Some properties of a class of sparse polynomials

We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic polynomials. After deriving some basic identities, we obtain properties concerning monotonicity and log-concavity, as well as identities involving derivatives. We also prove upper and lower bounds on the moduli of the zeros of these polynomials.     

Optimization of the thickness of layers of sandwich conical shells

We deal with an optimal control problem for variational inequalities, where the linear operators as well as the convex sets of possible states depend on the control parameter. The existence theorem for optimal control is applied to optimal design problems for sandwich conical shells where a variable thickness of given layers appears as a control variable.     

Game of two cars: Case of variable speed

A stochastic version of Isaacs's (Ref. 1) game of two cars is dealt with here. In this version, the pursuer, owing to thrust and drag forces, has a variable speed, whereas the evader's speed is constant. Also, the pursuer can maneuver as long as his speed is bounded by some lower and upper limits. The probability of interception, corresponding to optimal (saddle-point) feedback strategies, is computed and serves as a reference for evaluating the performance of four different versions of the proportional navigation pursuit law as well as two other strategies.     

Upper bounds of degree 3 for the probability of the union of events via linear programming

We consider the problem of generating upper bounds for the probability of the union of events when the individual probabilities of the events as well as the probabilities of pairs and triples of these events are known. By formulating the problem as a Linear Program, we can obtain bounds as objective function values corresponding to dual basic solutions. The upper bounds are based on underlying graph structures.     

The prediction of sequences

The problem of predicting the short-term future behavior of a sequence, after observing it as long as we please, so as to achieve a specified reliability against all possible sequences is considered. For a particular problem, namely, predicting when in a sequence of 0's and 1's the pair (1, 0) in that order is not coming next, a reliability of 3/4 can be approximated as closely as we please, but not achieved. This article appeared as a Research Memorandum of the Rand Corporation, RM-1570, 12 October 1955.     

Use of VNS heuristics for scheduling of patients in hospital

One of the important problems in hospital management is how to schedule the treatments of resident patients in hospital for a given day due to the restrictions imposed by their medical condition as well as restrictions on medical machines and qualified medical personnel availability. Patients are to be subjected to different kinds of treatments, each requiring a medical machine of a certain type as well as a physician being qualified to operate it. This is a highly complex problem not yet adequately addressed in the literature. At present in most hospitals the problem is being solved manually by specialized personnel. However, the resulting schedules are very often inaccurate and inefficient with patients waiting for a long time to be treated and medical personnel often working overtime. In this paper we formulate the model for this problem and develop a simple and efficient method based on Variable Neighbourhood Search for solving it. The heuristics has been tested on real-life as well as on generated instances. Numerical results show that the heuristics proposed outperform commercial software for optimization as well as manual solutions both in quality of solution and in computational time.     

Reactions of ketones on oxide surfaces. II. Nature of the active sites of alumina

The nature of the active sites of alumina for the reaction of cyclohexanone was investigated by the introduction of selective poisoning agents both on the surface of the alumina as well as in the reactant itself. Thus additives which are basic in nature such as sodium ions, ammonia, pyridine, as well as acidic additives such as carbon dioxide, carbon monoxide, cyclohexanol, and isoproponal were used. Dual acid-base sites seem to be responsible for the catalytic properties. The sites responsible for the aldol condensation reaction giving a dimer seem to be similar to those sites responsible for the formation of ethers from alcohols, while the sites responsible for the formation of cyclohexene from cyclohexanone seem to be similar to those sites responsible for the formation of a carboxylate species on the adsorption of alcohols on alumina surfaces. A mechanism is proposed for the reaction of cyclohexanone which does involve the intermediacy of cyclohexanol to account for the formation of cyclohexene.     

Convexity of the zeros of some orthogonal polynomials and related functions

We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well as functions related to them, using transformations under which the zeros remain unchanged. We give upper as well as lower bounds for the distance between consecutive zeros in several cases.     

Prototype modelling of irregular condition monitoring of production plant

The previously reported modelling of inspection-maintenanceproblems utilizing the delay-time concept is here extended toembrace condition-monitoring checks. The main requirements arethe relaxation of regular periodic inspections, and the requirementthat the initial point of a defect be measured as the time fromthe as-new condition and not, as previously, as the time sincea previous inspection. An expression for the probability ofa defect arising as a breakdown and as an inspection repairis constructed for a general inspection policy, under the assumptionof both perfect and imperfect repair. The revised role of theseprobability expressions in modelling the process of maintenancefor cost-effective production is discussed, various modellingalternatives are outlined, and a numerical example is presented.Parallels are drawn between condition monitoring in an industrialsetting and in the delivery of health care.     

On the Euler characteristic of finite unions of convex sets

The Euler characteristic plays an important role in many subjects of discrete and continuous mathematics. For noncompact spaces, its homological definition, being a homotopy invariant, seems not as important as its role for compact spaces. However, its combinatorial definition, as a finitely additive measure, proves to be more applicable in the study of singular spaces such as semialgebraic sets, finitely subanalytic sets, etc. We introduce an interesting integral by means of which the combinatorial Euler characteristic can be defined without the necessity of decomposition and extension as in the traditional treatment for polyhedra and finite unions of compact convex sets. Since finite unions of closed convex sets cannot be obtained by cutting convex sets as in the polyhedral case, a separate treatment of the Euler characteristic for functions generated by indicator functions of closed convex sets and relatively open convex sets is necessary, and this forms the content of this paper.     

风险非同质时索赔次数的分布拟合的估计与检验问题

在非寿险精算中 ,索赔次数的分布一般假设为泊松分布 P(λ) .风险非同质时 λ的分布称为混合分布 .本文考虑了混合分布为三参数伽玛分布时的参数估计以及位置参数的检验问题     

Bipolarization of posets and natural interpolation

The Choquet integral w.r.t. a capacity can be seen in the finite case as a parsimonious linear interpolator between vertices of n[0,1]. We take this basic fact as a starting point to define the Choquet integral in a very general way, using the geometric realization of lattices and their natural triangulation, as in the work of Koshevoy.A second aim of the paper is to define a general mechanism for the bipolarization of ordered structures. Bisets (or signed sets), as well as bisubmodular functions, bicapacities, bicooperative games, as well as the Choquet integral defined for them can be seen as particular instances of this scheme.Lastly, an application to multicriteria aggregation with multiple reference levels illustrates all the results presented in the paper.     

对应不同类型变分原理的旋转薄壳轴对称单元

本文在一定范围内,统一考察基于不同变分原理建立的有限元模型中,泛函约束条件的不同对单元性态的影响.文中以旋转薄壳轴对称单元(简称TSR单元)为例,采用相同的曲边单元几何描述,推导了七种TSR杂交单元和二种TSR位移协调元,它们分别对应于三类杂交变分原理及最小势能原理.通过单刚列式分析和波纹壳等数值算例比较,分析了不同模型的性态异同和应用上的适应性与局限性;讨论了两类模型间的相互关系;指出了TSR杂交位移元的—个发散条件,并推荐了二种性态较理想的TSR单元.     

Generators of an Abelian group of invertible measure-preserving transformations

A multi-dimensional counterpart of the asymptotic rate is introduced as a new conjugacy invariant for finitely generated Abelian groups of automorphisms of a fixed probability space. The generalization of the Kolmogorov generator theorem as well as some related results are established. It is not assumed that the underlying probability space is a Lebesgue one.     

Estimation of the eigenvalues of noncentrality parameter in matrix variate noncentral beta distribution

We consider the problem of estimating the eigenvalues of noncentrality parameter matrix in a matrix variate noncentral beta distribution, also known as multivariate noncentral F distribution. A decision theoretic approach is taken with square error as the loss function. We propose two types of new estimators and show their superior performance theoretically as well as numerically.     

On integral formulas for submanifolds of spaces of constant curvature and some applications

Integral formulas of Minkowski type, involving the higher mean curvatures as multilinear forms on the normal bundle, are proved for compact oriented immersed submanifolds with arbitrary codimension in a Riemannian manifold of constant curvature, and as application a generalization of the Liebmann-Süss theorem as well as upper bounds for the first positive eigenvalue of the Laplace operator are given.     

The Strength of Cartan's Version of Nevanlinna Theory

In 1933 Henri Cartan proved a fundamental theorem in Nevanlinnatheory, namely a generalization of Nevanlinna's second fundamentaltheorem. Cartan's theorem works very well for certain kindsof problems. Unfortunately, it seems that Cartan's theorem,its proof, and its usefulness, are not as widely known as theydeserve to be. To help give wider exposure to Cartan's theorem,the simple and general forms of the theorem are stated here.A proof of the general form is given, as well as several applicationsof the theorem. 2000 Mathematics Subject Classification 30D35.     

Computation of the best Diophantine approximations and of fundamental units of algebraic fields

A global generalization of continued fraction that yields the best Diophantine approximations of any dimension is considered. In the algebraic case, this generalization underlies a method for calculating the fundamental units of algebraic rings and the periods of best approximations, as well as the identification of the fundamental domain with respect to these periods. The units of an algebraic field are understood as the units of maximal order of this field.     

Asymptotic behavior of Shannon functions for the delays of schemes of functional elements

The behavior of Shannon functions for delays in the class of schemes of functional elements constructed over an arbitrary finite complete basis is studied. The case in which every basis element can have positive as well as zero delay is considered in this note. It is established that in this case the Shannon functions for delays asymptotically behave either as linear functions or as logarithmic functions or, starting from a certain number of arguments, remain constant.Translated from Matematicheskie Zametki, Vol. 19, No. 6, pp. 939–951, June, 1976.In conclusion the author thanks O. B. Lupanov for guidance during the preparation of this note.