We show that in the ground states of the infinite-volume limits of both the spin-1/2 anisotropic antiferromagnetic Heisenberg model (in dimensions d greater-than-or-equal-to 2), and the ferromagnetic Ising model in a ...
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We show that in the ground states of the infinite-volume limits of both the spin-1/2 anisotropic antiferromagnetic Heisenberg model (in dimensions d greater-than-or-equal-to 2), and the ferromagnetic Ising model in a strong transverse field (in dimensions d greater-than-or-equal-to 1) there is an interval in the spectrum above the mass gap which contains a continuous band of energy levels. We use the methods of Bricmont and Frohlich to develop our expansions, as well as a method of Kennedy and Tasaki to do the expansions in the quantum mechanical limit. Where the expansions converge, they are then shown to have spectral measures which have absolutely continuous parts on intervals above the mass gaps.
Although normal ordering (NO) is often used as a quantization procedure for classical problems that are based on complex mode amplitudes, an alternate choice, called ''symmetric ordering'' (SO), is cha...
Although normal ordering (NO) is often used as a quantization procedure for classical problems that are based on complex mode amplitudes, an alternate choice, called ''symmetric ordering'' (SO), is championed here. In normal form the SO operator is simply related to zero order Laguerre polynomials that are implied by Weyl's quantization rule. Thus SO is as convenient as NO, and it is more accurate when the rotating wave approximation is used for a mass-spring oscillator with a slightly nonlinear spring. The two quantization methods are compared for a Hartree analysis of the discrete self-trapping equation with an arbitrary power of the nonlinearity.
This paper addresses the possible connections between chaos, the unpredictable behavior of solutions of finite dimensional systems of ordinary differential and difference equations and turbulence, the unpredictable be...
This paper addresses the possible connections between chaos, the unpredictable behavior of solutions of finite dimensional systems of ordinary differential and difference equations and turbulence, the unpredictable behavior of solutions of partial differential equations. It is dedicated to Martin Kruskal on the occasion of his 60th birthday.
For a volume-preserving map, we show that the exit time averaged over the entry set of a region is given by the ratio of the measure of the accessible subset of the region to that of the entry set. This result is prim...
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For a volume-preserving map, we show that the exit time averaged over the entry set of a region is given by the ratio of the measure of the accessible subset of the region to that of the entry set. This result is primarily of interest to show two things: First, it gives a simple bound on the algebraic decay exponent of the survival probability. Second, it gives a tool for computing the measure of the accessible set. We use this to compute the measure of the bounded orbits for the Henon quadratic map. (C) 1997 American Institute of Physics.
For the quantum mechanical Ising model in a strong transverse field we show that the convergence of the ground-state energy per site as the volume goes to infinity has an Ornstein-Zernicke behavior. That is, if the di...
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For the quantum mechanical Ising model in a strong transverse field we show that the convergence of the ground-state energy per site as the volume goes to infinity has an Ornstein-Zernicke behavior. That is, if the diameter of the d-dimensional lattice is given by L, the absolute value of the difference of the ground-state energy per site and its limit is asymptotically exp(-xiL) L(-d/2) for some positive constant xi. We also show that the correlation function has the same behavior. Our results are derived by cluster expansions, using a method of Bricmont and Frohlich which we extend to the quantum mechanical case.
Background: Several studies show that large language models (LLMs) struggle with phenotype-driven gene prioritization for rare diseases. These studies typically use Human Phenotype Ontology (HPO) terms to prompt found...
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A new mechanism for the creation of structures in two-dimensional turbulence is investigated. The forced Navier-Stokes equations are solved numerically in a periodic square in the limit of zero viscosity. The force is...
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A new mechanism for the creation of structures in two-dimensional turbulence is investigated. The forced Navier-Stokes equations are solved numerically in a periodic square in the limit of zero viscosity. The force is a white-in-time random noise acting in a narrow band of high wavenumbers. The inverse-cascade process and the presence of the boundary lead ultimately to a pile-up of energy in the lowest wavenumber (Bose condensation). In the asymptotic limit where the enstrophy cascade range is negligible, Bose condensation is solely responsible for the generation of coherent vortices and intermittency in the system. We present the evolution of the velocity and vorticity fields through the later stages of the condensate state, and explore the possible implications for atmospheric turbulence constrained by the periodic domain about the earth.
We describe here a new technique and a package for rapid reconstruction of smooth surfaces from scattered data points. This method is based on a fast recurrent algorithm for the Delauney triangulation followed by rati...
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We describe here a new technique and a package for rapid reconstruction of smooth surfaces from scattered data points. This method is based on a fast recurrent algorithm for the Delauney triangulation followed by rational interpolation inside triangles. Preprocessing of data includes sorting and takes N log(N) time. Afterwards the computational cost is a linear function of the amount of data. This technique enables a user to construct a surface of any class of smoothness and degree of convergence. Our package reconstructs surfaces that can be uniquely projected either on a plane or on a sphere. The graphical section of this package includes three dimensional transformations, shading, hidden surface removal, interactive adding points into triangulation by mouse, etc. The graphics has been implemented on Iris-4D, SUN-4 and IBM-5080.
This work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the form u(t) = Lu + Nf(u), where L and N are linear differential operators and f(u) is a nonlinear f...
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This work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the form u(t) = Lu + Nf(u), where L and N are linear differential operators and f(u) is a nonlinear function. These equations are adaptively solved by projecting the solution u and the operators L and N into a wavelet basis. Vanishing moments of the basis functions permit a sparse representation of the solution and operators. Using these sparse representations fast and adaptive algorithms that apply operators to functions and evaluate nonlinear functions, are developed for solving evolution equations. For a wavelet representation of the solution u that contains N-s significant coefficients, the algorithms update the solution using O(N-s) operations. The approach is applied to a number of examples and numerical results are given. (C) 1997 Academic Press.
The dynamical system arising from the problem of billiards is a classical example where the theory of twist maps can be applied. In the case of an elliptic billiard table, the corresponding twist map is integrable and...
The dynamical system arising from the problem of billiards is a classical example where the theory of twist maps can be applied. In the case of an elliptic billiard table, the corresponding twist map is integrable and has a saddle connection between two hyperbolic period two points. Using a discrete analog to the Melnikov method, we are able to show that this saddle connection can be deformed into a transversal heteroclinic connection under certain analytic perturbations of the table. From the formulas that we get, we can show that the splitting of the separatrices is exponentially small as a function of the eccentricity of the original unperturbed elliptic table. In addition, we also include a characterization of the period two periodic points for any billiard table.
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