|
Hurwitz Matrix
In mathematics, a Hurwitz matrix, or Routh–Hurwitz matrix, in engineering stability matrix, is a structured real square matrix constructed with coefficients of a real polynomial. Hurwitz matrix and the Hurwitz stability criterion Namely, given a real polynomial :p(z)=a_z^n+a_z^+\cdots+a_z+a_n the n\times n square matrix : H= \begin a_1 & a_3 & a_5 & \dots & \dots & \dots & 0 & 0 & 0 \\ a_0 & a_2 & a_4 & & & & \vdots & \vdots & \vdots \\ 0 & a_1 & a_3 & & & & \vdots & \vdots & \vdots \\ \vdots & a_0 & a_2 & \ddots & & & 0 & \vdots & \vdots \\ \vdots & 0 & a_1 & & \ddots & & a_n & \vdots & \vdots \\ \vdots & \vdots & a_0 & & & \ddots & a_ & 0 & \vdots \\ \vdots & \vdots & 0 & & & & a_ & a_n & \vdots \\ \vdots & \vdots & \vdots & & & & a_ & a_ & 0 \\ 0 & 0 & 0 & \dots & \dots & \dots & a_ & a_ & a_n \end. is called Hurwitz matrix corresponding to the polynomial p. It was established by Adolf Hurwitz in 1895 that a real polynomial with a_0 > 0 is stable (that is, all its roots have ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical System (definition)
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interscience
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, technical, and engineering subject areas, abandoning its literary interests. Wiley's son John (born in Flatbush, New York, October 4, 1808; died in East Orange, New ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Approximation Theory
The ''Journal of Approximation Theory'' is "devoted to advances in pure and applied approximation theory and related areas." References External links ''Journal of Approximation Theory'' web site''Journal of Approximation Theory'' home page at Elsevier Mathematics journals Approximation theory Publications established in 1968 Elsevier academic journals English-language journals Monthly journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Applied Mathematics
The ''SIAM Journal on Applied Mathematics'' is a peer-reviewed academic journal in applied mathematics published by the Society for Industrial and Applied Mathematics (SIAM), with Paul A. Martin (Colorado School of Mines) as its editor-in-chief. It was founded in 1953 as SIAM's first journal, the ''Journal of the Society for Industrial and Applied Mathematics'', and was given its current name in 1966. In most years since 1999, it has been ranked by SCImago Journal Rank as a second-quartile journal in applied mathematics. Together with ''Communications on Pure and Applied Mathematics ''Communications on Pure and Applied Mathematics'' is a monthly peer-reviewed scientific journal which is published by John Wiley & Sons on behalf of the Courant Institute of Mathematical Sciences. It covers research originating from or solicited ...'' it has been called "one of the two greatest American entries in applied math".. References {{Society for Industrial and Applied Mathematics Mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Z-matrix (mathematics)
In mathematics, the class of ''Z''-matrices are those matrices whose off-diagonal entries are less than or equal to zero; that is, the matrices of the form: :Z=(z_);\quad z_\leq 0, \quad i\neq j. Note that this definition coincides precisely with that of a negated Metzler matrix or quasipositive matrix, thus the term ''quasinegative'' matrix appears from time to time in the literature, though this is rare and usually only in contexts where references to quasipositive matrices are made. The Jacobian of a competitive dynamical system is a ''Z''-matrix by definition. Likewise, if the Jacobian of a cooperative dynamical system is ''J'', then (−''J'') is a ''Z''-matrix. Related classes are ''L''-matrices, ''M''-matrices, ''P''-matrices, ''Hurwitz'' matrices and ''Metzler'' matrices. ''L''-matrices have the additional property that all diagonal entries are greater than zero. M-matrices have several equivalent definitions, one of which is as follows: a ''Z''-matrix is an ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perron–Frobenius Theorem
In matrix theory, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory ( ergodicity of Markov chains); to the theory of dynamical systems (subshifts of finite type); to economics ( Okishio's theorem, Hawkins–Simon condition); to demography ( Leslie population age distribution model); to social networks ( DeGroot learning process); to Internet search engines ( PageRank); and even to ranking of football teams. The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau. Statement Let positive and non-negative respectively describe matrices with exclusively positive real numbers as elements an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-matrix
In mathematics, a -matrix is a complex square matrix with every principal minor is positive. A closely related class is that of P_0-matrices, which are the closure of the class of -matrices, with every principal minor \geq 0. Spectra of -matrices By a theorem of Kellogg, the eigenvalues of - and P_0- matrices are bounded away from a wedge about the negative real axis as follows: :If \ are the eigenvalues of an -dimensional -matrix, where n>1, then ::, \arg(u_i), < \pi - \frac,\ i = 1,...,n :If , , are the eigenvalues of an -dimensional -matrix, then :: Remarks The class of nonsingular ''M''-matrices is a subset of the class of -matrices. More precisely, all matrices that are both -matrices and[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
M-matrix
In mathematics, especially linear algebra, an ''M''-matrix is a ''Z''-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular ''M''-matrices are a subset of the class of ''P''-matrices, and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of positive matrices). The name ''M''-matrix was seemingly originally chosen by Alexander Ostrowski in reference to Hermann Minkowski, who proved that if a Z-matrix has all of its row sums positive, then the determinant of that matrix is positive.. Characterizations An M-matrix is commonly defined as follows: Definition: Let be a real Z-matrix. That is, where for all . Then matrix ''A'' is also an ''M-matrix'' if it can be expressed in the form , where with , for all , where is at least as large as the maximum of the moduli of the eigenvalues of , and is an identity matrix. For the non-singularity of , according to the Perron–Frobenius theorem, it must be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liénard–Chipart Criterion
In control system theory, the Liénard–Chipart criterion is a stability criterion modified from the Routh–Hurwitz stability criterion, proposed by A. Liénard and M. H. Chipart. This criterion has a computational advantage over the Routh–Hurwitz criterion because it involves only about half the number of determinant computations. Algorithm The Routh–Hurwitz stability criterion says that a necessary and sufficient condition for all the roots of the polynomial with real coefficients ::f(z) = a_0 z^n + a_1 z^ + \cdots + a_n \, (a_0 > 0) to have negative real parts (i.e. f is Hurwitz stable A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...) is that :: \Delta_1 > 0,\, \Delta_2 > 0, \ldots, \Delta_n > 0, where \Delta_i is the ''i''-th leading principal minor of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Feedback
Positive feedback (exacerbating feedback, self-reinforcing feedback) is a process that occurs in a feedback loop which exacerbates the effects of a small disturbance. That is, the effects of a perturbation on a system include an increase in the magnitude of the perturbation. That is, ''A produces more of B which in turn produces more of A''.Keesing, R.M. (1981). Cultural anthropology: A contemporary perspective (2nd ed.) p.149. Sydney: Holt, Rinehard & Winston, Inc. In contrast, a system in which the results of a change act to reduce or counteract it has negative feedback. Both concepts play an important role in science and engineering, including biology, chemistry, and cybernetics. Mathematically, positive feedback is defined as a positive loop gain around a closed loop of cause and effect. That is, positive feedback is in phase with the input, in the sense that it adds to make the input larger. Positive feedback tends to cause system instability. When the loop gain is pos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negative Feedback
Negative feedback (or balancing feedback) occurs when some function of the output of a system, process, or mechanism is fed back in a manner that tends to reduce the fluctuations in the output, whether caused by changes in the input or by other disturbances. Whereas positive feedback tends to lead to instability via exponential growth, oscillation or chaotic behavior, negative feedback generally promotes stability. Negative feedback tends to promote a settling to equilibrium, and reduces the effects of perturbations. Negative feedback loops in which just the right amount of correction is applied with optimum timing can be very stable, accurate, and responsive. Negative feedback is widely used in mechanical and electronic engineering, and also within living organisms, and can be seen in many other fields from chemistry and economics to physical systems such as the climate. General negative feedback systems are studied in control systems engineering. Negative feedback ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
