A bound on the total size of a cut cover

A cycle cover (cut cover) of a graph G is a collection of cycles (cuts) of G that covers every edge of G at least once. The total size of a cycle cover (cut cover) is the sum of the number of edges of the cycles (cuts) in the cover.We discuss several results for cycle covers and the corresponding results for cut covers. Our main result is that every connected graph on n vertices and e edges has a cut cover of total size at most 2e-n+1 with equality precisely when every block of the graph is an odd cycle or a complete graph (other than K₄ or K₈). This corresponds to the result of Fan [J. Combin. Theory Ser. B 74 (1998) 353-367] that every graph without cut-edges has a cycle cover of total size at most e+n-1.     

The complexity of minimum difference cover

The complexity of searching minimum difference covers, both in Z⁺ and in Z_n, is studied. We prove that these two optimization problems are NP-hard. To obtain this result, we characterize those sets—called extrema—having themselves plus zero as minimum difference cover. Such a combinatorial characterization enables us to show that testing whether sets are not extrema, both in Z⁺ and in Z_n, is NP-complete. However, for these two decision problems we exhibit pseudo-polynomial time algorithms.     

Diagrams,orientations, and varieties

One of the central problems in the theory of ordered sets is to describe the orientations of the covering graph of an ordered set. We show that the particular operation called inversion, together with the classical constructions of retraction and product, provide a context for the classification of all such orientations. AMS subject classifications (1980). 06A10, 05C75, 05C20.     

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.     

Comparison of the Dynamics of Water Molecules in Crystals of Proton-Ordered Ice IX and Its Proton-Disordered Analog III

A computer simulation of proton-ordered ice IX and its proton-disordered analog III (768 molecules, 90 K) was carried out by the molecular dynamics method using Poltev–Malenkov's potential. For ice IX, the differences in the dynamic characteristics of molecules with O(1) and O(2) are much wider than those in the case of ice III. The libration spectrum of ice IX has a number of distinct acute peaks, and the spectrum of ice III is strongly smoothed. These peculiarities are explained by the proton ordering of ice IX and disordering of ice III. The latter is responsible for the great differences in the short- and especially long-range environment of water molecules in ice crystals and hence for the presence of many molecules with different dynamic characteristics. Thus averaging over a large number of different vibrational spectra of molecules leads to a smoothed total spectrum in the case of the proton-disordered crystal modification of ice.     

STUDIES ON TUNG OIL COATED REGENERATED CELLULOSE FILMS WITH WATERRESISTANCE

Regenerated cellulose films with water-resistance were obtained by an improved method ofpreparing cellulose cuoxam solution from pulps of agricultural wastes (linters, wheat straw, reedand Bamao). Experimental results showed that the mechanical properties of both the dry. and wetfilms were excellent. Data from IR, SEM and tensile strength measurements implied that thesignificant improvement of water-resistance of the films was due to the cohesion between the thinTung oil covers with hydrophobicity and the regenerated cellulose films. The films werecompletely biodegraded after being buried in soil for 100 days. The transmittance of the filmsderived from linter and reed in visible band range were 80-90%.     

The crystallographic structure of the natural air-hydrate in Greenland dye-3 deep ice core

We have carried out X-ray diffraction studies on single crystals of natural air-hydrate in deep ice cores recovered at Dye-3 Greenland. Integrated intensities for 470 diffracting planes were measured by an automated four-circle diffractometer. The space group determined is cubicFd3m and the lattice constant is 17.21(3) Å. These results indicate that the crystallographic structure is the Stackelberg's structure II, in contrast to the previously anticipated structure. This finding agrees with the recent results on the synthetic air-hydrate by Davidsonet al. It was also found by difference Fourier synthesis for guest molecules that electron density in a 16-hedral cage has multiple maxima displaced from the center of the cage while that in the 12-hedron was approximately spherical.Dedicated to Dr D. W. Davidson in honor of his great contributions to the sciences of inclusion phenomena.     

Covering with blocks in the non-symmetric case

Considering an infinite string of i.i.d. random letters drawn from a finite alphabet we define the cover timeW _n as the number of random letters needed until each pattern of lenghtn appears at least once as a substring. Sharp weak and a.s. limit results onW _n are known in the symmetric case, i.e., when the random letters are uniformly distributed over the alphabet. In this paper we determine the limit distribution ofW _n in the nonsymmetric case asn. Generalizations in terms of point processes are also proved.Dedicated to Endre Csáki on his 60th birthday.     

THE COMPARISON OF LANZHOU LOESS PROFILE WITH VOSTOK ICE CORE IN ANTARCTICA OVER THE LAST GLACIATION CYCLE

Loess near Lanzhou in the late Pleistocene is very sensitive to climatic fluctuations. The Beiyuan terrace profile in Linxia City, of which the curve of susceptibility tallies with the trend of isotopic curves of Vostok 2083 m ice core in Antarctica, is 35m thick. There are five layers of paleosols under Malan loess(L_1). Upper three layers (S_(1-a),S(1-b)and S_(1-c) correspond to three warm stages in last interglaciation, Interstadial of Last Glacial (C stage in Antarctic ice core)was clearly recorded in the Beiyuan profile, in which three layers of paleosols and two of loess were formed. The lowest section of the profile belongs to penultimate glaciation, in which fossils of cold-drought-resistant mammal and mollusc have been discovered.     

高分辨电感耦合等离子体质谱仪测定中亚山地冰川雪冰中超痕量元素

在超净实验条件下,利用高分辨电感耦合等离子体质谱仪直接测定雪冰样品中浓度在1ng/L~100μg/L之间的多种超痕量元素Al、Fe、Mn、Co、Cu、Zn、Sr、Sb、Cd、CsBa、Tl、Pb和Bi的实验方法进展。在测定过程中灵敏度的变化可通过监测ArAr质谱峰的变化进行校正,并建立了数学校正公式。以1%HNO3为空白溶液获得了各元素的检出限(ng/L)分别为:Al40、Mn6、Fe40、Co2.74、Cu9.6、Zn20、Sr0.03、Sb0.15、Cs0.04、Ba0.4、Tl0.07、Pb0.8和Bi0.05。利用标准参考物质SLRS-4对测试方法的准确度进行了评价。研究了在硝酸含量分别为0.5%、1%、2%和4%时雪冰中超痕量元素质谱信号强度的变化特点。测定结果发现,对于污化层样品,当硝酸的含量为1%时痕量元素的质谱信号强度最大;对于非污化层样品,痕量元素的质谱信号强度在硝酸含量为0.5%时较大,但与硝酸含量为1%时的质谱信号强度相比变化率小于10%。结合中亚山地冰川雪冰中痕量元素浓度差别大的特点,确定硝酸在样品中的含量为1%为合适的酸度。并对卓奥友峰和珠穆朗玛峰雪坑样品进行了分析方法的重复性研究。