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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... | Wikipedia/Yoneda_lemma |
In mathematics, an n-group, or n-dimensional higher group, is a special kind of n-category that generalises the concept of group to higher-dimensional algebra. Here,
n
{\displaystyle n}
may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoà... | Wikipedia/N-group_(category_theory) |
In mathematics, an automorphic L-function is a function L(s,π,r) of a complex variable s, associated to an automorphic representation π of a reductive group G over a global field and a finite-dimensional complex representation r of the Langlands dual group LG of G, generalizing the Dirichlet L-series of a Dirichlet cha... | Wikipedia/Automorphic_L-function |
Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field
Q
{\displaystyle \mathbb {Q} }
of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordel... | Wikipedia/Mordell_conjecture |
In mathematics, the Hardy–Ramanujan–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy, S. Ramanujan, and J. E. Littlewood, who developed it in a series of papers on Waring's problem.
== History ==
The initial idea is usually attributed to the work of Hardy with Srinivasa R... | Wikipedia/Hardy–Littlewood_method |
In mathematics, auxiliary functions are an important construction in transcendental number theory. They are functions that appear in most proofs in this area of mathematics and that have specific, desirable properties, such as taking the value zero for many arguments, or having a zero of high order at some point.
==... | Wikipedia/Auxiliary_function |
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.
There are various types of generati... | Wikipedia/Generating_function |
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic varieties (or a set of algebraic varieties) to the real numbers.
For ins... | Wikipedia/Height_function |
In number theory, the Elliott–Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter D. T. A. Elliott and Heini Halberstam, who stated a specific version of the conjecture in 1968.
One version of the conje... | Wikipedia/Elliott–Halberstam_conjecture |
Multiplicative number theory is a subfield of analytic number theory that deals with prime numbers and with factorization and divisors. The focus is usually on developing approximate formulas for counting these objects in various contexts. The prime number theorem is a key result in this subject. The Mathematics Subjec... | Wikipedia/Multiplicative_number_theory |
Maier's matrix method is a technique in analytic number theory by Helmut Maier that is used to demonstrate the existence of intervals of natural numbers within which the prime numbers are distributed with a certain property. In particular, it has been used to prove Maier's theorem (Maier 1985) and also the existence of... | Wikipedia/Maier's_matrix_method |
In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as
φ
(
n
)
{\displaystyle \varphi (n)}
or
ϕ
(
... | Wikipedia/Totient_function |
In mathematics, the Hardy–Ramanujan–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy, S. Ramanujan, and J. E. Littlewood, who developed it in a series of papers on Waring's problem.
== History ==
The initial idea is usually attributed to the work of Hardy with Srinivasa R... | Wikipedia/Hardy–Littlewood_circle_method |
In mathematics, Montgomery's pair correlation conjecture is a conjecture made by Hugh Montgomery (1973) that the pair correlation between pairs of zeros of the Riemann zeta function (normalized to have unit average spacing) is
1
−
(
... | Wikipedia/Montgomery's_pair_correlation_conjecture |
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables.
For example,
{
3
x
+
... | Wikipedia/Linear_equation_system |
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations f1 = 0, ..., fh = 0 where the fi are polynomials in several variables, say x1, ..., xn, over some field k.
A solution of a polynomial system is a set of values for the xis which belong to some algebraically closed... | Wikipedia/Nonlinear_equation_system |
A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a cycle. For example, the trigonometric functions, which repeat at intervals of
2
... | Wikipedia/Periodic_function |
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle i... | Wikipedia/Trigonometric_equation |
In mathematics, an implicit equation is a relation of the form
R
(
x
1
,
…
,
x
n
)
=
0
,
{\d... | Wikipedia/Implicit_equation |
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the triangle (the hypotenuse), ... | Wikipedia/Sine_function |
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... | Wikipedia/Elementary_functions |
An equation in mathematics is a formula stating that two expressions have the same value.
Equation may also refer to:
Chemical equation, a symbolic representation of a chemical reaction
Equation of time, the difference between solar time, as shown by a sundial, and mean time, as shown by a clock that runs at constant... | Wikipedia/Equation_(disambiguation) |
In applied mathematics, a transcendental equation is an equation over the real (or complex) numbers that is not algebraic, that is, if at least one of its sides describes a transcendental function.
Examples include:
x
... | Wikipedia/Transcendental_equation |
In mathematics, an extraneous solution (or spurious solution) is one which emerges from the process of solving a problem but is not a valid solution to it. A missing solution is a valid one which is lost during the solution process. Both situations frequently result from performing operations that are not invertible fo... | Wikipedia/Extraneous_solution |
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the triangle (the hypotenuse), ... | Wikipedia/Cosine_function |
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the po... | Wikipedia/Octic_equation |
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interactions of atoms and molecules. Electromagnetism can be thought of as a combinatio... | Wikipedia/Electromagnetic_theory |
Gauge theory gravity (GTG) is a theory of gravitation cast in the mathematical language of geometric algebra. To those familiar with general relativity, it is highly reminiscent of the tetrad formalism although there are significant conceptual differences. Most notably, the background in GTG is flat, Minkowski spacet... | Wikipedia/Gauge_theory_gravity |
Geometric algebra is an extension of vector algebra, providing additional algebraic structures on vector spaces, with geometric interpretations.
Vector algebra uses all dimensions and signatures, as does geometric algebra, notably 3+1 spacetime as well as 2 dimensions.
== Basic concepts and operations ==
Geometric al... | Wikipedia/Comparison_of_vector_algebra_and_geometric_algebra |
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
4
x
2
+
2
x
y
−
3
y
... | Wikipedia/Nondegenerate_quadratic_form |
In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product X ⧢ Y of two words X, Y: the sum of all ways of interlacing them. The interlacing is given by the riffle shuffle permutation.
The shuffle algebra on a finite set is the gra... | Wikipedia/Shuffle_algebra |
In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a widely studied branch of theoretical and mathematical physics. Penrose's idea was that twistor space should be the basic arena for physics from which space-time itself should emerge.... | Wikipedia/Twistor_theory |
Plane-based geometric algebra is an application of Clifford algebra to modelling planes, lines, points, and rigid transformations. Generally this is with the goal of solving applied problems involving these elements and their intersections, projections, and their angle from one another in 3D space. Originally growing o... | Wikipedia/Plane-based_geometric_algebra |
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.
The rigid transformations include rotations, translations, reflections, or any sequence of these. Re... | Wikipedia/Rigid_transformation |
In mathematics, a universal geometric algebra is a type of geometric algebra generated by real vector spaces endowed with an indefinite quadratic form. Some authors restrict this to the infinite-dimensional case.
The universal geometric algebra
G
... | Wikipedia/Universal_geometric_algebra |
In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product. That is, for each pair u, v of elements of V, we have
⟨
u
,
v
⟩
=
⟨
T
u
... | Wikipedia/Orthogonal_transformation |
In mathematics, a Grassmann–Cayley algebra is the exterior algebra with an additional product, which may be called the shuffle product or the regressive product.
It is the most general structure in which projective properties are expressed in a coordinate-free way.
The technique is based on work by German mathematicia... | Wikipedia/Grassmann–Cayley_algebra |
Plane-based geometric algebra is an application of Clifford algebra to modelling planes, lines, points, and rigid transformations. Generally this is with the goal of solving applied problems involving these elements and their intersections, projections, and their angle from one another in 3D space. Originally growing o... | Wikipedia/Plane-based_Geometric_Algebra |
Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an n-dimensional base space Rp,q to null vectors in Rp+1,q+1. This allows operations on the base space, including reflections, rotations and translations to be represented using versors of the geomet... | Wikipedia/Conformal_geometric_algebra |
Screw theory is the algebraic calculation of pairs of vectors, also known as dual vectors – such as angular and linear velocity, or forces and moments – that arise in the kinematics and dynamics of rigid bodies.
Screw theory provides a mathematical formulation for the geometry of lines which is central to rigid body ... | Wikipedia/Screw_theory |
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
The prefix super- comes from the theory of supersymmetry in theoretical phy... | Wikipedia/Even_subalgebra |
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.
The rigid transformations include rotations, translations, reflections, or any sequence of these. Re... | Wikipedia/Euclidean_transformation |
In mathematics, an extraneous solution (or spurious solution) is one which emerges from the process of solving a problem but is not a valid solution to it. A missing solution is a valid one which is lost during the solution process. Both situations frequently result from performing operations that are not invertible fo... | Wikipedia/Extraneous_and_missing_solutions |
Trial and error is a fundamental method of problem-solving characterized by repeated, varied attempts which are continued until success, or until the practicer stops trying.
According to W.H. Thorpe, the term was devised by C. Lloyd Morgan (1852–1936) after trying out similar phrases "trial and failure" and "trial and ... | Wikipedia/Trial_and_error |
In computer science, brute-force search or exhaustive search, also known as generate and test, is a very general problem-solving technique and algorithmic paradigm that consists of systematically checking all possible candidates for whether or not each candidate satisfies the problem's statement.
A brute-force algorith... | Wikipedia/Brute-force_search |
In mathematics, the inverse trigonometric functions (occasionally also called antitrigonometric, cyclometric, or arcus functions) are the inverse functions of the trigonometric functions, under suitably restricted domains. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant... | Wikipedia/Inverse_trigonometric_function |
Solving the geodesic equations is a procedure used in mathematics, particularly Riemannian geometry, and in physics, particularly in general relativity, that results in obtaining geodesics. Physically, these represent the paths of (usually ideal) particles with no proper acceleration, their motion satisfying the geodes... | Wikipedia/Solving_the_geodesic_equations |
In logic and computer science, specifically automated reasoning, unification is an algorithmic process of solving equations between symbolic expressions, each of the form Left-hand side = Right-hand side. For example, using x,y,z as variables, and taking f to be an uninterpreted function, the singleton equation set { f... | Wikipedia/Unification_(computer_science) |
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function f(w) = wew, where w is any complex number and ew is the exponential function. The function is named after Johann Lambert, who considered a ... | Wikipedia/Lambert's_W_function |
In mathematical optimization and computer science, a feasible region, feasible set, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem that satisfy the problem's constraints, potentially including inequalities, equalities, and integer constraints. Thi... | Wikipedia/Candidate_solutions |
In probability theory, an experiment or trial (see below) is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. An experiment is said to be random if it has more than one possible outcome, and deterministic if it has only one. A random experiment th... | Wikipedia/Experiment_(probability_theory) |
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with area or volume. In probability theory, they are used to define events with a we... | Wikipedia/Sigma-algebra |
In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity. The central concept of fuzzy measure theory is the fuzzy measure (also capacity, see ), which was introduced by Choquet in 1953 and independently defined by Sugeno in 1... | Wikipedia/Fuzzy_measure_theory |
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable
X
{\displaystyle X}
, or just distribution function of
X
{\displaystyle X}
, evaluated at
x
... | Wikipedia/Cumulative_distribution_function |
In probability theory, a tree diagram may be used to represent a probability space.
A tree diagram may represent a series of independent events (such as a set of coin flips) or conditional probabilities (such as drawing cards from a deck, without replacing the cards). Each node on the diagram represents an event and i... | Wikipedia/Tree_diagram_(probability_theory) |
A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the joint probability distribution of the random variable that (presumably) gen... | Wikipedia/Likelihood_function |
In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.
For a topological space X, the collection of al... | Wikipedia/Borel_algebra |
This page lists articles related to probability theory. In particular, it lists many articles corresponding to specific probability distributions. Such articles are marked here by a code of the form (X:Y), which refers to number of random variables involved and the type of the distribution. For example (2:DC) indicates... | Wikipedia/Catalog_of_articles_in_probability_theory |
In probability theory, an event is a subset of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of o... | Wikipedia/Event_(probability_theory) |
Predictive modelling uses statistics to predict outcomes. Most often the event one wants to predict is in the future, but predictive modelling can be applied to any type of unknown event, regardless of when it occurred. For example, predictive models are often used to detect crimes and identify suspects, after the crim... | Wikipedia/Predictive_modelling |
In abstract algebra, an automorphism of a Lie algebra
g
{\displaystyle {\mathfrak {g}}}
is an isomorphism from
g
{\displaystyle {\mathfrak {g}}}
... | Wikipedia/Automorphism_of_a_Lie_algebra |
In mathematics, a Lie algebra
g
{\displaystyle {\mathfrak {g}}}
is nilpotent if its lower central series terminates in the zero subalgebra. The lower central series is the sequence of subalgebras
... | Wikipedia/Nilpotent_Lie_algebra |
In mathematics, a restricted Lie algebra (or p-Lie algebra) is a Lie algebra over a field of characteristic p>0 together with an additional "pth power" operation. Most naturally occurring Lie algebras in characteristic p come with this structure, because the Lie algebra of a group scheme over a field of characteristic... | Wikipedia/Restricted_Lie_algebra |
In mathematics, a Lie algebra
g
{\displaystyle {\mathfrak {g}}}
is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie algebra
g
... | Wikipedia/Solvable_Lie_algebra |
In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension e is an enlargement of a given Lie algebra g by another Lie algebra h. Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extens... | Wikipedia/Lie_algebra_extension |
In mathematics, in particular abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain complex structures that are compatible. Such objects have applications in deformation theory and rational homotopy theory.
== Definition... | Wikipedia/Differential_graded_Lie_algebra |
In mathematics, a Lie algebra has been generalized in several ways.
== Graded Lie algebra and Lie superalgebra ==
A graded Lie algebra is a Lie algebra with grading. When the grading is
Z
/
2
{\displaystyle \mathbb {Z} /2... | Wikipedia/Quasi-Lie_algebra |
In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is
G
L
(
n
... | Wikipedia/Adjoint_representation_of_a_Lie_algebra |
In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operation. A choice of Cartan decomposition endows any semisimple Lie algebra w... | Wikipedia/Graded_Lie_algebra |
In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers.
Given a complex Lie algebra
g
{\displaystyle {\mathfrak {g}}}
, its conjugate
g
... | Wikipedia/Complex_Lie_algebra |
In mathematics, a Lie algebra has been generalized in several ways.
== Graded Lie algebra and Lie superalgebra ==
A graded Lie algebra is a Lie algebra with grading. When the grading is
Z
/
2
{\displaystyle \mathbb {Z} /2... | Wikipedia/Generalization_of_a_Lie_algebra |
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 Claude Chevalley and S... | Wikipedia/Lie_algebra_cohomology |
In mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five of them:
g
2
,
... | Wikipedia/Exceptional_Lie_algebra |
In mathematics, an orthogonal symmetric Lie algebra is a pair
(
g
,
s
)
{\displaystyle ({\mathfrak {g}},s)}
consisting of a real Lie algebra
g
... | Wikipedia/Orthogonal_symmetric_Lie_algebra |
In mathematics a Lie coalgebra is the dual structure to a Lie algebra.
In finite dimensions, these are dual objects: the dual vector space to a Lie algebra naturally has the structure of a Lie coalgebra, and conversely.
== Definition ==
Let
E
{\displaystyle E}
be a vector s... | Wikipedia/Lie_coalgebra |
In the mathematical field of Lie theory, the radical of a Lie algebra
g
{\displaystyle {\mathfrak {g}}}
is the largest solvable ideal of
g
.
... | Wikipedia/Radical_of_a_Lie_algebra |
In mathematics, an infinitesimal transformation is a limiting form of small transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 skew-symmetric matrix A. It is not the matrix of an actual rotation in space; but ... | Wikipedia/Infinitesimal_transformation |
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2n, F) and Sp(n) for positive integer n and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by
U
... | Wikipedia/Symplectic_Lie_algebra |
In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible.
It is a bialgebra where the multiplication is skew-symmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the com... | Wikipedia/Lie_bialgebra |
In mathematics, the Heisenberg group
H
{\displaystyle H}
, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
(
1
... | Wikipedia/Heisenberg_algebra |
This mathematics-related list provides Mubarakzyanov's classification of low-dimensional real Lie algebras, published in Russian in 1963. It complements the article on Lie algebra in the area of abstract algebra.
An English version and review of this classification was published by Popovych et al. in 2003.
== Mubarak... | Wikipedia/Classification_of_low-dimensional_real_Lie_algebras |
In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after Jean-Louis Loday, is a module L over a commutative ring R with a bilinear product [ _ , _ ] satisfying the Leibniz identity
[
[
a
,
b
... | Wikipedia/Leibniz_algebra |
In mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. It is named after Miguel Ángel Virasoro.
== Structure ==
The Virasoro algebra is spanned by generators Ln for... | Wikipedia/Virasoro_algebra |
The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types
A
n
{\displaystyle A_{n}}
,
B
n
... | Wikipedia/Classical_Lie_algebra |
In abstract algebra, the center of a group G is the set of elements that commute with every element of G. It is denoted Z(G), from German Zentrum, meaning center. In set-builder notation,
Z(G) = {z ∈ G | ∀g ∈ G, zg = gz}.
The center is a normal subgroup,
Z
(
G
)
... | Wikipedia/Center_(group_theory) |
In mathematics, a quasi-Frobenius Lie algebra
(
g
,
[
,
]
,
β
)
{\displaystyle ({\mathfrak {g}},[\,\,\,,\,\,\,],\beta... | Wikipedia/Quasi-Frobenius_Lie_algebra |
In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, hence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie algebra and an abelian Lie algebra:
g
=
... | Wikipedia/Reductive_Lie_algebra |
In mathematics, specifically linear algebra, a degenerate bilinear form f (x, y ) on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V ) given by v ↦ (x ↦ f (x, v )) is not an isomorphism. An equivalent definition when V is finite-dimensional is that it has a non-trivial kernel: t... | Wikipedia/Nondegenerate_form |
In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as rooted trees and vector fields on affine space.
The notion of pre-Lie algebra has been introduced by Murray Gerstenhaber in his work on deformations of algebras.
Pre-Lie algebras have been con... | Wikipedia/Pre-Lie_algebra |
In mathematics, a free Lie algebra over a field K is a Lie algebra generated by a set X, without any imposed relations other than the defining relations of alternating K-bilinearity and the Jacobi identity.
== Definition ==
The definition of the free Lie algebra generated by a set X is as follows:
Let X be a set an... | Wikipedia/Free_Lie_algebra |
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class; "degeneracy" is the condition of being a degenerate case.
The definitions of many classes of composite or structured objects often implicitly incl... | Wikipedia/Degenerate_(mathematics) |
This article summarizes several identities in exterior calculus, a mathematical notation used in differential geometry.
== Notation ==
The following summarizes short definitions and notations that are used in this article.
=== Manifold ===
M
{\displaystyle M}
,
... | Wikipedia/Exterior_calculus_identities |
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 Claude Chevalley and S... | Wikipedia/Lie_algebra_homology |
In the theory of multivariate polynomials, Buchberger's algorithm is a method for transforming a given set of polynomials into a Gröbner basis, which is another set of polynomials that have the same common zeros and are more convenient for extracting information on these common zeros. It was introduced by Bruno Buchber... | Wikipedia/Buchberger's_algorithm |
REDUCE is a general-purpose computer algebra system originally geared towards applications in physics.
The development of REDUCE was started in 1963 by Anthony C. Hearn; since then, many scientists from all over the world have contributed to its development. REDUCE was open-sourced in December 2008 and is available for... | Wikipedia/REDUCE_(computer_algebra_system) |
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects.
The fundamental objects of study in algebrai... | Wikipedia/Computational_algebraic_geometry |
In computer programming, a function (also procedure, method, subroutine, routine, or subprogram) is a callable unit of software logic that has a well-defined interface and behavior and can be invoked multiple times.
Callable units provide a powerful programming tool. The primary purpose is to allow for the decompositio... | Wikipedia/Function_(computer_science) |
Computer Science and Artificial Intelligence Laboratory (CSAIL) is a research institute at the Massachusetts Institute of Technology (MIT) formed by the 2003 merger of the Laboratory for Computer Science (LCS) and the Artificial Intelligence Laboratory (AI Lab). Housed within the Ray and Maria Stata Center, CSAIL is th... | Wikipedia/MIT_Computer_Science_and_Artificial_Intelligence_Laboratory |
In computational number theory and computational algebra, Pollard's kangaroo algorithm (also Pollard's lambda algorithm, see Naming below) is an algorithm for solving the discrete logarithm problem. The algorithm was introduced in 1978 by the number theorist John M. Pollard, in the same paper as his better-known Polla... | Wikipedia/Pollard's_kangaroo_algorithm |
In computer algebra, the Faugère F4 algorithm, by Jean-Charles Faugère, computes the Gröbner basis of an ideal of a multivariate polynomial ring. The algorithm uses the same mathematical principles as the Buchberger algorithm, but computes many normal forms in one go by forming a generally sparse matrix and using fast... | Wikipedia/Faugère_F4_algorithm |
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