Finite Abelian Groups of K3 Surfaces with Smooth Quotient

The quotient space of a K3 surface by a finite group is an Enriques surface or a rational surface if it is smooth. Finite groups where the quotient space are Enriques surfaces are known. In this paper, by analyzing effective divisors on smooth rational surfaces, the author will study finite groups which act faithfully on K3 surfaces such that the quotient space are smooth. In particular, he will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian co...     

Partitions of groups into absolutely dense subsets

It is proved that any infinite Abelian group with finitely many elements of order two can be partitioned into two subsets that are dense in any nondiscrete group topology, and hence contain no cosets of infinite subgroups. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 706–711, May, 2000.     

On the covering relation in the interpretability lattice of equational theories

We prove that the chapter in the interpretability lattice represented by the equational theory of Boolean algebras has a unique cover in the lattice. We conjecture that, among chapters represented by equational theories of two-element algebras, this is the only one to have a cover. Also, we prove that the chapter represented by the equational theory of Abelian groups has no cover.Presented by W. Taylor.Research supported by a Ulam Research Professorship at the University of Colorado and by NSF Grant DMS 89 04014.     

Noetherian rings of polynomial functions on Abelian groups

Polynomial functions on Abelian groups play a basic role in the theory of functional equations and spectral analysis. In this paper we investigate the ring-structure of polynomial functions on topological Abelian groups. We show that polynomial functions form a Noetherian ring if and only if the linear space of continuous homomorphisms of the group into the additive group of complex numbers is finite dimensional. In the case of discrete Abelian groups this is equivalent to the presence of spectral synthesis.     

Invariants of modular groups

For a faithful linear representation of a finite group G over a field of characteristic p, we study the ring of invariants. We especially study the polynomial and Cohen–Macaulay properties of the invariant ring. We first show that certain quotient rings of the invariant ring are polynomial rings by which we prove that the Hilbert ideal conjecture is true for a class of groups. In particular, we prove that the conjecture is true for vector invariant rings of Abelian reflection p-groups. Then we study the relationships between the invariant ring of G and that of a subgroup of G. Finally, we study the invariant rings of affine groups and show that, over a finite field, if an affine group contains all translations then the invariant ring is isomorphic to the invariant ring of a linear group.     

N-空间和度量空间的mssc-映象

本文用度量空间的mssc-映象给出了N-空间一些刻画,证明了空间X是N-空间当且仅当X是度量空间的序列覆盖(序列商)mssc-映象,肯定地回答了关于N-空间的一个猜想。     

Cubic Bridgeless Graphs and Braces

There are many long-standing open problems on cubic bridgeless graphs, for instance, Jaeger’s directed cycle double cover conjecture. On the other hand, many structural properties of braces have been recently discovered. In this work, we bijectively map the cubic bridgeless graphs to braces which we call the hexagon graphs, and explore the structure of hexagon graphs. We show that hexagon graphs are braces that can be generated from the ladder on 8 vertices using two types of McCuaig’s augmentations. In addition, we present a reformulation of Jaeger’s directed cycle double cover conjecture in the class of hexagon graphs.     

Parallel submanifolds of complex projective space and their normal holonomy

The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal holonomy approach. Indeed, we explain how these submanifolds can be regarded as the unique complex orbits of the (projectivized) isotropy representation of an irreducible Hermitian symmetric space. Moreover, we show how these important submanifolds are related to other areas of mathematics and theoretical physics. Finally, we state a conjecture about the normal holonomy group of a complete and full complex submanifold of the complex projective space. Research partially supported by GNSAGA (INdAM) and MIUR of Italy.     

Positive Ground States for the Fractional Klein–Gordon–Maxwell System with Critical Exponents

Mediterranean Journal of Mathematics - Let G be a locally compact Abelian (LCA) group which possesses a covering family. We define an atomic Hardy space $$H^1(G)$$ and $$\textrm{BMO}(G)$$ of...     

Connected finite loop spaces with maximal tori

Finite loop spaces are a generalization of compact Lie groups. However, they do not enjoy all of the nice properties of compact Lie groups. For example, having a maximal torus is a quite distinguished property. Actually, an old conjecture, due to Wilkerson, says that every connected finite loop space with a maximal torus is equivalent to a compact connected Lie group. We give some more evidence for this conjecture by showing that the associated action of the Weyl group on the maximal torus always represents the Weyl group as a crystallographic group. We also develop the notion of normalizers of maximal tori for connected finite loop spaces, and prove for a large class of connected finite loop spaces that a connected finite loop space with maximal torus is equivalent to a compact connected Lie group if it has the right normalizer of the maximal torus. Actually, in the cases under consideration the information about the Weyl group is sufficient to give the answer. All this is done by first studying the analogous local problems.

     

Induced orbit data and induced unitary representations for complex groups

We use induced orbit covers to define induced orbit data.By studying the space of regular functions on orbit cover,we know that the induced representation has close connection with the induced orbit datum under the meaning of Vogan’s conjecture.Therefore,when verifying Vogan’s conjecture,many cases can be reduced to the case of rigid orbit data.     

Zhang's conjecture and squares of Abelian surfaces

We give in this Note some squares of Abelian surfaces that are counterexamples to a conjecture formulated by Zhang about the intersection of subvarieties and preperiodic points.     

A note on finite-sheeted covering maps from 2-dimensional compact Abelian groups

We consider finite-sheeted covering maps from 2-dimensional compact connected abelian groups to Klein bottle weak solenoidal spaces, metric continua which are not groups. We show that whenever a group covers a Klein bottle weak solenoidal space it covers groups as well, moreover it covers the product of two solenoids. The converse is not true, we give an example of group which covers groups with any finite number of sheets, but does not cover any Klein bottle weak solenoidal space.     

On possible counterexamples to Negami's planar cover conjecture

A simple graph H is a cover of a graph G if there exists a mapping φ from H onto G such that φ maps the neighbors of every vertex υ in H bijectively to the neighbors of φ (υ) in G . Negami conjectured in 1986 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. The conjecture is still open. It follows from the results of Archdeacon, Fellows, Negami, and the first author that the conjecture holds as long as the graph K _1,2,2,2 has no finite planar cover. However, those results seem to say little about counterexamples if the conjecture was not true. We show that there are, up to obvious constructions, at most 16 possible counterexamples to Negami's conjecture. Moreover, we exhibit a finite list of sets of graphs such that the set of excluded minors for the property of having finite planar cover is one of the sets in our list. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 183–206, 2004     

Red-blue covering problems and the consecutive ones property

Set Cover problems are of core importance in many applications. In recent research, the “red-blue variants” where blue elements all need to be covered whereas red elements add further constraints on the optimality of a covering have received considerable interest. Application scenarios range from data mining to interference reduction in cellular networks. As a rule, these problem variants are computationally at least as hard as the original set cover problem. In this work we investigate whether and how the well-known consecutive ones property, restricting the structure of the input sets, makes the red-blue covering problems feasible. We explore a sharp border between polynomial-time solvability and NP-hardness for these problems.     

On the volume of the convex hull of two convex bodies

In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean $n$ -space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are translates, or reflected copies of each other about a common point or a hyperplane containing it. In particular, we give a proof of a related conjecture of Rogers and Shephard.     

Finite images and characters of topological groups

We investigate conditions under which every finite homomorphic image of a topological group is obtained as that of a continuous homomorphism. An Abelian topological group has such property if it is compact or if it has bounded exponent and sufficiently many characters.     

Torsion Abelian afi-Groups

This paper is devoted to the study of Abelian afi-groups. A subgroup A of an Abelian group G is called its absolute ideal if A is an ideal of any ring on G. We will call an Abelian group an afi-group if all of its absolute ideals are fully invariant subgroups. In this paper, we will describe afi-groups in the class of fully transitive torsion groups (in particular, separable torsion groups) and divisible torsion groups.     

On Finite Quantum Groups at −1

This is a contribution to the classification of finite-dimensional pointed Hopf algebras. We are concerned with the case when the group of group-like elements is Abelian of exponent 2. We attach to such a pointed Hopf algebra a generalized simply-laced Cartan matrix; we conjecture that the Hopf algebra is finite-dimensional if and only if the Cartan matrix is of finite type. We prove the conjecture for the types A_n and A_n⁽¹⁾. We obtain the classification of all possible Hopf algebras with Cartan matrix A_n. We use the lifting method developed by Hans-Jürgen Schneider and the first-named author. Presented by S. MontgomeryMathematics Subject Classifications (2000) Primary: 17B37; secondary: 16W30.This work was partially supported by CONICET, Agencia Córdoba Ciencia – CONICOR, FOMEC and Secyt (UNC).     

Covering finite fields with cosets of subspaces

If V is a vector space over a finite field F, the minimum number of cosets of k-dimensional subspaces of V required to cover the nonzero points of V is established. This is done by first regarding V as a field extension of F and then associating with each coset L of a subspace of V a polynomial whose roots are the points of L. A covering with cosets is then equivalent to a product of such polynomials having the minimal polynomial satisfied by all nonzero points of V as a factor.