On Factorization of Generalized Macdonald Polynomials

A remarkable feature of Schur functions -- the common eigenfunctions of cut-and-join operators from $W_\infty$ -- is that they factorize at the peculiar two-parametric topological locus in the space of time-variables, what is known as the hook formula for quantum dimensions of representations of $U_q(SL_N)$ and plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to {\it generalized} Macdonald polynomials (GMP), associated in the same way with the toroidal Ding-Iohara-Miki algebras, which play the central role in modern studies in Seiberg-Witten-Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time-variables, we discover a weak factorization -- on a codimension-one slice of the topological locus, what is already a very non-trivial property, calling for proof and better understanding.Comment: 8 page

The 2-leg vertex in K-theoretic DT theory

K-theoretic Donaldson-Thomas counts of curves in toric and many related threefolds can be computed in terms of a certain canonical 3-valent tensor, the K-theoretic equivariant vertex. In this paper we derive a formula for the vertex in the case when two out of three entries are nontrivial. We also discuss some applications of this result.Comment: 27 page

Modeling of the Influence of Energy Development on Different Branches of the National Economy

Within the Energy program, this paper contributes to the assessment, comparison. and evaluation of energy strategies, in particular the impact of the energy supply system on other branches of the economy. This impact is characterized by the demand for industrial products necessary for building and operating the energy supply system, the required putting into operation of industrial production capacities, capital investment in the energy supply system and related branches, and the direct and indirect expenses of limited domestic natural resources. The dynamic model for treating all these characteristics was constructed at the Siberian Power Institute and modified at IIASA