Equivalence of curves with respect to the action of the symplectic group

We establish an equivalence criterion for finite systems of curves with respect to the action of the symplectic group.     

Abelian groups as autocommutator groups

In this article, we establish that every finite abelian group is isomorphic to the autocommutator subgroup of some finite abelian group. Received: 21 September 2007     

Transitive partial actions of groups

J. Kellendonk and M. V. Lawson established that each partial action of a group G on a set Y can be extended to a global action of G on a set Y _G containing a copy of Y. In this paper we classify transitive partial group actions. When G is a topological group acting on a topological space Y partially and transitively we give a condition for having a Hausdorff topology on Y _G such that the global group action of G on Y _G is continuous and the injection Y into Y _G is an open dense equivariant embedding.      

Equations in finite fields with restricted solution sets. II (Algebraic equations)

Generalizing earlier results, it is shown that if are “large” subsets of a finite field F _q , then the equations a + b = cd, resp. ab + 1 = cd can be solved with . Other algebraic equations with solutions restricted to “large” subsets of F _q are also studied. The proofs are based on character sum estimates proved in Part I of the paper. Research partially supported by the Hungarian National Foundation for Scientific Research, Grants No. T 043623, T 043631 and T 049693.     

Two Turán type inequalities

Turán’s book [2], in Section 19.4, refers to the following result of Gábor Halász. Let a ₀, a ₁, ..., a _n−1 be complex numbers such that the roots α ₁, ⋯, α _n of the polynomial x ⁿ + a _n−1 x ⁿ⁻¹ + ⋯ + a ₁ x + a ₀ satisfy min _j Re α _j ≧ 0 and let function Y(t) be a solution of the linear differential equation Y ⁽ⁿ⁾ + a _n−1 Y ⁽ⁿ⁻¹⁾ + ⋯ + a ₁ Y′ + a ₀ Y = 0. Then

Multilinear Exponential Sums in Prime Fields Under Optimal Entropy Condition on the Sources

The main result of this paper is an exponential sum bound in prime fields for multilinear expressions of the type under nearly optimal conditions on . It provides the expected generalization of the well-known inequality for r = 2. We also establish a new result on Gauss sums for multiplicative subgroups H of , obtaining a nontrivial estimate provided . This is a further improvement on [BGK]. Received: May 2007, Revision: October 2007, Accepted: October 2007     

Phenomenologically symmetric local lie groups of transformations of the space <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript>

In this paper we define a phenomenologically symmetric local Lie group of transformations of an arbitrary-dimensional space. We take as a basis the axiom scheme of the theory of physical structures. Phenomenologically symmetric groups of transformations are nondegenerate both with respect to coordinates and to parameters. We obtain a multipoint invariant of this group of transformations and relate it with Ward quasigroups. We define a substructure of a physical structure as a certain phenomenologically symmetric subgroup of transformations. We establish a criterion for the phenomenological symmetry of the Lie group of transformations and prove the uniqueness of a structure with the minimal rank. We also introduce the notion of a phenomenologically symmetric product of physical structures.     

Some remarks on trigonometric sums

Let

On continuation of a topology from a system of generators of a group to a topology on the group

In this paper we establish conditions under which a topology is continued from a subset of a group to a topology on the group coordinated with the group operation.     

Type norms with respect to characters of compact abelian groups and algebraic relations

Let A _n =(a ₁,...,a _n) be a system of characters of a compact abelian group A _n with normalized Haar measure μ and let T be a bounded linear operator from a Banach space X into a Banach space Y. The type norm τ(T| A _n ) of T with respect to A _n is the least constant c such that

Asymptotics of polynomials orthogonal with respect to varying measures

Letd be a finite positive Borel measure on the interval [0, 2] such that >0 almost everywhere; andW _n be a sequence of polynomials, degW _n =n, whose zeros (w _n ,1,,w _n,n lie in [|z|1]. Let d _n <> for each_nN, whered _n =d/|W _n (e ⁱ )|². We consider the table of polynomials _n,m such that for each fixednN the system _n,m,mN, is orthonormal with respect tod _n . If

The operad of finite labeled tournaments

We study the operad of finite labeled tournaments. We describe the structure of suboperads of this operad generated by simple tournaments. We prove that a suboperad generated by a tournament with two vertices (i.e., the operad of finite linearly ordered sets) is isomorphic to the operad of symmetric groups, and a suboperad generated by a simple tournament with more that two vertices is isomorphic to the quotient operad of the free operad with respect to a certain congruence. We obtain this congruence explicitly.     

Polytopes passing through circles

Suppose a convex body wants to pass through a circular hole in a wall. Does its ability to do so depend on the thickness of the wall? In fact in most cases it does, and in this paper we present a sufficient criterion for a polytope to allow an affirmative answer to the question.     

On the computation of l- groups and natural maps

In this paper we compute surgery (resp. splitting) obstruction groups, here calledL-groups, and natural maps for many diagrams of oriented finite (not necessary abelian) 2-groups and homomorphisms which preserve orientations.     

Collapsing monoids consisting of permutations and constants

In this paper we determine all collapsing transformation monoids that contain at least one unary constant operation and whose nonconstant operations are permutations. Furthermore, we find an infinite family of transformation monoids that consist of at least three unary constant operations and some permutations for which the corresponding monoidal intervals are 2-element chains. This research is supported by Hungarian National Foundation for Scientific Research grant nos. T 37877 and K 60148.     

Flat laguerre planes admitting 4-dimensional groups of automorphisms that fix at least two parallel classes

We take a further step toward the classification of all flat (2-dimensional) Laguerre planes of group dimension 4 by determining, up to isomorphism, all such Laguerre planes that admit 4-dimensional groups of automorphisms that fix at least two parallel classes. It is shown that these planes occur among those flat Laguerre planes of generalised shear type that admit a 3-dimesional kernel.     

Isomorphism groups in Clifford algebras

A set of anticommuting multivectors in Clifford algebras can be taken as orthonormal basis set. The Clifford algebra generated by this basis is isomorphic to the original algebra. The non linear transformations between orthonormal basis sets form a group. In the four dimensionnal case six sets of five anticommuting multivectors are found. These sets yield 30 matrices defining basis sets. These matrices are representatives of left cosets, members of these cosets are related by permutation of rows. From the equivalence of all basis sets of multivectors it can be concluded that there is no canonical set of basis vectors in Clifford algebras.     

On some diophantine results related to Euler polynomials

In this paper we prove that there is at most one complex number b for which the shifted Euler polynomial E _n (x) + b has at most two zeros of odd multiplicity. Supported in part, by Grants T48791 and F68872 form HNFSR, and by the Hungarian Academy of Sciences.