Reliability bounds on HNBUE life distributions with known first two moments

In this paper, we study upper and lower bounds for the reliability function in harmonic new better than used in expectation (HNBUE) life distribution class with known first two moments. Here we say a life distribution has HNBUE property if the integral harmonic mean value of the residual life in any interval [0,t] is no more than its mean. By a constructive proof, we determine the lower and upper reliability bounds analytically and show that these bounds are all sharp.     

A study of tilt-stable optimality and sufficient conditions

Recent results by Eberhard et al. (2006) [4] and Eberhard and Wenczel (2009) [3] on the interaction of single- and double-envelope operations of nonsmooth functions and their interaction with second-order derivations have been used to study tilt-stability of local minima. This continues the study begun by Poliquin and Rockafellar (1998) [1] but now, armed with new tools we are able to make some new observations. We observe that tilt-stability entails a local density within the graph of the proximal subderivative of strict local minima order two of the tilted function. Indeed, it also entails the strict local minimality (order two) of the tilt-stable local minimum itself. For prox-regular, subdifferentially continuous functions this density property characterises tilt stability.     

On the Maximum Likelihood Estimation of a Covariance Matrix

For a multivariate normal set-up, it is well known that themaximumlikelihood estimator (MLE) of covariance matrix is neither admissible nor minimax under the Stein loss function. In this paper, we reveal that the MLE based on the Iwasawa parameterization leads to minimaxity with respect to the Stein loss function. Furthermore, a novel class of loss functions is proposed so that the minimum risks of the MLEs are identical in different coordinate systems, Cholesky parameterization and full Iwasawa parameterization. In other words, the MLEs based on these two different parameterizations are characterized by the property of minimaxity, without a Stein paradox. The application of our novel method to the high-dimensional covariance matrix problem is also discussed.     

Polynomial time solutions for scheduling problems on a proportionate flowshop with two competing agents

In scheduling problems with two competing agents, each one of the agents has his own set of jobs and his own objective function, but both share the same processor. The goal is to minimize the value of the objective function of one agent, subject to an upper bound on the value of the objective function of the second agent. In this paper we study two-agent scheduling problems on a proportionate flowshop. Three objective functions of the first agent are considered: minimum maximum cost of all the jobs, minimum total completion time, and minimum number of tardy jobs. For the second agent, an upper bound on the maximum allowable cost is assumed. We introduce efficient polynomial time solution algorithms for all cases.     

A Hermitian Cauchy formula on a domain with fractal boundary

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centered around the concept of monogenic functions, i.e. null solutions of a first order vector valued rotation invariant differential operator called Dirac operator, which factorizes the Laplacian; monogenic functions may thus also be seen as a generalization of holomorphic functions in the complex plane. Hermitian Clifford analysis offers yet a refinement of the Euclidean case; it focusses on the simultaneous null solutions, called Hermitian (or h-) monogenic functions, of two Hermitian Dirac operators which are invariant under the action of the unitary group. In Brackx et al. (2009) [8] a Clifford-Cauchy integral representation formula for h-monogenic functions has been established in the case of domains with smooth boundary, however the approach followed cannot be extended to the case where the boundary of the considered domain is fractal. At present, we investigate an alternative approach which will enable us to define in this case a Hermitian Cauchy integral over a fractal closed surface, leading to several types of integral representation formulae, including the Cauchy and Borel-Pompeiu representations.     

Convolutions for special classes of harmonic univalent functions

Ruscheweyh and Sheil-Small proved the PólyarSchoenberg conjecture that the class of convex analytic functions is closed under convolution or Hadamard product. They also showed that close-to-convexity is preserved under convolution with convex analytic functions. In this note, we investigate harmonic analogs. Beginning with convex analytic functions, we form certain harmonic functions which preserve close-to-convexity under convolution. An auxiliary function enables us to obtain necessary and sufficient convolution conditions for convex and starlike harmonic functions, which lead to sufficient coefficient bounds for inclusion in these classes.     

Two-dimensional steady-state general solution for isotropic thermoelastic materials with applications. II. Green’s function for two-phase infinite plane

Green’s function for isotropic thermoelastic two-phase infinite plane under a line heat source is established in this paper. By virtue of the fourth compact general solutions in Part I which is expressed in three harmonic functions, six new suitable harmonic functions with undetermined constants are constructed for the two semi-infinite planes of the two-phase infinite plane, respectively. The corresponding thermoelastic field can be obtained by substituting these harmonic functions into the general solution, and the undetermined constants can be determined by compatibility conditions and the equilibrium conditions. Numerical results are given graphically by contours.     

玛欣凯维奇函数与泊松核的一个新微分性质

将Stein[On the functions of Littlewood-Paley,Lusin,and Marcinkiewicz,Trans.Amer.Math.Soc.,1958,88:430-466]中的玛欣凯维奇函数的逆向不等式推广到一般情形.主要结果是对于n-维欧几里得空间k-阶球面调和函数空间的任意一基底,得到玛欣凯维奇函数的一般性的逆向不等式,即存在不依赖于函数f正常数C_p,使得||f||_p≤C_pΣ_(j=1)~N=1||μ_j(f)||_p,其中{μ_j(f)}_(j=1)~N是f的由这些球面调和函数生成的玛欣凯维奇函数.此外,对于任意的n-变元的k-阶调和多项式Q(x)以及泊松核P_t(x),有Q(D)P_t(x)=C_n k(tQ(x))/((|x|)~2+t~2~(n+2k+1)/2).     

Conservation laws for conformally invariant variational problems

We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations,..., etc.) in divergence form. These divergence-free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the classical regularity results known for 2-dimensional conformally invariant non-linear elliptic PDE (see [Hel]). It enables us also to establish new results. In particular we solve a conjecture by E. Heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by S. Hildebrandt [Hil1] claiming that critical points of continuously differentiable elliptic conformally invariant Lagrangian in two dimensions are continuous.     

Analysis of jump processes with nondegenerate jumping kernels

We prove regularity estimates for functions which are harmonic with respect to certain jump processes. The aim of this article is to extend the method of Bass–Levin (2002) [3] and Bogdan–Sztonyk (2005) [6] to more general processes. Furthermore, we establish a new version of the Harnack inequality that implies regularity estimates for corresponding harmonic functions.     

Positive Harmonic Functions that Vanish on a Subset of a Cylindrical Surface

This paper investigates positive harmonic functions on domains that are complementary to a subset of a cylindrical surface. It characterizes, both in terms of harmonic measure and of a Wiener-type criterion, those domains that admit minimal harmonic functions with exponential growth. Illustrative examples are provided. Two applications are also given. The first of these concerns minimal harmonic functions associated with an irregular boundary point, and amplifies a recent construction of Gardiner and Hansen. The second concerns the possible non-approximability of positive harmonic functions by integrable positive harmonic functions. This research was supported by Science Foundation Ireland under Grant 06/RFP/MAT057, and is also part of the programme of the ESF Network “Harmonic and Complex Analysis and Applications” (HCAA).     

Positional dimension-like functions of the type Ind

In Iliadis (2005) [4] positional dimension-like functions of the type ind are given. All these functions are studied only with respect to the property of universality. In a later paper by the present authors, and in two papers by V.V. Tkachuk (1981, 1982) (see [7] and [8]), these dimension-like functions have been studied with respect to the other standard properties of dimension theory. In R. Koga, Subspace-dimension with respect to total spaces, Master Thesis, Osaka Kyoiku University, 1998 (see also K.P. Hart, Jun-iti Nagata, J.E. Vaughan, Encyclopedia of General Topology, Elsevier Science Publishers, B.V., Amsterdam, 2004) a positional dimension-like function of the type Ind is given. Here we define new positional dimension-like functions of the type Ind, and present for all these functions, theorems concerning subspace theorems, partition theorems, sum theorems, and product theorems. Finally, we give some open questions concerning these functions.     

Periodic harmonic functions on lattices and points count in positive characteristic

This survey deals with pluri-periodic harmonic functions on lattices with values in a field of positive characteristic. We mention, as a motivation, the game “Lights Out” following the work of Sutner [20], Goldwasser- Klostermeyer-Ware [5], Barua-Ramakrishnan-Sarkar [2, 19], Hunzikel-Machiavello-Park [12] e.a.; see also [22, 23] for a more detailed account. Our approach uses harmonic analysis and algebraic geometry over a field of positive characteristic.      

A problem in the theory of mixing layers

The classical problem of the free steady mixing layer which is formed as the result of the interaction between two parallel homogeneous flows which move with different velocities and come into contact in a certain section is considered. Subject to the additional condition that the first derivative of the solution in a class of self-similar functions is positive, a boundary-value problem is studied, for values of the self-similarity index m > 0, which describes the mixing of two viscous streams of the same fluid for m = 1 [1] and for m = 2 [2]. The method of investigation used [3–5] enables the third-order non-linear equation to be reduced to a first-order equation and enables the corresponding solutions (Gz) to be constructed in a parametric form as a function of the values of m. A knowledge of the behaviour of the velocity profile of the main stream can be used to investigate the flow stability. The results obtained form the basis of the subsequent construction of the solution of Lock's problem [6] and the investigation of the uniqueness of the solutions obtained.     

Exact penalty functions in nonlinear programming

It is shown that the existence of a strict local minimum satisfying the constraint qualification of [16] or McCormick's [12] second order sufficient optimality condition implies the existence of a class of exact local penalty functions (that is ones with a finite value of the penalty parameter) for a nonlinear programming problem. A lower bound to the penalty parameter is given by a norm of the optimal Lagrange multipliers which is dual to the norm used in the penalty function.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation under Grant No. MCS74-20584 A02.     

On the shape of the mean residual lifetime function

Some general properties of the mean residual life (MRL) function are studied. The analysis is based on the shape of the corresponding failure rate. The conditions under which the failure rate and the reciprocal to the MRL function have asymptotically equivalent behaviour as t→∞ are discussed. The simplest non‐monotone shapes of the functions under consideration (bathtub and upside down bathtub) are also considered. The MRL functions for mixtures of distributions are described via the corresponding conditional probability density functions. The direct proportional model of mixing is characterized and some asymptotic results on the shape of the mixture MRL are obtained. Some simple examples are given. Copyright © 2002 John Wiley & Sons, Ltd.     

Some representations of midconvex set-valued functions

Summary In this note we establish conditions under which every midconvex set-valued function can be represented as sum of an additive function and a convex set-valued function. These results improve some theorems obtained in [8], [10] and [3]. Some results on local Jensen selections of midconvex set-valued functions are also given.     

Multiply Hyperharmonic Functions

We consider multiply hyperharmonic functions on the product space of two harmonic spaces in the sense of Constantinescu and Cornea. Earlier multiply superharmonic and harmonic functions have been studied in Brelot spaces notably by GowriSankaran. Important examples of Brelot spaces are solutions of elliptic differential equations. The theory of general harmonic spaces covers in addition to Brelot spaces also solution of parabolic differential equations. A locally lower bounded function is multiply hyperharmonic on the product space of two harmonic spaces if it is a hyperharmonic function in each variable for every fixed value of the other. We prove similar results as in Brelot spaces, but our approach is different. We study sheaf properties of multiply hyperharmonic functions. Our main theorem states that multiply hyperharmonic functions are lower semicontinuous and satisfy the axiom of completeness with respect to products of relatively compact sets. We also study nearly multiply hyperharmonic functions.     

Robust discrimination under a hierarchy on the scatter matrices

Under normality, Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244-261] investigated the asymptotic properties of the quadratic discrimination procedure under hierarchical models for the scatter matrices, that is: (i) arbitrary scatter matrices, (ii) common principal components, (iii) proportional scatter matrices and (iv) identical matrices. In this paper, we study the properties of robust quadratic discrimination rules based on robust estimates of the involved parameters. Our analysis is based on the partial influence functions of the functionals related to these parameters and allows to derive the asymptotic variances of the estimated coefficients under models (i)-(iv). From them, we conclude that the asymptotic variances verify the same order relations as those obtained by Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244-261] for the classical estimators. We also perform a Monte Carlo study for different sample sizes and different hierarchies which shows the advantage of using robust procedures over classical ones, when anomalous data are present. It also confirms that better rates of misclassification can be achieved if a more parsimonious model among all the correct ones is used instead of the standard quadratic discrimination.     

Some Spreading Properties of a Monotone Non-autonomous Parabolic Equation in a Periodic Cylinder

In our study of electrical networks we develop two themes: finding explicit formulas for special classes of functions defined on the vertices of a transient network, namely monopoles, dipoles, and harmonic functions. Secondly, our interest is focused on the properties of electrical networks supported on Bratteli diagrams. We show that the structure of Bratteli diagrams allows one to describe algorithmically harmonic functions as well as monopoles and dipoles. We also discuss some special classes of Bratteli diagrams (stationary, Pascal, trees), and we give conditions under which the harmonic functions defined on these diagrams have finite energy.