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Reciprocal 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. They 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 have several conn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 not greater 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 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 pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Totient Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Number Field Sieve
In number theory, a branch of mathematics, the special number field sieve (SNFS) is a special-purpose integer factorization algorithm. The general number field sieve (GNFS) was derived from it. The special number field sieve is efficient for integers of the form ''r''''e'' ± ''s'', where ''r'' and ''s'' are small (for instance Mersenne numbers). Heuristically, its complexity for factoring an integer n is of the form: :\exp\left(\left(1+o(1)\right)\left(\tfrac\log n\right)^\left(\log\log n\right)^\right)=L_n\left /3,(32/9)^\right/math> in O and L-notations. The SNFS has been used extensively by NFSNet (a volunteer distributed computing effort)NFS@Homeand others to factorise numbers of the Cunningham project; for some time the records for integer factorization have been numbers factored by SNFS. Overview of method The SNFS is based on an idea similar to the much simpler rational sieve; in particular, readers may find it helpful to read about the rational sieve first, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Polynomial (field Theory)
In field theory, a branch of mathematics, the minimal polynomial of an element of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the field, such that is a root of the polynomial. If the minimal polynomial of exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1, and the type for the remaining coefficients could be integers, rational numbers, real numbers, or others. More formally, a minimal polynomial is defined relative to a field extension and an element of the extension field . The minimal polynomial of an element, if it exists, is a member of , the ring of polynomials in the variable with coefficients in . Given an element of , let be the set of all polynomials in such that . The element is called a root or zero of each polynomial in . The set is so named because it is an ideal of . The zero polynomial, all of whose coefficients are 0, is in every since for all and . Thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as because it is a one-dimensional unit -sphere. If is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, and satisfy the equation x^2 + y^2 = 1. Since for all , and since the reflection of any point on the unit circle about the - or -axis is also on the unit circle, the above equation holds for all points on the unit circle, not only those in the first quadrant. The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk. One may also use other notions of "dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed more easily in polar coordinates—the magnitude or ''modulus'' of the product is the product of the two absolute values, or moduli, and the angle or ''argument'' of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is sometimes known as the Argand plane or Gauss plane. Notational conventions Complex numbers In complex analysis, the complex numbers are customarily represented by the symbol ''z'', which can be separated into its real (''x'') an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with rea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Therefore, a monic polynomial has the form: :x^n+c_x^+\cdots+c_2x^2+c_1x+c_0 Univariate polynomials If a polynomial has only one indeterminate ( univariate polynomial), then the terms are usually written either from highest degree to lowest degree ("descending powers") or from lowest degree to highest degree ("ascending powers"). A univariate polynomial in ''x'' of degree ''n'' then takes the general form displayed above, where : ''c''''n'' ≠ 0, ''c''''n''−1, ..., ''c''2, ''c''1 and ''c''0 are constants, the coefficients of the polynomial. Here the term ''c''''n''''x''''n'' is called the ''leading term'', and its coefficient ''c''''n'' the ''leading coefficient''; if the leading coefficient , the univariate polynomial is called monic. Properties Multiplicatively closed The s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
