text stringlengths 2.13k 184k | source stringlengths 31 108 |
|---|---|
In graph theory, a branch of mathematics, an indifference graph is an undirected graph constructed by assigning a real number to each vertex and connecting two vertices by an edge when their numbers are within one unit of each other. Indifference graphs are also the intersection graphs of sets of unit intervals, or of ... | https://en.wikipedia.org/wiki/Indifference_graph |
In mathematics, a handle decomposition of an m-manifold M is a union
$$
\emptyset = M_{-1} \subset M_0 \subset M_1 \subset M_2 \subset \dots \subset M_{m-1} \subset M_m = M
$$
where each
$$
M_i
$$
is obtained from
$$
M_{i-1}
$$
by the attaching of
$$
i
$$
-handles. A handle decomposition is to a manifold what a ... | https://en.wikipedia.org/wiki/Handle_decomposition |
A family tree, also called a genealogy or a pedigree chart, is a chart representing family relationships in a conventional tree structure. More detailed family trees, used in medicine and social work, are known as genograms.
## Representations of family history
Genealogical data can be represented in several formats, f... | https://en.wikipedia.org/wiki/Family_tree |
In information science, an upper ontology (also known as a top-level ontology, upper model, or foundation ontology) is an ontology (in the sense used in information science) that consists of very general terms (such as "object", "property", "relation") that are common across all domains. An important function of an u... | https://en.wikipedia.org/wiki/Upper_ontology |
In mathematics, the gamma function (represented by Γ, capital Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function
$$
\Gamma(z)
$$
is defined for all complex numbers
$$
z
$$
except non-positive integers, and for every positive... | https://en.wikipedia.org/wiki/Gamma_function |
The closed-world assumption (CWA), in a formal system of logic used for knowledge representation, is the presumption that a statement that is true is also known to be true. Therefore, conversely, what is not currently known to be true, is false. The same name also refers to a logical formalization of this assumption by... | https://en.wikipedia.org/wiki/Closed-world_assumption |
Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters (m ≥ n). It is used in some forms of nonlinear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by succ... | https://en.wikipedia.org/wiki/Non-linear_least_squares |
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
## Definitions
Formally, we start with a non-zero algebra D over a field. We call D a division algebra if for any element a in D and any non-zero e... | https://en.wikipedia.org/wiki/Division_algebra |
## Particle filters
, also known as sequential Monte Carlo methods, are a set of Monte Carlo algorithms used to find approximate solutions for filtering problems for nonlinear state-space systems, such as signal processing and Bayesian statistical inference.
## The filtering problem
consists of estimating the interna... | https://en.wikipedia.org/wiki/Particle_filter |
In number theory, the Padovan sequence is the sequence of integers P(n) defined by the initial values
$$
P(0) = P(1) = P(2) = 1,
$$
and the recurrence relation
$$
P(n) = P(n-2)+P(n-3).
$$
The first few values of P(n) are
1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ...
The Padovan ... | https://en.wikipedia.org/wiki/Padovan_sequence |
In the mathematical field of combinatorics, a bent function is a Boolean function that is maximally non-linear; it is as different as possible from the set of all linear and affine functions when measured by Hamming distance between truth tables. Concretely, this means the maximum correlation between the output of the ... | https://en.wikipedia.org/wiki/Bent_function |
In general topology, a branch of mathematics, a non-empty family
$$
A
$$
of subsets of a set
$$
X
$$
is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of
$$
A
$$
is non-empty. It has the strong finite intersection property (SFIP) if the intersection over any ... | https://en.wikipedia.org/wiki/Finite_intersection_property |
In computer science, graph reduction implements an efficient version of non-strict evaluation, an evaluation strategy where the arguments to a function are not immediately evaluated. This form of non-strict evaluation is also known as lazy evaluation and used in functional programming languages. The technique was first... | https://en.wikipedia.org/wiki/Graph_reduction |
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC.
## Induction by cases
Let
$$
P(\alpha)
$$
be a property defined for all ordinals
$$
\alpha
$$
. Suppose that whenever
$$
P(\beta)
$$
... | https://en.wikipedia.org/wiki/Transfinite_induction |
In mathematics, and more particularly in set theory, a cover (or covering) of a set
$$
X
$$
is a family of subsets of
$$
X
$$
whose union is all of
$$
X
$$
. More formally, if
$$
C = \lbrace U_\alpha : \alpha \in A \rbrace
$$
is an indexed family of subsets
$$
U_\alpha\subset X
$$
(indexed by the set
$$
A
$$
... | https://en.wikipedia.org/wiki/Cover_%28topology%29 |
Computational chemistry is a branch of chemistry that uses computer simulations to assist in solving chemical problems. It uses methods of theoretical chemistry incorporated into computer programs to calculate the structures and properties of molecules, groups of molecules, and solids. The importance of this subject s... | https://en.wikipedia.org/wiki/Computational_chemistry |
In physics and chemistry, a degree of freedom is an independent physical parameter in the chosen parameterization of a physical system. More formally, given a parameterization of a physical system, the number of degrees of freedom is the smallest number
$$
n
$$
of parameters whose values need to be known in order to ... | https://en.wikipedia.org/wiki/Degrees_of_freedom_%28physics_and_chemistry%29 |
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 [a, b] ⊂ R, its total variation on the interval of definition is a measure of the one... | https://en.wikipedia.org/wiki/Total_variation |
In computer science, an algorithm is called non-blocking if failure or suspension of any thread cannot cause failure or suspension of another thread; for some operations, these algorithms provide a useful alternative to traditional blocking implementations. A non-blocking algorithm is lock-free if there is guaranteed ... | https://en.wikipedia.org/wiki/Non-blocking_algorithm |
In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning... | https://en.wikipedia.org/wiki/Axiom_of_determinacy |
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topolog... | https://en.wikipedia.org/wiki/Discrete_space |
In physics, an ultraviolet divergence or UV divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with unbounded energy, or, equivalently, because of physical phenomena at infinitesimal distances.
## Overview
Since an infinite result is unphysical, u... | https://en.wikipedia.org/wiki/Ultraviolet_divergence |
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a pseudo-Riemannian manifold that preserves the metric tensor. Killing vector fields are the infinitesimal generators of isometries; that is, flows generated by Killing vector fields are continuous ... | https://en.wikipedia.org/wiki/Killing_vector_field |
The multiphase particle-in-cell method (MP-PIC) is a numerical method for modeling particle-fluid and particle-particle interactions in a computational fluid dynamics (CFD) calculation. The MP-PIC method achieves greater stability than its particle-in-cell predecessor by simultaneously treating the solid particles as ... | https://en.wikipedia.org/wiki/Multiphase_particle-in-cell_method |
The Immirzi parameter (also known as the Barbero–Immirzi parameter) is a numerical coefficient appearing in loop quantum gravity (LQG), a nonperturbative theory of quantum gravity. The Immirzi parameter measures the size of the quantum of area in Planck units. As a result, its value is currently fixed by matching the s... | https://en.wikipedia.org/wiki/Immirzi_parameter |
A small-world network is a graph characterized by a high clustering coefficient and low distances. In an example of the social network, high clustering implies the high probability that two friends of one person are friends themselves. The low distances, on the other hand, mean that there is a short chain of social con... | https://en.wikipedia.org/wiki/Small-world_network |
In statistics, response surface methodology (RSM) explores the relationships between several explanatory variables and one or more response variables. RSM is an empirical model which employs the use of mathematical and statistical techniques to relate input variables, otherwise known as factors, to the response. RSM be... | https://en.wikipedia.org/wiki/Response_surface_methodology |
Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even i... | https://en.wikipedia.org/wiki/Renormalization |
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dot... | https://en.wikipedia.org/wiki/Pentagonal_number |
In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function in terms of the derivative of . More precisely, if the inverse of
$$
f
$$
is denoted as
$$
f^{-1}
$$
, where
$$
f^{-1}(y) = x
$$
if and only if
$$
f(x) = y
$$
, then the inv... | https://en.wikipedia.org/wiki/Inverse_function_rule |
In geometric topology and differential topology, an (n + 1)-dimensional cobordism W between n-dimensional manifolds M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps
$$
M \hookrightarrow W \quad\mbox{and}\quad N \hookrightarrow W
$$
are homotopy equivalences.
The h-cobordism theore... | https://en.wikipedia.org/wiki/H-cobordism |
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more... | https://en.wikipedia.org/wiki/Equilateral_triangle |
In control theory, optimal projection equations constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller.
The linear-quadratic-Gaussian (LQG) control problem is one of the most fundamental optimal control problems. It concerns uncertain linear systems disturbed by additive whit... | https://en.wikipedia.org/wiki/Optimal_projection_equations |
The Aberth method, or Aberth–Ehrlich method or Ehrlich–Aberth method, named after Oliver Aberth and Louis W. Ehrlich, is a root-finding algorithm developed in 1967 for simultaneous approximation of all the roots of a univariate polynomial.
This method converges cubically, an improvement over the Durand–Kerner method, a... | https://en.wikipedia.org/wiki/Aberth_method |
In string theory, K-theory classification refers to a conjectured application of K-theory (in abstract algebra and algebraic topology) to superstrings, to classify the allowed Ramond–Ramond field strengths as well as the charges of stable D-branes.
In condensed matter physics K-theory has also found important applicati... | https://en.wikipedia.org/wiki/K-theory_%28physics%29 |
In geometry, the Euler line, named after Leonhard Euler ( ), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and ... | https://en.wikipedia.org/wiki/Euler_line |
In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as
$$
D_n(x)= \sum_{k=-n}^n e^{ikx} = \left(1+2\sum_{k=1}^n\cos(kx)\right)=\frac{\sin\left(\left(n +1/2\right) x \right)}{\sin(x/2)},
$$
where is any nonn... | https://en.wikipedia.org/wiki/Dirichlet_kernel |
In mathematics, the free group FS over a given set S consists of all words that can be built from members of S, considering two words to be different unless their equality follows from the group axioms (e.g. st = suu−1t but s ≠ t−1 for s,t,u ∈ S). The members of S are called generators of FS, and the number of generato... | https://en.wikipedia.org/wiki/Free_group |
In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the same base. In the case of numbers that are not square-free, the factorization is written without exponents, writing the repeated factor as ... | https://en.wikipedia.org/wiki/Smith_number |
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
$$
\varphi : \mathbb{R} \rightarrow G
$$
from the real line
$$
\mathbb{R}
$$
(as an additive group) to some other topological group
$$
G
$$
.
If
$$
\varphi
$$
is injective then
$$
\varphi(\mathbb{R})
$$
... | https://en.wikipedia.org/wiki/One-parameter_group |
In computer science, amortized analysis is a method for analyzing a given algorithm's complexity, or how much of a resource, especially time or memory, it takes to execute. The motivation for amortized analysis is that looking at the worst-case run time can be too pessimistic. Instead, amortized analysis averages the... | https://en.wikipedia.org/wiki/Amortized_analysis |
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the -algebra of Borel sets of a Hausdorff topological space that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "c... | https://en.wikipedia.org/wiki/Radon_measure |
In physics, action is a scalar quantity that describes how the balance of kinetic versus potential energy of a physical system changes with trajectory. Action is significant because it is an input to the principle of stationary action, an approach to classical mechanics that is simpler for multiple objects. Action and ... | https://en.wikipedia.org/wiki/Action_%28physics%29 |
In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below.
One of the most impo... | https://en.wikipedia.org/wiki/Non-analytic_smooth_function |
In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector.
A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, ab... | https://en.wikipedia.org/wiki/Barrelled_space |
In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values and (denoted as ), and likewise between and (denoted as ), but not between the v... | https://en.wikipedia.org/wiki/Relation_%28mathematics%29 |
In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an ℓ p space nor a c0 space can be embedded. The Tsirelson space is reflexive.
It was introduced by B. S. Tsirelson in 1974. The same year, Figiel and Johnson published a related article () whe... | https://en.wikipedia.org/wiki/Tsirelson_space |
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
## Definition
Let G be a Lie group with Lie algebra
$$
\mathfrak g
$$
, and P → B be a principal G-bundle. Let ω be an Ehresm... | https://en.wikipedia.org/wiki/Curvature_form |
In computer graphics, a triangle mesh is a type of polygon mesh. It comprises a set of triangles (typically in three dimensions) that are connected by their common edges or vertices.
Many graphics software packages and hardware devices can operate more efficiently on triangles that are grouped into meshes than on a si... | https://en.wikipedia.org/wiki/Triangle_mesh |
In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so electrons are subject to a regular potential inside the lattice. It is a gen... | https://en.wikipedia.org/wiki/Particle_in_a_one-dimensional_lattice |
In probability theory, the central limit theorem says that, under certain conditions, the sum of many independent identically-distributed random variables, when scaled appropriately, converges in distribution to a standard normal distribution. The martingale central limit theorem generalizes this result for random var... | https://en.wikipedia.org/wiki/Martingale_central_limit_theorem |
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules ... | https://en.wikipedia.org/wiki/Representation_theory_of_the_symmetric_group |
Packaging is the science, art and technology of enclosing or protecting products for distribution, storage, sale, and use. Packaging also refers to the process of designing, evaluating, and producing packages. Packaging can be described as a coordinated system of preparing goods for transport, warehousing, logistics, s... | https://en.wikipedia.org/wiki/Packaging |
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.
## Definition
In formal terms, a directed graph is an ordered pair where
- V is a set whose elements are called vertices, nodes, or points... | https://en.wikipedia.org/wiki/Directed_graph |
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally not effective) method for constructing models of any set of sentences that... | https://en.wikipedia.org/wiki/Compactness_theorem |
In graph theory and statistics, a graphon (also known as a graph limit) is a symmetric measurable function
$$
W:[0,1]^2\to[0,1]
$$
, that is important in the study of dense graphs. Graphons arise both as a natural notion for the limit of a sequence of dense graphs, and as the fundamental defining objects of exchangeab... | https://en.wikipedia.org/wiki/Graphon |
In recreational mathematics, a Keith number or repfigit number (short for repetitive Fibonacci-like digit) is a natural number
$$
n
$$
in a given number base
$$
b
$$
with
$$
k
$$
digits such that when a sequence is created such that the first
$$
k
$$
terms are the
$$
k
$$
digits of
$$
n
$$
and each subseque... | https://en.wikipedia.org/wiki/Keith_number |
Elementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of the hyperreal number system of Abraham Robinson and is sometimes given as An approach using infinitesimals. The book is available freely online and is currently published by Dover.
## ... | https://en.wikipedia.org/wiki/Elementary_Calculus%3A_An_Infinitesimal_Approach |
Perceptual computing is an application of Zadeh's theory of computing with words on the field of assisting people to make subjective judgments.
## Perceptual computer
The perceptual computer – Per-C – an instantiation of perceptual computing – has the architecture that is depicted in Fig. 1 [2]–[6]. It consists of thre... | https://en.wikipedia.org/wiki/Perceptual_computing |
Distributed ray tracing, also called distribution ray tracing and stochastic ray tracing, is a refinement of ray tracing that allows for the rendering of "soft" phenomena.
Conventional ray tracing uses single rays to sample many different domains. For example, when the color of an object is calculated, ray tracing mig... | https://en.wikipedia.org/wiki/Distributed_ray_tracing |
In mathematics, a simplicial set is a sequence of sets with internal order structure (abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs.
Every simplicial set gives rise to a "nice" topological space, known as its geometric realization. This realizatio... | https://en.wikipedia.org/wiki/Simplicial_set |
In statistics, the k-nearest neighbors algorithm (k-NN) is a non-parametric supervised learning method. It was first developed by Evelyn Fix and Joseph Hodges in 1951, and later expanded by Thomas Cover.
Most often, it is used for classification, as a k-NN classifier, the output of which is a class membership. An obje... | https://en.wikipedia.org/wiki/K-nearest_neighbors_algorithm |
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as
$$
\varphi(n)
$$
or
$$
\phi(n)
$$
, and may also be called Euler's phi function. In other words, it is the number of integers in the range for wh... | https://en.wikipedia.org/wiki/Euler%27s_totient_function |
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle... | https://en.wikipedia.org/wiki/Method_of_steepest_descent |
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics.
Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (... | https://en.wikipedia.org/wiki/Naive_set_theory |
In computational complexity theory, the linear speedup theorem for Turing machines states that given any real c > 0 and any k-tape Turing machine solving a problem in time f(n), there is another k-tape machine that solves the same problem in time at most , where k > 1.
If the original machine is non-deterministic, then... | https://en.wikipedia.org/wiki/Linear_speedup_theorem |
## In linear programming
, reduced cost, or opportunity cost, is the amount by which an objective function coefficient would have to improve (so increase for maximization problem, decrease for minimization problem) before it would be possible for a corresponding variable to assume a positive value in the optimal soluti... | https://en.wikipedia.org/wiki/Reduced_cost |
Information theory is the mathematical study of the quantification, storage, and communication of information. The field was established and formalized by Claude Shannon in the 1940s, though early contributions were made in the 1920s through the works of Harry Nyquist and Ralph Hartley. It is at the intersection of ele... | https://en.wikipedia.org/wiki/Information_theory |
Given random variables
$$
X,Y,\ldots
$$
, that are defined on the same probability space, the multivariate or joint probability distribution for
$$
X,Y,\ldots
$$
is a probability distribution that gives the probability that each of
$$
X,Y,\ldots
$$
falls in any particular range or discrete set of values specified ... | https://en.wikipedia.org/wiki/Joint_probability_distribution |
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative... | https://en.wikipedia.org/wiki/Saddle_point |
+Quaternion group multiplication table (simplified form)
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
$$
\{1,i,j,k,-1,-i,-j,-k\}
$$
of the quaternions under multiplication. It is given by the group presentation
$$... | https://en.wikipedia.org/wiki/Quaternion_group |
Robust optimization is a field of mathematical optimization theory that deals with optimization problems in which a certain measure of robustness is sought against uncertainty that can be represented as deterministic variability in the value of the parameters of the problem itself and/or its solution. It is related to,... | https://en.wikipedia.org/wiki/Robust_optimization |
In mathematics, a source for the representation theory of the group of diffeomorphisms of a smooth manifold M is the initial observation that (for M connected) that group acts transitively on M.
## History
A survey paper from 1975 of the subject by Anatoly Vershik, Israel Gelfand and M. I. Graev attributes the original... | https://en.wikipedia.org/wiki/Representation_theory_of_diffeomorphism_groups |
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
$$
A\times B = \{(a,b)\mid a \in A \ \mbox{ and } \ b \in B\}.
$$
A table can be created by taking the C... | https://en.wikipedia.org/wiki/Cartesian_product |
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and P... | https://en.wikipedia.org/wiki/Horner%27s_method |
In mathematics, a field of sets is a mathematical structure consisting of a pair
$$
( X, \mathcal{F} )
$$
consisting of a set
$$
X
$$
and a family
$$
\mathcal{F}
$$
of subsets of
$$
X
$$
called an algebra over that contains the empty set as an element, and is closed under the operations of taking complements i... | https://en.wikipedia.org/wiki/Field_of_sets |
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 topologi... | https://en.wikipedia.org/wiki/Measurable_function |
In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. They were introduced in 1981.
IFS fractals, as they are normally called, can be of any number of dimensions, but are c... | https://en.wikipedia.org/wiki/Iterated_function_system |
In physics, the Coriolis force is a pseudo force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the left of the motion of the object. In one with anticlockwise (or counterclockwise) rotation, the ... | https://en.wikipedia.org/wiki/Coriolis_force |
A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computation are mathematical equation solving and the execution of computer algorithms.
Mechanical or electronic devices (or, historically, people) that perform computations are known as computers. Computer sc... | https://en.wikipedia.org/wiki/Computation |
A Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers. This process is sometimes called the 196-algorithm, after the most famous number associated with the process. In base ten, no Lychrel numbers have been ye... | https://en.wikipedia.org/wiki/Lychrel_number |
Trajectory optimization is the process of designing a trajectory that minimizes (or maximizes) some measure of performance while satisfying a set of constraints. Generally speaking, trajectory optimization is a technique for computing an open-loop solution to an optimal control problem. It is often used for systems whe... | https://en.wikipedia.org/wiki/Trajectory_optimization |
{{DISPLAYTITLE:E8 (mathematics)}}
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification ... | https://en.wikipedia.org/wiki/E8_%28mathematics%29 |
The Bottleneck traveling salesman problem (bottleneck TSP) is a problem in discrete or combinatorial optimization. The problem is to find the Hamiltonian cycle (visiting each node exactly once) in a weighted graph which minimizes the weight of the highest-weight edge of the cycle. It was first formulated by with some... | https://en.wikipedia.org/wiki/Bottleneck_traveling_salesman_problem |
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the diamonds suit in playing cards whi... | https://en.wikipedia.org/wiki/Rhombus |
Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number
$$
\frac{1+\sqrt{5}}{2}
$$
≈ 1.61803399 symbolized by the Greek letter φ) as its base. It is sometimes referred to as base-φ, golden mean base, phi-base, or, colloquially, phinary. Any non-negative real numb... | https://en.wikipedia.org/wiki/Golden_ratio_base |
Destination dispatch is an optimization technique used for multi-elevator installations, in which groups of passengers heading to the same destinations use the same elevators, thereby reducing waiting and travel times. This contrasts with the traditional approach, in which each passenger calls for and enters the first ... | https://en.wikipedia.org/wiki/Destination_dispatch |
In mathematics, the fictitious domain method is a method to find the solution of a partial differential equations on a complicated domain
$$
D
$$
, by substituting a given problem
posed on a domain
$$
D
$$
, with a new problem posed on a simple domain
$$
\Omega
$$
containing
$$
D
$$
.
## General formulation
Assu... | https://en.wikipedia.org/wiki/Fictitious_domain_method |
Nonparametric regression is a form of regression analysis where the predictor does not take a predetermined form but is completely constructed using information derived from the data. That is, no parametric equation is assumed for the relationship between predictors and dependent variable. A larger sample size is neede... | https://en.wikipedia.org/wiki/Nonparametric_regression |
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right... | https://en.wikipedia.org/wiki/Adjoint_functors |
In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation.
## In set theory
, universes are often classes that contain (as elements) all sets for which one hopes ... | https://en.wikipedia.org/wiki/Universe_%28mathematics%29 |
In mathematical logic, the Peano axioms (, ), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including res... | https://en.wikipedia.org/wiki/Peano_axioms |
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The distribution of a Gaussian process is the joint distribution of all those ... | https://en.wikipedia.org/wiki/Gaussian_process |
Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers, whose corresponding number line has a "gap" at each irrational value. In the decimal number system, comp... | https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers |
In mathematics, given two groups, (G,∗) and (H, ·), a group homomorphism from (G,∗) to (H, ·) is a function h : G → H such that for all u and v in G it holds that
$$
h(u*v) = h(u) \cdot h(v)
$$
where the group operation on the left side of the equation is that of G and on the right side that of H.
From this property, o... | https://en.wikipedia.org/wiki/Group_homomorphism |
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.
More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motio... | https://en.wikipedia.org/wiki/Congruence_%28geometry%29 |
In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.
A torus in a compact Lie group G is a compact, connected, abelian Lie subgroup of G (and therefore isomorphic to the standard torus Tn). A maximal torus is one which is maximal amo... | https://en.wikipedia.org/wiki/Maximal_torus |
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the i-th approximation (called an "iterate") is derived from the previous ones.
A specific implementation with termination... | https://en.wikipedia.org/wiki/Iterative_method |
In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by a conservative force (where the work done is independent of path), with that of the total potential energy of the system. Mathematically, the the... | https://en.wikipedia.org/wiki/Virial_theorem |
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations.
## History
used Hecke operators on modular forms in a pap... | https://en.wikipedia.org/wiki/Hecke_operator |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.