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Sturm–Picone Comparison Theorem
In mathematics, in the field of ordinary differential equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the oscillation and non-oscillation of solutions of certain linear differential equations in the real domain. Let , for be real-valued continuous functions on the interval and let #(p_1(x) y^\prime)^\prime + q_1(x) y = 0 #(p_2(x) y^\prime)^\prime + q_2(x) y = 0 be two homogeneous linear second order differential equations in self-adjoint form with :0 < p_2(x) \le p_1(x) and : Let be a non-trivial solution of (1) with successive roots at and and let be a non-trivial solution of (2). Then one of the following properties holds. *There exists an in such that or *there exists a in such that . The first part of the conclusion is due to Sturm (1836), while the second (alternative) part of the theorem is due to Picone (19 ... [...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 Equation
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] |
Jacques Charles François Sturm
Jacques Charles François Sturm (29 September 1803 – 15 December 1855) was a French mathematician. Life and work Sturm was born in Geneva (then part of France) in 1803. The family of his father, Jean-Henri Sturm, had emigrated from Strasbourg around 1760—about 50 years before Charles-François's birth. His mother's name was Jeanne-Louise-Henriette Gremay. In 1818, he started to follow the lectures of the academy of Geneva. In 1819, the death of his father forced Sturm to give lessons to children of the rich in order to support his own family. In 1823, he became tutor to the son of Madame de Staël. At the end of 1823, Sturm stayed in Paris for a short time following the family of his student. He resolved, with his school-fellow Jean-Daniel Colladon, to try his fortune in Paris, and obtained employment on the ''Bulletin universel''. In 1829, he discovered the theorem that bears his name, and concerns real-root isolation, that is the determination of the number and the localiz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mauro Picone
Mauro Picone (2 May 1885 – 11 April 1977) was an Italian mathematician. He is known for the Picone identity, the Sturm-Picone comparison theorem and being the founder of the Istituto per le Applicazioni del Calcolo, presently named after him, the first applied mathematics institute ever founded.See , , and the references cited in this latter one. He was also an outstanding teacher of mathematical analysis: some of the best Italian mathematicians were among his pupils. Work Research activity Teaching activity Notable students: *Luigi Amerio *Renato Caccioppoli *Gianfranco Cimmino * Ennio de Giorgi *Gaetano Fichera *Carlo Miranda Selected publications * (Review of the whole volume I) (available from the "Edizione Nazionale Mathematica Italiana'"), reviewed by . *, (Review of the 2nd part of volume I) (available from the "Edizione Nazionale Mathematica Italiana'"). *, reviewed by and by . See also *Renato Caccioppoli *Lamberto Cesari * Ennio de Giorgi *Gaetano Fichera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oscillation Theory
In mathematics, in the field of ordinary differential equations, a nontrivial solution to an ordinary differential equation :F(x,y,y',\ \dots,\ y^)=y^ \quad x \in roots; otherwise it is called non-oscillating. The differential equation is called oscillating if it has an oscillating solution. The number of roots carries also information on the Spectrum (functional analysis)">spectrum of associated boundary value problems. Examples The differential equation :y'' + y = 0 is oscillating as sin(''x'') is a solution. Connection with spectral theory Oscillation theory was initiated by Jacques Charles François Sturm in his investigations of Sturm–Liouville problems from 1836. There he showed that the n'th eigenfunction of a Sturm–Liouville problem has precisely n-1 roots. For the one-dimensional Schrödinger equation the question about oscillation/non-oscillation answers the question whether the eigenvalues accumulate at the bottom of the continuous spectrum. Relative oscil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Differential Equation
In mathematics, 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) where and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable . Such an equation is an ordinary differential equation (ODE). A ''linear differential equation'' may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives. A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-adjoint Form
In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if x^*=y then since y^*=x^=x in a star-algebra, the set is a self-adjoint set even though ''x'' and ''y'' need not be self-adjoint elements. In functional analysis, a linear operator A : H \to H on a Hilbert space is called self-adjoint if it is equal to its own adjoint ''A''. See self-adjoint operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator ''A'' is self-adjoint if and only if the matrix describing ''A'' with respect to this basis is Hermitian, i.e. if it is equal to its own conjugate transpose. Hermitian matrices are also called self-adjoint. In a dagger cat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picone Identity
In the field of ordinary differential equations, the Picone identity, named after Mauro Picone, is a classical result about homogeneous linear second order differential equations. Since its inception in 1910 it has been used with tremendous success in association with an almost immediate proof of the Sturm comparison theorem, a theorem whose proof took up many pages in Sturm's original memoir of 1836. It is also useful in studying the oscillation of such equations and has been generalized to other type of differential equations and difference equations. The Picone identity is used to prove the Sturm–Picone comparison theorem. Picone identity Suppose that ''u'' and ''v'' are solutions of the two homogeneous linear second order differential equations in self-adjoint form In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturm Separation Theorem
In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of solutions of homogeneous second order linear differential equations. Basically the theorem states that given two linear independent solutions of such an equation the zeros of the two solutions are alternating. Sturm separation theorem If ''u''(''x'') and ''v''(''x'') are two non-trivial continuous linearly independent solutions to a homogeneous second order linear differential equation with ''x''0 and ''x''1 being successive roots of ''u''(''x''), then ''v''(''x'') has exactly one root in the open interval (''x''0, ''x''1). It is a special case of the Sturm-Picone comparison theorem. Proof Since \displaystyle u and \displaystyle v are linearly independent it follows that the Wronskian \displaystyle W ,v/math> must satisfy W ,vx)\equiv W(x)\neq 0 for all \displaystyle x where the differential equation is d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mingarelli Identity
In the field of ordinary differential equations, the Mingarelli identityThe locution was coined by Philip Hartman, according to is a theorem that provides criteria for the oscillation and non-oscillation of solutions of some linear differential equations in the real domain. It extends the Picone identity from two to three or more differential equations of the second order. The identity Consider the solutions of the following (uncoupled) system of second order linear differential equations over the –interval : :(p_i(t) x_i^\prime)^\prime + q_i(t) x_i = 0, \,\,\,\,\,\,\,\,\,\, x_i(a)=1,\,\, x_i^\prime(a)=R_i where i=1,2, \ldots, n. Let \Delta denote the forward difference operator, i.e. :\Delta x_i = x_-x_i. The second order difference operator is found by iterating the first order operator as in :\Delta^2 (x_i) = \Delta(\Delta x_i) = x_-2x_+x_,, with a similar definition for the higher iterates. Leaving out the independent variable for convenience, and assuming the on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joyce McLaughlin
Joyce Rogers McLaughlin (8 October 1939 – 23 October 2017) was an American mathematician, the Ford Foundation Professor of Mathematics at Rensselaer Polytechnic Institute. Her research interests were primarily in applied mathematics, and in particular in inverse problems. Academic career McLaughlin did her undergraduate studies at Kansas State University. After earning a master's degree from the University of Maryland, she moved to the University of California, Riverside for her doctoral studies, earning a Ph.D. in 1968 under the supervision of Joaquin Basilio Diaz.2004 AWM-SIAM Sonia Kovalevsky Lecturer: Joyce R. McLaughlin |
