Proceedings of the Steklov Institute of Mathematics最新文献

The paper presents an overview of methods for solving an ill-posed problem of constrained convex quadratic minimization based on the Fejér-type iterative methods, which widely use the ideas and approaches developed in the works of I. I. Eremin, the founder of the Ural research school of mathematical programming. Along with a problem statement of general form, we consider variants of the original problem with constraints in the form of systems of equalities and inequalities, which have numerous applications. In addition, particular formulations of the problem are investigated, including the problem of finding a metric projection and solving a linear program, which are of independent interest. A distinctive feature of these methods is that not only convergence but also stability with respect to errors in the input data are established for them; i.e., the methods generate regularizing algorithms in contrast to the direct methods, which do not have this property.

In this paper, we investigate a problem of optimal control over a finite time interval for a linear systemwith constant coefficients and a small parameter in the initial data in the class of piecewise continuous controlswith smooth geometric constraints. We consider a terminal convex performance index. We substantiate the limit relationsas the small parameter tends to zero for the optimal value of the performance index and for the vector generatingthe optimal control in the problem. We show that the asymptotics of the solution can be of complicated nature. Inparticular, it may have no expansion in the Poincaré sense in any asymptotic sequence of rational functions of thesmall parameter or its logarithms.

We develop a mathematical technique of Young dual variational inequalities, which are used to model market equilibrium in a network of production clusters that are heterogeneous from a technological point of view. Two formulations of the problem are considered: for a closed system with a given constraint on resources and for an open system in which resources can be supplied from outside at given prices. A theorem is proved on the existence of a solution to the variational inequality corresponding to market equilibrium in an open system.

The paper is concerned with sharp Carlson type inequalities of the form(|w(cdot)x(cdot)|_{L_{q}(T)}leq K|w_{0}(cdot)x(cdot)|_{L_{p}(T)}^{ gamma}max_{1leq jleq n}|w_{j}(cdot)x(cdot)|_{L_{r}(T)}^{1-gamma},)where (T) is a cone in (mathbb{R}^{d}) and the weight functions (w_{j}(cdot)), (j=1,mathinner{ldotpldotpldotp},n), are homogeneous with some symmetry property.

Earlier, the author described up to conjugacy all pairs ((A,B)) of nilpotent subgroups of a finite group (G) with socle (L_{2}(q)) for which (Acap B^{g}neq 1) for any element of (G). A similar description was obtained by the author later for primary subgroups (A) and (B) of a finite group (G) with socle (L_{n}(2^{m})). In this paper, we describe up to conjugacy all pairs ((A,B)) of nilpotent subgroups of a finite group (G) with simple socle from the “Atlas of Finite Groups” for which (Acap B^{g}neq 1) for any element (g) of (G). The results obtained in the considered cases confirm the hypothesis (Problem 15.40 from the “Kourovka Notebook”) that a finite simple nonabelian group (G) for any nilpotent subgroups (N) contains an element (g) such that (Ncap N^{g}=1).

We consider a decomposition approach to the problem of finding the orthogonal projection of a given point onto a convex polyhedral cone represented by a finite set of its generators. The reducibility of an arbitrary linear optimization problem to such projection problem potentially makes this approach one of the possible new ways to solve large-scale linear programming problems. Such an approach can be based on the idea of a recurrent dichotomy that splits the original large-scale problem into a binary tree of conical projections corresponding to a successive decomposition of the initial cone into the sum of lesser subcones. The key operation of this approach consists in solving the problem of projecting a certain point onto a cone represented as the sum of two subcones with the smallest possible modification of these subcones and their arbitrary selection. Three iterative algorithms implementing this basic operation are proposed, their convergence is proved, and numerical experiments demonstrating both the computational efficiency of the algorithms and certain challenges in their application are performed.

We consider an inclusion (widetilde{y}in F(x)) with a multivalued mapping acting in spaces with vector-valued metricswhose values are elements of cones in Banach spaces and can be infinite. A statement about the existence of a solution (xin X)and an estimate of its deviation from a given element (x_{0}in X) in a vector-valued metric are obtained. This result extendsthe known theorems on similar operator equations and inclusions in metric spaces and in the spaces with (n)-dimensional metricto a more general case and, applied to particular classes of functional equations and inclusions, allows to get less restrictive,compared to known, solvability conditions as well as more precise estimates of solutions. We apply this result to the integral inclusion(widetilde{y}(t)in f(t,intop_{a}^{b}varkappa(t,s)x(s),ds,x(t)), tin[a, b],)where the function (widetilde{y}) is measurable, the mapping (f) satisfies the Carathéodory conditions, and the solution (x) isrequired to be only measurable (the integrability of (x) is not assumed).

Let (mathfrak{F}) be a formation, and let (G) be a finite group. A subgroup (H) of (G) is called (mathrm{K}mathfrak{F})-subnormal (submodular) in (G) if there is a subgroup chain (H=H_{0}leq H_{1}leqmathinner{ldotpldotpldotp}leq H_{n-1}leq H_{n}=G) such that, for every (i) either (H_{i}) is normal in (H_{i+1}) or (H_{i+1}^{mathfrak{F}}leq H_{i}) ((H_{i}) is a modular subgroup of (H_{i+1}), respectively). We prove that, in a group, a primary subgroup is submodular if and only if it is (mathrm{K}mathfrak{U}_{1})-subnormal. Here (mathfrak{U}_{1}) is a formation of all supersolvable groups of square-free exponent. Moreover, for a solvable subgroup-closed formation (mathfrak{F}), every solvable (mathrm{K}mathfrak{F})-subnormal subgroup of a group (G) is contained in the solvable radical of (G). We also obtain a series of applications of these results to the investigation of groups factorized by (mathrm{K}mathfrak{F})-subnormal and submodular subgroups.

Abstract

A mathematical model is proposed for the replacement of methane with carbon dioxide in a hydrate contained in a reservoir in thermodynamic equilibrium with water and free methane. The substitution reaction region is assumed to be narrow enough to be approximated by the conversion front. A self-similar solution is found that reduces the problem to the numerical analysis of a system of transcendental equations. Numerical experiments show that there exist three characteristic regimes of injection, whose implementation depends on the amount of water in the free state in the reservoir.

Abstract

Laminar flows of fluids with a yield strength in pipes under the action of a periodically changing pressure drop are considered. The Herschel–Bulkley model is adopted to describe the rheological properties of moving fluids. The effect of pressure drop fluctuations on velocity profiles, as well as on the mean flow rates, friction on the pipe walls, and the thickness of the “quasi-solid” core, is studied numerically, depending on the amplitude and frequency of pressure drop fluctuations, the generalized Bingham number, and the fluid power index. It is shown that in flows of viscoplastic fluids with a power index greater than one, the effect of pressure drop fluctuations is qualitatively different in different ranges of shear rates. Additionally, flows are investigated in which the relative amplitudes of pressure drop oscillations are large. At large amplitudes and low frequencies of pressure drop oscillations, counter-flows periodically arise in the flow and two (rather than one) zones of “quasi-solid” flow are formed; moreover, finite time intervals periodically appear in which the flow rate is zero.