A Moore space with a -base which cannot be densely embedded in any Moore space with the Baire property

The author answers a question raised in the literature about twenty five years ago and raised again more recently in Open Problems in Topology, by G. M. Reed, concerning the conjecture that every Moore space with a -discrete -base can be densely embedded in a Moore space having the Baire property. Even though closely related results have made this conjecture seem likely to be true, the author shows that, surprisingly, the conjecture is false.

     

The ?-positive center figure

Based on Gage's notion of a “positive center” of a planar convex set, an ε-positive center figure is defined, constructed, and illustrated through example. The existence of an ε-positive center point, and the convexity of the ε-positive center figure, is conjectured.     

Use of non-adiabatic geometric phase for quantum computing by NMR

Geometric phases have stimulated researchers for its potential applications in many areas of science. One of them is fault-tolerant quantum computation. A preliminary requisite of quantum computation is the implementation of controlled dynamics of qubits. In controlled dynamics, one qubit undergoes coherent evolution and acquires appropriate phase, depending on the state of other qubits. If the evolution is geometric, then the phase acquired depend only on the geometry of the path executed, and is robust against certain types of error. This phenomenon leads to an inherently fault-tolerant quantum computation. Here we suggest a technique of using non-adiabatic geometric phase for quantum computation, using selective excitation. In a two-qubit system, we selectively evolve a suitable subsystem where the control qubit is in state |1, through a closed circuit. By this evolution, the target qubit gains a phase controlled by the state of the control qubit. Using the non-adiabatic geometric phase we demonstrate implementation of Deutsch-Jozsa algorithm and Grover's search algorithm in a two-qubit system.     

B-Splines with Arbitrary Connection Matrices

We consider a space of Chebyshev splines whose left and right derivatives satisfy linear constraints that are given by arbitrary nonsingular connection matrices. We show that for almost all knot sequences such spline spaces have basis functions whose support is equal to the support of the ordinary B-splines with the same knots. Consequently, there are knot insertion and evaluation algorithms analogous to de Boors algorithm for ordinary splines.     

A Classification of Regular Embeddings of Graphs of Order a Product of Two Primes

In this paper, we classify the regular embeddings of arc-transitive simple graphs of order pq for any two primes p and q (not necessarily distinct) into orientable surfaces. Our classification is obtained by direct analysis of the structure of arc-regular subgroups (with cyclic vertex-stabilizers) of the automorphism groups of such graphs. This work is independent of the classification of primitive permutation groups of degree p or degree pq for p q and it is also independent of the classification of the arc-transitive graphs of order pq for p q.     

Subspaces of Normed Riesz Spaces

It will be shown that a normed partially ordered vector space is linearly, norm, and order isomorphic to a subspace of a normed Riesz space if and only if its positive cone is closed and its norm p satisfies p(x)p(y) for all x and y with -yxy. A similar characterization of the subspaces of M-normed Riesz spaces is given. With aid of the first characterization, Krein's lemma on directedness of norm dual spaces can be directly derived from the result for normed Riesz spaces. Further properties of the norms ensuing from the characterization theorem are investigated. Also a generalization of the notion of Riesz norm is studied as an analogue of the r-norm from the theory of spaces of operators. Both classes of norms are used to extend results on spaces of operators between normed Riesz spaces to a setting with partially ordered vector spaces. Finally, a partial characterization of the subspaces of Riesz spaces with Riesz seminorms is given.     

Existence of Entire Solutions of a Singular Semilinear Elliptic Problem

In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang‘s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems.     

Some Limit Theorems in Geometric Processes

Geometric process (GP) was introduced by Lam[4,5], it is defined as a stochastic process {Xn, n = 1, 2,…} for which there exists a real number a > 0, such that {an-1 Xn, n = 1,2, …} forms a renewal process (RP). In this paper, we study some limit theorems in GP. We first derive the Wald equation for GP and then obtain the limit theorems of the age, residual life and the total life at t for a GP. A general limit theorem for Sn with a > 1 is also studied. Furthermore, we make a comparison between GP and RP, including the comparison of their limit distributions of the age, residual life and the total life at t.     

An Asymptotic Ratio in the Complete Binary Tree

Let T _n be the complete binary tree of height n considered as the Hasse-diagram of a poset with its root 1 _n as the maximum element. For a rooted tree T, define two functions counting the embeddings of T into T _n as follows A(n;T)=|{S T _n  : 1 _n ∈S, S≅T}|, and B(n;T)=|{S T _n :1 _n ∉S, S≅T}|. In this paper we investigate the asymptotic behavior of the ratio A(n;T)/B(n;T), and we show that lim  _n→∞[A(n;T)/B(n;T)]=2^ℓ;−1−1, for any tree T with ℓ leaves. This revised version was published online in June 2006 with corrections to the Cover Date.     

Differences of Convex Compact Sets in the Space of Directed Sets. Part I: The Space of Directed Sets

A normed and partially ordered vector space of so-called directed sets is constructed, in which the convex cone of all nonempty convex compact sets in R ⁿ is embedded by a positively linear, order preserving and isometric embedding (with respect to a new metric stronger than the Hausdorff metric and equivalent to the Demyanov one). This space is a Banach and a Riesz space for all dimensions and a Banach lattice for n=1. The directed sets in R ⁿ are parametrized by normal directions and defined recursively with respect to the dimension n by the help of a support function and directed supporting faces of lower dimension prescribing the boundary. The operations (addition, subtraction, scalar multiplication) are defined by acting separately on the support function and recursively on the directed supporting faces. Generalized intervals introduced by Kaucher form the basis of this recursive approach. Visualizations of directed sets will be presented in the second part of the paper.