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Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related struct... | Wikipedia/math |
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multip... | Wikipedia/algebra |
In mathematics, an algebraic expression is an expression built up from constants (usually, algebraic numbers), variables, and the basic algebraic operations:
addition (+), subtraction (-), multiplication (×), division (÷), whole number powers, and roots (fractional powers).. For example,
3
... | Wikipedia/Algebraic_expression |
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of nth roots (square roots, cube roots, etc.).
A well-known example is the quadratic ... | Wikipedia/Solution_in_radicals |
Multilinear algebra is the study of functions with multiple vector-valued arguments, with the functions being linear maps with respect to each argument. It involves concepts such as matrices, tensors, multivectors, systems of linear equations, higher-dimensional spaces, determinants, inner and outer products, and dual ... | Wikipedia/Multilinear_algebra |
In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems.
Some type theories serve as alternatives to set theory as a foundation of mathematics. Two influential type theories that have been proposed as foundat... | Wikipedia/Type_theory |
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a ... | Wikipedia/Polynomial_function |
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms i... | Wikipedia/Algebra_over_a_field |
In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is
a
x
4
+
b
x
3
... | Wikipedia/Quartic_equation |
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic int... | Wikipedia/Commutative_algebra |
An independent equation is an equation in a system of simultaneous equations which cannot be derived algebraically from the other equations. The concept typically arises in the context of linear equations. If it is possible to duplicate one of the equations in a system by multiplying each of the other equations by some... | Wikipedia/Independent_equation |
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied by various scientific disciplines—such as information theory, electrical ... | Wikipedia/Coding_theory |
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite f... | Wikipedia/Algebraic_number_theory |
In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is a commutative algebra over K that contains V, and is, in some sense, minimal for this property. Here, "minimal" means that S(V) satisfies the following universal property: for every linear map f from V to a commutativ... | Wikipedia/Symmetric_algebra |
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function
f
{\displaystyle f}
, is a member
x
{\displaystyle x}
of the domain of
f
{\displa... | Wikipedia/Zero_of_a_function |
In mathematics, and more specifically in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a ring, but without assuming the existence of a multiplicative identity. The term rng, pronounced like rung (IPA: ), is meant to suggest that it is a ring with... | Wikipedia/Rng_(algebra) |
In algebra, a sextic (or hexic) polynomial is a polynomial of degree six.
A sextic equation is a polynomial equation of degree six—that is, an equation whose left hand side is a sextic polynomial and whose right hand side is zero. More precisely, it has the form:
a
x
... | Wikipedia/Sextic_equation |
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.
The be... | Wikipedia/Field_theory_(mathematics) |
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand.
Galois int... | Wikipedia/Galois_theory |
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" alge... | Wikipedia/Tensor_algebra |
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its ... | Wikipedia/Fundamental_theorem_of_algebra |
In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a real-valued function... | Wikipedia/Newton–Raphson_method |
In mathematics, the graph of a function
f
{\displaystyle f}
is the set of ordered pairs
(
x
,
y
)
{\displaystyle (x,y)}
, where
f
(
x
)
=
... | Wikipedia/Graph_of_a_function |
In mathematics, a cubic function is a function of the form
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
... | Wikipedia/Cubic_function |
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing... | Wikipedia/Representation_theory |
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
Although algebraic topology primarily uses algebr... | Wikipedia/Algebraic_topology |
In mathematics, a basic algebraic operation is any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). These operations may be performed on numbers, in which case they are often called ar... | Wikipedia/Algebraic_operations |
Systems science, also referred to as systems research or simply systems, is a transdisciplinary field that is concerned with understanding simple and complex systems in nature and society, which leads to the advancements of formal, natural, social, and applied attributions throughout engineering, technology, and scienc... | Wikipedia/Systems_science |
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the geometric properties of smooth manifolds, including notions of... | Wikipedia/Differential_topology |
In linear algebra, an eigenvector ( EYE-gən-) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector
v
{\displaystyle \mathbf {v} }
of a linear transformation
... | Wikipedia/Eigenvalues_and_eigenvectors |
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral ... | Wikipedia/Calculus |
Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed two-person zero-sum games, in which a participant's gains or losses are exa... | Wikipedia/Game_theory |
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge ... | Wikipedia/Graph_(discrete_mathematics) |
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for exam... | Wikipedia/Number_theory |
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... | Wikipedia/Horner's_method |
In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (eve... | Wikipedia/Ideal_(ring_theory) |
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the "nondegen... | Wikipedia/Rank_(linear_algebra) |
In mathematics, the composition operator
takes two functions,
f
{\displaystyle f}
and
g
{\displaystyle g}
, and returns a new function
h
(
x
)
:=
(
g
∘... | Wikipedia/Function_composition |
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra... | Wikipedia/Abstract_algebra |
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/System_of_linear_equations |
In mathematics, an inequation is a statement that either an inequality (relations "greater than" and "less than", < and >) or a relation "not equal to" (≠) holds between two values. It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between the two sides, ... | Wikipedia/Inequation |
In mathematics, a constant function is a function whose (output) value is the same for every input value.
== Basic properties ==
As a real-valued function of a real-valued argument, a constant function has the general form y(x) = c or just y = c. For example, the function y(x) = 4 is the specific constant function w... | Wikipedia/Constant_function |
In mathematics, an algebraic structure or algebraic system consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities (known as axioms) that these operations must satis... | Wikipedia/Algebraic_structures |
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication ... | Wikipedia/Non-associative_algebra |
In mathematics, specifically in category theory, F-algebras generalize the notion of algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates all references to quantified elements from the axioms, and these algebraic laws may then be glued together in terms of a single functor F, the signature... | Wikipedia/F-algebra |
In algebra, the theory of equations is the study of algebraic equations (also called "polynomial equations"), which are equations defined by a polynomial. The main problem of the theory of equations was to know when an algebraic equation has an algebraic solution. This problem was completely solved in 1830 by Évariste ... | Wikipedia/Theory_of_equations |
Control theory is a field of control engineering and applied mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any delay, ... | Wikipedia/Control_theory |
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/Algebraic_geometry |
In abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set of all n × n matrices with entries in R is a matrix ring denoted Mn(R) (alternative notations: Matn(R) and Rn×n). Some sets of infinite matrices form infinite matri... | Wikipedia/Matrix_algebra |
In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.
The fundamental... | Wikipedia/General_topology |
In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called arcs, links or lines). A distinction is mad... | Wikipedia/Graph_theory |
In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables.
What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that cons... | Wikipedia/Algebraic_logic |
Linear algebra is the branch of mathematics concerning linear equations such as
a
1
x
1
+
⋯
+
a
n
... | Wikipedia/Linear_algebra |
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such rel... | Wikipedia/Differential_equation |
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... | Wikipedia/Information_theory |
In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory.
In its most basic form, the theorem asserts that given a field extension E/F that is finite ... | Wikipedia/Fundamental_theorem_of_Galois_theory |
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic t... | Wikipedia/Order_theory |
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication ... | Wikipedia/Non-associative_algebras |
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures. The historical ... | Wikipedia/Functional_analysis |
In mathematics, a loop in a topological space X is a continuous function f from the unit interval I = [0,1] to X such that f(0) = f(1). In other words, it is a path whose initial point is equal to its terminal point.
A loop may also be seen as a continuous map f from the pointed unit circle S1 into X, because S1 may be... | Wikipedia/Loop_(topology) |
In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates) is a non-vertical line in the plane.
The characteristic property of linear functions is that when the input variable is changed, the change in the output is pr... | Wikipedia/Linear_function_(calculus) |
Mathematical physics is the development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of ph... | Wikipedia/Mathematical_physics |
Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entails use of algebraic nota... | Wikipedia/Elementary_algebra |
In mathematics, a composition algebra A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N that satisfies
N
(
x
y
)
=
N
(
x
)
N
(
y
... | Wikipedia/Composition_algebra |
Physics is the scientific study of matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. It is one of the most fundamental scientific disciplines. A scientist who specializes in the field of physics is called a physicist.
Physics is one of th... | Wikipedia/Physics |
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemma or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebraic... | Wikipedia/Algebraic_closure |
Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word crystallography is derived from the Ancient Greek word κρύσταλλος (krústallos; "clear ice, rock-crystal"), and γράφειν (gráphein; "to write"). In July 2012, the United Nations recognised the imp... | Wikipedia/Crystallography |
The word 'algebra' is used for various branches and structures of mathematics. For their overview, see Algebra. The name comes from the famous 10th century book Al-Jabr by Al-Khwarizmi.
== The bare word "algebra" ==
The bare word "algebra" may refer to:
Elementary algebra
Abstract algebra
Algebra over a field
In uni... | Wikipedia/Algebra_(disambiguation) |
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which ar... | Wikipedia/Tensor |
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while a... | Wikipedia/Algebraic_variety |
In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation that must be closed by definition. No other properties are imposed.
== History and terminology ==
The term groupoid was introduced in 1927 b... | Wikipedia/Magma_(algebra) |
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout m... | Wikipedia/Group_theory |
Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into two or three major branches: the natural sciences (e.g., physics, chemistry, and biology), which study the physical world; and the social... | Wikipedia/Empirical_sciences |
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of stud... | Wikipedia/Universal_algebra |
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry ... | Wikipedia/Differential_geometry |
In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
"Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a comm... | Wikipedia/Subalgebra |
In mathematics, a linear equation is an equation that may be put in the form
a
1
x
1
+
…
+
a
n
... | Wikipedia/Linear_equation |
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
== History ==
The term "algebraic combinatorics" was introduced in the... | Wikipedia/Algebraic_combinatorics |
In mathematics, an algebraic equation or polynomial equation is an equation of the form
P
=
0
{\displaystyle P=0}
, where P is a polynomial with coefficients in some field, often the field of the rational numbers.
For example,
... | Wikipedia/Polynomial_equation |
In algebra, a septic equation is an equation of the form
a
x
7
+
b
x
6
+
c
x
5
... | Wikipedia/Septic_equation |
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mec... | Wikipedia/Computational_complexity_theory |
In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as
a
x
2
+
b
x
+
c
=
0
,
... | Wikipedia/Quadratic_equation |
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipline.
== Applications to other fields of mathematics ==
Besides algebraic topology, the theory has al... | Wikipedia/Homotopy_theory |
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in almost all areas of mathematics. In particular, many construction... | Wikipedia/Category_theory |
In algebra, a cubic equation in one variable is an equation of the form
a
x
3
+
b
x
2
+
c
x
+
d
=
0
... | Wikipedia/Cubic_equation |
In mathematics, a basic algebraic operation is any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). These operations may be performed on numbers, in which case they are often called ar... | Wikipedia/Algebraic_operation |
In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of ... | Wikipedia/Associative_algebra |
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 arithme... | Wikipedia/Analytic_number_theory |
In mathematics and particularly in algebra, a system of equations (either linear or nonlinear) is called consistent if there is at least one set of values for the unknowns that satisfies each equation in the system—that is, when substituted into each of the equations, they make each equation hold true as an identity. I... | Wikipedia/Consistent_and_inconsistent_equations |
In mathematics, a geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects ca... | Wikipedia/Geometric_algebra |
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probabi... | Wikipedia/Probability_theory |
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, wh... | Wikipedia/Equation |
In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as unknowns. A solution... | Wikipedia/Equation_solving |
In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebr... | Wikipedia/Variety_(universal_algebra) |
In mathematics, a Lie algebra (pronounced LEE) is a vector space
g
{\displaystyle {\mathfrak {g}}}
together with an operation called the Lie bracket, an alternating bilinear map
g
... | Wikipedia/Lie_algebra |
Quantum mechanics is the fundamental physical theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms.: 1.1 It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum... | Wikipedia/Quantum_mechanics |
In mathematics, an algebraic equation or polynomial equation is an equation of the form
P
=
0
{\displaystyle P=0}
, where P is a polynomial with coefficients in some field, often the field of the rational numbers.
For example,
... | Wikipedia/Polynomial_equations |
In mathematics, the exterior algebra or Grassmann algebra of a vector space
V
{\displaystyle V}
is an associative algebra that contains
V
,
{\displaystyle V,}
which has a product, called exterior product or wedge product and... | Wikipedia/Exterior_algebra |
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although computer algebra could be consider... | Wikipedia/Computer_algebra |
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number ... | Wikipedia/Model_theory |
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