Mathematical Physics

The following sections are included:

• ORGANIZING COMMITTEES
  • INTERNATIONAL ADVISORY COMMITTEE
  • LOCAL ORGANIZING COMMITTEE

• LIST OF PARTICIPANTS

• PREFACE

• CONTENTS

A discrete eigenvalue E_n of a Schrödinger operator H = −Δ + νf(r) is given, as a function F_n(ν) of the coupling parameter ν ≥ ν_c. It is shown how the potential shape f(x) can be reconstructed from F_n(ν): A constructive inversion algorithm and a functional inversion sequence are both discussed.

Most of the theoretical physics known today is described by using a small number of differential equations. If we study only linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe this problem. These equations have power series solutions with simple relations between consecutive coefficients and can be generally represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painlevé equation. There are important examples, however, where one has to use more complicated equations. An example often encountered in quantum mechanics is the hydrogen atom in an external electric field, the Stark effect. One often bypasses this difficulty by studying this problem using perturbation methods. If one studies certain problems in astronomy or general relativity, encounter with Heun equation is inevitable. This is a general equation whose special forms take names as Mathieu, Lamé and Coulomb spheroidal equations. Here the coeffcients in a power series expansions do not have two way recursion relations. We have a relation at least between three or four different coefficients. A simple integral transform solution also is not obtainable. Here I will try to introduce this equation and give some examples where the result can be expressed in terms of solutions of this equation. Although this equation was discovered more than hundred years ago, there is not a vast amount of literature on this topic and only advanced mathematical packages can identify it. Its popularity, however, increased recently, mostly among theoretical physicists, with ninety four papers in SCI in the last twenty five years. More than two thirds of the papers which use these functions in physical problems were written in the last decade.

Emergence of the experimental evidence of E₈ in the analysis of one dimensional Ising-model invokes further studies of the Coxeter-Weyl groups generated by reflections regarding their applications to polytopes. The Coxeter group W(H₄) which describes the Platonic Polytopes 600-cell and 120-cell in 4D singles out in the mass relations of the bound states of the Ising model for it is a maximal subgroup of the Coxeter-Weyl group W(E₈). There exists a one-to-one correspondence between the finite subgroups of quaternions and the Coxeter-Weyl groups of rank 4 which facilitates the study of the rank-4 Coxeter-Weyl groups. In this paper we study the systematic classifications of the 3D-polyhedra and 4D-polytopes through their symmetries described by the rank-3 and rank-4 Coxeter-Weyl groups represented by finite groups of quaternions. We also develop a technique on the constructions of the duals of the polyhedra and the polytopes and give a number of examples. Applications of the rank-2 Coxeter groups have been briefly mentioned.

In this work, we introduce and investigate the Hermite-Apostol-Bernoulli polynomials by means of a suitable generating function. We establish several interesting properties of these general polynomials. Furthermore, we prove Raabe relation and give some relations between 2-dimensional Hermite polynomials and Apostol-Bernoulli polynomials.

We review second-order system of coupled heat-diffusion equations. The system under consideration contains several arbitrary elements the forms of which are specified via the method of group classification based on the use of low-dimensional Lie algebras. We collect and present several results.

We calculate the Green’s function for the time-dependent linear harmonic oscillator with linear time-dependent terms added to the Hamiltonian and use it to calculate the transition amplitudes between states of the time-independent linear harmonic oscillator. In the absence of the linear terms, transitions are only allowed from even (odd) to even (odd) parity states. However, in the case when the linear terms are present, transitions from even (odd) to odd (even) states are also allowed.

We reduce the solution of the Schrödinger equation to the Riccati equation for the harmonic oscillators with time dependent parameters, the mass and the frequency. We also discuss the linear coupling of the harmonic oscillator with the external driving force. We solve the Riccati equation for three special cases: (i) A time dependent harmonic oscillator with M(t)ω(t) = const., (ii) a driven harmonic oscillator, (iii) a harmonic oscillator with exponentially increasing mass parameter. In the first and second cases, we obtain the coherent states. In third case, we find a quasi-coherent state.

We derive integrability conditions in the form of differential constraints for solving the conformal Ricci collineation equations for cylindrically symmetric static spacetimes. Once these constraint equations are solved one gets the Lie algebras for conformal collineation vectors for these spacetimes.

For flat spacetimes conformal Killing vectors are symmetries of Lagrangians for Geodesic equations. Here it is shown that in general conformal Killing vectors are not symmetries of Lagrangians for geodesic equations.

We study some universal features of gravity in higher dimensions and by universal we mean a feature that remains true in all dimensions ≥ 4. They include: (a) the gravitational dynamics always follow from the Bianchi derivative of a homogeneous polynomial in Riemann curvature and it thereby characterizes the Lovelock polynomial action, (b) all the Λ-vacuum solutions of the Einstein-Lovelock as well as pure Lovelock equation have the same asymptotic limit agreeing with the d dimensional Einstein solution and (c) gravity inside a uniform density sphere is independent of the spacetime dimension and it is always given by the Schwarzschild interior solution.

Recently, modified f(R) theory of gravity has attracted many people due to combined motivation coming from high-energy physics, cosmology and astrophysics. The purpose of this paper is two-fold: Firstly, we shall review some exact solutions of the modified field equations. Secondly, we present some new results on plane symmetric gravitational collapse in f(R) theory. Here, we take all higher order curvature terms on the matter side of the field equations and interpret them as a dark source. In this way, we find the effects of dark energy on the dynamics of plane symmetric gravitational collapse.

This talk summarizes recent studies on the gravitational properties of vacuum energy in a non-Riemannian geometry formed by the stress-energy tensor of vacuum, matter and radiation. Postulating that the gravitational effects of matter and radiation can be formulated by an appropriate modification of the spacetime connection, we obtain varied geometro-dynamical equations which properly comprise the usual gravitational field equations with, however, Planck-suppressed, non-local, higher-dimensional additional terms. The prime novelty brought about by the formalism is that, the vacuum energy does act not as the cosmological constant but as the source of the gravitational constant. The formalism thus deafens the cosmological constant problem by channeling vacuum energy to gravitational constant. Nevertheless, quantum gravitational effects, if any, restore the problem via the graviton and graviton-matter loops, and the mechanism proposed here falls short of taming such contributions to cosmological constant.

Great interest was generated when it was announced that Hawking had admitted that he had lost a bet with Susskind. The bet had been about the “information loss paradox” in which Hawking had tried to show that quantum theory, in its merger with gravity, is incomplete and both quantum theory and relativity need to be modified. Susskind had refuted the argument using quantum arguments why the information would not be lost. In the process he formulated a “black hole complementarity principle”. It turns out that work of Wheeler and I leads to the same conclusions qualitatively and there is reason to believe that introducing the uncertainty principle might give a quantitative result with no further quantum input.

The structure of the translationally invariant noncommutative field theory in determined and the corresponding gauge theory is presented.

We discuss new types of mechanisms to break CP symmetry and to realize flavor mixing in the scenario of gauge-Higgs unification, which guarantees a finite Higgs mass (from the viewpoint of hierarchy problem) and is an attractive scenario for the physics beyond the standard model. In the scenario the Higgs interactions are governed by gauge principle and it may shed some light on the arbitrariness problem of the Higgs interactions. On the other hand, since the Yukawa couplings are real to start with CP violation is a challenging issue. Two kinds of new mechanism of “spontaneous” CP violation are discussed to be possible, one by the complex structure of compactified extra space and one by the VEV of the Higgs. Also the Yukawa couplings are originally universal and hierarchical fermion mass spectrum and flavor mixing are other challenging issues. The hierarchical fermion masses are known to be naturally realized by the exponential suppression factor without fine tuning. Concerning the violation of flavor symmetry, non-degenerate bulk masses are new source of flavor violation beyond the argument of Glashow-Weinberg, and turns out to lead to FCNC at the tree level, though we find the existence of “GIM-like” mechanism in FCNC processes.

We study higher dimensional field theories with extra dimensions from a 4d spectrum point of view. It is shown that 4d mass spectra of spinor, gauge and gravity field theories are governed by quantum-mechanical supersymmetry. The 4d massless modes turn out to correspond to zero energy vacuum states of the supersymmetry. Allowed boundary conditions on extra dimensions compatible with the supersymmetry are found to be severely restricted.

In this talk I discuss and clarify some issues concerning chiral and nonchiral properties of the one-dimensional supermultiplets of the -Extended Supersymmetry. Quaternionic chirality can be defined for = 4, 5, 6, 7, 8. Octonionic chirality for = 8 and beyond. Inequivalent chiralities only arise when considering several copies of = 4 or = 8 supermultiplets.

In this paper, we demonstrate the applicability of the perturbation methods to different elastic models of DNA molecule. Two different kinds of perturbation methods are presented to find a first approximation for the force-extension characteristic of DNA in the anisotropic wormlike chain model, and the persistence length of DNA in the asymmetric elastic rod model. In both cases we show that it is meaningful to use the perturbation theory, and a first-order calculation is enough to find the result with an acceptable accuracy.

Here, we first analyze data from two GPS stations operating in Tashkent and Kitab and new Very Low Frequency (VLF) radio receiver operating in Tashkent for the possible earthquake ionospheric precursors. We find anomalous Total Electron Content (TEC) precursor signals and significant correlation in time between the TEC anomalies and the occurrence of the earthquake in Tashkent on August 22, 2008, M = 4.4. The obtained results have revealed a fine agreement with the TEC anomalies observed during the strong earthquake in Tashkent and we demonstrate the capabilities of the GPS technique to detect ionospheric perturbations caused by the local earthquakes. TEC decrease during the solar eclipse on August 1, 2008 is also obtained from the data at GPS stations in Tashkent and Kitab. We used tweek radio atmospherics (originating from lightning discharges) to estimate electron densities in the D-layer of ionosphere. The propagation characteristics of tweek atmospherics observed at the Tashkent station have been studied near the mode cut-off frequencies. It is shown that the height of the Earth-Ionosphere Wave Guide (EIWG) varies from 84 km to 91 km and nighttime electron density varies from 25 el/cm³ to 27 el/cm³. We have also studied VLF amplitude anomalies related to the earthquakes occurring on the path from the VLF transmitters to the Tashkent station. For analyzing narrowband data we have used the Nighttime Fluctuation method paying attention to the data obtained during the local nighttime (18:00 LT- 06:00 LT). The deathlines for rotating as well as oscillating magnetars are obtained for the different modes of oscillations and it is shown that the oscillations increase the region in the (P is the period) diagram of the magnetars which is allowed for the radio emission.