Unified Approach to Robust Performance of a Class of Transfer Functions with Multilinearly Correlated Perturbations

In this paper, we study the envelope of the Nyquist plots generated by a family of stable transfer functions with multilinearly correlated perturbations and show that the outer Nyquist envelope is generated by the Nyquist plots of the vertices of this family. We then apply this result to calculating the maximal H -norm and verifying the strict positive-realness condition for uncertain transfer function families. Vertex results for robust performance analysis are established. We also study the collection of Popov plots of this transfer function family and show that a large portion of its outer boundary comes from the vertices of this family. This result is then applied to the interval transfer function family to obtain a strong Kharitonov-like theorem.     

Robust statistical inference based on the C-divergence family

This paper describes a family of divergences, named herein as the C-divergence family, which is a generalized version of the power divergence family and also includes the density power divergence family as a particular member of this class. We explore the connection of this family with other divergence families and establish several characteristics of the corresponding minimum distance estimator including its asymptotic distribution under both discrete and continuous models; we also explore the use of the C-divergence family in parametric tests of hypothesis. We study the influence function of these minimum distance estimators, in both the first and second order, and indicate the possible limitations of the first-order influence function in this case. We also briefly study the breakdown results of the corresponding estimators. Some simulation results and real data examples demonstrate the small sample efficiency and robustness properties of the estimators.

     

Invariant families of sets and zero dynamics of linear nonstationary control systems with scalar input and output

We obtain conditions for the existence of a linear feedback providing the existence of a family (N _t ) of subspaces such that this family is invariant under the closed system and the output variable is zero on all motions lying in this family.     

r-convex functions

A family of real functions, calledr-convex functions, which represents a generalization of the notion of convexity is introduced. This family properly includes the family of convex functions and is included in the family of quasiconvex functions. Some properties ofr-convex functions are derived and relations with other generalizations of convex functions are discussed.Portions of this paper were presented at the 7th Mathematical Programming Symposium 1970, The Hague, The Netherlands.Research for this paper was supported in part by the Gerard Swope Fund at the Technion.     

Associated families of pluriharmonic maps and isotropy

Like minimal surface immersions in 3-space, pluriharmonic maps into symmetric spaces allow a one-parameter family of isometric deformations rotating the differential (“associated family”); in fact, pluriharmonic maps are characterized by this property. We give a geometric proof of this fact and investigate the “isotropic” case where this family is constant. It turns out that isotropic pluriharmonic maps arise from certain holomorphic maps into flag manifolds. Further, we also consider higher dimensional generalizations of constant mean curvature surfaces which are Kähler submanifolds with parallel (1,1) part of their soecond fundamental form; under certain restrictions there are also characterized by having some kind of (“weak”) associated family. Examples where this family is constant arise from extrinsic Kähler symmetric spaces.     

Enlargements of positive sets

In this paper we introduce the notion of enlargement of a positive set in SSD spaces. To a maximally positive set A we associate a family of enlargements E(A) and characterize the smallest and biggest element in this family with respect to the inclusion relation. We also emphasize the existence of a bijection between the subfamily of closed enlargements of E(A) and the family of so-called representative functions of A. We show that the extremal elements of the latter family are two functions recently introduced and studied by Stephen Simons. In this way we extend to SSD spaces some former results given for monotone and maximally monotone sets in Banach spaces.     

A characterization of the one parameter exponential family of distributions by monotonicity of likelihood ratios

Summary Any one parameter exponential family of distributions has monotone likelihood ratios. As the product probabilities of n identical distributions of an exponential family form again an exponential family, it has monotone likelihood ratios for arbitrary n. Furthermore, the members of an exponential family are mutually absolutely continuous. In Part 1, we show that these properties uniquely characterize the exponential family. The application of this result to the theory of testing hypotheses (Part 2) shows that if a family of mutually absolutely continuous distributions has uniformly most powerful tests for arbitrary levels of significance, and arbitrary sample sizes, then it is necessarily an exponential family.The research was done while this author was a Visiting Professor in the Department of Statistics at the University of Chicago. It was supported by Research Grants Nos. NSF-G10368 and NSF-G21058 from the Division of Mathematical, Physical and Engineering Sciences of the National Science Foundation.     

Associated families of pluriharmonic maps¶and isotropy

Like minimal surface immersions in 3-space, pluriharmonic maps into symmetric spaces allow a one-parameter family of isometric deformations rotating the differential (“associated family”); in fact, pluriharmonic maps are characterized by this property. We give a geometric proof of this fact and investigate the “isotropic” case where this family is constant. It turns out that isotropic pluriharmonic maps arise from certain holomorphic maps into flag manifolds. Further, we also consider higher dimensional generalizations of constant mean curvature surfaces which are K?hler submanifolds with parallel (1,1) part of their soecond fundamental form; under certain restrictions there are also characterized by having some kind of (“weak”) associated family. Examples where this family is constant arise from extrinsic K?hler symmetric spaces. Received: 8 July 1997     

Box ladders in a noninteger dimension

We construct a family of triangle-ladder diagrams that can be calculated using the Belokurov-Usyukina loop reduction technique in d=4?2? dimensions. The main idea of the approach we propose is to generalize this loop reduction technique existing in d=4 dimensions. We derive a recurrence relation between the result for an L-loop triangle-ladder diagram of this family and the result for an (L-1)-loop triangleladder diagram of the same family. Because the proposed method combines analytic and dimensional regularizations, we must remove the analytic regularization at the end of the calculation by taking the double uniform limit in which the parameters of the analytic regularization vanish. In the position space, we obtain a diagram in the left-hand side of the recurrence relations in which the rung indices are 1 and all other indices are 1 - ? in this limit. Fourier transforms of diagrams of this type give momentum space diagrams with rung indices 1 - ? and all other indices 1. By a conformal transformation of the dual space image of this momentum space representation, we relate such a family of triangle-ladder momentum diagrams to a family of box-ladder momentum diagrams with rung indices 1 - ? and all other indices 1. Because any diagram from this family is reducible to a one-loop diagram, the proposed generalization of the Belokurov-Usyukina loop reduction technique to a noninteger number of dimensions allows calculating this family of box-ladder diagrams in the momentum space explicitly in terms of Appell’s hypergeometric function F ₄ without expanding in powers of the parameter ? in an arbitrary kinematic region in the momentum space.     

Variable metric algorithms: Necessary and sufficient conditions for identical behavior of nonquadratic functions

Huang (Ref. 1) introduced a general family of variable metric updating formulas and showed that, for a convex quadratic function, all members of this family generate the same sequence of points and converge in at mostn steps. Huang and Levy (Ref. 2) published numerical data showing the behavior of this family for nonquadratic functions and concluded that this family could be divided into subsets that also generate sequences of identical points on more general functions. In this paper, the necessary and sufficient conditions for a group of algorithms to form part of one of these subsets are given.     

Third order iterative methods free from second derivative for nonlinear systems

In this work we present a family of predictor-corrector methods free from second derivative for solving nonlinear systems. We prove that the methods of this family are of third order convergence. We also perform numerical tests that allow us to compare these methods with Newton’s method. In addition, the numerical examples improve theoretical results, showing super cubic convergence for some methods of this family.     

A family of nonisomorphic Markov random fields

It was recently shown that there exists a family of ℤ² Markov random fields which areK but are not isomorphic to Bernoulli shifts [4]. In this paper we show that most distinct members of this family are not isomorphic. This implies that there is a two parameter family of ℤ² Markov random fields of the same entropy, no two of which are isomorphic.     

A family of (2+1)-dimensional hydrodynamic type systems possessing a pseudopotential

We construct a family of integrable hydrodynamic type systems with three independent and n ≥ 2 dependent variables in terms of solutions of a linear system of PDEs with rational coefficients. We choose the existence of a pseudopotential as a criterion of integrability. In the case n = 2 this family is a general solution of the classification problem for such systems. We also give an elliptic analog of this family in the case n > 2.     

Generalized Coherent States for the q-Oscillator Associated with Discrete q-Hermite Polynomials

We continue studying generalized coherent states of the Barut-Girardello type for oscillator-like systems related to a given set of orthogonal polynomials. In this paper we construct a family of coherent states associated with discrete q-Hermite polynomials of the II-type and prove the overcompleteness of this family by constructing the measure in the unity decomposition for this family of coherent states. Bibliography: 49 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 48–66.     

A Padé family of iterations for the matrix sign function and related problems

In this paper we consider the Pad'e family of iterations for computing the matrix sign function and the Padé family of iterations for computing the matrix p‐sector function. We prove that all the iterations of the Padé family for the matrix sign function have a common convergence region. It completes a similar result of Kenney and Laub for half of the Padé family. We show that the iterations of the Padé family for the matrix p‐sector function are well defined in an analogous common region, depending on p. For this purpose we proved that the Padé approximants to the function (1?z)^?σ, 0<σ<1, are a quotient of hypergeometric functions whose poles we have localized. Furthermore we proved that the coefficients of the power expansion of a certain analytic function form a positive sequence and in a special case this sequence has the log‐concavity property. Copyright © 2011 John Wiley & Sons, Ltd.     

On optimal superimposed codes

A (w,r) cover‐free family is a family of subsets of a finite set such that no intersection of w members of the family is covered by a union of r others. A binary (w,r) superimposed code is the incidence matrix of such a family. Such a family also arises in cryptography as a concept of key distribution patterns. In this paper, we develop a method of constructing superimposed codes and prove that some superimposed codes constructed in this way are optimal. © 2003 Wiley Periodicals, Inc. J Combin Designs 12: 79–71, 2004.     

On Mardia's and Song's measures of kurtosis in elliptical distributions

The main objective of this paper is the calculation and the comparative study of two general measures of multivariate kurtosis, namely Mardia's measure β_2,p and Song's measure S(f). In this context, general formulas for the said measures are derived for the broad family of the elliptically contoured symmetric distributions and also for specific members of this family, like the multivariate t-distribution, the multivariate Pearson type II, the multivariate Pearson type VII, the multivariate symmetric Kotz type distribution and the uniform distribution in the unit sphere. Analytic expressions for computing Shannon and Rényi entropies are obtained under the elliptic family. The behaviour of Mardia's and Song's measures, their similarities and differences, possible interpretations and uses in practice are investigated by comparing them in specific members of the elliptic family of multivariate distributions. An empirical estimator of Song's measure is moreover proposed and its asymptotic distribution is investigated under the elliptic family of multivariate distributions.     

Uncertainty principle and quantum Fisher information

A family of inequalities, related to the uncertainty principle, has been recently proved by S. Luo, Z. Zhang, Q. Zhang, H. Kosaki, K. Yanagi, S. Furuichi and K. Kuriyama. We show that the inequalities have a geometric interpretation in terms of quantum Fisher information. Using this formulation one may naturally ask if this family of inequalities can be further extendend, for example to the RLD quantum Fisher information. We show that this is impossible by producing a family of counterexamples.     

Family of multivariate generalized t distributions

In this paper, we introduce a new family of multivariate distributions as the scale mixture of the multivariate power exponential distribution introduced by Gómez et al. (Comm. Statist. Theory Methods 27(3) (1998) 589) and the inverse generalized gamma distribution. Since the resulting family includes the multivariate t distribution and the multivariate generalization of the univariate GT distribution introduced by McDonald and Newey (Econometric Theory 18 (11) (1988) 4039) we call this family as the “multivariate generalized t-distributions family”, or MGT for short. We show that this family of distributions belongs to the elliptically contoured distributions family, and investigate the properties. We give the stochastic representation of a random variable distributed as a multivariate generalized t distribution. We give the marginal distribution, the conditional distribution and the distribution of the quadratic forms. We also investigate the other properties, such as, asymmetry, kurtosis and the characteristic function.