Superconvergence of finite difference approximations for convection–diffusion problems

An Erratum has been published for this article in Numerical Linear Algebra with Applications 8 (4) 2001, iii–iv. We are concerned with numerical solutions of convection–diffusion equations. The convergence behaviour of numerical solutions is considered by using the finite difference approximation with respect to spatial variables and implicit method with respect to time variable. It is shown that superconvergence occurs near a part of the boundary which has Dirichlet's data. Copyright © 2001 John Wiley & Sons, Ltd.     

Estimation for the region of attraction of nonlinear difference equations

Summary. We consider nonlinear systems of difference equations with and , where A is any matrix, is a continuous vector-function, and is a numeral parameter. The spectrum of A belongs to the unit circle . We give the estimations for the region of attraction and the speed of convergence solutions to the zero solution of the systems. We indicate a set M such that for solutions of the system with parameter the limit is true. Received April 7, 2000 / Revised version received January 8, 2001 / Published online November 15, 2001     

Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford–Shah functional

The paper is concerned with the higher regularity properties of the minimizers of the Mumford–Shah functional. It is shown that, near to singular points where the scaled Dirichlet integral tends to 0, the discontinuity set is close to an Almgren area minimizing set. As a byproduct, the set of singular points of this type has Hausdorff dimension at most N-2, N being the dimension of the ambient space. Assuming higher integrability of the gradient this leads to an optimal estimate of the Hausdorff dimension of the full singular set. Received: 5 July 2001 / Accepted: 29 November 2001 / Published online: 23 May 2002     

Defining Integrality at Prime Sets of High Density over Function Fields

 Let F be a function field of characteristic over a finite field . Then for any finite set of F-primes S and any , there exists a set of F-primes W of density greater than such that and has a Diophantine definition over . (Here and is defined analogously.) Received 5 March 2001; in revised form 4 October 2001     

A singular limit arising in combustion theory: Fine properties of the free boundary

The equation where converges to the Dirac measure concentrated at with mass has been used as a model for the propagation of flames with high activation energy. For initial data that are bounded in and have a uniformly bounded support, we study non-negative solutions of the Cauchy problem in as We show that each limit of is a solution of the free boundary problem in on (in the sense of domain variations and in a more precise sense). For a.e. time t the graph of u(t) has a unique tangent cone at -a.e. The free boundary is up to a set of vanishing measure the sum of a countably n-1-rectifiable set and of the set on which vanishes in the mean. The non-degenerate singular set is for a.e. time a countably n-1-rectifiable set. As key tools we introduce a monotonicity formula and, on the singular set, an estimate for the parabolic mean frequency. Received: 8 August 2001 / Accepted: 8 May 2002 / Published online: 5 September 2002 RID="a" ID="a" Partially supported by a Grant-in-Aid for Scientific Research, Ministry of Education, Japan.     

Conditions Characterizing Minima of the Difference of Functions

 Optimization problems involving differences of functions arouse interest as generalizations of so-called d.c. problems, i.e. problems involving the difference of two convex functions. The class of d.c. functions is very rich, so d.c. problems are rather general optimization problems. Several global optimality conditions for these d.c. problems have been proposed in the optimization literature. We provide a survey of these conditions and try to detect their common basis. This enables us to give generalizations of the conditions to situations when the objective function is no longer a difference of convex functions, but the difference of two functions which are representable as the upper envelope of an arbitrary family of functions. (Received 6 February 2001; in revised form 11 October 2001)     

About 4-Isosceles Planar Sets

A finite planar set is k -isosceles for k \geq 3 , if every k -point subset of the set contains a point equidistant from the other two. This paper gives affirmative answers to a few open problems about 4-isosceles sets. Received August 14, 2001, and in revised form September 10, 2001. Online publication December 21, 2001.     

Aplicación del método level set para modelar el proceso de combustión premezclada en un motor Otto de dos tiempos

In this paper, the numerical method level set has been used to model the combustion process in an Otto two-stroke engine. The level set has been implemented in a CFD (Computational Fluid Dynamics) software based in finite volumes. The pressure and temperature fields have been obtained, such as the propagation of the flame front. In order to validate this model, the numerically obtained results have been compared with experimental data, verifying a satisfactory concordance between both of them. Besides, the level set method has been compared with other numerical procedure, showing the difference between both results.     

Independent perfect domination sets in Cayley graphs

In this paper, we show that a Cayley graph for an abelian group has an independent perfect domination set if and only if it is a covering graph of a complete graph. As an application, we show that the hypercube Q_n has an independent perfect domination set if and only if Q_n is a regular covering of the complete graph K_n+1 if and only if n = 2^m ? 1 for some natural number m. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 213–219, 2001     

Coupling vs. conductance for the Jerrum–Sinclair chain*

We address the following question: is the causal coupling method as strong as the conductance method in showing rapid mixing of Markov chains? A causal coupling is a coupling which uses only past and present information, but not information about the future. We answer the above question in the negative by showing that there exists a bipartite graph G such that any causal coupling argument on the Jerrum–Sinclair Markov chain for sampling almost uniformly from the set of perfect and near perfect matchings of G must necessarily take time exponential in the number of vertices in G. In contrast, the above Markov chain on G has been shown to mix in polynomial time using conductance arguments. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 1–17, 2001     

A Gibbs variational principle in space-time for infinite-dimensional diffusions

 We show that the set of stationary weak solutions for a class of infinite dimensional stochastic differential equations coincides with the set of shift invariant, space-time Gibbs fields for a certain potential. The key step consists in proving the Gibbs variational principle for space-time Gibbs fields. Received: 20 May 1999 / Revised version: 14 May 2001 / Published online: 11 December 2001     

Growth tightness for word hyperbolic groups

We show that non-elementary word hyperbolic groups are growth tight. This means that, given such a group G and a finite set A of its generators, for any infinite normal subgroup N of G, the exponential growth rate of G/N with respect to the natural image of A is strictly less than the exponential growth rate of G with respect to A. Received: 20 September 2001; in final form: 24 January 2002/ Published online: 5 September 2002 The work has been supported by the Swiss National Science Foundation. The second author has been supported also by INTAS, grant No. 97–1259     

Global optimization of the difference of affine increasing and positively homogeneous functions

The theory of increasing and positively homogeneous (IPH) functions defined on a real topological vector space X has well been developed. In this paper, we first give various characterizations for maximal elements of the support set of this class of functions. As an application, we present various characterizations for maximal elements of the support set of affine IPH functions. Finally, we investigate necessary and sufficient conditions for the global minimum of the difference of two strictly affine IPH functions.     

The Complexity of Embedding Orders into Small Products of Chains

Embedding a partially ordered set into a product of chains is a classical way to encode it. Such encodings have been used in various fields such as object oriented programming or distributed computing. The embedding associates with each element a sequence of integers which is used to perform comparisons between elements. A critical measure is the space required by the encoding, and several authors have investigated ways to minimize it, which comes to embedding partially ordered sets into small products of chains. The minimum size of such an encoding has been called the encoding dimension (Habib et al. 1995), and the string dimension (Garg and Skawratananand 2001) for a slightly different definition of embeddings. This paper investigates some new properties of the encoding dimension. We clear up the links with the string dimension and we answer the computational complexity questions raised in Garg and Skawratananand (2001) and Habib et al. (1995): both these parameters are NP{\ensuremath{\mathcal{NP}}}-hard to compute.     

Asymptotic and spectral analysis of non‐selfadjoint operators generated by a filament model with a critical value of a boundary parameter

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. In our previous paper (Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169), we have derived the asymptotic approximations for the eigenvalues and eigenfunctions of the aforementioned non‐selfadjoint operators when the boundary parameters were arbitrary complex numbers except for one specific value of one of the parameters. We call this value the critical value of the boundary parameter. It has been shown (in Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169) that the entire set of the eigenvalues is located in a strip parallel to the real axis. The latter property is crucial for the proof of the fact that the set of the root vectors of the operator forms a Riesz basis in the state space of the system. In the present paper, we derive the asymptotics of the spectrum exactly in the case of the critical value of the boundary parameter. We show that in this case, the asymptotics of the eigenvalues is totally different, i.e. both the imaginary and real parts of eigenvalues tend to ∞as the number of an eigenvalue increases. We will show in our next paper, that as an indirect consequence of such a behaviour of the eigenvalues, the set of the root vectors of the corresponding operator is not uniformly minimal (let alone the Riesz basis property). Copyright © 2003 John Wiley & Sons, Ltd.     

Chromatic numbers of quadrangulations on closed surfaces

It has been shown that every quadrangulation on any nonspherical orientable closed surface with a sufficiently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface N_k has chromatic number at least 4 if G has a cycle of odd length which cuts open N_k into an orientable surface. Moreover, we characterize the quadrangulations on the torus and the Klein bottle with chromatic number exactly 3. By our characterization, we prove that every quadrangulation on the torus with representativity at least 9 has chromatic number at most 3, and that a quadrangulation on the Klein bottle with representativity at least 7 has chromatic number at most 3 if a cycle cutting open the Klein bottle into an annulus has even length. As an application of our theory, we prove that every nonorientable closed surface N_k admits an eulerian triangulation with chromatic number at least 5 which has arbitrarily large representativity. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 100–114, 2001     

One (96,20,4)-symmetric Design and related Nonabelian Difference Sets

New (96,20,4)-symmetric design has been constructed, unique under the assumption of an automorphism group of order 576 action. The correspondence between a (96,20,4)-symmetric design having regular automorphism group and a difference set with the same parameters has been used to obtain difference sets in five nonabelian groups of order 96. None of them belongs to the class of groups that allow the application of so far known construction (McFarland, Dillon) for McFarland difference sets.AMS lassification: 05B05     

Actions and irreducible representations of the mapping class group

Let G be a countable discrete group. Call two subgroups and of G commensurable if has finite index in both and . We say that an action of G on a discrete set X has noncommensurable stabilizers if the stabilizers of any two distinct points of X are not commensurable. We prove in this paper that the action of the map ping class group on the complex of curves has noncommensurable stabilizers. Following a method due to Burger and de la Harpe, this action leads to constructions of irreducible unitary representations of the mapping class group. Received: 26 July 1999 / Revised version: 14 May 2001 / Published online: 19 October 2001     

Explicit Polynomial Generators for the Ring of Quasisymmetric Functions over the Integers

In (Hazewinkel in Adv. Math. 164:283–300, 2001, and CWI preprint, 2001) it has been proved that the ring of quasisymmetric functions over the integers is free polynomial. This is a matter that has been of great interest since 1972; for instance because of the role this statement plays in a classification theory for noncommutative formal groups that has been in development since then, see (Ditters in Invent. Math. 17:1–20, 1972; in Scholtens’ Thesis, Free Univ. of Amsterdam, 1996) and the references in the latter. Meanwhile quasisymmetric functions have found many more applications (see Gel’fand et al. in Adv. Math. 112:218–348, 1995). However, the proofs of the author in the aforementioned papers do not give explicit polynomial generators for QSymm over the integers. In this note I give a (really quite simple) set of polynomial generators for QSymm over the integers.     

On the -Discrepancy of the Hammersley Point Set

 We give a formula for the -discrepancy of the 2-dimensional Hammersley point set in base 2 for all integers p, . Received 18 May 2001; in revised form 18 December 2001