On a mapping of a projective space into a sphere

We obtain an exact estimate for the minimum multiplicity of a continuous finite-to-one mapping of a projective space into a sphere for all dimensions. For finite-to-one mappings of a projective space into a Euclidean space, we obtain an exact estimate for this multiplicity for n = 2, 3. For n ≥ 4, we prove that this estimate does not exceed 4. Several open questions are formulated.     

Refinement of a criterion for superfluidity of a classical liquid in a nanotube

The problem has a large parameter, the number N of particles (outside the condensate), and a small parameter, Planck’s constant ħ (more precisely, a dimensionless parameter containing ħ in the numerator). They turn out to be interrelated for Bose particles. Using the exact solution of the variational equation for a bosonic system in a capillary, we can obtain precise criteria restricting the superfluidity of a classical liquid in the capillary. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 3, pp. 524–528, June, 2008.     

Numerical solutions of a model for the propagation of a surface-catalysed flame in a tube

Numerical simulations of a surface-catalysed flame in a tubeare performed, corresponding to an experiment where a premixedfuel is fed into a tube whose inner surface is coated with acatalyst. In these experiments, subsequent to ignition, a reactionwave can be seen as a red-hot region which propagates back alongthe tube towards the inlet, and is due to low temperature combustionoccurring only on the inner surface of the tube where the catalystis present. The solutions of a mathematical model for this behaviourshow that initial-value problems do indeed result in such steadilypropagating waves. The numerically obtained wave speeds andsteady solution are compared to a previous large Damköhlernumber (D_a) asymptotic analysis using a simple reaction ratemodel, and agreement is very good even for moderately largevalues of D_a. However, for such Damköhler numbers, thewave speeds are found to be much larger than observed experimentally.Indeed, the simulations show that O(1) values of D_a are requiredto obtain the lower experimental wave speeds. Nevertheless,the wave speeds as a function of flow rate through the tubedo not agree well with the preliminary experimental resultsfor any choice of the parameters. A more realistic, Arrheniusreaction rate model is then considered. The Arrhenius modelpredicts a rapid change in temperature at the wave front, inmuch better agreement with the experiments than for the simplerreaction model.     

On stable estimation of a function of a parameter

Given the function f and the vector-statistic t_N which is a mean square consistent estimator of a parameter a, the problem is to estimate f(a). The criteria for the mean square consistency of the estimator f(t_N) are considered. In the case where the estimator f(t_N) is not mean square consistent, a class of estimators of f(a) is proposed, and it is proved that the estimators of the class are mean square consistent for all distribution of t_N. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 44–55, Perm, 1990.     

Lusternik-Schnirelmann category of a sphere-bundle over a sphere

We determine the Lusternik-Schnirelmann (L-S) category of a total space of a sphere-bundle over a sphere in terms of primary homotopy invariants of its characteristic map, and thus providing a complete answer to Ganea's Problem 4. As a result, we obtain a necessary and sufficient condition for a total space N to have the same L-S category as its ‘once punctured submanifold’ N\{P},P∈N. Also, necessary and sufficient conditions for a total space M to satisfy Ganea's conjecture are described.     

Complementing a finite subgroup of a hyperbolic group by a free factor

We give necessary and sufficient conditions for a finite subgroup H of a hyperbolic group G to contain a free subgroup F of rank two in G such that F and H generate a free product F ∗ H. A verification algorithm for these conditions is pointed out.     

On Blowup of a Solution to a Sobolev-Type Equation with a Nonlocal Source

We consider the initial-boundary value problem for a dissipative nonlinear wave equation of the Sobolev type with a nonlocal cubic source. For this problem we obtain a sufficient condition for blowup of a strong generalized solution. Moreover, we obtain a two-sided estimate for the blowup time of the solution.Original Russian Text Copyright © 2005 Korpusov M. O. and Sveshnikov A. G.The authors were supported by the Russian Foundation for Basic Research (Grants 02-01-00253; 02-01-06038) and a grant of the Project Junior Scientists of Russia.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 567–578, May–June, 2005.     

Interaction of a hollow cylinder of finite length and a plate with a cylindrical cavity with a rigid insert

Two problems of the interaction of a hollow circular cylinder with load-free ends and an unbounded plate with a cylindrical cavity and a symmetrically imbedded rigid insert are considered. Homogeneous solutions are found and the generalized orthogonality of these solutions is used when the modified boundary conditions are satisfied. As a result, we have a system of two integral equations in functions of the displacements of the outer and inner surfaces of the hollow cylinder. These functions are sought in the form of sums of a trigonometric series and a power function with a root singularity. The ill-posed infinite systems of linear algebraic equations obtained are regularized by the introduction of small positive parameters. Since the elements of the matrices of the systems as well as the contact stresses are defined by poorly converging numerical and functional series, an efficient method for calculating of the remainders of the above-mentioned series is developed. Formulae are found for the contact pressure distribution function and the integral characteristic. Examples of the calculation of the interaction of the cylinder and the plate with an insert are given.The method of solving contact problems described here has been used earlier1, 2 and the generalized orthogonality of the solutions found for bodies of finite dimensions, that is, for a rectangle and cylinders of finite length, is its basis. Problems for hollow cylinders with a band ² and an insert reduce to a system of two integral equations, and the problem for a rectangle¹ reduces to one integral equation. Solving these integral equations, ill-posed systems of linear algebraic equations are obtained which are subject to regularization³.     

Optimal cooling of a section of a solid body

The article considers the cooling of a section of a solid body subject to constant internal energy. The numerical method proposed for the solution of the problem is a generalization of Plotnikov's method for the solution of optimal control problems of parabolic systems.Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 3, pp. 58–64, 1993.     

On a class of isometric subgraphs of a graph

In a graphG, which has a loop at every vertex, a connected subgraphH=(V(H),E(H)) is a retract if, for anya, b ∈V(H) and for any pathsP, Q inG, both joininga tob, and satisfying |Q|≧ ≧|P|, thenP ⫅V(H) wheneverQ ⫅V(H). As such subgraphs can be described by a closure operator we are led to the investigation of the corresponding complete lattice of “closed” subgraphs. For example, in this complete lattice every element is the infimum of an irredundant family of infimum irreducible elements. The work presented here was supported in part by N.S.E.R.C. Operating Grant No. A4077.     

Finding a subdivision of a prescribed digraph of order 4

The problem of when a given digraph contains a subdivision of a fixed digraph F is considered. Bang‐Jensen et al. [4] laid out foundations for approaching this problem from the algorithmic point of view. In this article, we give further support to several open conjectures and speculations about algorithmic complexity of finding F‐subdivisions. In particular, up to five exceptions, we completely classify for which 4‐vertex digraphs F, the F‐subdivision problem is polynomial‐time solvable and for which it is NP‐complete. While all NP‐hardness proofs are made by reduction from some version of the 2‐linkage problem in digraphs, some of the polynomial‐time solvable cases involve relatively complicated algorithms.     

Investigation of a nonlinear difference equation in a Banach space in a neighborhood of a quasiperiodic solution

We investigate the behavior of a diserete dynamical system in a neighborhood of a quasiperiodic trajeetory for the case of an infinite-dimensional Banach space We find conditions sufficient for the system considered to reduce, in such a neighborhood, to a system with quasiperiodic coefficients. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1661–1676, December, 1997.     

Realization of a sum of sequences by a sum graph

It is shown that the realizability of the sequences ϕ=(a ₁,…, _a ), ψ=(b ₁,…,b _n ) and ϕ+ψ is a sufficient condition for the realizability of ϕ+ψ by a graph with a ϕ-factor ifb _i ≦1 fori=1,…,n. The condition is not sufficient in general. A necessary and sufficient condition for the realizability of ϕ+ψ by a graph with a ϕ-factor is given for the case that ϕ is realizable by a star and isolated vertices.     

Motion of a distant solid particle in a shear flow along a porous slab

The motion of a solid and no-slipping particle immersed in a shear flow along a sufficiently porous slab is investigated. The fluid flow outside and inside of the slab is governed by the Stokes and Darcy equations, respectively, and the so-called Beavers and Joseph slip boundary conditions are enforced on the slab surface. The problem is solved for a distant particle with length scale a in terms of the small parameter a/d where d designates the large particle–slab separation. This is achieved by asymptotically inverting a relevant boundary-integral equation on the particle surface, which has been recently proposed for any particle location (distant or close particle) in Khabthani et al. (J Fluid Mech 713:271–306, 2012). It is found that at order O(a/d) the slab behaves for any particle shape as a solid plane no-slip wall while the slab properties (thickness, permeability, associated slip length) solely enter at O((a/d)²). Moreover, for a spherical particle, the numerical results published in Khabthani et al. (J Fluid Mech 713:271–306, 2012) perfectly agree with the present asymptotic analysis.     

Analysis of a tandem network model of a single-router Network-on-Chip

We study a single-router Network-on-Chip modelled as a tandem queueing network. The first node is a geo ^K /D/1 queue (K fixed) representing a network interface, and the second node is a ./G/1 queue representing the packet switch. If K>1 we have train arrivals at the second node. If K=1 the arrival process of the second node reduces to a Bernoulli process. In the latter case, routers have been studied extensively as part of ATM and LAN networks under the assumption that the number of input ports N tends to infinity. In Networks-on-Chips N is usually 4 or 5 and results for ATM and LAN routers lead to inaccurate results. We introduce a new approximation scheme that yields accurate results for small switches. In addition to this we analyse the tandem network, both for K=1 and K>1, and we approximate the mean sojourn time in the switch and the mean end-to-end delay. If N=4 our approximation has a relative error of only 4.5% if K=6 and 1% if K=1.     

Study of a discrete dynamic system in a neighborhood of a quasi-periodic trajectory

We study a discrete system in a neighborhood of a quasi-periodic trajectory. We obtain conditions for reducing a system in this neighborhood to a system with quasi-periodic coefficients. We determine the behavior of this system under the action of small perturbations.This work was prepared with the financial support of the Ukrainian State Committee on Science and Technology.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 12, pp. 1702–1711, December, 1992.     

On the Genus of a Subfield of a Field

In 1977 Babai showed that if Γ₁ is a subgroup of a finite group Γ₂, then the genus of Γ₁ is at most the genus of Γ₂. We establish a similar result for finite fields.     

Optimal construction of a selection of a subpopulation

Summary The problem of selecting a subpopulation from a given populationII is to be, on the basis of measurements of members ofII, achieved by choosing those members ofII who satisfy the standards determined by a given selection cirterion and rejecting those who do not. Since the optimum selection depends on the unknown parameter of the probability distribution ofII, it is here considered how to construct a decision function from the space of subsidiary sample having infor-mation on θ to the space of selections. Thus the existence of Bayes and minimax decision functions under the constraint defined by the selection criterion is proved. A necessary and sufficient condition for a decision function satisfying the constraint to be a Bayes decision function is also obtained. The Institute of Statistical Mathematics