|
Principal Equation Form
In mathematics and, more specifically, in theory of equations, the principal form of an irreducible polynomial of degree at least three is a polynomial of the same degree ''n'' without terms of degrees ''n''−1 and ''n''−2, such that each root of either polynomial is a rational function of a root of the other polynomial. The principal form of a polynomial can be found by applying a suitable Tschirnhaus transformation to the given polynomial. Definition Let :f(x)=x^n+a_1x^+\cdots +a_x+a_n be an irreducible polynomial of degree at least three. Its ''principal form'' is a polynomial :g(y)=y^n+b_3y^+\cdots +b_y+b_n, together with a Tschirnhaus transformation of degree two :\varphi(x)=x^2 + \alpha x + \beta such that, if is a root of , \phi (r) is a root of . Expressing that does not has terms in and leads to a system of two equations in and , one of degree one and one of degree two. In general, this system has two solutions, giving two principal forms involving a square ... [...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] |
Theory Of Equations
In algebra, the theory of equations is the study of algebraic equations (also called "polynomial equations"), which are equation (mathematics), equations defined by a polynomial. The main problem of the theory of equations was to know when an algebraic equation has an algebraic solution. This problem was completely solved in 1830 by Évariste Galois, by introducing what is now called Galois theory. Before Galois, there was no clear distinction between the "theory of equations" and "algebra". Since then algebra has been dramatically enlarged to include many new subareas, and the theory of algebraic equations receives much less attention. Thus, the term "theory of equations" is mainly used in the context of the history of mathematics, to avoid confusion between old and new meanings of "algebra". History Until the end of the 19th century, "theory of equations" was almost synonymous with "algebra". For a long time, the main problem was to find the solutions of a single non-linear po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreducible Polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as \left(x - \sqrt\right)\left(x + \sqrt\right) if it is considered as a polynomial with real coefficients. One says that the polynomial is irreducible over the integers but not over the reals. Polynomial irreducibility can be considered for polynomials with coefficients in an integral domain, and there are two common definitions. Most often, a pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of A Polynomial
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is a solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6=(x-2)(x-3) has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field . In this case, one speaks of a rational function and a rational fraction ''over ''. The values of the variables may be taken in any field containing . Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is . The set of rational functions over a field is a field, the field of fractions of the ring of the polynomial functions over . Definitions A function f is called a rational function if it can be written in the form : f(x) = \frac where P and Q are polynomial functions of x and Q is not the zero function. The domain of f is the set of all values of x for which the denominator Q(x) is not zero. How ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tschirnhaus Transformation
In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683. Simply, it is a method for transforming a polynomial equation of degree n\ge2 with some nonzero intermediate coefficients, a_1, ..., a_, such that some or all of the transformed intermediate coefficients, a'_1, ..., a'_, are exactly zero. For example, finding a substitutiony(x)=k_1x^2 + k_2x+k_3for a cubic equation of degree n=3,f(x) = x^3+a_2x^2+a_1x+a_0such that substituting x=x(y) yields a new equationf'(y)=y^3+a'_2y^2+a'_1y+a'_0such that a'_1=0, a'_2=0, or both. More generally, it may be defined conveniently by means of field theory, as the transformation on minimal polynomials implied by a different choice of primitive element. This is the most general transformation of an irreducible polynomial that takes a root to some rational function applied to that root. Definition For a generic n^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor & Francis
Taylor & Francis Group is an international company originating in the United Kingdom that publishes books and academic journals. Its parts include Taylor & Francis, CRC Press, Routledge, F1000 (publisher), F1000 Research and Dovepress. It is a division of Informa, a United Kingdom-based publisher and conference company. Overview Founding The company was founded in 1852 when William Francis (chemist), William Francis joined Richard Taylor (editor), Richard Taylor in his publishing business. Taylor had founded his company in 1798. Their subjects covered agriculture, chemistry, education, engineering, geography, law, mathematics, medicine, and social sciences. Publications included the ''Philosophical Magazine''. Francis's son, Richard Taunton Francis (1883–1930), was sole partner in the firm from 1917 to 1930. Acquisitions and mergers In 1965, Taylor & Francis launched Wykeham Publications and began book publishing. T&F acquired Hemisphere Publishing in 1988, and the compa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
YouTube
YouTube is an American social media and online video sharing platform owned by Google. YouTube was founded on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim who were three former employees of PayPal. Headquartered in San Bruno, California, it is the second-most-visited website in the world, after Google Search. In January 2024, YouTube had more than 2.7billion monthly active users, who collectively watched more than one billion hours of videos every day. , videos were being uploaded to the platform at a rate of more than 500 hours of content per minute, and , there were approximately 14.8billion videos in total. On November 13, 2006, YouTube was purchased by Google for $1.65 billion (equivalent to $ billion in ). Google expanded YouTube's business model of generating revenue from advertisements alone, to offering paid content such as movies and exclusive content produced by and for YouTube. It also offers YouTube Premium, a paid subs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Equation
In algebra, a cubic equation in one variable is an equation of the form ax^3+bx^2+cx+d=0 in which is not zero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the coefficients , , , and of the cubic equation are real numbers, then it has at least one real root (this is true for all odd-degree polynomial functions). All of the roots of the cubic equation can be found by the following means: * algebraically: more precisely, they can be expressed by a ''cubic formula'' involving the four coefficients, the four basic arithmetic operations, square roots, and cube roots. (This is also true of quadratic (second-degree) and quartic (fourth-degree) equations, but not for higher-degree equations, by the Abel–Ruffini theorem.) * trigonometrically * numerical approximations of the roots can be found using root-finding algorithms such as Newton's method. The coefficients do not need to be real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nome (mathematics)
In mathematics, specifically the theory of elliptic functions, the nome is a special function that belongs to the non-elementary functions. This function is of great importance in the description of the elliptic functions, especially in the description of the modular identity of the Jacobi theta function, the Hermite elliptic transcendents and the Weber modular functions, that are used for solving equations of higher degrees. Definition The nome function is given by :q =\mathrm^ =\mathrm^ =\mathrm^ \, where K and iK' are the quarter periods, and \omega_1 and \omega_2 are the fundamental pair of periods, and \tau=\frac=\frac is the half-period ratio. The nome can be taken to be a function of any one of these quantities; conversely, any one of these quantities can be taken as functions of the nome. Each of them uniquely determines the others when 0 |
Jacobi Theta Function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. Theta functions are parametrized by points in a tube domain inside a complex Lagrangian Grassmannian, namely the Siegel upper half space. The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called ), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent. One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel Ruffini Theorem
Abel ( ''Hébel'', in pausa ''Hā́ḇel''; ''Hábel''; , ''Hābēl'') is a biblical figure in the Book of Genesis within the Abrahamic religions. Born as the second son of Adam and Eve, the first two humans created by God, he was a shepherd who offered his firstborn flock to God as a religious offering. God accepted Abel's offering but not the offering of his older brother Cain, leading Cain to stone Abel to death out of jealousy. This act marked the first death in biblical history, making Abel the first murder victim. Life and death Interpretations Jewish and Christian interpretations According to the narrative in Genesis, Abel is Eve's second son. His name in Hebrew is composed of the same three consonants as a root meaning "the air that remains after you exhale" also synonymous in Hebrew to "nothing", as stated in Ecclesiastes. Julius Wellhausen has proposed that the name is independent of the root. Eberhard Schrader had previously put forward the Akkadian (Old Assyrian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
