In this paper, for a specific kind of one-dimensional formal groups over the ring of integers of a local field in the case of small ramification we study the arithmetic of the formal module constructed on the maximal ideal of a local field, containing all the roots of the isogeny. This kind of formal groups is a little broader than Honda groups. The Shafarevich system of generators is constructed.

We solve some computational problems for triangulated closed three-dimensional manifolds using groups of simplicial homology and cohomology modulo 2. Two efficient algorithms for computing intersection numbers of 1- and 2-dimensional cycles are developed. Using these algorithms it is possible to construct a basis of the cohomology group from a given basis of the homology group of complementary dimension.

Many-core processor architecture is a promising paradigm for the development of modern supercomputers. In this paper, we consider the parallel implementation of the generic molecular dynamics algorithm for the many-core Epiphany architecture. This architecture implements a new type of many-core processor composed of 16 simple cores connected by a network on chip with mesh topology. New approaches to parallel programming are required to deploy this processor. We use LAMMPS running on one 64-bit ARMv8 Cortex-A53 CPU core for comparing the accuracy of the results of the presented variant of the molecular dynamics algorithm for Epiphany and its computational efficiency.

We study the signal extraction problemwhere a smooth signal is to be estimated against a long-range dependent noise. We consider an approach employing local estimates and derive a theoretically optimal (maximum likelihood) filter for a polynomial signal. On its basis, we propose a practical signal extraction algorithm and adapt it to the extraction of quasi-seasonal signals. We further study the performance of the proposed signal extraction scheme in comparison with conventional methods using the numerical analysis and real-world datasets.

Polynomial completeness of an operation guarantees that deciding solvability of equations over this operation is an NP-complete problem. Thus this property is beneficial from the viewpoint of cryptographic applications. We propose an algorithm for verification of polynomial completeness of quasigroups and analyse efficiency of its serial and parallel implementations.

We propose a construction that allows generating large families of Latin squares, i.e., Cayley tables of finite quasigroups. This construction generalizes proper families of functions over Abelian groups introduced by Nosov and Pankratiev. We also show that all quasigroups generated by the original construction contain at least one subquasigroup, while the generalized construction generates quasigroups free of subquasigroups.

We consider Bernoulli distribution algebras, i.e. sets of distributions that are closed under transformations achieved by substituting independent random variables for arguments of Boolean functions from a given system. We establish that, unless the transforming set contains only essentially unary functions, the set of algebra limit points is either empty, single-element or no less than countable.

The paper is devoted to modeling multi depot vehicle routing problem (VRP) with capacity constraints for petroleum products delivery. Applying efficient metaheuristics algorithms combined with local search procedures, we present how to get suboptimal solutions for this NP-hard problem in an acceptable time. Some parallel computing techniques are also used to reduce the execution time. Experimental results are performed by the case of VRP for petroleum products.

In the present paper we continue our study of non-commutative operator graphs in infinite-dimensional spaces. We consider examples of the non-commutative operator graphs generated by resolutions of identity corresponding to the Heisenberg–Weyl group of operators acting on the Fock space over one-particle state space. The problem of quantum error correction for such graphs is discussed.

The vertex 3-colourability problem is to verify whether it is possible to split the vertex set of a given graph into three subsets of pairwise nonadjacent vertices or not. This problem is known to be NP-complete for planar graphs of the maximum face length at most 4 (and, even, additionally, of the maximum vertex degree at most 5), and it can be solved in linear time for planar triangulations. Additionally, the vertex 3-colourability problem is NP-complete for planar graphs of the maximum vertex degree at most 4, but it can be solved in constant time for graphs of the maximum vertex degree at most 3. It would be interesting to investigate classes of planar graphs with simultaneously bounded length of faces and the maximum vertex degree and to find the threshold, for which the complexity of the vertex 3-colourability problem is changed from polynomial-time solvability to NP-completeness. In this paper, we prove NP-completeness of the vertex 3-colourability problem for planar graphs of the maximum vertex degree at most 4, whose faces are of length no more than 5.

Given a quantum channel it is possible to define the non-commutative operator graph whose properties determine a possibility of error-free transmission of information via this channel. The corresponding graph has a straight definition through Kraus operators determining quantum errors. We are discussing the opposite problem of a proper definition of errors that some graph corresponds to. Taking into account that any graph is generated by some POVM we give a solution to such a problem by means of the Naimark dilatation theorem. Using our approach we construct errors corresponding to the graphs generated by unitary dynamics of bipartite quantum systems. The cases of POVMs on the circle group \mathbb{Z}_n and the additive group \mathbb{R} are discussed. As an example we construct the graph corresponding to the errors generated by dynamics of two mode quantum oscillator.

We consider a class of gradient-like flows on three-dimensional closedmanifolds whose attractors and repellers belongs to a finite union of embedded surfaces and find conditions when the ambient manifold is Seifert.

We continue the study of non-commutative operator graphs generated by resolutions of identity covariant with respect to unitary actions of the circle group and the Heisenber-Weyl group as well. It is shown that the graphs generated by the circle group has the system of unitary generators fulfilling permutations of basis vectors. For the graph generated by the Heisenberg-Weyl group the explicit formula for a dimension is given. Thus, we found a new description of the linear structure for the operator graphs introduced in our previous works.

We consider transformations of random variables on finite sets by algebraic operations. A system of operations is said to be approximation complete if any random variable may be approximated with arbitrary precision by applying the given operations to mutually independent identically distributed random variables whose distributions have no zero components. We establish some necessary conditions for a function system to be approximation complete and construct examples of approximation incomplete systems.

In this article we apply methods of representation theory and combinatorial algebra to the different problems related to quantum tomography. For this purpose, we introduce the algebra generated by projectors satisfying some commutator relation. In this paper we study this commutator relation by combinatorialmethods and develop the representation theory of this algebra. Also, we apply our results to the case of mutually unbiased bases in dimension 7.

Solutions of functional differential equation of pointwise type (FDEPT) are in one-to-one correspondence with the traveling-wave type solutions for the canonically induced infinite-dimensional ordinary differential equation and vice versa. In particular, such infinite-dimensional ordinary differential equations are finite difference analogues of equations of mathematical physics. An important class of traveling-wave type solutions is made up of periodic and bounded traveling-wave type solutions. On the other hand, an important class of such systems is systems with strongly nonlinear potentials (polynomial potentials), for which periodic and bounded traveling wave solutions are studied. Such a problem is equivalent to the study of periodic and bounded solutions of the induced FDEPT to which the present work is devoted.

We consider a mathematical model of a spherical inverted pendulum on a movable cart. The cart moves on a horizontal plane under the influence of a planar bounded force. We study an optimal control problem related to this model. The control objective is to stabilize the inverted pendulum in the upright equilibrium position. For the linearized model it is shown that the optimal trajectories contains arcs with more and more frequent control switchings.

An algorithm of MPI processes mapping optimization is adapted for supercomputers with interconnect Angara. The mapping algorithm is based on partitioning of parallel program communication pattern. It is performed in such a way that the processes between which the most intensive exchanges take place are tied to the nodes/processors with the highest bandwidth. The algorithm finds a near-optimal distribution of its processes for processor cores to minimize the total execution time of exchanges between MPI processes. The analysis of results of optimized placement of processes using proposed method on small supercomputers is shown. The analysis of the dependence of the MPI program execution time on supercomputer parameters and task parameters is performed. A theoretical model is proposed for estimation of effect of mapping optimization on the execution time for several types of supercomputer topologies. The prospect of using implemented optimization library for large-scale supercomputers with the interconnect Angara is discussed.

We introduce a class of stochastic networks in which synchronization between nodes is modelled by a message passing mechanism with heterogeneous Markovian routing. We present a series of results about probability distribution related to steady states of such models.

The density functional theory (DFT) is a research tool of the highest importance for electronic structure calculations. It is often the only affordable method for ab initio calculations of complex materials. The pseudopotential approach allows reducing the total number of electrons in the model that speeds up calculations. However, there is a lack of pseudopotentials for heavy elements suitable for condensed matter DFT models. In this work, we present a pseudopotential for uranium developed in the Goedecker–Teter–Hutter form. Its accuracy is illustrated using several molecular and solid-state calculations.

We prove that a foliation (M;F) of codimension q on a ndimensional pseudo-Riemannian manifold is pseudo-Riemannian if and only if any geodesic that is orthogonal at one point to a leaf is orthogonal to every leaf it intersects.

We show that on the graph G = G(F) of a pseudo-Riemannian foliation there exists a unique pseudo-Riemannian metric such that canonical projections are pseudo-Riemannian submersions and the fibers of different projections are orthogonal at common points. Relatively this metric the induced foliation (G;F) on the graph is pseudo-Riemannian and the structure of the leaves of (G;F) is described. Special attention is given to the structure of graphs of transversally (geodesically) complete pseudo-Riemannian foliations which are totally geodesic pseudo-Riemannian ones.