On the cover time of random walks on graphs

This article deals with random walks on arbitrary graphs. We consider the cover time of finite graphs. That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once. We establish an upper bound ofO(n ²) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of (n logn) for the expected cover time for trees. We present examples showing all our bounds to be tight.Mike Saks was supported by NSF-DMS87-03541 and by AFOSR-0271. Jeff Kahn was supported by MCS-83-01867 and by AFOSR-0271.     

A Dynamic location problem for graphs

We introduce a class of optimization problems, calleddynamic location problems, involving the processing of requests that occur sequentially at the nodes of a graphG. This leads to the definition of a new parameter of graphs, called the window indexWX(G), that measures how large a window into the future is needed to solve every instance of the dynamic location problem onG optimally on-line. We completely characterize this parameter:WX(G)k if and only ifG is a weak retract of a product of complete graphs of size at mostk. As a byproduct, we obtain two (polynomially recognizable) structural characterizations of such graphs, extending a result of Bandelt.     

On the Local Structure of Generalized Elliptic Pseudodifferential Operators and the Gauss Method

In this paper, we study the class of operators whose dominant parts admit elliptic factorizations in a conic domain U from T'(X), i.e., they can be represented as the composition of diagonal operators and operators of zero order, elliptic in U. We denote this class by ELF ^°(U). It arises in the process of microlocalization of the notion of generalized ellipticity. We analyze the problem concerning the simplest factorization of the dominant part of the operator BAC, where A EFL ^°(U) and the operators B and C are chosen from the class EL ^°(U_q)(elliptic operators in a neighborhood U _q of the point q U). For A from the subclass denoted by BEL ^°(U), the dominant part BAC can be reduced to a single diagonal operator. In general, for operators from the class EFL ^°(U) such a representation may not exist. However, there always exists a representation whose dominant part BAC is composed of a finite number of diagonal operators, permutation matrices, and lower triangular matrices having units on the main diagonal. We prove this fact by using an analog of the Gauss method, which we introduce for the algebra of pseudodifferential operators. Bibliography: 5 titles.     

A parallel search game

We answer in negative a question of Gál and Miltersen [Proc 30th Int Coll Automata, Languages, and Programming (ICALP) 2003, pp. 332–344] about a combinatorial game arising in the study of time‐space trade‐offs for data structures. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Every poset has a central element

It is proved that there exists a constant , such that in every finite partially ordered set there is an element such that the fraction of order ideals containing that element is between δ and 1−δ. It is shown that δ can be taken to be at least (3−log₂ 5)/40.17. This settles a question asked independently by Colburn and Rival, and Rosenthal. The result implies that the information-theoretic lower bound for a certain class of search problems on partially ordered sets is tight up to a multiplicative constant.     

A short proof of the existence of k-saturated partitions of partially ordered sets

The theorem of Greene and Kleitman concerning the existence of k-saturated partitions for any partially ordered set P is proved by applying Dilworth's theorem to the product partial order of P with a chain of length k.     

Effect of alkylresorcinols on curing behaviour of phenol-formaldehyde resol resin

Thermal behaviour of cure-accelerated phenol-formaldehyde (PF) resins was studied using the addition of commercial mixture of water soluble oil shale alkylresorcinols (AR) to PF resin, 5-MR being as model compound. The acceleration effect of AR is based on the promotion of condensation of resin methylol groups and subsequent reaction of released formaldehyde with AR. Commercial PF resins SFŽ-3013VL and SFŽ-3014 from the Estonian factory VKG Resins have been used. The chemical structure of resins was characterised by ¹³C NMR spectroscopy. TG-DTA analysis was carried out using labsys^TM instrument Setaram. By TG-DTA measurements, the shift of exothermic and endothermic peaks and the changes of mass loss rate in the ranges of 1.5–10 g AR/100 g PF resin were studied. The effect of AR on the curing behaviour of PF resins was also followed by gel time. Testing of the plywood when using PF resin with 5 mass% of AR shows that the press time could be reduced by about 15%.     

TG-DTA study of melamine-urea-formaldehyde resins

The thermal behaviour of MUF resins from different suppliers with different content of melamine was studied, along with the ¹³C NMR spectroscopic analysis of resin structure and the testing of particleboards in current production at Estonian PB factory Pärnu Plaaditehas AS. The chemical structure of resins from DMSO-d₆ solutions was analysed by ¹³C NMR spectroscopy on a Bruker AMX500 NMR spectrometer. The melamine level in different MUF resins is compared by the ratios of carbonyl carbon of urea and triazine carbon of melamine in ¹³C NMR spectra. Curing behaviour of MUF resins was studied by stimultaneous TG-DTA techniques on the Labsys? instrument Setaram. The shape of DTA curves characterisises the resin synthesis procedure by the extent of polymerisation of UF and MF components and is in accordance with structural data.