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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applied Mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the 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 applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational ... [...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 1929–1934. 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 Fr ... [...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 An order-m linear differential operator is a map A from a function space \mathcal_1 to another function space \mathcal_2 that can be written as: A = \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. The operator D^\alpha is interpreted as D^\alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Continuous
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Piecewise definition is actually a way of expressing the function, rather than a characteristic of the function itself. A distinct, but related notion is that of a property holding piecewise for a function, used when the domain can be divided into intervals on which the property holds. Unlike for the notion above, this is actually a property of the function itself. A piecewise linear function (which happens to be also continuous) is depicted as an example. Notation and interpretation Piecewise functions can be defined using the common functional notation, where the body of the function is an array of functions and associated subdomains. These subdomains together must cover the whole domain; often it is also required that they are pair ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the 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 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 eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separated Boundary Conditions
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 In algebraic geometry, given a morphism of schemes p: X \to S, the diagonal morphism :\delta: X \to X \times_S X is a morphism determined by the universal property of the fiber product X \times_S X of ''p'' and ''p'' applied to the identity 1_X : X ..., 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), status of U.S. military personnel after release from active duty, but still having reserve obligations {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturm–Liouville Theory
In mathematics and its applications, classical Sturm–Liouville theory is the theory of ''real'' second-order ''linear'' ordinary differential equations of the form: for given coefficient functions , , and , an unknown function ''y = y''(''x'') of the free variable , and an unknown constant λ. All homogeneous (i.e. with the right-hand side equal to zero) second-order linear ordinary differential equations can be reduced to this form. In addition, the solution is typically required to satisfy some boundary conditions at extreme values of ''x''. Each such equation () together with its boundary conditions constitutes a Sturm–Liouville problem. In the simplest case where all coefficients are continuous on the finite closed interval and has continuous derivative, a function ''y = y''(''x'') is called a ''solution'' if it is continuously differentiable and satisfies the equation () at every x\in (a,b). In the case of more general , , , the solutions must be understood in a weak ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Integrable
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined. Long before the 20th century, mathematicians already understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the ''area under the curve'' could be defined as the integral, and computed using approximation techniques on the region by polygons. However, as the need to consider more irregular functions arose—e.g., as a result of the limiting processes of mathematical analysis and the mathematical theory of probability—it became clear that more careful approximation techniques were needed to define a suitable integral. Also, one might ... [...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 Heidelberg (; Palatine German language, Palatine German: ''Heidlberg'') is a c ... [...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] |
