Locally Closed Semirings and Iteration Semirings

Locally closed semirings, iteration semirings and Conway semirings play an important role in the algebraic theory of semirings and theoretical computer science. Z. ésik and W. Kuich showed that a locally closed commutative semiring is an iteration semiring (is also a Conway semiring). By study of polynomial semirings and matrix semirings, we obtain new expressions of certain polynomials and show that all matrix semirings over a locally closed semiring are also locally closed, and so a locally closed semiring (which need not be commutative) is an iteration semiring.     

Covering and packing in ${\Bbb Z}^n$ and ${\Bbb R}^n$, (I)

Given a finite subset of an additive group such as or , we are interested in efficient covering of by translates of , and efficient packing of translates of in . A set provides a covering if the translates with cover (i.e., their union is ), and the covering will be efficient if has small density in . On the other hand, a set will provide a packing if the translated sets with are mutually disjoint, and the packing is efficient if has large density. In the present part (I) we will derive some facts on these concepts when , and give estimates for the minimal covering densities and maximal packing densities of finite sets . In part (II) we will again deal with , and study the behaviour of such densities under linear transformations. In part (III) we will turn to . Authors’ address: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395, USA The first author was partially supported by NSF DMS 0074531.     

Constant angle surfaces in $ {\Bbb S}^2\times {\Bbb R} $

In this article we study surfaces in for which the unit normal makes a constant angle with the -direction. We give a complete classification for surfaces satisfying this simple geometric condition.     

Smooth threefolds in ${\Bbb P}^5$ without apparent triple or quadruple points and a quadruple-point formula

For a projective variety X of codimension 2 in defined over the complex number field , it is traditionally said that X has no apparent -ple points if the -secant lines of X do not fill up the ambient projective space , equivalently, the locus of -ple points of a generic projection of X to ${\Bbb P}^{n+1}$ is empty. We show that a smooth threefold in has no apparent triple points if and only if it is contained in a quadric hypersurface. We also obtain an enumerative formula counting the quadrisecant lines of X passing through a general point of and give necessary cohomological conditions for smooth threefolds in without apparent quadruple points. This work is intended to generalize the work of F. Severi [fSe] and A. Aure [Au], where it was shown that a smooth surface in has no triple points if and only if it is either a quintic elliptic scroll or contained in a hyperquadric. Furthermore we give open questions along these lines. Received: 24 January 2000 / Published online: 18 June 2001     

Some Examples in the Theory of Subgroup Growth

By estimating the subgroup numbers associated with various classes of large groups, we exhibit a number of new phenomena in the theory of subgroup growth.     

Homological Invariants for pro- p Grou ps and Some Finitely Presented pro-${\cal C}$ Grou ps

Let G be a finitely presented pro- group with discrete relations. We prove that the kernel of an epimorphism of G to is topologically finitely generated if G does not contain a free pro- group of rank 2. In the case of pro-p groups the result is due to J. Wilson and E. Zelmanov and does not require that the relations are discrete ([15], [17]).For a pro-p group G of type FP_m we define a homological invariant ^m(G) and prove that this invariant determines when a subgroup H of G that contains the commutator subgroup G is itself of type FP_m. This generalises work of J. King for ¹(G) in the case when G is metabelian [9].Both parts of the paper are linked via two conjectures for finitely presented pro-p groups G without free non-cyclic pro-p subgroups. The conjectures suggest that the above conditions on G impose some restrictions on ¹(G) and on the automorphism group of G.Both authors are partially supported by CNPq, Brazil.     

Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbb{R}^{2n}$

Let be a compact hypersurface in bounding a convex set with non-empty interior. In this paper it is proved that there always exist at least n geometrically distinct closed characteristics on if is symmetric with respect to the origin. Received: 18 May 1999 / Published online: 4 April 2002     

Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in ${\mathbb R}^3$

Summary. In this paper we describe and analyse a class of spectral methods, based on spherical polynomial approximation, for second-kind weakly singular boundary integral equations arising from the Helmholtz equation on smooth closed 3D surfaces diffeomorphic to the sphere. Our methods are fully discrete Galerkin methods, based on the application of special quadrature rules for computing the outer and inner integrals arising in the Galerkin matrix entries. For the outer integrals we use, for example, product-Gauss rules. For the inner integrals, a variant of the classical product integration procedure is employed to remove the singularity arising in the kernel. The key to the analysis is a recent result of Sloan and Womersley on the norm of discrete orthogonal projection operators on the sphere. We prove that our methods are stable for continuous data and superalgebraically convergent for smooth data. Our theory includes as a special case a method closely related to one of those proposed by Wienert (1990) for the fast computation of direct and inverse acoustic scattering in 3D. Received May 29, 2000 / Revised version received March 26, 2001/ Published online December 18, 2001     

Ellipticity on Manifolds With Edges and Boundary

Ellipticity of a manifold with edges and boundary is connected to boundary and edge conditions that complete corresponding operators to Fredholm operators between weighted Sobolev spaces. We study a new parameter-dependent calculus of elliptic operators, where the interior symbols have specific properties on the boundary. We construct elliptic operators with a prescribed number of edge conditions and obtain isomorphisms in the scale of edge Sobolev spaces. Supported by the Chinese-German Cooperation Program ‘Partial Differential Equations’, NSFC of China and DFG of Germany.     

Generalized Solutions of Semilinear Parabolic Equations

It is well-known that a semilinear parabolic equation has no unique solution in the classical sense. We study such equations from the viewpoint of generalized functions. By using approximations for generalized functions, we obtain results on existence and uniqueness of generalized solutions. Furthermore, we establish the relationship between generalized solutions and classical solutions. Current address: Institute of Mathematics, University of Tsukuba, Tsukuba 305-8571, Japan     

Comparing Semigroup and Monoid Presentations for Finite Monoids

 It is known that for any finite group G given by a finite group presentation there exists a finite semigroup presentation for G of the same deficiency, i.e. satisfying . It is also known that the analogous statement does not hold for all finite monoids. In this paper we give a necessary and sufficient condition for a finite monoid M, given by a finite monoid presentation, to have a finite semigroup presentation of the same deficiency. (Received 17 April 2001; in revised form 15 September 2001)     

A topological proof of cancellation theorem for ${\vec C}^2$

We give a new proof of the Cancellation Theorem for which says that if V is a smooth complex affine surface such that is isomorphic to then V is isomorphic to The proof is based on the topological method of Mumford-Ramanujam. Received: 2 November 2000 / in final form: 10 September 2001 / Published online: 1 February 2002     

Analysis of a Quantum Subband Model for the Transport of Partially Confined Charged Particles

This paper is devoted to the analysis of a quantum subband model, which is presented as an alternative to the standard 3D Schr?dinger-Poisson system for modeling the transport of electrons strongly confined along one direction. This subband model is composed of quasistatic 1D Schr?dinger equations in the direction of the confinement, coupled to 2D time-dependent Schr?dinger equations describing the transport in the non-confined directions. Selfconsistent electrostatic interactions are also taken into account via the Poisson equation. This system is studied in the framework of the strong partial confinement asymptotics introduced in the article “Adiabatic approximation of the Schr?dinger-Poisson system with a partial confinement”, by Ben Abdallah, Méhats and Pinaud (SIAM J. Math. Anal. 36 (2005), 986–1013).     

Certain classes of pluricomplex Green functions on ${\mathbb C}^n$

We consider (pluricomplex) Green functions defined on , with logarithmic poles in a finite set and with logarithmic growth at infinity. For certain sets, we describe all the corresponding Green functions. The set of these functions is large and it carries a certain algebraic structure. We also show that for some sets no such Green functions exist. Our results indicate the fact that the set of poles should have certain algebro-geometric properties in order for these Green functions to exist. Received November 24, 1998; in final form April 19, 1999 / Published online July 3, 2000     

A primal mixed formulation for exterior transmission problems in ${\bf R}^2$

Summary.   We combine a primal mixed finite element approach with a Dirichlet-to-Neumann mapping (arising from the boundary integral equation method) to study the weak solvability and Galerkin approximations of a class of linear exterior transmission problems in potential theory. Our results are mainly based on the Babuska-Brezzi theory for variational problems with constraints. We establish the uniqueness of solution for the continuous and discrete formulations, and show that finite element subspac es of Lagrange type satisfy the discrete compatibility conditions. In addition, we provide the error analysis, including polygonal approximations of the domain, and prove strong convergence of the Galerkin solutions. Moreover, under additional regularity assumptions on the solution of the continuous formulation, we obtain the asymptotic rate of convergence O(h). Received August 25, 1998 / Revised version received March 8, 2000 / Published online October 16, 2000     

A Type of Hyperelliptic Continued Fraction

Based on hints of Tschebychev, a continued fraction is described which gives an effective algorithm for calculating the torsion, if finite, of divisors D − D⁻ on a hyperelliptic curve of genus ≥2, where D is an effective divisor of degree 2 and D⁻ denotes the image of D under the hyperelliptic involution. The difficulties involved in extending the algorithm to divisors of degree ≥3 are briefly discussed.     

Two Remarks on Solutions of Gross-Pitaevskii Equations on Zhidkov Spaces

We consider the so-called Gross-Pitaevskii equations supplemented with non-standard boundary conditions. We prove two mathematical results concerned with the initial value problem for these equations in Zhidkov spaces.     

Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

A Quasi-Neutral Limit in a Hydrodynamic Model for Charged Fluids

 The combined quasineutral and relaxation time limit for a bipolar hydrodynamic model is considered. The resulting limit problem is a nonlinear diffusion equation describing a neutral fluid. We make use of various entropy functions and the related entropy productions in order to obtain strong enough uniform bounds. The necessary strong convergence of the densities is obtained by using a generalized version of the “div-curl” Lemma and monotonicity methods. Received September 27, 2001; in revised form February 25, 2002