Strong Subdifferentiability of Convex Functionals and Proximinality

Using strong subdifferentiability of convex functionals, we give a new sufficient condition for proximinality of closed subspaces of finite codimension in a Banach space. We apply this result to the Banach space K(l₂) of compact operators on l₂ and we show that a finite codimensional subspace Y of K(l₂) is strongly proximinal if and only if every linear form which vanishes on Y attains its norm.     

Metrizability of Spaces of ℝ-places of Function Fields of Transcendence Degree 1 Over Real Closed Fields

In this article we discuss the following question “When do different orderings of the rational function field R(X) (where R is a real closed field) induce the same ?-place?” We use this to show that if R contains a dense real closed subfield R′, then the spaces of ?-places of R(X) and R′(X) are homeomorphic. For the function field K = R(X), we prove that its space M(K) of ?-places is metrizible if and only if R contains a countable dense subfield. Moreover, we show that this condition is neccessary for the metrizability of M(F) for any function field F of transcendence degree 1 over R.     

On the codimension of the image of a derivation in a communicative field

Let δ be a derivation in a field K of characteristic 0, with fixed field k. We show that, if K/k is finitely generated, then the codimension, as k-vector space, of K δ + k in K is infinite, while in the infinitely generated case, the said codimension can be any finite non-negative integer.     

One-fibered ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal

Let (R;m) be a 2-dimensional rational singularity with algebraically closed residue field and whose associated graded ring is an integrally closed domain. Göhner has shown that for every prime divisor v of R, there exists a unique one-fibered complete m-primary ideal A _v in R with unique Rees valuation v and such that any complete m-primary ideal with unique Rees valuation v, is a power of A _v . We show that for v ≠ ord _R , A _v is the inverse transform of a simple complete ideal in an immediate quadratic transform of R, if and only if the degree coefficient d(A _v ; v) is 1. We then give a criterion for R to be regular.     

Completeness and compactness for varieties over a local field

This paper shows that for a local field K, a subfield k ⊂ K and a variety X over k, X is complete if and only if for every finite field extension Kʹ | K, the set X(Kʹ) is compact in its strong topology. The author likes to thank Florian Pop, Jakob Stix, Stefan Wewers, Gunther Cornelissen and his own parents for their support. Received: 13 April 2006     

Embedding Problems of Division Algebras

A finite group G is called admissible over a given field if there exists a central division algebra that contains a G-Galois field extension as a maximal subfield. We give a definition of embedding problems of division algebras that extends both the notion of embedding problems of fields as in classical Galois theory, and the question which finite groups are admissible over a field. In a recent work by Harbater, Hartmann, and Krashen, all admissible groups over function fields of curves over complete discretely valued fields with algebraically closed residue field of characteristic zero have been characterized. We show that also certain embedding problems of division algebras over such a field can be solved for admissible groups.     

A small generating set for the 3‐BD closed set of block sizes ≥ 6

Let v, k be positive integers and k ≥ 3, then K_k = : {v: v ≥ k} is a 3‐BD closed set. Two finite generating sets of 3‐BD closed sets K₄ and K₅ are obtained by H. Hanani [5] and Qiurong Wu [12] respectively. In this article we show that if v ≥ 6, then v ∈ B₃(K,1), where K = {6,7,…,41,45,46,47,51,52,53,83,84}\{22,26}; that is, we show that K is a generating set for K₆. Finally we show that v ∈ B₃(6,20) for all v ∈ K\{35,39,40,45}. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 128–136, 2008     

A short note on groups in separably closed valued fields

In this note we show that groups with definable generics in a separably closed valued field K of finite imperfection degree can be embedded into groups definable in the algebraic closure of K.     

Locally s‐distance transitive graphs

We give a unified approach to analyzing, for each positive integer s, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally s‐arc transitive graphs of diameter at least s. A graph is in the class if it is connected and if, for each vertex v, the subgroup of automorphisms fixing v acts transitively on the set of vertices at distance i from v, for each i from 1 to s. We prove that this class is closed under forming normal quotients. Several graphs in the class are designated as degenerate, and a nondegenerate graph in the class is called basic if all its nontrivial normal quotients are degenerate. We prove that, for s≥2, a nondegenerate, nonbasic graph in the class is either a complete multipartite graph or a normal cover of a basic graph. We prove further that, apart from the complete bipartite graphs, each basic graph admits a faithful quasiprimitive action on each of its (1 or 2) vertex‐orbits or a biquasiprimitive action. These results invite detailed additional analysis of the basic graphs using the theory of quasiprimitive permutation groups. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:176‐197, 2012     

On double and multiple interval graphs

In this paper we discuss a generalization of the familiar concept of an interval graph that arises naturally in scheduling and allocation problems. We define the interval number of a graph G to be the smallest positive integer t for which there exists a function f which assigns to each vertex u of G a subset f(u) of the real line so that f(u) is the union of t closed intervals of the real line, and distinct vertices u and v in G are adjacent if and only if f(u) and f(v)meet. We show that (1) the interval number of a tree is at most two, and (2) the complete bipartite graph K_{m, n} has interval number ?(mn + 1)/(m + n)?.     

On the equivariant structure of ideals in abelian extensions of local fields (with an appendix by W. Bley)

Let p be an odd rational prime and K a finite extension of \Bbb Q_p {\Bbb Q}_p . We give a complete classification of those finite abelian extensions L/K L/K in which any ideal of the valuation ring of L is free over its associated order in \Bbb Q_p[Gal(L/K)] {\Bbb Q}_p[Gal(L/K)] . In an appendix W. Bley describes an algorithm which can be used to determine the structure of Galois stable ideals in abelian extensions of number fields. The algorithm is applied to give several new and interesting examples.     

Potential semi-stability ofp-adic étale cohomology

We extend the methods of Faltings and Tsuji, and prove that ifK is a field of characteristic 0 with a complete, discrete valuation, and a perfect residue field of characteristicp, then thep-adic étale cohomology of a finite typeK-scheme is potentially semi-stable. We prove a similar result for cohomology with compact support, and for cohomology with support in a closed subspace ofX. We establish a relationship between these cohomology groups, and the de Rham cohomology ofX.     

Expansions of algebraically closed fields in o-minimal structures

We develop a notion of differentiability over an algebraically closed field K of characteristic zero with respect to a maximal real closed subfield R. We work in the context of an o-minimal expansion ? \cal {R} of the field R and obtain many of the standard results in complex analysis in this setting. In doing so we use the topological approach to complex analysis developed by Whyburn and others. We then prove a model theoretic theorem that states that the field R is definable in every proper expansion of the field K all of whose atomic relations are definable in ? \cal {R} . One corollary of this result is the classical theorem of Chow on projective analytic sets.     

Stable range one for rings with central units

Abstract

The purpose of this paper is to give a partial positive answer to a question raised by Khurana et al. as to whether a ring R with stable range one and central units is commutative. We show that this is the case under any of the following additional conditions: R is semiprime or R is one-sided Noetherian or R has unit-stable range 1 or R has classical Krull dimension 0 or R is an algebra over a field K such that K is uncountable and R has only countably many primitive ideals or R is affine and either K has characteristic 0 or has infinite transcendental degree over its prime subfield or is algebraically closed. However, the general question remains open.     

Elementary equivalence and codimension in p-adic fields

We give examples of fields elementarily equivalent to a given finite extension of the p-adic numbers but not containing a subfield of finite codimension elementarily equivalent to the p-adics.     

On point-halfspace graphs

The following definition is motivated by the study of circle orders and their connections to graphs. A graphs G is called a point-halfspace graph (in R ^k) provided one can assign to each vertex v ? (G) a point p _v R ^k and to each edge e ? E(G) a closed halfspace H_e ? R ^k so that v is incident with e if and only if p _v ? H_e. Let H ^k denote the set of point-halfspace graphs (in R ^k). We give complete forbidden subgraph and structural characterizations of the classes H ^k for every k. Surprisingly, these classes are closed under taking minors and we give forbidden minor characterizations as well. © 1996 John Wiley & Sons, Inc.     

The Linear Coordinate Preserving Problem

We prove that every K-endomorphism of a rank 2 polynomial algebra over an algebraically closed field K of positive characteristic taking all linear coordinates to coordinates is an automorphism. We give a new characterization of coordinates of K[t][x, y], where K is an algebraically closed field of any characteristic. We also explore the close connection between coordinates and permutation polynomials of finite fields.