Automorphic Functions,Poincaré Series,and Conformal Fields on Riemann Surfaces

Poincaré series and automorphic functions for SU(1, 1) and a discrete subgroup Γ are studied with harmonic analysis. We consider automorphic functions on the open unit circle with general “spin label” m and their decomposition into irreducible automorphic functions by means of the Plancherel formula. These automorphic functions are bijectively mapped onto automorphic distributions on the boundary of the unit circle by meam of the Poisson kernel. The exponent of convergence of Poincaré series is expressed in representation theory language. The results are applied to two-point functions of conformal fields.     

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     

The properties of liquid nitrogen

The two-electron density functions in p-space for H₂, LiH and BH are partially integrated to yield longitudinal and transverse distribution functions which, like their r-space analogues, admit of a simple physical interpretation. The forms of the corresponding one-electron distribution functions are also discussed and the identity of the longitudinal function with the directional Compton profile along the bond is noted. Finally, the effects of electron correlation are further examined by using the two-electron distribution functions to construct longitudinal and transverse hole functions in p-space.     

Molecular electronic distribution functions and correlation holes Theory and application to H2, LiH and BH

The one-electron density functions for the diatomic hydrides H₂, LiH and BH, defined with respect to a limited STO basis set, are partially integrated to yield longitudinal and transverse distribution functions which admit of a simple pictorial representation. The maxima in the longitudinal distribution functions occur in regions conventionally ascribed to core pairs, lone pairs or bond pairs.

The correlation between pairs of electrons of given spin is then analysed in terms of the analogous partially integrated pair density functions and in terms of the associated hole functions. In the case of LiH and BH, the Fermi hole functions are similar in form to the negatives of the one-electron distribution functions for the appropriate singly occupied molecular orbitals; but the differences between the two functions are more marked in BH, where there is increased spatial interpenetration of the valence electron pairs. The results also show, in a simple way, how the electron motion is over-correlated in H₂ when the Heitler-London wavefunction is used.     

On the structure of the conformal covariant N-point functions

The conformal covariant n-point functions for the fields with arbitrary spin and scale dimension are found. These functions are determined up to a set of arbitrary functions on $\text{n(n?3)}\text{2}$ independent conformal invariant variables. These arbitrary functions cannot be determined without dynamical assumptions.     

An examination of the steady-state solution of the differential equation governing indoor sound fields

The use of the characteristic functions to obtain a steady-state sound field is based on the completeness of the set of the characteristic functions. For absorptive boundaries, the characteristic functions are complex functions, and there is no proof in mathematics of whether the set of the characteristic functions is complete. In addition, the use of the characteristic functions to obtain a steady-state sound field of three-dimensional space requires the solving of a huge amount of the complex number k_nx The aims of this paper are to examine whether the set of the characteristic functions can still be complete for strong absorptive boundaries and to provide a practical solving method for k_nx.     

General formulae for polarization observables in deuteron electrodisintegration and linear relations

Formal expressions are derived for all possible polarization observables in deuteron electrodisintegration with longitudinally polarized incoming electrons, oriented deuteron targets and polarization analysis of outgoing nucleons. They are given in terms of general structure functions which can be determined experimentally. These structure functions are Hermitean forms of theT-matrix elements which, in principle, allow the determination of allT-matrix elements up to an arbitrary common phase. Since the set of structure functions is overcomplete, linear relations among various structure functions exist which are derived explicitly.Supported by the Deutsche Forschungsgemeinschaft (SFB 201)     

Discrete phase space II. Relativistic lattice wave equations and divergence-free green's functions

Therelativistic lattice Klein-Gordon equation, Dirac equation, electromagnetic equations, and gauge field equations are presented as partialdifference equations. Various lattice Green's functions are constructed (except for non-abelian gauge fields). It is proved that many of the lattice Green's functions are non-singular or divergence-free. Moreover, it is conjectured that all lattice Green's functions are non-singular.     

On a Mathematical Framework for the Constitutive Equations of Anisotropic Dielectric Relaxation

Three classes of time-domain non-relativistic anisotropic dielectric constitutive equations of increasing generality are discussed. In each class dissipativity is ensured by the choice of a class of convolution kernels in the D-to-E constitutive equation expressing the electric field E in terms of the electric displacement field D. Defining properties of the inverse (E-to-D) kernels and their Fourier-Laplace transforms (complex dielectric functions) are determined by inversion of the D-to-E constitutive equation. By this procedure it is shown that dielectric functions of the dipolar dielectrics are tensor-valued Bernstein functions while the dielectric functions of the Drude-Lorentz type are tensor-valued negative definite functions. The properties of the complex dielectric permittivities are also determined for either class. The theory is applied to an exhaustive review of empirical response functions of real dielectric materials encountered in the literature. Each class of convolution kernels is consistent with existence of a conserved energy, but in one case a strictly dissipative energy can be constructed.     

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.     

The role of Fermi motion and quark exchange as a nuclear medium effect in the nuclear structure function of 3He, 3H nuclei and their EMC effects

To extract the nuclear structure function of light nuclei like ³He and ³H, the convolution of proton and neutron structure functions in the conventional approach could not explain the modification of nuclear structure functions of these nuclei because of some nuclear medium effects. There are some models that have some success to explain the modification of nuclear structure functions in limited x range. So in this investigation the quark exchange model is considered to extract the nuclear medium effect of this phenomenon (quark exchange effect) for nuclear structure functions of ³He and ³H nuclei. The Fermi motion part of the nuclear structure functions for these nuclei is extracted by taking the GRV’s neutron and proton structure functions in the conventional approach. So the nuclear structure functions and the EMC ratios of ³He and ³H nuclei are calculated by considering both Fermi motion and the nuclear medium effect of quark exchange. Finally the extracted results are compared with available data.     

Quantum Group Approach to q-Special Functions

Quantum groups are used to define q-special functions. The Casimir operatorsof a variation of SU _q(2) and E _q(2) are derived. The proposed q-associated Legendreand q-Bessel functions are the eigenfunctions of the Casimirs. The results differfrom ordinary q-special functions, but this is expected since the q-generalizationis not unique.     

Gluon cascades and amplitudes in light-front perturbation theory

We construct the gluon wave functions, fragmentation functions and scattering amplitudes within the light-front perturbation theory. Recursion relations on the light-front are constructed for the wave functions and fragmentation functions, which in the latter case are the light-front analogs of the Berends–Giele recursion relations. Using general relations between wave functions and scattering amplitudes it is demonstrated how to obtain the maximally-helicity violating amplitudes, and explicit verification of the results is based on simple examples.     

Numerical results for the thermal scattering functions

A recent formulation in radiative transfer defined the thermal scattering functions that characterize radiative transfer from a general, plane-parallel, finite medium driven solely by an internal distribution of thermal sources. Exiting diffuse intensities are expressed as space convolutions of the thermal scattering functions with any thermal source distribution. A parametric study is presented to obtain the basic structure of these scattering functions. The independent variables of these azimuthally independent functions are the direction consine μ and source location t, while the parameters are the single scattering albedo ω, total optical depth t₀, and the asymmetry factor g in the Henyey-Greenstein phase function. The basic functional trends are discussed using various parametric plots, and selected tabular results are given to allow numerical checks. The computational method is invariant imbedding. As a particular application, these functions are used in the following companion paper to obtain exiting intensities from inhomogeneous and nonisothermal media.     

Analytical solutions and rogue waves in (3+1)-dimensional nonlinear Schrödinger equation

Analytical solutions in terms of rational-like functions are presented for a (3+1)-dimensional nonlinear Schrödinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz. Several free functions of time t are involved to generate abundant wave structures. Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.     

The representations of the Poincaré group as functions of the eigenvalues of casimir operators

In this paper explicit basis functions are defined for the Poincaré group. Both these functions and the representation matrix elements are continuous functions of the momentum variables for the whole real p² spectrum, including the p² = 0 point. The essence of our method is to enlarge previously obtained SL(2, C) basis functions and representations of a similar nature.     

Analyzing integrated-optical inhomogeneous planar waveguides by Galerkin's method: A detailed comparison of two different basis functions

Abstract

The trigonometric and Hermite-Gaussian basis functions for determining the modal characteristics of inhomogeneous optical waveguides by means of the Galerkin's method are presented and analyzed. The results obtained with each set of basis functions for mode spectra and field distributions are compared with other exact and approximate methods. The merits and problems arising with each set of basis functions are discussed.     

Structure functions at HERA

HERA provides the key facility for the measurement of proton structure functions. Formalism and methods are outlined for the measurement and interpretation of inclusive structure functions, including the use of polarised e ^± beams. The measurement of charm, beauty and photon structure functions is discussed, together with special runs at low proton energy for measurement of the longitudinal structure function. Finally, the functions accessed using polarised beams on polarised targets are indicated.     

Nonperturbative Green's function approach to quantum field theory and arbitrariness of solutions

Green's function equations are considered for interacting spinor and (pseudo) scalar fields with interactions . These equations do not determine higher many-point functions if two-point functions are given as “input.” If vertex parts are given as input, two-point functions are determined but higher many-point functions are not determined.