352 relations: Absolute convergence, Absolute value, Abstract algebra, Academic Press, Addison-Wesley, Addition, Adjacency matrix, Adjugate matrix, Admittance, Alfred North Whitehead, Alfred Tarski, Algebraically closed field, American Mathematical Society, Analysis of algorithms, APL (programming language), Arthur Cayley, Associative algebra, Associative property, Asterisk, Augustin-Louis Cauchy, Axiom, Axiom of reducibility, Basis (linear algebra), Bertrand Russell, Bijection, Bilinear form, Binary operation, Block matrix, Bracket, Cabibbo–Kobayashi–Maskawa matrix, Cambridge University Press, Carl Friedrich Gauss, Carl Gustav Jacob Jacobi, Category (mathematics), Cayley–Hamilton theorem, Characteristic polynomial, Chemical bond, Chemistry, Chinese mathematics, Classical mechanics, Clifford algebra, Closed set, Coding theory, Commutative property, Commutative ring, Complex conjugate, Complex number, Computer algebra system, Computer graphics, Computer language, ..., Condition number, Conformable matrix, Conjugate gradient method, Conjugate transpose, Continuous function, Control theory, Covariance matrix, Cramer's rule, Critical point (mathematics), Degree of a polynomial, Density matrix, Derivative, Descriptive statistics, Design matrix, Determinant, Diagonal matrix, Diagonalizable matrix, Differentiable function, Dimension, Dimension (vector space), Dimensionality reduction, Dimensionless quantity, Direct stiffness method, Distance matrix, Distributive property, Division (mathematics), Document-term matrix, Dot product, Dover Publications, Dual space, Dutch language, Economics, Eigendecomposition of a matrix, Eigenvalues and eigenvectors, Electrical impedance, Electromagnetism, Electronics, Elementary matrix, Elementary particle, Ellipse, Elliptic partial differential equation, Emphasis (typography), Empty product, Encryption, Endomorphism ring, Equations of motion, Equivalence of categories, Euclidean space, Euclidean vector, Exponential function, Expression (mathematics), Extension (predicate logic), Ferdinand Georg Frobenius, Fermion, Field (mathematics), Finite element method, Finite field, Finite group, Fock matrix, Free variables and bound variables, Function composition, Functional analysis, Gabriel Cramer, Game theory, Gamma matrices, Gauge theory, Gaussian elimination, Gell-Mann matrices, General linear group, Generalized inverse, Geometrical optics, Geometry, German language, Gerolamo Cardano, Gottfried Wilhelm Leibniz, Gotthold Eisenstein, Gram–Schmidt process, Graph (discrete mathematics), Graph theory, Group (mathematics), Group representation, Hadamard product (matrices), Hartree–Fock method, Hermitian matrix, Hessian matrix, Hilbert space, Hill cipher, Historia Mathematica, Houghton Mifflin Harcourt, HP 9800 series, Hyperbola, Hypercomplex number, Hyperlink, Identity matrix, If and only if, Image (mathematics), Imperial College London, Implicit function theorem, Independent equation, Indeterminate (variable), Infinitesimal, Infinity, Inner product space, Inverse element, Invertible matrix, Isomorphism, Jacobian matrix and determinant, James Joseph Sylvester, Japanese mathematics, Johan de Witt, John von Neumann, John Wiley & Sons, Jordan normal form, Karl Weierstrass, Kernel (linear algebra), Kronecker product, Laplace expansion, Leibniz formula for determinants, Lens (optics), Leopold Kronecker, Letter case, Light, Limit of a sequence, Line (geometry), Linear combination, Linear differential equation, Linear equation, Linear function, Linear independence, Linear least squares (mathematics), Linear map, Linear system, List of matrices, List of numerical analysis software, Logarithm of a matrix, Logical equivalence, Logical matrix, Lorentz group, LU decomposition, Main diagonal, Marcel Dekker, Markov chain, Mass, Mathematical analysis, Mathematical formulation of quantum mechanics, Mathematics, Matrix addition, Matrix calculus, Matrix equivalence, Matrix exponential, Matrix field, Matrix function, Matrix mechanics, Matrix multiplication, Matrix multiplication algorithm, Matrix norm, Matrix normal distribution, Matrix ring, Matrix splitting, Max Born, Maxima and minima, Mesh analysis, Minor (linear algebra), MIT Press, Module (mathematics), Molecular orbital, Molecule, Monic polynomial, Multiplication, Network theory, Nodal analysis, Normal matrix, Normal mode, Normal-form game, Number, Number theory, Numerical analysis, Numerical linear algebra, Numerical stability, O'Reilly Media, Optics, Orbital overlap, Orthogonal group, Orthogonal matrix, Orthogonality, Orthonormality, Oxford University Press, PageRank, Parallelogram, Partial derivative, Partial differential equation, Particle accelerator, Pascual Jordan, Pauli matrices, Periodic matrix set, Permutation matrix, Physics, Polynomial, Polynomial ring, Position (vector), Positive-definite matrix, Prentice Hall, Probability distribution, Probability theory, Probability vector, Quadratic form, Quadratic programming, Quantum chromodynamics, Quantum electrodynamics, Quantum field theory, Quantum mechanics, Quark, Quaternion, Random matrix, Random variable, Rank (linear algebra), Rank–nullity theorem, Rational number, Ray (optics), Ray transfer matrix analysis, Real number, Rectangle, Reflection (mathematics), Regular representation, Representation theory, Rigid body dynamics, Ring (mathematics), Roothaan equations, Rotation (mathematics), Rotation matrix, Row and column vectors, Row echelon form, Rule of Sarrus, S&P Global, S-matrix, Scalar (mathematics), Scalar multiplication, Scaling (geometry), Schur decomposition, Second derivative, Seki Takakazu, Sequence, Set (mathematics), Shear mapping, Singular-value decomposition, Skew-symmetric matrix, Sparse matrix, Special linear group, Special unitary group, Spectral theorem, Spectroscopy, Spin group, Spinor, Springer Science+Business Media, Square matrix, Square root of a matrix, Squeeze mapping, Statistics, Stochastic matrix, Strassen algorithm, Subgroup, Subtraction, Superalgebra, Supermatrix, Sylvester equation, Symbol (formal), Symmetric group, Symmetric matrix, Symmetry, System of linear equations, Taylor series, TED (conference), Tensor, Text mining, Tf–idf, The Nine Chapters on the Mathematical Art, Thesaurus, Trace (linear algebra), Transformation (function), Transformation matrix, Transpose, Transpose of a linear map, Triangular matrix, Truth table, Unit square, Unit vector, Unitary matrix, University of Minnesota, Upper and lower bounds, Variance, Vector space, Weak interaction, Website, Werner Heisenberg, Wilhelm Jordan (geodesist), William Rowan Hamilton, Wolfram Mathematica, Zero matrix, Zero object (algebra), 0, 2 × 2 real matrices, 3D modeling. Expand index (302 more) »

## Absolute convergence

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite.

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## Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

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## Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

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## Academic Press

Academic Press is an academic book publisher.

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## Addison-Wesley

Addison-Wesley is a publisher of textbooks and computer literature.

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## Addition

Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division.

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## Adjacency matrix

In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph.

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## Adjugate matrix

In linear algebra, the adjugate, classical adjoint, or adjunct of a square matrix is the transpose of its cofactor matrix.

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## Admittance

In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow.

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## Alfred North Whitehead

Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher.

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## Alfred Tarski

Alfred Tarski (January 14, 1901 – October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews,, School of Mathematics and Statistics, University of St Andrews.

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## Algebraically closed field

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

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## American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

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## Analysis of algorithms

In computer science, the analysis of algorithms is the determination of the computational complexity of algorithms, that is the amount of time, storage and/or other resources necessary to execute them.

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## APL (programming language)

APL (named after the book A Programming Language) is a programming language developed in the 1960s by Kenneth E. Iverson.

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## Arthur Cayley

Arthur Cayley F.R.S. (16 August 1821 – 26 January 1895) was a British mathematician.

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## Associative algebra

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.

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## Associative property

In mathematics, the associative property is a property of some binary operations.

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## Asterisk

An asterisk (*); from Late Latin asteriscus, from Ancient Greek ἀστερίσκος, asteriskos, "little star") is a typographical symbol or glyph. It is so called because it resembles a conventional image of a star. Computer scientists and mathematicians often vocalize it as star (as, for example, in the A* search algorithm or C*-algebra). In English, an asterisk is usually five-pointed in sans-serif typefaces, six-pointed in serif typefaces, and six- or eight-pointed when handwritten. It is often used to censor offensive words, and on the Internet, to indicate a correction to a previous message. The asterisk is derived from the need of the printers of family trees in feudal times for a symbol to indicate date of birth. The original shape was seven-armed, each arm like a teardrop shooting from the center. In computer science, the asterisk is commonly used as a wildcard character, or to denote pointers, repetition, or multiplication.

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## Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics.

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## Axiom

An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

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## Axiom of reducibility

The axiom of reducibility was introduced by Bertrand Russell in the early 20th century as part of his ramified theory of types.

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## Basis (linear algebra)

In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.

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## Bertrand Russell

Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, historian, writer, social critic, political activist, and Nobel laureate.

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## Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

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## Bilinear form

In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map, where K is the field of scalars.

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## Binary operation

In mathematics, a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set.

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## Block matrix

In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices.

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## Bracket

A bracket is a tall punctuation mark typically used in matched pairs within text, to set apart or interject other text.

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## Cabibbo–Kobayashi–Maskawa matrix

In the Standard Model of particle physics, the Cabibbo–Kobayashi–Maskawa matrix, CKM matrix, quark mixing matrix, or KM matrix is a unitary matrix which contains information on the strength of flavour-changing weak decays.

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## Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

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## Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

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## Carl Gustav Jacob Jacobi

Carl Gustav Jacob Jacobi (10 December 1804 – 18 February 1851) was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.

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## Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.

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## Cayley–Hamilton theorem

In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation.

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## Characteristic polynomial

In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots.

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## Chemical bond

A chemical bond is a lasting attraction between atoms, ions or molecules that enables the formation of chemical compounds.

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## Chemistry

Chemistry is the scientific discipline involved with compounds composed of atoms, i.e. elements, and molecules, i.e. combinations of atoms: their composition, structure, properties, behavior and the changes they undergo during a reaction with other compounds.

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## Chinese mathematics

Mathematics in China emerged independently by the 11th century BC.

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## Classical mechanics

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.

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## Clifford algebra

In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra.

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## Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

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## Coding theory

Coding theory is the study of the properties of codes and their respective fitness for specific applications.

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## Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

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## Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

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## Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

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## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

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## Computer algebra system

A computer algebra system (CAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists.

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## Computer graphics

Computer graphics are pictures and films created using computers.

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## Computer language

A computer language is a method of communication with a computer.

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## Condition number

In the field of numerical analysis, the condition number of a function with respect to an argument measures how much the output value of the function can change for a small change in the input argument.

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## Conformable matrix

In mathematics, a matrix is conformable if its dimensions are suitable for defining some operation (e.g. addition, multiplication, etc.).

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## Conjugate gradient method

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite.

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## Conjugate transpose

In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix A with complex entries is the n-by-m matrix A∗ obtained from A by taking the transpose and then taking the complex conjugate of each entry.

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## Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

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## Control theory

Control theory in control systems engineering deals with the control of continuously operating dynamical systems in engineered processes and machines.

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## Covariance matrix

In probability theory and statistics, a covariance matrix (also known as dispersion matrix or variance–covariance matrix) is a matrix whose element in the i, j position is the covariance between the i-th and j-th elements of a random vector.

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## Cramer's rule

In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution.

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## Critical point (mathematics)

In mathematics, a critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is 0.

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## Degree of a polynomial

The degree of a polynomial is the highest degree of its monomials (individual terms) with non-zero coefficients.

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## Density matrix

A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states.

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## Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

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## Descriptive statistics

A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features of a collection of information, while descriptive statistics in the mass noun sense is the process of using and analyzing those statistics.

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## Design matrix

In statistics, a design matrix, also known as model matrix or regressor matrix, is a matrix of values of explanatory variables of a set of objects, often denoted by X. Each row represents an individual object, with the successive columns corresponding to the variables and their specific values for that object.

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## Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

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## Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.

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## Diagonalizable matrix

In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P−1AP is a diagonal matrix.

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## Differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

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## Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

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## Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.

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## Dimensionality reduction

In statistics, machine learning, and information theory, dimensionality reduction or dimension reduction is the process of reducing the number of random variables under consideration by obtaining a set of principal variables.

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## Dimensionless quantity

In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned.

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## Direct stiffness method

As one of the methods of structural analysis, the direct stiffness method, also known as the matrix stiffness method, is particularly suited for computer-automated analysis of complex structures including the statically indeterminate type.

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## Distance matrix

In mathematics, computer science and especially graph theory, a distance matrix is a square matrix (two-dimensional array) containing the distances, taken pairwise, between the elements of a set.

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## Distributive property

In abstract algebra and formal logic, the distributive property of binary operations generalizes the distributive law from boolean algebra and elementary algebra.

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## Division (mathematics)

Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication.

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## Document-term matrix

A document-term matrix or term-document matrix is a mathematical matrix that describes the frequency of terms that occur in a collection of documents.

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## Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

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## Dover Publications

Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche.

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## Dual space

In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

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## Dutch language

The Dutch language is a West Germanic language, spoken by around 23 million people as a first language (including the population of the Netherlands where it is the official language, and about sixty percent of Belgium where it is one of the three official languages) and by another 5 million as a second language.

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## Economics

Economics is the social science that studies the production, distribution, and consumption of goods and services.

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## Eigendecomposition of a matrix

In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors.

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## Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.

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## Electrical impedance

Electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied.

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## Electromagnetism

Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles.

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## Electronics

Electronics is the discipline dealing with the development and application of devices and systems involving the flow of electrons in a vacuum, in gaseous media, and in semiconductors.

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## Elementary matrix

In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation.

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## Elementary particle

In particle physics, an elementary particle or fundamental particle is a particle with no substructure, thus not composed of other particles.

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## Ellipse

In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.

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## Elliptic partial differential equation

Second order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic.

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## Emphasis (typography)

In typography, emphasis is the strengthening of words in a text with a font in a different style from the rest of the text, to highlight them.

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## Empty product

In mathematics, an empty product, or nullary product, is the result of multiplying no factors.

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## Encryption

In cryptography, encryption is the process of encoding a message or information in such a way that only authorized parties can access it and those who are not authorized cannot.

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## Endomorphism ring

In abstract algebra, the endomorphism ring of an abelian group X, denoted by End(X), is the set of all endomorphisms of X (i.e., the set of all homomorphisms of X into itself) endowed with an addition operation defined by pointwise addition of functions and a multiplication operation defined by function composition.

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## Equations of motion

In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.

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## Equivalence of categories

In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same".

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## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

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## Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction.

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## Exponential function

In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.

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## Expression (mathematics)

In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.

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## Extension (predicate logic)

The extension of a predicatea truth-valued functionis the set of tuples of values that, used as arguments, satisfy the predicate.

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## Ferdinand Georg Frobenius

Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory.

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## Fermion

In particle physics, a fermion is a particle that follows Fermi–Dirac statistics.

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## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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## Finite element method

The finite element method (FEM), is a numerical method for solving problems of engineering and mathematical physics.

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## Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

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## Finite group

In abstract algebra, a finite group is a mathematical group with a finite number of elements.

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## Fock matrix

In the Hartree–Fock method of quantum mechanics, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of basis vectors.

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## Free variables and bound variables

In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place.

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## Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

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## Functional analysis

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 (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

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## Gabriel Cramer

Gabriel Cramer (31 July 1704 – 4 January 1752) was a Genevan mathematician.

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## Game theory

Game theory is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers".

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## Gamma matrices

In mathematical physics, the gamma matrices, \, also known as the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cℓ1,3(R).

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## Gauge theory

In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.

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## Gaussian elimination

In linear algebra, Gaussian elimination (also known as row reduction) is an algorithm for solving systems of linear equations.

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## Gell-Mann matrices

The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3x3 traceless Hermitian matrices used in the study of the strong interaction in particle physics.

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## General linear group

In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.

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## Generalized inverse

In mathematics, and in particular, algebra, a generalized inverse of an element x is an element y that has some properties of an inverse element but not necessarily all of them.

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## Geometrical optics

Geometrical optics, or ray optics, describes light propagation in terms of rays.

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## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

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## German language

German (Deutsch) is a West Germanic language that is mainly spoken in Central Europe.

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## Gerolamo Cardano

Gerolamo (or Girolamo, or Geronimo) Cardano (Jérôme Cardan; Hieronymus Cardanus; 24 September 1501 – 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged from being a mathematician, physician, biologist, physicist, chemist, astrologer, astronomer, philosopher, writer, and gambler.

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## Gottfried Wilhelm Leibniz

Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.

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## Gotthold Eisenstein

Ferdinand Gotthold Max Eisenstein (16 April 1823 – 11 October 1852) was a German mathematician.

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## Gram–Schmidt process

In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space Rn equipped with the standard inner product.

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## Graph (discrete mathematics)

In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related".

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## Graph theory

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.

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## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

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## Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.

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## Hadamard product (matrices)

In mathematics, the Hadamard product (also known as the Schur product or the entrywise product) is a binary operation that takes two matrices of the same dimensions, and produces another matrix where each element i,j is the product of elements i,j of the original two matrices.

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## Hartree–Fock method

In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.

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## Hermitian matrix

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and: Hermitian matrices can be understood as the complex extension of real symmetric matrices.

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## Hessian matrix

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field.

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## Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

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## Hill cipher

In classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra.

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## Historia Mathematica

Historia Mathematica: International Journal of History of Mathematics is an academic journal on the history of mathematics published by Elsevier.

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## Houghton Mifflin Harcourt

Houghton Mifflin Harcourt (HMH) is an educational and trade publisher in the United States.

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## HP 9800 series

The HP 9800 was a family of what were initially called programmable calculators and later desktop computers made by Hewlett-Packard, replacing their first HP 9100 calculator.

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## Hyperbola

In mathematics, a hyperbola (plural hyperbolas or hyperbolae) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set.

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## Hypercomplex number

In mathematics, a hypercomplex number is a traditional term for an element of a unital algebra over the field of real numbers.

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## Hyperlink

In computing, a hyperlink, or simply a link, is a reference to data that the reader can directly follow either by clicking, tapping, or hovering.

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## Identity matrix

In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.

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## If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

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## Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.

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## Imperial College London

Imperial College London (officially Imperial College of Science, Technology and Medicine) is a public research university located in London, United Kingdom.

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## Implicit function theorem

In mathematics, more specifically in multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables.

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## Independent equation

An independent equation is an equation in a system of simultaneous equations which cannot be derived algebraically from the other equations.

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## Indeterminate (variable)

In mathematics, and particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else but itself and is used as a placeholder in objects such as polynomials and formal power series.

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## Infinitesimal

In mathematics, infinitesimals are things so small that there is no way to measure them.

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## Infinity

Infinity (symbol) is a concept describing something without any bound or larger than any natural number.

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## Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

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## Inverse element

In abstract algebra, the idea of an inverse element generalises concepts of a negation (sign reversal) in relation to addition, and a reciprocal in relation to multiplication.

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## Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.

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## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

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## Jacobian matrix and determinant

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

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## James Joseph Sylvester

James Joseph Sylvester FRS (3 September 1814 – 15 March 1897) was an English mathematician.

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## Japanese mathematics

denotes a distinct kind of mathematics which was developed in Japan during the Edo period (1603–1867).

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## Johan de Witt

Johan de Witt or Jan de Witt, heer van Zuid- en Noord-Linschoten, Snelrewaard, Hekendorp and IJsselveere (24 September 1625 – 20 August 1672) was a key figure in Dutch politics in the mid-17th century, when its flourishing sea trade in a period of globalisation made the United Provinces a leading European power during the Dutch Golden Age.

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## John von Neumann

John von Neumann (Neumann János Lajos,; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, and polymath.

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## John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.

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## Jordan normal form

In linear algebra, a Jordan normal form (often called Jordan canonical form) of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis.

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## Karl Weierstrass

Karl Theodor Wilhelm Weierstrass (Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".

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## Kernel (linear algebra)

In mathematics, and more specifically in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map between two vector spaces V and W, is the set of all elements v of V for which, where 0 denotes the zero vector in W. That is, in set-builder notation,.

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## Kronecker product

In mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix.

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## Laplace expansion

In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression for the determinant |B| of an n × n matrix B that is a weighted sum of the determinants of n sub-matrices of B, each of size (n−1) × (n−1).

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## Leibniz formula for determinants

In algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements.

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## Lens (optics)

A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction.

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## Leopold Kronecker

Leopold Kronecker (7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic.

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## Letter case

Letter case (or just case) is the distinction between the letters that are in larger upper case (also uppercase, capital letters, capitals, caps, large letters, or more formally majuscule) and smaller lower case (also lowercase, small letters, or more formally minuscule) in the written representation of certain languages.

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## Light

Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum.

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## Limit of a sequence

As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.

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## Line (geometry)

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.

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## Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

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## Linear differential equation

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where,..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable.

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## Linear equation

In mathematics, a linear equation is an equation that may be put in the form where x_1, \ldots, x_n are the variables or unknowns, and c, a_1, \ldots, a_n are coefficients, which are often real numbers, but may be parameters, or even any expression that does not contain the unknowns.

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## Linear function

In mathematics, the term linear function refers to two distinct but related notions.

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## Linear independence

In the theory of vector spaces, a set of vectors is said to be if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be.

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## Linear least squares (mathematics)

In statistics and mathematics, linear least squares is an approach to fitting a mathematical or statistical model to data in cases where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model.

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## Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

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## Linear system

A linear system is a mathematical model of a system based on the use of a linear operator.

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## List of matrices

This page lists some important classes of matrices used in mathematics, science and engineering.

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## List of numerical analysis software

Listed here are end-user computer applications intended for use with numerical or data analysis.

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## Logarithm of a matrix

In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix.

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## Logical equivalence

In logic, statements p and q are logically equivalent if they have the same logical content.

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## Logical matrix

A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0,1) matrix is a matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets.

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## Lorentz group

In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (nongravitational) physical phenomena.

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## LU decomposition

In numerical analysis and linear algebra, LU decomposition (where "LU" stands for "lower–upper", and also called LU factorization) factors a matrix as the product of a lower triangular matrix and an upper triangular matrix.

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## Main diagonal

In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, or major diagonal) of a matrix A is the collection of entries A_ where i.

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## Marcel Dekker

Marcel Dekker was a journal and encyclopedia publishing company with editorial boards found in New York, New York.

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## Markov chain

A Markov chain is "a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event".

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## Mass

Mass is both a property of a physical body and a measure of its resistance to acceleration (a change in its state of motion) when a net force is applied.

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## Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

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## Mathematical formulation of quantum mechanics

The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics.

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## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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## Matrix addition

In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together.

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## Matrix calculus

In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.

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## Matrix equivalence

In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if for some invertible n-by-n matrix P and some invertible m-by-m matrix Q. Equivalent matrices represent the same linear transformation V → W under two different choices of a pair of bases of V and W, with P and Q being the change of basis matrices in V and W respectively.

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## Matrix exponential

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function.

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## Matrix field

In abstract algebra, a matrix field is a field with matrices as elements.

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## Matrix function

In mathematics, a matrix function is a function which maps a matrix to another matrix.

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## Matrix mechanics

Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.

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## Matrix multiplication

In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.

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## Matrix multiplication algorithm

Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms efficient.

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## Matrix norm

In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).

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## Matrix normal distribution

In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.

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## Matrix ring

In abstract algebra, a matrix ring is any collection of matrices over some ring R that form a ring under matrix addition and matrix multiplication.

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## Matrix splitting

In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices.

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## Max Born

Max Born (11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics.

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## Maxima and minima

In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema).

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## Mesh analysis

Mesh analysis (or the mesh current method) is a method that is used to solve planar circuits for the currents (and indirectly the voltages) at any place in the electrical circuit.

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## Minor (linear algebra)

In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns.

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## MIT Press

The MIT Press is a university press affiliated with the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts (United States).

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## Module (mathematics)

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.

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## Molecular orbital

In chemistry, a molecular orbital (MO) is a mathematical function describing the wave-like behavior of an electron in a molecule.

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## Molecule

A molecule is an electrically neutral group of two or more atoms held together by chemical bonds.

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## Monic polynomial

In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1.

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## Multiplication

Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division.

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## Network theory

Network theory is the study of graphs as a representation of either symmetric relations or asymmetric relations between discrete objects.

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## Nodal analysis

In electric circuits analysis, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage (potential difference) between "nodes" (points where elements or branches connect) in an electrical circuit in terms of the branch currents.

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## Normal matrix

In mathematics, a complex square matrix is normal if where is the conjugate transpose of.

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## Normal mode

A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation.

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## Normal-form game

In game theory, normal form is a description of a game.

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## Number

A number is a mathematical object used to count, measure and also label.

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## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

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## Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).

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## Numerical linear algebra

Numerical linear algebra is the study of algorithms for performing linear algebra computations, most notably matrix operations, on computers.

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## Numerical stability

In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms.

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## O'Reilly Media

O'Reilly Media (formerly O'Reilly & Associates) is an American media company established by Tim O'Reilly that publishes books and Web sites and produces conferences on computer technology topics.

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## Optics

Optics is the branch of physics which involves the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it.

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## Orbital overlap

In chemical bonds, an orbital overlap is the concentration of orbitals on adjacent atoms in the same regions of space.

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## Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.

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## Orthogonal matrix

In linear algebra, an orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e. where I is the identity matrix.

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## Orthogonality

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

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## Orthonormality

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and unit vectors.

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## Oxford University Press

Oxford University Press (OUP) is the largest university press in the world, and the second oldest after Cambridge University Press.

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## PageRank

PageRank (PR) is an algorithm used by Google Search to rank websites in their search engine results.

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## Parallelogram

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides.

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## Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).

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## Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

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## Particle accelerator

A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to nearly light speed and to contain them in well-defined beams.

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## Pascual Jordan

Ernst Pascual Jordan (18 October 1902 – 31 July 1980) was a theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory.

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## Pauli matrices

In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian and unitary.

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## Periodic matrix set

In mathematics, a periodic matrix set is a set of square matrices in which each square matrix is of a different size, and such that each cell within each matrix within a set contains data associated with some type of periodic distribution.

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## Permutation matrix

\pi.

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## Physics

Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

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## Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

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## Polynomial ring

In mathematics, especially in the field of abstract algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.

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## Position (vector)

In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point P in space in relation to an arbitrary reference origin O. Usually denoted x, r, or s, it corresponds to the straight-line from O to P. The term "position vector" is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus.

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## Positive-definite matrix

In linear algebra, a symmetric real matrix M is said to be positive definite if the scalar z^Mz is strictly positive for every non-zero column vector z of n real numbers.

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## Prentice Hall

Prentice Hall is a major educational publisher owned by Pearson plc.

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## Probability distribution

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

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## Probability theory

Probability theory is the branch of mathematics concerned with probability.

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## Probability vector

In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one.

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## Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables.

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## Quadratic programming

Quadratic programming (QP) is the process of solving a special type of mathematical optimization problem—specifically, a (linearly constrained) quadratic optimization problem, that is, the problem of optimizing (minimizing or maximizing) a quadratic function of several variables subject to linear constraints on these variables.

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## Quantum chromodynamics

In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks and gluons, the fundamental particles that make up composite hadrons such as the proton, neutron and pion.

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## Quantum electrodynamics

In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics.

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## Quantum field theory

In theoretical physics, quantum field theory (QFT) is the theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics.

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## Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

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## Quark

A quark is a type of elementary particle and a fundamental constituent of matter.

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## Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers.

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## Random matrix

In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables.

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## Random variable

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon.

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## Rank (linear algebra)

In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns.

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## Rank–nullity theorem

In mathematics, the rank–nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to the number of columns of the matrix.

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## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

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## Ray (optics)

In optics a ray is an idealized model of light, obtained by choosing a line that is perpendicular to the wavefronts of the actual light, and that points in the direction of energy flow.

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## Ray transfer matrix analysis

Ray transfer matrix analysis (also known as ABCD matrix analysis) is a type of ray tracing technique used in the design of some optical systems, particularly lasers.

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## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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## Rectangle

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles.

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## Reflection (mathematics)

In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.

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## Regular representation

In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation.

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## Representation theory

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.

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## Rigid body dynamics

Rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces.

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## Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

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## Roothaan equations

The Roothaan equations are a representation of the Hartree–Fock equation in a non orthonormal basis set which can be of Gaussian-type or Slater-type.

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## Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry.

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## Rotation matrix

In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space.

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## Row and column vectors

In linear algebra, a column vector or column matrix is an m × 1 matrix, that is, a matrix consisting of a single column of m elements, Similarly, a row vector or row matrix is a 1 × m matrix, that is, a matrix consisting of a single row of m elements Throughout, boldface is used for the row and column vectors.

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## Row echelon form

In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination.

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## Rule of Sarrus

Sarrus' rule or Sarrus' scheme is a method and a memorization scheme to compute the determinant of a 3×3 matrix.

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## S&P Global

S&P Global Inc. (prior to April 2016 McGraw Hill Financial, Inc., and prior to 2013 McGraw Hill Companies) is an American publicly traded corporation headquartered in New York City.

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## S-matrix

In physics, the S-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process.

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## Scalar (mathematics)

A scalar is an element of a field which is used to define a vector space.

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## Scalar multiplication

In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra).

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## Scaling (geometry)

In Euclidean geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions.

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## Schur decomposition

In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition.

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## Second derivative

In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of.

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## Seki Takakazu

, also known as,Selin, was a Japanese mathematician and author of the Edo period.

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## Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

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## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

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## Shear mapping

In plane geometry, a shear mapping is a linear map that displaces each point in fixed direction, by an amount proportional to its signed distance from a line that is parallel to that direction.

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## Singular-value decomposition

In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix.

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## Skew-symmetric matrix

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative; that is, it satisfies the condition In terms of the entries of the matrix, if aij denotes the entry in the and; i.e.,, then the skew-symmetric condition is For example, the following matrix is skew-symmetric: 0 & 2 & -1 \\ -2 & 0 & -4 \\ 1 & 4 & 0\end.

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## Sparse matrix

In numerical analysis and computer science, a sparse matrix or sparse array is a matrix in which most of the elements are zero.

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## Special linear group

In mathematics, the special linear group of degree n over a field F is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.

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## Special unitary group

In mathematics, the special unitary group of degree, denoted, is the Lie group of unitary matrices with determinant 1.

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## Spectral theorem

In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis).

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## Spectroscopy

Spectroscopy is the study of the interaction between matter and electromagnetic radiation.

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## Spin group

In mathematics the spin group Spin(n) is the double cover of the special orthogonal group, such that there exists a short exact sequence of Lie groups (with) As a Lie group, Spin(n) therefore shares its dimension,, and its Lie algebra with the special orthogonal group.

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## Spinor

In geometry and physics, spinors are elements of a (complex) vector space that can be associated with Euclidean space.

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## Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

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## Square matrix

In mathematics, a square matrix is a matrix with the same number of rows and columns.

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## Square root of a matrix

In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices.

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## Squeeze mapping

In linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping.

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## Statistics

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.

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## Stochastic matrix

In mathematics, a stochastic matrix (also termed probability matrix, transition matrix, substitution matrix, or Markov matrix) is a square matrix used to describe the transitions of a Markov chain.

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## Strassen algorithm

In linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm for matrix multiplication.

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## Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

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## Subtraction

Subtraction is an arithmetic operation that represents the operation of removing objects from a collection.

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## Superalgebra

In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra.

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## Supermatrix

In mathematics and theoretical physics, a supermatrix is a Z2-graded analog of an ordinary matrix.

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## Sylvester equation

In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form: Then given matrices A,B, and C, the problem is to find the possible matrices X that obey this equation.

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## Symbol (formal)

A logical symbol is a fundamental concept in logic, tokens of which may be marks or a configuration of marks which form a particular pattern.

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## Symmetric group

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.

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## Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.

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## Symmetry

Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance.

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## System of linear equations

In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same set of variables.

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## Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

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## TED (conference)

TED Conferences, LLC (Technology, Entertainment, Design) is a media organization that posts talks online for free distribution, under the slogan "ideas worth spreading".

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## Tensor

In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors.

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## Text mining

Text mining, also referred to as text data mining, roughly equivalent to text analytics, is the process of deriving high-quality information from text.

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## Tf–idf

In information retrieval, tf–idf or TFIDF, short for term frequency–inverse document frequency, is a numerical statistic that is intended to reflect how important a word is to a document in a collection or corpus.

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## The Nine Chapters on the Mathematical Art

The Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE.

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## Thesaurus

In general usage, a thesaurus is a reference work that lists words grouped together according to similarity of meaning (containing synonyms and sometimes antonyms), in contrast to a dictionary, which provides definitions for words, and generally lists them in alphabetical order.

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## Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii denotes the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis.

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## Transformation (function)

In mathematics, particularly in semigroup theory, a transformation is a function f that maps a set X to itself, i.e..

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## Transformation matrix

In linear algebra, linear transformations can be represented by matrices.

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## Transpose

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At).

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## Transpose of a linear map

In linear algebra, the transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces.

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## Triangular matrix

In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix.

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## Truth table

A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables (Enderton, 2001).

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## Unit square

In mathematics, a unit square is a square whose sides have length.

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## Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.

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## Unitary matrix

In mathematics, a complex square matrix is unitary if its conjugate transpose is also its inverse—that is, if where is the identity matrix.

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## University of Minnesota

The University of Minnesota, Twin Cities (often referred to as the University of Minnesota, Minnesota, the U of M, UMN, or simply the U) is a public research university in Minneapolis and Saint Paul, Minnesota.

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## Upper and lower bounds

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set (K, ≤) is an element of K which is greater than or equal to every element of S. The term lower bound is defined dually as an element of K which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound.

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## Variance

In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean.

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## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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## Weak interaction

In particle physics, the weak interaction (the weak force or weak nuclear force) is the mechanism of interaction between sub-atomic particles that causes radioactive decay and thus plays an essential role in nuclear fission.

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## Website

A website is a collection of related web pages, including multimedia content, typically identified with a common domain name, and published on at least one web server.

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## Werner Heisenberg

Werner Karl Heisenberg (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the key pioneers of quantum mechanics.

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## Wilhelm Jordan (geodesist)

Wilhelm Jordan (1 March 1842, Ellwangen, Württemberg – 17 April 1899, Hanover) was a German geodesist who conducted surveys in Germany and Africa and founded the German geodesy journal.

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## William Rowan Hamilton

Sir William Rowan Hamilton MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician who made important contributions to classical mechanics, optics, and algebra.

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## Wolfram Mathematica

Wolfram Mathematica (usually termed Mathematica) is a modern technical computing system spanning most areas of technical computing — including neural networks, machine learning, image processing, geometry, data science, visualizations, and others.

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## Zero matrix

In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero.

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## Zero object (algebra)

In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure.

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## 0

0 (zero) is both a number and the numerical digit used to represent that number in numerals.

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## 2 × 2 real matrices

In mathematics, the associative algebra of real matrices is denoted by M(2, R).

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## 3D modeling

In 3D computer graphics, 3D modeling (or three-dimensional modeling) is the process of developing a mathematical representation of any surface of an object (either inanimate or living) in three dimensions via specialized software.

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## Redirects here:

Applications of matrices, Array (mathematics), Complex matrices, Complex matrix, Empty matrix, Equal matrix, Infinite matrix, Matrix (Mathematics), Matrix (computer science), Matrix (math), Matrix (maths), Matrix Theory and Linear Algebra, Matrix equation, Matrix index, Matrix math, Matrix notation, Matrix operation, Matrix power, Matrix theory, Matrix(mathematics), Principal submatrix, Real matrices, Real matrix, Square (matrix), Square submatrix, Submatrices, Submatrix.

## References

[1] https://en.wikipedia.org/wiki/Matrix_(mathematics)