46 results on '"Mathematics"'
Search Results
2. Invertibility of nonnegative Hamiltonian operator with unbounded entries
- Author
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Wu, Deyu and Chen, Alatancang
- Subjects
- *
HAMILTONIAN operator , *SET theory , *MATHEMATICAL analysis , *DIFFERENTIAL operators , *MATHEMATICS , *QUANTUM theory - Abstract
Abstract: In this paper, the invertibility of nonnegative Hamiltonian operator with unbounded entries is studied, and the sufficient conditions for the everywhere defined bounded invertibility of nonnegative Hamiltonian operator are obtained. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
3. Self-Dual Codes and Modules for Finite Groups in Characteristic Two.
- Author
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Martínez-Pérez, Conchita and Willems, Wolfgang
- Subjects
- *
ALGEBRA , *MATHEMATICS , *MATHEMATICAL analysis , *GLEASON'S theorem (Quantum theory) , *QUANTUM theory , *GROUP theory - Abstract
Using representation theoretical methods we investigate self-dual group codes and their extensions In characteristic 2. We prove that the existence of a self-dual extended group code heavily depends on a particular structure of the group algebra KG which can be checked by an easy-to-handle criteria in elementary number theory. Surprisingly, in the binary case such a code Is doubly even If the convene of Gleason's theorem holds true, i.e., the length of the code Is divisible by 8. Furthermore, we give a short representation theoretical proof of an earlier result of Sloane and Thompson which states that a binary self-dual group code is never doubly even if the Sylow 2-subgroups of G are cyclic. It turns out that exactly In the case of a cyclic or Klein four group as Sylow 2-subgroup doubly even group codes do not exist. [ABSTRACT FROM AUTHOR]
- Published
- 2004
- Full Text
- View/download PDF
4. Deformation of Gabor systems
- Author
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Joaquim Ortega-Cerdà, Karlheinz Gröchenig, José Luis Romero, and Universitat de Barcelona
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Bandlimiting ,Anàlisi de Fourier ,Information theory ,General Mathematics ,010103 numerical & computational mathematics ,Teoria d'operadors ,Gabor frame ,01 natural sciences ,Harmonic analysis ,Set of uniqueness ,FOS: Mathematics ,Teoria quàntica ,0101 mathematics ,Mathematics ,Jitter ,Mathematics::Functional Analysis ,42C15, 42C30, 42C40 ,010102 general mathematics ,Mathematical analysis ,Gabor wavelet ,Operator theory ,Anàlisi harmònica ,Fourier analysis ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Nonlinear system ,Computer Science::Sound ,Quantum theory ,Computer Science::Computer Vision and Pattern Recognition ,Phase space ,Teoria de la informació - Abstract
We introduce a new notion for the deformation of Gabor systems. Such deformations are in general nonlinear and, in particular, include the standard jitter error and linear deformations of phase space. With this new notion we prove a strong deformation result for Gabor frames and Gabor Riesz sequences that covers the known perturbation and deformation results. Our proof of the deformation theorem requires a new characterization of Gabor frames and Gabor Riesz sequences. It is in the style of Beurling's characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of Gabor frames "without inequalities" from lattices to non-uniform sets., 31 pages, 2 figures
- Published
- 2015
5. Generalized spatial aliasing solution for the dispersion analysis of infinitely periodic multilayered composites using the finite element method
- Author
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Wael Alnahhal, Amjad J. Aref, Ratiba F. Ghachi, A. B. M. Tahidul Haque, and Jongmin Shim
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Finite element method ,Mathematical analysis ,General Engineering ,Metamaterial ,Shim (magnetism) ,02 engineering and technology ,Mixed finite element method ,021001 nanoscience & nanotechnology ,01 natural sciences ,Classical mechanics ,Dispersion relation ,Metamaterials ,Quantum theory ,0103 physical sciences ,Acoustic metamaterials ,Dispersion (waves) ,0210 nano-technology ,010301 acoustics ,Mathematics ,Extended finite element method - Abstract
The finite element (FE) method offers an efficient framework to investigate the evolution of phononic crystals which possess materials or geometric nonlinearity subject to external loading. Despite its superior efficiency, the FE method suffers from spectral distortions in the dispersion analysis of waves perpendicular to the layers in infinitely periodic multilayered composites. In this study, the analytical dispersion relation for sagittal elastic waves is reformulated in a substantially concise form, and it is employed to reproduce spatial aliasing-induced spectral distortions in FE dispersion relations. Furthermore, through an anti-aliasing condition and the effective elastic modulus theory, an FE modeling general guideline is provided to overcome the observed spectral distortions in FE dispersion relations of infinitely periodic multilayered composites, and its validity is also demonstrated. Qatar National Research Fund through Grant No. NPRP8-1568-2-666. Shim acknowledges start-up funds from the University at Buffalo (UB), and he is grateful to the support of UB Center for Computational Research.
- Published
- 2017
6. Exact Discrete Analogs of Canonical Commutation and Uncertainty Relations
- Author
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Vasily E. Tarasov
- Subjects
Discretization ,quantum theory ,canonical commutation relation ,uncertainty relations ,exact discretization ,finite difference ,010308 nuclear & particles physics ,General Mathematics ,lcsh:Mathematics ,Mathematical analysis ,Finite difference ,Canonical coordinates ,lcsh:QA1-939 ,01 natural sciences ,Canonical commutation relation ,0103 physical sciences ,Computer Science (miscellaneous) ,Applied mathematics ,Commutation ,010306 general physics ,Wave function ,Constant (mathematics) ,Engineering (miscellaneous) ,Mathematics - Abstract
An exact discretization of the canonical commutation and corresponding uncertainty relations are suggested. We prove that the canonical commutation relations of discrete quantum mechanics, which is based on standard finite difference, holds for constant wave functions only. In this paper, we use the recently proposed exact discretization of derivatives, which is based on differences that are represented by infinite series. This new mathematical tool allows us to build sensible discrete quantum mechanics based on the suggested differences and includes the correct canonical commutation and uncertainty relations.
- Published
- 2016
7. Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics
- Author
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Riccarda Rossi, Marita Thomas, Rodica Toader, and Giuliano Lazzaroni
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History ,Quantum theory ,Specific heat ,Damage variables ,Elasto-dynamics ,Heat equation ,Momentum balances ,Rate dependent ,Rate independents ,Time dependent ,Unidirectional flow ,Hysteresis ,media_common.quotation_subject ,Monotonic function ,energetic solutions ,Inertia ,01 natural sciences ,Heat capacity ,phase-field models ,Education ,Viscosity ,Physics and Astronomy (all) ,Partial damage ,Rheology ,Settore MAT/05 - Analisi Matematica ,Tensor ,0101 mathematics ,media_common ,Mathematics ,Variable (mathematics) ,heat equation ,010102 general mathematics ,Mathematical analysis ,74F05 ,rate-independent systems ,74H20 ,elastodynamics ,74C05 ,Computer Science Applications ,010101 applied mathematics ,Classical mechanics ,35Q74 ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,74R05 - Abstract
This note deals with the analysis of a model for partial damage, where the rate-independent, unidirectional flow rule for the damage variable is coupled with the rate-dependent heat equation, and with the momentum balance featuring inertia and viscosity according to Kelvin-Voigt rheology. The results presented here combine the approach from [Roubiček M2AS'09, SIAM'10] with the methods from Lazzaroni/Rossi/Thomas/Toader [WIAS Preprint 2025]. The present analysis encompasses, differently from [Roubiček SIAM'10], the monotonicity in time of damage and the dependence of the viscous tensor on damage and temperature, and, unlike [WIAS Preprint 2025], a nonconstant heat capacity and a time-dependent Dirichlet loading.
- Published
- 2016
8. On symmetries, conservation laws and invariant solutions of the foam-drainage equation
- Author
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Emrullah Yaşar, Teoman Özer, Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Anabilim Dalı., Yaşar, Emrullah, and AAG-9947-2021
- Subjects
Lagrange multipliers ,Infinitesimal ,Spacetime symmetries ,Solitary wave ,Conservation law ,Symmetry group ,Mechanics ,Tanh-coth method ,Laws of science ,Infinitesimal transformations ,Conservation Laws ,Lie Point Symmetries ,Self-Adjointness ,Differential-equations ,Invariant solutions ,Lie symmetries ,Similarity reductions ,Lie symmetry groups ,Nonlocal ,Conservation laws ,Mathematical physics ,Mathematics ,Physical properties ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Lie group method ,Lie group ,Invariant (physics) ,Lagrangian approaches ,Symmetry groups ,Condensed Matter::Soft Condensed Matter ,Similarity solution ,Mechanics of Materials ,Quantum theory ,Similarity solutions ,Homogeneous space ,Reductions - Abstract
This study deals with symmetry group properties and conservation laws of the foam-drainage equation. Firstly, we study the classical Lie symmetries, optimal systems, similarity reductions and similarity solutions of the foam-drainage equation which are obtained through the Lie group method of infinitesimal transformations. Secondly, using the new general theorem on non-local conservation laws and partial Lagrangian approach, local and non-local conservation laws are also studied and, finally, non-classical symmetries are derived.
- Published
- 2011
9. Almost quantum correlations
- Author
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Matty J. Hoban, Yelena Guryanova, Miguel Navascués, Antonio Acín, and Universitat Politècnica de Catalunya. Institut de Ciències Fotòniques
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Quantum correlations ,Quantum t-design ,probability ,FOS: Physical sciences ,General Physics and Astronomy ,01 natural sciences ,Article ,mathematical analysis ,General Biochemistry, Genetics and Molecular Biology ,010305 fluids & plasmas ,Quantum nonlocality ,Theoretical physics ,0103 physical sciences ,Quantum operation ,mathematical computing ,010306 general physics ,Set (psychology) ,Quantum ,Mathematics ,Quantum Physics ,Quantum discord ,Multidisciplinary ,Física [Àrees temàtiques de la UPC] ,quantum mechanics ,correlational study ,Quàntums, Teoria dels ,General Chemistry ,theoretical study ,measurement precision ,structure analysis ,correlation ,Quantum theory ,Quantum process ,conceptual framework ,Quantum Physics (quant-ph) - Abstract
There have been a number of attempts to derive the set of quantum non-local correlations from reasonable physical principles. Here we introduce $\tilde{Q}$, a set of multipartite supra-quantum correlations that has appeared under different names in fields as diverse as graph theory, quantum gravity and quantum information science. We argue that $\tilde{Q}$ may correspond to the set of correlations of a reasonable physical theory, in which case the research program to reconstruct quantum theory from device-independent principles is met with strong obstacles. In support of this conjecture, we prove that $\tilde{Q}$ is closed under classical operations and satisfies the physical principles of Non-Trivial Communication Complexity, No Advantage for Nonlocal Computation, Macroscopic Locality and Local Orthogonality. We also review numerical evidence that almost quantum correlations satisfy Information Causality., Comment: 15+2 pages, 1 figure
- Published
- 2015
10. A lattice kinetic scheme for incompressible viscous flows with heat transfer
- Author
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Takaji Inamuro
- Subjects
Hot Temperature ,General Mathematics ,Kinetic scheme ,Lattice Boltzmann methods ,General Physics and Astronomy ,Physics::Fluid Dynamics ,Diffusion ,Incompressible flow ,Pressure ,Computer Simulation ,Colloids ,Particle Size ,incompressible flow ,Mathematics ,Natural convection ,Models, Statistical ,Mathematical analysis ,General Engineering ,Kinetics ,Distribution function ,Classical mechanics ,Continuity equation ,Energy Transfer ,Models, Chemical ,lattice Boltzmann method ,Heat transfer ,Compressibility ,Quantum Theory ,Gases ,Crystallization ,Rheology ,kinetic scheme - Abstract
A lattice kinetic scheme for incompressible viscous flows with heat transfer is devel- oped based on the lattice Boltzmann method. In the new scheme, macroscopic vari- ables are calculated without velocity distribution functions. Thus, the scheme can save computer memory because there is no need to store the velocity distribution functions. Governing equations for the macroscopic variables are obtained by apply- ing the asymptotic theory. The continuity equation, the Navier-Stokes equations, and the convection-diffusion equation for fluid temperature are obtained with rel- ative errors of O(e 2 ), where e is a small parameter that is of the same order as a lattice spacing and is related to a relaxation parameter. In order to verify the accu- racy of the scheme, natural convection flows in a square cavity are simulated, and the calculated results are in good agreement with available standard results.
- Published
- 2002
11. Calculating work in adiabatic two-level quantum Markovian master equations: A characteristic function method
- Author
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Fei Liu
- Subjects
Work (thermodynamics) ,Characteristic function (probability theory) ,Statistical Mechanics (cond-mat.stat-mech) ,Mathematical analysis ,FOS: Physical sciences ,Probability density function ,Symmetry (physics) ,Markov Chains ,Master equation ,Quantum Theory ,Statistical physics ,Adiabatic process ,Quantum statistical mechanics ,Quantum ,Condensed Matter - Statistical Mechanics ,Mathematics - Abstract
We present a characteristic function method to calculate the probability density functions of the inclusive work in the adiabatic two-level quantum Markovian master equations. These systems are steered by some slowly varying parameters and the dissipations may depend on time. Our theory is based on the interpretation of the quantum jump for the master equations. In addition to the calculation, we also find that the fluctuation properties of the work can be described by the symmetry of the characteristic functions, which is exactly the same as the case of the isolated systems. A periodically driven two-level model is used to show the method., 2 figures
- Published
- 2014
12. Improved transfer matrix methods for calculating quantum transmission coefficient
- Author
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Vishal Kumar and Debabrata Biswas
- Subjects
Quantum Physics ,Condensed Matter - Mesoscale and Nanoscale Physics ,Mathematical analysis ,Transfer-matrix method (optics) ,FOS: Physical sciences ,Models, Theoretical ,Transfer matrix ,WKB approximation ,Domain (mathematical analysis) ,Schrödinger equation ,symbols.namesake ,Matrix (mathematics) ,Mesoscale and Nanoscale Physics (cond-mat.mes-hall) ,symbols ,Quantum Theory ,Boundary value problem ,Transmission coefficient ,Quantum Physics (quant-ph) ,Algorithms ,Mathematics - Abstract
Methods for calculating the transmission coefficient are proposed, all of which arise from improved non-reflecting WKB boundary conditions at the edge of the computational domain in 1-dimensional geometries. In the first, the Schr\"{o}dinger equation is solved numerically while the second is a transfer matrix (TM) algorithm where the potential is approximated by steps, but with the first and last matrix modified to reflect the new boundary condition. Both methods give excellent results with first order WKB boundary conditions. The third uses the transfer matrix method with third order WKB boundary conditions. For the the parabolic potential, the average error for the modified third order TM method reduces by factor of 4100 over the unmodified TM method.
- Published
- 2014
13. Few-cycle optical rogue waves:complex modified Korteweg-de Vries equation
- Author
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Robert Erdélyi, Lihong Wang, Jingsong He, Linjing Li, and K. Porsezian
- Subjects
Optics and Photonics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Electromagnetic Radiation ,Mathematical analysis ,Physics::Optics ,Nonlinear optics ,Optical rogue waves ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Models, Theoretical ,Nonlinear system ,Transformation (function) ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Nonlinear Dynamics ,Lax pair ,Quantum Theory ,Computer Simulation ,Rogue wave ,Exactly Solvable and Integrable Systems (nlin.SI) ,Korteweg–de Vries equation ,Ultrashort pulse ,Nonlinear Sciences::Pattern Formation and Solitons ,Mathematical Physics ,Mathematics - Abstract
In this paper, we consider the complex modified Korteweg-de Vries (mKdV) equation as a model of few-cycle optical pulses. Using the Lax pair, we construct a generalized Darboux transformation and systematically generate the first-, second- and third-order rogue wave solutions and analyze the nature of evolution of higher-order rogue waves in detail. Based on detailed numerical and analytical investigations, we classify the higher-order rogue waves with respect to their intrinsic structure, namely, fundamental pattern, triangular pattern, and ring pattern. We also present several new patterns of the rogue wave according to the standard and non-standard decomposition. The results of this paper explain the generalization of higher-order rogue waves in terms of rational solutions. We apply the contour line method to obtain the analytical formulas of the length and width of the first-order RW of the complex mKdV and the NLS equations. In nonlinear optics, the higher-order rogue wave solutions found here will be very useful to generate high-power few-cycle optical pulses which will be applicable in the area of ultra-short pulse technology., Comment: 31 pages, 22 figures, accepted by Phys.Rev.E
- Published
- 2014
- Full Text
- View/download PDF
14. Optimal shape and location of sensors or actuators in PDE models
- Author
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Yannick Privat, Enrique Zuazua, and Emmanuel Trélat
- Subjects
Optimization ,Observability ,Partial differential equations (PDE) ,Boundary (topology) ,Linear systems ,Domain (mathematical analysis) ,PDE surface ,Distributed parameter system ,Free boundary problem ,Boundary value problem ,Sensors or actuators ,Evolution systems ,Mathematics ,Partial differential equation ,Sensors and actuators ,Space dimensions ,Mathematical analysis ,Partial differential equations ,Spectrum analysis ,Partial observation ,Quantum theory ,Distributed parameter systems ,Quantum ergodicity ,Actuators - Abstract
We investigate the problem of optimizing the shape and location of sensors and actuators for evolution systems driven by distributed parameter systems or partial differential equations (PDE). We consider wave, Schrödinger and heat equations on an arbitrary domain Ω, in any space dimension, and with suitable boundary conditions (if there is a boundary) which can be of Dirichlet, Neumann, mixed or Robin type. This kind of problem is frequently encountered in applications where one aims, for instance, at maximizing the quality of reconstruction of the solution, using only a partial observation. From the mathematical point of view, using probabilistic considerations we model this problem as that of maximizing the so-called randomized observability constant, over all possible subdomains of Ω having a prescribed measure. The spectral analysis of this problem reveals intimate connections with the theory of quantum chaos. More precisely, we provide a solution to this problem when the domain Ω satisfies suitable quantum ergodicity assumptions.
- Published
- 2014
15. Logarithmic potential of Hermite polynomials and information entropies of the harmonic oscillator eigenstates
- Author
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Jorge Sánchez-Ruiz
- Subjects
Eigenvalues and eigenfunctions ,Harmonic oscillators ,Hermite polynomials ,Logarithm ,Matemáticas ,Entropy ,Mathematical analysis ,Statistical and Nonlinear Physics ,Polynomials ,Upper and lower bounds ,Quantum harmonic oscillator ,Quantum theory ,Entropy (information theory) ,Mathematical Physics ,Harmonic oscillator ,Stationary state ,Eigenvalues and eigenvectors ,Mathematical physics ,Mathematics - Abstract
13 pages, 1 figure.-- PACS nrs.: 03.65.Ge, 02.10.Nj, 02.10.Sp. MR#: MR1471913 (99c:81031) Zbl#: Zbl 0891.33007 The problem of calculating the information entropy in both position and momentum spaces for the nth stationary state of the one-dimensional quantum harmonic oscillator reduces to the evaluation of the logarithmic potential $V_n(t)=-\int e\sp {-x\sp 2}H_n\sp 2(x)\log -t x$ at the zeros of the Hermite polynomial Hn(x). Here, a closed analytical expression for Vn(t) is obtained, which in turn yields an exact analytical expression for the entropies when the exact location of the zeros of Hn(x) is known. An inequality for the values of Vn(t) at the zeros of Hn(x) is conjectured, which leads to a new, nonvariational, upper bound for the entropies. Finally, the exact formula for Vn(t) is written in an alternative way, which allows the entropies to be expressed in terms of the even-order spectral moments of the Hermite polynomials. The asymptotic (n>>1) limit of this alternative expression for the entropies is discussed, and the conjectured upper bound for the entropies is proved to be asymptotically valid The author gratefully acknowledges the financial support from the Fundació Aula (Barcelona, Spain). Publicado
- Published
- 1997
16. Continuous and discrete Schrödinger systems with parity-time-symmetric nonlinearities
- Author
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Ziad H. Musslimani, Amarendra K. Sarma, Mohammad-Ali Miri, and Demetrios N. Christodoulides
- Subjects
Models, Statistical ,Discretization ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Parity (physics) ,Mathematics::Spectral Theory ,Invariant (physics) ,Nonlinear system ,symbols.namesake ,Models, Chemical ,Nonlinear Dynamics ,Oscillometry ,symbols ,Quantum Theory ,Computer Simulation ,Soliton ,Nonlinear Sciences::Pattern Formation and Solitons ,Schrödinger's cat ,Mathematical physics ,Mathematics - Abstract
We investigate the dynamical behavior of continuous and discrete Schrödinger systems exhibiting parity-time (PT) invariant nonlinearities. We show that such equations behave in a fundamentally different fashion than their nonlinear Schrödinger counterparts. In particular, the PT-symmetric nonlinear Schrödinger equation can simultaneously support both bright and dark soliton solutions. In addition, we study a discretized version of this PT-nonlinear Schrödinger equation on a lattice. When only two elements are involved, by obtaining the underlying invariants, we show that this system is fully integrable and we identify the PT-symmetry-breaking conditions. This arrangement is unique in the sense that the exceptional points are fully dictated by the nonlinearity itself.
- Published
- 2013
17. Compressed Modes for Variational Problems in Mathematics and Physics
- Author
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Rongjie Lai, Vidvuds Ozoliņš, Russel E. Caflisch, and Stanley Osher
- Subjects
FOS: Physical sciences ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,Variational principle ,FOS: Mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Linear combination ,Eigenvalues and eigenvectors ,Mathematics ,Multidisciplinary ,Physics ,Mathematical analysis ,Numerical Analysis (math.NA) ,Computational Physics (physics.comp-ph) ,Models, Theoretical ,Eigenfunction ,Variational optimization ,010101 applied mathematics ,Formalism (philosophy of mathematics) ,Physical Sciences ,symbols ,Quantum Theory ,Physics - Computational Physics ,Schrödinger's cat - Abstract
This paper describes a general formalism for obtaining localized solutions to a class of problems in mathematical physics, which can be recast as variational optimization problems. This class includes the important cases of Schr\"odinger's equation in quantum mechanics and electromagnetic equations for light propagation in photonic crystals. These ideas can also be applied to develop a spatially localized basis that spans the eigenspace of a differential operator, for instance, the Laplace operator, generalizing the concept of plane waves to an orthogonal real-space basis with multi-resolution capabilities., Comment: 18 pages
- Published
- 2013
18. Spectral estimates on the sphere
- Author
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Jean Dolbeault, Maria J. Esteban, Ari Laptev, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Department of Mathematics [Imperial College London], and Imperial College London
- Subjects
Unit sphere ,n-sphere ,N-SPHERE ,NIRENBERG ,partial differential operators on manifolds ,Sobolev inequality ,Mathematics, Applied ,Inverse ,Gagliardo-Nirenberg-Sobolev inequalities ,01 natural sciences ,MATHEMATICS ,EIGENVALUES ,0102 Applied Mathematics ,one bound state Keller–Lieb–Thirring inequality ,81Q20 ,Schrodinger operator ,46E35 ,ground state ,spectral problems ,Mathematics ,Schrödinger operator ,Numerical Analysis ,one bound state Keller-Lieb-Thirring inequality ,estimation of eigenvalues ,Applied Mathematics ,Mathematical analysis ,Mathematics::Spectral Theory ,16. Peace & justice ,010101 applied mathematics ,81Q10 ,Physical Sciences ,symbols ,35P15 ,Interpolation ,Analysis of PDEs (math.AP) ,RIEMANNIAN-MANIFOLDS ,BOUNDS ,58E35 ,Gagliardo–Nirenberg–Sobolev inequalities ,58J50 ,0101 Pure Mathematics ,symbols.namesake ,Mathematics - Analysis of PDEs ,81Q35 ,47A75 ,26D10 ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,LOGARITHMIC SOBOLEV INEQUALITIES ,0101 mathematics ,logarithmic Sobolev inequality ,Eigenvalues and eigenvectors ,Science & Technology ,Euclidean space ,010102 general mathematics ,ROZENBLUM ,quantum theory ,interpolation ,SCHRODINGER-OPERATORS ,Analysis ,Schrödinger's cat - Abstract
In this article we establish optimal estimates for the first eigenvalue of Schrödinger operators on the [math] -dimensional unit sphere. These estimates depend on [math] norms of the potential, or of its inverse, and are equivalent to interpolation inequalities on the sphere. We also characterize a semiclassical asymptotic regime and discuss how our estimates on the sphere differ from those on the Euclidean space.
- Published
- 2013
19. Classical density functional theory and the phase-field crystal method using a rational function to describe the two-body direct correlation function
- Author
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Katsuyo Thornton, N. Pisutha-Arnond, V. W. L. Chan, Vikram Gavini, and Mrinal Iyer
- Subjects
Crystallography ,Phase field crystal ,Statistics as Topic ,Mathematical analysis ,Predictive capability ,Numerical Analysis, Computer-Assisted ,Rational function ,Models, Theoretical ,Correlation function (quantum field theory) ,Frequency domain ,Quantum Theory ,Thermodynamics ,Computer Simulation ,Density functional theory ,Statistical physics ,Material properties ,Representation (mathematics) ,Algorithms ,Mathematics - Abstract
We introduce a new approach to represent a two-body direct correlation function (DCF) in order to alleviate the computational demand of classical density functional theory (CDFT) and enhance the predictive capability of the phase-field crystal (PFC) method. The approach utilizes a rational function fit (RFF) to approximate the two-body DCF in Fourier space. We use the RFF to show that short-wavelength contributions of the two-body DCF play an important role in determining the thermodynamic properties of materials. We further show that using the RFF to empirically parametrize the two-body DCF allows us to obtain the thermodynamic properties of solids and liquids that agree with the results of CDFT simulations with the full two-body DCF without incurring significant computational costs. In addition, the RFF can also be used to improve the representation of the two-body DCF in the PFC method. Last, the RFF allows for a real-space reformulation of the CDFT and PFC method, which enables descriptions of nonperiodic systems and the use of nonuniform and adaptive grids.
- Published
- 2013
20. Approximate mean-field equations of motion for quasi-two-dimensional Bose-Einstein-condensate systems
- Author
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Michael Krygier, Charles W. Clark, Hadayat Seddiqi, Mark Edwards, and Brandon Benton
- Subjects
Gaussian ,01 natural sciences ,010305 fluids & plasmas ,law.invention ,symbols.namesake ,law ,0103 physical sciences ,Computer Simulation ,010306 general physics ,Wave function ,Mathematics ,Condensed Matter::Quantum Gases ,Condensed Matter::Other ,Plane (geometry) ,Fourth power ,Mathematical analysis ,Equations of motion ,Models, Theoretical ,Gross–Pitaevskii equation ,Classical mechanics ,Variational method ,symbols ,Quantum Theory ,Thermodynamics ,Algorithms ,Bose–Einstein condensate - Abstract
We present a method for approximating the solution of the three-dimensional, time-dependent Gross-Pitaevskii equation (GPE) for Bose-Einstein-condensate systems where the confinement in one dimension is much tighter than in the other two. This method employs a hybrid Lagrangian variational technique whose trial wave function is the product of a completely unspecified function of the coordinates in the plane of weak confinement and a Gaussian in the strongly confined direction having a time-dependent width and quadratic phase. The hybrid Lagrangian variational method produces equations of motion that consist of (1) a two-dimensional (2D) effective GPE whose nonlinear coefficient contains the width of the Gaussian and (2) an equation of motion for the width that depends on the integral of the fourth power of the solution of the 2D effective GPE. We apply this method to the dynamics of Bose-Einstein condensates confined in ring-shaped potentials and compare the approximate solution to the numerical solution of the full 3D GPE.
- Published
- 2012
21. Quantum calculus of classical vortex images, integrable models and quantum states
- Author
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Oktay K. Pashaev, TR57865, Pashaev, Oktay, and Izmir Institute of Technology. Mathematics
- Subjects
History ,Functional analysis ,Vortex flow ,Mathematical analysis ,Quantum calculus ,01 natural sciences ,Control nonlinearities ,010305 fluids & plasmas ,Computer Science Applications ,Education ,Vortex ,Schrödinger equation ,symbols.namesake ,Bernoulli's principle ,Nonlinear system ,Quantum state ,Quantum theory ,0103 physical sciences ,Euler's formula ,symbols ,Calculations ,010306 general physics ,Analytic function ,Mathematics - Abstract
International Conference on Quantum Science and Applications, ICQSA 2016; Eskisehir Osmangazi University Congress and Culture CentreEskisehir; Turkey; 25 May 2016 through 27 May 2016, From two circle theorem described in terms of q-periodic functions, in the limit q→1 we have derived the strip theorem and the stream function for N vortex problem. For regular N-vortex polygon we find compact expression for the velocity of uniform rotation and show that it represents a nonlinear oscillator. We describe q-dispersive extensions of the linear and nonlinear Schrodinger equations, as well as the q-semiclassical expansions in terms of Bernoulli and Euler polynomials. Different kind of q-analytic functions are introduced, including the pq-analytic and the golden analytic functions.
- Published
- 2016
22. Triangular rogue wave cascades
- Author
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David J. Kedziora, Nail Akhmediev, and Adrian Ankiewicz
- Subjects
Models, Statistical ,Mathematical analysis ,Nonlinear optics ,Nonlinear system ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Classical mechanics ,Transformation (function) ,Nonlinear Dynamics ,Cascade ,Cluster (physics) ,Quantum Theory ,Computer Simulation ,Rogue wave ,Rheology ,Nonlinear Sciences::Pattern Formation and Solitons ,Parametric statistics ,Mathematics ,Complement (set theory) - Abstract
By numerically applying the recursive Darboux transformation technique, we study high-order rational solutions of the nonlinear Schr\"odinger equation that appear spatiotemporally as triangular arrays of Peregrine solitons. These can be considered as rogue wave cascades and complement previously discovered circular cluster forms. In this analysis, we reveal a general parametric restriction for their existence and investigate the interplay between cascade and cluster forms. As a result, we demonstrate how to generate many more hybrid rogue wave solutions, including semicircular clusters that resemble claws.
- Published
- 2012
23. Relaxation in finite and isolated classical systems: An extension of Onsager's regression hypothesis
- Author
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Marcus V. S. Bonança
- Subjects
Thermal equilibrium ,Models, Statistical ,Statistical Mechanics (cond-mat.stat-mech) ,Dynamical systems theory ,Mathematical analysis ,Finite system ,FOS: Physical sciences ,Statistical mechanics ,Nonlinear Sciences - Chaotic Dynamics ,Regression ,Microcanonical ensemble ,Models, Chemical ,Reciprocity (electromagnetism) ,Quantum Theory ,Regression Analysis ,Thermodynamics ,Computer Simulation ,Statistical physics ,Chaotic Dynamics (nlin.CD) ,Condensed Matter - Statistical Mechanics ,Mathematics - Abstract
In order to derive the reciprocity relations, Onsager formulated a relation between thermal equilibrium fluctuations and relaxation widely known as regression hypothesis. It is shown in the present work how such relation can be extended to finite and isolated classical systems. This extension is derived from the fluctuation-dissipation theorem for the microcanonical ensemble. The results are exemplified with a nonintegrable system in order to motivate possible applications to dynamical systems and statistical mechanics of finite systems., 5 pages, 4 figures, final version
- Published
- 2012
24. Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits
- Author
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David J. Kedziora, Adrian Ankiewicz, and Nail Akhmediev
- Subjects
Breather ,Hyperbolic function ,Mathematical analysis ,Degenerate energy levels ,Nonlinear system ,symbols.namesake ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Models, Chemical ,Nonlinear Dynamics ,Inverse scattering problem ,symbols ,Quantum Theory ,Computer Simulation ,Rogue wave ,Nonlinear Sciences::Pattern Formation and Solitons ,Nonlinear Schrödinger equation ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We present an explicit analytic form for the two-breather solution of the nonlinear Schr\"odinger equation with imaginary eigenvalues. It describes various nonlinear combinations of Akhmediev breathers and Kuznetsov-Ma solitons. The degenerate case, when the two eigenvalues coincide, is quite involved. The standard inverse scattering technique does not generally provide an answer to this scenario. We show here that the solution can still be found as a special limit of the general second-order expression and appears as a mixture of polynomials with trigonometric and hyperbolic functions. A further restriction of this particular case, where the two eigenvalues are equal to $i$, produces the second-order rogue wave with two free parameters considered as differential shifts. The illustrations reveal a precarious dependence of wave profile on the degenerate eigenvalues and differential shifts. Thus we establish a hierarchy of second-order solutions, revealing the interrelated nature of the general case, the rogue wave, and the degenerate breathers.
- Published
- 2012
25. Operator solutions for fractional Fokker-Planck equations
- Author
-
D. Babusci, Gérard Duchamp, G. Dattoli, K. A. Penson, and Katarzyna Górska
- Subjects
Models, Molecular ,Statistical Mechanics (cond-mat.stat-mech) ,Fractional equations ,Mathematical analysis ,FOS: Physical sciences ,Order (ring theory) ,Type (model theory) ,01 natural sciences ,010305 fluids & plasmas ,Fractional calculus ,Set (abstract data type) ,Operator (computer programming) ,Exact results ,Models, Chemical ,0103 physical sciences ,Quantum Theory ,Fokker–Planck equation ,Computer Simulation ,010306 general physics ,Condensed Matter - Statistical Mechanics ,Mathematics - Abstract
We obtain exact results for fractional equations of Fokker-Planck type using evolution operator method. We employ exact forms of one-sided Levy stable distributions to generate a set of self-reproducing solutions. Explicit cases are reported and studied for various fractional order of derivatives, different initial conditions, and for different versions of Fokker-Planck operators., 4 pages, 3 figures
- Published
- 2011
26. Non-Hermitian Euclidean random matrix theory
- Author
-
Arthur Goetschy, Sergey E. Skipetrov, Laboratoire de physique et modélisation des milieux condensés (LPM2C), Université Joseph Fourier - Grenoble 1 (UJF)-Centre National de la Recherche Scientifique (CNRS), and ANR-06-BLAN-0096,CAROL,Cold Atoms for Random Optical Laser(2006)
- Subjects
Movement ,FOS: Physical sciences ,01 natural sciences ,010305 fluids & plasmas ,Diffusion ,Matrix (mathematics) ,0103 physical sciences ,Random compact set ,[PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech] ,010306 general physics ,Condensed Matter - Statistical Mechanics ,Eigenvalues and eigenvectors ,Eigendecomposition of a matrix ,Mathematics ,Probability ,Random field ,Models, Statistical ,Statistical Mechanics (cond-mat.stat-mech) ,Fourier Analysis ,Physics ,Mathematical analysis ,Random element ,Disordered Systems and Neural Networks (cond-mat.dis-nn) ,Condensed Matter - Disordered Systems and Neural Networks ,Models, Theoretical ,Hermitian matrix ,Quantum Theory ,Random matrix ,Algorithms - Abstract
We develop a theory for the eigenvalue density of arbitrary non-Hermitian Euclidean matrices. Closed equations for the resolvent and the eigenvector correlator are derived. The theory is applied to the random Green's matrix relevant to wave propagation in an ensemble of point-like scattering centers. This opens a new perspective in the study of wave diffusion, Anderson localization, and random lasing., Comment: 11 pages, 9 figures
- Published
- 2011
27. Mean-Field Approximation for Spacing Distribution Functions in Classical Systems
- Author
-
Theodore L. Einstein, Alberto Pimpinelli, and Diego Luis González
- Subjects
Models, Molecular ,Statistical Mechanics (cond-mat.stat-mech) ,Mathematical analysis ,FOS: Physical sciences ,Interpretation (model theory) ,Set (abstract data type) ,Interval approximation ,Distribution function ,Mean field theory ,Models, Chemical ,Quantum Theory ,Computer Simulation ,Colloids ,Stress, Mechanical ,Condensed Matter - Statistical Mechanics ,Mathematics - Abstract
We propose a mean-field method to calculate approximately the spacing distribution functions ${p}^{(n)}(s)$ in one-dimensional classical many-particle systems. We compare our method with two other commonly used methods, the independent interval approximation and the extended Wigner surmise. In our mean-field approach, ${p}^{(n)}(s)$ is calculated from a set of Langevin equations, which are decoupled by using a mean-field approximation. We find that in spite of its simplicity, the mean-field approximation provides good results in several systems. We offer many examples illustrating that the three previously mentioned methods give a reasonable description of the statistical behavior of the system. The physical interpretation of each method is also discussed.
- Published
- 2011
- Full Text
- View/download PDF
28. Complexity and nonseparability of classical Liouvillian dynamics
- Author
-
Tomaž Prosen
- Subjects
Models, Statistical ,Logarithm ,Schmidt decomposition ,Fourier Analysis ,Entropy ,Physics ,Mathematical analysis ,FOS: Physical sciences ,Models, Theoretical ,Nonlinear Sciences - Chaotic Dynamics ,Markov Chains ,Nonlinear system ,Exponential growth ,Nonlinear Dynamics ,Phase space ,Applied mathematics ,Quantum Theory ,Thermodynamics ,Chaotic Dynamics (nlin.CD) ,Linear growth ,Algorithms ,Mathematics - Abstract
We propose a simple complexity indicator of classical Liouvillian dynamics, namely the separability entropy, which determines the logarithm of an effective number of terms in a Schmidt decomposition of phase space density with respect to an arbitrary fixed product basis. We show that linear growth of separability entropy provides stricter criterion of complexity than Kolmogorov-Sinai entropy, namely it requires that dynamics is exponentially unstable, non-linear and non-markovian., Comment: Revised version, 5 pages (RevTeX), with 6 pdf-figures
- Published
- 2010
29. Modified path integral solution of fokker-planck equation: Response and bifurcation of nonlinear systems
- Author
-
Pankaj Kumar and S. Narayanan
- Subjects
New approaches ,White noise ,Harmonic analysis ,Probability density function ,Non-Gaussian ,Duffing oscillator ,Non-Linearity ,Mathematics ,Fokker Planck ,Stochastic systems ,Applied Mathematics ,Mathematical analysis ,General Medicine ,Path integration ,Nonlinear equations ,Control nonlinearities ,Bifurcation behavior ,Classical mechanics ,Stochastic jumps ,Fokker–Planck equation ,Harmonic excitation ,Reconnaissance aircraft ,Path integral ,Nonlinear stochastic dynamics ,Transitional probability ,Random vibrations ,Fokker Planck equation ,Duffing equation ,Harmonic (mathematics) ,White noise excitation ,Colored noise ,Kolmogorov equations ,Random response ,Nonlinear systems ,Delta functions ,Random excitations ,Mechanical Engineering ,PI method ,Modified path ,Nonlinear system ,Gauss-Legendre integration ,Probability distributions ,Control and Systems Engineering ,Colors of noise ,Bifurcation (mathematics) ,Quantum theory ,Structural dynamics ,Timing jitter ,Numerical implementation - Abstract
Response of nonlinear systems subjected to harmonic, parametric, and random excitations is of importance in the field of structural dynamics. The transitional probability density function (PDF) of the random response of nonlinear systems under white or colored noise excitation (delta correlated) is governed by both the forward Fokker–Planck (FP) and the backward Kolmogorov equations. This paper presents a new approach for efficient numerical implementation of the path integral (PI) method in the solution of the FP equation for some nonlinear systems subjected to white noise, parametric, and combined harmonic and white noise excitations. The modified PI method is based on a non-Gaussian transition PDF and the Gauss–Legendre integration scheme. The effects of white noise intensity, amplitude, and frequency of harmonic excitation and the level of nonlinearity on stochastic jump and bifurcation behaviors of a hardening Duffing oscillator are also investigated.
- Published
- 2010
30. Comparison among Various Expressions of Complex Admittance for Quantum System in Contact with Heat Reservoir
- Author
-
Mizuhiko Saeki, Chikako Uchiyama, Takashi Mori, and Seiji Miyashita
- Subjects
Hot Temperature ,Admittance ,Thermal reservoir ,Statistical Mechanics (cond-mat.stat-mech) ,Mathematical analysis ,Equations of motion ,FOS: Physical sciences ,Physics::Classical Physics ,Expression (mathematics) ,Closed and exact differential forms ,Energy Transfer ,Models, Chemical ,Kubo formula ,Quantum Theory ,Computer Simulation ,Born approximation ,Quantum statistical mechanics ,Condensed Matter - Statistical Mechanics ,Mathematics - Abstract
Relation among various expressions of the complex admittance for quantum systems in contact with heat reservoir is studied. Exact expressions of the complex admittance are derived in various types of formulations of equations of motion under contact with heat reservoir. Namely, the complex admittance is studied in the relaxation method and the external-field method. In the former method, the admittance is calculated using the Kubo formula for quantum systems in contact with heat reservoir in no external driving fields, while in the latter method the admittance is directly calculated from equations of motion with external driving terms. In each method, two types of equation of motions are considered, i.e., the time-convolution (TC) equation and time-convolutionless (TCL) equation. That is, the full of the four cases are studied. It is turned out that the expression of the complex admittance obtained by using the relaxation method with the TC equation exactly coincides with that obtained by using the external-field method with the TC equation, while other two methods give different forms. It is also explicitly demonstrated that all the expressions of the complex admittance coincide with each other in the lowest Born approximation for the systemreservoir interaction. The formulae necessary for the higher order expansions in powers of the system-reservoir interaction are derived, and also the expressions of the admittance in the n-th order approximation are given. To characterize the TC and TCL methods, we study the expressions of the admittances of two exactly solvable models. Each exact form of admittance is compared with the results of the two methods in the lowest Born approximation. It is found that depending on the model, either of TC and TCL would be the better method., 34pages, no figure
- Published
- 2009
31. Exactly solvable quantum Sturm-Liouville problems
- Author
-
Esra Tigrak-Ulas, Oktay K. Pashaev, Şirin A. Büyükaşık, Kapteyn Astronomical Institute, TR114692, TR57865, Atılgan Büyükaşık, Şirin, Pashaev, Oktay, and Izmir Institute of Technology. Mathematics
- Subjects
polynomials ,Lagrangian mechanics ,Quantum capacity ,FREQUENCY ,Quantum mechanics ,Polynomials ,Open quantum system ,Quantum error correction ,Sturm-Liouville equation ,Quantum operation ,Oscillators ,Quantum information ,FIELD ,DEPENDENT HARMONIC-OSCILLATOR ,Mathematical Physics ,Mathematics ,Mathematical physics ,Mathematical analysis ,Statistical and Nonlinear Physics ,parametric oscillators ,quantum theory ,COHERENT STATES ,harmonic oscillators ,Quantum process ,Quantum algorithm ,Quantum dissipation ,Sturm–Liouville problem - Abstract
The harmonic oscillator with time-dependent parameters covers a broad spectrum of physical problems from quantum transport, quantum optics, and quantum information to cosmology. Several methods have been developed to quantize this fundamental system, such as the path integral method, the Lewis-Riesenfeld time invariant method, the evolution operator or dynamical symmetry method, etc. In all these methods, solution of the quantum problem is given in terms of the classical one. However, only few exactly solvable problems of the last one, such as the damped oscillator or the Caldirola-Kanai model, have been treated. The goal of the present paper is to introduce a wide class of exactly solvable quantum models in terms of the Sturm-Liouville problem for classical orthogonal polynomials. This allows us to solve exactly the corresponding quantum parametric oscillators with specific damping and frequency dependence, which can be considered as quantum Sturm-Liouville problems., İzmir Institute of Technology Grant No. BAP 2008IYTE32
- Published
- 2009
32. Radiation boundary conditions for the numerical solution of the three-dimensional time-dependent Schrödinger equation with a localized interaction
- Author
-
Marco Heinen and H.-J. Kull
- Subjects
Mathematical analysis ,Laplace transforms ,Spherical harmonics ,Mixed boundary condition ,angular momentum ,quantum theory ,Singular boundary method ,Boundary knot method ,Schrodinger equation ,Poincaré–Steklov operator ,Robin boundary condition ,Green's function methods ,Neumann boundary condition ,ddc:530 ,Boundary value problem ,Mathematics - Abstract
Exact radiation boundary conditions on the surface of a sphere are presented for the single-particle time-dependent Schr\"odinger equation with a localized interaction. With these boundary conditions, numerical computations of spatially unbounded outgoing wave solutions can be restricted to the finite volume of a sphere. The boundary conditions are expressed in terms of the free-particle Green's function for the outside region. The Green's function is analytically calculated by an expansion in spherical harmonics and by the method of Laplace transformation. For each harmonic number a discrete boundary condition between the function values at adjacent radial grid points is obtained. The numerical method is applied to quantum tunneling through a spherically symmetric potential barrier with different angular-momentum quantum numbers $l$. Calculations for $l=0$ are compared to exact theoretical results.
- Published
- 2009
33. Linear augmented Slater-type orbital method for free standing clusters
- Author
-
James Glimm, K. S. Kang, Michael McGuigan, James W. Davenport, and David E. Keyes
- Subjects
Dirichlet problem ,Mathematical analysis ,Finite difference method ,General Chemistry ,Slater-type orbital ,Schrödinger equation ,Computational Mathematics ,symbols.namesake ,Multigrid method ,symbols ,Linear Models ,Quantum Theory ,Poisson's equation ,Multipole expansion ,Basis set ,Palladium ,Mathematics - Abstract
We have developed a Scalable Linear Augmented Slater-Type Orbital (LASTO) method for electronic-structure calculations on free-standing atomic clusters. As with other linear methods we solve the Schrodinger equation using a mixed basis set consisting of numerical functions inside atom-centered spheres and matched onto tail functions outside. The tail functions are Slater-type orbitals, which are localized, exponentially decaying functions. To solve the Poisson equation between spheres, we use a finite difference method replacing the rapidly varying charge density inside the spheres with a smoothed density with the same multipole moments. We use multigrid techniques on the mesh, which yields the Coulomb potential on the spheres and in turn defines the potential inside via a Dirichlet problem. To solve the linear eigen-problem, we use ScaLAPACK, a well-developed package to solve large eigensystems with dense matrices. We have tested the method on small clusters of palladium.
- Published
- 2008
34. Automated conformational energy fitting for force-field development
- Author
-
Alexander D. MacKerell and Olgun Guvench
- Subjects
Mean squared error ,Monte Carlo method ,Carbohydrates ,Molecular Conformation ,Dihedral angle ,Parameter space ,Catalysis ,Force field (chemistry) ,Article ,Inorganic Chemistry ,Computational chemistry ,Cyclohexanes ,Computer Simulation ,Physical and Theoretical Chemistry ,Mathematical Computing ,Mathematics ,Pyrans ,Organic Chemistry ,Mathematical analysis ,Potential energy ,Computer Science Applications ,Computational Theory and Mathematics ,Models, Chemical ,Potential energy surface ,Quantum Theory ,Peptides ,Parametrization ,Monte Carlo Method ,Algorithms - Abstract
We present a general conformational-energy fitting procedure based on Monte Carlo simulated annealing (MCSA) for application in the development of molecular mechanics force fields. Starting with a target potential energy surface and an unparametrized molecular mechanics potential energy surface, an optimized set of either dihedral or grid-based correction map (CMAP) parameters is produced that minimizes the root mean squared error RMSE between the parametrized and targeted energies. The fitting is done using an MCSA search in parameter space and consistently converges to the same RMSE irrespective of the randomized parameters used to seed the search. Any number of dihedral parameters can be simultaneously parametrized, allowing for fitting to multi-dimensional potential energy scans. Fitting options for dihedral parameters include non-uniform weighting of the target data, constraining multiple optimized parameters to the same value, constraining parameters to be no greater than a user-specified maximum value, including all or only a subset of multiplicities defining the dihedral Fourier series, and optimization of phase angles in addition to force constants. The dihedral parameter fitting algorithm's performance is characterized through multi-dimensional fitting of cyclohexane, tetrahydropyran, and hexopyranose monosaccharide energetics, with the latter case having a 30-dimensional parameter space. The CMAP fitting is applied in the context of polypeptides, and is used to develop a parametrization that simultaneously captures the phi,psi energetics of the alanine dipeptide and the alanine tetrapeptide. Because the dihedral energy term is common to many force fields, we have implemented the dihedral-fitting algorithm in the portable Python scripting language and have made it freely available as "fit_dihedral.py" for download at http://mackerell.umaryland.edu.
- Published
- 2008
35. Contracted auxiliary Gaussian basis integral and derivative evaluation
- Author
-
Darrin M. York and Timothy J. Giese
- Subjects
Gaussian ,Mathematical analysis ,Scalar (mathematics) ,Normal Distribution ,General Physics and Astronomy ,Models, Theoretical ,Article ,Gaussian random field ,law.invention ,symbols.namesake ,law ,Quantum mechanics ,symbols ,Gaussian function ,Quantum Theory ,Cartesian coordinate system ,Physical and Theoretical Chemistry ,Multipole expansion ,Linear combination ,Gaussian process ,Mathematics - Abstract
The rapid evaluation of two-center Coulomb and overlap integrals between contracted auxiliary solid harmonic Gaussian functions is examined. Integral expressions are derived from the application of Hobson's theorem and Dunlap's product and differentiation rules of the spherical tensor gradient operator. It is shown that inclusion of the primitive normalization constants greatly simplifies the calculation of contracted functions corresponding to a Gaussian multipole expansion of a diffuse charge density. Derivative expressions are presented and it is shown that chain rules are avoided by expressing the derivatives as a linear combination of auxiliary integrals involving no more than five terms. Calculation of integrals and derivatives requires the contraction of a single vector corresponding to the monopolar result and its scalar derivatives. Implementation of the method is discussed and comparison is made with a Cartesian Gaussian-based method. The current method is superior for the evaluation of both integrals and derivatives using either primitive or contracted functions.
- Published
- 2008
36. Radial rescaling approach for the eigenvalue problem of a particle in an arbitrarily shaped box
- Author
-
Erwin Lijnen, Arnout Ceulemans, and Liviu F. Chibotaru
- Subjects
boundary-value problems ,Mathematical analysis ,triangle ,Regular polygon ,Basis function ,Mixed boundary condition ,quantum theory ,Schrödinger equation ,symbols.namesake ,quantum ,schrodinger-equation ,ground-state ,Dirichlet boundary condition ,symbols ,Boundary value problem ,Dynamical billiards ,eigenvalues and eigenfunctions ,billiard ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In the present work we introduce a new methodology for solving a quantum billiard with Dirichlet boundary conditions. The procedure starts from the exactly known solutions for the particle in a circular disk, which are subsequently radially rescaled in such a way that they obey the new boundary conditions. In this way one constructs a complete basis set which can be used to obtain the eigenstates and eigenenergies of the corresponding quantum billiard to a high level of precision. Test calculations for several regular polygons show the efficiency of the method which often requires one or two basis functions to describe the lowest eigenstates with high accuracy. ispartof: Physical Review E, Statistical, Nonlinear and Soft Matter Physics vol:77 issue:1 ispartof: location:United States status: published
- Published
- 2008
37. Multifractal eigenstates of quantum chaos and the Thue-Morse sequence
- Author
-
N. Meenakshisundaram and Arul Lakshminarayan
- Subjects
Sequence ,Quantum Physics ,Quantum chaotic states ,Mathematical analysis ,Chaotic ,Approximation theory ,FOS: Physical sciences ,Thue–Morse sequence ,Multifractal system ,Nonlinear Sciences - Chaotic Dynamics ,Quantum chaos ,Homoclinic excursions ,Nonlinear Sciences::Chaotic Dynamics ,Condensed Matter - Other Condensed Matter ,Quasiperiodicity ,Fractals ,Quantum theory ,Statistical physics ,Homoclinic orbit ,Chaotic Dynamics (nlin.CD) ,Quantum Physics (quant-ph) ,Quantum ,Other Condensed Matter (cond-mat.other) ,Mathematics - Abstract
We analyze certain eigenstates of the quantum baker's map and demonstrate, using the Walsh-Hadamard transform, the emergence of the ubiquitous Thue-Morse sequence, a simple sequence that is at the border between quasi-periodicity and chaos, and hence is a good paradigm for quantum chaotic states. We show a family of states that are also simply related to Thue-Morse sequence, and are strongly scarred by short periodic orbits and their homoclinic excursions. We give approximate expressions for these states and provide evidence that these and other generic states are multifractal., Substantially modified from the original, worth a second download. To appear in Phys. Rev. E as a Rapid Communication
- Published
- 2005
38. Determination of a Wave Function Functional
- Author
-
Xiao-Yin Pan, Viraht Sahni, and Lou Massa
- Subjects
Chemical Physics (physics.chem-ph) ,Atomic Physics (physics.atom-ph) ,Mathematical analysis ,FOS: Physical sciences ,General Physics and Astronomy ,Observable ,Function (mathematics) ,Space (mathematics) ,Upper and lower bounds ,Physics - Atomic Physics ,Quantum mechanics ,Physics - Chemical Physics ,Quantum Theory ,Thermodynamics ,Calculus of variations ,Sum rule in quantum mechanics ,Wave function ,Subspace topology ,Mathematics - Abstract
In this paper we propose the idea of expanding the space of variations in standard variational calculations for the energy by considering the wave function $\psi$ to be a functional of a set of functions $\chi: \psi = \psi[\chi]$, rather than a function. In this manner a greater flexibility to the structure of the wave function is achieved. A constrained search in a subspace over all functions $\chi$ such that the wave function functional $\psi[\chi]$ satisfies a constraint such as normalization or the Fermi-Coulomb hole charge sum rule, or the requirement that it lead to a physical observable such as the density, diamagnetic susceptibility, etc. is then performed. A rigorous upper bound to the energy is subsequently obtained by variational minimization with respect to the parameters in the approximate wave function functional. Hence, the terminology, the constrained-search variational method. The \emph{rigorous} construction of such a constrained-search--variational wave function functional is demonstrated by example of the ground state of the Helium atom., Comment: 10 pages, 2 figures, changes made, references added
- Published
- 2004
39. Quasisteady aero-acoustic response of orifices
- Author
-
G. Ajello, G.C.J. Hofmans, M. Peters, Yves Aurégan, R.J.J. Boot, A Avraham Hirschberg, P.P.J.M. Durrieu, and Technisch Physische Dienst TNO - TH
- Subjects
Acoustics and Ultrasonics ,Anechoic chamber ,Acoustics ,Flow (psychology) ,Tubes (components) ,Geometry ,Mathematical analysis ,Physics::Fluid Dynamics ,symbols.namesake ,Mathematical model ,Arts and Humanities (miscellaneous) ,Mean flow ,Orifices ,Mathematics ,Priority journal ,Vena contracta ,Mach number ,Scattering ,Mechanics ,Quantum theory ,symbols ,Aeroacoustics ,Body orifice - Abstract
The low frequency response of orifices (slit, circular diaphragm, and perforated plate) in the presence of mean flow is well predicted by a quasisteady theory. A refinement is brought to the theory by considering a Mach number dependent vena contracta coefficient. The measurements of the vena contracta coefficient of a slit agree well with the simple analytical expression existing in the case of the Borda tube orifice. The measured scattering matrix coefficients do not depend strongly on the geometry of the element. If the frequency is increased the moduli remain relatively unaffected while the arguments exhibit a complex behavior which depends on the geometry. From these considerations an anechoic termination efficient at high mass flow is designed.
- Published
- 2001
40. Playing Quantum Physics Jeopardy with zero-energy eigenstates.
- Author
-
Gilbert, L. P., Belloni, M., Doncheski, M. A., and Robinett, R. W.
- Subjects
- *
QUANTUM theory , *FORCE & energy , *EQUATIONS , *MATHEMATICS , *STUDENTS , *PROBLEM solving , *PHYSICS education , *MATHEMATICAL analysis , *GEOPHYSICS - Abstract
We describe an example of an exact, quantitative Jeopardy-type quantum mechanics problem. This problem type is based on the conditions in one-dimensional quantum systems that allow an energy eigenstate for the infinite square well to have zero curvature and zero energy when suitable Dirac delta functions are added. This condition and its solution are not often discussed in quantum mechanics texts and have interesting pedagogical consequences. [ABSTRACT FROM AUTHOR]
- Published
- 2006
- Full Text
- View/download PDF
41. Numerical investigation of iso-spectral cavities built from trinagles
- Author
-
Donald W. L. Sprung, Hua Wu, J. Martorell, and Universitat de Barcelona
- Subjects
Special relativity (Physics) ,Field theory (Physics) ,Integrable system ,Iterative method ,Gaussian ,Física matemàtica ,Relativitat especial (Física) ,FOS: Physical sciences ,Electromagnetisme ,symbols.namesake ,Electromagnetism ,Quantum mechanics ,Termodinàmica ,Differentiable dynamical systems ,Teoria quàntica ,Òptica electrònica ,Gaussian process ,Mathematics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Electron optics ,Mathematical analysis ,Spectrum (functional analysis) ,Finite difference method ,Teoria de camps (Física) ,Sistemes dinàmics diferenciables ,Isospectral ,Fourier analysis ,Mathematical physics ,Quantum theory ,symbols ,Thermodynamics ,Exactly Solvable and Integrable Systems (nlin.SI) - Abstract
We present computational approaches as alternatives to the recent microwave cavity experiment by S. Sridhar and A. Kudrolli (Phys. Rev. Lett. {\bf 72}, 2175 (1994)) on iso-spectral cavities built from triangles. A straightforward proof of iso-spectrality is given based on the mode matching method. Our results show that the experiment is accurate to 0.3% for the first 25 states. The level statistics resemble those of GOE when the integrable part of the spectrum is removed., Comment: 15 pages, revtex, 5 postscript figures
- Published
- 1995
42. Receiver operating characteristic analysis. Application to the study of quantum fluctuation effects in optic nerve of Rana pipiens
- Author
-
Daniel Green, Theodore E. Cohn, and Wilson P. Tanner
- Subjects
Physiology ,Models, Neurological ,Context (language use) ,Poisson distribution ,Signal ,Luminance ,Synaptic Transmission ,symbols.namesake ,Optics ,Animals ,Scotopic vision ,Quantum fluctuation ,Probability ,Physics ,Receiver operating characteristic ,business.industry ,Mathematical analysis ,Optic Nerve ,Articles ,symbols ,Optic nerve ,Visual Perception ,Quantum Theory ,sense organs ,Anura ,business ,Mathematics - Abstract
Receiver operating characteristic (ROC) analysis of nerve messages is described. The hypothesis that quantum fluctuations provide the only limit to the ability of frog ganglion cells to signal luminance change information is examined using ROC analysis. In the context of ROC analysis, the quantum fluctuation hypothesis predicts (a) the detectability of a luminance change signal should rise proportionally to the size of the change, (b) detectability should decrease as the square root of background, an implication of which is the deVries-Rose law, and (c) ROC curves should exhibit a shape particular to underlying Poisson distributions. Each of these predictions is confirmed for the responses of dimming ganglion cells to brief luminance decrements at scotopic levels, but none could have been tested using classical nerve message analysis procedures.
- Published
- 1975
43. Poincar-Cartan integral invariant and canonical transformation for singular lagrangians
- Author
-
Joaquim Gomis, Daniele Dominici, and Universitat de Barcelona
- Subjects
Pure mathematics ,Equacions en derivades parcials ,Mathematical analysis ,Equations of motion ,Statistical and Nonlinear Physics ,Canonical transformation ,Invariant (physics) ,Partial differential equations ,Dynamics ,symbols.namesake ,Formalism (philosophy of mathematics) ,Quantum theory ,Poincaré conjecture ,Dinàmica ,symbols ,Teoria quàntica ,Finite set ,Mathematical Physics ,Lagrangian ,Mathematics - Abstract
In this work we develop the canonical formalism for constrained systems with a finite number of degrees of freedom by making use of the Poincare–Cartan integral invariant method. A set of variables suitable for the reduction to the physical ones can be obtained by means of a canonical transformation. From the invariance of the Poincare–Cartan integral under canonical transformations we get the form of the equations of motion for the physical variables of the system.
44. Lee Hwa Chung theorem for presymplectic manifolds. Canonical transformations for constrained systems
- Author
-
J. Llosa, N. Román, Joaquim Gomis, and Universitat de Barcelona
- Subjects
Pure mathematics ,Dynamical systems theory ,Canonical system ,Lee Hwa Chung theorem ,Dirac (video compression format) ,Física matemàtica ,Mathematical analysis ,Statistical and Nonlinear Physics ,Canonical transformation ,Dynamics ,symbols.namesake ,Dirac equation ,Phase space ,Mathematical physics ,Quantum theory ,Dinàmica ,symbols ,Camps vectorials ,Vector field ,Teoria quàntica ,Mathematics::Symplectic Geometry ,Vector fields ,Mathematical Physics ,Mathematics - Abstract
We generalize the analogous of Lee Hwa Chung’s theorem to the case of presymplectic manifolds. As an application, we study the canonical transformations of a canonical system (M, S, Ω). The role of Dirac brackets as a test of canonicity is clarified.
45. Coisotropic regularization of singular Lagrangians
- Author
-
Jesús Marín-Solano, A. Ibort, and Universitat de Barcelona
- Subjects
Hamiltonian mechanics ,Camps de galga (Física) ,Field theory (Physics) ,Mathematical analysis ,Teoria de camps (Física) ,Statistical and Nonlinear Physics ,Gauge fields (Physics) ,Mechanics ,Mecànica ,symbols.namesake ,Phase space ,Regularization (physics) ,Quantum theory ,symbols ,Embedding ,Covariant Hamiltonian field theory ,Teoria quàntica ,Configuration space ,Gauge theory ,Hamiltonian (quantum mechanics) ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics ,Mathematical physics - Abstract
We present an alternative approach to the usual treatments of singular Lagrangians. It is based on a Hamiltonian regularization scheme inspired on the coisotropic embedding of presymplectic systems. A Lagrangian regularization of a singular Lagrangian is a regular Lagrangian defined on an extended velocity phase space that reproduces the original theory when restricted to the initial configuration space. A Lagrangian regularization does not always exists, but a family of singular Lagrangians is studied for which such a regularization can be described explicitly. These regularizations turn out to be essentially unique and provide an alternative setting to quantize the corresponding physical systems. These ideas can be applied both in classical mechanics and field theories. Several examples are discussed in detail. 1995 American Institute of Physics.
46. Canonical transformations theory for presymplectic systems
- Author
-
N. Roman, Joaquim Gomis, José F. Cariñena, Luis A. Ibort, and Universitat de Barcelona
- Subjects
Camps de galga (Física) ,Pure mathematics ,Geometry ,Canonical transformation ,01 natural sciences ,symbols.namesake ,Gauge group ,0103 physical sciences ,Sistemes hamiltonians ,Teoria quàntica ,Canonical form ,Hamiltonian systems ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics ,Generating function (physics) ,Hamiltonian mechanics ,010102 general mathematics ,Mathematical analysis ,Canonical coordinates ,Gauge fields (Physics) ,Statistical and Nonlinear Physics ,Quantum theory ,Camps vectorials ,symbols ,010307 mathematical physics ,Vector fields ,Group theory ,Symplectic geometry - Abstract
We develop a theory of canonical transformations for presymplectic systems, reducing this concept to that of canonical transformations for regular coisotropic canonical systems. In this way we can also link these with the usual canonical transformations for the symplectic reduced phase space. Furthermore, the concept of a generating function arises in a natural way as well as that of gauge group.
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