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Normalized Solution (mathematics)
In mathematics, a normalized solution to an Ordinary differential equation, ordinary or partial differential equation is a solution with prescribed norm, that is, a solution which satisfies a condition like \int_ , u(x), ^2 \, dx = 1. In this article, the normalized solution is introduced by using the nonlinear Schrödinger equation. The nonlinear Schrödinger equation (NLSE) is a fundamental equation in quantum mechanics and other various fields of physics, describing the evolution of complex wave functions. In Quantum Physics, normalization means that the total probability of finding a quantum particle anywhere in the universe is unity. Definition and variational framework In order to illustrate this concept, consider the following nonlinear Schrödinger equation with prescribed norm: : -\Delta u + \lambda u = f(u), \quad \int_ , u, ^2 \, dx = 1, where \Delta is a Laplacian operator, N\ge1, \lambda\in \mathbb is a Lagrange multiplier and f is a nonlinearity. If we want to fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Particle Probability Density
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an fundamental interaction, interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantization (physics), quantization". This means that the magnitude of the physical property can take on only Wiktionary:discrete, discrete values consisting of integer multiples of one quantum. For example, a photon is a single quantum of light (or of any other form of electromagnetic radiation). Similarly, the energy of an electron bound within an atom is quantized and can exist only in certain discrete values. (Atoms and matter in general are stable because electrons can exist only at discrete energy levels within an atom.) Quantization is one of the foundations of the much broader physics of quantum mechanics. Quantization of energy and its influence on how energy and matter interact (quantum electrodynamics) is part of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emilio Gagliardo
Emilio Gagliardo (5 November 1930, Genoa – 15 August 2008, Genoa) was an Italian mathematician working in the field of Analysis. Life He did his PhD in Algebraic Geometry at the University of Genoa with Eugenio Togliatti and graduated in 1953.He then became an assistant of Guido Stampacchia and started to study partial differential equations. In 1959 he got his Habilitation and spend some time abroad with Nachman Aronszajn at the University of Kansas and with Jacques-Louis Lions in Nancy. In 1961 he became Professor in Genoa. From 1968 to 1975 he was at the University of Oregon and since 1975 at the University of Padua. His main contributions are to the field of parabolic partial differential equations, interpolation in Banach spaces, and Sobolev spaces. 1964 was awarded the Caccioppoli Prize. Selected works * Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat., 8, 1959, 24–52 * Caratterizzazioni delle tracce sulla frontiera relative ad al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stephen Smale
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics faculty of the University of California, Berkeley (1960–1961 and 1964–1995), where he currently is Professor Emeritus, with research interests in algorithms, numerical analysis and global analysis. Education and career Smale was born in Flint, Michigan and entered the University of Michigan in 1948. Initially, he was a good student, placing into an honors calculus sequence taught by Bob Thrall and earning himself A's. However, his sophomore and junior years were marred with mediocre grades, mostly Bs, Cs and even an F in nuclear physics. However, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department. Yet again, Smale performed poorly in his first years, earning a C average as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Palais
Richard Sheldon Palais (born May 22, 1931) is a mathematician working in geometry who introduced the principle of symmetric criticality, the Mostow–Palais theorem, the Lie–Palais theorem, the Morse–Palais lemma, and the Palais–Smale compactness condition. From 1965 to 1967 Palais was a Sloan Fellow. In 1970 he was an invited speaker (''Banach manifolds of fiber bundle sections'') at the International Congress of Mathematicians in Nice. From 1965 to 1982 he was an editor for the ''Journal of Differential Geometry'' and from 1966 to 1969 an editor for the ''Transactions of the American Mathematical Society''. In 2010 he received a Lester R. Ford Award. In 2012 he became a fellow of the American Mathematical Society. retrieved 2013-05-05. He obtained his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the hydrogen spectrum in a completely rigorous way. The equation also implied the existence of a new form of matter, ''antimatter'', previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a ''theoretical'' justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin. The wave functions in the Dirac t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Domain
In mathematical analysis, a domain or region is a non-empty connected open set in a topological space, in particular any non-empty connected open subset of the real coordinate space or the complex coordinate space . This is a different concept than the domain of a function, though it is often used for that purpose, for example in partial differential equations and Sobolev spaces. The basic idea of a connected subset of a space dates from the 19th century, but precise definitions vary slightly from generation to generation, author to author, and edition to edition, as concepts developed and terms were translated between German, French, and English works. In English, some authors use the term ''domain'', some use the term ''region'', some use both terms interchangeably, and some define the two terms slightly differently; some avoid ambiguity by sticking with a phrase such as ''non-empty connected open subset''. One common convention is to define a ''domain'' as a connected open set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pokhozhaev's Identity
Pokhozhaev's identity is an integral relation satisfied by stationary localized solutions to a nonlinear Schrödinger equation or nonlinear Klein–Gordon equation. It was obtained by S.I. Pokhozhaev and is similar to the virial theorem. This relation is also known as G.H. Derrick's theorem. Similar identities can be derived for other equations of mathematical physics. The Pokhozhaev identity for the stationary nonlinear Schrödinger equation Here is a general form due to H. Berestycki and P.-L. Lions. Let g(s) be continuous and real-valued, with g(0)=0. Denote G(s)=\int_0^s g(t)\,dt. Let :u\in L^\infty_(\R^n), \qquad \nabla u\in L^2(\R^n), \qquad G(u)\in L^1(\R^n), \qquad n\in\N, be a solution to the equation :-\nabla^2 u=g(u), in the sense of distributions. Then u satisfies the relation :\frac\int_, \nabla u(x), ^2\,dx=n\int_G(u(x))\,dx. The Pokhozhaev identity for the stationary nonlinear Dirac equation There is a form of the virial identity for the stationary non ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compactness
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nehari Manifold
In the calculus of variations, a branch of mathematics, a Nehari manifold is a manifold of functions, whose definition is motivated by the work of . It is a differentiable manifold associated to the Dirichlet problem for the semilinear elliptic partial differential equation : -\triangle u = , u, ^u,\textu\mid_ = 0. Here Δ is the Laplacian on a bounded domain Ω in R''n''. There are infinitely many solutions to this problem. Solutions are precisely the critical points for the energy functional :J(v) = \frac12\int_-\frac1\int_ on the Sobolev space . The Nehari manifold is defined to be the set of such that :\, \nabla v\, ^2_ = \, v\, ^_ > 0. Solutions to the original variational problem that lie in the Nehari manifold are (constrained) minimizers of the energy, and so direct methods in the calculus of variations can be brought to bear. More generally, given a suitable functional ''J'', the associated Nehari manifold is defined as the set of functions ''u'' in an appropr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bifurcation Theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems (described by ordinary, delay or partial differential equations) and discrete systems (described by maps). The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior. Henri Poincaré also later named various types of stationary points and classified them . Bifurcation types It is useful to divide bifurcations into two principal classes: * Local bifurcations, which can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
