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Cyclotomic Polynomial
In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primitive roots of unity e^ , where ''k'' runs over the positive integers less than ''n'' and coprime to ''n'' (and ''i'' is the imaginary unit). In other words, the ''n''th cyclotomic polynomial is equal to : \Phi_n(x) = \prod_\stackrel \left(x-e^\right). It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive ''n''th-root of unity ( e^ is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is :\prod_\Phi_d(x) = x^n - 1, showing that x is a root of x^n - 1 if and only if it is a ''d''th primitive root of unity for some ''d'' that divides ''n''. Examples If ''n'' is a prim ... [...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] |
Palindromic Polynomial
In algebra, given a polynomial :p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n, with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial,* denoted by or , is the polynomial :p^*(x) = a_n + a_x + \cdots + a_0x^n = x^n p(x^). That is, the coefficients of are the coefficients of in reverse order. Reciprocal polynomials arise naturally in linear algebra as the characteristic polynomial of the inverse of a matrix. In the special case where the field is the complex numbers, when :p(z) = a_0 + a_1z + a_2z^2 + \cdots + a_nz^n, the conjugate reciprocal polynomial, denoted , is defined by, :p^(z) = \overline + \overlinez + \cdots + \overlinez^n = z^n\overline, where \overline denotes the complex conjugate of a_i, and is also called the reciprocal polynomial when no confusion can arise. A polynomial is called self-reciprocal or palindromic if . The coefficients of a self-reciprocal polynomial satisfy for all . Properties Reciprocal polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maple (software)
Maple is a symbolic and numeric computing environment as well as a multi-paradigm programming language. It covers several areas of technical computing, such as symbolic mathematics, numerical analysis, data processing, visualization, and others. A toolbox, MapleSim, adds functionality for multidomain physical modeling and code generation. Maple's capacity for symbolic computing include those of a general-purpose computer algebra system. For instance, it can manipulate mathematical expressions and find symbolic solutions to certain problems, such as those arising from ordinary and partial differential equations. Maple is developed commercially by the Canadian software company Maplesoft. The name 'Maple' is a reference to the software's Canadian heritage. Overview Core functionality Users can enter mathematics in traditional mathematical notation. Custom user interfaces can also be created. There is support for numeric computations, to arbitrary precision, as well as symbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SageMath
SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, group theory, differentiable manifolds, numerical analysis, number theory, calculus, and statistics. The first version of SageMath was released on 24 February 2005 as free and open-source software under the terms of the GNU General Public License version 2, with the initial goals of creating an "open source alternative to Magma, Maple, Mathematica, and MATLAB". The originator and leader of the SageMath project, William Stein, was a mathematician at the University of Washington. SageMath uses a syntax resembling Python's, supporting procedural, functional, and object-oriented constructs. Development Stein realized when designing Sage that there were many open-source mathematics software packages already written in different languages, namely C, C++, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Algebra Systems
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "computer algebra" or "symbolic computation", which has spurred work in algorithms over mathematical objects such as polynomials. Computer algebra systems may be divided into two classes: specialized and general-purpose. The specialized ones are devoted to a specific part of mathematics, such as number theory, group theory, or teaching of elementary mathematics. General-purpose computer algebra systems aim to be useful to a user working in any scientific field that requires manipulation of mathematical expressions. To be useful, a general-purpose computer algebra system must include various features such as: *a user interface all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Division Of Polynomials
In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers. In the important case of univariate polynomials over a field the polynomial GCD may be computed, like for the integer GCD, by the Euclidean algorithm using long division. The polynomial GCD is defined only up to the multiplication by an invertible constant. The similarity between the integer GCD and the polynomial GCD allows extending to univariate polynomials all the properties that may be deduced from the Euclidean algorithm and Euclidean division. Moreover, the polynomial GCD has specific properties that make it a fundamental notion in various areas of algebra. Typically, the roots of the GCD of two polynomials are the common roots of the two polynomials, and this provides information on the roots without comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Factorization
In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number. For example, is a composite number because , but is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example . Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers , , , and so on, up to the square root of . For larger numbers, especially when using a computer, various more sophis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Long Division
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones. Sometimes using a shorthand version called synthetic division is faster, with less writing and fewer calculations. Another abbreviated method is polynomial short division (Blomqvist's method). Polynomial long division is an algorithm that implements the Euclidean division of polynomials, which starting from two polynomials ''A'' (the ''dividend'') and ''B'' (the ''divisor'') produces, if ''B'' is not zero, a '' quotient'' ''Q'' and a ''remainder'' ''R'' such that :''A'' = ''BQ'' + ''R'', and either ''R'' = 0 or the degree of ''R'' is lower than the degree of ''B''. These conditions uniquely define ''Q'' and ''R'', which means that ''Q'' and ''R'' do not depend o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursive Definition
In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set ( Aczel 1977:740ff). Some examples of recursively definable objects include factorials, natural numbers, Fibonacci numbers, and the Cantor ternary set. A recursive definition of a function defines values of the function for some inputs in terms of the values of the same function for other (usually smaller) inputs. For example, the factorial function is defined by the rules :\begin & 0! = 1. \\ & (n+1)! = (n+1) \cdot n!. \end This definition is valid for each natural number , because the recursion eventually reaches the base case of 0. The definition may also be thought of as giving a procedure for computing the value of the function , starting from and proceeding onwards with etc. The recursion theorem states that such a definition indeed defines a function that is unique. The proof uses mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Function
The Möbius function \mu(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted \mu(x). Definition The Möbius function is defined by :\mu(n) = \begin 1 & \text n = 1 \\ (-1)^k & \text n \text k \text \\ 0 & \text n \text > 1 \end The Möbius function can alternatively be represented as : \mu(n) = \delta_ \lambda(n), where \delta_ is the Kronecker delta, \lambda(n) is the Liouville function, Prime omega function, \omega(n) is the number of distinct prime divisors of n, and Prime omega function, \Omega(n) is the number of prime factors of n, counted with multiplicity. Another characterization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Inversion Formula
In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius. A large generalization of this formula applies to summation over an arbitrary locally finite partially ordered set, with Möbius' classical formula applying to the set of the natural numbers ordered by divisibility: see incidence algebra. Statement of the formula The classic version states that if and are arithmetic functions satisfying : g(n)=\sum_f(d)\quad\textn\ge 1 then :f(n)=\sum_\mu(d)\,g\!\left(\frac\right)\quad\textn\ge 1 where is the Möbius function and the sums extend over all positive divisors of (indicated by d \mid n in the above formulae). In effect, the original can be determined given by using the inversion formula. The two sequences are said to be Möbius transforms of each other. The formula is also correct if and are f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eisenstein's Criterion
In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials with rational coefficients. This criterion is not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but it does allow in certain important cases for irreducibility to be proved with very little effort. It may apply either directly or after transformation of the original polynomial. This criterion is named after Gotthold Eisenstein. In the early 20th century, it was also known as the Schönemann–Eisenstein theorem because Theodor Schönemann was the first to publish it. Criterion Suppose we have the following polynomial with integer coefficients: Q(x) = a_n x^n + a_ x^ + \cdots + a_1 x + a_0. If there exists a prime number such that the following three conditions all apply: * divid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
