On characterizing the bivariate exponential and geometric distributions

In this note, a characterization of the Gumbel's bivariate exponential distribution based on the properities of the conditional moments is discussed. The result forms a sort of bivariate analogue of the characterization of the univariate exponential distribution given by Sahobov and Geshev (1974) (cited in Lau and Rao ((1982), Sankhy Ser. A, 44, 87)). A discrete version of the property provides a similar conclusion relating to a bivariate geometric distribution.     

Recurrence relations for single and product moments of progressive Type-II right censored order statistics from exponential and truncated exponential distributions

In this paper, we establish several recurrence relations satisfied by the single and product moments of progressive Type-II right censored order statistics from an exponential distribution. These relations may then be used, for example, to compute all the means, variances and covariances of exponential progressive Type-II right censored order statistics for all sample sizes n and all censoring schemes (R ₁, R ₂, ..., R _m ), mn. The results presented in the paper generalize the results given by Joshi (1978, Sankhy Ser. B, 39, 362–371; 1982, J. Statist. Plann. Inference, 6, 13–16) for the single moments and product moments of order statistics from the exponential distribution.To further generalize these results, we consider also the right truncated exponential distribution. Recurrence relations for the single and product moments are established for progressive Type-II right censored order statistics from the right truncated exponential distribution.     

Adaptive choice of trimming proportions

We consider Jaeckel's (1971,Ann. Math. Statist.,42, 1540–1552) proposal for choosing the trimming proportion of the trimmed mean in the more general context of choosing a trimming proportion for a trimmedL-estimator of location. We obtain higher order expansions which enable us to evaluate the effect of the estimated trimming proportion on the adaptive estimator. We find thatL-estimators with smooth weight functions are to be preferred to those with discontinuous weight functions (such as the trimmed mean) because the effect of the estimated trimming proportion on the estimator is of ordern ^–1 rather thann ^–3/4. In particular, we find that valid inferences can be based on a particular smooth trimmed mean with its asymptotic standard error and the Studentt distribution with degrees of freedom given by the Tukey and McLaughlin (1963,Sankhy Ser. A,25, 331–352) proposal.     

Posterior Sensitivity to the Sampling Distribution and the Prior: More than One Observation

Sensitivity of a posterior quantity (f, P) to the choice of the sampling distribution f and prior P is considered. Sensitivity is measured by the range of (f, P) when f and P vary in nonparametric classes ^f and ^P respectively. Direct and iterative methods are described which obtain the range of (f, P) over f ^f when prior P is fixed, and also the overall range over f ^f and P ^P . When multiple i.i.d. observations X ₁,...,X _k are observed from f, the posterior quantity (f, P) is not a ratio-linear function of f. A method of steepest descent is proposed to obtain the range of (f, P). Several examples illustrate applications of these methods.     

On Geometric-Stable Laws, a Related Property of Stable Processes, and Stable Densities of Exponent One

Klebanov et al. (1985, Theory Probab. Appl., 29, 791-794) introduced a class of probability laws termed by them "geometrically-infinitely-divisible" laws, and studied in detail the sub-class of "geometrically-strictly-stable" laws. In Section 2 of the present paper, the larger sub-class of "geometric-stable" laws is (defined and) studied. In Section 3, a characterization of stable processes involving (stochastic integrals and) geometric-stable laws is presented. In Section 4, the asymptotic behaviour of stable densities of exponent one (and || < 1) is studied using only real analysis methods. In an Appendix, "geometric domains of attraction" to geometric-stable laws are investigated, motivated by the work of Mohan et al. (1993, Sankhy Ser. A, 55, 171-179).     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

Approximation of the viability kernel

We study recursive inclusionsx ⁿ⁺¹ G(x ⁿ ). For instance, such systems appear for discrete finite-difference inclusionsx ⁿ⁺¹ G (x ⁿ) whereG :=1+F. The discrete viability kernel ofG , i.e., the largest discrete viability domain, can be an internal approximation of the viability kernel ofK underF. We study discrete and finite dynamical systems. In the Lipschitz case we get a generalization to differential inclusions of the Euler and Runge-Kutta methods. We prove first that the viability kernel ofK underF can be approached by a sequence of discrete viability kernels associated withx ⁿ⁺¹ (xⁿ) where (x) =x + F(x) + (ML/2) ². Secondly, we show that it can be approached by finite viability kernels associated withx _h ⁿ⁺¹ ( (x _h ⁿ⁺¹ ) +(h) X _h .     

Volumes des représentations sur un corps local

Let be a locally compact second countable group, F a local field of characteristic zero and G an F-almost-simple F-algebraic group. In this paper we study the space X(,G) of Zariski-dense representations : G = G(F) using the natural morphism of cohomological functors * : H*(G, ·) H*(, ·) (where H denotes the continuous cohomology).First let F be a p-adic field. We completely describe the relations between the geometry and the cohomology of G : using geometric properties of the Bruhat-Tits building of G we construct natural cocycles for any irreducible cohomological representation of G. We then adapt these results to the case where the field F is archimedean.Using these cocycles we obtain a simple cohomological characterization of representations with bounded image.Our main result is then the construction, using the previous cocycles and dynamical properties at infinity of , of cohomological invariants (called volumes) on the space X(,G). These volumes describe how the image () goes to infinity in G. They have coefficients in the natural universal infinite-dimensional representation L(, )$\mathbb{C}$ of .In the case where is a cocompact lattice of SO(n, 1) or SU(n, 1), we use these volumes to produce new non-trivial numerical invariants on X(,G), which refine previously known invariants.

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Volumes des représentations sur un corps local

     

Thermoelastic Equilibrium of Bodies in Generalized Cylindrical Coordinates

Using the method of separation of variables, an exact solution is constructed for some boundary value and boundary-contact problems of thermoelastic equilibrium of one- and multilayer bodies bounded by the coordinate surfaces of generalized cylindrical coordinates , , z. , are the orthogonal coordinates on the plane and z is the linear coordinate. The body, occupying the domain = {₀ < < ₁, ₀ < < ₁, 0 < z < z ₁}, is subjected to the action of a stationary thermal field and surface disturbances (such as stresses, displacements, or their combinations) for z = 0 and z = z ₁. Special type homogeneous conditions are given on the remainder of the surface. The elastic body is assumed to be transversally isotropic with the plane of isotropy z = const and nonhomogeneous along z. The same assumption is made for the layers of the multilayer body which contact along z = const.     

Robust facility location

Let A be a nonempty finite subset of the plane representing the geographical coordinates of a set of demand points (towns, ), to be served by a facility, whose location within a given region S is sought. Assuming that the unit cost for aA if the facility is located at xS is proportional to dist(x,a) — the distance from x to a — and that demand of point a is given by _a , minimizing the total transportation cost TC(,x) amounts to solving the Weber problem. In practice, it may be the case, however, that the demand vector is not known, and only an estimator circ; can be provided. Moreover the errors in such estimation process may be non-negligible. We propose a new model for this situation: select a threshold value B>0 representing the highest admissible transportation cost. Define the robustness of a location x as the minimum increase in demand needed to become inadmissible, i.e. (x)=min{|–circ;|:TC(,x)>B,0} and find the x maximizing to get the most robust location. Acknowledgment.The authors acknowledge the constructive remarks of the referees. The research of the first author has been supported in part by grant BFM2002-04525-C02-02 of MCYT, Spain. The research of the second author has been supported in part by a grant of the Deutsche Forschungsgemeinschaft.     

Newton polyhedra and the Bergman kernel

The purpose of this paper is to study singularities of the Bergman kernel at the boundary for pseudoconvex domains of finite type from the viewpoint of the theory of singularities. Under some assumptions on a domain in ⁿ⁺¹ , the Bergman kernel B(z) of takes the form near a boundary point p: where (w,) is some polar coordinates on a nontangential cone with apex at p and means the distance from the boundary. Here admits some asymptotic expansion with respect to the variables ^{1/ m} and log(1/) as 0 on . The values of d _F >0, m _F ₊ and m are determined by geometrical properties of the Newton polyhedron of defining functions of domains and the limit of as 0 on is a positive constant depending only on the Newton principal part of the defining function. Analogous results are obtained in the case of the Szegö kernel. Mathematics Subject Classification (2000):32A25, 32A36, 32T25, 14M25.     

Balanced ternary designs with block size three,any Λ and anyR

A balanced ternary design onV elements is a collection ofB blocks (which are multisets) of sizeK, such that each element occurs 0, 1 or 2 times per block andR times altogether, and such that each unordered pair of distinct elements occurs times. (For example, in the blockxxyyz, the pairxy is said to occur four times and the pairsxz, yz twice each.) It is straightforward to show that each element has to occur singly in a constant number of blocks, say ₁, and so each element also occurs twice in a constant number of blocks, say ₂, whereR= ₁+2 ₂. If ₂=0 the design is a balanced incomplete block design (binary design), so we assume ₂>0, andK<2V (corresponding to incompleteness in the binary case). Necessarily >1 if ₂>0 (andK>2).In 1980 and 1982 the author gave necessary and sufficient conditions for the existence of balanced ternary designs withK=3, =2 and ₂=1, 2 or 3. In this paper work on the existence of balanced ternary designs with block size three is concluded, in that necessary and sufficient conditions for the existence of a balanced ternary design withK=3, any >1 and any ₂ are given.     

On Bures Distance and *-Algebraic Transition Probability between Inner Derived Positive Linear Forms over W*-Algebras

On a W^*-algebra M, for given two positive linear forms , M ₊ ^* and algebra elements a, b M, a variational expression for the Bures distance d _B( ^a , ^b ) between the inner derived positive linear forms ^a =(a _*·a) and ^b =(b ^*·b) is obtained. Along with the proof of the formula, also an earlier result of S. Gudder on noncommutative probability will be slighly extended. Also, the given expression of the Bures distance relates nicely to the system of seminorms proposed by D. Buchholz which occurs, along with the problem of estimating the so-called `weak intertwiners", in algebraic quantum field theory. In the last section, some optimization problem will be considered.     

Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

For the classB _p , 0 < 1, 1p , of 2-periodic functions of the form f(t)=u(,t), whereu (,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K(t) of the convolution f= K *g, gl, with respect to the metric of L₁. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1549–1557, November, 1995.     

Polynomial operations on complex K-theory

We adapt here the results of the author concerning polynomial operations on the 0th stable cohomotopy to the case of the 0th complex K-theory and consider polynomial operations : Kh, where h is a ring-valued contravariant functor, defined on finite CW-complexes, satisfying some properties. We construct a family of generating operations for the ring Pol(K,h) of all polynomial operations : Kh and doing so, we describe the additive structure of this ring in terms of the h(BU(n)'s. As an illustration of how polynomiality could be used to study operations in the setting of algebraic K-theory, we consider, from our point of view, the well known situation operations : KK on complex K-theory.     

Scaling limit of anticommuting self-adjoint operators and applications to Dirac operators

LetA andB be two anticommuting self-adjoint operators andV() be a symmetric operator in a Hilbert space, where >0 is a parameter. It is proven that, under some conditions forV(), the resolvents of A+² B±²|B|+V() converge as . Applications to the nonrelativistic-limit problem of Dirac operators and supersymmetry are discussed.This work is supported by the Grant-In-Aid 0560139 for science research from the Ministry of Education, Japan.     

Taut Distance-Regular Graphs of Odd Diameter

Let denote a bipartite distance-regular graph with diameter D 4, valency k 3, and distinct eigenvalues ₀ > ₁ > ··· > _D. Let M denote the Bose-Mesner algebra of . For 0 i D, let E _i denote the primitive idempotent of M associated with _i . We refer to E ₀ and E _D as the trivial idempotents of M. Let E, F denote primitive idempotents of M. We say the pair E, F is taut whenever (i) E, F are nontrivial, and (ii) the entry-wise product E F is a linear combination of two distinct primitive idempotents of M. We show the pair E, F is taut if and only if there exist real scalars , such that _{i + 1 i + 1} – _{i – 1 i – 1} = _i ( _{i + 1} – _{i – 1}) + _i ( _{i + 1} – _{i – 1}) + (1 i D – 1)where ₀, ₁, ..., _D and ₀, ₁, ..., _D denote the cosine sequences of E, F, respectively. We define to be taut whenever has at least one taut pair of primitive idempotents but is not 2-homogeneous in the sense of Nomura and Curtin. Assume is taut and D is odd, and assume the pair E, F is taut. We show

Über eine Verallgemeinerung der Nicoletti-Aufgabe für Funktional-Differentialgleichungen mit voreilendem Argument

We consider a functional differential equation (1) (t)=F(t,) fort[0,+) together with a generalized Nicoletti condition (2)H()=. The functionF: [0,+)×C ⁰[0,+)B is given (whereB denotes the Banach space) and the value ofF (t, ) may depend on the values of (t) fort[0,+);H: C ⁰[0,+)B is a given linear operator and B. Under suitable assumptions we show that when the solution :[0,+)B satisfies a certain growth condition, then there exists exactly one solution of the problem (1), (2).     

Zeros of the densities of infinitely divisible measures on R n

Let be an infinitely divisible probability measure onR ⁿ without Gaussian component and let be its Lévy measure. Suppose that is absolutely continuous with respect to the Lebesgue measure . We investigate the structure of the set ⁿ of admissible translates of . This yields a unified presentation of previously known results. We also show that if(S)>0 then is equivalent to , under the assumption that supp =R ⁿ , whereS is the closure of the semigroup generated by the support of .The research of this author is supported by KBN Grant.The research of this author is supported by AFSOR Grant No. 90-0168, and the University of Tennessee Science Alliance, a State of Tennessee Center of Excellence.     

Parametric stochastic convexity and concavity of stochastic processes

A collection of random variables {X(), } is said to be parametrically stochastically increasing and convex (concave) in if X() is stochastically increasing in , and if for any increasing convex (concave) function , E(X()) is increasing and convex (concave) in whenever these expectations exist. In this paper a notion of directional convexity (concavity) is introduced and its stochastic analog is studied. Using the notion of stochastic directional convexity (concavity), a sufficient condition, on the transition matrix of a discrete time Markov process {X _n(), n=0,1,2,...}, which implies the stochastic monotonicity and convexity of {X _n(), }, for any n, is found. Through uniformization these kinds of results extend to the continuous time case. Some illustrative applications in queueing theory, reliability theory and branching processes are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205. Reproduction in whole or in part is permitted for any purpose by the United States Government.