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Vector space span

The span of subspace generated by vectors and isA set of vectors can be tested to see if they span -dimensional space using the following Wolfram Language function: SpanningVectorsQ[m_List?MatrixQ] := (NullSpace[m] == {})

Huffman coding

A lossless data compression algorithm which uses a small number of bits to encode common characters. Huffman coding approximates the probability for each character as a power of 1/2 to avoid complications associated with using a nonintegral number of bits to encode characters using their actual probabilities.Huffman coding works on a list of weights by building an extended binary tree with minimum weighted external path length and proceeds by finding the two smallest s, and , viewed as external nodes, and replacing them with an internal node of weight . The procedure is them repeated stepwise until the root node is reached. An individual external node can then be encoded by a binary string of 0s (for left branches) and 1s (for right branches).The procedure is summarized below for the weights 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, and 41 given by the first 13 primes, and the resulting tree is shown above (Knuth 1997, pp. 402-403). As is clear..

Random permutation

A random permutation is a permutation containing a fixed number of a random selection from a given set of elements. There are two main algorithms for constructing random permutations. The first constructs a vector of random real numbers and uses them as keys to records containing the integers 1 to . The second starts with an arbitrary permutation and then exchanges the th element with a randomly selected one from the first elements for , ..., (Skiena 1990).A random permutation on the integers can be implemented in the Wolfram Language as RandomSample[Range[n]]. A random permutation in the permutation graph pg can be computed using RandomPermutation[pg], and such random permutations by RandomPermutation[pg, n]. random permutations in the symmetric group of order can be computed using RandomPermutation[d, n].There are an average of permutation inversions in a permutation on elements (Skiena 1990, p. 29). The expected number of permutation..

Generalized vandermonde matrix

A generalized Vandermonde matrix of two sequences and where is an increasing sequence of positive integers and is an increasing sequence of nonnegative integers of the same length is the outer product of and with multiplication operation given by the power function. The generalized Vandermonde matrix can be implemented in the Wolfram Language as Vandermonde[a_List?VectorQ, b_List?VectorQ] := Outer[Power, a, b] /; Equal @@ Length /@ {a, b}A generalized Vandermonde matrix is a minor of a Vandermonde matrix. Alternatively, it has the same form as a Vandermonde matrix , where is an increasing sequence of positive integers, except now is any increasing sequence of nonnegative integers. In the special case of a Vandermonde matrix, .While there is no general formula for the determinant of a generalized Vandermonde matrix, its determinant is always positive. Since any minor of a generalized Vandermonde matrix is also a generalized Vandermonde..

Random matrix

A random matrix is a matrix of given type and size whoseentries consist of random numbers from some specified distribution.Random matrix theory is cited as one of the "modern tools" used in Catherine'sproof of an important result in prime number theory in the 2005 film Proof.For a real matrix with elements having a standard normal distribution, the expected number of real eigenvalues is given by(1)(2)where is a hypergeometric function and is a beta function (Edelman et al. 1994, Edelman and Kostlan 1994). has asymptotic behavior(3)Let be the probability that there are exactly real eigenvalues in the complex spectrum of the matrix. Edelman (1997) showed that(4)which is the smallest probability of all s. The entire probability function of the number of expected real eigenvalues in the spectrum of a Gaussian real random matrix was derived by Kanzieper and Akemann (2005) as(5)where(6)(7)In (6), the summation runs over all partitions..

Gaussian elimination

Gaussian elimination is a method for solving matrixequations of the form(1)To perform Gaussian elimination starting with the system of equations(2)compose the "augmented matrix equation"(3)Here, the column vector in the variables is carried along for labeling the matrix rows. Now, perform elementary row operations to put the augmented matrix into the upper triangular form(4)Solve the equation of the th row for , then substitute back into the equation of the st row to obtain a solution for , etc., according to the formula(5)In the Wolfram Language, RowReduce performs a version of Gaussian elimination, with the equation being solved by GaussianElimination[m_?MatrixQ, v_?VectorQ] := Last /@ RowReduce[Flatten /@ Transpose[{m, v}]]LU decomposition of a matrix is frequently usedas part of a Gaussian elimination process for solving a matrix equation.A matrix that has undergone Gaussian elimination is said to be in echelonform.For..

Unitary matrix

A square matrix is a unitary matrix if(1)where denotes the conjugate transpose and is the matrix inverse. For example,(2)is a unitary matrix.Unitary matrices leave the length of a complex vectorunchanged.For real matrices, unitary is the same as orthogonal. In fact, there are some similarities between orthogonal matrices and unitary matrices. The rows of a unitary matrix are a unitary basis. That is, each row has length one, and their Hermitian inner product is zero. Similarly, the columns are also a unitary basis. In fact, given any unitary basis, the matrix whose rows are that basis is a unitary matrix. It is automatically the case that the columns are another unitary basis.A matrix can be tested to see if it is unitary using the Wolfram Language function: UnitaryQ[m_List?MatrixQ] := ([email protected] @ m . m == IdentityMatrix @ Length @ m)The definition of a unitary matrix guarantees that(3)where is the identity matrix. In particular,..

Permanent

The permanent is an analog of a determinant where all the signs in the expansion by minors are taken as positive. The permanent of a matrix is the coefficient of in(1)(Vardi 1991). Another equation is the Ryser formula(2)where the sum is over all subsets of , and is the number of elements in (Vardi 1991). Muir (1960, p. 19) uses the notation to denote a permanent.The permanent can be implemented in the WolframLanguage as Permanent[m_List] := With[{v = Array[x, Length[m]]}, Coefficient[Times @@ (m.v), Times @@ v] ]The computation of permanents has been studied fairly extensively in algebraic complexity theory. The complexity of the best-known algorithms grows as the exponent of the matrix size (Knuth 1998, p. 499), which would appear to be very surprising, given the permanent's similarity to the tractable determinant. Computation of the permanent is P-complete (i.e, sharp-P complete; Valiant 1979).If is a unitary matrix, then(3)(Minc..

Affine variety

An affine variety is an algebraic variety contained in affine space. For example,(1)is the cone, and(2)is a conic section, which is a subvariety of the cone. The cone can be written to indicate that it is the variety corresponding to . Naturally, many other polynomials vanish on , in fact all polynomials in . The set is an ideal in the polynomial ring . Note also, that the ideal of polynomials vanishing on the conic section is the ideal generated by and .A morphism between two affine varieties is given by polynomial coordinate functions. For example, the map is a morphism from to . Two affine varieties are isomorphic if there is a morphism which has an inverse morphism. For example, the affine variety is isomorphic to the cone via the coordinate change .Many polynomials may be factored, for instance , and then . Consequently, only irreducible polynomials, and more generally only prime ideals are used in the definition of a variety. An affine variety is..

The adjacency list representation of a graph consists of lists one for each vertex , , which gives the vertices to which is adjacent. The adjacency lists of a graph may be computed in the Wolfram Language using AdjacencyList[g, ]& /@ VertexList[g]and a graph may be constructed from adjacency lists using Graph[UndirectedEdge @@@ Union[ Sort /@ Flatten[ MapIndexed[{, 2[[1]]}&, l, {2}], 1]]]

Game of life

The game of life is the best-known two-dimensional cellular automaton, invented by John H. Conway and popularized in Martin Gardner's Scientific American column starting in October 1970. The game of life was originally played (i.e., successive generations were produced) by hand with counters, but implementation on a computer greatly increased the ease of exploring patterns.The life cellular automaton is run by placing a number of filled cells on a two-dimensional grid. Each generation then switches cells on or off depending on the state of the cells that surround it. The rules are defined as follows. All eight of the cells surrounding the current one are checked to see if they are on or not. Any cells that are on are counted, and this count is then used to determine what will happen to the current cell. 1. Death: if the count is less than 2 or greater than 3, the current cell is switched off. 2. Survival: if (a) the count is exactly 2, or (b) the count..

Bootstrap percolation

A two-dimensional binary () totalistic cellular automaton with a von Neumann neighborhood of range . It has a birth rule that at least 2 of its 4 neighbors are alive, and a survival rule that all cells survive. steps of bootstrap percolation on an grid with random initial condition of density can be implemented in the Wolfram Language asWith[{n = 10, p = 0.1, s = 20}, CellularAutomaton[ {1018, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, Table[If[Random[Real] < p, 1, 0], {s}, {s}], n ]]If the initial condition consists of a random sparse arrangement of cells with density , then the system seems to quickly converge to a steady state of rectangular islands of live cells surrounded by a sea of dead cells. However, as crosses some threshold on finite-sized grids, the behavior appears to change so that every cell becomes live. Several examples are shown above on three grids with random initial conditions and different starting densities.However, this..

Vector space tensor product

The tensor product of two vector spaces and , denoted and also called the tensor direct product, is a way of creating a new vector space analogous to multiplication of integers. For instance,(1)In particular,(2)Also, the tensor product obeys a distributive law with the directsum operation:(3)The analogy with an algebra is the motivation behind K-theory. The tensor product of two tensors and can be implemented in the Wolfram Language as: TensorProduct[a_List, b_List] := Outer[List, a, b]Algebraically, the vector space is spanned by elements of the form , and the following rules are satisfied, for any scalar . The definition is the same no matter which scalar field is used.(4)(5)(6)One basic consequence of these formulas is that(7)A vector basis of and of gives a basis for , namely , for all pairs . An arbitrary element of can be written uniquely as , where are scalars. If is dimensional and is dimensional, then has dimension .Using tensor products,..

Lagrange's identity

Lagrange's identity is the algebraic identity(1)(Mitrinović 1970, p. 41; Marsden and Tromba 1981, p. 57; Gradshteyn and Ryzhik 2000, p. 1049).Lagrange's identity is a special case of the Binet-Cauchy identity, and Cauchy's inequality in dimensions follows from it.It can be coded in the Wolfram Languageas follow. LagrangesIdentity[n_] := Module[ {aa = Array[a, n], bb = Array[b, n]}, Total[(aa^2) Plus @@ (bb^2)] == Total[(a[1]b[2] - a[2]b[1])^2& @@@ Subsets[Range[n], {2}]] + (aa.bb)^2 ]Plugging in gives the and identities (2)(3)(4)A vector quadruple product formula knownas Lagrange's identity given by(5)(Bronshtein and Semendyayev 2004, p. 185).A related identity also known as Lagrange's identity is given by defining and to be -dimensional vectors for , ..., . Then(6)(Greub 1978, p. 155), where denotes a cross product, denotes a dot product, and is the determinant of the matrix..

Fundamental discriminant

An integer is a fundamental discriminant if it is not equal to 1, not divisible by any square of any odd prime, and satisfies or . The function FundamentalDiscriminantQ[d] in the Wolfram Language version 5.2 add-on package NumberTheory$NumberTheoryFunctions$ tests if an integer is a fundamental discriminant.It can be implemented as: FundamentalDiscriminantQ[n_Integer] := n != 1&& (Mod[n, 4] == 1 \[Or] ! Unequal[Mod[n, 16], 8, 12])&& SquareFreeQ[n/2^IntegerExponent[n, 2]]The first few positive fundamental discriminants are 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, ... (OEIS A003658). Similarly, the first few negative fundamental discriminants are , , , , , , , , , , , ... (OEIS A003657).

Prime factorization

The factorization of a number into its constituent primes, also called prime decomposition. Given a positive integer , the prime factorization is writtenwhere the s are the prime factors, each of order . Each factor is called a primary. Prime factorization can be performed in the Wolfram Language using the command FactorInteger[n], which returns a list of pairs.Through his invention of the Pratt certificate, Pratt (1975) became the first to establish that prime factorization lies in the complexity class NP.The following Wolfram Language code can be used to give a nicely typeset form of a number : FactorForm[n_?NumberQ, fac_:Automatic] := Times @@ (HoldForm[Power[]]& @@@ FactorInteger[n, fac])The first few prime factorizations (the number 1, by definition, has a prime factorization of "1") are given in the following table.prime factorizationprime factorization11111122123313134145515616771717818919191020The..

Power tower

The power tower of order is defined as(1)where is Knuth up-arrow notation (Knuth 1976), which in turn is defined by(2)together with(3)(4)Rucker (1995, p. 74) uses the notation(5)and refers to this operation as "tetration."A power tower can be implemented in the WolframLanguage as PowerTower[a_, k_Integer] := Nest[Power[a, ]&, 1, k]or PowerTower[a_, k_Integer] := Power @@ Table[a, {k}]The following table gives values of for , 2, ... for small .OEIS1A0000271, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...2A0003121, 4, 27, 256, 3125, 46656, ...3A0024881, 16, 7625597484987, ...41, 65536, ...The following table gives for , 2, ... for small .OEIS1A0000121, 1, 1, 1, 1, 1, ...2A0142212, 4, 16, 65536, , ...3A0142223, 27, 7625597484987, ...44, 256, , ...Consider and let be defined as(6)(Galidakis 2004). Then for , is entire with series expansion:(7)Similarly, for , is analytic for in the domain of the principal branch of , with series expansion:(8)For..

Power set

Given a set , the power set of , sometimes also called the powerset, is the set of all subsets of . The order of a power set of a set of order is . Power sets are larger than the sets associated with them. The power set of is variously denoted or .The power set of a given set can be found in the Wolfram Language using Subsets[s].

Composite number

A composite number is a positive integer which is not prime (i.e., which has factors other than 1 and itself). The first few composite numbers (sometimes called "composites" for short) are 4, 6, 8, 9, 10, 12, 14, 15, 16, ... (OEIS A002808), whose prime decompositions are summarized in the following table. Note that the number 1 is a special case which is considered to be neither composite nor prime.prime factorizationprime factorization420621822924102512261427152816301832The th composite number can be generated using the Wolfram Language code Composite[n_Integer] := FixedPoint[n + PrimePi[] + 1&, n]The Dirichlet generating function of the characteristic function of the composite numbers is given by(1)(2)(3)where is the Riemann zeta function, is the prime zeta function, and is an Iverson bracket. There are an infinite number of composite numbers.The composite number problem asks if there exist positive integers..

Lattice reduction

The process of finding a reduced set of basis vectors for a given lattice having certain special properties. Lattice reduction algorithms are used in a number of modern number theoretical applications, including in the discovery of a spigot algorithm for pi. Although determining the shortest basis is possibly an NP-complete problem, algorithms such as the LLL algorithm can find a short basis in polynomial time with guaranteed worst-case performance.The LLL algorithm of lattice reduction is implemented in the Wolfram Language using the function LatticeReduce. RootApproximant[x, n] also calls this routine in order to find a algebraic number of degree at most such that is an approximate zero of the number.When used to find integer relations, a typical input to the algorithm consists of an augmented identity matrix with the entries in the last column consisting of the elements (multiplied by a large positive constant to penalize vectors that..

Suborder function

The multiplicative suborder of a number (mod ) is the least exponent such that (mod ), or zero if no such exists. An always exists if and .This function is denoted and can be implemented in the Wolfram Language as: Suborder[a_,n_] := If[n>1&& GCD[a,n] == 1, Min[MultiplicativeOrder[a, n, {-1, 1}]], 0 ]The following table summarizes for small values of and .OEIS for , 1, ...20, 0, 0, 1, 0, 2, 0, 3, 0, 3, 0, 5, 0, 6, 0, ...3A1034890, 0, 1, 0, 1, 2, 0, 3, 2, 0, 2, 5, 0, 3, 3, ...40, 0, 0, 1, 0, 1, 0, 3, 0, 3, 0, 5, 0, 3, 0, ...5A1034910, 0, 1, 1, 1, 0, 1, 3, 2, 3, 0, 5, 2, 2, 3, ...

Cofactor

Given a factor of a number , the cofactor of is .A different type of cofactor, sometimes called a cofactor matrix, is a signed version of a minor defined byand used in the computation of the determinant of a matrix according toThe cofactor can be computed in the WolframLanguage using Cofactor[m_List?MatrixQ, {i_Integer, j_Integer}] := (-1)^(i+j) Det[Drop[Transpose[ Drop[Transpose[m], {j}]], {i} ]]which is the equivalent of the th component of the CofactorMatrix defined below. MinorMatrix[m_List?MatrixQ] := Map[Reverse, Minors[m], {0, 1}] CofactorMatrix[m_List?MatrixQ] := MapIndexed[1 (-1)^(Plus @@ 2)&, MinorMatrix[m],{2}]Cofactors can be computed using Cofactor[m, i, j] in the Wolfram Language package Combinatorica` .

Squarefree part

That part of a positive integer left after all square factors are divided out. For example, the squarefree part of is 6, since . For , 2, ..., the first few are 1, 2, 3, 1, 5, 6, 7, 2, 1, 10, ... (OEIS A007913). The squarefree part function can be implemented in the Wolfram Language as SquarefreePart[n_Integer?Positive] := Times @@ Power @@@ ({[[1]], Mod[[[2]], 2]}& /@ FactorInteger[n])

Base

The word "base" in mathematics is used to refer to a particular mathematical object that is used as a building block. The most common uses are the related concepts of the number system whose digits are used to represent numbers and the number system in which logarithms are defined. It can also be used to refer to the bottom edge or surface of a geometric figure.A real number can be represented using any integer number as a base (sometimes also called a radix or scale). The choice of a base yields to a representation of numbers known as a number system. In base , the digits 0, 1, ..., are used (where, by convention, for bases larger than 10, the symbols A, B, C, ... are generally used as symbols representing the decimal numbers 10, 11, 12, ...).The digits of a number in base (for integer ) can be obtained in the Wolfram Language using IntegerDigits[x, b].Let the base representation of a number be written(1)(e.g., ). Then, for example, the number 10 is..

Series reversion

Series reversion is the computation of the coefficients of the inverse function given those of the forward function. For a function expressed in a series with no constant term (i.e., ) as(1)the series expansion of the inverse series is given by(2)By plugging (2) into (1), the following equationis obtained(3)Equating coefficients then gives(4)(5)(6)(7)(8)(9)(10)(Dwight 1961, Abramowitz and Stegun 1972, p. 16).Series reversion is implemented in the Wolfram Language as InverseSeries[s, x], where is given as a SeriesData object. For example, to obtain the terms shown above, With[{n = 7}, CoefficientList[ InverseSeries[SeriesData[x, 0, Array[a, n], 1, n + 1, 1]], x] ]A derivation of the explicit formula for the th term is given by Morse and Feshbach (1953),(11)where(12)

Secant method

A root-finding algorithm which assumes a function to be approximately linear in the region of interest. Each improvement is taken as the point where the approximating line crosses the axis. The secant method retains only the most recent estimate, so the root does not necessarily remain bracketed. The secant method is implemented in the Wolfram Language as the undocumented option Method -> Secant in FindRoot[eqn, x, x0, x1].When the algorithm does converge, its order of convergenceis(1)where is a constant and is the golden ratio.(2)(3)(4)so(5)The secant method can be implemented in the WolframLanguage as SecantMethodList[f_, {x_, x0_, x1_}, n_] := NestList[Last[] - {0, (Function[x, f][Last[]]* Subtract @@ )/Subtract @@ Function[x, f] /@ }&, {x0, x1}, n]

Robbins axiom

The logical axiomwhere denotes NOT and denotes OR, that, when taken together with associativity and commutativity, is equivalent to the axioms of Boolean algebra.The Robbins operator can be defined in the WolframLanguage by Robbins := Function[{x, y}, ! (! (! y \[Or] x) \[Or] ! (x \[Or] y))]That the Robbins axiom is a true statement in Booleanalgebra can be verified by examining its truth table.TTTTFTFTFFFF

Newton's method

Newton's method, also called the Newton-Raphson method, is a root-finding algorithm that uses the first few terms of the Taylor series of a function in the vicinity of a suspected root. Newton's method is sometimes also known as Newton's iteration, although in this work the latter term is reserved to the application of Newton's method for computing square roots.For a polynomial, Newton's method is essentially the same as Horner's method.The Taylor series of about the point is given by(1)Keeping terms only to first order,(2)Equation (2) is the equation of the tangent line to the curve at , so is the place where that tangent line intersects the -axis. A graph can therefore give a good intuitive idea of why Newton's method works at a well-chosen starting point and why it might diverge with a poorly-chosen starting point.This expression above can be used to estimate the amount of offset needed to land closer to the root starting from an initial guess..

Wedge product

The wedge product is the product in an exterior algebra. If and are differential k-forms of degrees and , respectively, then(1)It is not (in general) commutative, but it is associative,(2)and bilinear(3)(4)(Spivak 1999, p. 203), where and are constants. The exterior algebra is generated by elements of degree one, and so the wedge product can be defined using a basis for :(5)when the indices are distinct, and the product is zero otherwise.While the formula holds when has degree one, it does not hold in general. For example, consider :(6)(7)(8)If have degree one, then they are linearly independent iff .The wedge product is the "correct" type of product to use in computinga volume element(9)The wedge product can therefore be used to calculate determinants and volumes of parallelepipeds. For example, write where are the columns of . Then(10)and is the volume of the parallelepiped spanned by ...

Circulant matrix

An matrix whose rows are composed of cyclically shifted versions of a length- list . For example, the circulant matrix on the list is given by(1)Circulant matrices are very useful in digital image processing, and the circulant matrix is implemented as CirculantMatrix[l, n] in the Mathematica application package Digital Image Processing.Circulant matrices can be implemented in the WolframLanguage as follows. CirculantMatrix[l_List?VectorQ] := NestList[RotateRight, RotateRight[l], Length[l] - 1] CirculantMatrix[l_List?VectorQ, n_Integer] := NestList[RotateRight, RotateRight[Join[Table[0, {n - Length[l]}], l]], n - 1] /; n >= Length[l]where the first input creates a matrix with dimensions equal to the length of and the second pads with zeros to give an matrix. A special type of circulant matrix is defined as(2)where is a binomial coefficient. The determinant of is given by the beautiful formula(3)where , , ..., are the th..

P&oacute;lya plot

A Pólya plot is a plot of the vector field of of a complex function . Several examples are shown above.Pólya plots can be created in the WolframLanguage using the following code: PolyaFieldPlot[f_, {x_, xmin_, xmax_}, {y_, ymin_, ymax_}, opts : OptionsPattern[]] := VectorPlot[Evaluate @ {Re[f], -Im[f]}, {x, xmin, xmax}, {y, ymin, ymax}, VectorScale -> {Automatic, Automatic, Log[5 + 1]&}, opts ]

Cumulative product

A cumulative product is a sequence of partial products of a given sequence. For example, the cumulative products of the sequence , are , , , .... Cumulative products can be implemented in the Wolfram Language as Rest[FoldList[Times, 1, list]]

Gr&ouml;bner basis

A Gröbner basis for a system of polynomials is an equivalence system that possesses useful properties, for example, that another polynomial is a combination of those in iff the remainder of with respect to is 0. (Here, the division algorithm requires an order of a certain type on the monomials.) Furthermore, the set of polynomials in a Gröbner basis have the same collection of roots as the original polynomials. For linear functions in any number of variables, a Gröbner basis is equivalent to Gaussian elimination.The algorithm for computing Gröbner bases is known as Buchberger's algorithm. Calculating a Gröbner basis is typically a very time-consuming process for large polynomial systems (Trott 2006, p. 37).Gröbner bases are pervasive in the construction of symbolic algebra algorithms, and Gröbner bases with respect to lexicographic order are very useful for solving equations and for elimination..

Sylvester matrix

For two polynomials and of degrees and , respectively, the Sylvester matrix is an matrix formed by filling the matrix beginning with the upper left corner with the coefficients of , then shifting down one row and one column to the right and filling in the coefficients starting there until they hit the right side. The process is then repeated for the coefficients of .The Sylvester matrix can be implemented in the WolframLanguage as: SylvesterMatrix1[poly1_, poly2_, var_] := Function[{coeffs1, coeffs2}, With[ {l1 = Length[coeffs1], l2 = Length[coeffs2]}, Join[ NestList[RotateRight, PadRight[coeffs1, l1 + l2 - 2], l2 - 2], NestList[RotateRight, PadRight[coeffs2, l1 + l2 - 2], l1 - 2] ] ] ][ Reverse[CoefficientList[poly1, var]], Reverse[CoefficientList[poly2, var]] ]For example, the Sylvester matrix for and isThe determinant of the Sylvester matrix of two polynomialsis the resultant of the polynomials.SylvesterMatrix is an (undocumented)..

Polynomial

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. A polynomial in one variable (i.e., a univariate polynomial) with constant coefficients is given by(1)The individual summands with the coefficients (usually) included are called monomials (Becker and Weispfenning 1993, p. 191), whereas the products of the form in the multivariate case, i.e., with the coefficients omitted, are called terms (Becker and Weispfenning 1993, p. 188). However, the term "monomial" is sometimes also used to mean polynomial summands without their coefficients, and in some older works, the definitions of monomial and term are reversed. Care is therefore needed in attempting to distinguish these conflicting usages.The highest power in a univariate polynomial is calledits order, or sometimes its degree.Any polynomial with can be expressed as(2)where the product..

Stable polynomial

A real polynomial is said to be stable if all its roots lie in the left half-plane. The term "stable" is used to describe such a polynomial because, in the theory of linear servomechanisms, a system exhibits unforced time-dependent motion of the form , where is the root of a certain real polynomial . A system is therefore mechanically stable iff is a stable polynomial.The polynomial is stable iff , and the irreducible polynomial is stable iff both and are greater than zero. The Routh-Hurwitz theorem can be used to determine if a polynomial is stable.Given two real polynomials and , if and are stable, then so is their product , and vice versa (Séroul 2000, p. 280). It therefore follows that the coefficients of stable real polynomials are either all positive or all negative (although this is not a sufficient condition, as shown with the counterexample ). Furthermore, the values of a stable polynomial are never zero for and have..

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