Note on induced subgraphs of the unit distance graphE n

The unit distance graphE ⁿ is the graph whose vertices are the points in Euclideann-space, and two vertices are adjacent if and only if the distance between them is 1. We prove that for anyn there is a finite bipartite graph which cannot be embedded inE ⁿ as an induced subgraph and that every finite graph with maximum degreed can be embedded inE ^N ,N=(d ³ –d)/2, as an induced subgraph.     

On the dimension to represent a graph by a unit distance graph

It is proved that if a graphG has maximum degreed, then its vertices can be represented by distinct unit vectors inR ^2d so that two vectors are orthogonal if and only if the corresponding vertices are adjacent. As a corollary it follows that if a graph has maximum degreed, then it is isomorphic to a unit distance graph inR ^2d.     

A threshold for the size of random caps to cover a sphere

Consider a unit sphere on which are placed N random spherical caps of area 4p(N). We prove that if % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% qGSbGaaeyAaiaab2gaaaWaaeWaaeaacaWGWbWaaeWaaeaacaWGobaa% caGLOaGaayzkaaGaai4Taiaad6eacaGGVaGaaeiBaiaab+gacaqGNb% Gaaeiiaiaad6eaaiaawIcacaGLPaaacqGH8aapcaaIXaaaaa!454E!\[\overline {{\rm{lim}}} \left( {p\left( N \right)\cdotN/{\rm{log }}N} \right) < 1\], then the probability that the sphere is completely covered by N caps tends to 0 as N , and if % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaaca% qGSbGaaeyAaiaab2gaaaWaaeWaaeaacaWGWbWaaeWaaeaacaWGobaa% caGLOaGaayzkaaGaai4Taiaad6eacaGGVaGaaeiBaiaab+gacaqGNb% Gaaeiiaiaad6eaaiaawIcacaGLPaaacqGH+aGpcaaIXaaaaa!4551!\[\underline {{\rm{lim}}} \left( {p\left( N \right)\cdotN/{\rm{log }}N} \right) > 1\], then for any integer n>0 the probability that each point of the sphere is covered more than n times tends to 1 as N .     

On the contact dimensions of graphs

Every simple graphG=(V, E) can be represented by a family of equal nonoverlapping spheres {S _v :v V} in a Euclidean spaceR ⁿ in such a way thatuv E if and only ifS _u andS _v touch each other. The smallest dimensionn needed to representG in such a way is called the contact dimension ofG and it is denoted by cd(G). We prove that (1) cd(T)<(7.3) log |T| for every treeT, and (2)

Transformation of Cinchona alkaloids into 1-N-oxide derivatives by endophytic Xylaria sp isolated from Cinchona pubescens

The microbial transformation of four Cinchona alkaloids (quinine, quinidine, cinchonidine, and cinchonine) by endophytic fungi isolated from Cinchona pubescens was investigated. The endophytic filamentous fungus Xylaria sp. was found to transform the Cinchona alkaloids into their 1-N-oxide derivatives.     

Temperature-Responsiveness of A-B-A Block Telomers at Solid-Liquid Interfaces

Macroinitiators were prepared by coupling disuccinimidyl ester of 4,4'-azobis(cyanovaleric acid) with poly(N-isopropylacrylamide) (PIPA), which had an amino group at its end. Styrene was telomerized with the initiators in THF. When the styrene content in the A-B-A block telomer obtained (PIPA-b-PSt-b-PIPA) was high, the telomer formed an irreversible aggregation resulting in microspheres, whereas the telomer with a much shorter styrene block could be dispersed monomolecularly. The telomers dispersed in water were aggregated by raising the temperature above 32 degrees C due to a coil-globule transition of PIPA moieties. The PIPA-b-PSt-b-PIPA could be strongly adsorbed to polystyrene (PSt) solid surfaces to form a layer, and the PSt blocks might lay on the PSt surface and the PIPA blocks might direct to the solution phase. The contact angle of air bubbles on the surface of telomer-coated PSt in the air-in-water system was dependent on temperature; that is, with the increase in temperature the contact angle of air bubbles largely decreased and leveled off above the coil-globule transition temperature (Tc). Correspondingly, the amount of protein Concanavalin A adsorbed to the telomer layer deposited on the PSt surface increased gradually with an increase in temperature and leveled off above the Tc. These phenomena were attributed to the changes in hydrophobicity of the telomer layer below and above the Tc. The usefulness of macroinitiators in preparing various kinds of block telomers which have responsiveness to external stimuli was strongly suggested. Copyright 1999 Academic Press.     

DNA topology on an increase in positive writhing number of DNA: conformation changes in the time course of cis-diamminedichloroplatinum(II)-DNA adducts

We show that the topological significance of the gel mobility of cis-diamminedichloroplatinum(II) (DDP)-closed circular DNA (ccDNA) adducts decreases with reaction time, until a point at which it joins relaxed DNA, and that the mobility of the adducts increases again. There is no relationship between the relative length of the adducts and the gel mobility. Although the significance of the decrease of gel mobility is due to the unwinding of cis-DDP-DNA (or trans-DDP-DNA) adducts, the conformational significance of the subsequent increase in mobility is unclear. To elucidate the conformational significance for unwinding of the adducts, we measured the writhing number (Wk) of the adducts using electron microscopy and analyzed the topological states of cis-DDP (or trans-DDP) adducts based on the White rule, Lk=Wk+Tk. Where, Lk and Tk represent the linking and twisting number in the ring, respectively. From the data, we found that the Wk of cis-DDP-ccDNA adducts in comparison with trans-DDP-ccDNA adducts increases from a negative to a positive number with time. This suggests that cis-DDP plays a role in the change of the topological state of ccDNA. In the abstraction of platinum from the adducts with CN- ion, the differences in both topological states may explain why Pt in trans-DDP is abstracted more easily than in cis-DDP. To explain the abstraction of Pt ion, we also discuss the findings based on the thermodynamic cycle in a intermolecular crosslink model Pt(NH3)2(guanine)2(2+)-->Pt(CN)4(2-) using the Pt parametrized PM3 method.     

Biooxidation of (+)-catechin and (-)-epicatechin into 3,4-dihydroxyflavan derivatives by the endophytic fungus Diaporthe sp. isolated from a tea plant

The microbial transformation of (+)-catechin (1) and (-)-epicatechin (2) by endophytic fungi isolated from a tea plant was investigated. It was found that the endophytic filamentous fungus Diaporthe sp. transformed them (1, 2) into the 3,4-cis-dihydroxyflavan derivatives, (+)-(2R,3S,4S)-3,4,5,7,3',4'-hexahydroxyflavan (3) and (-)-(2R,3R,4R)-3,4,5,7,3',4'-hexahydroxyflavan (7), respectively, whereas (-)-catechin (ent-1) and (+)-epicatechin (ent-2) with a 2S-phenyl group resisted the biooxidation.