Discontinuous extensions and almost resolvability

It is proven that in large classes of topological spaces each real-valued continuous function on aG _δ-setG has an extension to the whole space which is continuous exactly at the points ofG. AmongG _δ-spaces this property characterizes the almost resolvable normal spaces.     

On the resolvability of locally connected spaces

We solve a problem of Padmavally about resolvability of locally connected spaces, in the case where the space under consideration is regular.

     

Resolvability vs. almost resolvability

A space X is κ-resolvable (resp. almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets of X).Answering a problem raised by Juhász, Soukup, and Szentmiklóssy, and improving a consistency result of Comfort and Hu, we prove, in ZFC, that for every infinite cardinal κ there is an almost κ²-resolvable but not ω₁-resolvable space of dispersion character κ.     

On the maximal resolvability of monotonically normal spaces

We continue the work started in [6] and show that all monotonically normal (in short: MN) spaces are maximally resolvable if and only if all uniform ultrafilters are maximally decomposable. As a consequence we get that the existence of an MN space which is not maximally resolvable is equi-consistent with the existence of a measurable cardinal. We also show that it is consistent (modulo the consistency of a measurable cardinal) that there is an MN space X with |X| = Δ(X) = ? _ω which is not ω ₁-resolvable. It follows from the results of [6] that this is best possible.     

Tychonoff expansions with prescribed resolvability properties

The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) [3] and to Comfort and García-Ferreira (2001) [5]: (1) Is every ω-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of KID expansion, the authors show that every suitably restricted Tychonoff topological space (X,T) admits a larger Tychonoff topology (that is, an “expansion”) witnessing such failure. Specifically the authors show in ZFC that if (X,T) is a maximally resolvable Tychonoff space with S(X,T)?Δ(X,T)=κ, then (X,T) has Tychonoff expansions U=Ui (1?i?5), with Δ(X,Ui)=Δ(X,T) and S(X,Ui)?Δ(X,Ui), such that (X,Ui) is: (i=1) ω-resolvable but not maximally resolvable; (i=2) [if κ^′ is regular, with S(X,T)?κ^′?κ] τ-resolvable for all τ<κ^′, but not κ^′-resolvable; (i=3) maximally resolvable, but not extraresolvable; (i=4) extraresolvable, but not maximally resolvable; (i=5) maximally resolvable and extraresolvable, but not strongly extraresolvable.     

带可分解性质的幂等拟群大集

具有幂单正交侣的幂等拟群称为可分解的. 具有幂等正交侣的幂等拟群称为几乎可分解的. 若v 元集合上的所有分量互不相同的3-向量能够分拆成互不相交(幂等3-向量除外) 的v-2 个v 阶幂等拟群, 则称之为v 阶幂等拟群大集. 本文使用t-平衡设计(t=2; 3) 的方法给出了可分解幂等拟群大集、几乎可分解幂等拟群大集及可分解对称幂等拟群大集(即可分解高尔夫设计) 的构造方法, 给出了其存在性的若干结果.     

Cyclic resolvability of cyclic Steiner 2-designs

In this article, direct and recursive constructions for a cyclically resolvable cyclic Steiner 2-design are given. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 177–187, 1997     

Borel resolvability of compact spaces and their subspaces

The presence of disjoint dense (Borel) subsets in Tychonoff cubes, Borel subspaces of Tychonoff cubes, and dyadic compacta is examined. Several problems are stated. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 701–712, November, 1998. The author wishes to express his gratitude to V. Shchigolev for fruitful discussion of some questions touched upon in this paper and to the referee for valuable comments and suggestions concerning the style and the content. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01619.     

Global smooth resolvability for a model equations of fluid flow in a pipe

1. IntroductionWe consider here the existence of the global smooth (continuous) solution to the Cauchyproblem for the following quasilinear hyperbolic system modeling fluid flow in a pipe:Pt G., (1.1)Gi (D). P(p)' % = 0, (1 2)where p, G and p = p(p) denote, respectively, the mass density, the momentum density andthe pressure, f is the "Moody" friction factor (see [11 for details).The term --M accounts for the momentum loss due to viscous friction between thefluid and the pipe:all. S…     

H-designs with the properties of resolvability or (1, 2)-resolvability

An H-design is said to be (1, α)-resolvable, if its block set can be partitioned into α-parallel classes, each of which contains every point of the design exactly α times. When α = 1, a (1, α)-resolvable H-design of type g ⁿ is simply called a resolvable H-design and denoted by RH(g ⁿ ), for which the general existence problem has been determined leaving mainly the case of g ≡ 0 (mod 12) open. When α = 2, a (1, 2)-RH(1 ⁿ ) is usually called a (1, 2)-resolvable Steiner quadruple system of order n, for which the existence problem is far from complete. In this paper, we consider these two outstanding problems. First, we prove that an RH(12 ⁿ ) exists for all n ≥ 4 with a small number of possible exceptions. Next, we give a near complete solution to the existence problem of (1, 2)-resolvable H-designs with group size 2. As a consequence, we obtain a near complete solution to the above two open problems.     

About weak resolvability of initial-boundary value problem for Foight model of liquid motion in domain with time-dependent boundary

We prove the existence and uniqueness of a weak solution of the initial-boundary value problem for the Foight model of liquid motion in a domain with time-dependent boundary. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vertex Cover: Further Observations and Further Improvements

Recently, there has been increasing interest and progress in lowering the worst-case time complexity for well-known NP-hard problems, particularly for the Cover problem. In this paper, new properties for the problem are indicated, and several simple and new techniques are introduced, which lead to an improved algorithm of time O(kn + 1.2852^k) for the problem. Our algorithm also induces improvement on previous algorithms for the problem on graphs of small degree.     

Further characterization of ellipsoids

In this paper the following result is proved: Let K be a planar convex body and let γ be a differentiable closed convex curve. If for every point p in the interior of K the set of the midpoints, of all the chords through p, are forming a curve α(p) which is directly homothetic to γ, then K is an ellipse.     

Further gossip problems

n people have distinct bits of information. They can communicate via k-party conference calls. How many such calls are needed to inform everyone of everyone else's information? Let f(n,k) be this minimum number. Then we give a simple proof that f(n,k)= [(n?k)(k?1)]+[nk] for 1?n?k², f(n,k)=2[(n?k)(k?1)] for n>k².In the 2-party case we consider the case in which certain of the calls may permit information flow in only one direction. We show that any 2n-4 call scheme that conveys everone's information to all must contain a 4-cycle, each of whose calls is “two way”, along with some other results. The method follows that of Bumby who first proved the 4-cycle conjecture.