Spectral Asymptotic for the High-Dimensional Euler Top

In this paper, we compute the leading coefficient in the asymptotic expansion of the joint eigenvalues for the high-dimensional Euler Top. We also prove a central limit theorem for the same eigenvalues.      

Asymptotic Lower Bound for the Relative Disparities of Truncated-Path-Integral Partition Functions

Any truncated-path-integral partition function of a nonrelativistic quantum system in thermodynamic equilibrium—one obtained by means of the Feynman path-integral-procedure using a finite number of such integrals—is known to have a value not less than that of the exact one corresponding to it. A rigorous asymptotic lower bound obtained for the relative disparity in their values—the difference in their values divided by that of the exact partition function— confirms asymptotic positive-definiteness of the original upper bound. Values determined directly for a linear harmonic oscillator agree asymptotically with values of they bound.     

Proof of an Asymptotic Property of Self-Similar Solutions of the Boltzmann Equation for Granular Materials

We consider a question related to the kinetic theory of granular materials. The model of hard spheres with inelastic collisions is replaced by a Maxwell model, characterized by a collision frequency independent of the relative speed of colliding particles. Our main result is that, in the space-homogeneous case, a self-similar asymptotics holds, as conjectured by Ernst–Brito. The proof holds for any initial distribution function with a finite moment of some order greater than two.     

A Closed Formula for the Asymptotic Expansion of the Bergman Kernel

We prove a graph theoretic closed formula for coefficients in the Tian-Yau-Zelditch asymptotic expansion of the Bergman kernel. The formula is expressed in terms of the characteristic polynomial of the directed graphs representing Weyl invariants. The proof relies on a combinatorial interpretation of a recursive formula due to M. Engliš and A. Loi.     

Asymptotic Energy Expansion for Rational Power Polynomial Potentials

Asymptotic energy expansion method is extended for polynomial potentials having rational powers. New types of recurrence relations are derived for the potentials of the form V(x)= x2n/m+b1xn1/m1+b2x n2/m2+ …+ bNxnN/mN where n, m, n1, m1, . . . , nN, mN are positive integers while coefficients bk∈C. As in the case of even degree polynomial potentials with integer powers, all the integrals in the expansion can be evaluated analytically in terms of Γ functions. With the help of two examples, we demonstrate the usefulness of these expansions in getting analytic insight into the quantum systems having rational power polynomial potentials.     

Asymptotic Energy Expansion for Rational Power Polynomial Potentials

Asymptotic energy expansion method is extended for polynomial potentials having rational powers. New types of recurrence relations are derived for the potentials of the form V(x)=x^2n/m+b₁x^n1/m1+b₂x^n2/m2 +··· + b_Nx^n_N/m_N where n,m,n₁,m₁,...,n_N,m_N are positive integers while coefficients b_k∈ C. As in the case of even degree polynomial potentials with integer powers, all the integrals in the expansion can be evaluated analytically in terms of Γ functions. With the help of two examples, we demonstrate the usefulness of these expansions in getting analytic insight into the quantum systems having rational power polynomial potentials.     

An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem

Let λ _d (p) be the p monomer-dimer entropy on the d-dimensional integer lattice ℤ ^d , where p∈[0,1] is the dimer density. We give upper and lower bounds for λ _d (p) in terms of expressions involving λ _d−1(q). The upper bound is based on a conjecture claiming that the p monomer-dimer entropy of an infinite subset of ℤ ^d is bounded above by λ _d (p). We compute the first three terms in the formal asymptotic expansion of λ _d (p) in powers of  \frac1d\frac{1}{d}. We prove that the lower asymptotic matching conjecture is satisfied for λ _d (p). Converted to a power series in p, our “formal” expansion shows remarkable validity in low dimensions, d=1,2,3, in which dimensions we give some numerical studies.     

Approximating the Partition Function of Morse Oscillators

A simple method is proposed leading to reasonable approximations of the partition function of Morse oscillators in terms of elementary functions.     

Asymptotic Heat Kernel Expansion in the Semi-Classical Limit

Let H_(^h/2p) = (^h/_2p)²L +V{H_\hbar = \hbar^{2}L +V}, where L is a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold and V is a symmetric endomorphism field. We derive an asymptotic expansion for the heat kernel of H_(^h/2p){H_\hbar} as (^h/_2p) \searrow 0{\hbar \searrow 0}. As a consequence we get an asymptotic expansion for the quantum partition function and we see that it is asymptotic to the classical partition function. Moreover, we show how to bound the quantum partition function for positive (^h/_2p){\hbar} by the classical partition function.     

Region Graph Partition Function Expansion and Approximate Free Energy Landscapes: Theory and Some Numerical Results

Graphical models for finite-dimensional spin glasses and real-world combinatorial optimization and satisfaction problems usually have an abundant number of short loops. The cluster variation method and its extension, the region graph method, are theoretical approaches for treating the complicated short-loop-induced local correlations. For graphical models represented by non-redundant or redundant region graphs, approximate free energy landscapes are constructed in this paper through the mathematical framework of region graph partition function expansion. Several free energy functionals are obtained, each of which use a set of probability distribution functions or functionals as order parameters. These probability distribution function/functionals are required to satisfy the region graph belief-propagation equation or the region graph survey-propagation equation to ensure vanishing correction contributions of region subgraphs with dangling edges. As a simple application of the general theory, we perform region graph belief-propagation simulations on the square-lattice ferromagnetic Ising model and the Edwards-Anderson model. Considerable improvements over the conventional Bethe-Peierls approximation are achieved. Collective domains of different sizes in the disordered and frustrated square lattice are identified by the message-passing procedure. Such collective domains and the frustrations among them are responsible for the low-temperature glass-like dynamical behaviors of the system.     

Asymptotic Behavior of Spectral Energy Distribution Function of Equilibrium Radiation in Maxwell Plasma at Low Frequencies

The low-frequency asymptotic behavior of the spectral distribution density of equilibrium radiation in collisionless non-degenerate electron plasma is studied. It is shown that the consideration of the spatial dispersion in the electron gas permittivity leads to a logarithmic singularity in the spectral distribution density at low frequencies. In this case, the total radiation energy remains finite. The results of the analytical consideration are identical to numerical calculations.     

Spectral Simplicity and Asymptotic Separation of Variables

We describe a method for comparing the spectra of two real-analytic families, (a _t ) and (q _t ), of quadratic forms that both degenerate as a positive parameter t tends to zero. We suppose that the family (a _t ) is amenable to ‘separation of variables’ and that each eigenspace of a _t is 1-dimensional for some t. We show that if (q _t ) is asymptotic to (a _t ) at first order as t → 0, then the eigenspaces of (q _t ) are also 1-dimensional for all but countably many t. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian acting on Dirichlet functions is 1-dimensional.     

Asymptotic Expansion of the Pressure in the Inverse Interaction Range

We consider an Ising system in d2 dimensions with a ferromagnetic Kac potential whose scaling parameter is denoted by . We derive an asymptotic series for the thermodynamic pressure P _, in powers of . The 0th-order term of the expansion is the mean-field pressure of the Lebowitz and Penrose theory.     

An Improved Singularity Free Self-Similar Model of Proton Structure Function

In this paper we make re-analysis of a self-similarity based model of the proton structure function pursued in recent years. The additional assumption is that it should be singularity free in the entire kinematic range of x:0 x 1. We also suggest a plausible singularity free form of the model, which has a wider phenomenological range of validity in Q~2 than the earlier ones.     

Topological Entanglement Entropy from the Holographic Partition Function

We study the entropy of chiral 2+01-dimensional topological phases, where there are both gapped bulk excitations and gapless edge modes. We show how the entanglement entropy of both types of excitations can be encoded in a single partition function. This partition function is holographic because it can be expressed entirely in terms of the conformal field theory describing the edge modes. We give a general expression for the holographic partition function, and discuss several examples in depth, including abelian and non-abelian fractional quantum Hall states, and $p+ip$ superconductors. We extend these results to include a point contact allowing tunneling between two points on the edge, which causes thermodynamic entropy associated with the point contact to be lost with decreasing temperature. Such a perturbation effectively breaks the system in two, and we can identify the thermodynamic entropy loss with the loss of the edge entanglement entropy. From these results, we obtain a simple interpretation of the non-integer ‘ground state degeneracy’ which is obtained in 1+1-dimensional quantum impurity problems: its logarithm is a 2+1-dimensional topological entanglement entropy.     

Airy Equation for the Topological String Partition Function in a Scaling Limit

We use the polynomial formulation of the holomorphic anomaly equations governing perturbative topological string theory to derive the free energies in a scaling limit to all orders in perturbation theory for any Calabi–Yau threefold. The partition function in this limit satisfies an Airy differential equation in a rescaled topological string coupling. One of the two solutions of this equation gives the perturbative expansion and the other solution provides geometric hints of the non-perturbative structure of topological string theory. Both solutions can be expanded naturally around strong coupling.     

按波长分区的LCD颜色特征化模型

为了建立液晶显示器颜色计算模型,分别对EIZO CG19, IBM 19, DELL 19和HP 19 LCD显示器进行了通道独立性和颜色叠加性的检验。通过检验单通道和双通道数字驱动值与颜色三刺激值之间的关系可以验证各通道之间的独立性,从而证明了上述显示器的显示色与三原色光谱辐亮度函数之间存在线性叠加关系。实验中发现三原色光谱辐亮度与数字驱动值的关系是波长的非线性函数,提出了一种按波长分区法计算三原色光谱辐亮度函数的方法,用RGB三原色的测量光谱拟合出各波长辐亮度值与数字驱动值之间的三次多项式关系,利用这个关系可计算得到任意数字驱动值对应的三原色光谱辐亮度曲线,然后可按颜色相加关系由三原色光谱辐亮度曲线计算出任意数字驱动值下的显示色。实验结果表明,只需要测量少量几个数字驱动值下的光谱曲线就可以准确计算出任意数字驱动值下的三原色光谱辐亮度曲线,进而计算得到任意混合色的光谱辐亮度曲线,大大减少了测量样品数量。这种计算方法所需的测量样品数量少、计算准确度高,可以满足工业应用的精度要求,能作为LCD显示器颜色特征化的一种方法。