Analytic mappings between two ultrahyperelliptic surfaces

Aequationes mathematicae -     

Universalfunktionen in einfach zusammenhängenden Gebieten

Ohne Zusammenfassung

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On averaging multisets

We give sufficient conditions to ensure that, given a set , everyxint convM can be represented as a convex combination,x = _{i = 1} ⁿ _i x _i , wherex _i M, _i rational, andn=2s orn=2s–1, respectively.     

Generalized Jordan algebras

We study commutative algebras which are generalizations of Jordan algebras. The associator is defined as usual by (x, y, z) = (x y)z − x(y z). The Jordan identity is (x², y, x) = 0. In the three generalizations given below, t, β, and γare scalars. ((x x)y)x + t((x x)x)y = 0, ((x x)x)(y x) − (((x x)x)y)x = 0, β((x x)y)x + γ((x x)x)y − (β + γ)((y x)x)x = 0. We show that with the exception of a few values of the parameters, the first implies both the second and the third. The first is equivalent to the combination of ((x x)x)x = 0 and the third. We give examples to show that our results are in some reasonable sense, the best possible.     

The prison yard problem

Given a polygon II withn vertices whose sides arewalls. Guards, located at vertices can see all directions, but cannot see beyond walls. We prove that at most [n/2] guards suffice to see everywhere the whole plane. If II is not convex, then [n/2] suffice.The research was done while this author visited the Department of Mathematics at Rutgers University. Research supported in part by the Hungarian National Science Foundation under grant No. 1812Supported in part by NSF grant DMS 86-06225 and AF grant OSR-86-0078     

On the norm of singular integral operator on curves with cusps

Applying the results on singular integral operators with the complex conjugation on curves with cusps (see R. Duduchava, T. Latsabidze, A. Saginashvili, 1992, 1994) the explicit formula for the local norm of the Cauchy singular integral operator on a curve with cusps in a Lebesgue space with an exponential weightL ₂ (, ) is obtained. For curves with angles the formula was already known (see R. Avedanio, N. Krupnik, 1988).     

The maximum dimension of a subspace of nilpotent matrices of index 2

A matrix M is nilpotent of index 2 if M²=0. Let V be a space of nilpotent n×n matrices of index 2 over a field k where and suppose that r is the maximum rank of any matrix in V. The object of this paper is to give an elementary proof of the fact that . We show that the inequality is sharp and construct all such subspaces of maximum dimension. We use the result to find the maximum dimension of spaces of anti-commuting matrices and zero subalgebras of special Jordan Algebras.     

Complements and the Krull-Schmidt Theorem in arbitrary categories

We study direct product decompositions of objects in a finitely complete and cocomplete category with zero object and certain axioms for a coimage factorization of morphisms. Direct productsC=A×B can be characterized by inner properties ofC and its subobjectsA andB. We also show that the Fitting Lemma and the Krull-Schmidt Theorem hold.     

Intersections of maximal staircase sets

A compact set is staircase connected if every two points a, b ∈ S can be connected by an x-monotone and y-monotone polygonal path with sides parallel to the coordinate axes. In [5] we have introduced the concepts of staircase k-stars and kernels. In this paper we prove that if the staircase k-kernel is not empty, then it can be expressed as the intersection of a covering family of maximal subsets of staircase diameter k of S.      

Simplifying equations in Arabic algebra

Historians have always seen jabr (restoration) and muqābala (confrontation) as technical terms for specific operations in Arabic algebra. This assumption clashes with the fact that the words were used in a variety of contexts. By examining the different uses of jabr, muqābala, ikmāl (completion), and radd (returning) in the worked-out problems of several medieval mathematics texts, we show that they are really nontechnical words used to name the immediate goals of particular steps. We also find that the phrase al-jabr wa'l-muqābala was first used within the solutions of problems to mean al-jabr and/or al-muqābala, and from there it became the name of the art of algebra.     

Staircase kernels for orthogonald-polytopes

LetS be a finite union of boxes inR ^d . Forx inS, defineA _x ={yx is clearly visible fromy via staircase paths inS}, and let KerS denote the staircase kernel ofS. Then KerS={A _x x is a point of local nonconvexity ofS}. A similar result holds with clearly visible replaced by visible and points of local nonconvexity ofS replaced by boundary points ofS.Supported in part by NSF grant DMS-9207019.     

On the matrices of given rank in a large subspace

Let V be a linear subspace of M_n,p(K) with codimension lesser than n, where K is an arbitrary field and n?p. In a recent work of the author, it was proven that V is always spanned by its rank p matrices unless n=p=2 and K?F₂. Here, we give a sufficient condition on codim V for V to be spanned by its rank r matrices for a given r∈?1,p-1?. This involves a generalization of the Gerstenhaber theorem on linear subspaces of nilpotent matrices.     

Two kinds of Novikov algebras and their realizations

In this paper, we construct two kinds of Novikov algebras, characterize some properties of them and give their realizations by triangle functions, respectively.     

CLASSES OF CONNECTIONS ASSOCIATED WITH DISTRIBUTIONS ON PRODUCT SPACES

Abstract

It is assumed that an n-dimensional distribution is given on an (n+M)-dimensional product space. The latter is endowed with a connection, by means of which the covariant exterior derivatives of the functions that specify the distribution are defined. It is postulated that the connection be such that these derivatives vanish identically. This gives rise to an analysis of the integrability conditions associated with the distribution in terms of appropriate torsion and curvature 2-forms. A further specialization of the connection leads to Edelen's theory [1] of distributions on spaces of fibres.     

Congruence lattices of locally finite algebras

It is shown that there exist algebraic lattices that cannot be represented as the congruence lattice of a locally finite algebra. Received August 11, 2004; accepted in final form January 6, 2005.     

An algorithm for associative bilinear forms

We establish the equivalence between the problem of existence of associative bilinear forms and the problem of solvability in commutative power-associative finite-dimensional nil-algebras. We use the tensor product to find sufficient and necessary conditions to assure the existence of associative bilinear forms in an algebra A. The result gives us an algorithm to describe the space of associative bilinear forms for an algebra when its constants of structure γ_i,j,k for i,j,k=1,…,n are known.