Properties of scalar perturbations generated by conformal scalar field

Primordial scalar perturbations may be generated when complex conformal scalar field rolls down its negative quartic potential. We begin with the discussion of peculiar infrared properties of this scenario. We then consider the statistical anisotropy inherent in the model. Finally, we discuss the non-Gaussianity of scalar perturbations. Because of symmetries, the bispectrum vanishes identically. We present a general expression for the trispectrum and give its explicit form in the folded limit.Comment: Prepared for "Gravity and Cosmology (GC2010)" workshop, Kyoto, Japan, May,24-July,16, 2010; 15 pages, 1 figur

Octonic Electrodynamics

In this paper we present eight-component values "octons", generating associative noncommutative algebra. It is shown that the electromagnetic field in a vacuum can be described by a generalized octonic equation, which leads both to the wave equations for potentials and fields and to the system of Maxwell's equations. The octonic algebra allows one to perform compact combined calculations simultaneously with scalars, vectors, pseudoscalars and pseudovectors. Examples of such calculations are demonstrated by deriving the relations for energy, momentum and Lorentz invariants of the electromagnetic field. The generalized octonic equation for electromagnetic field in a matter is formulated.Comment: 12 pages, 1 figur

Superpolynomials for toric knots from evolution induced by cut-and-join operators

The colored HOMFLY polynomials, which describe Wilson loop averages in Chern-Simons theory, possess an especially simple representation for torus knots, which begins from quantum R-matrix and ends up with a trivially-looking split W representation familiar from character calculus applications to matrix models and Hurwitz theory. Substitution of MacDonald polynomials for characters in these formulas provides a very simple description of "superpolynomials", much simpler than the recently studied alternative which deforms relation to the WZNW theory and explicitly involves the Littlewood-Richardson coefficients. A lot of explicit expressions are presented for different representations (Young diagrams), many of them new. In particular, we provide the superpolynomial P_[1]^[m,km\pm 1] for arbitrary m and k. The procedure is not restricted to the fundamental (all antisymmetric) representations and the torus knots, still in these cases some subtleties persist.Comment: 23 pages + Tables (51 pages

New method of verifying cryptographic protocols based on the process model

A cryptographic protocol (CP) is a distributed algorithm designed to provide a secure communication in an insecure environment. CPs are used, for example, in electronic payments, electronic voting procedures, database access systems, etc. Errors in the CPs can lead to great financial and social damage, therefore it is necessary to use mathematical methods to justify the correctness and safety of the CPs. In this paper, a new mathematical model of a CP is introduced, which allows one to describe both the CPs and their properties. It is shown how, on the base of this model, it is possible to solve the problems of verification of CPs