Optimal control of the system of coupled cylinders

We consider the problem of optimal control of a system consisting of two coupled cylinders. Such a system is a mathe-matical model of the nuclear fuel transfer mechanism at the nuclear power plant reactor. And also, such models are found in various robotic systems. We have obtained optimal control under certain assumptions on a controllable system. © 2018 Author(s).The research was supported by Russian Foundation for Basic Research, project no. 17-08-01385

Optimal and heuristic algorithms of planning of low-rise residential buildings

The problem of the optimal layout of low-rise residential building is considered. Each apartment must be no less than the corresponding apartment from the proposed list. Also all requests must be made and excess of the total square over of the total square of apartment from the list must be minimized. The difference in the squares formed due to with the discreteness of distances between bearing walls and a number of other technological limitations. It shown, that this problem is NP-hard. The authors built a linear-integer model and conducted her qualitative analysis. As well, authors developed a heuristic algorithm for the solution tasks of a high dimension. The computational experiment was conducted which confirming the efficiency of the proposed approach. Practical recommendations on the use the proposed algorithms are given. © 2017 Author(s).Russian Foundation for Basic Research, RFBR: 16-01-00649The work was supported by Act 211 Government of the Russian Federation, contract No 02.A03.21.0006, and by the Russian Foundation for Basic Research, project No. 16-01-00649

On the BPS Spectrum at the Root of the Higgs Branch

We study the BPS spectrum and walls of marginal stability of the $\mathcal{N}=2$ supersymmetric theory in four dimensions with gauge group SU(n) and $n\le N_{f}<2n$ fundamental flavours at the root of the Higgs branch. The strong-coupling spectrum of this theory was conjectured in hep-th/9902134 to coincide with that of the two-dimensional supersymmetric $\mathbb{CP}^{2n-N_{f}-1}$ sigma model. Using the Kontsevich--Soibelman wall-crossing formula, we start with the conjectured strong-coupling spectrum and extrapolate it to all other regions of the moduli space. In the weak-coupling regime, our results precisely agree with the semiclassical analysis of hep-th/9902134: in addition to the usual dyons, quarks, and $W$ bosons, if the complex masses obey a particular inequality, the resulting weak-coupling spectrum includes a tower of bound states consisting of a dyon and one or more quarks. In the special case of $\mathbb{Z}_{n}$-symmetric masses, there are bound states with one quark for odd $n$ and no bound states for even $n$.Comment: 11 pages, 4 figure

Routing problems: constraints and optimality

We consider the issues of routing under constraints and formulate a mathematical problem of visiting megalopolises. The order of visits is subject to precedence constraints. In addition, the cost functions depend on the set of pending tasks. The quality criterion is a variety of the additive criterion. The problem is established within the dynamic programming framework, however, a heuristic is proposed and implemented to solve practical problems of large dimensionality. © 201

Moduli Space and Wall-Crossing Formulae in Higher-Rank Gauge Theories

We study the interplay between wall-crossing in four-dimensional gauge theory and instanton contributions to the moduli space metric of the same theory on $\mathbb{R}^{3}\times S^{1}$. We consider $\mathcal{N}=2$ SUSY Yang--Mills with gauge group SU(n) and focus on walls of marginal stability which extend to weak coupling. By comparison with explicit field theory results we verify the Kontsevich--Soibelman formula for the change in the BPS spectrum at these walls and check the smoothness of the metric in the corresponding compactified theory. We also verify in detail the predictions for the one instanton contribution to the metric coming from the non-linear integral equations of Gaiotto, Moore and Nietzke.Comment: 26 pages, no figure