This is a list of integrable models as well as classes of integrable models in physics.
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Integrable algorithms are numerical algorithms that rely on basic ideas from the mathematical theory of integrable systems.
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The KAM theorem is usually stated in terms of trajectories in phase space of an integrable Hamiltonian system. The motion of an integrable system is confined to an invariant torus (a doughnut-shaped surface). Different initial conditions of the integrable Hamiltonian system will trace different invariant tori in phase space. Plotting the coordinates of an integrable system would show that they are quasiperiodic.
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Hamiltonian systems can be classified as integrable and nonintegrable. SAM is integrable when the mass ratio \mu = M/m = 3. The system also looks pretty regular for \mu = 4 n^2 - 1 = 3, 15, 35, ..., but the \mu = 3 case is the only known integrable mass ratio. It has been shown that the system is not integrable for \mu \in (0,1) \cup (3,\infty).
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The OIF has produced several generations of tunable laser IAs including the Micro Integrable Tunable Laser Assembly and the Integrable Tunable Laser Assembly Multi Source Agreement.
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Complex-valued functions can be similarly integrated, by considering the real part and the imaginary part separately. If h=f+ig for real-valued integrable functions f, g, then the integral of h is defined by : \int h \, d \mu = \int f \, d \mu + i \int g \, d \mu. The function is Lebesgue integrable if and only if its absolute value is Lebesgue integrable (see Absolutely integrable function).
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A new integrable case for the problem of three dimensional Swinging Atwood Machine (3D-SAM) was announced in 2016. Like the 2D version, the problem is integrable when M = 3m.
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In general case LLE (2) is nonintegrable. But it admits the two integrable reductions: : a) in the 1+1 dimensions, that is Eq. (3), it is integrable : b) when J=0. In this case the (1+1)-dimensional LLE (3) turns into the continuous classical Heisenberg ferromagnet equation (see e.g. Heisenberg model (classical)) which is already integrable.
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When such systems are integrable, they are studied as Hitchin systems.
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In general, an integrable system has constants of motion other than the energy. By contrast, energy is the only constant of motion in a non-integrable system; such systems are termed chaotic. In general, a classical mechanical system can be quantized only if it is integrable; as of 2006, there is no known consistent method for quantizing chaotic dynamical systems.
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Birkhoff showed that a billiard system with an elliptic table is integrable.
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Note that the notation L2(R) usually denotes the Kolmogorov quotient, the set of equivalence classes of square integrable functions that differ on sets of measure zero, rather than simply the vector space of square integrable functions that the notation suggests.
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The theory of integrable systems has advanced with the connection between numerical analysis. For example, the discovery of solitons came from the numerical experiments to the KdV equation by Norman Zabusky and Martin David Kruskal. Today, various relations between numerical analysis and integrable systems have been found (Toda lattice and numerical linear algebra, discrete soliton equations and series acceleration), and studies to apply integrable systems to numerical computation are rapidly advancing.
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He later constructed an example of an integrable function whose Fourier series diverges everywhere .
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Generally, it is hard to accurately compute the solutions of nonlinear differential equations due to its non-linearity. In order to overcome this difficulty, R. Hirota has made discrete versions of integrable systems with the viewpoint of "Preserve mathematical structures of integrable systems in the discrete versions". At the same time, Mark J. Ablowitz and others have not only made discrete soliton equations with discrete Lax pair but also compared numerical results between integrable difference schemes and ordinary methods. As a result of their experiments, they have found that the accuracy can be improved with integrable difference schemes at some cases.
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If the integrals at hand are Lebesgue integrals, we may use the bounded convergence theorem (valid for these integrals, but not for Riemann integrals) in order to show that the limit can be passed through the integral sign. Note that this proof is weaker in the sense that it only shows that fx(x,t) is Lebesgue integrable, but not that it is Riemann integrable. In the former (stronger) proof, if f(x,t) is Riemann integrable, then so is fx(x,t) (and thus is obviously also Lebesgue integrable). Let :u(x) = \int_a^b f(x, t) \,dt.
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The Nekhoroshev estimates are an important result in the theory of Hamiltonian systems concerning the long-time stability of solutions of integrable systems under a small perturbation of the Hamiltonian. The first paper on the subject was written by Nikolay Nekhoroshev in 1971. The theorem complements both the KAM theorem and the phenomenon of instability for nearly integrable Hamiltonian systems, sometimes called Arnold Diffusion, in the following way: The KAM Theorem tells us that many solutions to nearly integrable Hamiltonian systems persist under a perturbation for all time, while, as Vladimir Arnold first demonstrated in 1964, some solutions do not stay close to their integrable counterparts for all time. The Nekhoroshev estimates tell us that, nonetheless, all solutions stay close to their integrable counterparts for an exponentially long time.
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Let be an integrable, -valued random variable, which is independent of the integrable, real-valued random variable with . Define for all . Then assumptions (), (), (), and () with are satisfied, hence also () and (), and Wald's equation applies. If the distribution of is not symmetric, then () does not hold.
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In mathematics, a locally integrable function (sometimes also called locally summable function)According to . is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to spaces, but its members are not required to satisfy any growth restriction on their behavior at the boundary of their domain (at infinity if the domain is unbounded): in other words, locally integrable functions can grow arbitrarily fast at the domain boundary, but are still manageable in a way similar to ordinary integrable functions.
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He is considered an authority on quantum field theory, quantum statistics, anyons, integrable systems,Polychronakos, A.P., (1992). "Exchange operator formalism for integrable systems of particles", Physical Review Letters 69 (5), p.703. and quantum fluids, having authored over 110 refereed papers.Nair, V. P., & Polychronakos, A. P. (2001).
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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 arise when considering slowly modulated long waves of an integrable dispersive PDE system.
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It provides a framework for solving two- dimensional integrable models by using the quantum inverse scattering method.
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The question is whether this complex structure can be defined globally. An almost complex structure that comes from a complex structure is called integrable, and when one wishes to specify a complex structure as opposed to an almost complex structure, one says an integrable complex structure. For integrable complex structures the so-called Nijenhuis tensor vanishes. This tensor is defined on pairs of vector fields, X, Y by :N_J(X,Y) = [X,Y] + J[JX,Y] + J[X,JY]-[JX,JY]\ .
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Olivier Babelon, Pierre Cartier, Yvette Kosmann- Schwarzbach: Integrable systems. The Verdier memorial colloquium. Birkhäuser, Progress in Mathematics, 1993.
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In mathematics, the AKNS system is an integrable system of partial differential equations, introduced by and named after .
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In more than one dimension, the equation is not integrable, it allows for a collapse and wave turbulence.
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In formal terms, this representation is a wavelet series representation of a square-integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set or frame of a vector space, for the Hilbert space of square integrable functions. This is accomplished through coherent states.
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Staudacher suggests that the integrable spin chains of condensed matter physics may form the link between the two approaches.
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To have a defined L2-norm, the integral must be bounded, which restricts the functions to being square-integrable.
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In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies on the crossroads of algebraic geometry, the theory of Lie algebras and integrable system theory. It also plays an important role in geometric Langlands correspondence over the field of complex numbers; related to conformal field theory. A genus zero analogue of the Hitchin system arises as a certain limit of the Knizhnik- Zamolodchikov equations.
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Evgeny Konstantinovich Sklyanin (, born May 24, 1955 in Leningrad, Soviet Union) is a mathematical physicist, currently a professor of mathematics at the University of York. His research is in the fields of integrable systems and quantum groups. His major contributions are in the theory of quantum integrable systems, separation of variables, special functions.
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His research concerns asymptotic representation theory, relations with random matrices and integrable systems, and the difference equation formulation of monodromy.
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These results remain true for the Henstock–Kurzweil integral, which allows a larger class of integrable functions . In higher dimensions Lebesgue's differentiation theorem generalizes the Fundamental theorem of calculus by stating that for almost every x, the average value of a function f over a ball of radius r centered at x tends to f(x) as r tends to 0. Part II of the theorem is true for any Lebesgue integrable function f, which has an antiderivative F (not all integrable functions do, though). In other words, if a real function F on admits a derivative f(x) at every point x of and if this derivative f is Lebesgue integrable on then :F(b) - F(a) = \int_a^b f(t) \, dt.
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In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line (-\infty,+\infty) is defined as follows. One may also speak of quadratic integrability over bounded intervals such as [a,b] for a \leq b. An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable.
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The stated theorem is actually valid more generally for coefficient functions 1/p,\, q,\, w that are Lebesgue integrable over I.
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A central-force problem is said to be "integrable" if this final integration can be solved in terms of known functions.
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Warning: E(z, s) is not a square-integrable function of z with respect to the invariant Riemannian metric on H.
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Via the Penrose-Ward transform these solutions give the holomorphic vector bundles often seen in the context of algebraic integrable systems.
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Therefore we have an opportunity sometimes to find Poincaré map of the non-integrable in quadrature systems even in elementary functions.
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In differential geometry, a hypercomplex manifold is a manifold with the tangent bundle equipped with an action by the algebra of quaternions in such a way that the quaternions I, J, K define integrable almost complex structures. If the almost complex structures are instead not assumed to be integrable, the manifold is called quaternionic, or almost hypercomplex.
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The special Kähler geometry on the moduli space of vacua in Seiberg–Witten theory can be identified with the geometry of the base of complex completely integrable system. The total phase of this complex completely integrable system can be identified with the moduli space of vacua of the 4d theory compactified on a circle. See Hitchin system.
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The study of dynamical systems overlaps with that of integrable systems; there one has the idea of a normal form (dynamical systems).
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Hazewinkel, Michiel; ed. (2012). Encyclopaedia of Mathematics, pp. 271–2. Springer. . It has been proven that no other holonomic integrable tops exist .
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IE admits an important reduction: in 1+1 dimensions it reduces to the continuous classical Heisenberg ferromagnet equation (CCHFE). The CCHFE is integrable.
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For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part. The vector space of square integrable functions (with respect to Lebesgue measure) form the Lp space with p=2. Among the Lp spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a Hilbert space, since all of the Lp spaces are complete under their respective p-norms.
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In classical mechanics, the precession of a rigid body such as a top under the influence of gravity is not, in general, an integrable problem. There are however three (or four) famous cases that are integrable, the Euler, the Lagrange, and the Kovalevskaya top..Whittaker, E. T. (1952). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press. .
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He developed the method of separation of variables in the theory of integrable systems. He was elected Fellow of the Royal Society in 2008.
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Let be an integrable function. The span of translations = is dense in if and only if the Fourier transform of has no real zeros.
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Tonelli's theorem, introduced by Leonida Tonelli in 1909, is similar, but applies to a non-negative measurable function rather than one integrable over its domain.
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The inverse scattering transform may be applied to many of the so-called exactly solvable models, that is to say completely integrable infinite dimensional systems.
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As discovered by Courant and parallel by Dorfman, the graph of a 2-form ω ∈ Ω2(M) is maximally isotropic and moreover integrable iff dω = 0, i.e. the 2-form is closed under the de Rham differential, i.e. a presymplectic structure. A second class of examples arises from bivectors \Pi\in\Gamma(\wedge^2 TM) whose graph is maximally isotropic and integrable iff [Π,Π] = 0, i.e.
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Therefore, it was thought that no progress could be made for such a difficult problem. Yet, many breakthroughs have been made since the 1990s. It must be stressed again that the chiral Potts model was not invented because it was integrable but the integrable case was found, after it was introduced to explain experimental data. In a very profound way physics is here far ahead of mathematics.
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In mathematics, the Gelfand–Zeitlin system (also written Gelfand–Zetlin system, Gelfand–Cetlin system, Gelfand–Tsetlin system) is an integrable system on conjugacy classes of Hermitian matrices. It was introduced by , who named it after the Gelfand–Zeitlin basis, an early example of canonical basis, introduced by I. M. Gelfand and M. L. Cetlin in 1950s. introduced a complex version of this integrable system.
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If M is a continuous local martingale then a predictable process H is M-integrable if and only if :\int_0^t H^2 d[M] <\infty, for each t, and H · M is always a local martingale. The most general statement for a discontinuous local martingale M is that if (H2 · [M])1/2 is locally integrable then H · M exists and is a local martingale.
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The Volterra lattice is an integrable system, and is related to the Toda lattice. It is also used as a model for Langmuir waves in plasmas.
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Like the KdV equation, the KP equation is completely integrable. It can also be solved using the inverse scattering transform much like the nonlinear Schrödinger equation.
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Percy Alec Deift (born September 10, 1945) is a mathematician known for his work on spectral theory, integrable systems, random matrix theory and Riemann–Hilbert problems.
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In 1983 he was an invited speaker at the International Congress of Mathematicians in Warsaw and gave a talk Integrable models in classical and quantum field theory.
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Every differentiable manifold has a cotangent bundle. That bundle can always be endowed with a certain differential form, called the canonical one-form. This form gives the cotangent bundle the structure of a symplectic manifold, and allows vector fields on the manifold to be integrated by means of the Euler-Lagrange equations, or by means of Hamiltonian mechanics. Such systems of integrable differential equations are called integrable systems.
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If and are compactly supported continuous functions, then their convolution exists, and is also compactly supported and continuous . More generally, if either function (say ) is compactly supported and the other is locally integrable, then the convolution is well-defined and continuous. Convolution of and is also well defined when both functions are locally square integrable on and supported on an interval of the form (or both supported on ).
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In the mathematics department of the University of Illinois at Urbana-Champaign, she was from 2001 to 2006 an assistant professor and from 2006 to 2012 an associate professor and is since 2012 a full professor. Kedem's research deals with mathematical physics, Lie algebras, integrable models, and cluster algebras. In 2014 she was an invited speaker with talk Fermionic spectra in integrable systems at the International Congress of Mathematicians in Seoul.
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His research interests are Yang-Mills gauge theories, supersymmetry, supergravity, quaternion and octonion algebras, spin structures, generalised theories of gravity, cosmological solutions, integrable systems and phase space quantisation.
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PDEs that arise from integrable systems are often the easiest to study, and can sometimes be completely solved. A well-known example is the Korteweg–de Vries equation.
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This relation help to study Correlation function (statistical mechanics). The correlation functions can be described by Integrable system. In a simple case, it is Painlevé transcendents. A textbookV.
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His research concerns probability, mathematical physics, quantum integrable systems, stochastic PDEs, and random matrix theory. He is particularly known for work related to the Kardar–Parisi–Zhang equation.
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In Khovanova's earlier mathematical research, she studied representation theory, the theory of integrable systems, quantum group theory, and superstring theory. Her later work explores combinatorics and recreational mathematics.
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Now they are referred to as completely integrable integral operators. They have multiple applications not only to quantum exactly solvable models, but also to random matrices and algebraic combinatorics.
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The perturbative treatment of the Stark effect has some problems. In the presence of an electric field, states of atoms and molecules that were previously bound (square-integrable), become formally (non-square-integrable) resonances of finite width. These resonances may decay in finite time via field ionization. For low lying states and not too strong fields the decay times are so long, however, that for all practical purposes the system can be regarded as bound.
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When its weights satisfy the Yang–Baxter equation, (star–triangle relation), it is integrable. For the integrable chiral Potts model, its weights are parametrized by a high genus curve, the chiral Potts curve.Au-Yang H., McCoy B. M., Perk J. H. H., Tang S. and Yan M-L. (1987), "Commuting transfer matrices in the chiral Potts models: Solutions of the star-triangle equations with genus > 1", Physics Letters A 123 219–23.
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Freelance member of the Department of Geometry and Topology of the Steklov Institute of Mathematics. Area of scientific interests: theory of integrable systems in geometry and physics: Frobenius manifolds, Gromov–Witten invariants, singularity theory, normal forms of integrable partial differential equations, Hamiltonian perturbations of hyperbolic systems, geometry of isomonodromic deformations, theta functions on Riemann surfaces, and nonlinear waves. In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin.
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In the field of real analysis, he discovered the Riemann integral in his habilitation. Among other things, he showed that every piecewise continuous function is integrable. Similarly, the Stieltjes integral goes back to the Göttinger mathematician, and so they are named together the Riemann–Stieltjes integral. In his habilitation work on Fourier series, where he followed the work of his teacher Dirichlet, he showed that Riemann- integrable functions are "representable" by Fourier series.
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All of the above limits are cases of the indeterminate form ∞ − ∞. These pathologies do not affect "Lebesgue- integrable" functions, that is, functions the integrals of whose absolute values are finite.
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I, pp. 118–124. Moscow: Gostekhizdat. . Cited in Bechlivanidis & van Moerbek (1987) and Hazewinkel (2012).) is also integrable (I_1=I_2=4I_3). Its center of gravity lies in the equatorial plane.
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Let Z be a cadlag submartingale of class D. Then there exists a unique, increasing, predictable process A with A_0 =0 such that M_t = Z_t + A_t is a uniformly integrable martingale.
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The special case of Fubini's theorem for continuous functions on a product of closed bounded subsets of real vector spaces was known to Leonhard Euler in the 18th century. extended this to bounded measurable functions on a product of intervals. conjectured that the theorem could be extended to functions that were integrable rather than bounded, and this was proved by . Reprinted in gave a variation of Fubini's theorem that applies to non-negative functions rather than integrable functions.
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They are only revealed when one does the pentagram map and watches a point move round and round, filling up one of the tori. Roughly speaking, when dynamical systems have these invariant tori, they are called integrable systems. Most of the results in this article have to do with establishing that the pentagram map is an integrable system, that these tori really exist. The monodromy invariants, discussed below, turn out to be the equations for the tori.
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For example, the objects and are equal everywhere except at yet have integrals that are different. According to Lebesgue integration theory, if f and g are functions such that almost everywhere, then f is integrable if and only if g is integrable and the integrals of f and g are identical. A rigorous approach to regarding the Dirac delta function as a mathematical object in its own right requires measure theory or the theory of distributions.
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Herman Flaschka (born 25 March 1945) is an Austrian-American mathematical physicist and Professor of Mathematics at the University of Arizona, known for his important contributions in completely integrable systems (soliton equations).
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Whereas the central limit theorem for sums of random variables requires the condition of finite variance, the corresponding theorem for products requires the corresponding condition that the density function be square-integrable.
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The integral of an odd function from −A to +A is zero, provided that A is finite and that the function is integrable (e.g., has no vertical asymptotes between −A and A). The integral of an even function from −A to +A is twice the integral from 0 to +A, provided that A is finite and the function is integrable (e.g., has no vertical asymptotes between −A and A). This also holds true when A is infinite, but only if the integral converges.
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The bundle is involutive if, for each point and pair of sections and Y of defined in a neighborhood of p, the Lie bracket of and Y evaluated at p, lies in : : [X,Y]_p \in E_p On the other hand, is integrable if, for each , there is an immersed submanifold whose image contains p, such that the differential of is an isomorphism of TN with . The Frobenius theorem states that a subbundle is integrable if and only if it is involutive.
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Nikolai Nikolaevich Nekhoroshev (; 2 October 1946 – 18 October 2008) was a prominent Soviet Russian mathematician specializing in classical mechanics and dynamical systems. His research concerned Hamiltonian mechanics, perturbation theory, celestial mechanics, integrable systems, dynamical systems, the quasiclassical approximation, and singularity theory. He proved, in particular, a stability result in KAM-theory stating that, under certain conditions, solutions of nearly integrable systems stay close to invariant tori for exponentially long times . Nekhoroshev was professor of the Moscow State University and University of Milan.
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His research focuses on integrable systems, examining asymptotic analysis of matrix models using Riemann–Hilbert and isomonodromy methods, asymptotic analysis of correlation functions related to aspects of theoretical Fredholm and Toeplitz operators, and the theory of integrable nonlinear partial and ordinary differential equations of KdV and Painlevé types. Awards received by Its over the course of his career include the Prize of the Moscow Mathematical Society (1976), the Prize of the Leningrad Mathematical Society (1981), the Hardy Fellowship of the London Mathematical Society (2002), the Batsheva de Rothschild Fellowship of the Israel Academy of Sciences and Humanities (2009), and Fellowship in the American Mathematical Society (2012). In 2012, a conference on "Integrable Systems and Random Matrices" was held in his honor at the Institut Henri Poincaré in Paris.
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The Ishimori equation (IE) is a partial differential equation proposed by the Japanese mathematician . Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable .
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In mathematics, the Novikov–Veselov equation (or Veselov–Novikov equation) is a natural (2+1)-dimensional analogue of the Korteweg–de Vries (KdV) equation. Unlike another (2+1)-dimensional analogue of KdV, the Kadomtsev–Petviashvili equation, it is integrable via the inverse scattering transform for the 2-dimensional stationary Schrödinger equation. Similarly, the Korteweg–de Vries equation is integrable via the inverse scattering transform for the 1-dimensional Schrödinger equation. The equation is named after S.P. Novikov and A.P. Veselov who published it in .
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The Lin–Tsien equation (named after C. C. Lin and H. S. Tsien) is an integrable partial differential equation : 2u_{tx}+u_xu_{xx}-u_{yy} = 0. Integrability of this equation follows from its being, modulo an appropriate linear change of dependent and independent variables, a potential form of the dispersionless KP equation. Namely, if u satisfies the Lin–Tsien equation, then v=u_x satisfies, modulo the said change of variables, the dispersionless KP equation. The Lin-Tsien equation admits a (3+1)-dimensional integrable generalization, see.
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She then became interested > in submanifold geometry. Her main contributions are developing a structure > theory for isoparametric submanifolds in Rn and constructing soliton > equations from special submanifolds. Recently, Terng and Karen Uhlenbeck > (University of Texas at Austin) have developed a general approach to > integrable PDEs that explains their hidden symmetries in terms of loop group > actions. She is co-author of the book Submanifold Geometry and Critical > Point Theory and an editor of the Journal of Differential Geometry survey > volume 4 on "Integrable systems".
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This promptly lead to the study of "nearly integrable systems", of which the KAM torus is the canonical example. At the same time, it was also discovered that many (rather special) non-linear systems, which were previously approachable only through perturbation theory, are in fact completely integrable. This discovery was quite dramatic, as it allowed exact solutions to be given. This, in turn, helped clarify the meaning of the perturbative series, as one could now compare the results of the series to the exact solutions.
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The quantum inverse scattering method relates two different approaches: #the Bethe ansatz, a method of solving integrable quantum models in one space and one time dimension; #the Inverse scattering transform, a method of solving classical integrable differential equations of the evolutionary type. An important concept in the Inverse scattering transform is the Lax representation; the quantum inverse scattering method starts by the quantization of the Lax representation and reproduces the results of the Bethe ansatz. In fact, it allows the Bethe ansatz to be written in a new form: the algebraic Bethe ansatz.cf. e.g. the lectures by N.A. Slavnov, This led to further progress in the understanding of quantum Integrable systems, for example: a) the Heisenberg model (quantum), b) the quantum Nonlinear Schrödinger equation (also known as the Lieb–Liniger model or the Tonks–Girardeau gas) and c) the Hubbard model.
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In mathematics, a real- valued function K(x,y) is said to fulfill Mercer's condition if for all square-integrable functions g(x) one has : \iint g(x)K(x,y)g(y)\,dx\,dy \geq 0.
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In physics, completely integrable systems, especially in the infinite-dimensional setting, are often referred to as exactly solvable models. This obscures the distinction between integrability in the Hamiltonian sense, and the more general dynamical systems sense. There are also exactly solvable models in statistical mechanics, which are more closely related to quantum integrable systems than classical ones. Two closely related methods: the Bethe ansatz approach, in its modern sense, based on the Yang–Baxter equations and the quantum inverse scattering method provide quantum analogs of the inverse spectral methods.
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These symmetry groups are often infinite dimensional, but this is not always a useful feature. Emmy Noether showed that a slight but profound generalization of Lie's notion of symmetry can result in an even more powerful method of attack. This turns out to be closely related to the discovery that some equations, which are said to be completely integrable, enjoy an infinite sequence of conservation laws. Quite remarkably, both the Ernst equation (which arises several ways in the studies of exact solutions) and the NLS turn out to be completely integrable.
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If we solve the time-independent Schrödinger equation for an energy E>V_0, the solutions will be oscillatory both inside and outside the well. Thus, the solution is never square integrable; that is, it is always a non-normalizable state. This does not mean, however, that it is impossible for a quantum particle to have energy greater than V_0, it merely means that the system has continuous spectrum above V_0. The non-normalizable eigenstates are close enough to being square integrable that they still contribute to the spectrum of the Hamiltonian as an unbounded operator.
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He worked on the deformation theory for mappings to groups, which led to the solution of the Novikov problem on multiplicative subgroups in operator doubles, and to construction of the quantum group of complex cobordisms. He went on to treat problems related both with algebraic geometry and integrable systems. He is also well known for his work on sigma-functions on universal spaces of Jacobian varieties of algebraic curves that give effective solutions of important integrable systems. Buchstaber created an algebro-functional theory of symmetric products of spaces and described algebraic varieties of polysymmetric polynomials.
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His construction leads to what Edmond Bonan called the Obata connection. which is torsion free, if and only if, "two" of the almost complex structures I, J, K are integrable and in this case the manifold is hypercomplex.
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Jarvis works at the School of Mathematics and Physics, at the University of Tasmania. His main focus is on algebraic structures in mathematical physics and their applications, especially combinatorial Hopf algebras in integrable systems and quantum field theory.
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If and are positive, the eigenvalues are all positive, and the solutions are trigonometric functions. A solution that satisfies square-integrable initial conditions for and can be obtained from expansion of these functions in the appropriate trigonometric series.
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In mathematics, a Lagrangian foliation or polarization is a foliation of a symplectic manifold, whose leaves are Lagrangian submanifolds. It is one of the steps involved in the geometric quantization of a square-integrable functions on a symplectic manifold.
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There are various types of RF MEMS components, such as CMOS integrable RF MEMS resonators and self-sustained oscillators with small form factor and low phase noise, RF MEMS tunable inductors, and RF MEMS switches, switched capacitors and varactors.
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A theorem of Gábor Szegő states that if f is in H^1, the Hardy space with integrable norm, and if f is not identically zero, then the zeroes of f (certainly countable in number) satisfy the Blaschke condition.
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If g : I → R is a Lebesgue- integrable function on some interval I = [a,b], and if :f(x) = \int_a^x g(t)\,dt is its Lebesgue indefinite integral, then the following assertions are true: #f is absolutely continuous (see below) #f is differentiable almost everywhere #Its derivative coincides almost everywhere with g(x). (In fact, all absolutely continuous functions are obtained in this manner.) The Lebesgue integral could be defined as follows: g is Lebesgue-integrable on I iff there exists a function f that is absolutely continuous whose derivative coincides with g almost everywhere. However, even if f : I → R is differentiable everywhere, and g is its derivative, it does not follow that f is (up to a constant) the Lebesgue indefinite integral of g, simply because g can fail to be Lebesgue-integrable, i.e., f can fail to be absolutely continuous.
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This result may fail for continuous functions F that admit a derivative f(x) at almost every point x, as the example of the Cantor function shows. However, if F is absolutely continuous, it admits a derivative F′(x) at almost every point x, and moreover F′ is integrable, with equal to the integral of F′ on Conversely, if f is any integrable function, then F as given in the first formula will be absolutely continuous with F′ = f a.e. The conditions of this theorem may again be relaxed by considering the integrals involved as Henstock–Kurzweil integrals. Specifically, if a continuous function F(x) admits a derivative f(x) at all but countably many points, then f(x) is Henstock–Kurzweil integrable and is equal to the integral of f on The difference here is that the integrability of f does not need to be assumed.
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In probability theory, a pregaussian class or pregaussian set of functions is a set of functions, square integrable with respect to some probability measure, such that there exists a certain Gaussian process, indexed by this set, satisfying the conditions below.
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The Z_N model is a simplified statistical mechanical spin model. It is a generalization of the Ising model. Although it can be defined on an arbitrary graph, it is integrable only on one and two-dimensional lattices, in several special cases.
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Following the reaction, any remaining transition metal is removed by chemical etching, leaving silicide contacts in only the active regions of the device. A fully integrable manufacturing process may be more complex, involving additional anneals, surface treatments, or etch processes.
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Pavel Winternitz is a Canadian Czech-born mathematical physicist. He did his undergraduate studies at Prague University and his doctorate at Leningrad University (Ph.D. 1962) under the supervision of J. A. Smorodinsky. His research is on integrable systems and symmetries.
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The following examples show how tensor products arise naturally. Given two measure spaces X and Y, with measures \mu and u respectively, one may look at L^2(X\times Y), the space of functions on X\times Y that are square integrable with respect to the product measure \mu\times u. If f is a square integrable function on X, and g is a square integrable function on Y, then we can define a function h on X\times Y by h(x,y)=f(x)g(y). The definition of the product measure ensures that all functions of this form are square integrable, so this defines a bilinear mapping L^2(X)\times L^2(Y)\to L^2(X\times Y). Linear combinations of functions of the form f(x)g(y) are also in L^2(X\times Y). It turns out that the set of linear combinations is in fact dense in L^2(X\times Y), if L^2(X) and L^2(Y) are separable. This shows that L^2(X)\otimes L^2(Y) is isomorphic to L^2(X\times Y), and it also explains why we need to take the completion in the construction of the Hilbert space tensor product.
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In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced ), Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral – is one of a number of definitions of the integral of a function. It is a generalization of the Riemann integral, and in some situations is more general than the Lebesgue integral. In particular, a function is Lebesgue integrable if and only if the function and its absolute value are Henstock–Kurzweil integrable. This integral was first defined by Arnaud Denjoy (1912).
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We don't need to assume continuity of f on the whole interval. Part I of the theorem then says: if f is any Lebesgue integrable function on and x0 is a number in such that f is continuous at x0, then :F(x) = \int_a^x f(t)\, dt is differentiable for with We can relax the conditions on f still further and suppose that it is merely locally integrable. In that case, we can conclude that the function F is differentiable almost everywhere and almost everywhere. On the real line this statement is equivalent to Lebesgue's differentiation theorem.
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A real-valued stochastic process is a submartingale if and only if it has a Doob decomposition into a martingale and an integrable predictable process that is almost surely increasing. It is a supermartingale, if and only if is almost surely decreasing.
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In 1967, he found a consistent condition for a one dimensional factorized scattering many body system, the equation was later named the Yang–Baxter equation, it plays an important role in integrable models and has influenced several branches of physics and mathematics.
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Gerald Teschl (born May 12, 1970 in Graz) is an Austrian mathematical physicist and professor of mathematics. He works in the area of mathematical physics; in particular direct and inverse spectral theory with application to completely integrable partial differential equations (soliton equations).
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In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for Henri Lebesgue.
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In the continuum- limit approximation the FK model reduces to the exactly integrable sine-Gordon equation or SG equation which allows for soliton solutions. For this reason the FK model is also known as the 'discrete sine-Gordon' or 'periodic Klein- Gordon' equation.
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For example, her 1936 paper. proves a version of Rolle's theorem for Denjoy–Perron integrable functions using different techniques from the standard proofs:. Her 1953 paper. established several important results on summability kernels and is referenced in two textbooks on functional analysis.
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Viewed as conjugate harmonic functions, the Cauchy–Riemann equations are a simple example of a Bäcklund transform. More complicated, generally non-linear Bäcklund transforms, such as in the sine-Gordon equation, are of great interest in the theory of solitons and integrable systems.
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The dominated convergence theorem applies also to measurable functions with values in a Banach space, with the dominating function still being non-negative and integrable as above. The assumption of convergence almost everywhere can be weakened to require only convergence in measure.
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The function is zero everywhere, except on a finite set of points. Hence its Riemann integral is zero. Each is non- negative, and this sequence of functions is monotonically increasing, but its limit as is , which is not Riemann integrable. Unsuitability for unbounded intervals.
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Since 2005 he is a Professor at MIT. He is married to Tanya Javits-Etingof (1997–present) and has two daughters; Miriam (1998) and Ariela (2004). Etingof does research on the intersection of mathematical physics (exactly integrable systems) and representation theory, e.g. quantum groups.
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The authors also constructed a hierarchy of evolution equations integrable via the inverse scattering transform for the 2-dimensional Schrödinger equation at fixed energy. This set of evolution equations (which is sometimes called the hierarchy of the Novikov–Veselov equations) contains, in particular, the equation ().
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In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the Lp space L^1([a,b]). See distributions for a more general definition.
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In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb. It is also related to integrable models, knot theory and the braid group, quantum groups and subfactors of von Neumann algebras.
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Only when these constants can be reinterpreted, within the full phase space setting, as the values of a complete set of Poisson commuting functions restricted to the leaves of a Lagrangian foliation, can the system be regarded as completely integrable in the Liouville sense.
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However, this does not imply any special dynamical structure. To explain quantum integrability, it is helpful to consider the free particle setting. Here all dynamics are one-body reducible. A quantum system is said to be integrable if the dynamics are two-body reducible.
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Au-Yang H and Perk J. H. H. (2011), "Spontaneous magnetization of the integrable chiral Potts model", Journal of Physics A 44, 445005 (20pp), arXiv:1003.4805. an algebraic (Ising-like) way of obtaining order parameter has been given, giving more insight into the algebraic structure.
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The winding number is closely related with the (2 + 1)-dimensional continuous Heisenberg ferromagnet equations and its integrable extensions: the Ishimori equation etc. Solutions of the last equations are classified by the winding number or topological charge (topological invariant and/or topological quantum number).
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In mathematics, Ward's conjecture is the conjecture made by that "many (and perhaps all?) of the ordinary and partial differential equations that are regarded as being integrable or solvable may be obtained from the self-dual gauge field equations (or its generalizations) by reduction".
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Due to Bronshtein and Semendyayev containing a comprehensive table of analytically solvable integrals, integrals are sometimes referred to as being "Bronshtein- integrable" in German universities if they can be looked up in the book (in playful analogy to terms like Riemann-integrability and Lebesgue- integrability).
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Let be a real-valued function defined on a closed interval [] that admits an antiderivative on . That is, and are functions such that for all in , :f(x) = F'(x). If is integrable on then :\int_a^b f(x)\,dx = F(b) - F(a).
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In real analysis, a branch of mathematics, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal. The definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral. Consequently, introductory textbooks on calculus and real analysis often develop Riemann integration using the Darboux integral, rather than the true Riemann integral.
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6, A. Engel, trans., Princeton U. Press, Princeton, NJ (1997), p. 434 now known as Einstein–Brillouin–Keller method. In 1971, Martin Gutzwiller took into account that this method only works for integrable systems and derived a semiclassical way of quantizing chaotic systems from path integrals.
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Nekrasov is mostly known for his work on supersymmetric gauge theory, quantum integrability, and string theory. The Nekrasov partition function, which he introduced in his 2002 paper, relates in an intricate way the instantons in gauge theory, integrable systems, and representation theory of infinite-dimensional algebras.
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The Poisson summation formula may be used to derive Landau's asymptotic formula for the number of lattice points in a large Euclidean sphere. It can also be used to show that if an integrable function, f and \hat f both have compact support then f = 0 .
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Locally integrable functions play a prominent role in distribution theory and they occur in the definition of various classes of functions and function spaces, like functions of bounded variation. Moreover, they appear in the Radon–Nikodym theorem by characterizing the absolutely continuous part of every measure.
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Since the sequence is uniformly bounded, there is a real number M such that for all and for all n. Define for all . Then the sequence is dominated by g. Furthermore, g is integrable since it is a constant function on a set of finite measure.
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John Harnad (born Hernád János) is a Hungarian-born Canadian mathematical physicist. He did his undergraduate studies at McGill University and his doctorate at the University of Oxford (D.Phil. 1972) under the supervision of John C. Taylor. His research is on integrable systems, gauge theory and random matrices.
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Benettin, G., Christodoulidi, H., and Ponno, A. (2013). The Fermi–Pasta–Ulam Problem and Its Underlying Integrable Dynamics. Journal of Statistical Physics, 1–18Casetti, L., Cerruti-Sola, M., Pettini, M., and Cohen, E. G. D. (1997). The Fermi–Pasta–Ulam problem revisited: stochasticity thresholds in nonlinear Hamiltonian systems.
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Involutive distributions are important in that they satisfy the conditions of the Frobenius theorem, and thus lead to integrable systems. A related idea occurs in Hamiltonian mechanics: two functions f and g on a symplectic manifold are said to be in mutual involution if their Poisson bracket vanishes.
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An interesting result for algorithms of this general type (splitting based on the radix, then comparison-based sorting) is that they are O(n) for any bounded and integrable probability density function. This result can be obtained by forcing Spreadsort to always iterate one more time if the bin size after the first iteration is above some constant value. If the key density function is known to be Riemann integrable and bounded, this modification of Spreadsort can attain some performance improvement over the basic algorithm, and will have better worst-case performance. If this restriction cannot usually be depended on, this change will add a little extra runtime overhead to the algorithm and gain little.
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The eigenmode of a classically integrable system (e.g. the circular cavity on the left) can be very confined even for high mode number. On the contrary the eigenmodes of a classically chaotic system (e.g. the stadium-shaped cavity on the right) tend to become gradually more uniform with increasing mode number.
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A Lie conformal algebra is in some sense a generalization of a Lie algebra in that it too is a "Lie algebra," though in a different pseudo-tensor category. Lie conformal algebras are very closely related to vertex algebras and have many applications in other areas of algebra and integrable systems.
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This representation has several important realizations, or models. In the Schrödinger model, the Heisenberg group acts on the space of square integrable functions. In the theta representation, it acts on the space of holomorphic functions on the upper half-plane; it is so named for its connection with the theta functions.
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E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, 1993 explains in detail the description of quantum correlation functions of Tonks–Girardeau gas by means of classical completely integrable differential equations. Thermodynamics of Tonks–Girardeau gas was described by Chen Ning Yang.
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This idea of oscillation is sufficient to, for example, characterize Riemann-integrable functions as continuous except on a set of measure zero. Note that points of nonzero oscillation (i.e., points at which f is "badly behaved") are discontinuities which, unless they make up a set of zero, are confined to a negligible set.
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In mathematics, the quantum inverse scattering method is a method for solving integrable models in 1+1 dimensions, introduced by L. D. Faddeev in about 1979. This method led to the formulation of quantum groups. Especially interesting is the Yangian, and the center of the Yangian is given by the quantum determinant.
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The Heisenberg group also occurs in Fourier analysis, where it is used in some formulations of the Stone–von Neumann theorem. In this case, the Heisenberg group can be understood to act on the space of square integrable functions; the result is a representation of the Heisenberg groups sometimes called the Weyl representation.
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88, pp. 85-139 was influential for the Chicago School of hard analysis. The Calderón-Zygmund decomposition lemma, invented to prove the weak-type continuity of singular integrals of integrable functions, became a standard tool in analysis and probability theory. The Calderón-Zygmund Seminar at the University of Chicago ran for decades.
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In mathematics, a regulated function, or ruled function, is a certain kind of well-behaved function of a single real variable. Regulated functions arise as a class of integrable functions, and have several equivalent characterisations. Regulated functions were introduced by Nicolas Bourbaki in 1949, in their book "Livre IV: Fonctions d'une variable réelle".
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Recently, he organized a minisymposium on ´´Integrable Dynamical Systems and Their Applications´´ for the 1995 International Congress on Industrial and Applied Mathematics in Hamburg, Germany. Blackmore is a member of the National Honor Societies Sigma Xi and Tau Beta Pi, and was awarded the Harlan Perlis Research Award from NJIT in 1993.
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This was later formalized by Banach and Hilbert, around 1920.. At that time, algebra and the new field of functional analysis began to interact, notably with key concepts such as spaces of p-integrable functions and Hilbert spaces., . Also at this time, the first studies concerning infinite- dimensional vector spaces were done.
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In mathematical analysis, many generalizations of Fourier series have proved to be useful. They are all special cases of decompositions over an orthonormal basis of an inner product space. Here we consider that of square-integrable functions defined on an interval of the real line, which is important, among others, for interpolation theory.
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The symbol is separated from the integrand by a space (as shown). A function is said to be integrable if the integral of the function over its domain is finite. The points and are called the limits of the integral. An integral where the limits are specified is called a definite integral.
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There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but also occasionally for pedagogical reasons. The most commonly used definitions of integral are Riemann integrals and Lebesgue integrals.
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The collection of Riemann-integrable functions on a closed interval forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integration : f \mapsto \int_a^b f(x) \; dx is a linear functional on this vector space. Thus, firstly, the collection of integrable functions is closed under taking linear combinations; and, secondly, the integral of a linear combination is the linear combination of the integrals, : \int_a^b (\alpha f + \beta g)(x) \, dx = \alpha \int_a^b f(x) \,dx + \beta \int_a^b g(x) \, dx. \, Similarly, the set of real-valued Lebesgue-integrable functions on a given measure space with measure is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral : f\mapsto \int_E f \, d\mu is a linear functional on this vector space, so that : \int_E (\alpha f + \beta g) \, d\mu = \alpha \int_E f \, d\mu + \beta \int_E g \, d\mu. More generally, consider the vector space of all measurable functions on a measure space , taking values in a locally compact complete topological vector space over a locally compact topological field .
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Let U be an open set in a manifold , be the space of smooth, differentiable 1-forms on U, and F be a submodule of of rank r, the rank being constant in value over U. The Frobenius theorem states that F is integrable if and only if for every in the stalk Fp is generated by r exact differential forms. Geometrically, the theorem states that an integrable module of -forms of rank r is the same thing as a codimension-r foliation. The correspondence to the definition in terms of vector fields given in the introduction follows from the close relationship between differential forms and Lie derivatives. Frobenius' theorem is one of the basic tools for the study of vector fields and foliations.
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Let L be an elliptic operator of order 2k with coefficients having 2k continuous derivatives. The Dirichlet problem for L is to find a function u, given a function f and some appropriate boundary values, such that Lu = f and such that u has the appropriate boundary values and normal derivatives. The existence theory for elliptic operators, using Gårding's inequality and the Lax–Milgram lemma, only guarantees that a weak solution u exists in the Sobolev space Hk. This situation is ultimately unsatisfactory, as the weak solution u might not have enough derivatives for the expression Lu to even make sense. The elliptic regularity theorem guarantees that, provided f is square-integrable, u will in fact have 2k square-integrable weak derivatives.
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His research is on integrable systems of mathematical physics (such as the theory of solitons) and applications of quantum field theories and models of string theory to algebraic geometry and complex analysis and includes quantum field theories on algebraic curves and associated reciprocity laws, two- dimensional quantum gravity and Weil–Petersson geometry of moduli spaces, the Kähler geometry of universal Teichmüller space, and trace formulas. His major contributions are in theory of classical and quantum integrable systems, quantum groups and Weil–Petersson geometry of moduli spaces. Together with Ludvig Faddeev and Evgeny Sklyanin he formulated the algebraic Bethe Ansatz and quantum inverse scattering method. Together with Ludvig Faddeev and Nicolai Reshetikhin he proposed a method of quantization of Lie groups and algebras, the FRT construction.
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The Kundu equation is a completely integrable system, allowing Lax pair representation, exact solutions, and higher conserved quantity. Along with its different particular cases, this equation has been investigated for finding its exact travelling wave solutions, exact solitary wave solutions via bilinearization, and Darboux transformation together with the orbital stability for such solitary wave solutions. The Kundu equation has been applied to various physical processes such as fluid dynamics, plasma physics, and nonlinear optics. It is linked to the mixed nonlinear Schrödinger equation through a gauge transformation and is reducible to a variety of known integrable equations such as the nonlinear Schrödinger equation (NLSE), derivative NLSE, higher nonlinear derivative NLSE, Chen–Lee–Liu, Gerjikov-Vanov, and Kundu–Eckhaus equations, for different choices of the parameters.
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Let be a probability space, } with or a finite or an infinite index set, a filtration of , and an adapted stochastic process with for all . Then there exists a martingale and an integrable predictable process starting with such that for every . Here predictable means that is -measurable for every }. This decomposition is almost surely unique.
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Almost all integrable systems of classical mechanics can be obtained as particular cases of the Hitchin system (or its meromorphic generalization or in a singular limit). The Hitchin fibration is the map from the moduli space of Hitchin pairs to characteristic polynomials. used Hitchin fibrations over finite fields in his proof of the fundamental lemma.
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Paul B. Wiegmann (Павел Борисович Вигман) is a Russian physicist. He is the Robert W. Reneker Distinguished Service Professor in the Department of Physics at the University of Chicago, James Franck Institute and Enrico Fermi Institute. He specializes in theoretical condensed matter physics. He made pioneering contributions to the field of quantum integrable systems.
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In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense. They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.
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Most Riemann–Hilbert factorization problems studied in the literature are 2-dimensional, i.e., the unknown matrices are of dimension 2. Higher-dimensional problems have been studied by Arno Kuijlaars and collaborators, see e.g. . The numerical analysis of Riemann-Hilbert problems can also provide a most effective way for numerically solving integrable PDEs, see eg.
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A spin model is a mathematical model used in physics primarily to explain magnetism. Spin models may either be classical or quantum mechanical in nature. Spin models have been studied in quantum field theory as examples of integrable models. Spin models are also used in quantum information theory and computability theory in theoretical computer science.
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Quantum tunneling oscillations of probability in an integrable double well of potential, seen in phase space. The concept of quantum tunneling can be extended to situations where there exists a quantum transport between regions that are classically not connected even if there is no associated potential barrier, this phenomenon is known as dynamical tunneling.
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The wavefunction for a quantum-mechanical particle in a box whose walls have arbitrary shape is given by the Helmholtz equation subject to the boundary condition that the wavefunction vanishes at the walls. These systems are studied in the field of quantum chaos for wall shapes whose corresponding dynamical billiard tables are non-integrable.
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Similarly, foliations correspond to G-structures coming from block matrices, together with integrability conditions so that the Frobenius theorem applies. A flat G-structure is a G-structure P having a global section (V1,...,Vn) consisting of commuting vector fields. A G-structure is integrable (or locally flat) if it is locally isomorphic to a flat G-structure.
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Jacques-Claude Hurtubise FRSC (born March 12, 1957) is a Canadian mathematician who works as a professor of mathematics and chair of the mathematics department at McGill University. His research interests include moduli spaces, integrable systems, and Riemann surfaces.Curriculum vitae, retrieved 2015-03-01. Among other contributions, he is known for proving the Atiyah–Jones conjecture.
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There is also the artifact "normalization to a delta function" that is frequently employed for notational convenience, see further down. The delta functions themselves aren't square integrable either. The above description of the function space containing the wave functions is mostly mathematically motivated. The function spaces are, due to completeness, very large in a certain sense.
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In this field, Severini proved a generalized version of the Weierstrass approximation theorem. Precisely, he extended the original result of Karl Weierstrass to the class of bounded locally integrable functions, which is a class including particular discontinuous functions as members.According to , the result is given in various papers, source perhaps being the most accessible of them.
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Nalini Joshi is an Australian mathematician. She is a professor in the School of Mathematics and Statistics at the University of Sydney, the first woman in the School to hold this position, and is a past-president of the Australian Mathematical Society. Joshi is a member of the School's Applied Mathematics Research Group. Her research concerns integrable systems.
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The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the continuous limit of the pentagram map is the classical Boussinesq equation. This equation is a classical example of an integrable partial differential equation. Here is a description of the geometric action of the Boussinesq equation.
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The eigenvalues of an almost complex structure are ±i and the eigenspaces form sub-bundles denoted by T0,1M and T1,0M. The Newlander- Nirenberg theorem shows that an almost complex structure is actually a complex structure precisely when these subbundles are involutive, i.e., closed under the Lie bracket of vector fields, and such an almost complex structure is called integrable.
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In 2006, he was recipient of the CAP-CRM Prize in Theoretical and Mathematical Physics 2006 CAP/CRM Prize in Theoretical and Mathematical Physics CAP-CRM Prize in Theoretical and Mathematical Physics – Previous Winners "For his deep and lasting contributions to the theory of integrable systems with connections to gauge theory, inverse scattering and random matrices".
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In mathematics, noncommutative residue, defined independently by M. and , is a certain trace on the algebra of pseudodifferential operators on a compact differentiable manifold that is expressed via a local density. In the case of the circle, the noncommutative residue had been studied earlier by M. and Y. in the context of one-dimensional integrable systems.
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He is also the adjunct professor of physics at Saha Institute of Nuclear Physics in India. Das' research is in the area of theoretical high energy physics. He works on supersymmetry and supergravity. In recent years, he has worked extensively on non-linear integrable systems, which are systems which in spite of their complicated appearance can be exactly solved.
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Consider the Hilbert space of complex- valued square integrable functions on the interval . With , define the operator :T_f\varphi(x) = x^2 \varphi (x) \quad for any function in . This will be a self-adjoint bounded linear operator, with domain all of with norm . Its spectrum will be the interval (the range of the function defined on .
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A given is integrable iff everywhere. There is a global foliation theory, because topological constraints exist. For example, in the surface case, an everywhere non-zero vector field can exist on an orientable compact surface only for the torus. This is a consequence of the Poincaré–Hopf index theorem, which shows the Euler characteristic will have to be 0.
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In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the Lax equation. Lax pairs were introduced by Peter Lax to discuss solitons in continuous media. The inverse scattering transform makes use of the Lax equations to solve such systems.
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The concept of dynamical tunneling is particularly suited to address the problem of quantum tunneling in high dimensions (d>1). In the case of Integrable system, where bounded classical trajectories are confined onto tori in phase space, tunneling can be understood as the quantum transport between semi-classical states built on two distinct but symmetric tori.
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Example: A complex structure on the tangent bundle of a real manifold M is usually called an almost complex structure. A theorem of Newlander and Nirenberg says that an almost complex structure J is "integrable" in the sense it is induced by a structure of a complex manifold if and only if a certain tensor involving J vanishes.
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361 in . Haar used these functions to give an example of an orthonormal system for the space of square-integrable functions on the unit interval [0, 1]. The study of wavelets, and even the term "wavelet", did not come until much later. As a special case of the Daubechies wavelet, the Haar wavelet is also known as Db1.
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In mathematics -- specifically, in integration theory -- the Alexiewicz norm is an integral norm associated to the Henstock-Kurzweil integral. The Alexiewicz norm turns the space of Henstock-Kurzweil integrable functions into a topological vector space that is barrelled but not complete. The Alexiewicz norm is named after the Polish mathematician Andrzej Alexiewicz, who introduced it in 1948.
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The following conditions on a real-valued function f on a compact interval [a,b] are equivalent:; also and . :(1) f is absolutely continuous; :(2) f has a derivative f ′ almost everywhere, the derivative is Lebesgue integrable, and :: f(x) = f(a) + \int_a^x f'(t) \, dt :for all x on [a,b]; :(3) there exists a Lebesgue integrable function g on [a,b] such that :: f(x) = f(a) + \int_a^x g(t) \, dt :for all x in [a,b]. If these equivalent conditions are satisfied then necessarily g = f ′ almost everywhere. Equivalence between (1) and (3) is known as the fundamental theorem of Lebesgue integral calculus, due to Lebesgue.. For an equivalent definition in terms of measures see the section Relation between the two notions of absolute continuity.
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A variant not having this requirement is: :If G : [a, b] → R is a monotonic (not necessarily decreasing and positive) function and φ : [a, b] → R is an integrable function, then there exists a number x in (a, b) such that :: \int_a^b G(t)\varphi(t)\,dt = G(a^+) \int_a^x \varphi(t)\,dt + G(b^-) \int_x^b \varphi(t)\,dt.
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There is also a continuous version of Chebyshev's sum inequality: If f and g are real- valued, integrable functions over [0,1], both non-increasing or both non- decreasing, then : \int_0^1 f(x)g(x) \, dx \geq \int_0^1 f(x) \, dx \int_0^1 g(x) \, dx, with the inequality reversed if one is non-increasing and the other is non-decreasing.
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Theoretical and Mathematical Physics (Russian: Теоретическая и Математическая Физика) is a Russian scientific journal. It was founded in 1969 by Nikolai Bogolubov. Currently handled by the Russian Academy of Sciences, it appears in 12 issues per year. The journal publishes papers on mathematical aspects of quantum mechanics, quantum field theory, statistical physics, supersymmetry, and integrable models (in any areas of physics).
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They are typical of later application of generalized function methods. An influential book on operational calculus was Oliver Heaviside's Electromagnetic Theory of 1899. When the Lebesgue integral was introduced, there was for the first time a notion of generalized function central to mathematics. An integrable function, in Lebesgue's theory, is equivalent to any other which is the same almost everywhere.
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C. Alcaraz and A. Lima Santos, Nuclear Physics B 275. a more general self-dual case of the integrable chiral Potts model was discovered. The weight are given in product formH. Au-Yang, B. M. McCoy, J. H. H. Perk, and S. Tang (1988), "Solvable models in statistical mechanics and Riemann surfaces of genus greater than one", in Algebraic Analysis, Vol.
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A Kähler manifold is a symplectic manifold equipped with an integrable almost-complex structure J which is compatible with the symplectic form ω, meaning that the bilinear form :g(u,v)=\omega(u,Jv) on the tangent space of X at each point is symmetric and positive definite (and hence a Riemannian metric on X).Cannas da Silva (2001), Definition 16.1.
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A similar theorem can be proven for signed and complex measures: namely, that if is a nonnegative σ-finite measure, and is a finite-valued signed or complex measure such that , i.e. is absolutely continuous with respect to , then there is a -integrable real- or complex-valued function on such that for every measurable set , : u(A) = \int_A g \, d\mu.
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Continuous group theory, Lie algebras and differential geometry are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform and finally finding exact analytic solutions to the PDE. Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.
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The Koopman–von Neumann mechanics is a description of classical mechanics in terms of Hilbert space, introduced by Bernard Koopman and John von Neumann in 1931 and 1932, respectively. As Koopman and von Neumann demonstrated, a Hilbert space of complex, square integrable wavefunctions can be defined in which classical mechanics can be formulated as an operatorial theory similar to quantum mechanics.
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In mathematics, in the area of functional analysis and operator theory, the Volterra operator, named after Vito Volterra, is a bounded linear operator on the space L2[0,1] of complex-valued square-integrable functions on the interval [0,1]. On the subspace C[0,1] of continuous functions it represents indefinite integration. It is the operator corresponding to the Volterra integral equations.
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Friedrich "Fritz" Gesztesy (born 5 November 1953 in Austria) is a well-known Austrian-American mathematical physicist and Professor of Mathematics at Baylor University, known for his important contributions in spectral theory, functional analysis, nonrelativistic quantum mechanics (particularly, Schrödinger operators), ordinary and partial differential operators, and completely integrable systems (soliton equations). He has authored more than 270 publications on mathematics and physics.
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In 1977 he became a professor at Yale University. Beals works on inverse problems in scattering theory, integrable systems, pseudodifferential operators, complex analysis, global analysis and transport theory. He has been married since 1962 and has three children. He should not be confused with the mathematics professor at Rutgers University named R. Michael Beals (born in 1954), who is Richard Beals's brother.
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See Dirac spinor for details of solutions to the Dirac equation. Note that since the Dirac operator acts on 4-tuples of square-integrable functions, its solutions should be members of the same Hilbert space. The fact that the energies of the solutions do not have a lower bound is unexpected – see the hole theory section below for more details.
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An example of this is given by the derivative g of the (differentiable but not absolutely continuous) function f(x)=x²·sin(1/x²) (the function g is not Lebesgue-integrable around 0). The Denjoy integral corrects this lack by ensuring that the derivative of any function f that is everywhere differentiable (or even differentiable everywhere except for at most countably many points) is integrable, and its integral reconstructs f up to a constant; the Khinchin integral is even more general in that it can integrate the approximate derivative of an approximately differentiable function (see below for definitions). To do this, one first finds a condition that is weaker than absolute continuity but is satisfied by any approximately differentiable function. This is the concept of generalized absolute continuity; generalized absolutely continuous functions will be exactly those functions which are indefinite Khinchin integrals.
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"Quantum mechanics on the noncommutative plane and sphere", Physics Letters B505(1-4), 267-274. He is a Fellow of the American Physical Society (2012), cited for "For important contributions to the field of statistical mechanics and integrable systems, including the Polychronakos model and the exchange operator formalism, fractional statistics, matrix model description of quantum Hall systems as well as other areas such as noncommutative geometry".
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This step was taken by Sommerfeld, who proposed the general quantization rule for an integrable system, : J_k = h n_k. \, Each action variable is a separate integer, a separate quantum number. This condition reproduces the circular orbit condition for two dimensional motion: let be polar coordinates for a central potential. Then is already an angle variable, and the canonical momentum conjugate is , the angular momentum.
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As the perturbation increases and the smooth curves disintegrate we move from KAM theory to Aubry–Mather theory which requires less stringent hypotheses and works with the Cantor-like sets. The existence of a KAM theorem for perturbations of quantum many-body integrable systems is still an open question, although it is believed that arbitrarily small perturbations will destroy integrability in the infinite size limit.
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So the determinant in Yangian theory has natural interpretation via Manin matrices. For the sake of quantum integrable systems it is important to construct commutative subalgebras in Yangian. It is well known that in the classical limit expressions Tr(Tk(z)) generate Poisson commutative subalgebra. The correct quantization of these expressions has been first proposed by the use of Newton identities for Manin matrices: Proposition.
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That means its value at a given point is (in a sense) not its most important feature. In functional analysis a clear formulation is given of the essential feature of an integrable function, namely the way it defines a linear functional on other functions. This allows a definition of weak derivative. During the late 1920s and 1930s further steps were taken, basic to future work.
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There is also a notion of quantum integrable systems. In the quantum setting, functions on phase space must be replaced by self-adjoint operators on a Hilbert space, and the notion of Poisson commuting functions replaced by commuting operators. The notion of conservation laws must be specialized to local conservation laws . Every Hamiltonian has an infinite set of conserved quantities given by projectors to its energy eigenstates.
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In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer. For many authors, the Riesz–Fischer theorem refers to the fact that the Lp spaces from Lebesgue integration theory are complete.
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Among other things they demonstrated that in 1-dimensional space solitons exist, e.g. in the form of two unidirectionally propagating pulses with different size and speed and exhibiting the remarkable property that number, shape and size are the same before and after collision. Gardner et al. introduced the inverse scattering technique for solving the KdV equation and proved that this equation is completely integrable.
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Poisson structures are one instance of Jacobi structures introduced by André Lichnerowicz in 1977. They were further studied in the classical paper of Alan Weinstein, where many basic structure theorems were first proved, and which exerted a huge influence on the development of Poisson geometry — which today is deeply entangled with non-commutative geometry, integrable systems, topological field theories and representation theory, to name a few.
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The Fokas method, or unified transform, is an algorithmic procedure for analysing boundary value problems for linear partial differential equations and for an important class of nonlinear PDEs belonging to the so-called integrable systems. It is named after Greek mathematician Athanassios S. Fokas. Traditionally, linear boundary value problems are analysed using either integral transforms and infinite series, or by employing appropriate fundamental solutions.
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He is currently Director of the Mathematical Physics group at the Centre de recherches mathématiques (CRM), a national research centre in mathematics at the Université de Montréal and Professor in the Department of Mathematics and Statistics at Concordia University. He is an affiliate member of the Perimeter Institute for Theoretical Physics Perimeter Institute of Theoretical Physics and was a long-time visiting member of the Princeton Institute for Advanced Study .Advanced Study Scholars His work has had a strong impact in several domains of mathematical physics, and his publications are very widely cited Scientific publications of John Harnad on INSPIRE-HEP Citations of scientific publications of John Harnad on Google Scholar. He has made fundamental contributions on: geometrical and topological methods in gauge theory, classical and quantum integrable systems, the spectral theory of random matrices, isomonodromic deformations, the bispectral problem, integrable random processes, transformation groups and symmetries.
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It was realized that very complicated behavior is possible in dynamical systems with only a few degrees of freedom. This complexity cannot be adequately described in terms of individual trajectories and requires statistical methods. Typical Hamiltonian systems are not integrable but chaotic, and this chaos is not homogeneous. At the same values of the control parameters, there coexist regions in the phase space with regular and chaotic motion.
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Not all functions are realistic descriptions of any physical system. For instance, in the function space one can find the function that takes on the value for all rational numbers and for the irrationals in the interval . This is square integrable,As is explained in a later footnote, the integral must be taken to be the Lebesgue integral, the Riemann integral is not sufficient. but can hardly represent a physical state.
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Solitons are traditionally a key object of study in the theory of nonlinear integrable equations. The solitons of the Novikov–Veselov equation at positive energy are transparent potentials, similarly to the one-dimensional case (in which solitons are reflectionless potentials). However, unlike the one-dimensional case where there exist well-known exponentially decaying solitons, the Novikov–Veselov equation (at least at non-zero energy) does not possess exponentially localized solitons .
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The Capelli identity from 19th century gives one of the first examples of determinants for matrices with non-commuting elements. Manin matrices give a new look on this classical subject. This example is related to Lie algebra gln and serves as a prototype for more complicated applications to loop Lie algebra for gln, Yangian and integrable systems. Take Eij be matrices with 1 at position (i,j) and zeros everywhere else.
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In 2004 he became a Fellow of the American Physical Society. Independently of Paul Wiegmann, he succeeded in 1980 in finding the exact solution of the Kondo problem.In 2017, both were awarded the Lars Onsager Prize.. With John H. Lowenstein, he solved the Chiral Gross–Neveu model using Bethe ansatz technique. He deals with the relations between conformal and exactly integrable field theories and string theory in loop space.
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In mathematical physics, the Ehlers group, named after Jürgen Ehlers, is a finite-dimensional transformation group of stationary vacuum spacetimes which maps solutions of Einstein's field equations to other solutions. It has since found a number of applications, from use as a tool in the discovery of previously unknown solutions to a proof that solutions in the stationary axisymmetric case form an integrable system.The original articles are and ; for the applications, .
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The title of his dissertation with Raphael Høegh-Krohn was Point Interactions and the Short-Range Expansion. A Solvable Model in Quantum Mechanics and Its Approximation. He was appointed professor at the Norwegian Institute of Technology (now: the Norwegian University of Science and Technology ) in 1991. His research interests are Differential equations, mathematical physics (in particular hyperbolic conservation laws and completely integrable systems), Stochastic analysis, and flow in porous media.
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Riemann–Hilbert problems have applications to several related classes of problems. A. Integrable models. The inverse scattering or inverse spectral problem associated to the Cauchy problems for 1+1 dimensional partial differential equations on the line, or to periodic problems, or even to initial-boundary value problems (), can be stated as a Riemann–Hilbert problem. Likewise the inverse monodromy problem for Painlevé equations can be stated as a Riemann–Hilbert problem.
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A pair of real valued random functions \textstyle X(t) and \textstyle Y(t) with t \in T, a compact interval, can be viewed as realizations of square-integrable stochastic process in a Hilbert space. Since both X and Y are infinite dimensional, some kind of dimension reduction is required to explore their relationship. Notions of correlation for functional data include the following.Wang JL, Chiou JM, Müller HG. (2015).
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Before collapse, the wave function may be any square-integrable function. This function is expressible as a linear combination of the eigenstates of any observable. Observables represent classical dynamical variables, and when one is measured by a classical observer, the wave function is projected onto a random eigenstate of that observable. The observer simultaneously measures the classical value of that observable to be the eigenvalue of the final state.
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In Hamiltonian dynamics, Liouville also introduced the notion of action-angle variables as a description of completely integrable systems. The modern formulation of this is sometimes called the Liouville–Arnold theorem, and the underlying concept of integrability is referred to as Liouville integrability. In 1851, he was elected a foreign member of the Royal Swedish Academy of Sciences. The crater Liouville on the Moon is named after him.
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In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (1894–1964). The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent Schwartz.
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Alekseev does research on representation theory of Lie groups and algebras, moment theory, symplectic geometry and mathematical physics. In 2006 he, with Eckhard Meinrenken, published in Inventiones Mathematicae a proof of the Kashiwara-Vergne conjecture. arXiv preprint In 2008 he gave a new proof with Charles Torossian. 2014 Alekseev was an invited speaker with talk Three lives of the Gelfand-Zeitlin integrable system at the International Congress of Mathematicians in Seoul.
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In 1934, Egon Orowan, Michael Polanyi and Geoffrey Ingram Taylor, roughly simultaneously, realized that the plastic deformation of ductile materials could be explained in terms of the theory of dislocations. The mathematical theory of plasticity, flow plasticity theory, uses a set of non-linear, non-integrable equations to describe the set of changes on strain and stress with respect to a previous state and a small increase of deformation.
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In these models, functional predictors ( X ) are paired with responses ( Y ) that can be either scalar or functional. The response can follow a continuous or discrete distribution and this distribution may be in the exponential family. In the latter case, there would be a canonical link that connects predictors and responses. Functional predictors (or responses) can be viewed as random trajectories generated by a square-integrable stochastic process.
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Fellows list, John Simon Guggenheim Memorial Foundation. Accessed January 13, 2010. He gave an invited address at the International Congress of Mathematicians in Berlin in 1998Professor Percy Deift, Integrable Systems, Rigorous Asymptotics and Applications Workshop, August 22–23, 2004, University of Melbourne. Accessed January 13, 2010 and plenary addresses in 2006 at the International Congress of Mathematicians in Madrid and at the International Congress on Mathematical Physics in Rio de Janeiro.
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A resurgence of interest in classical integrable systems came with the discovery, in the late 1960s, that solitons, which are strongly stable, localized solutions of partial differential equations like the Korteweg–de Vries equation (which describes 1-dimensional non-dissipative fluid dynamics in shallow basins), could be understood by viewing these equations as infinite-dimensional integrable Hamiltonian systems. Their study leads to a very fruitful approach for "integrating" such systems, the inverse scattering transform and more general inverse spectral methods (often reducible to Riemann–Hilbert problems), which generalize local linear methods like Fourier analysis to nonlocal linearization, through the solution of associated integral equations. The basic idea of this method is to introduce a linear operator that is determined by the position in phase space and which evolves under the dynamics of the system in question in such a way that its "spectrum" (in a suitably generalized sense) is invariant under the evolution, cf. Lax pair.
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University of Rochester. Physics professor Ashok Das Receives Fulbright to Teach in Brazil He is known for his teaching and has received university and department awards for his teaching including the Department Award for Excellence in Undergraduate Teaching, Department of Physics and Astronomy, University of Rochester four times (1987, 1990, 1997 and 2006), the Edward Peck Curtis Award for Excellence in Undergraduate Teaching (1991), and the 2006 William H. Riker University Award for Excellence in Graduate teaching. He has written numerous books and monographs on various disciplines of theoretical physics in advanced and undergraduate and graduate level, like A Path Integral Approach (World Scientific publishers), Finite Temperature Field Theory (World Scientific publishers), Integrable Models (World Scientific Lecture Notes in Physics), Lectures on Gravitation (World Scientific publishers), and Lectures on Electromagnetism: second edition (World Scientific publishers) etc. In 2002 Das was made a fellow of the American Physical Society "For contributions in the areas of supergravity, integrable models and finite temperature field theory".
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The Witten conjecture is a special case of a more general relation between integrable systems of Hamiltonian PDEs and the geometry of certain families of 2D topological field theories (axiomatized in the form of the so-called cohomological field theories by Kontsevich and Manin), which was explored and studied systematically by B. Dubrovin and Y. Zhang, A. Givental, C. Teleman and others. The Virasoro conjecture is a generalization of the Witten conjecture.
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In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale and a predictable process (or "drift") starting at zero. The theorem was proved by and is named for Joseph L. Doob., see The analogous theorem in the continuous-time case is the Doob–Meyer decomposition theorem.
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In mathematics, a summability kernel is a family or sequence of periodic integrable functions satisfying a certain set of properties, listed below. Certain kernels, such as the Fejér kernel, are particularly useful in Fourier analysis. Summability kernels are related to approximation of the identity; definitions of an approximation of identity vary, but sometimes the definition of an approximation of the identity is taken to be the same as for a summability kernel.
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In 1963 along with his father Lester R. Ford, he published an innovative textbook on calculus.Lester Ford Sr. & Jr. (1963) Calculus, McGraw-Hill via HathiTrust. For a given function f and point x, they defined a frame as a rectangle containing (x, f(x)) with sides parallel to the axes of the plane (page 9). Frames are then exploited to define continuous functions (page 10) and to describe integrable functions (page 148).
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These matrices have applications in representation theory in particular to Capelli's identity, Yangian and quantum integrable systems. Manin matrices are particular examples of Manin's general construction of "non-commutative symmetries" which can be applied to any algebra. From this point of view they are "non-commutative endomorphisms" of polynomial algebra C[x1, ...xn]. Taking (q)-(super)-commuting variables one will get (q)-(super)-analogs of Manin matrices, which are closely related to quantum groups.
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In a different approach, the EP quantum mechanics formulate on the basis of an Equivalence Principle (EP), a quantum potential is written as:Alon E. Faraggi, M. Matone: The Equivalence Postulate of Quantum Mechanics, International Journal of Modern Physics A, vol. 15, no. 13, pp. 1869–2017. arXiv hep-th/9809127 of 6 August 1999Robert Carroll: Aspects of quantum groups and integrable systems, Proceedings of Institute of Mathematics of NAS of Ukraine, vo.
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He then also considered conformal field theories such as the Liouville field theory, string theories and two-dimensional quantum gravity from the point of view of exactly integrable systems. With André Neveu, he investigated in the 1980s also non-critical string theories.J-L Gervais and A Neveu: Nuclear Physics B 209 (1982) p. 125 In 1997 he was awarded the highly reputed Prix Créé par l'État from the French Académie des sciences.
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The terms the holonomic and nonholonomic systems were introduced by Heinrich Hertz in 1894. In 1897, S. A. Chaplygin first suggested to form the equations of motion without Lagrange multipliers. Under certain linear constraints, he introduced on the left-hand side of the equations of motion a group of extra terms of the Lagrange-operator type. The remaining extra terms characterise the nonholonomicity of system and they become zero when the given constrains are integrable.
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In 1972 Zakharov and Shabat found another integrable equation and finally it turned out that the inverse scattering technique can be applied successfully to a whole class of equations (e.g. the nonlinear Schrödinger and sine-Gordon equations). From 1965 up to about 1975, a common agreement was reached: to reserve the term soliton to pulse-like solitary solutions of conservative nonlinear partial differential equations that can be solved by using the inverse scattering technique.
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Moser and Arnold expanded the ideas of Kolmogorov (who was inspired by questions of Poincaré) and gave rise to what is now known as Kolmogorov–Arnold–Moser theorem (or "KAM theory"), which concerns the persistence of some quasi- periodic motions (nearly integrable Hamiltonian systems) when they are perturbed. KAM theory shows that, despite the perturbations, such systems can be stable over an infinite period of time, and specifies what the conditions for this are.
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Their universality as the simplest genuinely nonlinear integrable systems means that the isomonodromy equations have an extremely diverse range of applications. Perhaps of greatest practical importance is the field of random matrix theory. Here, the statistical properties of eigenvalues of large random matrices are described by particular transcendents. The initial impetus for the resurgence of interest in isomonodromy in the 1970s was the appearance of transcendents in correlation functions in Bose gases.
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Xin Zhou is a mathematician known for his contributions in scattering theory, integrable systems, random matrices and Riemann–Hilbert problems. He is Professor Emeritus of Mathematics at Duke University. Zhou had obtained M.Sc. from the University of the Chinese Academy of Sciences in 1982 and then got his Ph.D. in 1988 from the University of Rochester. He received the Pólya prize in 1998 and was awarded with the Guggenheim Fellowship in 1999.
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Supersymmetry may be considered a possible "loophole" of the theorem because it contains additional generators (supercharges) that are not scalars but rather spinors. This loophole is possible because supersymmetry is a Lie superalgebra, not a Lie algebra. The corresponding theorem for supersymmetric theories with a mass gap is the Haag–Łopuszański–Sohnius theorem. Quantum group symmetry, present in some two- dimensional integrable quantum field theories like the sine-Gordon model, exploits a similar loophole.
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In other words, the Fourier transform of Tf at a frequency ξ is given by the Fourier transform of f at that frequency, multiplied by the value of the multiplier at that frequency. This explains the terminology "multiplier". Note that the above definition only defines Tf implicitly; in order to recover Tf explicitly one needs to invert the Fourier transform. This can be easily done if both f and m are sufficiently smooth and integrable.
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He graduated from the Department of Physics, Leningrad State University (USSR) in 1978 and earned PhD (Candidate) in 1980 and DrSci (Habilitation) degree in 1989, both at Steklov Mathematical Institute, St. Petersburg. He then held various research positions at Steklov until 2001, when he moved to the University of York. He provided, via particular examples, ideas that led to the discovery of quantum groups and Yangians. He pioneered the investigation of quantum integrable systems with boundaries.
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The Hessenberg operator is an infinite dimensional Hessenberg matrix. It commonly occurs as the generalization of the Jacobi operator to a system of orthogonal polynomials for the space of square-integrable holomorphic functions over some domain -- that is, a Bergman space. In this case, the Hessenberg operator is the right-shift operator S, given by :[Sf](z)=zf(z). The eigenvalues of each principle submatrix of the Hessenberg operator are given by the characteristic polynomial for that submatrix.
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After original Manin's works there were only a few papers on Manin matrices until 2003. But around and some after this date Manin matrices appeared in several not quite related areas: obtained certain noncommutative generalization of the MacMahon master identity, which was used in knot theory; applications to quantum integrable systems, Lie algebras has been found in; generalizations of the Capelli identity involving Manin matrices appeared in. Directions proposed in these papers has been further developed.
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In mathematics, the Dirichlet conditions are sufficient conditions for a real- valued, periodic function f to be equal to the sum of its Fourier series at each point where f is continuous. Moreover, the behavior of the Fourier series at points of discontinuity is determined as well (it is the midpoint of the values of the discontinuity). These conditions are named after Peter Gustav Lejeune Dirichlet. The conditions are: #f must be absolutely integrable over a period.
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An extension of the notion of integrability is also applicable to discrete systems such as lattices. This definition can be adapted to describe evolution equations that either are systems of differential equations or finite difference equations. The distinction between integrable and nonintegrable dynamical systems thus has the qualitative implication of regular motion vs. chaotic motion and hence is an intrinsic property, not just a matter of whether a system can be explicitly integrated in exact form.
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Not every Pfaffian system is completely integrable in the Frobenius sense. For example, consider the following one- form : :\theta=z\,dx +x\,dy+y\,dz. If dθ were in the ideal generated by θ we would have, by the skewness of the wedge product :\theta\wedge d\theta=0. But a direct calculation gives :\theta\wedge d\theta=(x+y+z)\,dx\wedge dy\wedge dz which is a nonzero multiple of the standard volume form on R3.
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In this case, the diagonal ensemble is precisely the same as the microcanonical ensemble, and there is no mystery as to why their predictions are identical. However, this explanation is disfavored for much the same reasons as the first. # Integrable quantum systems are proved to thermalize under condition of simple regular time-dependence of parameters, suggesting that cosmological expansion of the Universe and integrability of the most fundamental equations of motion are ultimately responsible for thermalization.
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The theory of correlation functions was developed : determinant representations, descriptions by differential equations and the Riemann–Hilbert problem. Asymptotics of correlation functions (even for space, time and temperature dependence) were evaluated in 1991. Explicit expressions for the higher conservation laws of the integrable models were obtained in 1989. Essential progress was achieved in study of ice-type models: the bulk free energy of the six vertex model depends on boundary conditions even in the thermodynamic limit.
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Let I = [a,b] be an interval and g : I → R be a real-valued function on I. The function g is said to be Khinchin-integrable on I iff there exists a function f that is generalized absolutely continuous whose approximate derivative coincides with g almost everywhere; in this case, the function f is determined by g up to a constant, and the Khinchin-integral of g from a to b is defined as f(b) − f(a).
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In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions. It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian physicist and geometer, who published independent proofs respectively in 1910 and 1911. Egorov's theorem can be used along with compactly supported continuous functions to prove Lusin's theorem for integrable functions.
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If is a -finite signed measure, then it can be Hahn–Jordan decomposed as where one of the measures is finite. Applying the previous result to those two measures, one obtains two functions, , satisfying the Radon–Nikodym theorem for and respectively, at least one of which is -integrable (i.e., its integral with respect to is finite). It is clear then that satisfies the required properties, including uniqueness, since both and are unique up to -almost everywhere equality.
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The given functions (f, g) may be discontinuous, provided that they are locally integrable (on the given interval). In this case, Lebesgue integration is meant, the conclusions hold almost everywhere (thus, in all continuity points), and differentiability of g is interpreted as local absolute continuity (rather than continuous differentiability). Sometimes the given functions are assumed to be piecewise continuous, in which case Riemann integration suffices, and the conclusions are stated everywhere except the finite set of discontinuity points.
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At the University of California, Los Angeles (UCLA), he was an associate professor from 1988 to 1990 and was appointed a full professor in 1990 and a distinguished professor in 2009. From the 1980s onwards, he collaborated extensively with Duong H. Phong on the geometry underlying superstring perturbation theory, among other topics in the mathematics of supersymmetry and superstring theory. Another topic of D'Hoker's research is integrable systems. In 1997 he was at the Institute for Advanced Study.
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In mathematics, a Novikov–Shubin invariant, introduced by , is an invariant of a compact Riemannian manifold related to the spectrum of the Laplace operator acting on square-integrable differential forms on its universal cover. The Novikov–Shubin invariant gives a measure of the density of eigenvalues around zero. It can be computed from a triangulation of the manifold, and it is a homotopy invariant. In particular, it does not depend on the chosen Riemannian metric on the manifold.
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Since Fourier series have such good convergence properties, many are often surprised by some of the negative results. For example, the Fourier series of a continuous T-periodic function need not converge pointwise. The uniform boundedness principle yields a simple non-constructive proof of this fact. In 1922, Andrey Kolmogorov published an article titled Une série de Fourier- Lebesgue divergente presque partout in which he gave an example of a Lebesgue- integrable function whose Fourier series diverges almost everywhere.
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When the definite integral exists (in the sense of either the Riemann integral or the more advanced Lebesgue integral), this ambiguity is resolved as both the proper and improper integral will coincide in value. Often one is able to compute values for improper integrals, even when the function is not integrable in the conventional sense (as a Riemann integral, for instance) because of a singularity in the function or because one of the bounds of integration is infinite.
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During 1986/87, he was a department representative at the University of Düsseldorf. Since 1987, he is a full professor of mathematics at the ETH Zurich. Knörrer studies algebraic geometry and its connection to mathematical physics, for example, for integrable systems, as well as mathematical theory of many-particle systems in statistical mechanics and solid state physics (Fermi liquids). Together with Brieskorn, he wrote an extensive and rich illustrated textbook on algebraic curves, which also was translated into English.
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It has important applications in mathematical finance and stochastic differential equations. The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes: :Y_t=\int_0^t H_s\,dX_s, where H is a locally square-integrable process adapted to the filtration generated by X , which is a Brownian motion or, more generally, a semimartingale. The result of the integration is then another stochastic process.
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Lu et al. proved that if the width of a deep neural network with ReLU activation is strictly larger than the input dimension, then the network can approximate any Lebesgue integrable function; If the width is smaller or equal to the input dimension, then deep neural network is not a universal approximator. The probabilistic interpretation derives from the field of machine learning. It features inference, as well as the optimization concepts of training and testing, related to fitting and generalization, respectively.
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The essential idea underlying the complex absorbing potentials to calculate the resonances is to introduce an absorbing boundary condition in the exterior region of the molecular scattered target which results in a non-Hermitian Hamiltonian, one of the square-integrable eigenfunctions of which corresponds to the resonant state. The associated complex eigen-value then gives the position and width of the resonance or the auto-ionizing state. The important relaxation and correlation effects are included in the coupled-cluster method.
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The Fourier inversion theorem holds for all Schwartz functions (roughly speaking, smooth functions that decay quickly and whose derivatives all decay quickly). This condition has the benefit that it is an elementary direct statement about the function (as opposed to imposing a condition on its Fourier transform), and the integral that defines the Fourier transform and its inverse are absolutely integrable. This version of the theorem is used in the proof of the Fourier inversion theorem for tempered distributions (see below).
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The conditions of the Frobenius theorem depend on whether the underlying field is or . If it is R, then assume F is continuously differentiable. If it is , then assume F is twice continuously differentiable. Then (1) is completely integrable at each point of if and only if :D_1F(x,y)\cdot(s_1,s_2) + D_2F(x,y)\cdot(F(x,y)\cdot s_1,s_2) = D_1F(x,y) \cdot (s_2,s_1) + D_2F(x,y)\cdot(F(x,y)\cdot s_2,s_1) for all .
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As a mere representation change, however, Weyl's map underlies the alternate Phase space formulation of conventional quantum mechanics. A more geometric approach to quantization, in which the classical phase space can be a general symplectic manifold, was developed in the 1970s by Bertram Kostant and Jean- Marie Souriau. The method proceeds in two stages. Chapters 22 and 23 First, once constructs a "prequantum Hilbert space" consisting of square-integrable functions (or, more properly, sections of a line bundle) over the phase space.
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This part is sometimes referred to as the second fundamental theorem of calculus or the Newton–Leibniz axiom. Let f be a real- valued function on a closed interval [a,b] and F an antiderivative of f in [a,b]: :F'(x) = f(x). If f is Riemann integrable on [a,b] then :\int_a^b f(x)\,dx = F(b) - F(a). The second part is somewhat stronger than the corollary because it does not assume that f is continuous.
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In addition to compactly supported functions and integrable functions, functions that have sufficiently rapid decay at infinity can also be convolved. An important feature of the convolution is that if f and g both decay rapidly, then f∗g also decays rapidly. In particular, if f and g are rapidly decreasing functions, then so is the convolution f∗g. Combined with the fact that convolution commutes with differentiation (see #Properties), it follows that the class of Schwartz functions is closed under convolution .
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More generally, the circumcenter of mass and center of mass coincide for a simplicial polytope for which each face has the sum of squares of its edges a constant. The circumcenter of mass is invariant under the operation of "recutting" of polygons. and the discrete bicycle (Darboux) transformation; in other words, the image of a polygon under these operations has the same circumcenter of mass as the original polygon. The generalized Euler line makes other appearances in the theory of integrable systems.
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A set of problems formulated in this thesis for a long time attracted attention from mathematicians. For example, the first problem in the list, on the convergence of the Fourier series for a square-integrable function, was solved by Lennart Carleson in 1966 (Carleson's theorem). In the theory of boundary properties of analytic functions he proved an important result on the invariance of sets of boundary points under conformal mappings (1919). Luzin was one of the founders of descriptive set theory.
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Rectangle, triangle, and other functions can also be used. A rectangular window does not modify the data segment at all. It's only for modelling purposes that we say it multiplies by 1 inside the window and by 0 outside. A more general definition of window functions does not require them to be identically zero outside an interval, as long as the product of the window multiplied by its argument is square integrable, and, more specifically, that the function goes sufficiently rapidly toward zero.
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In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of on the space of square- integrable functions, where is a Lie group and a cofinite discrete group. The character is given by the trace of certain functions on . The simplest case is when is cocompact, when the representation breaks up into discrete summands. Here the trace formula is an extension of the Frobenius formula for the character of an induced representation of finite groups.
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In calculus and mathematical analysis the limits of integration of the integral : \int_a^b f(x) \, dx of a Riemann integrable function f defined on a closed and bounded [interval] are the real numbers a and b . The region that is bounded can be seen as the area inside a and b . For example, the function f(x)=x^3 is bounded on the interval [2, 4] \int_2^4 x^3 \, dx with the limits of integration being 2 and 4.
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In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit diskOne must use the open unit disk in Cn as the model space instead of Cn because these are not isomorphic, unlike for real manifolds. in Cn, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a complex manifold in the sense above (which can be specified as an integrable complex manifold), and an almost complex manifold.
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This led to a representation of two-dimensional quantum gravity by random fluctuating surfaces or closed bosonic strings, in terms of random matrices (E Brezin et S. Hikami, Random matrix theory with an external source, Springer, 2016). He showed that the continuous boundary of such models is linked to integrable hierarchies such as KdV flows. He has also worked on establishing the universality of eigenvalue correlations for random matrices (« Edoaud Brezin, membre de l'Académie des sciences » [archive], sur Académie des sciences (consulté le 9 novembre 2018)).
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Godement started as a student at the École normale supérieure in 1940, where he became a student of Henri Cartan. He started research into harmonic analysis on locally compact abelian groups, finding a number of major results; this work was in parallel but independent of similar investigations in the USSR and Japan. Work on the abstract theory of spherical functions published in 1952 proved very influential in subsequent work, particularly that of Harish-Chandra. The isolation of the concept of square-integrable representation is attributed to him.
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This model is out of the class of all previously known models and raises a host of unsolved questions which are related to some of the most intractable problems of algebraic geometry which have been with us for 150 years. The chiral Potts models are used to understand the commensurate-incommensurate phase transitions.S. Howes, L.P. Kadanoff and M. den Nijs (1983), Nuclear Physics B 215, 169. For N = 3 and 4, the integrable case was discovered in 1986 in Stonybrook and published the following year.
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The semi- classical limit corresponds to \hbar \;\to\; 0 which can be seen to be equivalent to m \;\to\; \infty, the mass increasing so that it behaves classically. As a general statement, one may say that whenever the classical equations of motion are integrable (e.g. rectangular or circular billiard tables), then the quantum-mechanical version of the billiards is completely solvable. When the classical system is chaotic, then the quantum system is generally not exactly solvable, and presents numerous difficulties in its quantization and evaluation.
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Suppose we want to show that the function f(x) = x is Darboux-integrable on the interval [0,1] and determine its value. To do this we partition [0,1] into n equally sized subintervals each of length 1/n. We denote a partition of n equally sized subintervals as Pn. Now since f(x) = x is strictly increasing on [0,1], the infimum on any particular subinterval is given by its starting point. Likewise the supremum on any particular subinterval is given by its end point.
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An abstract CR structure on a real manifold M of real dimension n consists of a complex subbundle L of the complexified tangent bundle which is formally integrable, in the sense that [L,L] ⊂ L, which has zero intersection with its complex conjugate. The CR codimension of the CR structure is k = n - 2 \dim L, where dim L is the complex dimension. In case k = 1, the CR structure is said to be of hypersurface type. Most examples of abstract CR structures are of hypersurface type.
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Trubowitz, who was born in 1951, received his doctorate in 1977 under the supervision of Henry McKean at New York University, with thesis titled The inverse problem for periodic potentials. Since 1983, he is a full professor of mathematics at the Swiss Federal Institute of Technology Zurich. As of 2016, he has retired from his position at ETH. Trubowitz studies scattering theory (some with Percy Deift, and inverse scattering theory), integrable systems and their connection to algebraic geometry, mathematical theory of Fermi liquids in the statistical mechanics.
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In mathematics, Bäcklund transforms or Bäcklund transformations (named after the Swedish mathematician Albert Victor Bäcklund) relate partial differential equations and their solutions. They are an important tool in soliton theory and integrable systems. A Bäcklund transform is typically a system of first order partial differential equations relating two functions, and often depending on an additional parameter. It implies that the two functions separately satisfy partial differential equations, and each of the two functions is then said to be a Bäcklund transformation of the other.
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Verdier called the condition (w) for Whitney, as at the time he thought (w) might be equivalent to Whitney's condition (b). Real algebraic examples for which the Whitney conditions (b) hold but Verdier's condition (w) fails, were constructed by David Trotman who has obtained many geometric properties of (w)-regular stratifications. Work of Bernard Teissier, aided by Jean-Pierre Henry and Michel Merle at the École Polytechnique, led to the 1982 result that Verdier's condition (w) is equivalent to the Whitney conditions for complex analytic stratifications. Verdier later worked on the theory of integrable systems.
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The solutions are one-electron functions and are referred to as hydrogen-like atomic orbitals.In quantum chemistry an orbital is synonymous with "a one-electron function", a square integrable function of x, y, z. Other systems may also be referred to as "hydrogen-like atoms", such as muonium (an electron orbiting an antimuon), positronium (an electron and a positron), certain exotic atoms (formed with other particles), or Rydberg atoms (in which one electron is in such a high energy state that it sees the rest of the atom practically as a point charge).
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The Weierstrass transform is intimately related to the heat equation (or, equivalently, the diffusion equation with constant diffusion coefficient). If the function describes the initial temperature at each point of an infinitely long rod that has constant thermal conductivity equal to 1, then the temperature distribution of the rod t = 1 time units later will be given by the function F. By using values of t different from 1, we can define the generalized Weierstrass transform of . The generalized Weierstrass transform provides a means to approximate a given integrable function arbitrarily well with analytic functions.
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The Yang–Baxter equation is a consequence of this reducibility and leads to trace identities which provide an infinite set of conserved quantities. All of these ideas are incorporated into the quantum inverse scattering method where the algebraic Bethe ansatz can be used to obtain explicit solutions. Examples of quantum integrable models are the Lieb–Liniger model, the Hubbard model and several variations on the Heisenberg model. Some other types of quantum integrability are known in explicitly time-dependent quantum problems, such as the driven Tavis-Cummings model..
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Essentially, these distinctions correspond to the dimensions of the leaves of the foliation. When the number of independent Poisson commuting invariants is less than maximal (but, in the case of autonomous systems, more than one), we say the system is partially integrable. When there exist further functionally independent invariants, beyond the maximal number that can be Poisson commuting, and hence the dimension of the leaves of the invariant foliation is less than n, we say the system is superintegrable. If there is a regular foliation with one-dimensional leaves (curves), this is called maximally superintegrable.
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13, pages 87–132 (freely available on-line from Google Books here): Riemann's definition of the integral is given in section 4, "Über der Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit" (On the concept of a definite integral and the extent of its validity), pp. 101–103, and analyzes this paper. he also gave an example of a meagre set which is not negligible in the sense of measure theory, since its measure is not zero:See . a function which is everywhere continuous except on this set is not Riemann integrable.
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For many years there was a gap between these results: the solutions that could be constructed were known to have square integrable second derivatives, which was not quite strong enough to feed into the machinery that could prove they were analytic, which needed continuity of first derivatives. This gap was filled independently by , and . They were able to show the solutions had first derivatives that were Hölder continuous, which by previous results implied that the solutions are analytic whenever the differential equation has analytic coefficients, thus completing the solution of Hilbert's nineteenth problem.
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Here, the rapid-oscillation part of the integrand is taken into account via specialized methods for W_k, whereas the unknown function f(x) is usually better behaved. Another case where weight functions are especially useful is if the integrand is unknown but has a known singularity of some form, e.g. a known discontinuity or integrable divergence (such as 1/) at some point. In this case the singularity can be pulled into the weight function w(x) and its analytical properties can be used to compute W_k accurately beforehand.
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Like symplectic geometry, contact geometry has broad applications in physics, e.g. geometrical optics, classical mechanics, thermodynamics, geometric quantization, integrable systems and to control theory. Contact geometry also has applications to low-dimensional topology; for example, it has been used by Kronheimer and Mrowka to prove the property P conjecture, by Michael Hutchings to define an invariant of smooth three- manifolds, and by Lenhard Ng to define invariants of knots. It was also used by Yakov Eliashberg to derive a topological characterization of Stein manifolds of dimension at least six.
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As a result, while SWT is completely described by statistical methods, in DWT both integrable and chaotic dynamics are accounted for. A graphical representation of a resonant cluster of wave components is given by the corresponding NR-diagram (nonlinear resonance diagram). In some wave turbulent systems both discrete and statistical layers of turbulence are observed simultaneously, this wave turbulent regime have been described in and is called mesoscopic. Accordingly, three wave turbulent regimes can be singled out—kinetic, discrete and mesoscopic described by KZ-spectra, resonance clustering and their coexistence correspondingly.
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In the 1940s, Werner Heisenberg developed, independently, and substantiated the idea of the S-matrix. Because of the problematic divergences present in quantum field theory at that time, Heisenberg was motivated to isolate the essential features of the theory that would not be affected by future changes as the theory developed. In doing so, he was led to introduce a unitary "characteristic" S-matrix. Today, however, exact S-matrix results are a crowning achievement of conformal field theory, integrable systems, and several further areas of quantum field theory and string theory.
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A (properly speaking) nonlinear steepest descent method was introduced by Kamvissis, K. McLaughlin and P. Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and Zhou. As in the linear case, steepest descent contours solve a min-max problem. In the nonlinear case they turn out to be "S-curves" (defined in a different context back in the 80s by Stahl, Gonchar and Rakhmanov). The nonlinear stationary phase/steepest descent method has applications to the theory of soliton equations and integrable models, random matrices and combinatorics.
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In quantum mechanics, each physical system is associated with a Hilbert space. The approach codified by John von Neumann represents a measurement upon a physical system by a self- adjoint operator on that Hilbert space termed an “observable”. These observables play the role of measurable quantities familiar from classical physics: position, momentum, energy, angular momentum and so on. The dimension of the Hilbert space may be infinite, as it is for the space of square- integrable functions on a line, which is used to define the quantum physics of a continuous degree of freedom.
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In the mathematical theory of automorphic representations, a multiplicity-one theorem is a result about the representation theory of an adelic reductive algebraic group. The multiplicity in question is the number of times a given abstract group representation is realised in a certain space, of square- integrable functions, given in a concrete way. A multiplicity one theorem may also refer to a result about the restriction of a representation of a group G to a subgroup H. In that context, the pair (G, H) is called a strong Gelfand pair.
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Fourier's original formulation of the transform did not use complex numbers, but rather sines and cosines. Statisticians and others still use this form. An absolutely integrable function for which Fourier inversion holds good can be expanded in terms of genuine frequencies (avoiding negative frequencies, which are sometimes considered hard to interpret physically.) by :f(t) = \int_0^\infty \bigl( a(\lambda ) \cos( 2\pi \lambda t) + b(\lambda ) \sin( 2\pi \lambda t)\bigr) \, d\lambda. This is called an expansion as a trigonometric integral, or a Fourier integral expansion.
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Modified Korteweg–de Vries (MKdV) Equation , tosio.math.toronto.edu With these conservation laws, Miura showed a connection (called the Miura transformation) between solutions of the KdV and MKdV equations. This was a clue that enabled Kruskal, with Clifford S. Gardner, John M. Greene, and Miura (GGKM), to discover a general technique for exact solution of the KdV equation and understanding of its conservation laws. This was the inverse scattering method, a surprising and elegant method that demonstrates that the KdV equation admits an infinite number of Poisson- commuting conserved quantities and is completely integrable.
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A (properly speaking) nonlinear steepest descent method was introduced by Kamvissis, K. McLaughlin and P. Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and Zhou. As in the linear case, "steepest descent contours" solve a min-max problem. In the nonlinear case they turn out to be "S-curves" (defined in a different context back in the 80s by Stahl, Gonchar and Rakhmanov). The nonlinear stationary phase/steepest descent method has applications to the theory of soliton equations and integrable models, random matrices and combinatorics.
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Luigi Chierchia (born 1957) is an Italian mathematician, specializing in nonlinear differential equations, mathematical physics, and dynamical systems (celestial mechanics and Hamiltonian systems). Chierchia studied physics and mathematics at the Sapienza University of Rome with Laurea degree in 1981 with supervisor Giovanni Gallavotti. After a year of military service, Chierchia studied mathematics at the Courant Institute of New York University and received his PhD there in 1985. His doctoral dissseration Quasi-Periodic Schrödinger Operators in One Dimension, Absolutely Continuous Spectra, Bloch Waves and integrable Hamiltonian Systems was supervised by Henry P. McKean.
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13 (1960), 457–468. Yau proved new rigidity results for functions on complete Riemannian manifolds, for instance showing that if is a smooth and positive function on a complete Riemannian manifold, then together with the Lp integrability of implies that must be constant. Similarly, on a complete Kähler manifold, every holomorphic complex-valued function which is Lp-integrable must be constant. Via an extension of Hermann Weyl's differential identity used in the solution of the Weyl isometric embedding problem, Cheng and Yau produced novel rigidity theorems characterizing hypersurfaces of space forms by their intrinsic geometry.
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It now remains to construct a global (approximate) solution to the Schrödinger equation. For the wave function to be square-integrable, we must take only the exponentially decaying solution in the two classically forbidden regions. These must then "connect" properly through the turning points to the classically allowed region. For most values of E, this matching procedure will not work: The function obtained by connecting the solution near +\infty to the classically allowed region will not agree with the function obtained by connecting the solution near -\infty to the classically allowed region.
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The theory has applications to both ordinary and partial differential equations. A general solution approach uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras, and differential geometry are used to understand the structure of linear and nonlinear (partial) differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform, and finally finding exact analytic solutions to DE. Symmetry methods have been applied to differential equations that arise in mathematics, physics, engineering, and other disciplines.
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The Einstein–Brillouin–Keller method (EBK) is a semiclassical method (named after Albert Einstein, Léon Brillouin, and Joseph B. Keller) used to compute eigenvalues in quantum-mechanical systems. EBK quantization is an improvement from Bohr-Sommerfeld quantization which did not consider the caustic phase jumps at classical turning points. This procedure is able to reproduce exactly the spectrum of the 3D harmonic oscillator, particle in a box, and even the relativistic fine structure of the hydrogen atom. In 1976–1977, Berry and Tabor derived an extension to Gutzwiller trace formula for the density of states of an integrable system starting from EBK quantization.
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In 2009 he received the Academy Award of the Berlin-Brandenburg Academy of Sciences and Humanities and became a mathematical physics professor at Humboldt University of Berlin in 2010. Some of his publications have been instrumental in developing an understanding of the so-called AdS/CFT correspondence,Matthias Staudacher, "Integrable Spin Chains and the AdS/CFT Correspondence: Geometry and Physics after 100 Years of Einstein's Relativity", Potsdam, April 5–8, 2005. Retrieved 2010-02-05. a duality between the Yang-Mills-type quantum theory and supersymmetric string theory first suggested in the 1990s by Juan Martín Maldacena.
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Similarly, we can show that L^2(X;H), denoting the space of square integrable functions X\to H, is isomorphic to L^2(X)\otimes H if this space is separable. The isomorphism maps f(x)\otimes\phi\in L^2(X)\otimes H to f(x)\phi\in L^2(X;H) We can combine this with the previous example and conclude that L^2(X)\otimes L^2(Y) and L^2(X\times Y) are both isomorphic to L^2(X;L^2(Y)). Tensor products of Hilbert spaces arise often in quantum mechanics.
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This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential. Space, in this construction, still has the ordinary Euclidean geometry. However, spacetime as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable.
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Let π:E→M be a smooth fiber bundle over a smooth manifold M. The vertical bundle is the kernel VE := ker(dπ) of the tangent map dπ : TE → TM. (page 77) Since dπe is surjective at each point e, it yields a regular subbundle of TE. Furthermore, the vertical bundle VE is also integrable. An Ehresmann connection on E is a choice of a complementary subbundle HE to VE in TE, called the horizontal bundle of the connection. At each point e in E, the two subspaces form a direct sum, such that TeE = VeE ⊕ HeE.
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Nicolai Yuryevich Reshetikhin (, born October 10, 1958 in Leningrad, Soviet Union) is a mathematical physicist, currently a professor of mathematics at the University of California, Berkeley and a professor of mathematical physics at the University of Amsterdam. His research is in the fields of low- dimensional topology, representation theory, and quantum groups. His major contributions are in the theory of quantum integrable systems, in representation theory of quantum groups and in quantum topology. He and Vladimir Turaev constructed invariants of 3-manifolds which are expected to describe quantum Chern-Simons field theory introduced by Edward Witten.
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In the mathematics of the nineteenth century, aspects of generalized function theory appeared, for example in the definition of the Green's function, in the Laplace transform, and in Riemann's theory of trigonometric series, which were not necessarily the Fourier series of an integrable function. These were disconnected aspects of mathematical analysis at the time. The intensive use of the Laplace transform in engineering led to the heuristic use of symbolic methods, called operational calculus. Since justifications were given that used divergent series, these methods had a bad reputation from the point of view of pure mathematics.
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They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.
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The factorization of a linear partial differential operator (LPDO) is an important issue in the theory of integrability, due to the Laplace-Darboux transformations,Weiss (1986) which allow construction of integrable LPDEs. Laplace solved the factorization problem for a bivariate hyperbolic operator of the second order (see Hyperbolic partial differential equation), constructing two Laplace invariants. Each Laplace invariant is an explicit polynomial condition of factorization; coefficients of this polynomial are explicit functions of the coefficients of the initial LPDO. The polynomial conditions of factorization are called invariants because they have the same form for equivalent (i.e.
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Rathke stated, "However, while solutions of the Schrödinger equation with n<1 indeed exist, they are not square integrable. This violates not only an axiom of quantum mechanics, but in practical terms prohibits that these solutions can in any way describe the probability density of a particle." In the same year, the Journal of Applied Physics published a critique by A.V. Phelps of the 2004 article, "Water bath calorimetric study of excess heat generation in resonant transfer plasmas" by J. Phillips, R. Mills and X. Chen. Phelps criticized both the calorimetric techniques and the underlying theory described in the Phillips/Mills/Chen article.
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Zabusky and Kruskal argued that it was the fact that soliton solutions of the KdV equation can pass through one another without affecting the asymptotic shapes that explained the quasi-periodicity of the waves in the FPUT experiment. In short, thermalization could not occur because of a certain "soliton symmetry" in the system, which broke ergodicity. A similar set of manipulations (and approximations) lead to the Toda lattice, which is also famous for being a completely integrable system. It, too, has soliton solutions, the Lax pairs, and so also can be used to argue for the lack of ergodicity in the FPUT model.
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This larger symmetry group has since become known as the Ehlers group. Its discovery led to further generalizations, notably the infinite-dimensional Geroch group (the Geroch group is generated by two non-commuting subgroups, one of which is the Ehlers group). These so-called hidden symmetries play an important role in the Kaluza–Klein reduction of both general relativity and its generalizations, such as eleven-dimensional supergravity. Other applications include their use as a tool in the discovery of previously unknown solutions and their role in a proof that solutions in the stationary axi-symmetric case form an integrable system.
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Generalizing a theory of Bruns (1887), Poincaré showed that the three-body problem is not integrable. In other words, the general solution of the three-body problem can not be expressed in terms of algebraic and transcendental functions through unambiguous coordinates and velocities of the bodies. His work in this area was the first major achievement in celestial mechanics since Isaac Newton.J. Stillwell, Mathematics and its history, page 254 These monographs include an idea of Poincaré, which later became the basis for mathematical "chaos theory" (see, in particular, the Poincaré recurrence theorem) and the general theory of dynamical systems.
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He turned next to trigonometric functions with his 1903 paper "Sur les séries trigonométriques". He presented three major theorems in this work: that a trigonometrical series representing a bounded function is a Fourier series, that the nth Fourier coefficient tends to zero (the Riemann–Lebesgue lemma), and that a Fourier series is integrable term by term. In 1904-1905 Lebesgue lectured once again at the Collège de France, this time on trigonometrical series and he went on to publish his lectures in another of the "Borel tracts". In this tract he once again treats the subject in its historical context.
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Washek F. Pfeffer (born 1936) is a Czech-born US mathematician and Emeritus Professor at the University of California, Davis. Pfeffer is one of the world's pre-eminent authorities on real integration and has authored several books on the topic of integration, and numerous papers on these topics and others related to many areas of real analysis and measure theory. Pfeffer gave his name to the Pfeffer integral, which extends a Riemann-type construction for the integral of a measurable function both to higher-dimensional domains and, in the case of one dimension, to a superset of the Lebesgue integrable functions.
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More precisely, he demonstrated that the Hilbert space of states is always finite-dimensional and can be canonically identified with the space of conformal blocks of the G WZW model at level k. For example, when Σ is a 2-sphere, this Hilbert space is one-dimensional and so there is only one state. When Σ is a 2-torus the states correspond to the integrable representations of the affine Lie algebra corresponding to g at level k. Characterizations of the conformal blocks at higher genera are not necessary for Witten's solution of Chern–Simons theory.
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His work has included the solution of some outstanding problems, using techniques from combinatorics and probability theory (especially stopping times). In the theory of Hardy spaces, Carleson's contributions include the corona theorem (1962), and establishing the almost everywhere convergence of Fourier series for square-integrable functions (now known as Carleson's theorem). It was a famous old problem by Joseph Fourier when he invented Fourier analysis in 1807 and formalised by Nikolai Luzin in 1913 as the Lusin's conjecture. Kolmogorov proved a famous negative result of the conjecture for L1 function in 1928 and stated that the conjecture must be false.
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The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space M the dynamical system is integrable. In most cases the patch cannot be extended to the entire phase space. There may be singular points in the vector field (where v(x) = 0); or the patches may become smaller and smaller as some point is approached. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one.
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Null sets play a key role in the definition of the Lebesgue integral: if functions f and g are equal except on a null set, then f is integrable if and only if g is, and their integrals are equal. A measure in which all subsets of null sets are measurable is complete. Any non-complete measure can be completed to form a complete measure by asserting that subsets of null sets have measure zero. Lebesgue measure is an example of a complete measure; in some constructions, it is defined as the completion of a non-complete Borel measure.
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In mathematics, in particular in the field of the representation theory of groups, a representative function is a function f on a compact topological group G obtained by composing a representation of G on a vector space V with a linear map from the endomorphisms of V into V 's underlying field. Representative functions arise naturally from finite-dimensional representations of G as the matrix-entry functions of the corresponding matrix representations. It follows from the Peter–Weyl theorem that the representative functions on G are dense in the Hilbert space of square- integrable functions on G.
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Murtazin was born in the village Aznash in Uchalinsky District, now in Bashkortostan. He graduated from the Department of Mathematics of Bashkir State University and defended his doctoral thesis in 1994. Since 1978 until the present day he is the head of the Mathematical Analysis chair of the department Scientific activity is devoted to problems of quantum mechanics. Murtazin investigated the asymptotic behavior of the discrete spectrum of the Schrödinger operator, the spectrum of perturbations of partial differential operators, results on the two-particle operators in the class of integrable potentials, conditions for the existence of virtual particles 4.
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It is a generalization and refinement of Fourier analysis, for the case when the signal frequency characteristics are varying with time. Since many signals of interest – such as speech, music, images, and medical signals – have changing frequency characteristics, time–frequency analysis has broad scope of applications. Whereas the technique of the Fourier transform can be extended to obtain the frequency spectrum of any slowly growing locally integrable signal, this approach requires a complete description of the signal's behavior over all time. Indeed, one can think of points in the (spectral) frequency domain as smearing together information from across the entire time domain.
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These new transcendental functions, solving the remaining six equations, are called the Painlevé transcendents, and interest in them has revived recently due to their appearance in modern geometry, integrable systems Ablowitz, M. J. and Clarkson, P.A. (1991) Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press and statistical mechanics. In 1895 he gave a series of lectures at Stockholm University on differential equations, at the end stating the Painlevé conjecture about singularities of the n-body problem. In the 1920s, Painlevé briefly turned his attention to the new theory of gravitation, general relativity, which had recently been introduced by Albert Einstein.
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In mathematics, the equations governing the isomonodromic deformation of meromorphic linear systems of ordinary differential equations are, in a fairly precise sense, the most fundamental exact nonlinear differential equations. As a result, their solutions and properties lie at the heart of the field of exact nonlinearity and integrable systems. Isomonodromic deformations were first studied by Richard Fuchs, with early pioneering contributions from Lazarus Fuchs, Paul Painlevé, René Garnier, and Ludwig Schlesinger. Inspired by results in statistical mechanics, a seminal contribution to the theory was made by Michio Jimbo, Tetsuji Miwa and Kimio Ueno, who studied cases with arbitrary singularity structure.
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Antoni Zygmund wrote a classic two-volume set of books entitled Trigonometric Series, which discusses many different aspects of trigonometric series. The first edition was a single volume, published in 1935 (under the slightly different title Trigonometrical Series). The second edition of 1959 was greatly expanded, taking up two volumes, though it was later reprinted as a single volume paperback. The third edition of 2002 is similar to the second edition, with the addition of a preface by Robert A. Fefferman on more recent developments, in particular Carleson's theorem about almost everywhere pointwise convergence for square integrable functions.
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In the same year with Paul Dupuis, they established the necessary Sobolev smoothness conditions requiring vector fields to have strictly greater than 2.5 square-integrable, generalized derivatives (in the space of 3-dimensions) to ensure that smooth submanifold shapes are carried smoothly via integration of the flows. The Computational anatomy framework via diffeomorphisms at the 1mm morphological scale is one of the de facto standards for cross-section analyses of populations. Codes now exist for diffeomorphic template or atlas mapping, including ANTS, DARTEL, DEMONS, LDDMM, StationaryLDDMM, all actively used codes for constructing correspondences between coordinate systems based on sparse features and dense images.
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In mathematics, the Calderón–Zygmund lemma is a fundamental result in Fourier analysis, harmonic analysis, and singular integrals. It is named for the mathematicians Alberto Calderón and Antoni Zygmund. Given an integrable function , where denotes Euclidean space and denotes the complex numbers, the lemma gives a precise way of partitioning into two sets: one where is essentially small; the other a countable collection of cubes where is essentially large, but where some control of the function is retained. This leads to the associated Calderón–Zygmund decomposition of , wherein is written as the sum of "good" and "bad" functions, using the above sets.
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Although all bounded piecewise continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly in the context of Fourier analysis—to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure theory (a subfield of real analysis). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the standard part of an infinite Riemann sum, based on the hyperreal number system.
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If the spectrum of the operator is reduced to one single eigenvalue, its corresponding motion is that of a single bump that propagates at constant velocity and without deformation, a solitary wave called a "soliton". A perfect signal and its generalizations for the Korteweg–de Vries equation or other integrable nonlinear partial differential equations are of great interest, with many possible applications. This area has been studied as a branch of mathematical physics since the 1970s. Nonlinear inverse problems are also currently studied in many fields of applied science (acoustics, mechanics, quantum mechanics, electromagnetic scattering - in particular radar soundings, seismic soundings, and nearly all imaging modalities).
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The old quantum theory was instigated by the 1900 work of Max Planck on the emission and absorption of light, and began in earnest after the work of Albert Einstein on the specific heats of solids. Einstein, followed by Debye, applied quantum principles to the motion of atoms, explaining the specific heat anomaly. In 1913, Niels Bohr identified the correspondence principle and used it to formulate a model of the hydrogen atom which explained the line spectrum. In the next few years Arnold Sommerfeld extended the quantum rule to arbitrary integrable systems making use of the principle of adiabatic invariance of the quantum numbers introduced by Lorentz and Einstein.
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The concept of function spaces enters naturally in the discussion about wave functions. A function space is a set of functions, usually with some defining requirements on the functions (in the present case that they are square integrable), sometimes with an algebraic structure on the set (in the present case a vector space structure with an inner product), together with a topology on the set. The latter will sparsely be used here, it is only needed to obtain a precise definition of what it means for a subset of a function space to be closed. It will be concluded below that the function space of wave functions is a Hilbert space.
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Yvette Kosmann-Schwarzbach (born 30 April 1941)Birth date from Library of Congress and French National Library, retrieved 2019-10-13 is a French mathematician and professor. She has been teaching mathematics at the Lille University of Science and Technology and at the École polytechnique since 1993. Kosmann-Schwarzbach obtained her doctoral degree in 1970 at the University of Paris under supervision of André Lichnerowicz on a dissertation titled Dérivées de Lie des spineurs (Lie derivatives of spinors). She is the author of over fifty articles on differential geometry, algebra and mathematical physics, as well as the co-editor of several books concerning the theory of integrable systems.
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The point process depends on a single mathematical object, which, depending on the context, may be a constant, a locally integrable function or, in more general settings, a Radon measure. In the first case, the constant, known as the rate or intensity, is the average density of the points in the Poisson process located in some region of space. The resulting point process is called a homogeneous or stationary Poisson point process. In the second case, the point process is called an inhomogeneous or nonhomogeneous Poisson point process, and the average density of points depend on the location of the underlying space of the Poisson point process.
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Peter David Lax (born Lax Péter Dávid; 1 May 1926) is a Hungarian-born American mathematician working in the areas of pure and applied mathematics. Lax has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields. In a 1958 paper Lax stated a conjecture about matrix representations for third order hyperbolic polynomials which remained unproven for over four decades. Interest in the "Lax conjecture" grew as mathematicians working in several different areas recognized the importance of its implications in their field, until it was finally proven to be true in 2003.
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The incorporation of radiation corrections was difficult, because it required finding action-angle coordinates for a combined radiation/atom system, which is difficult when the radiation is allowed to escape. The whole theory did not extend to non-integrable motions, which meant that many systems could not be treated even in principle. In the end, the model was replaced by the modern quantum mechanical treatment of the hydrogen atom, which was first given by Wolfgang Pauli in 1925, using Heisenberg's matrix mechanics. The current picture of the hydrogen atom is based on the atomic orbitals of wave mechanics which Erwin Schrödinger developed in 1926.
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In functional analysis, the Dunford–Pettis property, named after Nelson Dunford and B. J. Pettis, is a property of a Banach space stating that all weakly compact operators from this space into another Banach space are completely continuous. Many standard Banach spaces have this property, most notably, the space C(K) of continuous functions on a compact space and the space L1(μ) of the Lebesgue integrable functions on a measure space. Alexander Grothendieck introduced the concept in the early 1950s , following the work of Dunford and Pettis, who developed earlier results of Shizuo Kakutani, Kōsaku Yosida, and several others. Important results were obtained more recently by Jean Bourgain.
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His research areas include gauge field theories, supersymmetry, quantum algebra, integrable systems and combinatorics. At the Université de Montréal, Vinet held the position of director of the Centre de recherches mathématiques (CRM) from 1993 to 1999. During his term as director, the CRM succeeded in rallying the forces of quantitative research by forming a network of centers of excellence in computing from the association of seven major Montreal research centers (CERCA, CIRANO, CRIM, CRM, CRI, GERAD and INRS–Télécom) under the banner of the Network for Computing and Mathematical Modeling (NCM2). The research network provides "one-stop" access to expertise calculation and modeling for more than 20 partner enterprises.
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Dolan is responsible for several important discoveries which have furthered the study of elementary particle physics. She co-authored "Symmetry Behavior at Finite Temperature", now regularly cited, in 1974.Phys. Rev. D 9, 3320-41 (1974), Symmetry behavior at finite temperature This paper became a part of the foundation of quantitative analysis of phase transitions in the early universe in cosmological theories and is widely recognized as a seminal work. In 1981 she pioneered the uses of affine algebras in particle physics and her coruscant contributions to string theory have included symmetries in the Type II superstring and integrable structures in super conformal non-abelian gauge theories.
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In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. There are several inequivalent definitions of Baire sets, but in the most widely used, the Baire sets of a locally compact Hausdorff space form the smallest σ-algebra such that all compactly supported continuous functions are measurable. Thus, measures defined on this σ-algebra, called Baire measures, are a convenient framework for integration on locally compact Hausdorff spaces. In particular, any compactly supported continuous function on such a space is integrable with respect to any finite Baire measure.
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It is also known to explicitly fail in certain integrable systems, in which the presence of a large number of constants of motion prevent thermalization. It is also important to note that the ETH makes statements about specific observables on a case by case basis - it does not make any claims about whether every observable in a system will obey ETH. In fact, this certainly cannot be true. Given a basis of energy eigenstates, one can always explicitly construct an operator which violates the ETH, simply by writing down the operator as a matrix in this basis whose elements explicitly do not obey the conditions imposed by the ETH.
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In quantum mechanics, the 1D NLSE is a special case of the classical nonlinear Schrödinger field, which in turn is a classical limit of a quantum Schrödinger field. Conversely, when the classical Schrödinger field is canonically quantized, it becomes a quantum field theory (which is linear, despite the fact that it is called ″quantum nonlinear Schrödinger equation″) that describes bosonic point particles with delta-function interactions — the particles either repel or attract when they are at the same point. In fact, when the number of particles is finite, this quantum field theory is equivalent to the Lieb–Liniger model. Both the quantum and the classical 1D nonlinear Schrödinger equations are integrable.
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A robot is holonomic system if all the constraints that it is subjected to are integrable into positional constraints of the form: The variables q_i are the system coordinates. When a system contains constraints that cannot be written in this form, it is said to be nonholonomic. In simpler terms, a holonomic system is when the number of controllable degrees of freedom is equal to the total degrees of freedom. e.g. A holonomic robot can drive straight to a goal that is not in-line with its orientation, where as a non-holonomic robot must either rotate to the desired orientation before moving forward or rotate as it moves.
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Given an oriented "contour" Σ (technically: an oriented union of smooth curves without points of infinite self-intersection in the complex plane). A Birkhoff factorization problem is the following. Given a matrix function V defined on the contour Σ, to find a holomorphic matrix function M defined on the complement of Σ, such that two conditions be satisfied: # If M+ and M− denote the non-tangential limits of M as we approach Σ, then M+ = M−V, at all points of non-intersection in Σ. #As z tends to infinity along any direction outside Σ, M tends to the identity matrix. In the simplest case V is smooth and integrable.
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The exact nature of this Hilbert space is dependent on the system – for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes. Each observable is represented by a maximally Hermitian (precisely: by a self-adjoint) linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the operator's spectrum is discrete, the observable can attain only those discrete eigenvalues.
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These conditions can be local (like demanding that the sections in the domain vanish at the boundary) or more complicated global conditions (like requiring that the sections in the domain solve some differential equation). The local case was worked out by Atiyah and Bott, but they showed that many interesting operators (e.g., the signature operator) do not admit local boundary conditions. To handle these operators, Atiyah, Patodi and Singer introduced global boundary conditions equivalent to attaching a cylinder to the manifold along the boundary and then restricting the domain to those sections that are square integrable along the cylinder, and also introduced the Atiyah–Patodi–Singer eta invariant.
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Chaos-assisted tunneling oscillations between two regular tori embedded in a chaotic sea, seen in phase space In real life, most system are not integrable and displays various degree of chaos. Classical dynamics is then said to be mixed and the system phase space is typically composed of islands of regular orbits surrounded by a large sea of chaotic orbits. The existence of the chaotic sea, where transport is classically allowed, between the two symmetric tori then assists the quantum tunneling between them. This phenomenon is referred as chaos-assisted tunneling and is characterized by sharp resonances of the tunneling rate when varying any parameter of the system.
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It is sometimes called the nucleus of the integral, whence the term nuclear operator arises. In the general theory, and may be points on any manifold; the real number line or -dimensional Euclidean space in the simplest cases. The general theory also often requires that the functions belong to some given function space: often, the space of square-integrable functions is studied, and Sobolev spaces appear often. The actual function space used is often determined by the solutions of the eigenvalue problem of the differential operator; that is, by the solutions to :L\psi_n(x)=\omega_n \psi_n(x) where the are the eigenvalues, and the are the eigenvectors.
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McCoy in 2002 Barry Malcolm McCoy (born 14 December 1940 in Trenton, New Jersey) is an American physicist, known for his contributions to classical statistical mechanics, integrable models and conformal field theories. He earned a B.Sc. from California Institute of Technology (1963), and a Ph.D. from Harvard University (1967), the thesis entitled Spin Correlations of the Two Dimensional Ising Model advised by Tai Tsun Wu.. The two of them also wrote the book The Two Dimensional Ising Model (Harvard University Press, 1973). He then joined the institute for theoretical physics at State University of New York at Stony Brook (1967). where he has since been, now as a distinguished professor.
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With Boris Chirikov, Italo Guarneri and Dima Shepelyansky Casati also discovered that quantum localization deeply affects the excitation of hydrogen atom in strong monochromatic fields. Further major contributions considered the connections between quantization of non integrable systems and the statistical theory of spectra. With the advent of quantum computing Casati and his coworkers studied the efficient quantum computing of complex dynamics.G. Benenti, G. Casati and G. Strini, Principles of quantum computation and information, Volume I: Basic concepts (World Scientific, Singapore, 2004).G. Benenti, G. Casati and G. Strini, Principles of quantum computation and information, Volume II: Basic tools and special topics (World Scientific, Singapore, 2007).
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Jointly with Boris Feigin, Frenkel constructed the free field realizations of affine Kac–Moody algebras (these are also known as Wakimoto modules), defined the quantum Drinfeld-Sokolov reduction, and described the center of the universal enveloping algebra of an affine Kac–Moody algebra. The last result, often referred to as Feigin–Frenkel isomorphism, has been used by Alexander Beilinson and Vladimir Drinfeld in their work on the geometric Langlands correspondence. Together with Nicolai Reshetikhin, Frenkel introduced deformations of W-algebras and q-characters of representations of quantum affine algebras. Frenkel's recent work has focused on the Langlands program and its connections to representation theory, integrable systems, geometry, and physics.
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Moeglin is a Directeur de recherche at the Centre national de la recherche scientifique and is currently working at the Institut de mathématiques de Jussieu. She was a speaker at the 1990 International Congress of Mathematicians, on decomposition into distinguished subspaces of certain spaces of square-integral automorphic forms. She was a recipient of the Jaffé prize of the French Academy of Sciences in 2004, "for her work, most notably on the topics of enveloping algebras of Lie algebras, automorphic forms and the classification of square- integrable representations of reductive classical p-adic groups by their cuspidal representations". She was the chief editor of the Journal of the Institute of Mathematics of Jussieu from 2002 to 2006.
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A key aspect of GFLM is estimation and inference for the smooth parameter function \beta which is usually obtained by dimension reduction of the infinite dimensional functional predictor. A common method is to expand the predictor function X in an orthonormal basis of L2 space, the Hilbert space of square integrable functions with the simultaneous expansion of the parameter function in the same basis. This representation is then combined with a truncation step to reduce the contribution of the parameter function \beta in the linear predictor to a finite number of regression coefficients. Functional principal component analysis (FPCA) that employs the Karhunen–Loève expansion is a common and parsimonious approach to accomplish this.
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With Jimbō he then examined general isomonodromic deformations of linear differential equations. (This mathematical approach to linear differential equations was begun during the early years of the 20th century by Ludwig Schlesinger.) Miwa studied, with Jimbō and Etsuro Date, the role of affine Lie algebras in soliton equations and, with Jimbō, the role of quantum groups in exactly solvable grid models of statistical mechanics. Miwa and Michio Jimbō were jointly awarded in 1987 the autumn prize of the Mathematical Society of Japan and in 1999 the Asahi Prize. In 1986 he was an Invited Speaker with talk Integrable lattice models and branching coefficients at the International Congress of Mathematicians (ICM) in Berkeley.
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Yang Chen-Ning or Yang Zhenning (; born October 1, 1922) is a Chinese theoretical physicist who made significant contributions to statistical mechanics, integrable systems, gauge theory, and both particle physics and condensed matter physics. He and Tsung-Dao Lee received the 1957 Nobel Prize in Physics for their work on parity nonconservation of weak interaction. The two proposed that one of the basic quantum-mechanics laws, the conservation of parity, is violated in the so-called weak nuclear reactions, those nuclear processes that result in the emission of beta or alpha particles. Yang is also well known for his collaboration with Robert Mills in developing non-Abelian gauge theory, widely known as the Yang–Mills theory.
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In 1978 Dekkers had shown that for systems of size 2 Plemelj's claim is true. and independently showed that for any size, an irreducible monodromy group can be realised by a Fuchsian system. The codimension of the variety of monodromy groups of regular systems of size n with p+1 poles which cannot be realised by Fuchsian systems equals 2(n-1)p ().) Parallel to this the Grothendieck school of algebraic geometry had become interested in questions of 'integrable connections on algebraic varieties', generalising the theory of linear differential equations on Riemann surfaces. Pierre Deligne proved a precise Riemann–Hilbert correspondence in this general context (a major point being to say what 'Fuchsian' means).
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A continuous function fails to be absolutely continuous if it fails to be uniformly continuous, which can happen if the domain of the function is not compact – examples are tan(x) over [0, /2), x2 over the entire real line, and sin(1/x) over (0, 1]. But a continuous function f can fail to be absolutely continuous even on a compact interval. It may not be "differentiable almost everywhere" (like the Weierstrass function, which is not differentiable anywhere). Or it may be differentiable almost everywhere and its derivative f ′ may be Lebesgue integrable, but the integral of f ′ differs from the increment of f (how much f changes over an interval).
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This can be recast in terms of the Hilbert space consisting of functions such that , along with its weak partial derivatives, are square integrable on , and vanish on the boundary. The question then reduces to finding in this space such that for all in this space : a(u, v) = b(v) where is a continuous bilinear form, and is a continuous linear functional, given respectively by :a(u, v) = \int_\Omega abla u\cdot abla v,\quad b(v)= \int_\Omega gv\,. Since the Poisson equation is elliptic, it follows from Poincaré's inequality that the bilinear form is coercive. The Lax–Milgram theorem then ensures the existence and uniqueness of solutions of this equation.
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When a finite-dimensional Hamiltonian system is completely integrable in the Liouville sense, and the energy level sets are compact, the flows are complete, and the leaves of the invariant foliation are tori. There then exist, as mentioned above, special sets of canonical coordinates on the phase space known as action-angle variables, such that the invariant tori are the joint level sets of the action variables. These thus provide a complete set of invariants of the Hamiltonian flow (constants of motion), and the angle variables are the natural periodic coordinates on the torus. The motion on the invariant tori, expressed in terms of these canonical coordinates, is linear in the angle variables.
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Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.
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In this case Σ consists of finitely many points. To a single point we associate a vector space V = Z(point) and to n-points the n-fold tensor product: V⊗n = V ⊗ … ⊗ V. The symmetric group Sn acts on V⊗n. A standard way to get the quantum Hilbert space is to start with a classical symplectic manifold (or phase space) and then quantize it. Let us extend Sn to a compact Lie group G and consider "integrable" orbits for which the symplectic structure comes from a line bundle, then quantization leads to the irreducible representations V of G. This is the physical interpretation of the Borel–Weil theorem or the Borel–Weil–Bott theorem.
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His main contributions to those topics and are the papers and . In the first one he proves that a condition on a sequence of integrable functions previously introduced by Mauro Picone is both necessary and sufficient in order to assure that limit process and the integration process commute, both in bounded and unbounded domains: the theorem is similar in spirit to the dominated convergence theorem, which however only states a sufficient condition. The second paper contains an extension of the Lebesgue's decomposition theorem to finitely additive measures: this extension required him to generalize the Radon–Nikodym derivative, requiring it to be a set function belonging to a given class and minimizing a particular functional.
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Ludvig Dmitrievich Faddeev (also Ludwig Dmitriyevich; ; 23 March 1934 – 26 February 2017) was a Soviet and Russian theoretical physicist and mathematician. He is known for the discovery of the Faddeev equations in the theory of the quantum mechanical three-body problem and for the development of path integral methods in the quantization of non-abelian gauge field theories, including the introduction (with Victor Popov) of Faddeev–Popov ghosts. He led the Leningrad School, in which he along with many of his students developed the quantum inverse scattering method for studying quantum integrable systems in one space and one time dimension. This work led to the invention of quantum groups by Drinfeld and Jimbo.
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It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t, the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows. Defined on the real line, the Poisson process can be interpreted as a stochastic process, among other random objects.
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Kandrup's work led to greater understanding in the fields of stellar dynamics, chaos, and plasma physics. Much of Kandrup's research was directed toward developing a more refined mathematical description of dynamical relaxation in stellar systems. In a series of papers from the early 1990s, Henry developed the idea of chaotic phase mixing, the process by which an ensemble of points evolves toward a uniform coarse-grained population of phase space. Among his other contributions were a demonstration of the equivalence of Landau damping and phase mixing; a proof (with J. F. Sygnet) of the linear stability of a broad class of stellar systems; and a generalization of Jeans's theorem to non-integrable systems.
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The most celebrated example is the theorem of on the distribution of the length of the longest increasing subsequence of a random permutation. Together with the study of B above, it is one of the original rigorous investigations of so-called "integrable probability". But the connection between the theory of integrability and various classical ensembles of random matrices goes back to the work of Dyson (e.g.). In particular, Riemann–Hilbert factorization problems are used to extract asymptotics for the three problems above (say, as time goes to infinity, or as the dispersion coefficient goes to zero, or as the polynomial degree goes to infinity, or as the size of the permutation goes to infinity).
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In collaboration with Bott and Lars Gårding, Atiyah wrote three papers updating and generalizing Petrovsky's work. Atiyah showed how to extend the index theorem to some non- compact manifolds, acted on by a discrete group with compact quotient. The kernel of the elliptic operator is in general infinite-dimensional in this case, but it is possible to get a finite index using the dimension of a module over a von Neumann algebra; this index is in general real rather than integer valued. This version is called the L2 index theorem, and was used by Atiyah and Schmid to give a geometric construction, using square integrable harmonic spinors, of Harish-Chandra's discrete series representations of semisimple Lie groups.
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The memoir pointed out Cauchy's mistake and introduced Dirichlet's test for the convergence of series. It also introduced the Dirichlet function as an example of a function that is not integrable (the definite integral was still a developing topic at the time) and, in the proof of the theorem for the Fourier series, introduced the Dirichlet kernel and the Dirichlet integral. Dirichlet also studied the first boundary value problem, for the Laplace equation, proving the uniqueness of the solution; this type of problem in the theory of partial differential equations was later named the Dirichlet problem after him. A function satisfying a partial differential equation subject to the Dirichlet boundary conditions must have fixed values on the boundary.
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These terms can be replaced by dots, lines, squiggles and similar marks, each standing for a term, a denominator, an integral, and so on; thus complex integrals can be written as simple diagrams, with absolutely no ambiguity as to what they mean. The one-to-one correspondence between the diagrams, and specific integrals is what gives them their power. Although originally developed for quantum field theory, it turns out the diagrammatic technique is broadly applicable to all perturbative series (although, perhaps, not always so useful). In the second half of the 20th century, as chaos theory developed, it became clear that unperturbed systems were in general completely integrable systems, while the perturbed systems were not.
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In mathematics, Anderson's theorem is a result in real analysis and geometry which says that the integral of an integrable, symmetric, unimodal, non- negative function f over an n-dimensional convex body K does not decrease if K is translated inwards towards the origin. This is a natural statement, since the graph of f can be thought of as a hill with a single peak over the origin; however, for n ≥ 2, the proof is not entirely obvious, as there may be points x of the body K where the value f(x) is larger than at the corresponding translate of x. Anderson's theorem also has an interesting application to probability theory.
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Antiderivatives can be used to compute definite integrals, using the fundamental theorem of calculus: if is an antiderivative of the integrable function over the interval [a,b], then: :\int_a^b f(x)\,dx = F(b) - F(a). Because of this, each of the infinitely many antiderivatives of a given function is sometimes called the "general integral" or "indefinite integral" of f, and is written using the integral symbol with no bounds: :\int f(x)\, dx. If is an antiderivative of , and the function is defined on some interval, then every other antiderivative of differs from by a constant: there exists a number such that G(x) = F(x)+c for all . is called the constant of integration.
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Suppose f : [a, b] → R is continuous and g is a nonnegative integrable function on [a, b]. By the extreme value theorem, there exists m and M such that for each x in [a, b], m\leqslant f(x) \leqslant M and f[a,b] = [m, M]. Since g is nonnegative, :m \int_a^b g(x) \, dx \leqslant \int^b_a f(x)g(x) \, dx \leqslant M \int_a^b g(x) \, dx. Now let :I = \int_a^b g(x) \, dx. If I = 0, we're done since :0 \leqslant \int_a^b f(x) g(x)\, dx \leqslant 0 means :\int_a^b f(x)g(x)\, dx=0, so for any c in (a, b), :\int_a^b f(x)g(x)\, dx = f(c) I = 0.
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The generalized functional linear model (GFLM) is an extension of the generalized linear model (GLM) that allows one to regress univariate responses of various types (continuous or discrete) on functional predictors, which are mostly random trajectories generated by a square-integrable stochastic processes. Similarly to GLM, a link function relates the expected value of the response variable to a linear predictor, which in case of GFLM is obtained by forming the scalar product of the random predictor function X with a smooth parameter function \beta . Functional Linear Regression, Functional Poisson Regression and Functional Binomial Regression, with the important Functional Logistic Regression included, are special cases of GFLM. Applications of GFLM include classification and discrimination of stochastic processes and functional data.
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A complete metric space along with the additional structure of an inner product (a conjugate symmetric sesquilinear form) is known as a Hilbert space, which is in some sense a particularly well-behaved Banach space. Functional analysis applies the methods of linear algebra alongside those of mathematical analysis to study various function spaces; the central objects of study in functional analysis are Lp spaces, which are Banach spaces, and especially the L2 space of square integrable functions, which is the only Hilbert space among them. Functional analysis is of particular importance to quantum mechanics, the theory of partial differential equations, digital signal processing, and electrical engineering. It also provides the foundation and theoretical framework that underlies the Fourier transform and related methods.
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Theorem. Given a Riemann surface X and two distinct points A and B on X, there is a holomorphic 1-form on X with simple poles at the two points with non-zero residues having sum zero such that the 1-form is square integrable on the complement of any open neighbourhoods of the two points. The proof is similar to the proof of the result on holomorphic 1-forms with a single double pole. The result is first proved when A and B are close and lie in a parametric disk. Indeed, once this is proved, a sum of 1-forms for a chain of sufficiently close points between A and B will provide the required 1-form, since the intermediate singular terms will cancel.
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In quantum chaos, a branch of mathematical physics, quantum ergodicity is a property of the quantization of classical mechanical systems that are chaotic in the sense of exponential sensitivity to initial conditions. Quantum ergodicity states, roughly, that in the high-energy limit, the probability distributions associated to energy eigenstates of a quantized ergodic Hamiltonian tend to a uniform distribution in the classical phase space. This is consistent with the intuition that the flows of ergodic systems are equidistributed in phase space. By contrast, classical completely integrable systems generally have periodic orbits in phase space, and this is exhibited in a variety of ways in the high-energy limit of the eigenstates: typically that some form of concentration or "scarring" occurs in the limit.
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The theory of distributions (generalized functions) eliminates analytic problems with the symmetry. The derivative of an integrable function can always be defined as a distribution, and symmetry of mixed partial derivatives always holds as an equality of distributions. The use of formal integration by parts to define differentiation of distributions puts the symmetry question back onto the test functions, which are smooth and certainly satisfy this symmetry. In more detail (where f is a distribution, written as an operator on test functions, and φ is a test function), : \left(D_1 D_2 f\right)[\phi] = -\left(D_2f\right)\left[D_1\phi\right] = f\left[D_2 D_1\phi\right] = f\left[D_1 D_2\phi\right] = -\left(D_1 f\right)\left[D_2\phi\right] = \left(D_2 D_1 f\right)[\phi].
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This provides, in certain cases, enough invariants, or "integrals of motion" to make the system completely integrable. In the case of systems having an infinite number of degrees of freedom, such as the KdV equation, this is not sufficient to make precise the property of Liouville integrability. However, for suitably defined boundary conditions, the spectral transform can, in fact, be interpreted as a transformation to completely ignorable coordinates, in which the conserved quantities form half of a doubly infinite set of canonical coordinates, and the flow linearizes in these. In some cases, this may even be seen as a transformation to action-angle variables, although typically only a finite number of the "position" variables are actually angle coordinates, and the rest are noncompact.
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From the seriesM. Henkel and J. Lack, preprint Bonn-He- 85–22 the order parameter is conjecturedAlbertini G., McCoy B. M., Perk J. H. H. and Tang S. (1989), "Excitation spectrum and order parameter for the integrable N-state chiral Potts model", Nuclear Physics B 314, 741–763 to have the simple form :\langle \sigma^n\rangle=(1-k'^2)^\beta,\quad \beta=n(N-n)/2N^2. It took many years to prove this conjecture, as the usual corner transfer matrix technique could not be used, because of the higher genus curve. This conjecture was finally proven by Baxter in 2005Baxter R. J. (2005), "Derivation of the order parameter of the chiral Potts model", Physical Review Letters, 94 130602 (3 pp) arXiv:cond- mat/0501227.
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In mathematics and theoretical physics, fusion rules are rules that determine the exact decomposition of the tensor product of two representations of a group into a direct sum of irreducible representations. The term is often used in the context of two-dimensional conformal field theory where the relevant group is generated by the Virasoro algebra, the relevant representations are the conformal families associated with a primary field and the tensor product is realized by operator product expansions. The fusion rules contain the information about the kind of families that appear on the right hand side of these OPEs, including the multiplicities. More generally, integrable models in 2 dimensions which aren't conformal field theories are also described by fusion rules for their charges.
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One infinite-dimensional generalization is as follows. Let and be Banach spaces, and a pair of open sets. Let :F:A\times B \to L(X,Y) be a continuously differentiable function of the Cartesian product (which inherits a differentiable structure from its inclusion into X × Y) into the space of continuous linear transformations of into Y. A differentiable mapping u : A → B is a solution of the differential equation :(1) \quad y' = F(x,y) if :\forall x \in A: \quad u'(x) = F(x, u(x)). The equation (1) is completely integrable if for each (x_0, y_0)\in A\times B, there is a neighborhood U of x0 such that (1) has a unique solution defined on U such that u(x0)=y0.
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Calogero's scientific publications in English include five books and over 400 papers (about half with co-authors). His main research concerns integrable many-body problems. Several solvable many-body models and nonlinear evolution partial differential equations (PDEs) are named after Calogero in the mathematical physics literature. He also formulated the Calogero conjecture that quantum behavior is caused by the stochastic component of the local gravitational field due to the chaotic component of the motion of all particles of the Universe due to their mutual gravitational interaction.. He also introduced a novel differential algorithm to evaluate all the zeros of any generic polynomial of arbitrary degree [F. Calogero, “Novel differential algorithm to evaluate all the zeros of any generic polynomial”, J. Nonlinear Math. Phys.
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In this case, the orthonormal basis is sometimes called a Hilbert basis for H. Note that an orthonormal basis in this sense is not generally a Hamel basis, since infinite linear combinations are required. Specifically, the linear span of the basis must be dense in H, but it may not be the entire space. If we go on to Hilbert spaces, a non-orthonormal set of vectors having the same linear span as an orthonormal basis may not be a basis at all. For instance, any square-integrable function on the interval [−1, 1] can be expressed (almost everywhere) as an infinite sum of Legendre polynomials (an orthonormal basis), but not necessarily as an infinite sum of the monomials xn.
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The theory continues by introducing the concept of reducible measure, meaning that the quotient ρ/μ is element of L2(I, R, μ). The following results are then established: The reducer φ of ρ is an antecedent of ρ/μ for the operator Tρ. (In fact the only antecedent which belongs to Hρ). For any function square integrable for ρ, there is an equality known as the reducing formula: : \langle f/\varphi \rangle_\rho = \langle T_\rho (f)/1 \rangle_\rho. The operator :f\mapsto \varphi\times f -T_\rho (f) defined on the polynomials is prolonged in an isometry Sρ linking the closure of the space of these polynomials in L2(I, R, ρ2μ−1) to the hyperplane Hρ provided with the norm induced by ρ.
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Vladimir Igorevich Arnold (alternative spelling Arnol'd, , 12 June 1937 – 3 June 2010)Mort d'un grand mathématicien russe, AFP (Le Figaro) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, he made important contributions in several areas including dynamical systems theory, algebra, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics and singularity theory, including posing the ADE classification problem, since his first main result—the solution of Hilbert's thirteenth problem in 1957 at the age of 19. He co-founded two new branches of mathematics—KAM theory, and topological Galois theory (this, with his student Askold Khovanskii). Arnold was also known as a popularizer of mathematics.
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Define f to be holomorphic if it is analytic at each point in its domain. Osgood's lemma shows (using the multivariate Cauchy integral formula) that, for a continuous function f, this is equivalent to f being holomorphic in each variable separately (meaning that if any coordinates are fixed, then the restriction of f is a holomorphic function of the remaining coordinate). The much deeper Hartogs' theorem proves that the continuity hypothesis is unnecessary: f is holomorphic if and only if it is holomorphic in each variable separately. More generally, a function of several complex variables that is square integrable over every compact subset of its domain is analytic if and only if it satisfies the Cauchy–Riemann equations in the sense of distributions.
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In his Théorie Analytique de la Chaleur, Fourier claimed that an arbitrary function could be represented by a Fourier series.Contemporary mathematicians, with much broader and more precise conceptions of functions, integration, and different notions of convergence than was possible in Fourier's time (including examples of functions that were regarded as pathological and referred to as "monsters" until as late as the turn of the 20th century), would not agree with Fourier that a completely arbitrary function can be expanded in Fourier series, even if its Fourier coefficients are well-defined. For example, Kolmogorov (1922) constructed a Lebesgue integrable function whose Fourier series diverges pointwise almost everywhere. Nevertheless, a very wide class of functions can be expanded in Fourier series, especially if one allows weaker forms of convergence, such as convergence in the sense of distributions.
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These are equally important in the study of solvable models in statistical mechanics. An imprecise notion of "exact solvability" as meaning: "The solutions can be expressed explicitly in terms of some previously known functions" is also sometimes used, as though this were an intrinsic property of the system itself, rather than the purely calculational feature that we happen to have some "known" functions available, in terms of which the solutions may be expressed. This notion has no intrinsic meaning, since what is meant by "known" functions very often is defined precisely by the fact that they satisfy certain given equations, and the list of such "known functions" is constantly growing. Although such a characterization of "integrability" has no intrinsic validity, it often implies the sort of regularity that is to be expected in integrable systems.
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More precisely, a nonholonomic system, also called an anholonomic system, is one in which there is a continuous closed circuit of the governing parameters, by which the system may be transformed from any given state to any other state. Because the final state of the system depends on the intermediate values of its trajectory through parameter space, the system cannot be represented by a conservative potential function as can, for example, the inverse square law of the gravitational force. This latter is an example of a holonomic system: path integrals in the system depend only upon the initial and final states of the system (positions in the potential), completely independent of the trajectory of transition between those states. The system is therefore said to be integrable, while the nonholonomic system is said to be nonintegrable.
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In July 2019 Glassbox announced a strategic partnership with Microsoft , and in September 2019 Glassbox announced that its toolkit was integrable with the Salesforce platform . Glassbox is SOC 2 and ISO 27001 certified. In addition, in order to prevent customer-sensitive information from being intercepted by man-in-the-middle attacks, the Glassbox SDK provides the capability of `blacking-out' sensitive data fields from any part of a client's user interface (such as credit card details, home addresses and Social Security Numbers) in order to obfuscate visual data if it is being transmitted across an unsecured network. In February 2019, the company CTO issued statements on the company website suggested that going forward, all Glassbox clients should both ensure that this functionality is used correctly, and explicitly request user consent .
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They are therefore susceptible to solution by techniques resembling the inverse scattering transform which was originally developed to solve the Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation which arises in the theory of solitons, and which is also completely integrable. Unfortunately, the solutions obtained by these methods are often not as nice as one would like. For example, in a manner analogous to the way that one obtains a multiple soliton solution of the KdV from the single soliton solution (which can be found from Lie's notion of point symmetry), one can obtain a multiple Kerr object solution, but unfortunately, this has some features which make it physically implausible. A monograph on the use of soliton methods to produce stationary axisymmetric vacuum solutions, colliding gravitational plane waves, and so forth.
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Total variation can be seen as a non-negative real-valued functional defined on the space of real-valued functions (for the case of functions of one variable) or on the space of integrable functions (for the case of functions of several variables). For signals, especially, total variation refers to the integral of the absolute gradient of the signal. In signal and image reconstruction, it is applied as total variation regularization where the underlying principle is that signals with excessive details have high total variation and that removing these details, while retaining important information such as edges, would reduce the total variation of the signal and make the signal subject closer to the original signal in the problem. For the purpose of signal and image reconstruction, l1 minimization models are used.
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In March 2019, Uhlenbeck became the first woman to receive the Abel Prize, with the award committee citing the decision for "her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics." Hans Munthe-Kaas, who chairs the award committee, stated that "Her theories have revolutionised our understanding of minimal surfaces, such as more general minimisation problems in higher dimensions". Uhlenbeck also won the National Medal of Science in 2000,. and the Leroy P. Steele Prize for Seminal Contribution to Research of the American Mathematical Society in 2007, "for her foundational contributions in analytic aspects of mathematical gauge theory", based on her 1982 papers "Removable singularities in Yang–Mills fields" and "Connections with bounds on curvature".
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Pointwise multiplication determines a representation of this algebra on the Hilbert space of square integrable functions on X. An early observation of John von Neumann was that this correspondence also worked in reverse: Given some mild technical hypotheses, a commutative von Neumann algebra together with a representation on a Hilbert space determines a measure space, and these two constructions (of a von Neumann algebra plus a representation and of a measure space) are mutually inverse. Von Neumann then proposed that non- commutative von Neumann algebras should have geometric meaning, just as commutative von Neumann algebras do. Together with Francis Murray, he produced a classification of von Neumann algebras. The direct integral construction shows how to break any von Neumann algebra into a collection of simpler algebras called factors.
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If the resolution is not limited by the rectangular sampling rate of either the source or target image, then one should ideally use rotationally symmetrical filter or interpolation functions, as though the data were a two dimensional function of continuous x and y. The sinc function of the radius has too long a tail to make a good filter (it is not even square-integrable). A more appropriate analog to the one-dimensional sinc is the two-dimensional Airy disc amplitude, the 2D Fourier transform of a circular region in 2D frequency space, as opposed to a square region. Gaussian plus differential function One might consider a Gaussian plus enough of its second derivative to flatten the top (in the frequency domain) or sharpen it up (in the spatial domain), as shown.
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This theory has proven to be a central element in structural analysis and recently also in the construction of concrete quantum field theoretical modelsAn overview of the construction of a large number of models using these methods can be found in: Gandalf Lechner, Algebraic Constructive Quantum Field Theory: Integrable Models and Deformation Techniques, pp. 397–449 in: Advances in Algebraic Quantum Field Theory, Springer, 2015.. Together with Daniel Kastler and Ewa Trych-Pohlmeyer, Haag also succeeded in deriving the KMS condition from the stability properties of thermal equilibrium states. Together with Huzihiro Araki, Daniel Kastler and Masamichi Takesaki, he also developed a theory of chemical potential in this context. The framework created by Haag and Kastler for studying quantum field theories in Minkowski space can be transferred to theories in curved spacetime.
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The following conditions on a finite measure μ on Borel subsets of the real line are equivalent:Equivalence between (1) and (2) is a special case of (fails for σ-finite measures); equivalence between (1) and (3) is a special case of the Radon–Nikodym theorem, see or (still holds for σ-finite measures). :(1) μ is absolutely continuous; :(2) for every positive number ε there is a positive number δ such that for all Borel sets A of Lebesgue measure less than δ; :(3) there exists a Lebesgue integrable function g on the real line such that :: \mu(A) = \int_A g \,d\lambda :for all Borel subsets A of the real line. For an equivalent definition in terms of functions see the section Relation between the two notions of absolute continuity. Any other function satisfying (3) is equal to g almost everywhere.
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In contrast to the theory of deformation quantization described above, geometric quantization seeks to construct an actual Hilbert space and operators on it. Starting with a symplectic manifold M, one first constructs a prequantum Hilbert space consisting of the space of square-integrable sections of an appropriate line bundle over M. On this space, one can map all classical observables to operators on the prequantum Hilbert space, with the commutator corresponding exactly to the Poisson bracket. The prequantum Hilbert space, however, is clearly too big to describe the quantization of M. One then proceeds by choosing a polarization, that is (roughly), a choice of n variables on the 2n-dimensional phase space. The quantum Hilbert space is then the space of sections that depend only on the n chosen variables, in the sense that they are covariantly constant in the other n directions.
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Shatashvili has made several discoveries in the fields of theoretical and mathematical physics. He is mostly known for his work with Ludwig Faddeev on quantum anomalies, with Anton Alekseev on geometric methods in two-dimensional conformal field theories, for his work on background independent open string field theory, with Cumrun Vafa on superstrings and manifolds of exceptional holonomy, with Anton Gerasimov on tachyon condensation, with Andrei Losev, Nikita Nekrasov and Greg Moore on four dimensional analogs of two dimensional conformal field theories, as well as for his work with Nikita Nekrasov on quantum integrable systems. In particular, Shatashvili and Nikita Nekrasov discovered the gauge/Bethe correspondence. In 1995 he received an Outstanding Junior Investigator Award of the Department of Energy (DOE) and a NSF Career Award and from 1996 to 2000 he was a Sloan Fellow.
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A major mathematical difficulty in symbolic integration is that in many cases, a closed formula for the antiderivative of a rather simple-looking function does not exist. For instance, it is known that the antiderivatives of the functions and cannot be expressed in the closed form involving only rational and exponential functions, logarithm, trigonometric functions and inverse trigonometric functions, and the operations of multiplication and composition; in other words, none of the three given functions is integrable in elementary functions, which are the functions which may be built from rational functions, roots of a polynomial, logarithm, and exponential functions. The Risch algorithm provides a general criterion to determine whether the antiderivative of an elementary function is elementary, and, if it is, to compute it. Unfortunately, it turns out that functions with closed expressions of antiderivatives are the exception rather than the rule.
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Robin K. Bullough (21 November 1929 – 30 August 2008) was a British mathematical physicist known for his contributions to the theory of solitons, in particular for his role in the development of the theory of the optical soliton, now commonly used, for example, in the theory of trans-oceanic optical fibre communication theory, but first recognised in Bullough's work on ultra-short (nano- and femto-second) optical pulses. He is also known for deriving exact solutions to the nonlinear equations describing these solitons and for associated work on integrable systems, infinite-dimensional Hamiltonian systems (both classical and quantum), and the statistical mechanics for these systems. Bullough also contributed to nonlinear mathematical physics, including Bose–Einstein condensation in magnetic traps. Bullough obtained his first academic position in the Mathematics Department at UMIST in 1960 and was appointed chair of Mathematical Physics in 1973 where he remained until his retirement in 1995.
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The duality between the circle group S1 and the integers Z, or more generally, between a torus Tn and Zn is well known in analysis as the theory of Fourier series, and the Fourier transform similarly expresses the fact that the space of characters on a real vector space is the dual vector space. Thus unitary representation theory and harmonic analysis are intimately related, and abstract harmonic analysis exploits this relationship, by developing the analysis of functions on locally compact topological groups and related spaces. A major goal is to provide a general form of the Fourier transform and the Plancherel theorem. This is done by constructing a measure on the unitary dual and an isomorphism between the regular representation of G on the space L2(G) of square integrable functions on G and its representation on the space of L2 functions on the unitary dual.
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Fomenko is a full member (Academician) of the Russian Academy of Sciences (1994), the International Higher Education Academy of Sciences (1993) and Russian Academy of Technological Sciences (2009), as well as a doctor of physics and mathematics (1972), a professor (1980), and head of the Differential Geometry and Applications Department of the Faculty of Mathematics and Mechanics in Moscow State University (1992). Fomenko is the author of the theory of topological invariants of an integrable Hamiltonian system. He is the author of 180 scientific publications, 26 monographs and textbooks on mathematics, a specialist in geometry and topology, variational calculus, symplectic topology, Hamiltonian geometry and mechanics, and computational geometry. Fomenko is also the author of a number of books on the development of new empirico-statistical methods and their application to the analysis of historical chronicles as well as the chronology of antiquity and the Middle Ages.
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She has done work both in the pure representation theory of Lie groups real or p-adic (the study of unitary representations of those groups) and in the study of the "automorphic spectrum" of arithmetic groups (the study of those unitary representations which have an arithmetic significance), especially in the area of the Langlands programme. A prominent example of her achievements in the former is her classification, obtained with Jean-Loup Waldspurger, of the non- cuspidal discrete factors in the decomposition into irreducible components of the spaces of square-integrable invariant functions on adelic general linear groups. For this purpose it was first necessary to write down in a rigorous form the general theory of Eisenstein series laid down years earlier by Langlands, which they did in a seminar in Paris the content of which was later published in book form. Another notable work in the domain, with Waldspurger and Marie-France Vignéras, is a book on the Howe correspondence.
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During the last years groups of scientists from Yerevan Physics Institute have actively participated in intermediate and high energy physics experiments abroad (JLAB, DESY, CERN-LHC, MAX-lab, MAMI), exploring the meson and nucleon structures, electromagnetic interactions of the nucleon, quark-hadron duality, short range nucleon-nucleon correlations, quark hadronization in nuclear medium, physics beyond standard model, Higgs boson searches, quark-gluon plasma, fission and fragmentation of nuclei and hypernuclei and many other topics, as well as constructing experimental hardware and develop the software for data acquisition and analysis. The theoretical department assure major achievements in the following areas: B-meson physics, QCD and Related Phenomenology, Neutrino physics, Quantum Field Theory, String/M-theory, Integrable Models, Statistical physics, Condensed Matter and Quantum Information. These results are internationally recognized and highly cited. In the mid-1980s in YerPhI was developed the concept of stereoscopic approach in Very High Energy gamma-ray astronomy using multiple Imaging Atmospheric Cherenkov Telescopes (IACT).
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In the Travaux, Cartan breaks down his work into 15 areas. Using modern terminology, they are: # Lie theory # Representations of Lie groups # Hypercomplex numbers, division algebras # Systems of PDEs, Cartan–Kähler theorem # Theory of equivalence # Integrable systems, theory of prolongation and systems in involution # Infinite- dimensional groups and pseudogroups # Differential geometry and moving frames # Generalised spaces with structure groups and connections, Cartan connection, holonomy, Weyl tensor # Geometry and topology of Lie groups # Riemannian geometry # Symmetric spaces # Topology of compact groups and their homogeneous spaces # Integral invariants and classical mechanics # Relativity, spinors Cartan's mathematical work can be described as the development of analysis on differentiable manifolds, which many now consider the central and most vital part of modern mathematics and which he was foremost in shaping and advancing. This field centers on Lie groups, partial differential systems, and differential geometry; these, chiefly through Cartan's contributions, are now closely interwoven and constitute a unified and powerful tool.
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Ideally one would like to describe the (moduli) space of all solutions explicitly, and for some very special PDEs this is possible. (In general this is a hopeless problem: it is unlikely that there is any useful description of all solutions of the Navier–Stokes equation for example, as this would involve describing all possible fluid motions.) If the equation has a very large symmetry group, then one is usually only interested in the moduli space of solutions modulo the symmetry group, and this is sometimes a finite- dimensional compact manifold, possibly with singularities; for example, this happens in the case of the Seiberg–Witten equations. A slightly more complicated case is the self dual Yang–Mills equations, when the moduli space is finite-dimensional but not necessarily compact, though it can often be compactified explicitly. Another case when one can sometimes hope to describe all solutions is the case of completely integrable models, when solutions are sometimes a sort of superposition of solitons; this happens e.g.
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Uhlenbeck is one of the founders of the field of geometric analysis, a discipline that uses differential geometry to study the solutions to differential equations and vice versa. She has also contributed to topological quantum field theory and integrable systems.. Together with Jonathan Sacks in the early 1980s, Uhlenbeck established regularity estimates that have found applications to studies of the singularities of harmonic maps and the existence of smooth local solutions to the Yang–Mills–Higgs equations in gauge theory. In particular, Donaldson describes their joint 1981 paper The existence of minimal immersions of 2-spheres as a "landmark paper... which showed that, with a deeper analysis, variational arguments can still be used to give general existence results" for harmonic map equations. Building on these ideas, Uhlenbeck initiated a systematic study of the moduli theory of minimal surfaces in hyperbolic 3-manifolds (also called minimal submanifold theory) in her 1983 paper, Closed minimal surfaces in hyperbolic 3-manifolds.
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Connes, H. Moscovici; The Local Index Formula in Noncommutative Geometry expresses the pairing of the K-group of the manifold with this K-cycle in two ways: the 'analytic/global' side involves the usual trace on the Hilbert space and commutators of functions with the phase operator (which corresponds to the 'index' part of the index theorem), while the 'geometric/local' side involves the Dixmier trace and commutators with the Dirac operator (which corresponds to the 'characteristic class integration' part of the index theorem). Extensions of the index theorem can be considered in cases, typically when one has an action of a group on the manifold, or when the manifold is endowed with a foliation structure, among others. In those cases the algebraic system of the 'functions' which expresses the underlying geometric object is no longer commutative, but one may able to find the space of square integrable spinors (or, sections of a Clifford module) on which the algebra acts, and the corresponding 'Dirac' operator on it satisfying certain boundedness of commutators implied by the pseudo-differential calculus.
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Sofʹja Vasilʹevna Kovalevskaja, Mémoire sur un cas particulier du problème de la rotation d'un corps pesant autour d'un point fixe où l'intégration s'effectue à l'aide de fonctions ultraelliptiques du temps, IMprimerie nationale, 1894 Her submission featured the celebrated discovery of what is now known as the "Kovalevskaya top", which was subsequently shown to be the only other case of rigid body motion that is "completely integrable" other than the tops of Euler and Lagrange. In 1889 Kovalevskaya was appointed Ordinary Professor (full professor) at Stockholm University, the first woman in Europe in modern times to hold such a position. After much lobbying on her behalf (and a change in the Academy's rules) she was made a Corresponding Member of the Russian Academy of Sciences, but she was never offered a professorship in Russia. Kovalevskaya, who was involved in the vibrant, politically progressive and feminist currents of late nineteenth-century Russian nihilism, wrote several non-mathematical works as well, including a memoir, A Russian Childhood, two plays (in collaboration with Duchess Anne Charlotte Edgren-Leffler) and a partly autobiographical novel, Nihilist Girl (1890).
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Amongst his notable discoveries are the Hitchin-Thorpe inequality; Hitchin's projectively flat connection over Teichmüller space; the Atiyah–Hitchin monopole metric; the Atiyah–Hitchin–Singer theorem; the ADHM construction of instantons (of Michael Atiyah, Vladimir Drinfeld, Hitchin, and Yuri Manin); the hyperkähler quotient (of Hitchin, Anders Karlhede, Ulf Lindström and Martin Roček); Higgs bundles, which arise as solutions to the Hitchin equations, a 2-dimensional reduction of the self-dual Yang–Mills equations; and the Hitchin system, an algebraically completely integrable Hamiltonian system associated to the data of an algebraic curve and a complex reductive group. He and Shoshichi Kobayashi independently conjectured the Kobayashi–Hitchin correspondence. Higgs bundles, which are also developed in the work of Carlos Simpson, are closely related to the Hitchin system, which has an interpretation as a moduli space of semistable Higgs bundles over a compact Riemann surface or algebraic curve. This moduli space has emerged as a focal point for deep connections between algebraic geometry, differential geometry, hyperkähler geometry, mathematical physics, and representation theory.
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In the same paper, he applied such criterion to explain some puzzling experimental results on plasma confinement in open mirror traps, that had just been obtained at the Kurchatov Institute. This was the very first physical theory of chaos, which succeeded in explaining a concrete experiment, and which was developed long before computers made the icons of chaos familiar to everyone. Other results obtained by his group include: analysis of the transition to strong chaos in the Fermi-Pasta-Ulam problem; the derivation of the chaos border for the Fermi acceleration model; the numerical computation of the Kolmogorov-Sinai entropy in area-preserving maps; the investigations of weak instabilities in many-dimensional Hamiltonian systems (Arnold diffusion and modulational diffusion); the demonstration that the homogeneous models of classical Yang-Mills field have positive Kolmogorov-Sinai entropy, and therefore are generally not integrable; the discovery of the power law decay of Poincaré recurrences in Hamiltonian systems with divided phase space; the demonstration that the dynamics of the Halley comet is chaotic, and is described by a simple map. Yuzhnoye Cemetery.
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