machine can solve more decision problems in s... | https://en.wikipedia.org/wiki/Space_hierarchy_theorem |
In statistics, canonical-correlation analysis (CCA), also called canonical variates analysis, is a way of inferring information from cross-covariance matrices. If we have two vectors X = (X1, ..., Xn) and Y = (Y1, ..., Ym) of random variables, and there are correlations among the variables, then canonical-correlation ... | https://en.wikipedia.org/wiki/Canonical_correlation |
In mathematical analysis, a bump function (also called a test function) is a function
$$
f : \Reals^n \to \Reals
$$
on a Euclidean space
$$
\Reals^n
$$
which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump functions with domain
$$
\Reals^n
... | https://en.wikipedia.org/wiki/Bump_function |
In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence , does there exist a positive Borel measure (for instance, the measure determined by the cumulative distribution function of a random variable) on the real line such that
$$
m_n = \int_{-\infty}^... | https://en.wikipedia.org/wiki/Hamburger_moment_problem |
In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used t... | https://en.wikipedia.org/wiki/Modular_curve |
In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal simila... | https://en.wikipedia.org/wiki/Ricci_flow |
In computer science, model checking or property checking is a method for checking whether a finite-state model of a system meets a given specification (also known as correctness). This is typically associated with hardware or software systems, where the specification contains liveness requirements (such as avoidance of... | https://en.wikipedia.org/wiki/Model_checking |
Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M. Kan, who constructed certain (Kan) extensions using limits in 1960.
An early use of (what is now known as) a Kan extension fr... | https://en.wikipedia.org/wiki/Kan_extension |
Information geometry is an interdisciplinary field that applies the techniques of differential geometry to study probability theory and statistics. It studies statistical manifolds, which are Riemannian manifolds whose points correspond to probability distributions.
## Introduction
Historically, information geometry c... | https://en.wikipedia.org/wiki/Information_geometry |
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied obj... | https://en.wikipedia.org/wiki/Abelian_variety |
In computational complexity theory, the Cook–Levin theorem, also known as Cook's theorem, states that the Boolean satisfiability problem is NP-complete. That is, it is in NP, and any problem in NP can be reduced in polynomial time by a deterministic Turing machine to the Boolean satisfiability problem.
The theorem is ... | https://en.wikipedia.org/wiki/Cook%E2%80%93Levin_theorem |
The Mersenne Twister is a general-purpose pseudorandom number generator (PRNG) developed in 1997 by and .E.g. Marsland S. (2011) Machine Learning (CRC Press), §4.1.1. Also see the section "Adoption in software systems". Its name derives from the choice of a Mersenne prime as its period length.
The Mersenne Twister was... | https://en.wikipedia.org/wiki/Mersenne_Twister |
Automatic label placement, sometimes called text placement or name placement, comprises the computer methods of placing labels automatically on a map or chart. This is related to the typographic design of such labels.
The typical features depicted on a geographic map are line features (e.g. roads), area features (count... | https://en.wikipedia.org/wiki/Automatic_label_placement |
In the theory of algebraic plane curves, a general quartic plane curve has 28 bitangent lines, lines that are tangent to the curve in two places. These lines exist in the complex projective plane, but it is possible to define quartic curves for which all 28 of these lines have real numbers as their coordinates and ther... | https://en.wikipedia.org/wiki/Bitangents_of_a_quartic |
In 3D computer graphics and computer vision, a depth map is an image or image channel that contains information relating to the distance of the surfaces of scene objects from a viewpoint. The term is related (and may be analogous) to depth buffer, Z-buffer, Z-buffering, and Z-depth. The "Z" in these latter terms relate... | https://en.wikipedia.org/wiki/Depth_map |
In complex analysis, a complex-valued function
$$
f
$$
of a complex variable
$$
z
$$
:
- is said to be holomorphic at a point
$$
a
$$
if it is differentiable at every point within some open disk centered at
$$
a
$$
, and
- is said to be analytic at
$$
a
$$
if in some open disk centered at
$$
a
$$
it can be ex... | https://en.wikipedia.org/wiki/Analyticity_of_holomorphic_functions |
In economics, a network effect (also called network externality or demand-side economies of scale) is the phenomenon by which the value or utility a user derives from a good or service depends on the number of users of compatible products. Network effects are typically positive feedback systems, resulting in users deri... | https://en.wikipedia.org/wiki/Network_effect |
In mathematics, the linear span (also called the linear hull or just span) of a set
$$
S
$$
of elements of a vector space
$$
V
$$
is the smallest linear subspace of
$$
V
$$
that contains
$$
S.
$$
It is the set of all finite linear combinations of the elements of , and the intersection of all linear subspaces th... | https://en.wikipedia.org/wiki/Linear_span |
Cognition is the "mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses all aspects of intellectual functions and processes such as: perception, attention, thought, imagination, intelligence, the formation of knowledge, memory and working memory, ... | https://en.wikipedia.org/wiki/Cognition |
In statistics, multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. with more than two possible discrete outcomes. That is, it is a model that is used to predict the probabilities of the different possible outcomes of a categorically distributed d... | https://en.wikipedia.org/wiki/Multinomial_logistic_regression |
Event generators are software libraries that generate simulated high-energy particle physics events.M. A. Dobbs et al., hep-ph/0403045.
They randomly generate events as those produced in particle accelerators, collider experiments or the early universe.
Events come in different types called processes as discussed in th... | https://en.wikipedia.org/wiki/Event_generator |
The Playfair cipher or Playfair square or Wheatstone–Playfair cipher is a manual symmetric encryption technique and was the first literal digram substitution cipher. The scheme was invented in 1854 by Charles Wheatstone, but bears the name of Lord Playfair for promoting its use.
The technique encrypts pairs of letters ... | https://en.wikipedia.org/wiki/Playfair_cipher |
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring is a special case of graph labeling. In its simplest form, it is a way of c... | https://en.wikipedia.org/wiki/Graph_coloring |
In algebra, a domain is a nonzero ring in which implies or . (Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative domain is called an integral domain.Rowen (1994), ... | https://en.wikipedia.org/wiki/Domain_%28ring_theory%29 |
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship among quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and other algebraic ... | https://en.wikipedia.org/wiki/Isomorphism_theorems |
In mathematics, particularly in functional analysis, a seminorm is like a norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm.
A topological... | https://en.wikipedia.org/wiki/Seminorm |
In mathematics, the total derivative of a function 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, thi... | https://en.wikipedia.org/wiki/Total_derivative |
A clipping path (or "deep etch") is a closed vector path, or shape, used to cut out a 2D image in image editing software. Anything inside the path will be included after the clipping path is applied; anything outside the path will be omitted from the output. Applying the clipping path results in a hard (aliased) or sof... | https://en.wikipedia.org/wiki/Clipping_path |
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under ... | https://en.wikipedia.org/wiki/Invariant_theory |
In mathematics, a set of n functions f1, f2, ..., fn is unisolvent (meaning "uniquely solvable") on a domain Ω if the vectors
$$
\begin{bmatrix}f_1(x_1) \\ f_1(x_2) \\ \vdots \\ f_1(x_n)\end{bmatrix}, \begin{bmatrix}f_2(x_1) \\ f_2(x_2) \\ \vdots \\ f_2(x_n)\end{bmatrix}, \dots, \begin{bmatrix}f_n(x_1) \\ f_n(x_2) \\ \... | https://en.wikipedia.org/wiki/Unisolvent_functions |
In the theory of computation, a generalized nondeterministic finite automaton (GNFA), also known as an expression automaton or a generalized nondeterministic finite state machine, is a variation of a
nondeterministic finite automaton (NFA) where each transition is labeled with any regular expression. The GNFA reads b... | https://en.wikipedia.org/wiki/Generalized_nondeterministic_finite_automaton |
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