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.     

Metal components analysis of metallothionein-III in the brain sections of metallothionein-I and metallothionein-II null mice exposed to mercury vapor with HPLC/ICP-MS

Mercury vapor is effectively absorbed via inhalation and easily passes through the blood–brain barrier; therefore, mercury poisoning with primarily central nervous system symptoms occurs. Metallothionein (MT) is a cysteine-rich metal-binding protein and plays a protective role in heavy-metal poisoning and it is associated with the metabolism of trace elements. Two MT isoforms, MT-I and MT-II, are expressed coordinately in all mammalian tissues, whereas MT-III is a brain-specific member of the MT family. MT-III binds zinc and copper physiologically and is seemed to have important neurophysiological and neuromodulatory functions. The MT functions and metal components of MTs in the brain after mercury vapor exposure are of much interest; however, until now they have not been fully examined. In this study, the influences of the lack of MT-I and MT-II on mercury accumulation in the brain and the changes of zinc and copper concentrations and metal components of MTs were examined after mercury vapor exposure by using MT-I, II null mice and 129/Sv (wild-type) mice as experimental animals. MT-I, II null mice and wild-type mice were exposed to mercury vapor or an air stream for 2 h and were killed 24 h later. The brain was dissected into the cerebral cortex, the cerebellum, and the hippocampus. The concentrations of mercury in each brain section were determined by cold vapor atomic absorption spectrometry. The concentrations of mercury, copper, and zinc in each brain section were determined by inductively coupled plasma mass spectrometry (ICP-MS). The mercury accumulated in brains after mercury vapor exposure for MT-I, II null mice and wild-type mice. The mercury levels of MT-I, II null mice in each brain section were significantly higher than those of wild-type mice after mercury vapor exposure. A significant change of zinc concentrations with the following mercury vapor exposure for MT-I, II null mice was observed only in the cerebellum analyzed by two-way analysis of variance. As for zinc, the copper concentrations only changed significantly in the cerebellum. Metal components of metal-binding proteins of soluble fractions in the brain sections were analyzed by size-exclusion high-performance liquid chromatography (HPLC) connected with ICP-MS. From the results of HPLC/ICP-MS analyses, it was concluded that the mercury components of MT-III and high molecular weight metal-binding proteins in the cerebellum of MT-I, II null mice were much higher than those of wild-type mice. It was suggested that MT-III is associated with the storage of mercury in conditions lacking MT-I, and MT-II. It was also suggested that the physiological role of MT-III and some kind of high molecular weight proteins might be impaired by exposure to mercury vapor and lack of MT-I and MT-II.     

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.     

Structural determination of carvone, a component of spearmint, by means of gas electron diffraction augmented by theoretical calculations

The molecular structure and conformation of carvone, a compound with a minty odor, were investigated by means of gas electron diffraction supported by theoretical calculations. Electron diffraction patterns were recorded by heating the nozzle up to 128 °C to obtain enough scattering intensity. The infrared spectrum was also measured by using an absorption cell with a path length of 10 m. The obtained molecular scattering intensities were analyzed with the aid of theoretical calculations and infrared spectroscopy. It was revealed that the experimental data are well reproduced by assuming that carvone consists of a mixture of three conformers that have the isopropenyl group in the equatorial position and mutually differ in the torsional angle around the single bond connecting the ring and the isopropenyl group. It was also found that the puckering amplitude of the ring of carvone is close to those of menthol and isomenthol, a minty compound and its nonminty isomer. The determined structural parameters (r_g and ∠_α) of the most abundant conformer of carvone are as follows: 〈r(C-C)〉=1.520(3) Å; 〈r(CC)〉=1.360(5) Å; r(CO)=1.225(5) Å; 〈r(C-H)〉=1.104(4)Å; 〈∠CC-C〉=121.1(5)°; 〈∠C-C-C〉=110.4(5)°; ∠C-CO-C=117.1(14)°; 〈∠C-C-H〉=111.1(13)°. Angle brackets denote average values and parenthesized values are the estimated limits of error (3σ) referring to the last significant digit.     

Enhancement of electronic excitation energy transfer in the colloidal crystals of colloidal silica suspensions doped with fluorescent dyes

The efficiency of electronic excitation energy transfer from photo-excited rhodamine 110 (Rh110, energy donor) to rhodamine B (RhB, energy acceptor) in an exhaustively deionized colloidal silica suspension has been studied. This colloidal suspension shows Bragg reflection due to the formation of colloidal crystals and the Bragg-peak wavelength is controllable by the volume fraction of the silica spheres. When the Bragg-peak wavelength matches with the fluorescence band of Rh110, a depletion was observed in the Rh110 fluorescence spectrum. This means the fluorescence of Rh110 is partially trapped due to the Bragg reflection inside the crystal lattice. In the coexistence of RhB, the enhancement of RhB fluorescence intensity was observed. These facts clearly indicate the trapped photon energy of Rh110 is efficiently transferred to RhB within the colloidal crystals. The quantitative measurements showed that the enhancement of the transfer efficiency is 20% (or slightly more) in the present experimental conditions.     

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.