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Atkinson–Mingarelli Theorem
In applied mathematics, the Atkinson–Mingarelli theorem, named after Frederick Valentine Atkinson and A. B. Mingarelli, concerns eigenvalues of certain Sturm–Liouville differential operators. In the simplest of formulations let ''p'', ''q'', ''w'' be real-valued piecewise continuous functions defined on a closed bounded real interval, . The function ''w''(''x''), which is sometimes denoted by ''r''(''x''), is called the "weight" or "density" function. Consider the Sturm–Liouville differential equation where ''y'' is a function of the independent variable ''x''. In this case, ''y'' is called a ''solution'' if it is continuously differentiable on (''a'',''b'') and (''p'' ''y''′)(''x'') is piecewise continuously differentiable and ''y'' satisfies the equation () at all except a finite number of points in (''a'',''b''). The unknown function ''y'' is typically required to satisfy some boundary conditions at ''a'' and ''b''. The boundary conditions under consideration here ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applied Mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the profession, professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics. History Historically, applied mathematics consisted principally of Mathematical analysis, applied analysis, most notably differential equations; approximation theory (broadly construed, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frederick Valentine Atkinson
Frederick Valentine "Derick" Atkinson (25 January 1916 – 13 November 2002) was a British mathematician, formerly of the University of Toronto, Canada, where he spent most of his career. Atkinson's theorem and Atkinson–Wilcox theorem are named after him. His PhD advisor at Oxford was Edward Charles Titchmarsh. Early life and education The following synopsis is condensed (with permission) from Mingarelli's tribute to Atkinson. He attended St Paul's School, London from 1929 to 1934. The High Master of St. Paul's once wrote of Atkinson: "Extremely promising: He should make a brilliant mathematician"! Atkinson attended The Queen's College, Oxford in 1934 with a scholarship. During his stay at Queen's, he was secretary of the Chinese Student Society, and a member of the Indian Student Society. Auto-didactic when it came to languages, he taught himself and became fluent in Latin, Ancient Greek, Urdu, German, Hungarian, and Russian with some proficiency in Spanish, Italian, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative. Definition Given a nonnegative integer ''m'', an order-m linear differential operator is a map P from a function space \mathcal_1 on \mathbb^n to another function space \mathcal_2 that can be written as: P = \sum_a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative integers, , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n, and for each \alpha, a_\alpha(x) is a function on some open domain in ''n''-dimensional space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Continuous
In mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain is partitioned into several intervals ("subdomains") on which the function may be defined differently. Piecewise definition is actually a way of specifying the function, rather than a characteristic of the resulting function itself, as every function whose domain contains at least two points can be rewritten as a piecewise function. The first three paragraphs of this article only deal with this first meaning of "piecewise". Terms like piecewise linear, piecewise smooth, piecewise continuous, and others are also very common. The meaning of a function being piecewise P, for a property P is roughly that the domain of the function can be partitioned into pieces on which the property P holds, but is used slightly differently by different authors. Unlike the first meaning, this is a property of the function itself and not on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Condition
In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems, in the linear case, involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devote ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separated Boundary Conditions , status of U.S. military personnel after release from active duty, but still having reserve obligations
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Separated can refer to: *Marital separation of spouses **Legal separation of spouses * "Separated" (song), song by Avant *Separated sets, a concept in mathematical topology *Separated space, a synonym for Hausdorff space, a concept in mathematical topology *Separated morphism, a concept in algebraic geometry analogous to that of separated space in topology *Separation of conjoined twins, a procedure that allows them to live independently. *Separation (United States military) In the United States Armed Forces, separation means that a person is leaving active duty but not necessarily the service entirely. Separation typically occurs when someone reaches the date of their Expiration of Term of Service and are released fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturm–Liouville Theory
In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form \frac \left (x) \frac\right+ q(x)y = -\lambda w(x) y for given functions p(x), q(x) and w(x), together with some boundary conditions at extreme values of x. The goals of a given Sturm–Liouville problem are: * To find the for which there exists a non-trivial solution to the problem. Such values are called the ''eigenvalues'' of the problem. * For each eigenvalue , to find the corresponding solution y = y(x) of the problem. Such functions y are called the '' eigenfunctions'' associated to each . Sturm–Liouville theory is the general study of Sturm–Liouville problems. In particular, for a "regular" Sturm–Liouville problem, it can be shown that there are an infinite number of eigenvalues each with a unique eigenfunction, and that these eigenfunctions form an orthonormal basis of a certain Hilbert space of functions. This theory is important ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Integrable
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, named after france, French mathematician Henri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions. The Lebesgue integral is more general than the Riemann integral, which it largely replaced in mathematical analysis since the first half of the 20th century. It can accommodate functions with discontinuities arising in many applications that are pathological from the perspective of the Riemann integral. The Lebesgue integral also has generally better analytical properties. For instance, under mild conditions, it is possible to exchange limits and Lebesgue integration, while the conditions for doing this with a Riemann integral are comparatively baroque. Furthermore, the Lebesgue integral can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Konrad Jörgens
Konrad Jörgens (3 December 1926 – 28 April 1974) was a German mathematician. He made important contributions to mathematical physics, in particular to the foundations of quantum mechanics, and to the theory of partial differential equations and integral operators. Career He studied at Karlsruhe (1949–51) and Göttingen (1951–54) where he received his doctorate in 1954 under Franz Rellich, with a thesis on the Monge-Ampere equation. From 1954-1958 he was at the Max Planck Institute for Physics and Astrophysics at Göttingen, with an interim stay at New York University (1956–57). From 1958 he was at the Institute of Applied Mathematics at Heidelberg, where he received his habilitation in July, 1959. In June 1961 he was appointed to the newly created professorship of applied and practical mathematics at the same institute. In 1966 he became professor of applied mathematics at Heidelberg. Selected publications * * * References * External links Konrad Jörgensat th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equations
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with ''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with ''stochastic'' differential equations (SDEs) where the progression is random. 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 a_0(x),\ldots,a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y',\ldots, y^ are the successive derivatives of the unknown function y of the variable x. Among ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
