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Descartes' Rule Of Signs
In mathematics, Descartes' rule of signs, described by René Descartes in his ''La Géométrie'', counts the roots of a polynomial by examining sign changes in its coefficients. The number of positive real roots is at most the number of sign changes in the sequence of the polynomial's coefficients (omitting zero coefficients), and the difference between the root count and the sign change count is always even. In particular, when the number of sign changes is zero or one, then there are exactly zero or one positive roots. A linear fractional transformation of the variable makes it possible to use the rule of signs to count roots in any interval. This is the basic idea of Budan's theorem and the Budan–Fourier theorem. Repeated division of an interval in two results in a set of disjoint intervals, each containing one root, and together listing all the roots. This approach is used in the fastest algorithms today for computer computation of real roots of polynomials (see real-root is ... [...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] |
Factorization Of Polynomials
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of computer algebra systems. The first polynomial factorization algorithm was published by Theodor von Schubert in 1793. Leopold Kronecker rediscovered Schubert's algorithm in 1882 and extended it to multivariate polynomials and coefficients in an algebraic extension. But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra systems: When the long-known finite step algorithms were first put on computers, they turned out to be highly inefficient. The fact that almost any uni- or multivariate polynomial of degree up to 100 and with coefficients of a moderate size (up to 100 bits) can be factored by modern algorithms in a few minutes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendental Function
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction, multiplication, and division (without the need of taking limits). This is in contrast to an algebraic function. Examples of transcendental functions include the exponential function, the logarithm function, the hyperbolic functions, and the trigonometric functions. Equations over these expressions are called transcendental equations. Definition Formally, an analytic function f of one real or complex variable is transcendental if it is algebraically independent of that variable. This means the function does not satisfy any polynomial equation. For example, the function f given by :f(x)=\frac for all x is not transcendental, but algebraic, because it satisfies the polynomial equation :(ax+b)-(cx+d)f(x)=0. Similarly, the functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fewnomial
In mathematics, a sparse polynomial (also lacunary polynomial or fewnomial) is a polynomial that has far fewer terms than its degree and number of variables would suggest. For example, x^ + 3x^3 + 1 is a sparse polynomial, as it is a trinomial with a degree of 10. The motivation for studying sparse polynomials is to concentrate on the structure of a polynomial's monomials instead of its degree, as one can see, for instance, by comparing Bernstein–Kushnirenko theorem with Bezout's theorem. Research on sparse polynomials has also included work on algorithms whose running time grows as a function of the number of terms rather than on the degree, for problems including polynomial multiplication, division, root-finding algorithms, and polynomial greatest common divisors. Sparse polynomials have also been used in pure mathematics, especially in the study of Galois groups, because it has been easier to determine the Galois groups of certain families of sparse polynomials than it is f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Askold Khovanskii
Askold Georgievich Khovanskii (; born 3 June 1947, Moscow) is a Russian and Canadian mathematician currently a professor of mathematics at the University of Toronto, Canada. His areas of research are algebraic geometry, commutative algebra, singularity theory, differential geometry and differential equations. His research is in the development of the theory of toric varieties and Newton polyhedra in algebraic geometry. He is also the inventor of the theory of fewnomials, and the Bernstein–Khovanskii–Kushnirenko theorem is named after him. He obtained his Ph.D. from Steklov Mathematical Institute in Moscow under the supervision of Vladimir Arnold. In his Ph.D. thesis, he developed a topological version of Galois theory. He studies the theory of Newton–Okounkov bodies, or Okounkov bodies for short. Among his graduate students are Olga Gel'fond, Feodor Borodich, H. Petrov-Tan'kin, Kiumars Kaveh, Farzali Izadi, Ivan Soprunov, Jenya Soprunova, Vladlen Timorin, Valentina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Part
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 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 have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in 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 real or complex coefficients h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - bi. The complex conjugate of z is often denoted as \overline or z^*. In polar form, if r and \varphi are real numbers then the conjugate of r e^ is r e^. This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a^2 + b^2 (or r^2 in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root. Notation The complex conjugate of a complex number z is written as \overline z or z^*. The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even Number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows for a geometric interpretation of complex numbers. Under addition, they add like vector (geometry), vectors. The multiplication of two complex numbers can be expressed more easily in polar coordinates: the magnitude or ' of the product is the product of the two absolute values, or moduli, and the angle or ' 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 called the Argand plane or Gauss plane. Notational conventions Complex numbers In complex analysis, the complex numbers are customarily represented by the symbol , which can be sepa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rolle's Theorem
In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one point, somewhere between them, at which the slope of the tangent line is zero. Such a point is known as a stationary point. It is a point at which the first derivative of the function is zero. The theorem is named after Michel Rolle. Standard version of the theorem If a real-valued function is continuous on a proper closed interval , differentiable on the open interval , and , then there exists at least one in the open interval such that f'(c) = 0. This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of Taylor's theorem. History Although the theorem is named after Michel Rolle, Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods of di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
