New and Old Results in Resultant Theory

Resultants are getting increasingly important in modern theoretical physics: they appear whenever one deals with non-linear (polynomial) equations, with non-quadratic forms or with non-Gaussian integrals. Being a subject of more than three-hundred-year research, resultants are of course rather well studied: a lot of explicit formulas, beautiful properties and intriguing relationships are known in this field. We present a brief overview of these results, including both recent and already classical. Emphasis is made on explicit formulas for resultants, which could be practically useful in a future physics research.Comment: 50 pages, 15 figure

Realistic interatomic potential for MD simulations

The coefficients of interatomic potential of simple form Exp-6 for neon are obtained. Repulsive part is calculated ab-initio in the Hartree-Fock approximation using the basis of atomic orbitals orthogonalized exactly on different lattice sites. Attractive part is determined empirically using single fitting parameter. The potential obtained describes well the equation of state and elastic moduli of neon crystal in wide range of interatomic distances and it is appropriate for molecular dynamic simulations of high temperature properties and phenomena in crystals and liquids.Comment: MikTex v.2.1 (AMS-TEX),11 pages, 3 EPS figure

Dynamical lattice instability versus spin liquid state in a frustrated spin chain system

The low-dimensional s=1/2 compound (NO)[Cu(NO3)3] has recently been suggested to follow the Nersesyan-Tsvelik model of coupled spin chains. Such a system shows unbound spinon excitations and a resonating valence bond ground state due spin frustration. Our Raman scattering study demonstrates phonon anomalies as well as the suppression of a broad magnetic scattering continuum for temperatures below a characteristic temperature, T<T*=100K. We interpret these effects as evidence for a dynamical interplay of spin and lattice degrees of freedom that might lead to a further transition into a dimerized or structurally distorted phase at lower temperatures.Comment: 5 pages, 6 figure

M-Theory of Matrix Models

Small M-theories unify various models of a given family in the same way as the M-theory unifies a variety of superstring models. We consider this idea in application to the family of eigenvalue matrix models: their M-theory unifies various branches of Hermitean matrix model (including Dijkgraaf-Vafa partition functions) with Kontsevich tau-function. Moreover, the corresponding duality relations look like direct analogues of instanton and meron decompositions, familiar from Yang-Mills theory.Comment: 12 pages, contribution to the Proceedings of the Workshop "Classical and Quantum Integrable Systems", Protvino, Russia, January, 200

Application of Johnson disribution to the problemof aerospace images classification

Solving the problem of aerospace images classification it was suggested to approximate distribution density of image characteristics by Johnson distribution. The possibilities of such approach were investigated and its availability was show

Partition Functions of Matrix Models as the First Special Functions of String Theory I. Finite Size Hermitean 1-Matrix Model

Even though matrix model partition functions do not exhaust the entire set of tau-functions relevant for string theory, they seem to be elementary building blocks for many others and they seem to properly capture the fundamental symplicial nature of quantum gravity and string theory. We propose to consider matrix model partition functions as new special functions. This means they should be investigated and put into some standard form, with no reference to particular applications. At the same time, the tables and lists of properties should be full enough to avoid discoveries of unexpected peculiarities in new applications. This is a big job, and the present paper is just a step in this direction. Here we restrict our consideration to the finite-size Hermitean 1-matrix model and concentrate mostly on its phase/branch structure arising when the partition function is considered as a D-module. We discuss the role of the CIV-DV prepotential (as generating a possible basis in the linear space of solutions to the Virasoro constraints, but with a lack of understanding of why and how this basis is distinguished) and evaluate first few multiloop correlators, which generalize semicircular distribution to the case of multitrace and non-planar correlators.Comment: 64 pages, LaTe