Unary operations with long pre-periods

It is well known that the congruence lattice ConA of an algebra A is uniquely determined by the unary polynomial operations of A (see e.g. [K. Denecke, S.L. Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman & Hall, CRC Press, Boca Raton, London, New York, Washington DC, 2002 [2]]). Let A be a finite algebra with |A|=n. If Imf=A or |Imf|=1 for every unary polynomial operation f of A, then A is called a permutation algebra. Permutation algebras play an important role in tame congruence theory [D. Hobby, R. McKenzie, The structure of finite algebras, Contemporary Mathematics, vol. 76, Providence, Rhode Island, 1988 [3]]. If f:A→A is not a permutation then A⊃Imf and there is a least natural number λ(f) with Imf^λ(f)=Imf^λ(f)+1. We consider unary operations with λ(f)=n-1 for n?2 and λ(f)=n-2 for n?3 and look for equivalence relations on A which are invariant with respect to such unary operations. As application we show that every finite group which has a unary polynomial operation with one of these properties is simple or has only normal subgroups of index 2.     

The polylogarithm in the field of two irreducible quintics

A brief review of the polylogarithmic ladder and its cyclotomic equation is followed by an application to the roots of two irreducible quintic equations. The first is unique in its class and possesses four accessible and two inaccessible valid ladders. The other equation gives rise to a single ladder determinable, at the present time, only by numerical computation, and is a member of a completely new category.     

Enumeration results for Runge–Kutta methods for index 2 DAEs

Recursive formulas are presented that give the number of order conditions for single-step Runge–Kutta methods for index 2 DAEs.     

Internal normality and internal compactness

This text contains an example which presents a way to modify any Dowker space to get a normal space X such that X×[0,1] is not κ-normal, and a theorem implying the existence of a non-Tychonoff space which is internally compact in a larger regular space. It gives answers to several questions by Arhangel'skii [A.V. Arhangel'skii, Relative normality and dense subspaces, Topology Appl. 123 (2002) 27-36].     

Iterative solvers and stabilisation for mixed electrostatic and magnetostatic formulations

Mixed electrostatic and magnetostatic finite element formulations are considered. Solution methods for the resulting indefinite algebraic systems are investigated. Methods developed for the mixed formulations of the Stokes equations are modified in order to apply to the Maxwell equations: an efficient block preconditioner is proposed and a stabilised formulation is described. The different methods are applied to 2D and 3D examples.     

A convergent adaptive finite element method for an optimal design problem

The optimal design problem for maximal torsion stiffness of an infinite bar of given geometry and unknown distribution of two materials of prescribed amounts is one model example in topology optimisation. It eventually leads to a degenerate convex minimisation problem. The numerical analysis is therefore delicate for possibly multiple primal variables u but unique derivatives σ : = DW(D u). Even fine a posteriori error estimates still suffer from the reliability-efficiency gap. However, it motivates a simple edge-based adaptive mesh-refining algorithm (AFEM) that is not a priori guaranteed to refine everywhere. Its convergence proof is therefore based on energy estimates and some refined convexity control. Numerical experiments illustrate even nearly optimal convergence rates of the proposed AFEM. Supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin.     

Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators

We study uniformly elliptic fully nonlinear equations of the type F(D²u,Du,u,x)=f(x). We show that convex positively 1-homogeneous operators possess two principal eigenvalues and eigenfunctions, and study these objects; we obtain existence and uniqueness results for nonproper operators whose principal eigenvalues (in some cases, only one of them) are positive; finally, we obtain an existence result for nonproper Isaac's equations.     

A non-analytic proof of the newman—znám result for disjoint covering systems

A direct combinatorial proof is given to a generalization of the fact that the largest modulusN of a disjoint covering system appears at leastp times in the system, wherep is the smallest prime dividingN. The method is based on geometric properties of lattice parallelotopes. This research was supported by grant 85-00368 from the United States-Is rael Binational Science Foundation, Jerusalem, Israel.     

Superconvergence for a mixed finite elmenent method for elastic wave propagation in a plane domain

Summary A higher order mixed finite element method is introduced to approximate the solution of wave propagation in a plane elastic medium. A quasi-projection analysis is given to obtain error estimates in Sobolev spaces of nonpositive index. Estimates are given for difference quotients for a spatially periodic problem and superconvergence results of the same type as those of Bramble and Schatz for Galerkin methods are derived.     

On the enumeration of a class of plane multigraphs

A class of plane multigraphs having a series-parallel structure is defined and enumerated. The enumeration is carried out both for the rooted graphs as originally defined, and also for those where no distinctions are made between the vertices. Some other related problems are also discussed.