Identification of a source factor in a control problem for the heat equation with a boundary memory term

We consider a material with thermal memory occupying a bounded region Ω with boundary Γ. The evolution of the temperature u(t,x) is described by an integrodifferential parabolic equation containing a heat source of the form f(t)z₀(x). We formulate an initial and boundary value control problem based on a feedback device located on Γ and prescribed by means of a quite general memory operator. Assuming both u and the source factor f are unknown, we study the corresponding inverse and control problem on account of an additional information. We prove a result of existence and uniqueness of the solution (u,f). Copyright © 2015 John Wiley & Sons, Ltd.     

Approximation of a function of two variables by a product of functions of one variable on a given domain

We are concerned with the problem of uniform approximation of a continuous function of two variables by a product of continuous functions of one variable on some domain D. This problem have been examined so far only on a rectangular domain D = U × V, where U and V are compact sets. An algorithm to give a solution of this problem in the discrete case is available. We put forward an algorithm which in certain cases allows one to construct an approximate solution of the problem on a given domain (not necessarily rectangular). This approximate solution is built in the form of interpolating natural splines, which in turn are constructed by means of discrete approximation. Depending on the degree of the splines, the problem can be solved in classes of functions with appropriate degree of smoothness.     

Bound states of a system of two fermions on a one-dimensional lattice

We consider the Hamiltonian of a system of two fermions on a one-dimensional integer lattice. We prove that the number of bound states N(k) is a nondecreasing function of the total quasimomentum of the system k ∈ [0, π]. We describe the set of discontinuity points of N(k) and evaluate the jump N(k +0) − N(k) at the discontinuity points. We establish that the bound-state energy z _n (k) increases as the total quasimomentum k ∈ [0, π] increases. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 47–57, April, 2006.     

Stability of a nonlocal difference problem with a complex parameter

We consider a difference scheme for the heat equation with nonlocal boundary conditions containing a complex parameter γ. We single out stability domains in the complex γ-plane and derive a criterion for stability with respect to the initial data in the form of a condition on γ and the grid increments.     

Transformation semigroups preserving a binary relation

We investigate the semigroups of full and partial transformations of a set X which preserve a binary relation σ defined on X. We consider in detail the case where σ is an order or a quasi-order relation. There are conditions of regularity of such semigroups. We introduce two definitions of preservation of σ for the semigroup of binary relations. It is proved that subsets of B(X) preserving σ are semigroups in each case. We give the condition of regularity of B _σ (X) in the case where σ(X) is a quasi-order.     

Permutation Groups, a Related Algebra and a Conjecture of Cameron

We consider the permutation group algebra defined by Cameron and show that if the permutation group has no finite orbits, then no homogeneous element of degree one is a zero-divisor of the algebra. We proceed to make a conjecture which would show that the algebra is an integral domain if, in addition, the group is oligomorphic. We go on to show that this conjecture is true in certain special cases, including those of the form H Wr S and H Wr A, and show that in the oligormorphic case, the algebras corresponding to these special groups are polynomial algebras. In the H Wr A case, the algebra is related to the shuffle algebra of free Lie algebra theory.     

Bounded orbits and measures on a group

We consider a definable group G acting on the space of complete types over G, in a monster model of a theory T. We discuss the notion of a bounded orbit of this action. We prove that some boundedness assumptions imply definable amenability of G.     

Optimally Cutting a Surface into a Disk

We consider the problem of cutting a subset of the edges of a polyhedral manifold surface, possibly with boundary, to obtain a single topological disk, minimizing either the total number of cut edges or their total length. We show that this problem is NP-hard in general, even for manifolds without boundary and for punctured spheres. We also describe an algorithm with running time n ^O(g+k), where n is the combinatorial complexity, g is the genus, and k is the number of boundary components of the input surface. Finally, we describe a greedy algorithm that outputs a O(log² g)-approximation of the minimum cut graph in O(g ² n log n) time.     

Covering a set with homothets of a convex body

We consider two problems mentioned in the book “Research Problems in Discrete Geometry” (Brass et al. in research problems in discrete geometry, vol xii+499. Springer, New York, pp ISBN: 978-0387-23815-8; 0-387-23815-8, 2005). First, let K and L be given convex bodies in \mathbbR^d{\mathbb{R}^{d}} . We prove that if the total volume of a family of positive homothets of K is sufficiently large then they permit a translative covering of L. This problem, in the case when K = L and the dimension is two, was originally posed by L. Fejes Tóth. The previously known bound (Januszewski in proc. of the International scientific conference on mathematics, pp 29–34. Žilina, 1998) on the total volume (in the case when K = L) was of order d ^d vol(K), we prove a bound that is exponential in the dimension. The second problem is the following: Find a condition, in terms of the coefficients of homothety, that is necessary for a family of positive homothets of K to cover K. The problem was phrased by V. Soltan, who conjectured that the sum of the coefficients is at least d. We confirm an asymptotic version of this conjecture.     

Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation

We employ a piecewise-constant, discontinuous Galerkin method for the time discretization of a sub-diffusion equation. Denoting the maximum time step by k, we prove an a priori error bound of order k under realistic assumptions on the regularity of the solution. We also show that a spatial discretization using continuous, piecewise-linear finite elements leads to an additional error term of order h ² max (1,logk ^− 1). Some simple numerical examples illustrate this convergence behaviour in practice. We thank the University of New South Wales for financial support provided by a Faculty Research Grant.     

Stability and localization of unbounded solutions of a nonlinear heat equation in a plane

The article investigates unbounded solutions of the equation u _t = div (u ^σgrad u) + u ^β in a plane. We numerically analyze the stability of two-dimensional self-similar solutions (structures) that increase with blowup. We confirm structural stability of the simple structure with a single maximum and metastability of complex structures. We prove structural stability of the radially symmetrical structure with a zero region at the center and investigate its attraction region. We study the effect of various perturbations of the initial function on the evolution of self-similar solutions. We further investigate how arbitrary compact-support initial distributions attain the self-similar mode, including distributions whose support is different from a disk. We show that the self-similar mode described by a simple radially symmetrical structure is achieved only in the central region, while the entire localization region does not have enough time to transform into a disk during blowup. We show for the first time that simple structures may merge into a complex structure, which evolves for a long time according to self-similar law.     

Cryptanalysis of a Matrix-based MOR System

We cryptanalyze a recently proposed matrix-based MOR cryptosystem. The security of the system depends on the difficulty of solving the following discrete logarithm problem: given an inner automorphism φ of SL(d, 𝔽 _q ) and φ ^a (each given in terms of their images on generators of SL(d, 𝔽 _q )), find a. We show that this problem can be reduced to a small number of similar problems in quotients of polynomial rings and solved in subexponential-time.     

On Powerful Normal Subgroups of a p-Group

 In this paper we investigate the action of a p-group G on its powerful normal subgroup N. We shall mainly be concerned to find conditions which guarantee that G acts uniserially on N. We shall also study what consequences uniserial action on N has on the structure of N. (Received 21 August 1998; in revised form 28 June 1999)     

Pasch trades with a negative block

A Steiner triple system of order v, STS(v), may be called equivalent to another STS(v) if one can be converted to the other by a sequence of three simple operations involving Pasch trades with a single negative block. It is conjectured that any two STS(v)s on the same base set are equivalent in this sense. We prove that the equivalence class containing a given system S on a base set V contains all the systems that can be obtained from S by any sequence of well over one hundred distinct trades, and that this equivalence class contains all isomorphic copies of S on V. We also show that there are trades which cannot be effected by means of Pasch trades with a single negative block.     

On the vibrations of a plate with a concentrated mass and very small thickness

We consider the vibrations of an elastic plate that contains a small region whose size depends on a small parameter ε. The density is of order O(ε^–m) in the small region, the concentrated mass, and it is of order O(1) outside; m is a positive parameter. The thickness plate h being fixed, we describe the asymptotic behaviour, as ε→O, of the eigenvalues and eigenfunctions of the corresponding spectral problem, depending on the value of m: Low‐ and high‐frequency vibrations are studied for m>2. We also consider the case where the thickness plate h depends on ε; then, different values of m are singled out. Copyright © 2003 John Wiley & Sons, Ltd.     

Groups Admitting a Kantor Family and a Factorized NormalSubgroup

We study the structure of a finite groupG admitting a Kantor family (F, F ^*) of type (s, t) and a nontrivial normal subgroupX which isfactorized byF F ^*. The most interesting cases, giving necessary conditions on the structure ofG and the parameterss andt, are those where a further Kantor family is induced inX, or where a partial congruence partition is induced in the factor groupG/X. Most of the known finite generalized quadrangles can be constructed as coset geometries with respect to a Kantor family. We show that the parameters of a skew translation generalized quadrangle necessarily are powers of the same prime. Furthermore, the structure of nonabelian groups admitting a Kantor family consisting only of abelian members is considered.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

Bruhat Order for Two Flags and a Line

The classical Ehresmann-Bruhat order describes the possible degenerations of a pair of flags in a linear space V under linear transformations of V; or equivalently, it describes the closure of an orbit of GL(V acting diagonally on the product of two flag varieties.We consider the degenerations of a triple consisting of two flags and a line, or equivalently the closure of an orbit of GL(V) acting diagonally on the product of two flag varieties and a projective space. We give a simple rank criterion to decide whether one triple can degenerate to another. We also classify the minimal degenerations, which involve not only reflections (i.e., transpositions) in the Weyl group S_VSn = dim(V, but also cycles of arbitrary length. Our proofs use only elementary linear algebra and combinatorics.     

Approximation of a nonlinear degenerate parabolic equation via a linear relaxation scheme

We study a nonlinear degenerate parabolic equation of the type accompanied by an initial datum and mixed boundary conditions. The symbol [ · ]₊ denotes the usual cutoff function. The problem represents a model of a reactive solute transport in porous media. The exponent p fulfills p ∈ (0, 1). This limits the regularity of a solution and leads to inconveniences in the error analysis. We design a new robust linear numerical scheme for the time discretization. This is based on a suitable combination of the backward Euler method and a linear relaxation scheme. We prove the convergence of relaxation iterations on each time point t_i. We derive the error estimates in suitable function spaces for all values of p ∈ (0, 1). © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Resonance phenomena in a semi-infinite string with a periodic end

We study the propagation of linear waves, generated by a compactly supported time-harmonic force distribution, in a semi-infinite string under the assumption that the material properties depend p-period-ically on the space variable outside a sufficiently large interval [0, a]. The spectrum of the self-adjoint extension A of the spatial part of the differential operator consists of a finite or countable number of bands and a (possibly empty) discrete set of eigenvalues located in the gaps of the continuous spectrum. We show that resonances of order t or t^½, respectively, occur if either ω² is an eigenvalue of A or (i) ω² is a boundary point of the continuous spectrum of A and (ii) the corresponding time-independent homogeneous problem has a non-trivial solution which is p-periodic or p-semiperiodic for x > a (‘standing wave’).