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 .     

The upper and lower second order directional derivatives of a sup-type function

The purpose of this paper is to give a formula for expressing the second order directional derivatives of the sup-type functionS(x) = sup{f(x, t); t T} in terms of the first and second derivatives off(x, t), whereT is a compact set in a metric space and we assume thatf, f/x and ² f/x ² are continuous on ⁿ × T. We will give a geometrical meaning of the formula. We will moreover give a sufficient condition forS(x) to be directionally twice differentiable.     

Decomposition of multiattribute expected-utility functions

This paper integrates and extends the theory of the decomposition of multiattribute expected-utility functions based on utility independence. In a preliminary section, the standard decision model of expected utility is briefly discussed, including the fact that the decision maker's preference forlotteries with two outcomes determines the utility function uniquely. The decomposition possibilities of a utility function are captured by the concept ofautonomous sets of attributes, an affine separability of some kind known as generalized utility independence.Overlapping autonomous sets lead to biaffine-associative, i.e.multiplicative oradditive decompositions. The multiplicative representation shows that autonomy has strongerclosure properties than utility independence, for instance with respect to set-theoretic difference. Autonomy is also a concept with a wider scope since it applies to the decomposition of Boolean functions, games and a number of other topics in combinatorial optimization. This relationship to the well-known theory ofsubstitution decomposition in discrete mathematics also reveals a kind of discrete core behind the decomposition of utility functions. The entirety of autonomous sets can be represented by a compact data structure, the so-calledcomposition tree, which frequently corresponds to a natural hierarchy of attributes. Multiplicative/additive ormulti-affine functions correspond to the hierarchy steps. The known representation of multi-affine functions is shown to be given by aMoebius inversion formula. The entire approach has the advantage that it allows the application of more sophisticated representation methods on a detailed level, whereas it employs onlyfinite set theory andarithmetic on the main levels in the hierarchy.     

Solvable isotherms for a two-component system of charged rods on a line

The Coulomb system consisting of an equal number of positive and negative charged rods confined to a one-dimensional lattice is studied. The grand partition function can be calculated exactly at two values of the coupling constant=q ²/k _B T (q denoting the magnitude of the charges). The exact results lead to the conjecture that in the complex scaled fugacity plane, all the zeros of the grand partition function lie on the negative real axis for<2, on the point=–1 for=2, and on the unit circle for>2. In addition, for>4, we conjecture in general and prove at=4 that the zeros pinch the real axis in the thermodynamic limit, with an essential singularity in the pressure at the reduced density 1/2.     

How likely is Polya's drunkard to stay in x⩾ y⩾z?

In his celebrated paper, Polya has considered the random walk in the three-dimensional (cubic) lattice and showed that the probability of return to the origin is less than 1. Subsequent authors have shown that the probability is %34.053.... Here we consider the same random walk, with the restriction that the drunkard is only allowed to stay inxyz. It is shown that his probability of returning to the originand staying in the allowed region is %6.4844....     

A neutral gas model for electron swarms

A BGK-type Boltzmann equation for a neutral gas is considered as a model for electron swarms, because the gas and the electron Boltzmann equation have a common diffusion approximation. Both full- and half-range theory are developed using orthogonality methods of solution. Preliminary comparisons with diffusion theory are presented.     

Computer simulation of some dynamical properties of the Lorentz gas

We carried out molecular dynamics simulations of a Lorentz gas, consisting of a lone hydrogen molecule moving in a sea of stationary argon atoms. A Lennard-Jones form was assumed for the H₂-Ar potential. The calculations were performed at a reduced temperatureK ^* =kT/_{H 2–Ar} = 4.64 and at reduced densities ^*= _{Ar Ar} ³ in the range 0.074–0.414. The placement of Ar atoms was assumed to be random rather than dictated by equilibrium considerations. We followed the trajectories of many H₂ molecules, each of which is assigned in turn a velocity given by the Maxwell-Boltzmann distribution at the temperature of the simulation. Solving the equations of motion classically, we obtained the translational part of the incoherent dynamic structure factor for the H₂ molecule,S _tr(q, ). This was convoluted with the rotational structure factorS _rot(q, ) calculated assuming unhindered rotation to obtain the total structure factorS(q, ). Our results agree well with experimental data on this function obtained by Egelstaffet al. At the highest density ( ^*=0.414) we studied the dependence ofS(q, ) on system size (number of Ar atoms), number of H₂ molecules for which trajectories are generated, and the length of time over which these trajectories are followed.     

On the spectrum of the Heisenberg Hamiltonian

The quantum, antiferromagnetic, spin-1/2 Heisenberg Hamiltonian on thed-dimensional cubic lattice ^d is considered for any dimensiond. First the anisotropic case is considered for small transversal coupling and a convergent expansion is given for a family of eigenprojections which is complete in all finite-volume truncations. Then the general case is considered, for which an upper bound to the ground-state energy is given which is optimal for strong enough anisotropy. This bound is expressed through a functional involving the statistical expectation value at finite temperature of a certain correlation function of an Ising model defined on the lattice ^d itself.     

异烟肼及其代谢产物的串联质谱分析

报道了血样中异烟肼的串联质谱快速分析，利用这种分析方法，证实了一男性死者血样中异烟肼及其代谢产物的存在，从而为法庭诉讼提供了可靠的证据。这种方法具有快速性，高灵敏度和高选择性，适合于需要快速分析的场合。     

苯部分加氢制环己烯的非晶态Ru-M-B/ZrO2催化剂的表征

采用化学还原法，制备了高活性，高选择性非晶态RU-M-B/ZRO2催化剂，并将其用于催化苯部分加氢制环己烯，在140℃、5.0Mpa氢压下，苯转化40%时，环己烯选择性达到85%左右。环己烯最高收率达到52.1%，用XRD、SEM、BET比表面积测定等手段对摧化剂进行表征，XRD和SEM测试表明，RU-U-B/ZRO2属于非晶态，活性组分高度分散，XRD结果证实，在加氢过程中，非晶分解，RU晶化；温度愈高，RU晶化愈快，催化剂的活性、选择性与RU微晶的粒径有关，RU微晶粒径应控制在5nm左右，BET比表面积测定表明，ZRO2的负载提高了催化剂的比表面积，从而有利于活性组分的高度分散，并可阻止RU微晶的长大，讨论了B和ZRO2对提高选择性的作用。