Strichartz Estimate
In mathematical analysis, Strichartz estimates are a family of inequalities for linear dispersive partial differential equations. These inequalities establish size and decay of solutions in mixed norm Lebesgue spaces. They were first noted by Robert Strichartz and arose out of connections to the Fourier restriction problem. Examples Consider the linear Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ... in \mathbb^d with ''h'' = ''m'' = 1. Then the solution for initial data u_0 is given by e^u_0. Let ''q'' and ''r'' be real numbers satisfying 2\leq q, r \leq \infty; \frac+\frac=\frac; and (q,r,d)\neq(2,\infty,2). In this case the homogeneous Strichartz estimates take the form: :\, e^ u_0\, _\leq C_ \, u_0\, _. Further suppose that \t ...
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Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ...
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Dispersive Partial Differential Equation
In mathematics, a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive. In this context, dispersion means that waves of different wavelength propagate at different phase velocities. Examples Linear equations * Euler–Bernoulli beam equation with time-dependent loading *Airy equation *Schrödinger equation *Klein–Gordon equation Nonlinear equations *nonlinear Schrödinger equation * Korteweg–de Vries equation (or KdV equation) *Boussinesq equation (water waves) * sine–Gordon equation See also *Dispersion (optics) *Dispersion (water waves) *Dispersionless equation Dispersionless (or quasi-classical) limits of integrable partial differential equations (PDE) arise in various problems of mathematics and physics and have been intensively studied in recent literature (see e.g. references below). They typically a ... External links *ThDispersive PDE Wiki {{mathanalysis-stub Partial differential equations Nonlin ...
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Mixed Norm
Mixed is the past tense of ''mix''. Mixed may refer to: * Mixed (United Kingdom ethnicity category), an ethnicity category that has been used by the United Kingdom's Office for National Statistics since the 1991 Census * ''Mixed'' (album), a compilation album of two avant-garde jazz sessions featuring performances by the Cecil Taylor Unit and the Roswell Rudd Sextet See also * Mix (other) * Mixed breed, an animal whose parents are from different breeds or species * Mixed ethnicity Mixed race people are people of more than one race or ethnicity. A variety of terms have been used both historically and presently for mixed race people in a variety of contexts, including ''multiethnic'', ''polyethnic'', occasionally ''bi-eth ...

Lp Space
In mathematics, the spaces are function spaces defined using a natural generalization of the Norm (mathematics)#p-norm, -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Nicolas Bourbaki, Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Applications Statistics In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of metrics, and measures of central tendency can be characterized as Central tendency#Solutions to variational problems, solutions to ...
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Robert Strichartz
Robert "Bob" Stephen Strichartz (October 14, 1943 – December 19, 2021) was an American mathematician who specialized in mathematical analysis. He was born in New York City on October 14, 1943. Bob graduated from Bronx High School of Science in 1961 and later earned his B.A. from Dartmouth College in 1963. As an undergraduate, he was notably part of two successful Putnam campaigns for Dartmouth. The Dartmouth team finished fifth in 1961 and second in 1962. To date, no Dartmouth Putnam team has replicated a top five finish. Individually, Bob was also recognized as a Putnam fellow in 1962. He was one of the five highest-ranking individual competitors and was eligible for a $3,000 William Lowell Putnam Scholarship at Harvard University. In 1966 Bob received his PhD from Princeton University under Elias Stein with thesis ''Multipliers on generalized Sobolev spaces''. Bob was a NATO postdoctoral fellow at Paris-Sud University, University of Paris Sud (Orsay) from 1966 to 1967. Bob' ...
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Schrödinger Equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of a wave function, the quantum-mechanical characterization of an isolated physical system. The equation can be derived from the fact that the time-evolution operator must be unitary, and must therefore be generated by t ...
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Theorems In Analysis
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ...
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