49 results on '"FOURIER transforms"'
Search Results
2. Multipartite quantum systems: an approach based on Markov matrices and the Gini index.
- Author
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Vourdas, A
- Subjects
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MATRICES (Mathematics) , *HILBERT space , *QUANTUM statistics , *FOURIER transforms , *STOCHASTIC matrices , *PERMUTATIONS , *RANDOM matrices - Abstract
An expansion of row Markov matrices in terms of matrices related to permutations with repetitions, is introduced. It generalises the Birkhoff–von Neumann expansion of doubly stochastic matrices in terms of permutation matrices (without repetitions). An interpretation of the formalism in terms of sequences of integers that open random safes described by the Markov matrices, is presented. Various quantities that describe probabilities and correlations in this context, are discussed. The Gini index is used to quantify the sparsity (certainty) of various probability vectors. The formalism is used in the context of multipartite quantum systems with finite dimensional Hilbert space, which can be viewed as quantum permutations with repetitions or as quantum safes. The scalar product of row Markov matrices, the various Gini indices, etc, are novel probabilistic quantities that describe the statistics of multipartite quantum systems. Local and global Fourier transforms are used to define locally dual and also globally dual statistical quantities. The latter depend on off-diagonal elements that entangle (in general) the various components of the system. Examples which demonstrate these ideas are also presented. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
3. Secure multi-party computation with a quantum manner.
- Author
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Lu, Changbin, Miao, Fuyou, Hou, Junpeng, Su, Zhaofeng, and Xiong, Yan
- Subjects
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QUANTUM computing , *QUANTUM cryptography , *QUANTUM information science , *FINITE fields , *FOURIER transforms , *CRYPTOGRAPHY - Abstract
Quantum information processing protocols have great advantages over their classical counterparts, especially on cryptography. Secure multi-party computation is one of the most important issues and has been extensively studied in cryptography. It is of both theoretical and practical significance to develop the quantum information processing protocols for secure multi-party computation. In this paper, we consider the secure multi-party computation for n-variable polynomial functions over the finite field GF(d). We propose two protocols using quantum resources to compute the function within a one-time execution. One is based on d-level mutually unbiased (orthonormal) bases with cyclic property and the other takes advantage of quantum Fourier transform. Analytical results show that the proposed protocols are secure against a passive adversary with unlimited computing power, including colluding attack mounted by n − 2 parties. We also implement the second protocol of the special case d = 2 on the IBM Q Experience. In principle, our proposals can be experimentally realized in the arbitrary d dimension with the advances in realizations and controls of high-dimensional quantum computation. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
4. Repeated differentiation suppresses superoscillations.
- Author
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Berry, M V
- Subjects
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SMOOTHNESS of functions , *FOURIER transforms , *OSCILLATIONS , *PHYSICS - Abstract
Two mathematical phenomena with applications in physics are: superoscillations, in which band-limited functions oscillate more rapidly than their fastest Fourier component; and the transformation of almost any smooth function into a monochromatic oscillation under repeated differentiation. These are opposite phenomena, and one mutates into the other, i.e. superoscillations are destroyed, as the number of derivatives increases. This behaviour is explained, and illustrated with an example. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
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5. Harmonic inversion analysis of exceptional points in resonance spectra.
- Author
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Fuchs, Jacob, Main, Jörg, Cartarius, Holger, and Wunner, Günter
- Subjects
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HARMONIC analysis (Mathematics) , *NUCLEAR magnetic resonance spectroscopy , *QUANTUM mechanics , *LORENTZIAN function , *FREQUENCY-domain analysis , *FOURIER transforms - Abstract
The spectra of, e.g. open quantum systems are typically given as the superposition of resonances with a Lorentzian line shape, where each resonance is related to a simple pole in the complex energy domain. However, at exceptional points two or more resonances are degenerate and the resulting non-Lorentzian line shapes are related to higher order poles in the complex energy domain. In the Fourier-transform time domain an nth order exceptional point is characterized by a non-exponentially decaying time signal given as the product of an exponential function and a polynomial of degree n − 1. The complex positions and amplitudes of the non-degenerate resonances can be determined with high accuracy by application of the nonlinear harmonic inversion method to the real-valued resonance spectra. We extend the harmonic inversion method to include the analysis of exceptional points. The technique yields, in the energy domain, the amplitudes of the higher order poles contributing to the spectra, and, in the time domain, the coefficients of the polynomial characterizing the non-exponential decay of the time signal. The extended harmonic inversion method is demonstrated on two examples, viz. the analysis of exceptional points in resonance spectra of the hydrogen atom in crossed magnetic and electric fields, and an exceptional point occurring in the dynamics of a single particle in a time-dependent harmonic trap. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
6. S-duality as Fourier transform for arbitrary ϵ1, ϵ2.
- Author
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Nemkov, N
- Subjects
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FOURIER transforms , *DUALITY theory (Mathematics) , *MATHEMATICAL analysis , *PERTURBATION theory , *MATHEMATICAL physics - Abstract
The Alday–Gaiotto–Tachikawa relations reduce S-duality to the modular transformations of conformal blocks. It was recently conjectured that, for the four-point conformal block, the modular transform up to the non-perturbative contributions can be written in the form of the ordinary Fourier transform when β ≡ −ϵ1/ϵ2 = 1. Here I extend this conjecture to general values of ϵ1, ϵ2. Namely, I argue that, for a properly normalized four-point conformal block the S-duality is perturbatively given by the Fourier transform for arbitrary values of the deformation parameters ϵ1, ϵ2. The conjecture is based on explicit perturbative computations in the first few orders of the string coupling constant g2 ≡ −ϵ1ϵ2 and hypermultiplet masses. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
7. On the nonlinear Fourier transform associated with periodic AKNS-ZS systems and its inverse.
- Author
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Saksida, Pavle
- Subjects
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NONLINEAR theories , *FOURIER transforms , *STATISTICAL association , *SYSTEMS theory , *INVERSE functions , *PARTIAL differential equations - Abstract
We study the nonlinear Fourier transform F associated with the integrable nonlinear partial differential equations of AKNS-ZS type. We show that F is a real analytic operator between the appropriate Hilbert spaces, and that it has a real analytic local inverse near the origin. We construct a convergent iterative scheme by means of which one can calculate the inverse F-1 to any desired degree of accuracy. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
8. Bunches of random cross-correlated sequences.
- Author
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Maystrenko, A. A., Melnik, S. S., Pritula, G. M., and Usatenko, O. V.
- Subjects
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MATHEMATICAL convolutions , *FOURIER transforms , *CROSS correlation , *WHITE noise , *GAUSSIAN processes , *STATISTICAL correlation - Abstract
The statistical properties of random cross-correlated sequences constructed by the convolution method (likewise referred to as the Rice or the inverse Fourier transformation) are examined. We clarify the meaning of the filtering function--the kernel of the convolution operator--and show that it is the value of the cross-correlation function which describes correlations between the initial white noise and constructed correlated sequences. The matrix generalization of this method for constructing a bunch of N cross-correlated sequences is presented. Algorithms for their generation are reduced to solving the problem of decomposition of the Fourier transform of the correlationmatrix into a product of two mutually conjugate matrices. Different decompositions are considered. The limits of weak and strong correlations for the one-point probability and pair correlation functions of sequences generated by themethod under consideration are studied. Special cases of heavy-tailed distributions of the generated sequences are analyzed. We show that, if the filtering function is rather smooth, the distribution function of generated variables has the Gaussian or L'evy form depending on the analytical properties of the distribution (or characteristic) functions of the initial white noise. Anisotropic properties of statistically homogeneous random sequences related to the asymmetry of a filtering function are revealed and studied. These asymmetry properties are expressed in terms of the third- or fourth-order correlation functions. Several examples of the construction of correlated chains with a predefined correlation matrix are given. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
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9. Noise cascades and L'evy correlations.
- Author
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Eliazar, Iddo I. and Shlesinger, Michael F.
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CASCADES (Fluid dynamics) , *MATHEMATICAL convolutions , *STOCHASTIC models , *RAINDROPS , *PAIRING correlations (Nuclear physics) , *FOURIER transforms - Abstract
We explore a general model of stochastic noise cascades which can be illustrated by the example of rain dropping down on the earth and then seeping through layers of ground--pouring down layer by layer. The rain represents an input noise that is assumed to be spatially uncorrelated, and each ground layer represents a stochastic convolution filter. As the input noise percolates through the layered filters spatial correlations--which are initially nonexistent--build up. We study this build-up of correlations and focus on the following question: are there universally emergent forms of spatial correlations? The answer is proved affirmative, and is shown to be uniquely characterized by power spectra that coincide with the Fourier transform of the spherically symmetric L'evy distribution. We term these universally emergent spatial correlations 'L'evy correlations'. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
10. Unveiling and exemplifying the unitary equivalence of discrete time quantum walk models.
- Author
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Venancio, B. F., Andrade, F. M., and da Luz, M. G. E.
- Subjects
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QUANTUM theory , *AMPLITUDE estimation , *ALGEBRAIC topology , *HARTLEY transforms , *HADAMARD matrices , *FOURIER transforms - Abstract
The two major discrete time formulations for quantum walks, coined and scattering, are unitarily equivalent for arbitrary position-dependent transition amplitudes and any topology (Andrade et al 2009 Phys. Rev. A 80 052301). Although the proof explicitly describes the mapping obtention, its high technicality may hinder relevant physical aspects involved in the equivalence. Discussing concrete examples-the most general constructions for the line, square and honeycomb lattices-here we unveil the similarities and differences of these two versions of quantum walks. We moreover show how to derive the dynamics of one from the other by means of proper projections. We perform calculations for different probability amplitudes such as Hadamard, Grover, discrete Fourier transform and the uncommon in the area (but interesting) discrete Hartley transform, comparing the evolutions. Our study illustrates the models' interplay, an important issue for implementations and applications of such systems. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
11. Riemann zeros in radiation patterns.
- Author
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Berry, M. V.
- Subjects
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RIEMANNIAN manifolds , *ANTENNA radiation patterns , *FOURIER transforms , *PHYSICS experiments , *ZETA functions , *GENERATING functions - Abstract
Propagation into the far field changes initialwaves into their Fourier transforms. This implies that the Riemann zeros could be observed experimentally in the radiation pattern generated by an initial wave whose Fourier transform is proportional to the Riemann zeta function on the critical line. Two such waves are examined, generating the Riemann &b.Xi;(t) function (pattern 1) and the function ζ (1/2 + it)/(1/2 + it) (pattern 2). For pattern 1, the radiation side lobes are probably too weak to allow detection of the zeros, but for pattern 2 the lobes are stronger, suggesting a feasible experiment. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
12. Almost complete coherent state subsystems and partial reconstruction of wavefunctions in the Fock-Bargmann phase-number representation.
- Author
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Calixto, M., Guerrero, J., and Sánchez-Monreal, J. C.
- Subjects
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COHERENT states , *WAVE functions , *PHASE equilibrium , *MATHEMATICAL formulas , *FOURIER transforms , *HOLOMORPHIC functions , *APPROXIMATION theory - Abstract
We provide (partial) reconstruction formulas and discrete Fourier transforms for wavefunctions in standard Fock-Bargmann (holomorphic) phase-number representation from a finite number N of phase samples {&thetas;k = 2πk/N}N-1 k=0 for a given mean number p of particles. The resulting coherent state subsystem S = {|zk = p1/2 ei&thetas;k 〉} is complete (a frame) for truncated Hilbert spaces (finite number of particles) and reconstruction formulas are exact. For an unbounded number of particles, S is 'almost complete' (a pseudo-frame) and partial reconstruction formulas are provided along with a study of the accuracy of the approximation, which tends to be exact when p < N and/or N →∞. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
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13. The Fourier transform of tubular densities.
- Author
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Prior, C. B. and Goriely, A.
- Subjects
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FOURIER transforms , *DENSITY , *BIOPOLYMERS , *DISTRIBUTION (Probability theory) , *CURVATURE , *MANIFOLDS (Mathematics) , *SUPERPOSITION principle (Physics) - Abstract
We consider the Fourier transform of tubular volume densities, with arbitrary axial geometry and (possibly) twisted internal structure. This density can be used to represent, among others, magnetic flux or the electron density of biopolymer molecules. We consider tubes of both finite radii and unrestricted radius. When there is overlap of the tube structure the net density is calculated using the super-position principle. The Fourier transform of this density is composed of two expressions, one for which the radius of the tube is less than the curvature of the axis and one for which the radius is greater (which must have density overlap). This expression can accommodate an asymmetric density distribution and a tube structure which has non-uniform twisting. In addition we give several simpler expressions for isotropic densities, densities of finite radius, densities which decay at a rate sufficient to minimize local overlap and finally individual surfaces of the tube manifold. These simplified cases can often be expressed as arclength integrals and can be evaluated using a system of first-order ODEs. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
14. Generalization of Bochner's theorem for functions of the positive type.
- Subjects
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FOURIER transforms , *INTEGRAL transforms , *SCHRODINGER equation , *INTEGRAL representations , *INVERSE scattering transform , *MATHEMATICAL physics - Abstract
We generalize Bochner's theorem for functions of the positive type--theorem 1--to more general integral transforms using the Jost solution of the radial Schrödinger equation. The generalized theorem is theorem 2. We then use Bochner's theorem to obtain an integral representation for the phase shift, shown in theorem 4. In a forthcoming paper, this theorem will be used in inverse scattering theory. The proofs are simple, and make use of well-known theorems of real analysis and Fourier transforms of L1, L1[?]L2, ... functions. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
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15. On q-extended eigenvectors of the integral and finite Fourier transforms.
- Author
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N M Atakishiyev, J P Rueda, and K B Wolf
- Subjects
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EIGENVECTORS , *INTEGRAL transforms , *FOURIER transforms , *HERMITE polynomials , *EIGENFUNCTIONS , *HARMONIC oscillators , *LIMIT theorems - Abstract
Mehta has shown that eigenvectors of the N × N finite Fourier transform can be written in terms of the standard Hermite eigenfunctions of the quantum harmonic oscillator (1987 J. Math. Phys. 28 781). Here, we construct a one-parameter family of q-extensions of these eigenvectors, based on the continuous q-Hermite polynomials of Rogers. In the limit when q - 1 these q-extensions coincide with Mehta's eigenvectors, and in the continuum limit as N - [?] they give rise to q-extensions of eigenfunctions of the Fourier integral transform. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
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16. The Tasaki-Crooks quantum fluctuation theorem.
- Author
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Peter Talkner and Peter H
- Subjects
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FLUCTUATIONS (Physics) , *QUANTUM theory , *FOURIER transforms , *CHARACTERISTIC functions , *MATHEMATICAL physics - Abstract
Starting out from the recently established quantum correlation function expression of the characteristic function for the work performed by a force protocol on the system in Talkner et al(2007 Phys. Rev.E 75050102 (Preprintcond-mat/0703213)) the quantum version of the Crooks fluctuation theorem is shown to emerge almost immediately by the mere application of an inverse Fourier transformation. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
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17. Discrete Painlevé system for the partition function of N f = 2 SU(2) supersymmetric gauge theory and its double scaling limit.
- Author
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H Itoyama, T Oota, and Katsuya Yano
- Subjects
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PARTITION functions , *DISCRETE systems , *ORTHOGONAL polynomials , *GENERATING functions , *FOURIER transforms - Abstract
We continue to study the matrix model of the Nf = 2 case that represents the irregular conformal block. What provides us with the Painlevé system is not the instanton partition function per se but rather a finite analog of its Fourier transform that can serve as a generating function. The system reduces to the extension of the Gross–Witten–Wadia unitary one-matrix model by the logarithmic potential while keeping the planar critical behavior intact. The double scaling limit to this critical point is a constructive way to study Argyres–Douglas type theory from IR. We elaborate upon the method of orthogonal polynomials and its relevance to these problems, developing it further for the case of a generic unitary matrix model and that of a special case with the logarithmic potential. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
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18. Nonlinear Fourier transform—towards the construction of nonlinear Fourier modes.
- Author
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Pavle Saksida
- Subjects
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FOURIER transforms , *NONLINEAR theories , *SUPERPOSITION principle (Physics) - Abstract
We study a version of the nonlinear Fourier transform associated with ZS-AKNS systems. This version is suitable for the construction of nonlinear analogues of Fourier modes, and for the perturbation-theoretic study of their superposition. We provide an iterative scheme for computing the inverse of our transform. The relevant formulae are expressed in terms of Bell polynomials and functions related to them. In order to prove the validity of our iterative scheme, we show that our transform has the necessary analytic properties. We show that up to order three of the perturbation parameter, the nonlinear Fourier mode is a complex sinusoid modulated by the second Bernoulli polynomial. We describe an application of the nonlinear superposition of two modes to a problem of transmission through a nonlinear medium. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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19. Form factors of bound states in the XXZ chain.
- Author
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Karol K Kozlowski
- Subjects
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BOUND states , *SPIN excitations , *FOURIER transforms - Abstract
This work focuses on the calculation of the large-volume behaviour of form factors of local operators in the XXZ spin-1/2 chain taken between the ground state and an excited state containing bound states. The analysis is rigorous and builds on various fine properties of the string solutions to the Bethe equations and certain technical hypotheses. These technical hypotheses are satisfied for a generic excited state. The results obtained in this work pave the way for extracting, starting from the first principles, the large-distance and long-time asymptotic behaviour of the XXZ chain’s two-point functions just as the so-called edge singularities of their Fourier transforms. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
20. Evolution of superoscillatory initial data in several variables in uniform electric field.
- Author
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Y Aharonov, F Colombo, I Sabadini, D C Struppa, and J Tollaksen
- Subjects
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ELECTRIC fields , *FOURIER transforms , *HAMILTON'S principle function - Abstract
A superoscillating function is defined by the property that it oscillates faster than its fastest Fourier components. This is mathematically possible because the coefficients of the linear combinations of the band limited components depend on the number of components. This phenomenon was discovered in the context of quantum physics, but it has important applications in a variety of areas, including metrology, antenna theory, and a new theory of superresolution in optics. In this paper we study the evolution of superoscillatory functions in uniform electric field by the Schrödinger equation where we assume that the Hamiltonian contains a even polynomial of the linear momentum p. This includes the classical case but also relativistic corrections of any order. Moreover, we extend our results to the case of several variables using the theory of superoscillating functions in several variables. We conclude by discussing a comparison of our work with the existing literature. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
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21. A note on the Fourier transform.
- Author
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M L Glasser
- Subjects
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FOURIER transforms , *INTEGRAL transforms , *SPECIAL functions , *INTEGRAL equations , *FREDHOLM equations - Abstract
The Fourier transforms of and are shown to be related in terms of a Fredholm integral equation. Several examples are discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
22. The limit distribution in the q-CLT for is unique and can not have a compact support.
- Author
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Sabir Umarov and Constantino Tsallis
- Subjects
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FOURIER transforms , *CENTRAL limit theorem , *GAUSSIAN processes , *SYMMETRY (Physics) , *RANDOM variables - Abstract
In a paper by Umarov et al (2008 Milan J. Math.76 307–28), a generalization of the Fourier transform, called the q-Fourier transform, was introduced and applied for the proof of a q-generalized central limit theorem (q-CLT). Subsequently, Hilhorst illustrated (2009 Braz. J. Phys.39 371–9; 2010 J. Stat. Mech. P10023) that the q-Fourier transform for , is not invertible in the space of density functions. Indeed, using an invariance principle, he constructed a family of densities with the same q-Fourier transform and noted that ‘as a consequence, the q-CLT falls short of achieving its stated goal’. The distributions constructed there have compact support. We prove now that the limit distribution in the q-CLT is unique and can not have a compact support. This result excludes all the possible counterexamples which can be constructed using the invariance principle and fills the gap mentioned by Hilhorst. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
23. Efficient quantum circuits for Toeplitz and Hankel matrices.
- Author
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A Mahasinghe and J B Wang
- Subjects
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TOEPLITZ matrices , *HANKEL functions , *QUANTUM computing , *HAMILTONIAN systems , *FOURIER transforms , *DIFFERENTIAL equations - Abstract
Toeplitz and Hankel matrices have been a subject of intense interest in a wide range of science and engineering related applications. In this paper, we show that quantum circuits can efficiently implement sparse or Fourier-sparse Toeplitz and Hankel matrices. This provides an essential ingredient for solving many physical problems with Toeplitz or Hankel symmetry in the quantum setting with deterministic queries. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
24. The Green's function for the three-dimensional linear Boltzmann equation via Fourier transform.
- Author
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Manabu Machida
- Subjects
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BOLTZMANN'S equation , *LINEAR equations , *FOURIER transforms , *GREEN'S functions , *SCATTERING (Mathematics) - Abstract
The linear Boltzmann equation with constant coefficients in the three-dimensional infinite space is revisited. It is known that the Green's function can be calculated via the Fourier transform in the case of isotropic scattering. In this paper, we show that the three-dimensional Green's function can be computed with the Fourier transform even in the case of arbitrary anisotropic scattering. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
25. Representation of superoperators in double phase space.
- Author
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Marcos Saraceno and Alfredo M Ozorio de Almeida
- Subjects
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QUANTUM mechanics , *WIGNER-Weyl transform , *CANONICAL transformations , *ISOMORPHISM (Mathematics) , *FOURIER transforms , *LAGRANGIAN mechanics - Abstract
Operators in quantum mechanics—either observables, density or evolution operators, unitary or not—can be represented by c-numbers in operator bases. The position and momentum bases are in one-to-one correspondence with lagrangian planes in double phase space, but this is also true for the well known Wigner–Weyl correspondence based on translation and reflection operators. These phase space methods are here extended to the representation of superoperators. We show that the Choi–Jamiolkowsky isomorphism between the dynamical matrix and the linear action of the superoperator constitutes a ‘double’ Wigner or chord transform when represented in double phase space. As a byproduct several previously unknown integral relationships between products of Wigner and chord distributions for pure states are derived. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
26. Riemann zeros in radiation patterns: II. Fourier transforms of zeta.
- Author
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M V Berry
- Subjects
- *
FOURIER transforms , *ZETA functions , *NUMBER theory , *HELMHOLTZ equation , *WAVENUMBER - Abstract
This extends a previous study (2012 J. Phys. A: Math. Theor. 45 302001) of two initial waveforms whose far-field radiation patterns possess sidelobes separated by the Riemann zeros. The analysis suffered from the disadvantage that the sidelobes were very weak, making it difficult to detect the zeros between them. To overcome this, new Fourier pairs are derived, whose sidelobes are not weak. These are transforms of the zeta function on the critical line, modulated by functions with no zeros on the line. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
27. Traveling waves and their tails in locally resonant granular systems.
- Author
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H Xu, P G Kevrekidis, and A Stefanov
- Subjects
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TRAVELING waves (Physics) , *RESONANCE , *GRANULAR materials , *FOURIER transforms , *STRAINS & stresses (Mechanics) - Abstract
In the present study, we revisit the theme of wave propagation in locally resonant granular crystal systems, also referred to as mass-in-mass systems. We use three distinct approaches to identify relevant traveling waves. The first consists of a direct solution of the traveling wave problem. The second one consists of the solution of the Fourier tranformed variant of the problem, or, more precisely, of its convolution reformulation (upon an inverse Fourier transform) in real space. Finally, our third approach will restrict considerations to a finite domain, utilizing the notion of Fourier series for important technical reasons, namely the avoidance of resonances, which will be discussed in detail. All three approaches can be utilized in either the displacement or the strain formulation. Typical resulting computations in finite domains result in the solitary waves bearing symmetric non-vanishing tails at both ends of the computational domain. Importantly, however, a countably infinite set of anti-resonance conditions is identified for which solutions with genuinely rapidly decaying tails arise. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
28. Quantum Bochner’s theorem for phase spaces built on projective representations.
- Author
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Ninnat Dangniam and Christopher Ferrie
- Subjects
- *
BOCHNER'S theorem , *WIGNER distribution , *QUANTUM correlations , *QUANTUM theory , *NUMERICAL solutions to functional equations , *FOURIER transforms - Abstract
Bochner’s theorem gives the necessary and sufficient conditions on a function such that its Fourier transform corresponds to a true probability density function. In the Wigner phase space picture, quantum Bochner’s theorem gives the necessary and sufficient conditions on a function such that it is a quantum characteristic function of a valid (and possibly mixed) quantum state and such that its Fourier transform is a true probability density. We extend this theorem to discrete phase space representations which possess enough symmetry. More precisely, we show that discrete phase space representations that are built on projective unitary representations of abelian groups, with a slight restriction on admissible two-cocycles, enable a quantum Bochner’s theorem. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
29. Bivariate raising and lowering differential operators for eigenfunctions of a 2D Fourier transform.
- Author
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Iván Area, Natig Atakishiyev, Eduardo Godoy, and Kurt Bernardo Wolf
- Subjects
- *
FOURIER transforms , *EIGENFUNCTIONS , *DIFFERENTIAL operators , *HERMITE polynomials , *PARTIAL differential equations , *HAMILTONIAN systems - Abstract
We define a two-dimensional (2D) Fourier transform that self-reproduces a one-parameter family of bivariate Hermite functions; these are eigenfunctions of a Hamiltonian differential operator of second order, whose exponential is that transform. We find explicit forms of the bivariate raising and lowering partial differential operators of first degree for the eigenfunctions of this 2D Fourier transform. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
30. Renormalization of correlations in a quasiperiodically forced two-level system for a general class of modulation function.
- Author
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L N C Adamson and A H Osbaldestin
- Subjects
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RENORMALIZATION (Physics) , *STATISTICAL correlation , *MODERATION (Statistics) , *MODULES (Algebra) , *FOURIER transforms , *CHAOS theory , *GOLDEN ratio - Abstract
We provide a renormalization analysis of correlations in a quasi-periodically forced two-level system in a time dependent field with periodic kicks whose amplitude is given by a general class of discontinuous modulation function. For certain intensities of modulation, we give a complete understanding of the autocorrelation function. Furthermore, once the locations of the discontinuities of the modulation function are known, aperiodic orbits lead to correlations on renormalization strange sets which are determined by two specified features of the modulation function of which there are only a finite number of variations. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
31. Spin glass field theory with replica Fourier transforms.
- Author
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I R Pimentel and C De Dominicis
- Subjects
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SPIN glasses , *FIELD theory (Physics) , *FOURIER transforms , *EIGENVALUES , *FREE energy (Thermodynamics) , *INTEGERS - Abstract
We develop a field theory for spin glasses using replica Fourier transforms (RFT). We present the formalism for the case of replica symmetry and the case of replica symmetry breaking on an ultrametric tree, with the number of replicas n and the number of replica symmetry breaking steps R generic integers. We show how the RFT applied to the two-replica fields allows one to construct a new basis which block-diagonalizes the four-replica mass-matrix, into the replicon, anomalous and longitudinal modes. The eigenvalues are given in terms of the mass RFT and the propagators in the RFT space are obtained by inversion of the block-diagonal matrix. The formalism allows one to express any i-replica vertex in the new RFT basis and hence enables one to perform a standard perturbation expansion. We apply the formalism to calculate the contribution of the Gaussian fluctuations around the Parisi solution for the free-energy of an Ising spin glass. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
32. A class of symmetric Bell diagonal entanglement witnesses—a geometric perspective.
- Author
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Dariusz Chruściński
- Subjects
- *
BELL'S theorem , *QUANTUM entanglement , *HERMITIAN operators , *SYMMETRIC operators , *FOURIER transforms - Abstract
We provide a class of Bell diagonal entanglement witnesses displaying an additional local symmetry—a maximal commutative subgroup of the unitary group U(n). Remarkably, this class of witnesses is parameterized by a torus being a maximal commutative subgroup of an orthogonal group . It is shown that a generic element from the class defines an indecomposable entanglement witness. The paper provides a geometric perspective for some aspects of the entanglement theory and an interesting interplay between group theory and block-positive operators in .This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell’s theorem’. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
33. Long-time limit of correlation functions.
- Author
-
Franosch, Thomas
- Subjects
- *
STOCHASTIC processes , *BOCHNER'S theorem , *FOURIER transforms , *BOREL sets , *GLASS transitions - Abstract
Auto-correlation functions in an equilibrium stochastic process are well-characterized by Bochnerʼs theorem as Fourier transforms of a finite symmetric Borel measure. The existence of a long-time limit of these correlation functions depends on the spectral properties of the measure. Here we provide conditions applicable to a wide class of dynamical theories guaranteeing the existence of the long-time limit. We discuss the implications in the context of the mode-coupling theory of the glass transition where a non-trivial long-time limit signals an idealized glass state. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
34. Superoscillations and supershifts in phase space: Wigner and Husimi function interpretations.
- Author
-
Berry, M V and Moiseyev, N
- Subjects
- *
OSCILLATIONS , *PHASE space , *WAVENUMBER , *FOURIER transforms , *COHERENT states - Abstract
Superoscillations, namely regions where a band-limited function f (x) varies faster than the fastest of its Fourier components k, generate the illusion that the Fourier content is ‘supershifted’ so as to lie outside the spectrum of the function. The relation between supershifts and superoscillations, central to the quantum weak measurements scheme, is explored in terms of two different representations of the local Fourier transform in the ‘phase space’ (x, k). The Wigner function W(x, k), regarded as a function of k for fixed x, inherits the band-limited property of f (x). Neverthless, its local k average can lie outside the spectrum because W, although real, posesses negative values. The local Wigner average of k equals the local wavenumber at x (local weak value of momentum), defined as the phase variation kloc(x) = ∂xarg f (x). By contrast, the Husimi function H(x, k), i.e. the windowed Fourier transform with window width L, corresponding to squeezing of the coherent state associated with (x, k) (and representing the pointer wavefunction after a weak measurement), is positive-definite. But it is not band-limited, and the local Husimi average of k equals kloc if L is small enough. These properties are illustrated numerically with two superoscillatory functions. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
35. Growth of a grain-boundary groove by surface superdiffusion.
- Author
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Hamed, M Abu and Nepomnyashchy, A A
- Subjects
- *
KIRKENDALL effect , *SURFACE diffusion , *NANOCRYSTALS , *FRACTIONAL calculus , *RIESZ spaces , *FOURIER transforms - Abstract
In the case of normal surface diffusion, an exact solution describing self-similar growth of a grain-boundary groove was obtained by Mullins. In some systems, surface superdiffusion takes place, e.g. due to long jumps of molecules above the surface. In the present paper, the problem of the groove growth is solved in the case of surface superdiffusion. An exact self-similar solution has been obtained, and its basic properties are described. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
36. The $\mathfrak {su}(2)_\alpha $ Hahn oscillator and a discrete Fourier-Hahn transform.
- Author
-
E I Jafarov, N I Stoilova, J Van, and der Jeugt
- Subjects
- *
HARMONIC oscillators , *FOURIER transforms , *QUADRATIC fields , *LIE algebras , *OPERATOR theory , *REPRESENTATIONS of algebras , *WAVE functions , *POLYNOMIALS - Abstract
We define the quadratic algebra $\mathfrak {su}(2)_\alpha$ which is a one-parameter deformation of the Lie algebra $\mathfrak {su}(2)$ extended by a parity operator. The odd-dimensional representations of $\mathfrak {su}(2)$ (with representation label j, a positive integer) can be extended to representations of $\mathfrak {su}(2)_\alpha$. We investigate a model of the finite one-dimensional harmonic oscillator based upon this algebra $\mathfrak {su}(2)_\alpha$. It turns out that in this model the spectrum of the position and momentum operator can be computed explicitly, and that the corresponding (discrete) wavefunctions can be determined in terms of Hahn polynomials. The operation mapping position wavefunctions into momentum wavefunctions is studied, and this so-called discrete Fourier-Hahn transform is computed explicitly. The matrix of this discrete Fourier-Hahn transform has many interesting properties, similar to those of the traditional discrete Fourier transform. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
37. Extremely compact formulas for the Fourier transform of a product of two-centre Slater-type orbitals.
- Author
-
T Vukovic and S Dmitrovic
- Subjects
- *
FOURIER transforms , *MOLECULAR orbitals , *FOURIER series , *HYPERGEOMETRIC functions , *BESSEL functions , *INTEGRAL calculus , *MATHEMATICAL formulas - Abstract
A compact formula for the Fourier transform of a product of Slater-type orbitals on different centres is derived. The integral is reduced to a finite one-dimensional integration over non-oscillatory hypergeometric functions of type {}_1F_2(\mathbf {x};\mathbf {y};z). The formula is valid for all quantum numbers and does not involve the reduced Bessel functions that are usually used to evaluate these integrals. Reduced formulas are calculated for some special directions in the reciprocal space. Also, some useful identities for the Fourier transforms of a product of Slater-type orbitals with correlated sets of parameters are obtained. In order to illustrate simple and efficient use of the presented results, we have applied them to graphene. [ABSTRACT FROM AUTHOR]
- Published
- 2010
38. The quantum state vector in phase space and Gabor's windowed Fourier transform.
- Author
-
A J Bracken and P Watson
- Subjects
- *
QUANTUM field theory , *FOURIER transforms , *PHASE space , *LINEAR algebra , *CONJUGATE direction methods , *SIGNAL processing , *MATHEMATICAL analysis - Abstract
Representations of quantum state vectors by complex phase space amplitudes, complementing the description of the density operator by the Wigner function, have been defined by applying the Weyl-Wigner transform to dyadic operators, linear in the state vector and anti-linear in a fixed 'window state vector'. Here aspects of this construction are explored, and a connection is established with Gabor's 'windowed Fourier transform'. The amplitudes that arise for simple quantum states from various choices of windows are presented as illustrations. Generalized Bargmann representations of the state vector appear as special cases, associated with Gaussian windows. For every choice of window, amplitudes lie in a corresponding linear subspace of square-integrable functions on phase space. A generalized Born interpretation of amplitudes is described, with both the Wigner function and a generalized Husimi function appearing as quantities linear in an amplitude and anti-linear in its complex conjugate. Schrodinger's time-dependent and time-independent equations are represented on phase space amplitudes, and their solutions described in simple cases. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
39. Simultaneous dense coding.
- Author
-
Haozhen Situ and Daowen Qiu
- Subjects
- *
CODING theory , *RADIO transmitter-receivers , *INFORMATION processing , *QUANTUM theory , *FOURIER transforms - Abstract
We present a dense coding scheme between one sender and two receivers, which guarantees that the receivers simultaneously achieve their respective information. In our scheme, the sender first performs a locking operation to entangle the particles from two independent quantum entanglement channels, so that the receivers cannot achieve their information unless they collaborate to perform the unlocking operation. We also show that the quantum Fourier transform can act as the locking operator both in simultaneous dense coding and teleportation. Finally we compare simultaneous dense coding with quantum secret sharing of classical messages. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
40. Cubic string boundary value problems and Cauchy biorthogonal polynomials.
- Author
-
M Bertola, M Gekhtman, and J Szmigielski
- Subjects
- *
BOUNDARY value problems , *CAUCHY problem , *BIORTHOGONAL systems , *ORTHOGONAL polynomials , *NUMERICAL solutions to nonlinear differential equations , *RANDOM matrices , *FOURIER transforms - Abstract
Cauchy biorthogonal polynomials appear in the study of special solutions to the dispersive nonlinear partial differential equation called the Degasperis-Procesi (DP) equation, as well as in certain two-matrix random matrix models. Another context in which such biorthogonal polynomials play a role is the cubic string; a third-order ODE boundary value problem -f ''' = zgf which is a generalization of the inhomogeneous string problem studied by Krein. A general class of such boundary value problems going beyond the original cubic string problem associated with the DP equation is discussed under the assumption that the source of inhomogeneity g is a discrete measure. It is shown that by a suitable choice of a generalized Fourier transform associated with these boundary value problems one can establish a Parseval type identity which aligns Cauchy biorthogonal polynomials with certain natural orthogonal systems on L2g. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
41. q-Extension of Mehta's eigenvectors of the finite Fourier transform for q, a root of unity.
- Author
-
Mesuma K Atakishiyeva, Natig M Atakishiyev, and Tom H Koornwinder
- Subjects
- *
EIGENVECTORS , *FINITE fields , *FOURIER transforms , *HERMITE polynomials , *MATHEMATICAL analysis - Abstract
It is shown that the continuous q-Hermite polynomials for q, a root of unity, have simple transformation properties with respect to the classical Fourier transform. This result is then used to construct q-extended eigenvectors of the finite Fourier transform in terms of these polynomials. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
42. An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, the unitary group and the Pauli group.
- Subjects
- *
ANGULAR momentum (Mechanics) , *QUADRATIC transformations , *FOURIER transforms , *UNITARY operators , *HILBERT space , *REPRESENTATIONS of algebras , *MATHEMATICAL decomposition , *GROUP theory - Abstract
The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained from a polar decomposition of SU(2) and is analyzed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
43. Efficient evaluation of the Fourier transform over products of Slater-type orbitals on different centers.
- Author
-
T A Niehaus, R L, ópez and, and J F Rico
- Subjects
- *
FOURIER transforms , *MOLECULAR orbitals , *OPERATOR theory , *MATHEMATICAL formulas , *QUANTUM theory , *MATHEMATICAL analysis , *ALGORITHMS - Abstract
Using the shift-operator technique, a compact formula for the Fourier transform of a product of two Slater-type orbitals located on different atomic centers is derived. The result is valid for arbitrary quantum numbers and was found to be numerically stable for a wide range of geometrical parameters and momenta. Details of the implementation are presented together with benchmark data for representative integrals. We also discuss the assets and drawbacks of alternative algorithms available and analyze the numerical efficiency of the new scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
44. The 't/2 law' for quantum random walks on the line starting in the classical state.
- Subjects
- *
RANDOM walks , *MATRICES (Mathematics) , *FOURIER transforms , *DISTRIBUTION (Probability theory) , *EIGENVALUES , *STANDARD deviations - Abstract
Based on the theory of unitary matrices, our treatment of the theory of quantum random walks simplifies and clarifies certain prior derivations based on Fourier transform methods. Given a quantum random walk on the line determined by a 2× 2 unitary matrix U, we show how the first two moments of the position probability distribution are determined by the eigenvalues of U. By varying the 'coin operator' A, we show that the leading term of the standard deviation of the position probability distribution is ct, where t denotes time and 0 [?] c [?] 1. However, it turns out that the maximum value of c, namely c = 1, is achievable when and only when the coin operator A is diagonal, and the initial state is unbiased. Starting in the classical state |0[?] [?] |1[?], our approach confirms that the maximum value of the leading term of the standard deviation of the position probability distribution is \frac{t}{2} , which, by way of known examples, is verified to be achievable. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
45. The D-bar method, inversion of certain integrals and integrability in 4 + 2 and 3 + 1 dimensions.
- Subjects
- *
INTEGRAL transforms , *DIRICHLET forms , *VON Neumann algebras , *HARMONIC functions , *CURVILINEAR coordinates , *FOURIER transforms , *CAUCHY problem , *SPECTRAL theory - Abstract
We first review a method for deriving linear and nonlinear transform pairs, which is based on the spectral analysis of an eigenvalue equation and on the formulation of a d-bar problem. Then, we present two applications of this method: (a) we derive a certain linear transform pair in one dimension, which appears in the characterization of the Dirichlet-to-Neumann map of the Laplace equation in the interior of a convex two-dimensional curvilinear domain. (b) We derive a nonlinear Fourier transform pair in four dimensions, which can be used for the solution of the Cauchy problem of an integrable generalization of the Kadomtsev-Petviashvilli equation in 4 + 2, i.e. in four spatial and two temporal dimensions. The question of reducing this equation form 4 + 2 to 3 + 1 dimensions is also discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
46. Stochastic volatility models and Kelvin waves.
- Author
-
Alex Lipton and Artur Sepp
- Subjects
- *
STOCHASTIC analysis , *MARKET volatility , *PRICE regulation , *MONTE Carlo method , *FLUID dynamics , *FOURIER transforms , *FINITE differences - Abstract
We use stochastic volatility models to describe the evolution of an asset price, its instantaneous volatility and its realized volatility. In particular, we concentrate on the Stein and Stein model (SSM) (1991) for the stochastic asset volatility and the Heston model (HM) (1993) for the stochastic asset variance. By construction, the volatility is not sign definite in SSM and is non-negative in HM. It is well known that both models produce closed-form expressions for the prices of vanilla option via the Lewis-Lipton formula. However, the numerical pricing of exotic options by means of the finite difference and Monte Carlo methods is much more complex for HM than for SSM. Until now, this complexity was considered to be an acceptable price to pay for ensuring that the asset volatility is non-negative. We argue that having negative stochastic volatility is a psychological rather than financial or mathematical problem, and advocate using SSM rather than HM in most applications. We extend SSM by adding volatility jumps and obtain a closed-form expression for the density of the asset price and its realized volatility. We also show that the current method of choice for solving pricing problems with stochastic volatility (via the affine ansatz for the Fourier-transformed density function) can be traced back to the Kelvin method designed in the 19th century for studying wave motion problems arising in fluid dynamics. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
47. The Signum function method for the generation of correlated dichotomic chains.
- Author
-
S S Apostolov, F M Izrailev, N M Makarov, Z A Mayzelis, S S Melnyk, and O V Usatenko
- Subjects
- *
FUNCTION algebras , *MATHEMATICAL sequences , *STATISTICAL correlation , *FOURIER transforms , *MATHEMATICAL convolutions , *WAVEGUIDES , *MATHEMATICAL physics - Abstract
We analyze the signum-generation method for creating random dichotomic sequences with prescribed correlation properties. The method is based on a binary mapping of the convolution of continuous random numbers with some function originated from the Fourier transform of a binary correlator. The goal of our study is to reveal conditions under which one can construct binary sequences with a given pair correlator. Our results can be used in the construction of superlattices and waveguides with selective transport properties. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
48. On continuous q-Hermite polynomials and the classical Fourier transform.
- Author
-
M K Atakishiyeva and N M Atakishiyev
- Subjects
- *
HERMITE polynomials , *FOURIER transforms , *OPERATOR theory , *DIFFERENCE equations , *MATHEMATICS - Abstract
We prove that the classical Fourier-transform operator \widehat{\cal F} intertwines two q-difference equations for the continuous q-Hermite polynomials Hn(x|q) of Rogers, which are associated with the two distinct sets of values for the parameter q: 0 < q < 1 and 1 < q < [?]. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
49. A discrete quantum model of the harmonic oscillator.
- Author
-
Natig M Atakishiyev, Anatoliy U Klimyk, and Kurt Bernardo
- Subjects
- *
HARMONIC oscillators , *QUANTUM theory , *MATHEMATICAL models , *HERMITE polynomials , *WAVE functions , *HILBERT space , *FOURIER transforms - Abstract
We construct a new model of the quantum oscillator, whose energy spectrum is equally-spaced and lower-bound, whereas the spectra of position and of momentum are a denumerable non-degenerate set of points in [[?] 1, 1] that depends on the deformation parameter q [?] (0, 1). We provide its explicit wavefunctions, both in position and momentum representations, in terms of the discrete q-Hermite polynomials. We build a Hilbert space with a unique measure, where an analogue of the fractional Fourier transform is defined in order to govern the time evolution of this discrete oscillator. In the limit when q - 1[?], one recovers the ordinary quantum harmonic oscillator. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
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