EveryL 1-predual is complemented in a simplex space

We show that everyL ₁-predual space is complemented in a simplex space. This answers a question raised by Lazar and Lindenstrauss.     

On a nonnegative irreducible matrix that is similar to a positive matrix

Let A be an n×n irreducible nonnegative (elementwise) matrix. Borobia and Moro raised the following question: Suppose that every diagonal of A contains a positive entry. Is A similar to a positive matrix? We give an affirmative answer in the case n = 4.     

Finite p-groups not characteristic in any p-group in which they are properly contained

Answering a question raised by Y. Berkovich, we give examples of finite p-groups G with the property that the only finite p-group K with G char K, is G itself. We also prove a theorem stating that every finite p-group is contained in such a group G.     

On the generalized Busemann-Petty problem

The generalized Busemann-Petty problem asks whether the origin-symmetric convex bodies in ℝ ⁿ with a larger volume of all i-dimensional sections necessarily have a larger volume. As proved by Bourgain and Zhang, the answer to this question is negative if i > 3. The problem is still open for i = 2, 3. In this article we prove two specific affirmative answers to the generalized Busemann-Petty problem if the body with a smaller i-dimensional volume belongs to given classes. Our results generalize Zhang’s specific affirmative answer to the generalized Busemann-Petty problem. This work was supported, in part, by the National Natural Science Foundation of China (Grant No. 10671117)     

Local and mean k-Ramsey numbers for complete graphs

This paper establishes that the local k-Ramsey number R(K_m, k — loc) is identical with the mean k-Ramsey number R(K_m, k — mean). This answers part of a question raised by Caro and Tuza. © 1997 John Wiley & Sons, Inc.     

Generic noncommutative surfaces

We study a class of noncommutative surfaces, and their higher dimensional analogs, which come from generic subalgebras of twisted homogeneous coordinate rings of projective space. Such rings provide answers to several open questions in noncommutative projective geometry. Specifically, these rings R are the first known graded algebras over a field k which are noetherian but not strongly noetherian: in other words, R⊗_kB is not noetherian for some choice of commutative noetherian extension ring B. This answers a question of Artin, Small, and Zhang. The rings R are also maximal orders, but they do not satisfy all of the χ conditions of Artin and Zhang. In particular, they satisfy the χ₁ condition but not χ_i for i?2, answering a question of Stafford and Zhang and a question of Stafford and Van den Bergh. Finally, we show that the noncommutative scheme R-proj has finite global dimension.     

Continuous tree-like scales

Answering a question raised by Luis Pereira, we show that a continuous tree-like scale can exist above a supercompact cardinal. We also show that the existence of a continuous tree-like scale at ℵ _ω is consistent with Martin’s Maximum.     

On a problem posed by Orey and Pruitt related to the range of the N-parameter wiener process in R d

Let W ^(N,d) be the N-parameter Wiener process with values in R ^d .It is shown that the range of W^(N,d) is all of R^d when d<2N, answering a question raised earlier by Orey and Pruitt.     

On the problem ofk structure

For everyk and everyp, 1<p<2, we construct a Banach space having ak structure and being of typep. This is an answer to a question raised by W. Davis and J. Lindenstrauss in [1].     

Discrepancies between metric dimension and partition dimension of a connected graph

Let and be graphs where the set of vertices is the set of points of the integer lattice and the set of edges consists of all pairs of vertices whose city block and chessboard distances, respectively, are 1.In this paper it is shown that the partition dimensions of these graphs are 3 and 4, respectively, while their metric dimensions are not finite. Also, for every n3 there exists an induced subgraph of of order 3n-1 with metric dimension n and partition dimension 3. These examples will answer a question raised by Chartrand, Salehi and Zhang. Furthermore, graphs of order n9 having partition dimension n-2 are characterized, thus completing the characterization of graphs of order n having partition dimension 2, n, or n-1 given by Chartrand, Salehi and Zhang. The list of these graphs includes 23 members.     

Multi-symbolic Rees algebras and strong F-regularity

Let I be a divisorial ideal of a strongly F-regular ring A. K.-i. Watanabe raised the question whether the symbolic Rees algebra is Cohen-Macaulay whenever it is Noetherian. We develop the notion of multi-symbolic Rees algebras and use this to show that is indeed Cohen-Macaulay whenever a certain auxiliary ring is finitely generated over A. Received August 10, 1998 / in final form October 18, 1999 / Published online July 20, 2000     

Energy of flows on Z2 percolation clusters

We show that if p>p_c( Z ²), then the unique infinite percolation cluster supports a nonzero flow f with finite q energy for all q>2. This extends the work of Grimmett, Kesten, and Zhang (Probab Th. Relat Fields 96 (1993), 33–44) and Levin and Peres (Preprint, 1998) in dimensions d≥3. As an application of our techniques we exhibit a graph that has transient percolation clusters, but does not admit exponential intersection tails. This answers a question asked by Benjamini, Pemantle, and Peres (Ann Probab.) 26(3) (1998) 1198–1211. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 143–155, 2000     

Intersecting curves and algebraic subgroups: Conjectures and more results

This paper solves in the affirmative, up to dimension , a question raised in an earlier paper by the authors. The equivalence of the problem with a conjecture of Shou-Wu Zhang is proved in the Appendix.

     

An Oscillation Theorem for Groups of Isometries

This paper answers a question raised by Vladimir Pestov. Let G be a non-trivial group of isometries of a complete, separable metric space. Let Ĝ be the semi-group arising from its completion with respect to the left uniformity. Then there is a uniformly continuous function from Ĝ to the unit interval which assumes both extreme points on any non-empty right ideal. Received: July 2006, Revision: March 2007, Accepted: May 2007     

AnF-space with trivial dual where the Krein-Milman theorem holds

We show that in certain non-locally convex Orlicz function spacesL _ϕ with trivial dual every compact convex set is locally convex and hence the Krein-Milman theorem holds. This complements the example constructed by Roberts of a compact convex set without extreme points inL _p (0<p<1) and answers a question raised by Shapiro.     

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.     

<Emphasis Type="Italic">K</Emphasis>th Roots of <Emphasis Type="Italic">p</Emphasis>-Hyponormal Operators are Subscalar Operators of Order 4<Emphasis Type="Italic">k</Emphasis>

In this paper, we consider the special case of the question raised by Halmos (see below). In particular, we show that if T^k is p-hyponormal, then T is a subscalar operator of order 4k. As a corollary, we obtain that if T^k is p-hyponormal and σ(T) has nonempty interior in the plane, then T has a nontrivial invariant subspace.     

Instability of nondiscrete free subgroups in lie groups

We study finitely-generated nondiscrete free subgroups in Lie groups. We address the following question first raised by étienne Ghys: is it always possible to make arbitrarily small perturbations of the generators of the free subgroup in such a way that the new group formed by the perturbed generators be not free? In other words, is it possible to approximate generators of a free subgroup by elements satisfying a nontrivial relation? We prove that the answer to Ghys’ question is positive and generalize this result to certain nonfree subgroups. We also consider the question on the best approximation rate in terms of the minimal length of relation in the approximating group. We give an upper bound on the optimal approximation rate as e ^{- clk} {e^{ - c{l^\kappa }}} , where c > 0 is a constant, l the minimal length of relation and 0.19 < κ < 0.2.     

Partition of unity and density: A counterexample

By providing a counterexample we show that there exists a shift-invariant spaceS generated by a piecewise linear function such that the union of the corresponding scaled spacesS ^h (h>0) is dense inC ₀(R ²) butS does not contain a stable and locally supported partition of unity. This settles a question raised by C. de Boor and R. DeVore a decade ago.     

A note on some chain conditions

We answer a question raised by Othman Echi: Is an E ₁ (resp., a C ₁) ring an E (resp., a C) ring? We construct a C ₁ (thus E ₁) ring which is not an E ₂ (thus not a C ₂) ring. Received: 11 June 2007