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Algebraic structure in mathematical physics
In mathematics and mathematical physics, a factorization algebra is an algebraic structure first introduced by Beilinson and Drinfel'd in an algebro-geometric
Factorization_algebra
(Mathematical) decomposition into a product
example, 3 × 5 is an integer factorization of 15, and (x − 2)(x + 2) is a polynomial factorization of x2 − 4. Factorization is not usually considered meaningful
Factorization
Computational method
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field
Factorization_of_polynomials
Branch of number theory
arithmetic, that every (positive) integer has a factorization into a product of prime numbers, and this factorization is unique up to the ordering of the factors
Algebraic_number_theory
Field of mathematics
numerical linear algebra include obtaining matrix decompositions like the singular value decomposition, the QR factorization, the LU factorization, or the eigendecomposition
Numerical_linear_algebra
Algebra, a branch of mathematics
In homological algebra, a branch of mathematics, a matrix factorization is a tool used to study infinitely long resolutions, generally over commutative
Matrix factorization (algebra)
Matrix_factorization_(algebra)
Type of integral domain
unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields Formally, a unique factorization domain
Unique_factorization_domain
Algebraic structure with addition and multiplication
⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields A ring is a set
Ring_(mathematics)
Algorithms for matrix decomposition
matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a
Non-negative matrix factorization
Non-negative_matrix_factorization
Algebraic structure
factorization, as there are factorization algorithms that have a polynomial complexity. They are implemented in most general purpose computer algebra
Polynomial_ring
Polynomial with no repeated root
fractions. Square-free factorization is the first step of the polynomial factorization algorithms that are implemented in computer algebra systems. Therefore
Square-free_polynomial
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition
Factorization of polynomials over finite fields
Factorization_of_polynomials_over_finite_fields
Finite extension of the rationals
study of rings of algebraic integers. For general Dedekind rings, in particular rings of integers, there is a unique factorization of ideals into a product
Algebraic_number_field
Matrix decomposition
In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of
QR_decomposition
algebra Factorization algebra Genetic algebra Geometric algebra Gerstenhaber algebra Graded algebra Griess algebra Group algebra Group algebra of a locally
List_of_algebras
Algebra used in 2D conformal field theories and string theory
D-module-theoretic objects called chiral algebras introduced by Alexander Beilinson and Vladimir Drinfeld and factorization algebras, also introduced by Beilinson
Vertex_operator_algebra
Integers have unique prime factorizations
fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 is
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Statement in complex analysis
mathematics, and particularly in the field of complex analysis, the Hadamard factorization theorem asserts that every entire function with finite order can be
Hadamard factorization theorem
Hadamard_factorization_theorem
Every polynomial has a real or complex root
proof: https://mizar.org/version/current/html/polynom5.html#T74 Prime Factorization Method — Prime Factorization Method explained in detail with Example.
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Matrix decomposition method
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced /ʃəˈlɛski/ shə-LES-kee) is a decomposition of a Hermitian, positive-definite
Cholesky_decomposition
Representation of a matrix as a product
the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices
Matrix_decomposition
Branch of mathematics
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems
Algebra
Scientific area at the interface between computer science and mathematics
differentiation using the chain rule, polynomial factorization, indefinite integration, etc. Computer algebra is widely used to experiment in mathematics and
Computer_algebra
Chiral algebras can also be reformulated as factorization algebras. Chiral homology Chiral Lie algebra Beilinson, Alexander (2004). Chiral algebras. Colloquium
Chiral_algebra
Accomplishments in factoring large integers
Integer factorization is the process of determining which prime numbers divide a given positive integer. Doing this quickly has applications in cryptography
Integer_factorization_records
Concept in numerical linear algebra
numerical linear algebra, an incomplete LU factorization (abbreviated as ILU) of a matrix is a sparse approximation of the LU factorization often used as
Incomplete_LU_factorization
Number divisible only by 1 and itself
although there are many different ways of finding a factorization using an integer factorization algorithm, they all must produce the same result. Primes
Prime_number
Exact sequence used to describe the structure of an object
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact
Resolution_(algebra)
Dimension of the column space of a matrix
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
Rank_(linear_algebra)
Branch of mathematics
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations
Abstract_algebra
Decomposition of a number into a product
called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer
Integer_factorization
Type of matrix factorization
In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix
LU_decomposition
Algebraic construction
integers, every element has a factorization into irreducible elements, but the ring need not have the property of unique factorization: for example, in the ring
Ring_of_integers
Mathematical polynomial factorization
irreducible polynomial, so this factorization of infinitely many of its values cannot be extended to a factorization of Φ 4 {\displaystyle \Phi _{4}}
Sophie_Germain's_identity
Algebra based on a vector space with a quadratic form
mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure
Clifford_algebra
Commutative ring with no zero divisors other than zero
⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields An integral domain
Integral_domain
Theorem in complex analysis
fundamental theorem of algebra: any polynomial function p ( z ) {\displaystyle p(z)} in the complex plane has a factorization p ( z ) = a ∏ n ( z − c
Weierstrass factorization theorem
Weierstrass_factorization_theorem
mathematics, a noncommutative unique factorization domain is a noncommutative ring with the unique factorization property. The ring of Hurwitz quaternions
Noncommutative unique factorization domain
Noncommutative_unique_factorization_domain
Concept in number theory
In number theory, an aurifeuillean factorization, named after Léon-François-Antoine Aurifeuille, is factorization of certain integer values of the cyclotomic
Aurifeuillean_factorization
Mathematical software
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in
Computer_algebra_system
Matrix decomposition
In linear algebra, eigendecomposition (also known as eigenvalue decomposition or EVD) is a factorization of a matrix A {\displaystyle A} into a canonical
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Algebraic structure where all polynomials have roots
⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields As an example,
Algebraically_closed_field
Algebraic structure used in analysis
In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket
Lie_algebra
Prime number of the form 2^n – 1
– Factorization of Mersenne numbers Mn (n up to 1280) Factorization of completely factored Mersenne numbers The Cunningham project, factorization of
Mersenne_prime
Algebraic structure with only one element
In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton
Zero_object_(algebra)
the factorization p ( t ) = q ( t ) q ¯ ( t ) {\displaystyle p(t)=q(t){\bar {q}}(t)} called the spectral factorization (or Wiener-Hopf factorization) of
Polynomial matrix spectral factorization
Polynomial_matrix_spectral_factorization
Algebraic ring without a multiplicative identity
⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields Formally, a rng
Rng_(algebra)
Algebraic study of differential equations
Horowitz-Ostrogradsky algorithm, squarefree factorization and splitting factorization to special and normal polynomials. Differential algebra can determine if a set of
Differential_algebra
Theorem of mathematics
mathematics, the Cohen–Hewitt factorization theorem states that if V {\displaystyle V} is a left module over a Banach algebra B {\displaystyle B} with a
Cohen–Hewitt factorization theorem
Cohen–Hewitt_factorization_theorem
Polynomial without nontrivial factorization
essentially unique factorization into prime or irreducible factors. When the coefficient ring is a field or other unique factorization domain, an irreducible
Irreducible_polynomial
Number in {..., –2, –1, 0, 1, 2, ...}
numbers. In algebraic number theory, integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In
Integer
About products of primitive polynomials
integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to the fundamental theorem
Gauss's_lemma_(polynomials)
Approach to public-key cryptography
in several integer factorization algorithms that have applications in cryptography, such as Lenstra elliptic-curve factorization. The use of elliptic
Elliptic-curve_cryptography
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
Free_Lie_algebra
Matrix equal to its transpose
symmetric form", Linear Algebra Appl., 57: 215–226, doi:10.1016/0024-3795(84)90189-7 Bosch, A. J. (1986). "The factorization of a square matrix into two
Symmetric_matrix
Lenstra elliptic-curve factorization Galbraith, Steven (2012). "Primality Testing and Integer Factorisation using Algebraic Groups". Mathematics of Public
Algebraic-group factorisation algorithm
Algebraic-group_factorisation_algorithm
Integer having a non-trivial divisor
a number is prime or composite, which do not necessarily reveal the factorization of a composite input. Grimm's conjecture states that, for every set
Composite_number
computer algebra abilities. PARI/GP is a widely used computer algebra system designed for fast computations in number theory (factorizations, algebraic number
List of numerical-analysis software
List_of_numerical-analysis_software
Field of knowledge
including number theory (the study of integers and their properties), algebra (the study of operations and the structures they form), geometry (the study
Mathematics
Algebraic structure
Dedekind domains, which allows replacing unique factorization of elements with unique factorization of ideals. In particular, many Z [ ζ p ] , {\displaystyle
Principal_ideal_domain
In algebra, the content of a nonzero polynomial with integer coefficients (or, more generally, with coefficients in a unique factorization domain) is
Primitive_part_and_content
comparison of linear algebra software libraries, either specialized or general purpose libraries with significant linear algebra coverage. Matrix types
Comparison of linear algebra libraries
Comparison_of_linear_algebra_libraries
Concept in linear algebra
\mathbb {F} ^{m\times n}} , a rank decomposition or rank factorization of A is a factorization of A of the form A = CF, where C ∈ F m × r {\displaystyle
Rank_factorization
since become a tool for constructing interesting C*-algebras whose structure reflects the factorization rules. Some compact quantum groups like S U q ( 3
K-graph_C*-algebra
Middle-school math class in the U.S.
for the study of algebra. Usually, Algebra I is taught in the 8th or 9th grade. As an intermediate stage after arithmetic, pre-algebra helps students pass
Pre-algebra
Orthonormalization of a set of vectors
In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a way of finding a set of two
Gram–Schmidt_process
Computer system for solving algebra problems
polynomials. Integer factorization algorithms include the Elliptic Curve Method, the Quadratic sieve and the Number field sieve. Algebraic number theory Magma
Magma (computer algebra system)
Magma_(computer_algebra_system)
1966 mathematics textbook by Serge Lang
introduces the polynomial ideal as an algebraic structure, proving basic results about division and factorization before applying ideals in the decomposition
Linear_Algebra_(book)
Elements taken to zero by a homomorphism
In algebra, the kernel of a homomorphism is the relation describing how elements in the domain of the homomorphism become related in the image. A homomorphism
Kernel_(algebra)
Submodule of a mathematical ring
theorem can be generalized to ideals. There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number
Ideal_(ring_theory)
In number theory, measure of non-unique factorization
domain, and hence from satisfying unique prime factorization (Dedekind domains are unique factorization domains if and only if they are principal ideal
Ideal_class_group
Group of mathematical theorems
groups is (normal epi, mono)-factorizable; in other words, the normal epimorphisms and the monomorphisms form a factorization system for the category. This
Isomorphism_theorems
Branch of pure mathematics
composite numbers. Factorization is a method of expressing a number as a product. Specifically in number theory, integer factorization is the decomposition
Number_theory
Theorem
mathematics, Stinespring's dilation theorem, also called Stinespring's factorization theorem, named after W. Forrest Stinespring,[when?] is a result from
Stinespring_dilation_theorem
Ring that is also a vector space or a module
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
Associative_algebra
Algebraic structure with addition, multiplication, and division
and factorization, Cambridge University Press, ISBN 0-521-33718-6, Zbl 0674.13008 Steinitz, Ernst (1910), "Algebraische Theorie der Körper" [Algebraic Theory
Field_(mathematics)
Mathematical object studied in the field of algebraic geometry
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as
Algebraic_variety
Every nonzero proper ideal in the ring of integers of a number field factorizes uniquely
is a Dedekind domain. Keith Conrad, Ideal factorization Hilbert, D. (20 August 1998). The Theory of Algebraic Number Fields. Trans. by Iain T. Adamson
Fundamental theorem of ideal theory in number fields
Fundamental_theorem_of_ideal_theory_in_number_fields
Basic concepts of algebra
{b^{2}-4ac}}}{2a}}}}}} Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted
Elementary_algebra
divisors). Every unique factorization domain obviously satisfies these two conditions, but neither implies unique factorization. Cohn, P. M. (1968). "Bezout
Atomic_domain
In algebraic geometry, the Stein factorization, introduced by Karl Stein (1956) for the case of complex spaces, states that a proper morphism of schemes
Stein_factorization
Function in algebra
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size
Valuation_(algebra)
Infinite dimensional Lie group
Birkhoff factorization of loops in the character group of the associated Hopf algebra. The models considered by Kreimer (1999) had Hopf algebra H and character
Butcher_group
Algorithm in number theory
theory, Dixon's factorization method (also Dixon's random squares method or Dixon's algorithm) is a general-purpose integer factorization algorithm; it
Dixon's_factorization_method
Mathematical structure in abstract algebra
mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra; read as "star-algebra") is a mathematical structure consisting of
*-algebra
Algebraic structure
⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields A ring is a set
Commutative_ring
Method in computational algebra
Berlekamp, Elwyn R. (1968). Algebraic Coding Theory. McGraw Hill. ISBN 0-89412-063-8. Knuth, Donald E (1997). "4.6.2 Factorization of Polynomials". Seminumerical
Berlekamp's_algorithm
Algorithm for computing greatest common divisors
essential step in several integer factorization algorithms, such as Pollard's rho algorithm, Shor's algorithm, Dixon's factorization method and the Lenstra elliptic
Euclidean_algorithm
Algorithm for factoring polynomials over finite fields
the PARI/GP computer algebra system as the factormod() function (formerly factorcantor()). Polynomial factorization Factorization of polynomials over finite
Cantor–Zassenhaus_algorithm
Type of mathematical expression
form, called factorization is, in general, too difficult to be done by hand-written computation. However, efficient polynomial factorization algorithms
Polynomial
Natural number
and any other power of 1 is always equal to 1 itself. More generally, in algebra, it denotes the multiplicative identity in any unital ring or field. An
1
This glossary of linear algebra is a list of definitions and terms relevant to the field of linear algebra, the branch of mathematics concerned with linear
Glossary_of_linear_algebra
Complex number whose real and imaginary parts are both integers
every unique factorization domain, every Gaussian integer may be factored as a product of a unit and Gaussian primes, and this factorization is unique up
Gaussian_integer
Irish mathematician
recent work on formalism for quantum field theory uses the idea of a factorization algebra to describe the local structure of quantum observables, such as
Kevin_Costello
Topics referred to by the same term
coalition (French: Union des forces démocratiques) Unique factorization domain, in abstract algebra United Front Department, a North Korean government body
UFD
Generalization of vector spaces from fields to rings
central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. In a vector space, the
Module_(mathematics)
(PARI/GP is a widely used computer algebra system designed for fast computations in number theory (factorizations, algebraic number theory, elliptic curves
List of arbitrary-precision arithmetic software
List_of_arbitrary-precision_arithmetic_software
Reduction of a ring by one of its ideals
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite
Quotient_ring
Mathematical technique
In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that
Matrix factorization of a polynomial
Matrix_factorization_of_a_polynomial
Factorization method based on the difference of two squares
difference is algebraically factorable as ( a + b ) ( a − b ) {\displaystyle (a+b)(a-b)} ; if neither factor equals one, it is a proper factorization of N. Each
Fermat's_factorization_method
FACTORIZATION ALGEBRA
FACTORIZATION ALGEBRA
FACTORIZATION ALGEBRA
FACTORIZATION ALGEBRA
Girl/Female
Biblical
The house of corn, or of fish.
Boy/Male
Indian, Sanskrit
Sprung from Twisted Hair; Spring; Fountain
Female
Greek
(Άνθεια) Greek name ANTHEIA means "flower." In mythology, this is the name of a goddess of flowers, gardens, love, marshes, and swamps. She was worshiped on Crete.
Boy/Male
Indian, Punjabi, Sikh
One Absorbed in Eternal One
Female
Basque
, rose.
Boy/Male
Arabic, Muslim
A Person who Upholds the Truth; Just
Boy/Male
Australian, French, German, Greek, Italian
Place Name
Boy/Male
Arabic, Muslim
Sample; Specimen
Girl/Female
Gujarati, Hindu, Indian, Kannada, Marathi, Sanskrit, Sindhi, Telugu
Motivate for Truth; Vigilance
Boy/Male
Indian, Punjabi, Sikh
Godly Face
FACTORIZATION ALGEBRA
FACTORIZATION ALGEBRA
FACTORIZATION ALGEBRA
FACTORIZATION ALGEBRA
FACTORIZATION ALGEBRA
n.
Any particular system of characters, symbols, or abbreviated expressions used in art or science, to express briefly technical facts, quantities, etc. Esp., the system of figures, letters, and signs used in arithmetic and algebra to express number, quantity, or operations.
n.
Anything which is required to be done; as, in geometry, to bisect a line, to draw a perpendicular; or, in algebra, to find an unknown quantity.
n.
That branch of algebra which treats of quadratic equations.
a.
That may be sqyared, or reduced to an equivalent square; -- said of a surface when the area limited by a curve can be exactly found, and expressed in a finite number of algebraic terms.
adv.
By algebraic process.
n.
A single algebraic expression; that is, an expression unconnected with any other by the sign of addition, substraction, equality, or inequality.
n.
A homogeneous algebraic function of two or more variables, in general containing only positive integral powers of the variables, and called quadric, cubic, quartic, etc., according as it is of the second, third, fourth, fifth, or a higher degree. These are further called binary, ternary, quaternary, etc., according as they contain two, three, four, or more variables; thus, the quantic / is a binary cubic.
v. t.
To perform by algebra; to reduce to algebraic form.
n.
One of the terms in an algebraic expression.
a.
A branch of algebra which relates to the direct search for unknown quantities.
n.
Either of the two parts of an algebraic equation, connected by the sign of equality.
v. t.
To change, as an algebraic expression or geometrical figure, into another from without altering its value.
a.
Susceptible of being solved; as, a soluble algebraic problem; susceptible of being disentangled, unraveled, or explained; as, the mystery is perhaps soluble.
a.
That can be passed over in a single course; -- said of a curve when the coordinates of the point on the curve can be expressed as rational algebraic functions of a single parameter /.
a.
Alt. of Algebraical
n.
An algebraic curve, so called from its resemblance to a heart.
n.
An expression of the condition of equality between two algebraic quantities or sets of quantities, the sign = being placed between them; as, a binomial equation; a quadratic equation; an algebraic equation; a transcendental equation; an exponential equation; a logarithmic equation; a differential equation, etc.
n.
One versed in algebra.
n.
The quotient of two vectors, or of two directed right lines in space, considered as depending on four geometrical elements, and as expressible by an algebraic symbol of quadrinomial form.
a.
Of or pertaining to algebra; containing an operation of algebra, or deduced from such operation; as, algebraic characters; algebraical writings.