|
Routh–Hurwitz Theorem
In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left half-plane. Polynomials with this property are called Hurwitz stable polynomials. The Routh-Hurwitz theorem is important in dynamical systems and control theory, because the characteristic polynomial of the differential equations of a stable linear system has roots limited to the left half plane (negative eigenvalues). Thus the theorem provides a test to determine whether a linear dynamical system is stable without solving the system. The Routh–Hurwitz theorem was proved in 1895, and it was named after Edward John Routh and Adolf Hurwitz. Notations Let ''f''(''z'') be a polynomial (with complex coefficients) of degree ''n'' with no roots on the imaginary axis (i.e. the line ''Z'' = ''ic'' where ''i'' is the imaginary unit and ''c'' is a real number). Let us define P_0(y) (a polynomial of degree ''n'') and P_1(y) (a nonzero polynomial o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Polynomial
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of ''degree'' but, nowadays, may refer to several other concepts (see order of a polynomial (other)). For example, the polynomial 7x^2y^3 + 4x - 9, which can also be written as 7x^2y^3 + 4x^1y^0 - 9x^0y^0, has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (x+1)^2 - (x-1)^2, one can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems About Polynomials
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press
Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books by decree in 1586, it is the second oldest university press after Cambridge University Press. It is a department of the University of Oxford and is governed by a group of 15 academics known as the Delegates of the Press, who are appointed by the vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, opposite Somerville College, in the inner suburb of Jericho. For the last 500 years, OUP has primarily focused on the publication of pedagogical texts and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Polynomial
In the context of the characteristic polynomial of a differential equation or difference equation, a polynomial is said to be stable if either: * all its roots lie in the open left half-plane, or * all its roots lie in the open unit disk. The first condition provides stability for continuous-time linear systems, and the second case relates to stability of discrete-time linear systems. A polynomial with the first property is called at times a Hurwitz polynomial and with the second property a Schur polynomial. Stable polynomials arise in control theory and in mathematical theory of differential and difference equations. A linear, time-invariant system (see LTI system theory) is said to be BIBO stable if every bounded input produces bounded output. A linear system is BIBO stable if its characteristic polynomial is stable. The denominator is required to be Hurwitz stable if the system is in continuous-time and Schur stable if it is in discrete-time. In practice, stability is det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a point. The integers are often shown as specially-marked points evenly spaced on the line. Although the image only shows the integers from –3 to 3, the line includes all real numbers, continuing forever in each direction, and also numbers that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers. In advanced mathematics, the number line can be called as a real line or real number line, formally defined as the set (mathematics), set of all real numbers, viewed as a geometry, geometric space (mathematics), space, namely the Euclidean space of dimension one. It can be thought of as a vector space (or affine space), a metric space, a topological ... [...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 ''K''. In this case, one speaks of a rational function and a rational fraction ''over K''. The values of the variables may be taken in any field ''L'' containing ''K''. 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 ''L''. The set of rational functions over a field ''K'' is a field, the field of fractions of the ring of the polynomial functions over ''K''. Definitions A function f(x) is called a rational function if and only 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\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Index
In mathematical analysis, the Cauchy index is an integer associated to a real rational function over an interval. By the Routh–Hurwitz theorem, we have the following interpretation: the Cauchy index of :''r''(''x'') = ''p''(''x'')/''q''(''x'') over the real line is the difference between the number of roots of ''f''(''z'') located in the right half-plane and those located in the left half-plane. The complex polynomial ''f''(''z'') is such that :''f''(''iy'') = ''q''(''y'') + ''ip''(''y''). We must also assume that ''p'' has degree less than the degree of ''q''. Definition * The Cauchy index was first defined for a pole ''s'' of the rational function ''r'' by Augustin-Louis Cauchy in 1837 using one-sided limits as: : I_sr = \begin +1, & \text \displaystyle\lim_r(x)=-\infty \;\land\; \lim_r(x)=+\infty, \\ -1, & \text \displaystyle\lim_r(x)=+\infty \;\land\; \lim_r(x)=-\infty, \\ 0, & \text \end * A generalization over the compact interval 'a'',''b''is direct (when neither ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his ''Elements'' (c. 300 BC). It is an example of an ''algorithm'', a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations. The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturm's Theorem
In mathematics, the Sturm sequence of a univariate polynomial is a sequence of polynomials associated with and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of located in an interval in terms of the number of changes of signs of the values of the Sturm sequence at the bounds of the interval. Applied to the interval of all the real numbers, it gives the total number of real roots of . Whereas the fundamental theorem of algebra readily yields the overall number of complex roots, counted with multiplicity, it does not provide a procedure for calculating them. Sturm's theorem counts the number of distinct real roots and locates them in intervals. By subdividing the intervals containing some roots, it can isolate the roots into arbitrarily small intervals, each containing exactly one root. This yields the oldest real-root isolation algorithm, and arbitrary-precision root-finding algorithm for univariate ... [...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 number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.PNG "Click for more on -> Theorems About Polynomials")
.jpg "Click for more on -> Oxford University Press")
