An efficient series approximation for the Lévy α-stable symmetric distribution

A relevant problem in the statistical and mathematical physics literature is to derive numerically accurate expressions to calculate Lévy α-stable distributions $P_{\alpha }(x;\beta )$. On the formal side, important exact results rely on special functions, such as Meijer-G, Fox-H and finite sums of hypergeometric functions, with only a few exceptional cases expressed in terms of elementary functions. Hence, from a more practical point of view, methods such as series expansions are in order, e.g., to allow for the estimation of the Lévy distribution with high numerical precision, even though most of the existing approaches are restricted to a subset of the distribution parameters and/or usually demand relatively time-consuming sophisticated algorithms. Here we present a rather simple truncated expansion for the case of symmetric Lévy distributions $P_{\alpha }(x)$ ($\beta =0$). This is achieved by dividing the full range of integration into windows, performing proper series expansion inside each, and then calculating the integrals term by term. The obtained representation is convergent for any $0<\alpha \le 2$. Moreover, its accuracy is directly controlled by the number of terms in the truncated expression, being straightforward to implement numerically. As we show with different examples, for almost all allowable α's the calculations lead to $P_{\alpha }(x)$ with reasonable low absolute error for computationally inexpensive simulations.     

Low‐energy collision‐induced dissociation (low‐energy CID), collision‐induced dissociation (CID), and higher energy collision dissociation (HCD) mass spectrometry for structural elucidation of saccharides and clarification of their dissolution mechanism in DMAc/LiCl

The dissolution mechanism of oligosaccharides in N,N‐dimethylacetamide/lithium chloride (DMAc/LiCl), a solvent used for cellulose dissolution, and the capabilities of low‐energy collision‐induced dissociation (low‐energy CID), collision‐induced dissociation (CID), and higher energy collision dissociation (HCD) for structural analysis of carbohydrates were investigated. Comparing the spectra obtained using 3 techniques shows that, generally, when working with monolithiated sugars, CID spectra provide more structurally informative fragments, and glycosidic bond cleavage is the main pathway. However, when working with dilithiated sugars, HCD spectra can be more informative providing predominately cross‐ring cleavage fragments. This is because HCD is a nonresonant activation technique, and it allows a higher amount of energy to be deposited in a short time, giving access to more endothermic decomposition pathways as well as consecutive fragmentations. The difference in preferred dissociation pathways of monolithiated and dilithiated sugars indicates that the presence of the second lithium strongly influences the relative rate constants for cross‐ring cleavages vs glycosidic bond cleavages, and disfavors the latter. Regarding the dissolution mechanism of sugars in DMAc/LiCl, CID and HCD experiments on dilithiated and trilithiated sugars reveal that intensities of product ions containing 2 Li⁺ or 3 Li⁺, respectively, are higher than those bearing only 1 Li⁺. In addition, comparing the fragmentation spectra (both HCD and CID) of LiCl‐adducted lithiated sugar and NaCl‐adducted sodiated sugar shows that while, in the latter case, loss of NaCl is dominant, in the former case, loss of HCl occurs preferentially. The compiled evidence implies that there is a strong and direct interaction between lithium and the saccharide during the dissolution process in the DMAc/LiCl solvent system.     

Convergence of multipole expanded intermolecular interaction energies for Gaussian-type-function and Slater-type-function basis sets

 It is shown that the multipole expansion of each order of the polarization series converges for large enough intermolecular distances when finite basis sets of Gaussian or Slater-type functions are used to approximate molecular response properties. Convergence of the multipole expansion for each order of the polarization series does not imply convergence of the polarization series itself. A corresponding convergence condition is extracted from the general perturbation theory in a finite-dimensional space and is applied to the H + H⁺ problem. Received: 29 September 1999 / Accepted: 22 May 2000 / Published online: 18 August 2000     

混合狄雷克利分布和高维表的贝叶斯估计

本文讨论多项分布情况下的高维列联表使用混合狄雷克利分布为先验分布时，贝叶斯估计的表达，以及独立性条件的表述．将文献［４］和［５］的结论推广到高维列联表中．     

Some properties of subexponential distributions

The nonnegative random variableX is said to have a subexponential distribution if we have (1-G(t))/(1-F(t))→2 ast→∞, whereF(t)=P{X≤t} andG(t) is the convolution ofF(t) with itself. Conditions on the distribution of independent nonnegative random variablesX andY such that max(X, Y) and min(X, Y) have a subexponential distribution are given. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 138–144, July, 1997. Translated by N. K. Kulman     

索赔服从Phase-type分布的风险模型破产概率

本文研究了索赔服从Phase-type分布的风险模型在第n次索赔时破产的概率问题.利用Phasetype分布的性质及索赔时刻的盈余与净收入之间的关系,得到盈余密度函数的Laplace变换递推关系,进而得出风险过程在第n次索赔时的破产概率,最后举例说明之.     

Nonlinear generalized functions and jump conditions for a standard one pressure liquid–gas model

The mixture of a liquid and a gas is classically represented by one pressure models. These models are a system of PDEs in nonconservative form and shock wave solutions do not make sense within the theory of distributions: they give rise to products of distributions that are not defined within distribution theory. But they make sense by applying a theory of nonlinear generalized functions to these equations. In contrast to the familiar case of conservative systems the jump conditions cannot be calculated a priori. Jump conditions for these nonconservative systems can be obtained using the theory of nonlinear generalized functions by inserting some adequate physical information into the equations. The physical information that we propose to insert for the one pressure models of a mixture of a liquid and a gas is a natural mathematical expression in the theory of nonlinear generalized functions of the fact that liquids are practically incompressible while gases are very compressible, and so they do not satisfy equally well their respective state laws on the shock waves. This modelization gives well defined explicit jump conditions. The great numerical difficulty for solving numerically nonconservative systems is due to the fact that slightly different numerical schemes can give significantly different results. The jump conditions obtained above permit to select the numerical schemes and validate those that give numerical solutions that satisfy these jump conditions, which can be an important piece of information in the absence of other explicit discontinuous solutions and of precise observational results. We expose with care the mathematical originality of the theory of nonlinear generalized functions (an original abstract analysis issued by the Leopoldo Nachbin team on infinite dimensional holomorphy) that permits to state mathematically physical facts that cannot be formulated within distribution theory, and are the key for the removal of “ambiguities” that classically appear when one tries to calculate on “multiplications of distributions” that occur in the differential equations of physics.     

The two-state random walk

We develop asymptotic results for the two-state random walk, which can be regarded as a generalization of the continuous-time random walk. The two-state random walk is one in which a particle can be in one of two states for random periods of time, each of the states having different spatial transition probabilities. When the sojourn times in each of the states and the second moments of transition probabilities are finite, the state probabilities have an asymptotic Gaussian form. Several known asymptotic results are reproduced, such as the Gaussian form for the probability density of position in continuous-time random walks, the time spent in one of these states, and the diffusion constant of a two-state diffusing particle.     

Reflection of Progressing Waves for Quasilinear Hyperbolic Systems

It is proved that when a progressing wave described by conormal distribution for a quasilinear hyperbolic 2 × 2 system hits a solid wall transvenally, the reflected wave remains conormal. In contrast to the semilinear case, such conormal distributions had to be defined inductively to take into account of the fact that the relevant characteristic surfaces are not necessarily smooth. The argument involves a suitable coordinate change to reduce the problem to a simple form and an iterative induction on the tangential regularity of the solution as well as that of the characteristic surfaces at which the wave fronts are situated.     

带负顾客和强占优先权的不耐烦信元排队

考虑带有负顾客的两类信元的强占优先权M/M/1排队系统.两类信元及负顾客的到达过程均为泊松过程.两类信元到达后分别在各自有限的缓冲器内排队,第一类信元较第二类信元有强占优先权,同时第一类信元是不耐烦的.负顾客一对一抵消队尾的第一类信元(若有),若系统中无第一类信元,到达的负顾客就自动消失.负顾客不接受服务.采用矩阵分析的方法得到了两类信元各自的稳态分布,并作了相应的性能分析.