|
Monodromy Operator
In mathematics, and particularly ordinary differential equations (ODEs), a monodromy matrix is the fundamental matrix of a system of ODEs evaluated at the period of the coefficients of the system. It is used for the analysis of periodic solutions of ODEs in Floquet theory. See also *Floquet theory *Monodromy *Riemann–Hilbert problem In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems h ... References * * Ordinary differential equations {{mathanalysis-stub ... [...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] |
Ordinary Differential Equations
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Matrix (linear Differential Equation)
In mathematics, a fundamental matrix of a system of ''n'' homogeneous linear ordinary differential equations \dot(t) = A(t) \mathbf(t) is a matrix-valued function \Psi(t) whose columns are linearly independent solutions of the system. Then every solution to the system can be written as \mathbf(t) = \Psi(t) \mathbf, for some constant vector \mathbf (written as a column vector of height ). One can show that a matrix-valued function \Psi is a fundamental matrix of \dot(t) = A(t) \mathbf(t) if and only if \dot(t) = A(t) \Psi(t) and \Psi is a non-singular matrix for all Control theory The fundamental matrix is used to express the state-transition matrix, an essential component in the solution of a system of linear ordinary differential equations. See also *Linear differential equation *Liouville's formula In mathematics, Liouville's formula, also known as the Abel-Jacobi-Liouville Identity, is an equation that expresses the determinant of a square-matrix solution of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floquet Theory
Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form :\dot = A(t) x, with \displaystyle A(t) a Piecewise#Continuity, piecewise continuous periodic function with period T and defines the state of the stability of solutions. The main theorem of Floquet theory, Floquet's theorem, due to , gives a canonical form for each Fundamental solution, fundamental matrix solution of this common linear system. It gives a Change of coordinates, coordinate change \displaystyle y=Q^(t)x with \displaystyle Q(t+2T)=Q(t) that transforms the periodic system to a traditional linear system with constant, real coefficients. When applied to physical systems with periodic potentials, such as crystals in condensed matter physics, the result is known as Bloch's theorem. Note that the solutions of the linear differential equation form a vector space. A matrix \phi\,(t) is called a ''Fundamental ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''monodromy'' comes from "running round singly". It is closely associated with covering maps and their degeneration into ramification; the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be ''single-valued'' as we "run round" a path encircling a singularity. The failure of monodromy can be measured by defining a monodromy group: a group of transformations acting on the data that encodes what happens as we "run round" in one dimension. Lack of monodromy is sometimes called ''polydromy''. Definition Let be a connected and locally connected based topological space with base point , and let p: \tilde \to X be a covering with fiber F = p^(x). For a loop based at , denote a lift under the covering ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Hilbert Problem
In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems have been produced by Mark Krein, Israel Gohberg and others (see the book by Clancey and Gohberg (1981)). The Riemann problem Suppose that \Sigma is a closed simple contour in the complex plane dividing the plane into two parts denoted by \Sigma_ (the inside) and \Sigma_ (the outside), determined by the index of the contour with respect to a point. The classical problem, considered in Riemann's PhD dissertation (see ), was that of finding a function :M_+(z) = u(z) + i v(z) analytic inside \Sigma_ such that the boundary values of ''M''+ along \Sigma satisfy the equation :a(z)u(z) - b(z)v(z) = c(z) for all z\in \Sigma, where ''a'', ''b'', and ''c'' are given real-valued functions . By the Riemann mapping theorem, it suffices to consider ... [...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 was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to 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 influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Providence, Rhode Island
Providence is the capital and most populous city of the U.S. state of Rhode Island. One of the oldest cities in New England, it was founded in 1636 by Roger Williams, a Reformed Baptist theologian and religious exile from the Massachusetts Bay Colony. He named the area in honor of "God's merciful Providence" which he believed was responsible for revealing such a haven for him and his followers. The city developed as a busy port as it is situated at the mouth of the Providence River in Providence County, at the head of Narragansett Bay. Providence was one of the first cities in the country to industrialize and became noted for its textile manufacturing and subsequent machine tool, jewelry, and silverware industries. Today, the city of Providence is home to eight hospitals and List of colleges and universities in Rhode Island#Institutions, eight institutions of higher learning which have shifted the city's economy into service industries, though it still retains some manufacturin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
