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.     

Travelling Waves in a Nonlocal Reaction-Diffusion Equation as a Model for a Population Structured by a Space Variable and a Phenotypic Trait

We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed c* > 0, and prove the existence of waves when c ≥ c* and the nonexistence when 0 ≤ c < c*.     

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.     

The Terwilliger Algebra Associated with a Set of Vertices in a Distance-Regular Graph

Let Γ be a distance-regular graph of diameter D. Let X denote the vertex set of Γ and let Y be a nonempty subset of X. We define an algebra τ = τ(Y). This algebra is finite dimensional and semisimple. If Y consists of a single vertex then τ is the corresponding subconstituent algebra defined by P. Terwilliger. We investigate the irreducible τ-modules. We define endpoints and thin condition on irreducible τ-modules as a generalization of the case when Y consists of a single vertex. We determine when an irreducible module is thin. When the module is generated by the characteristic vector of Y, it is thin if and only if Y is a completely regular code of Γ. By considering a suitable subset Y, every irreducible τ(x)-module of endpoint i can be regarded as an irreducible τ(Y)-module of endpoint 0.This research was partially supported by the Grant-in-Aid for Scientific Research (No. 12640039), Japan Society of the Promotion of Science. A part of the research was done when the author was visiting the Ohio State University.     

Singular and nonsingular modules relative to a torsion theory

In this paper, we introduce and study torsion-theoretic generalizations of singular and nonsingular modules by using the concept of τ-essential submodule for a hereditary torsion theory τ. We introduce two new module classes called τ-singular and non-τ-singular modules. We investigate some properties of these module classes and present some examples to show that these new module classes are different from singular and nonsingular modules. We give a characterization of τ-semisimple rings via non-τ-singular modules. We prove that if M∕τ(M) is non-τ-singular for a module M, then every submodule of M has a unique τ-closure. We give some properties of the torsion theory generated by the class of all τ-singular modules. We obtain a decomposition theorem for a strongly τ-extending module by using non-τ-singular modules.     

Structure of the Particle Population for a Branching Random Walk with a Critical Reproduction Law

We consider a continuous-time symmetric branching random walk on the d-dimensional lattice, d ≥?1, and assume that at the initial moment there is one particle at every lattice point. Moreover, we assume that the underlying random walk has a finite variance of jumps and the reproduction law is described by a continuous-time Markov branching process (a continuous-time analog of a Bienamye-Galton-Watson process) at every lattice point. We study the structure of the particle subpopulation generated by the initial particle situated at a lattice point x. We replay why vanishing of the majority of subpopulations does not affect the convergence to the steady state and leads to clusterization for lattice dimensions d =?1 and d =?2.

     

Surfaces of order three with a peak! I

A surfaceF of order three in the real projective three space is met by every line, not inF, in at most three points. We present a synthetic theory of these surfaces based upon a concept of differentiability. A pointv inF is a peak ifF has a unique tangent atvs and it does not lie inF. We classify theF with a peak by determining the number of lines inF and their configuration. We describe each surface by determining the tangent plane sections ofF at the differentiable points.     

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.     

Notes on growing a tree in a graph

We study the height of a spanning tree T of a graph G obtained by starting with a single vertex of G and repeatedly selecting, uniformly at random, an edge of G with exactly one endpoint in T and adding this edge to T.     

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.     

On polynomial rings over a ring with a selfduality

We prove that a ringR has a self duality induced by a leftR-moduleM if and only if its polynomial ringR[x] has a graded self duality induced by a graded leftR[x]-moduleM[x ⁻¹]. Supported by the Natural Science Foundation of Fujian Province (1994–1997)     

Measures of affinity of a sequence for a number

We define strong and weak affinities of a number a for a sequence (x_k ) denoted by L (a,(x_k )) and U (a, (x_k )) respectively. We show U (a,(x_k )) > 0 if and only if the number a is a statistical limit point of the sequence (x_k ). We consider the distribution of sequences with positive weak and strong measures of affinity within the space l ^∞ of bounded sequences. The main result is that the set of bounded sequences with U (a,(x_k )) > 0, that is, the set of sequences with statistical limit points, is a dense subset in l ^∞ of the first category. We also show the set of sequences with positive strong affinities is a nowhere dense subset of l ^∞.     

Approximation of a bending plate problem with a boundary unilateral constraint

Summary We study a variational formulation of the unilaterally supported bent plate problem and we analyze the approximation of the problem by a mixed finite element method. We proveO(h) andO(h|lnh|^1/2) error bounds respectively for the moments and the displacement.Work partially supported by M.P.I., by G.N.I.M. of C.N.R. and by I.A.N. of C.N.R. of Pavia     

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.     

Symplectic Automorphisms and the Picard Group of a K3 Surface

We consider the symplectic action of a finite group G on a K3 surface X. The Picard group of X has a primitive sublattice determined by G. We show how to compute the rank and discriminant of this sublattice. We then investigate the classification of symplectic actions by a fixed finite group, using moduli spaces of K3 surfaces with symplectic G-action.     

Scheduling by positional completion times: Analysis of a two-stage flow shop problem with a batching machine

We consider a scheduling problem introduced by Ahmadi et al., Batching and scheduling jobs on batch and discrete processors, Operation Research 40 (1992) 750–763, in which each job has to be prepared before it can be processed. The preparation is performed by a batching machine; it can prepare at mostc jobs in each run, which takes an amount of time that is independent of the number and identity of the jobs under preparation. We present a very strong Lagrangian lower bound by using the new concept of positional completion times. This bound can be computed in O(n logn) time only, wheren is the number of jobs. We further present two classes of O(n logn) heuristics that transform an optimal schedule for the Lagrangian dual problem into a feasible schedule. Any heuristic in one class has performance guarantee of 3/2. We further show that even the most naive heuristic in this class has a compelling empirical performance. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.An earlier draft of this paper has appeared in the Proceedings of the Fourth International IPCO Conference, Lecture Notes in Computer Science, vol. 920, Springer, Berlin.     

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.     

Extrema of a Real Polynomial

In this paper, we investigate critical point and extrema structure of a multivariate real polynomial. We classify critical surfaces of a real polynomial f into three classes: repeated, intersected and primal critical surfaces. These different critical surfaces are defined by some essential factors of f, where an essential factor of f means a polynomial factor of f–c ₀, for some constant c ₀. We show that the degree sum of repeated critical surfaces is at most d–1, where d is the degree of f. When a real polynomial f has only two variables, we give the minimum upper bound for the number of other isolated critical points even when there are nondegenerate critical curves, and the minimum upper bound of isolated local extrema even when there are saddle curves. We show that a normal polynomial has no odd degree essential factors, and all of its even degree essential factors are normal polynomials, up to a sign change. We show that if a normal quartic polynomial f has a normal quadratic essential factor, a global minimum of f can be either easily found, or located within the interior(s) of one or two ellipsoids. We also show that a normal quartic polynomial can have at most one local maximum.     

The Guessing Secrets problem: a probabilistic approach

We introduce a probabilistic variant of the Guessing Secrets problem proposed by Chung et al. in [Electron. J. Combin. 8 (2001) R13]. In our variation, a player tries to discover the identity of a set S of n unknown secrets drawn by a second player, from a space Ω of N secrets. The first player tries to learn as much as possible about the elements of S by asking binary questions. For each question asked, the second player randomly chooses one of the n secrets of S that he uses in supplying the answer, which in any case must be truthful. We define a simple probabilistic guessing algorithm that allows us to guess all secrets of S with probability one. We show that the expected number of questions needed to guess all secrets is 2n²log₂N and the expected time complexity of the algorithm is . We also propose a generalization of this probabilistic guessing secrets problem, and provide some similar results for this generalization.     

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.