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Dickson Polynomial
In mathematics, the Dickson polynomials, denoted , form a polynomial sequence introduced by . They were rediscovered by in his study of Brewer sums and have at times, although rarely, been referred to as Brewer polynomials. Over the complex numbers, Dickson polynomials are essentially equivalent to Chebyshev polynomials with a change of variable, and, in fact, Dickson polynomials are sometimes called Chebyshev polynomials. Dickson polynomials are generally studied over finite fields, where they sometimes may not be equivalent to Chebyshev polynomials. One of the main reasons for interest in them is that for fixed , they give many examples of '' permutation polynomials''; polynomials acting as permutations of finite fields. Definition First kind For integer and in a commutative ring with identity (often chosen to be the finite field ) the Dickson polynomials (of the first kind) over are given by :D_n(x,\alpha)=\sum_^\frac \binom (-\alpha)^i x^ \,. The first few Dickson pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematics), function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equation, ''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with stochastic differential equations, ''stochastic'' differential equations (SDEs) where the progression is random. Differential equations 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 :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where a_0(x),\ldots,a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called a power product or primitive monomial, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, x^2yz^3=xxyzzz is a monomial. The constant 1 is a primitive monomial, being equal to the empty product and to x^0 for any variable x. If only a single variable x is considered, this means that a monomial is either 1 or a power x^n of x, with n a positive integer. If several variables are considered, say, x, y, z, then each can be given an exponent, so that any monomial is of the form x^a y^b z^c with a,b,c non-negative integers (taking note that any exponent 0 makes the corresponding factor equal to 1). # A monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A primitive monomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotent (ring Theory)
In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element such that . That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that for any positive integer . For example, an idempotent element of a matrix ring is precisely an idempotent matrix. For general rings, elements idempotent under multiplication are involved in decompositions of modules, and connected to homological properties of the ring. In Boolean algebra, the main objects of study are rings in which all elements are idempotent under both addition and multiplication. Examples Quotients of Z One may consider the ring of integers modulo , where is square-free. By the Chinese remainder theorem, this ring factors into the product of rings of integers modulo , where is prime. Now each of these factors is a field, so it is clear that the factors' only idempotents will be and . That is, each factor h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas Polynomials
In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials. Definition These Fibonacci polynomials are defined by a recurrence relation:Benjamin & Quinn p. 141 :F_n(x)= \begin 0, & \mbox n = 0\\ 1, & \mbox n = 1\\ x F_(x) + F_(x),& \mbox n \geq 2 \end The Lucas polynomials use the same recurrence with different starting values: :L_n(x) = \begin 2, & \mbox n = 0 \\ x, & \mbox n = 1 \\ x L_(x) + L_(x), & \mbox n \geq 2. \end They can be defined for negative indices bySpringer :F_(x)=(-1)^F_(x), :L_(x)=(-1)^nL_(x). The Fibonacci polynomials form a sequence of orthogonal polynomials with A_n=C_n=1 and B_n=0. Examples The first few Fibonacci polynomials are: :F_0(x)=0 \, :F_1(x)=1 \, :F_2(x)=x \, :F_3(x)=x^2+1 \, :F_4(x)=x^3+2x \, :F_5(x)=x^4+3x^2+1 \, :F_6(x)=x^5+4x^3+3x \, The first few Lucas polynomia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas Sequence
In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation : x_n = P \cdot x_ - Q \cdot x_ where P and Q are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences U_n(P, Q) and V_n(P, Q). More generally, Lucas sequences U_n(P, Q) and V_n(P, Q) represent sequences of polynomials in P and Q with integer coefficients. Famous examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Lucas numbers, Jacobsthal numbers, and a superset of Fermat numbers (see below). Lucas sequences are named after the French mathematician Édouard Lucas. Recurrence relations Given two integer parameters P and Q, the Lucas sequences of the first kind U_n(P,Q) and of the second kind V_n(P,Q) are defined by the recurrence relations: :\begin U_0(P,Q)&=0, \\ U_1(P,Q)&=1, \\ U_n(P,Q)&=P\cdot U_(P,Q)-Q\cdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed. Generating functions are sometimes called generating series, in that a series of terms can be said to be the generator of its sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Sequence
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in enumerative combinatorics and algebraic combinatorics, as well as applied mathematics. Examples Some polynomial sequences arise in physics and approximation theory as the solutions of certain ordinary differential equations: * Laguerre polynomials * Chebyshev polynomials * Legendre polynomials * Jacobi polynomials Others come from statistics: * Hermite polynomials Many are studied in algebra and combinatorics: * Monomials * Rising factorials * Falling factorials * All-one polynomials * Abel polynomials * Bell polynomials * Bernoulli polynomials * Cyclotomic polynomials * Dickson polynomials * Fibonacci polynomials * Lagrange polynomials * Lucas polynomials * Spread polynomials * Touchard polynomials * Rook polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Commutative rings appear in the following chain of subclass (set theory), class inclusions: Definition and first examples Definition A ''ring'' is a Set (mathematics), set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings) of the set : written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is factorial, us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
