Finite sheeted covering maps over 2-dimensional connected, compact Abelian groups

Let G be a 2-dimensional connected, compact Abelian group and s be a positive integer. We prove that a classification of s-sheeted covering maps over G is reduced to a classification of s-index torsionfree supergroups of the Pontrjagin dual . Using group theoretic results from earlier paper we demonstrate its consequences. We also prove that for a connected compact group Y:

(1)

Every finite-sheeted covering map from a connected space over Y is equivalent to a covering homomorphism from a compact, connected group.

(2)

If two finite-sheeted covering homomorphisms over Y are equivalent, then they are equivalent as topological homomorphisms.

     

Flag transitive planes of dimension four over GF(3)

Two non desarguesian flag transitive planes of order 3⁴ whose Kernel is GF(3) are constructed. These planes are distinct from the planes of the same order contained in the class constructed by Narayana Rao M. L. (Proceedings of American Mathematical Society 39 (1973) 51–56) and Ebert, G.L. and Baker, R. (Enumeration of two dimensional Flag-Transitive planes, Algebras, Groups and Geometries 3 (1985) 248–257). The Flag Transitive group modulo the scalar collineations of these planes is generated by two elements and is of order 328.     

Matroids algebraic overF(t) are algebraic overF

In his thesis [3] M. J. Piff conjectured that a matroid, which is algebraic over a fieldFit) witht transcendent overF, must be algebraic overF. Two proofs have appeared, one by Shameeva [5] and another one by the author [2], but both are unsatisfactory. In this paper I will settle conjecture by applying a theorem of Seidenberg.     

Densely ball remotal subspaces of C(K)

We call a subspace Y of a Banach space X a DBR subspace if its unit ball By admits farthest points from a dense set of points of X. In this paper, we study DBR subspaces of C(K). In the process, we study boundaries, in particular, the Choquet boundary of any general subspace of C(K). An infinite compact Hausdorff space K has no isolated point if and only if any finite co-dimensional subspace, in particular, any hyperplane is DBR in C(K). As a consequence, we show that a Banach space X is reflexive if and only if X is a DBR subspace of any superspace. As applications, we prove that any M-ideal or any closed *-subalgebra of C(K) is a DBR subspace of C(K). It follows that C(K) is ball remotal in C(K)^**.     

Fermat-type configurations of lines in P3 and the containment problem

The purpose of this note is to show a new series of examples of homogeneous ideals I in $\mathbb{K}[x,y,z,w]$ for which the containment $I^{(3)}?I^2$ fails. These ideals are supported on certain arrangements of lines in ${\mathbb{P}}^3$, which resemble Fermat configurations of points in ${\mathbb{P}}^2$, see [14]. All examples exhibiting the failure of the containment $I^{(3)}?I^2$ constructed so far have been supported on points or cones over configurations of points. Apart from providing new counterexamples, these ideals seem quite interesting on their own.     

Growth functions for semi-regular tilings of the hyperbolic plane

This paper studies the growth function, with respect to the generating set of edge identifications, of a surface group with fundamental domainD in the hyperbolic plane ann-gon whose angles alternate between /p and /q. The possibilities ofn,p andq for which a torsion-free surface group can have such a fundamental polygon are classified, and the growth functions are computed. Conditions are given for which the denominator of the growth function is a product of cyclotomic polynomials and a Salem polynomial.This work was supported in part by NSF Research Grants.     

The Whitehead group of (almost) extra-special p-groups with p odd

Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P, the subgroup $C{l}_1(\mathbb{Z}P)$ of $S{K}_1(\mathbb{Z}P)$, in terms of a genetic basis of P. We also introduce a deflation map $C{l}_1(\mathbb{Z}P)\to C{l}_1(\mathbb{Z}(P/N))$, for a normal subgroup N of P, and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of $S{K}_1(\mathbb{Z}P)$, when P is an elementary abelian p-group.     

The t#-property for integral domains

Generalizing work of Gilmer and Heinzer, we define a t#-domain to be a domain R in which for any two distinct subsets and of the set of maximal t-ideals of R. We provide characterizations of these domains, and we show that polynomial rings over t#-domains are again t#-domains. Finally, we study overrings of t#-domains.     

Sections of zero dimensional ideals over a Noetherian ring

Let A be a commutative Noetherian ring and be an ideal containing a monic polynomial such that A[T]/I is zero dimensional. Suppose the conormal module I/I ² is generated by r elements over A[T]/I. Then a set of r generators of can be lifted to a r generating set of I. A part of this work is done at the Abdus Salam, International Centre for Theoretical Physics, Trieste, Italy. Received: 12 March 2006 Revised: 29 January 2007     

Some (2,n) combinatorial Stein groupoids

A Stein groupoid (quasigroup) is a groupoid (quasigroup) satisfying the identityx(xy)=yx. We show that, for certain two variable identities, the variety of Stein groupoids defined by any one of these identities has the properties that every groupoid in the variety is a quasigroup and that the free groupoid generated by two elements is of finite (small) order which we determine. These results provide characterizations of some Stein quasigroups of small order and we give some further characterizations involving other identities.     

Convergence of block iterative methods for linear systems with generalized H-matrices

The paper studies the convergence of some block iterative methods for the solution of linear systems when the coefficient matrices are generalized H$H$-matrices. A truth is found that the class of conjugate generalized H$H$-matrices is a subclass of the class of generalized H$H$-matrices and the convergence results of R. Nabben [R. Nabben, On a class of matrices which arises in the numerical solution of Euler equations, Numer. Math. 63 (1992) 411–431] are then extended to the class of generalized H$H$-matrices. Furthermore, the convergence of the block AOR iterative method for linear systems with generalized H$H$-matrices is established and some properties of special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper.     

Lelek maps and n-dimensional maps from compacta to polyhedra

For a natural number m?0, a map from a compactum X to a metric space Y is an m-dimensional Lelek map if the union of all non-trivial continua contained in the fibers of f is of dimension ?m. In [M. Levin, Certain finite-dimensional maps and their application to hyperspaces, Israel J. Math. 105 (1998) 257-262], Levin proved that in the space C(X,I) of all maps of an n-dimensional compactum X to the unit interval I=[0,1], almost all maps are (n−1)-dimensional Lelek maps. Moreover, he showed that in the space C(X,Ik) of all maps of an n-dimensional compactum X to the k-dimensional cube Ik (k?1), almost all maps are (n−k)-dimensional Lelek maps. In this paper, we generalize Levin's result. For any (separable) metric space Y, we define the piecewise embedding dimension ped(Y) of Y and we prove that in the space C(X,Y) of all maps of an n-dimensional compactum X to a complete metric ANR Y, almost all maps are (n−k)-dimensional Lelek maps, where k=ped(Y). As a corollary, we prove that in the space C(X,Y) of all maps of an n-dimensional compactum X to a Peano curve Y, almost all maps are (n−1)-dimensional Lelek maps and in the space C(X,M) of all maps of an n-dimensional compactum X to a k-dimensional Menger manifold M, almost all maps are (n−k)-dimensional Lelek maps. It is known that k-dimensional Lelek maps are k-dimensional maps for k?0.     

Almost splitting sets in integral domains, II

Let D be an integral domain. A saturated multiplicative subset S of D is an almost splitting set if, for each 0≠d∈D, there exists a positive integer n=n(d) such that dⁿ=st for some s∈S and t∈D which is v-coprime to each element of S. We show that every upper to zero in D[X] contains a primary element if and only if D?{0} is an almost splitting set in D[X], if and only if D is a UMT-domain and Cl(D[X]) is torsion. We also prove that D[X] is an almost GCD-domain if and only if D is an almost GCD-domain and Cl(D[X]) is torsion. Using this result, we construct an integral domain D such that Cl(D) is torsion, but Cl(D[X]) is not torsion.     

Finite projective planes admitting a projective linear group PSL (2, q)

Let S be a projective plane, and let G?Aut(S) and PSL(2, q) ? G ? PΓL(2, q) with q > 3. If G acts point-transitively on S, then q = 7 and S is of order 2.     

Exceptional collections of line bundles on projective homogeneous varieties

We construct new examples of exceptional collections of line bundles on the variety of Borel subgroups of a split semisimple linear algebraic group G$G$ of rank 2 over a field. We exhibit exceptional collections of the expected length for types _A2$A_2$ and _B2=_C2$B_2=C_2$ and prove that no such collection exists for type _G2$G_2$. This settles the question of the existence of full exceptional collections of line bundles on projective homogeneous G$G$-varieties for split linear algebraic groups G$G$ of rank at most 2.     

An Abstract Cogalois Theory for profinite groups

The aim of this paper is to develop an abstract group theoretic framework for the Cogalois Theory of field extensions.     

A pair of spaces of upper semi-continuous maps and continuous maps

For a Tychonoff space X, we use ↓USC(X) and ↓C(X) to denote the families of the regions below all upper semi-continuous maps and of the regions below all continuous maps from X to I=[0,1], respectively. In this paper, we consider the spaces ↓USC(X) and ↓C(X) topologized as subspaces of the hyperspace Cld(X×I) consisting of all non-empty closed sets in X×I endowed with the Vietoris topology. We shall prove that ↓USC(X) is homeomorphic (≈) to the Hilbert cube Q=ω[−1,1] if and only if X is an infinite compact metric space. And we shall prove that (↓USC(X),↓C(X))≈(Q,c₀), where , if and only if ↓C(X)≈c₀ if and only if X is a compact metric space and the set of isolated points is not dense in X.