Painlevé transcendents and the Hankel determinants generated by a discontinuous Gaussian weight

This paper studies the Hankel determinants generated by a discontinuous Gaussian weight with one and two jumps. It is an extension in a previous study, in which they studied the discontinuous Gaussian weight with a single jump. By using the ladder operator approach, we obtain a series of difference and differential equations to describe the Hankel determinant for the single jump case. These equations include the Chazy II equation, continuous and discrete Painlevé IV. In addition, we consider the large n behavior of the corresponding orthogonal polynomials and prove that they satisfy the biconfluent Heun equation. We also consider the jump at the edge under a double scaling, from which a Painlevé XXXIV appeared. Furthermore, we study the Gaussian weight with two jumps and show that a quantity related to the Hankel determinant satisfies a two variables' generalization of the Jimbo‐Miwa‐Okamoto σ‐form of the Painlevé IV.     

经典风险模型中破产概率的一个渐近式

本文研究经典风险模型中破产概率的渐近行为．利用几何和的方法，获得了索赔额的分布属于S（γ）．γ〉0。时破产概率的一个局部渐近式．同时．给出了一个具体的数值的例子．     

New 3π‐2Spiro Ladder‐Type Phenylene Materials: Synthesis,Physicochemical Properties and Applications in OLEDs

New routes to ladder‐type phenylene materials 1 and 2 are described. The oligomers 1 and 2 , which possess a “3π‐2spiro” architecture, have been synthesized by using extended diketone derivatives 3 and 10 as key intermediates. The physicochemical properties of the new blue‐light emitter 2 were studied in detail and compared with those of the less‐extended 1 . Owing to the recent development of fluorenone derivatives and their corresponding more conjugated analogues as potential electron‐transport materials in organic light‐emitting diodes (OLEDs) and as n‐type materials for photovoltaic applications, we also report herein the thermal, optical and electrochemical behavior of the key intermediates, diketones 3 and 10 . Finally, the application of dispiro 2 as a new light‐emitting material in OLEDs is reported.     

On the exact asymptotic behaviour of the distribution of ladder epochs

Let T₊ denote the first increasing ladder epoch in a random walk with a typical step-length X. It is known that for a large class of random walks with E(X)=0,E(X²)=∞, and the right-hand tail of the distribution function of X asymptotically larger than the left-hand tail, $\text{P}\text{T}{}_{\mathrm{+}}\text{?n∽n}{}^{\mathrm{<mtext>1</mtext><mtext>\beta ?1</mtext>}}\text{L}{}_{\mathrm{+}}\text{(n)}$ as n→∞, with 1<β<2 and L₊ slowly varying, if and only if$\text{P}\text{{X?x}∽ 1/{x}{}^{\mathrm{\beta }}\text{L(x)}}$ as x→+∞, with L slowly varying. In this paper it is shown how the asymptotic behaviour of L determines the asymptotic behaviour of L₊ and vice versa. As a by-product, it follows that a certain class of random walks which are in the domain of attraction of one-sided stable laws is such that the down-going ladder height distribution has finite mean.     

Local asymptotic behavior of the survival probability of the equilibrium renewal model with heavy tails

Recently, Tang established a local asymptotic relation for the ruin probability in the Cramer-Lundberg risk model. In this short note we extend the corresponding result to the equilibrium renewal risk model.     

The Category of Lattices over a Lattice-Finite Ring

It is well-known that for a lattice-finite order Λ over a complete discrete valuation domain, the radical of Λ-lat (the category of Λ-lattices) is nilpotent modulo projectives. Iyama has shown that this property merely depends on the combinatorial data given by the Auslander–Reiten quiver of Λ. Moreover, he established a criterion for a finite (symmetrizable) translation quiver Q to be the Auslander–Reiten quivers of an order Λ. We improve his characterization by showing that the remaining conditions on Q can be replaced by the existence of an additive function on the vertices of Q (Theorem 4). Our proof rests on a functorial theory of ladders, expressing the Auslander–Reiten structure of Λ-lat by means of an adjoint pair of functors L⊣L⁻ in the homotopy category of two-termed complexes over Λ-lat. Presented by I. Reiten Mathematics Subject Classifications (2000) Primary: 16G70, 16G30; secondary: 16G60.     

Chain ladder method: Bayesian bootstrap versus classical bootstrap

The intention of this paper is to estimate a Bayesian distribution-free chain ladder (DFCL) model using approximate Bayesian computation (ABC) methodology. We demonstrate how to estimate quantities of interest in claims reserving and compare the estimates to those obtained from classical and credibility approaches. In this context, a novel numerical procedure utilizing a Markov chain Monte Carlo (MCMC) technique, ABC and a Bayesian bootstrap procedure was developed in a truly distribution-free setting. The ABC methodology arises because we work in a distribution-free setting in which we make no parametric assumptions, meaning we cannot evaluate the likelihood point-wise or in this case simulate directly from the likelihood model. The use of a bootstrap procedure allows us to generate samples from the intractable likelihood without the requirement of distributional assumptions; this is crucial to the ABC framework. The developed methodology is used to obtain the empirical distribution of the DFCL model parameters and the predictive distribution of the outstanding loss liabilities conditional on the observed claims. We then estimate predictive Bayesian capital estimates, the value at risk (VaR) and the mean square error of prediction (MSEP). The latter is compared with the classical bootstrap and credibility methods.     

基于2,4′-二羧基二苯甲酮构筑的具有一维梯状链和一维隧道的三维超分子体系

采用水热法设计合成了两个新型三维超分子化合物H₂L·H₂O (1)和[Ag(bpy)₂]·HL·H₂O (2) (其中bpy=2,2'-联吡啶, H₂L=2,4′-二羧基二苯甲酮),晶体结构分析表明,它们均是通过氢键采用不同的连接方式拓展而成。其中,化合物1 是2,4′-二羧基二苯甲酮和水分子通过O–H···O氢键形成的一维梯状链扩展构筑的三维超分子体系;化合物2 则是2,4′-二羧基二苯甲酮和水分子通过两种氢键形成含有一维隧道的三维超分子体系。有趣的是,[Ag(bpy)₂]⁺ 阳离子通过π–π 堆积和弱的Ag···Ag相互作用连在一起,进而以客体形式填充其中。荧光性质研究表明,由于存在bpy的螯合与堆积效应,化合物2相比配体和化合物1,其荧光发射峰发生红移。     

Steady state Behaviour of a Three level Ladder Atomic System with Incoherent Population Pumping in a Squeezed Vacuum

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