Pólya number and first return of bursty random walk: Rigorous solutions

The recurrence properties of random walks can be characterized by Pólya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we investigate Pólya number and first return for bursty random walk on a line, in which the walk has different step size and moving probabilities. Using the concept of the Catalan number, we obtain exact results for first return probability, the average first return time and Pólya number for the first time. We show that Pólya number displays two different functional behavior when the walk deviates from the recurrent point. By utilizing the Lagrange inversion formula, we interpret our findings by transferring Pólya number to the closed-form solutions of an inverse function. We also calculate Pólya number using another approach, which corroborates our results and conclusions. Finally, we consider the recurrence properties and Pólya number of two variations of the bursty random walk model.     

Generalized classical, quantum and intermediate statistics and the Pólya urn model

Generalized probability distributions for Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics, with unequal source (“prior”) probabilities qi for each level i, are obtained by combinatorial reasoning. For equiprobable degenerate sublevels, these reduce to those given by Brillouin in 1930, more commonly given as a statistical weight for each statistic. These distributions and corresponding cross-entropy (divergence) functions are shown to be special cases of the Pólya urn model, involving neither independent nor identically distributed (“ninid”) sampling. The most probable Pólya distribution is shown to contain the Acharya-Swamy intermediate statistic.     

Random walks on the Bethe lattice

We obtain random walk statistics for a nearest-neighbor (Pólya) walk on a Bethe lattice (infinite Cayley tree) of coordination numberz, and show how a random walk problem for a particular inhomogeneous Bethe lattice may be solved exactly. We question the common assertion that the Bethe lattice is an infinite-dimensional system.Supported in part by the U.S. Department of Energy.     

Inelastic phonon interactions at solid–graphite interfaces

The presented model predicts thermal boundary conductance at interfaces where one material comprising the junction is characterized by high elastic anisotropy. In contrast to previous approaches, the current methodology accounts for contributions from inelastic scattering through consideration of multiple-phonon interactions. Inelastic contributions become significant as the temperature, as well as the degree of acoustic mismatch between the materials, increases. Inclusion of the inelastic interactions is necessary for a variety of interfacial systems including the metal–graphite boundary examined here. Improvement is shown over existing approaches that address only elastic scattering as both three- and four-phonon interactions significantly augment the transport.     

Land cover change using an energy transition paradigm in a statistical mechanics approach

This paper explores a statistical mechanics approach as a means to better understand specific land cover changes on a continental scale. Integrated assessment models are used to calculate the impact of anthropogenic emissions via the coupling of technoeconomic and earth/atmospheric system models and they have often overlooked or oversimplified the evolution of land cover change. Different time scales and the uncertainties inherent in long term projections of land cover make their coupling to integrated assessment models difficult. The mainstream approach to land cover modelling is rule-based methodology and this necessarily implies that decision mechanisms are often removed from the physical geospatial realities, therefore a number of questions remain: How much of the predictive power of land cover change can be linked to the physical situation as opposed to social and policy realities? Can land cover change be understood using a statistical approach that includes only economic drivers and the availability of resources? In this paper, we use an energy transition paradigm as a means to predict this change. A cost function is applied to developed land covers for urban and agricultural areas. The counting of area is addressed using specific examples of a Pólya process involving Maxwell–Boltzmann and Bose–Einstein statistics. We apply an iterative counting method and compare the simulated statistics with fractional land cover data with a multi-national database. An energy level paradigm is used as a basis in a flow model for land cover change. The model is compared with tabulated land cover change in Europe for the period 1990–2000. The model post-predicts changes for each nation. When strong extraneous factors are absent, the model shows promise in reproducing data and can provide a means to test hypothesis for the standard rules-based algorithms.     

Multienergetic continuous time random walk

We have extended the CTRW theory of Montroll and Weiss including the effect of extra variables, like the energy. This MCTRW scheme can be written in a simple matrix notation, that simplifies its solution. As an example of their usefulness we have studied a two-energy-group neutron diffusion problem. This has shown the peculiarities of the transient behaviour for the variance of the probability distribution, due to the coupling between the groups.Comisión Nacional de Energiá AtómicaComisión Nacional de Energía Atómica and Universidad Nacional de Cuyo     

Comparison Theorems for Eigenvalues of Elliptic Operators and the Generalized Pólya Conjecture

We establish comparison theorems for eigenvalues between higher order elliptic equations on compact manifolds with boundary. As an application, it follows that if the Pólya conjecture is true then so is the generalized Pólya conjecture proposed by Ku et al. (J Differ Equ 97:127–139, 1992). We also obtain new lower bound for the eigenvalues of higher order elliptic equations on bounded domains in a Euclidean space.     

Properties of Generalized Univariate Hypergeometric Functions

Based on Spiridonov’s analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E ₇ (elliptic, hyperbolic) and of type E ₆ (trigonometric) using the appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous relations using fundamental addition formulas for theta and sine functions. The top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with Ruijsenaars’ relativistic hypergeometric function and the Askey-Wilson function, respectively. We show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions. We also describe an intimate connection between the hyperbolic and trigonometric theory, which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions.     

Nonclassical degrees of freedom in the Riemann Hamiltonian

The Hilbert-Pólya conjecture states that the imaginary parts of the zeros of the Riemann zeta function are eigenvalues of a quantum Hamiltonian. If so, conjectures by Katz and Sarnak put this Hamiltonian in the Altland-Zirnbauer universality class?C. This implies that the system must have a nonclassical two-valued degree of freedom. In such a system, the dominant primitive periodic orbits contribute to the density of states with a phase factor of -1. This resolves a previously mysterious sign problem with the oscillatory contributions to the density of the Riemann zeros.     

Coupled continuous-time random-walk: Pseudo diffusion with anisotropic scattering

A generalization of pseudo-diffusion processes in media with anisotropic scattering, described by a coupled RW, is done by means of the CTRW theory. The behaviour of the probability distribution has been studied. A kind of phase breaking in such a behaviour arises as a function of the anisotropic parameter. Some examples, with different models for the waiting time density, are presented. In the limit of strong forward or backward scattering, we have a new way to analyze quasi-unidirectional or quasi-collapsed processes respectively.Also at Universidad Nacional del Comahue, Centro Regional BarilocheComisión Nacional de Energia AtómicaComisión Nacional de Energía Atómica and Universidad Nacional de Cuyo     

四氯化碳和二硫化碳中乙醇构象异构体布居的羟基伸缩振动拉曼光谱

综合利用拉曼光谱和密度泛函理论研究了乙醇在四氯化碳、二硫化碳溶液中乙醇构象异构体的布居. 首先确定了trans和gauche构象乙醇在OH伸缩振动拉曼光谱中的归属,然后结合理论计算的拉曼散射截面估计了两个异构体的能级差。 可以看出在四氯化碳和二硫化碳中trans乙醇更稳定。通过分析不同温度的拉曼光谱,发现范霍夫方程在这里并不适用. 利用玻尔兹曼分布律和理论拉曼散射截面,发现了两个异构体的能级差随着温度升高而增大,这反映了溶剂与乙醇之间越弱的分子间相互作用更有利于trans构象乙醇的布居。     

Hypergeometric modes

A new family of paraxial laser beams that form an orthogonal basis is discussed. When propagated in uniform space, these beams preserve their structure to scale. The intensity distribution profile for such beams is similar to that for the Bessel modes, representing a set of alternating bright and dark concentric rings. The complex amplitude of these beams is proportional to the degenerate (confluent) hypergeometric function, and therefore we term such beams hypergeometric (HyG) modes. The HyG modes are generated with a liquid-crystal microdisplay.     

Two-body problem for Weber-like interactions

The problem of two moving bodies interacting through a Weber-like force is presented. Trajectories are obtained analytically once relativistic and quantic considerations are neglected. The main results are that in the case of limited trajectories, in general, they are not closed and in the case of open trajectories, the deflection angles are not the same for similar particles with given energies and angular momenta but opposite potentials. This last feature suggests the possibility of a direct verification of the validity of Weber's law of force for electromagnetic interactions.Departamento de Eletrônica Quântica/Grupo de Plasmas (on leave of absence from Comisión Nacional de Energía Atómica, Argentina).     

电子碰撞类氢离子电离非相对论跃迁矩阵元

利用BBK模型,导出了电子-类氢离子碰撞电离跃迁矩阵元的解析形式。出射道用包含三个适当的合流超几何函数,且满足适当的三体库仑边界条件的关联BBK波函数来描述,入射道考虑了入射电子与靶离子的长程库仑相互作用而包括一库仑波。含有四个合流超几何函数在内的跃迁T-矩阵元的六维空间积分已约化为一包含高斯-超几何函数的三维实参数积分。忽略入射电子与靶离子的库仑相互作用时,表示用平面波描述的入射电子与靶离子的碰撞,再取靶离子电荷数Z=1时,表示氢原子的电离;略去排斥因子(α₁₂=0)时,又表示不考虑关联的电离。从而可分析各种情况的电离性质。在目前离子的(e,2e)反应三微分截面实验数据还不具备的情况下,可先提供一些必要的理论结果。进而可广泛地应用于处理散射问题之中。     

NONCOMMUTATIVE HYPERGEOMETRIC AND BASIC HYPERGEOMETRIC EQUATIONS

Recently, J. A. Tirao [Proc. Nat. Acad. Sci. 100(14) (2003) 8138–8141] considered a matrix-valued analogue of the ₂F₁ Gauß hypergeometric function and showed that it is the unique solution of a matrix-valued hypergeometric equation analytic at z = 0 with value I, the identity matrix, at z = 0. We give an independent proof of Tirao's result, extended to the more general setting of hypergeometric functions over an abstract unital Banach algebra. We provide a similar (but more complicated-looking) result for a second type of noncommutative ₂F₁ Gauß hypergeometric function. We further give q-analogues for both types of noncommutative hypergeometric equations.     

Random matrix theory of singular values of rectangular complex matrices I: Exact formula of one-body distribution function in fixed-trace ensemble

The fixed-trace ensemble of random complex matrices is the fundamental model that excellently describes the entanglement in the quantum states realized in a coupled system by its strongly chaotic dynamical evolution [see H. Kubotani, S. Adachi, M. Toda, Phys. Rev. Lett. 100 (2008) 240501]. The fixed-trace ensemble fully takes into account the conservation of probability for quantum states. The present paper derives for the first time the exact analytical formula of the one-body distribution function of singular values of random complex matrices in the fixed-trace ensemble. The distribution function of singular values (i.e. Schmidt eigenvalues) of a quantum state is so important since it describes characteristics of the entanglement in the state. The derivation of the exact analytical formula utilizes two recent achievements in mathematics, which appeared in 1990s. The first is the Kaneko theory that extends the famous Selberg integral by inserting a hypergeometric type weight factor into the integrand to obtain an analytical formula for the extended integral. The second is the Petkovšek–Wilf–Zeilberger theory that calculates definite hypergeometric sums in a closed form.     

The instrumental complexity of states

We apply a formalism developed previously to study the notion of the complexity of states in a general statistical theory. We identify the extreme points of the instrument sets as those instruments that view an intrinsic complexity of states and are uncontaminated by stochastic contributions of the experimenter. Various notions of entropy of a state are investigated.This work was financially supported by CNP_q (Conselho Nacional de Desenvolvimento Científico e Tecnológico) and FINEP (Financiadora de Estudos e Projetos) of the Brazilian government.     

Closed expressions for matrix elements of the trigonometric Pöschl-Teller potential

Analytical matrix elements of the xn (n>0) and m[tan(x)]m^′[cos(x)]dn/dxn operators are derived using the eigenfunctions of the symmetric trigonometric Pöschl-Teller potential. The closed formulas are written in terms of Gauss hypergeometric functions and could be used in variational calculations to describe vibrational energy levels associated with bending modes. Multiprecision computational packages are considered in order to obtain an arbitrary level of precision.     

Effective Lagrangians for points,strings and membranes from the BFV formalism

We derive an expression for the charge radius and anapole moment of a free fermion induced at one loop in the standard Glashow-Salam-Weinberg model of electroweak interactions. The result, despite earlier claims to the contrary, is demonstrably gauge-invariant and observable in principle.On leave from Instituto de Física, Universidad National Autónoma de México