Generalized Value Distribution for Herglotz Functions and Spectral Theory

We generalize the theory of value distribution for a class of functions defined as boundary values of Herglotz functions, by considering other measures than Lebesgue measure. The link with compositions of Herglotz functions is presented, and precise relations for the associated measures are obtained. We also consider uniformly convergent sequences of Herglotz functions on compact subsets of the upper half-plane, and prove that the corresponding sequence of Herglotz measures and the generalized value distribution of these functions also converge.     

How many constitutive functions are necessary to determine a thermoelastic material?

The independence of the constitutive functions of a thermoelastic material compatible with the Clausius-Duhem inequality is analysed. The constitutive functions for the absolute temperature and entropy are shown to be determined (to within a change of a unit for the absolute temperature and the entropy constant) by the constitutive functions for the stress tensor and the heat flux vector. Hence two constitutive functions are sufficient to specify a thermoelastic material.     

Estimates for Entries of Matrix Valued Functions of Infinite Matrices

Sharp upper estimates for the absolute values of entries of matrix valued functions of infinite matrices, as well as two sided estimates for the entries of matrix valued functions of infinite M-matrices (monotone matrices) are derived. They give us bounds for the lattice norms of matrix valued functions and positivity conditions for functions of M-matrices. In addition, some results on perturbations and comparison of matrix functions are proved. Applications of the obtained estimates to the Hille-Tamarkin matrices and differential equations are also discussed. This research was supported by the Kamea fund of the Israel     

Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions

Considering the equations for some functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four- and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma functions are presented.     

Analytic continuation and reciprocity relation for collinear splitting in QCD

It is well-known that direct analytic continuation of the DGLAP evolution kernel(splitting functions)from space-like to time-like kinematics breaks down at three loops.We identify the origin of this breakdown as due to splitting functions not being analytic functions of external momenta.However,splitting functions can be constructed from the squares of(generalized)splitting amplitudes.We establish the rules of analytic continuation for splitting amplitudes,and use them to determine the analytic continuation of certain holomorphic and anti-holomorphic part of splitting functions and transverse-momentum dependent distributions.In this way we derive the time-like splitting functions at three loops without ambiguity.We also propose a reciprocity relation for singlet splitting functions,and provide non-trivial evidence that it holds in QCD at least through three loops.     

Method for breaking chains of equations for green's functions

This article presents a new method for closing infinite systems of equations for Green's functions. Its basis is a procedure for simultaneously splitting two higher-order Green's functions. There is a particular relationship between the three Green's functions that result. These Green's functions are self-consistent. Known Green's functions are used to reconstruct the corresponding correlation functions. An example is provided by the problem of magnetization for a spin system in the Heisenberg model with first-and secondorder approximations in the spin-spin interaction. The first-order approximation corresponds to the molecular-field approximation. The second-order computation corresponds to obtaining general relations determining the magnetization of a system. V. I. Lenin Moscow State Pedagogical University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 42–47, August, 1996.     

Correlated basis functions for the many-electron problem

The method of correlated basis functions is applied to the electron system in a metal. To overcome the twofold difficulty of large particle number and long-range Coulomb interaction in metals, a new optimal cluster decomposition for arbitrary correlated wave functions is derived. With this method, not only ground-state properties, but also thermal averages and response functions can be calculated from a given set of correlated basis functions. The appropriate synthesis of correlated wave functions including physically expected properties of subsystems as for instance partially filled inner shells in transition metals is discussed in detail.     

Two-particle Green's functions in non-equilibrium matter

The method of the derivation of two-particle Green's functions in non-equilibrium matter is developed. The closed set of equations for the vertex functions and also for the two-particle Green's functions is obtained by means of the summation of the series of irreducible diagrams. The solution of such equations completely defines the two-particle Green's functions in matter.     

Comparing the evolution of canalyzing and threshold networks

We study the influence of the type of update functions on the evolution of Boolean networks under selection for dynamical robustness. The chosen types of functions are canalyzing functions and threshold functions. Starting from a random initial network, we evolve the network by an adaptive walk. During the first time period, where the networks evolve to the plateau of 100 percent fitness, we find that both type of update functions give the same behavior, albeit for different network sizes and connectedness. However, on the long run, as the networks continue to evolve on the fitness plateau, the different types of update functions give rise to different network structure, due to their different mutational robustness. When both types of update functions occur together, none of them is preferred under long-term evolution.     

Plasma response functions,fluctuation-dissipation relations and the velocity-average-approximation

We derive and list relationships for quadratic and cubic static response functions and connect them with three- and four-point functions through generalized fluctuation-dissipation relations. We also introduce the useful concept of “response function of the second kind.” to describe the reponse of a system through the perturbation of its two-point functions. Next, we point out that the VAA (velocity-average-approximation) introduced earlier as a part of a dynamical approximation scheme for strongly coupled one-component plasmas is exact in the static limit. This observation, combined with the fluctuation-dissipation relations, allows one to derive a hierarchy of equations for response functions of increasing nonlinearity, which is equivalent to the BGY hierarchy for particle distribution functions.     

Two-time correlation functions: stochastic and conventional quantum mechanics

An investigation of two-time correlation functions is reported within the framework of (i) stochastic quantum mechanics and (ii) conventional Heisenberg-Schr?dinger quantum mechanics. The spectral functions associated with the two-time electric dipole moment correlation functions are worked out in detail for the case of the hydrogen atom. While the single time averages are identical for stochastic and conventional quantum mechanics, differences arise in the two approaches for multiple time correlation functions.     

On the representations of the groups SO(3), O(2, 1), and M(2) defined in terms of Lamé, Mathieu,and parabolic functions

Products of Mathieu, parabolic-cylinder and Lamé functions appear as basis functions for the irreducible representations of the groups M(2), O(2,1) and SO(3). Addition formulas concerning the Mathieu functions, parabolic cylinder functions, and Lamé polynomials are recovered or obtained.     

Explicit spheroidal wave functions of the hydrogen atom

A simple and straightforward calculating scheme is suggested for finding wave functions of the hydrogen atom in prolate spheroidal coordinates. The wave functions are found in an explicit form by the direct solution of appropriate one-dimensional equations. The suggested calculating scheme allows us to carry out simple calculations and to obtain spheroidal wave functions in principle for arbitrary eigenstates of the hydrogen atom. Expansions are found for the obtained spheroidal wave functions over a spherical basis.     

Exponential localization of Wannier functions in insulators

The exponential localization of Wannier functions in two or three dimensions is proven for all insulators that display time-reversal symmetry, settling a long-standing conjecture. Our proof relies on the equivalence between the existence of analytic quasi-Bloch functions and the nullity of the Chern numbers (or of the Hall current) for the system under consideration. The same equivalence implies that Chern insulators cannot display exponentially localized Wannier functions. An explicit condition for the reality of the Wannier functions is identified.     

Two-particle Green's functions in non-equilibrium matter

The method of derivation of two-particle Green's functions in the non-equilibrium matter has been developed. The closed set of equations for the vertex functions, as well as for the two-particle Green's functions, is obtained by means of the summation of the series of indecomposable diagrams. The solution of such equations completely determines the two-particle Green's functions in the matter.     

Limiting Distributions for a Polynuclear Growth Model with External Sources

The purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by Prähofer and Spohn. Depending on the strength of the sources, the limiting distribution functions are either the Tracy–Widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero. Moreover, we obtain transition functions between pairs of the above distribution functions in suitably scaled limits. There are also similar results for a discrete totally asymmetric exclusion process.     

Numerical calculation of localized (Wannier) functions from ab-initio Hartree-Fock wave functions

The Fourier transformation necessary to obtain localized (Wannier) functions from delocalized ab-initio crystal Hartree-Fock (Bloch) functions is carried out for the one-dimensional models of solids. Combinations of Wannier functions from different bands lead to even more localized functions in close analogy to localization in molecules.     

Radial Distribution Functions for Molecules in the Universal Equation of State Model for Gaseous/Fluid/Condensed Systems

Analytical expressions and a numerical method for calculation of distribution functions of hard spheres gij(r) based on inverting the Laplace transform for functions rgij(r) obtained from the Percus—Yevick equation are obtained. The method for calculation of radial distribution functions is applicable for any distances between hard spheres; it is verified by comparison of numerical results and Monte Carlo simulations. The application of the developed method for calculation of the radial distribution functions of metal atoms is demonstrated. Distribution functions are required to construct a universal theoretical model of equation of state capable of describing both dense multicomponent gas and condensed substances (liquid or solid phases) with high accuracy which is substantially faster than computer experiments (Monte Carlo and molecular dynamics methods).