Multiplicative symmetry and related functional equations

Summary Let (G, *) be a commutative monoid. Following J. G. Dhombres, we shall say that a functionf: G G is multiplicative symmetric on (G, *) if it satisfies the functional equationf(x * f(y)) = f(y * f(x)) for allx, y inG. (1)Equivalently, iff: G G satisfies a functional equation of the following type:f(x * f(y)) = F(x, y) (x, y G), whereF: G × G G is a symmetric function (possibly depending onf), thenf is multiplicative symmetric on (G, *).In Section I, we recall the results obtained for various monoidsG by J. G. Dhombres and others concerning the functional equation (1) and some functional equations of the formf(x * f(y)) = F(x, y) (x, y G), (E) whereF: G × G G may depend onf. We complete these results, in particular in the case whereG is the field of complex numbers, and we generalize also some results by considering more general functionsF. In Section II, we consider some functional equations of the formf(x * f(y)) + f(y * f(x)) = 2F(x, y) (x, y K), where (K, +, ·) is a commutative field of characteristic zero, * is either + or · andF: K × K K is some symmetric function which has already been considered in Section I for the functional equation (E). We investigate here the following problem: which conditions guarantee that all solutionsf: K K of such equations are multiplicative symmetric either on (K, +) or on (K, ·)? Under such conditions, these equations are equivalent to some functional equations of the form (E) for which the solutions have been given in Section I. This is a partial answer to a question asked by J. G. Dhombres in 1973.     

Actions of Finite Hypergroups

This paper is concerned with actions of finite hypergroups on sets. After introducing the definitions in the first section, we use the notion of maximal actions to characterise those hypergroups which arise from association schemes, introduce the natural sub-class of *-actions of a hypergroup and introduce a geometric condition for the existence of *-actions of a Hermitian hypergroup. Following an insightful suggestion of Eiichi Bannai we obtain an example of the surprising phenomenon of a 3-element hypergroup with infinitely many pairwise inequivalent irreducible *-actions.     

Estimates of the solution of the Dirichlet problem for the laplace operator in exterior domains

Let u be the solution of the equation u=f with a finite function f in the exterior domain R^u and satisfying the conditions u¦=0, u0 as ¦x¦ . It is shown that the coercive estimate is valid only for p < n/2. For p n/2 it holds for the solution of the exterior Dirichlet problem, which does not vanish at infinity, and may tend to constant or even to a linear (for p > n) function.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 146, pp. 92–101, 1985.     

Chemical trees enumeration algorithms

In the chemical community the need for representing chemical structures within a given family and of efficiently enumerating these structures suggested the use of computers and the implementation of fast enumeration algorithms. This paper considers the isomeric acyclic structures focusing on the enumeration of the alkane molecular family. For this family, Trinajsti et al. (1991) devised an enumeration algorithm which is the most widely known and utilized nowadays. Kvasnika and Pospichal (1991) have proposed an algorithmic scheme which, from the computational complexity point of view, can prove to be more efficient than the Trinajsti one, nevertheless, this algorithm, to the best of our knowledge, has never been implemented. Indeed an efficient implementation requires the introduction of non trivial data structures and other computational tricks. The main contribution of this paper consists of the definition of the implementation details of Kvasnika-Pospichals algorithm, in a comparison of Trinajstis, Kvasnika-Pospichals and two new algorithms, proposed here, in terms of both computational complexity analysis and running times.AMS classification: 05A15, 05C05, 05C30, 05C90Part of this work has been developed during a visit of the first two authors at the EPFL of Lausanne     

The lower algebraicK-theory of virtually infinite cyclic groups

This paper gives a proof of a conjecture of W.-C. Hsiang for the negativeK-theory of integral grouprings , when the group is a subgroup of a uniform lattice in a Lie group. The authors' earlier paper reduced this result to the very special cases where either is finite or is virtually infinite cyclic. The finite case was done much earlier by Carter extending results of Bass and Murthy. The major work of the present paper consists of proving the conjecture when is virtually infinite cyclic.Both authors were supported in part by the National Science Foundation.     

On term by term dyadic differentiability of Walsh series

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Functions with a finite dirichlet integral in a domain with cusp points on the boundary

Let be a domain in ⁿ, n >2, the boundary of which has a cusp point, pointing inside or outside the domain. The purpose of the paper is to characterize the traces on of the elements of the space H¹() of functions with a finite Dirichlet integral. As a consequence one establishes the existence of a linear continuous extension operator H¹ () H¹(ⁿ) under the presence of an interior cusp point on . Theorems on domains with cusps are proved with the aid of results on cylindrical domains. In the space of functions with a finite Dirichlet integral in the exterior or the interior of the cylinder one introduces the norm, depending on a small parameter and generating a norm of the trace on as an element of the quotient space. The latter is placed in correspondence with an explicitly described norm of functions on the boundary, uniformly equivalent relative to . One constructs an operator of extension of functions from the exterior of the cylinder to Rⁿ, preserving H¹, whose norm is uniformly bounded relative to . For the optimal operator of extension from the inside of the cylinder one finds the asymptotic behavior of the norm as 0. From these results there follow similar theorems on functions with a finite Dirichlet integral inside and outside a thin closed tube (of width ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 117–137, 1983.     

Linearly ordered groups whose system of convex subgroups is central

The order P on a group G is called rigid if for p P the relation p¦[x, p]¦ P holds for every x G, =±1. In this note we give criteria for the existence of a rigid linear order, for the extendability of a rigid partial order to a rigid linear order, and for the extendability of each rigid partial order to a rigid linear order on a group. It is proved that the class of groups each of whose rigid partial orders can be extended to a rigid linear order is closed with respect to direct products. A new proof of the theorem of M. I. Kargapolov which states that if a group G can be approximated by finite p-groups for infinite number of primes p, then it has a central system of subgroups with torsion-free factors is presented.Translated from Matematicheskie Zametki, Vol. 19, No. 1, pp. 85–90, January, 1976.     

Traces of functions belonging to Sobolev and Besov spaces and extensions from subsets of the Euclidean space

It is proved that from the existence of the trace operator, 0n and E is a Borel subset of ⁿ.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, AN SSSR, Vol. 157, pp. 137–145, 1987.     

Construction of po-groups with quasi-divisors theory

A method is presented making it possible to construct po-groups with a strong theory of quasi-divisors of finite character and with some prescribed properties as subgroups of restricted Hahn groups H(, ), where are finitely atomic root systems. Some examples of these constructions are presented.     

Quasiexact Solution of a Relativistic Finite-Difference Analogue of the Schrödinger Equation for a Rectangular Potential Well

We consider a well-posed formulation of the spectral problem for a relativistic analogue of the one-dimensional Schrödinger equation with differential operators replaced with operators of finite purely imaginary argument shifts exp(±id/dx). We find effective solution methods that permit determining the spectrum and investigating the properties of wave functions in a wide parameter range for this problem in the case of potentials of the type of a rectangular well. We show that the properties of solutions of these equations depend essentially on the relation between and the parameters of the potential and a situation in which the solution for 1 is nevertheless fundamentally different from its Schrödinger analogue is quite possible.     

A family of noetherian rings with their finite length modules under control

We investigate the category mod of finite length modules over the ring =A _k , where is a V-ring, i.e. a ring for which every simple module is injective, k a subfield of its centre and A an elementary k-algebra. Each simple module E _j gives rise to a quasiprogenerator P _j = A E _j . By a result of K. Fuller, P _j induces a category equivalence from which we deduce that mod _j mod EndP _j . As a consequence we can(1) construct for each elementary k-algebra A over a finite field k a nonartinian noetherian ring such that modA mod(2) find twisted versions of algebras of wild representation type such that itself is of finite or tame representation type (in mod)(3) describe for certain rings the minimal almost split morphisms in mod and observe that almost all of these maps are not almost split in Mod.     

Sets of uniqueness and closed subgroups in Vilenkin groups

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This research was partially supported by National Science Foundation under grant INT-8400708.     

On the Riemann derivatives ofC sP-integrable functions

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Approximation of a nondifferentiable nonlinear problem related to MHD equilibria

Summary We present a simple method, based on a variant of the implicit function theorem, which leads to the existence of (a part of) a nontrivial solution branch of the nonlinear eigenvalue problem –u=u ⁺ in ,u=–1 on , where is a two-dimensional domain with boundary . The advantage of this method is that we can apply it for analysing the approximation of the above problem by a finite element method; the error analysis of the discrete problem appears immediately. We give also an iteration scheme which allows to solve the approximate problem.     

Ghost ideals in uniform Roe algebras of coarse spaces

Let be the uniform Roe algebra of a coarse space X with uniformly locally finite coarse structure. We show that an operator G in is a ghost element if and only if the finite propagation operators in the principal ideal G are all compact operators. In contrast, if X is a discrete metric space with Yus property (A), then any ideal in is the closure of the finite propagation operators in the ideal.Received: 9 June 2004     

Vector measure range duality and factorizations of (<Emphasis Type="Italic">D,p</Emphasis>)-summing operators from Banach function spaces

We characterize the relationship between the space L₁() and the dual L₁() of the space L₁(), where (, ) is a dual pair of vector measures with associated spaces of integrable functions L₁() and L₁() respectively. Since the result is rather restrictive, we introduce the notion of range duality in order to obtain factorizations of operators from Banach function spaces that are dominated by the integration map associated to the vector measure . We obtain in this way a generalization of the Grothendieck-Pietsch Theorem for p-summing operators.*The research was partially supported by MCYT DGI project BFM 2001-2670.**The research was partially supported by MCYT DGI project BFM 2000-1111.     

The Cantor-Lebesgue and Denjoy-Luzin properties for double systems of functions

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On Gibbs' phenomenon for Riesz spherical means of multiple Fourier integrals and Fourier series

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On a problem of Tandori

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