Contemporary Problems in Mathematical Physics

Foreword.

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Contents.

We review the basic notions of group theory, in particular Lie groups and Lie algebras, and of representations of the latter. Then we examine briefly their occurrence in classical physics for the description of invariance properties of physical systems and the concomitant conservation laws resulting from Noether's theorem. In the last section, finally, we give an overview of the applications of group representation theory in quantum physics, with special emphasis on the proper mathematical description of symmetry properties, both in quantum mechanics and in quantum field theory.

These lectures provide an introduction to the AdS/CFT correspondence. The goal is to review the recent progress which provides the gravitational duals of certain marginal deformations of super Yang–Mills theory. An attempt has been made to keep the lectures as self contained as possible.

A discussion is presented on aspects of supersymmetry in the context of quantum mechanical models. Rather than dealing with the details of specific models, the presentation aims to draw out general features of such systems.

Bosonization of the Schwinger model with noncommutative chiral bosons is considered on a spacetime of cylinder topology. Using point splitting regularization, manifest gauge invariance is maintained throughout. Physical consequences are discussed.

A short review of Bol algebras and the suggestion of their relevance to physics is given. The classification of trivial Bol algebras is obtained.

We have developed a random phase approximation with exchange (RPAE) method to study the inner-shell electron excitation and photoionization of an open-shell atom or ion. The equations for the dipole matrix element and Coulomb matrix element are derived using the second quantization method and checked against previous formulas by reducing them to those for the case of a single open shell or a closed shell. A computer code has been written to perform the RPAE calculation of the inner-shell photoionization processes. Our RPAE method has been used to calculate the photoionization cross sections of the 4d–ɛf, so called Giant resonance for the atom I and ions Xe⁺ and I⁺. The ground states and core wave functions of Xe⁺, I and I⁺ are obtained through self-consistent Hartree–Fock calculations. The radial functions of the continuum electron are determined by solving the linear HF equations. Finally the following states for Xe⁺ and I are included in the calculations,

In this contribution, we provide new analytical relations finely tuned to describe various atmospheric elements and based on the determination of the corresponding backscattered optical radiation. The novelty of this work consists in the fact that the cloud is not considered anymore as a well delimited and isolated layer in the ambient air with a constant density of particles, but as an open structure mixing with the ambient air and possessing a non-constant number of particles. The proposed model gives rise to a unique formulation of ice crystallite clouds as well as of fog or aerosol concentration. Application to a real signal is considered as an illustration.

The nonabelian global chiral symmetries of the two-dimensional N flavour massless Schwinger model are realised through bosonisation and a vertex operator construction.

No abstract received.

We generalize a diffusion equation modeling a brain cancer treatment and investigate its Lie symmetry group. Analytical solutions are derived.

A physical model of the O(3)-invariant coupled integrable dispersionless equations that describe the dynamics of a focused system within the background of a plane gravitational field is studied. The investigation is carried out both numerically and analytically, and is pursued both in the Lie algebra and quasigroup theory contexts in which the structure constants of the former are extended into the structure functions of the latter. The energy density and topological structures such as loop solitons are examined.

The adsorption of CH₃ on Pt(100) and Ni(111) surfaces at the coverage of 0.25 ML has been investigated using the density functional theory with the generalised gradient approximation of Perdew–Wang for exchange correlation energy. The metallic surface was represented with a slab formed by four layers of Pt(100) or Ni(111). We optimized the adsorbed CH₃ on all high symmetry sites of Pt(100) (on top, hollow, bridge) and Ni(111) (on top, bridge, hcp, fcc) surfaces. On Pt(100) surface, CH₃ is found to adsorb only on the top site, the hollow and bridge sites being unstable. CH₃ is found to adsorb at all four high symmetry sites on the Ni(111) surface with similar adsorption energies for fcc, hcp and bridge sites, whereas the on top site is much less stable.

Following the convincing experimental evidence of oscillatory exchange interaction between antiferromagnetic FeMn layers across a Cu spacer recently found by Cai et al., we have performed ab-initio spin-polarized density functional theory calculations on (FeMn)_n/Cu/(FeMn)_n with n = 1, 2, 3 in the (001) crystallographic face. For n = 1 we have investigated all possible magnetic configurations in the self-consistent procedure and we end up with five solutions. The ground state is found to be of ferromagnetic type: i) there is an intrinsic ferromagnetic configuration in the FeMn plane, and ii) there is a ferromagnetic coupling between the two FeMn planes separated by the Cu spacer. At 20.17 mRyd/cell a solution with complex magnetic behavior is observed: ferromagnetic coupling between Fe and Mn, in one FeMn plane, and antiferromagnetic coupling between Fe and Mn in the second FeMn plane separated by Cu. Another solution at 41.26 mRyd above the ferromagnetic ground state presents in-plane antiferromagnetic configurations in each FeMn layer. For n = 2 the ground state presents in-plane ferromagnetism for the two FeMn layers adjacent to Cu; the other two FeMn layers are clearly of in-plane antiferromagnetic type. Contrary to n = 1, the ferromagnetic coupling through the Cu spacer is now absent. We report on this ground state and various metastable states with small differences of energy with the ground state. For n = 3 the magnetic map is more complex. Explanation of the experimental results of Cai et al. is tentatively made.

We present a prescription for forming matrices with specified eigenvalues and known eigenvectors. With this method, we can form Hermitian, anti-Hermitian, symmetric and general matrices with arbitrary eigenvalues with great ease. Certain functions are required for the implementation of this method. Probability amplitudes connecting observables with discrete eigenvalue spectra perform the task, and they can be obtained from spin theory. For the example case of 5 × 5 matrices, these functions are given, and various illustrative matrices are generated together with their normalized eigenvectors.

We use SU(1,1) potential group methods to deduce exact solutions of Sturm–Liouville equations.

The time required to achieve pregnancy among married couples termed fecundability has been proposed to follow a beta-geometric distribution. The accuracy of the method used in estimating the parameters of the model has an implication on the goodness of fit of the model. In this study, the parameters of the model are estimated using the Method of Moments and Newton–Raphson estimation procedure. The goodness of fit of the model was considered, using estimates from the two methods of estimation, as well as the asymptotic relative efficiency of the estimates. A noticeable improvement in the fit of the model to the data on time to conception was observed, when the parameters are estimated by Newton–Raphson procedure, and thereby estimating reasonable expectations of fecundability for married female population in the country.

In this work, we coupled the filtration model at a decreasing rate to the heat equation and determined the condition for the existence and uniqueness of a solution to the coupled equation.

In this paper, we provide a general overview of the basic set of polynomials {P_n(z)}_n≥0. The domain of effectiveness is related to the radius r of convergence of the associated basic series of {P_n(z)}_n≥0 The Cannon condition is given. The application in the case of Chebychev polynomials leads to the improvement of the Whittaker constant up to fourteen digits.

We review the construction of the continuous wavelet transform (CWT) on the 2-sphere by two methods, the group-theoretical approach and the geometrical method based on conformal invariance. Then we discuss the discretization of the spherical CWT and build associated discrete wavelet frames, first half-continuous ones (only the scale is discretized), then fully discrete ones. To that effect, we generalize the notion of frame, introducing weighted and controlled frames.

In this paper we prove, among other things, that a semi-simple and topologically simple Banach algebra which has an identity and contains a non-zero finite-dimensional element is itself a finite-dimensional algebra.

In this paper, the Adomian Decomposition Method (ADM) is used to study the Duffing equation. The series solution is constructed and compared with the solution obtained by the perturbation method.

In this paper, we define the Hermite interpolation polynomial of several real variables.

By using the table of Jones polynomials we compute the braid index with up to ten crossings.

We consider a nonparametric estimation of the hazard rate function based on right-censored data using the wavelet method. Asymptotic properties and strong uniform consistency rates are established under suitable conditions.

In this paper, we give Laguerre–Freud equations for the recurrence coefficients of discrete semi-classical orthogonal polynomials of class two, when the polynomials in the Pearson equation are of the same degree. The case of generalized Charlier polynomials is also presented.

We give existence results for unilateral contact between a certain special case of a class of nonlinear viscoelastic bodies and a proposed harmonic deformable foundation under Tresca friction. The harmonic foundation is defined in terms of a subdifferential and incorporates in some limits the cases of a rigid foundation and a Winkler foundation. Our results are obtained using tools from the theories of nonlinear functional analysis and variational inequalities.

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