Optical properties of a disordered metallic film: local vs. collective phenomena

We apply the dual-varibles approach to the problem of the optical response of an disordered film of metal particles with dipole-dipole interaction. Long range dipole-dipole interaction makes the effect of spatial correlations significant, so that dual-variables technique provides a desirable improvement of the coherent-potential results. It is shown that the effect of nonlocality is more pronounced for a medium-range concentration of the particles. The result is compared with the non-local cluster approach. It is shown that short-range correlations accounted in the cluster method reveal themselves in the spectral properties of the response, whereas long-range phenomena kept in the dual technique are more pronounced in the k-dependence of the Green's function.Comment: 8 pages, 3 figure

On computational complexity of Set Automata

We consider a computational model which is known as set automata. The set automata are one-way finite automata with an additional storage---the set. There are two kinds of set automata---the deterministic and the nondeterministic ones. We denote them as DSA and NSA respectively. The model was introduced by M. Kutrib, A. Malcher, M. Wendlandt in 2014. It was shown that DSA-languages look similar to DCFL due to their closure properties and NSA-languages look similar to CFL due to their undecidability properties. In this paper we show that this similarity is natural: we prove that languages recognizable by NSA form a rational cone, so as CFL. The main topic of this paper is computational complexity: we prove that - languages recognizable by DSA belong to P and there are P-complete languages among them; - languages recognizable by NSA are in NP and there are NP-complete languages among them; - the word membership problem is P-complete for DSA without epsilon-loops and PSPACE-complete for general DSA; - the emptiness problem is in PSPACE for NSA and, moreover, it is PSPACE-complete for DSA.Comment: 31 pages, an extended version of the conference paper (DLT 2017), includes new results and omitted proof

Hilbert Schemes, Separated Variables, and D-Branes

We explain Sklyanin's separation of variables in geometrical terms and construct it for Hitchin and Mukai integrable systems. We construct Hilbert schemes of points on $T^{*}\Sigma$ for \Sigma = {\IC}, {\IC}^{*} or elliptic curve, and on ${\bf C}^{2}/{\Gamma}$ and show that their complex deformations are integrable systems of Calogero-Sutherland-Moser type. We present the hyperk\"ahler quotient constructions for Hilbert schemes of points on cotangent bundles to the higher genus curves, utilizing the results of Hurtubise, Kronheimer and Nakajima. Finally we discuss the connections to physics of $D$-branes and string duality.Comment: harvmac, 27 pp. big mode; v2. typos and references correcte

Duality in Integrable Systems and Gauge Theories

We discuss various dualities, relating integrable systems and show that these dualities are explained in the framework of Hamiltonian and Poisson reductions. The dualities we study shed some light on the known integrable systems as well as allow to construct new ones, double elliptic among them. We also discuss applications to the (supersymmetric) gauge theories in various dimensions.Comment: harvmac 45 pp.; v4. minor corrections, to appear in JHE