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 .     

Minimum concave cost production system: A further generalization of multi-echelon model

We will consider a concave minimization problem associated with a series production system in which raw material is processed inm consecutive facilities. The products at some facility are either sent to the next facility or stocked in the warehouse. The amount of demand for the final products during periodi, i = 1,,n, are known in advance. Our problem is to minimize the sum of processing, holding and backlogging cost, all of which are assumed to be concave.The origin of this model is the classical economic lot size problem of Wagner and Whitin and was extensively studied by Zangwill. This model is very important from the theoretical as well as practical point of view and this is one of the very rare instances in which polynomial time algorithm has been constructed for concave minimization problems.The purpose of this paper is to extend the model further to the situation in which time lag is associated with processing at each facility. We will propose an efficient O(n ⁴ m) algorithm for this class of problems.     

Senecicannabine,a new pyrrolizidine alkaloid from senecio cannabifolius

The structure of senecicannabine($\text{1}$), a new mcrocyclic pyrrolizidine alkaloid isolated from $\text{Senecio cannabifolius}$, has been established by chemical study and X-ray analysis.     

Substituent effects on the kinetics for the chemiluminescence reaction of 6-arylimidazo[1,2-a]pyrazin-3(7H)-ones (Cypridina luciferin analogues): support for the single electron transfer (SET)-oxygenation mechanism with triplet molecular oxygen

Kinetics of chemiluminescence reactions of 2-methyl-6-phenylimidazo[1,2-a]pyrazin-3(7H)-one (1c, Cypridina luciferin analogue) and substituent effects of the 6-aryl group of derivatives 1 strongly suggest that the rate-determining step is a single electron transfer from an anion derived from 1 to a triplet molecular oxygen (O₂) in the oxygenation process.     

Solubility of α-amino acids in water under high pressure: glycine, -alanine, -valine, -leucine, and -isoleucine

The solubility of glycine, -alanine, -valine, -leucine, and -isoleucine in water was measured at 298.15 K and pressures up to 400 MPa. The standard deviation of the logarithm of the solubility is 0.001–0.003, equal to or better than the accuracy of atmospheric pressure measurement in the literature (0.001–0.05). A variety of solubility phenomena were observed. The solubility of glycine decreased with increasing pressure, whereas that of -alanine increased. The -valine and -isoleucine have a solubility maximum at around 100 MPa, and -leucine seems to exhibit a solid-phase phase transition at around 200 MPa. Pressure coefficient of the solubilities at 0.10 MPa is compared with that thermodynamically estimated in reference to aqueous density measurements of glycine and -alanine at 298.15 K and 0.10 MPa, supporting a reliability of our high-pressure measurements.     

Synthesis of Molecular Wires of Linear and Branched Bis(terpyridine)‐Complex Oligomers and Electrochemical Observation of Through‐Bond Redox Conduction

Films of linear and branched oligomer wires of Fe(tpy)₂ (tpy=2,2′:6′,2′′‐terpyridine) were constructed on a gold‐electrode surface by the interfacial stepwise coordination method, in which a surface‐anchoring ligand, (tpy? C₆H₄N?NC₆H₄? S)₂ ( 1 ), two bridging ligands, 1,4‐(tpy)₂C₆H₄ ( 3 ) and 1,3,5‐(C?C? tpy)₃C₆H₃ ( 4 ), and metal ions were used. The quantitative complexation of the ligands and Fe^II ions was monitored by electrochemical measurements in up to eight complexation cycles for linear oligomers of 3 and in up to four cycles for branched oligomers of 4 . STM observation of branched oligomers at low surface coverage showed an even distribution of nanodots of uniform size and shape, which suggests the quantitative formation of dendritic structures. The electron‐transport mechanism and kinetics for the redox reaction of the films of linear and branched oligomer wires were analyzed by potential‐step chronoamperometry (PSCA). The unique current‐versus‐time behavior observed under all conditions indicates that electron conduction occurs not by diffusional motion but by successive electron hopping between neighboring redox sites within a molecular wire. Redox conduction in a single molecular wire in a redox‐polymer film has not been reported previously. The analysis provided the rate constant for electron transfer between the electrode and the nearest redox‐complex moiety, k₁ (s^?1), as well as that for intrawire electron transfer between neighboring redox‐complex moieties, k₂ (cm² mol^?1 s^?1). The strong effect of the electrolyte concentration on both k₁ and k₂ indicates that the counterion motion limits the electron‐hopping rate at lower electrolyte concentrations. Analysis of the dependence of k₁ and k₂ on the potential gave intrinsic kinetic parameters without overpotential effects: k₁⁰=110 s^?1, k₂⁰=2.6×10¹² cm² mol^?1 s^?1 for [n Fe 3 ], and k₁⁰=100 s^?1, k₂⁰=4.1×10¹¹ cm² mol^?1 s^?1 for [n Fe 4 ] (n=number of complexation cycles).     

Structure of Sol-Gel-Derived Sodium Dititanate Glass

The sodium dititanate, Na₂O·2TiO₂ glass was prepared by the sol-gel method. The structure of the glass, especially local environment of Ti⁴⁺ ions was examined using X-ray diffraction and X-ray absorption fine structure (XAFS) analyses, and was compared with that of the melt-derived glass with the same composition. It was found that Ti⁴⁺ ions are rather in five-fold coordination state, forming TiO₅ pyramids with one doubly bonded Ti=O in the gel-glass, while they were in lower coordination state or four-fold coordination in the melt-derived glass.