Tests for specific nonparametric relations between two distribution functions with applications

Let (X,Y) be a random vector and let G and H be the marginal distributions of X and Y, respectively. In this paper, we propose two tests, one of Kolmogorov‐Smirnov type and the other of Wilcoxon type, for the null hypothesis Ψ(G) = H against the alternative Ψ(G) < H, where Ψ() is a function such that Ψ(G) is a distribution function. The tests are based on the empirical distribution functions of the observations on X and Y, which are dependent. We obtain their asymptotic null distributions. A suspected relationship between the distribution functions of two dependent outcomes can be specified as a hypothesis to be tested in examples like the load sharing models, record values, and auction bidding models. As an application, we consider in detail the problem of testing the effect of load sharing in two component parallel systems.     

Boundedness, regularity and smoothness of universal Taylor series

Let Ω be an unbounded simply connected domain in satisfying some topological assumptions; for example let Ω be an open half-plane. We show that there exists a bounded holomorphic function on Ω which extends continuously on and is a universal Taylor series in Ω in the sense of Luh and Chui–Parnes with respect to any center. Our proof uses Arakeljan’s Approximation Theorem. Further we strengthen results of G. Costakis [2] concerning universal Taylor series with respect to one center in the sense of Luh and Chui–Parnes in the complement G of a compact connected set. We prove that such functions can be smooth on the boundary of G and be zero at ∞. If the universal approximation is also valid on ∂G, then the function can not be smooth on ∂G, but it may vanish at ∞. Our results are generic in natural Fréchet spaces of holomorphic functions. Received: 29 September 2005; revised: 21 February 2006     

The Chebyshev solution of certain matrix equations

Summary The problem of approximating ann x m matrixC by matrices of the formX A+B Y whereA, B, X andY are of appropriate size is considered. The measure of error is the supremum of the absolute values of the individual entries in the error matrix. The problem is closely related to that of approximating a bivariate functionf by sums of functions of the formxh+gy wherex andg are functions of the first variable alone andh andy are functions of the second variable. An old algorithm for constructing best approximations is described, and some of the properties of its convergence are discussed.     

$L^2$-Approximations of power and logarithmic functions with applications to numerical conformal mapping

Summary. For a bounded Jordan domain G with quasiconformal boundary L, two-sided estimates are obtained for the error in best polynomial approximation to functions of the form , and , where . Furthermore, Andrievskii's lemma that provides an upper bound for the norm of a polynomial in terms of the norm of is extended to the case when a finite linear combination (independent of n) of functions of the above form is added to . For the case when the boundary of G is piecewise analytic without cusps, the results are used to analyze the improvement in rate of convergence achieved by using augmented, rather than classical, Bieberbach polynomial approximants of the Riemann mapping function of G onto a disk. Finally, numerical results are presented that illustrate the theoretical results obtained. Received September 1, 1999 / Published online August 17, 2001     

The fraction of the bijections generating the near-ring of 0-preserving functions

Let 〈G, +〉 be a finite (not necessarily abelian) group. Then M₀(G) := {f : G → G| f (0) = 0} is a near-ring, i.e., a group which is also closed under composition of functions. In Theorem 4.1 we give lower and upper bounds for the fraction of the bijections which generate the near-ring M₀(G). From these bounds we conclude the following: If G has few involutions and the order of G is large, then a high fraction of the bijections generate the near-ring M₀(G). Also the converse holds: If a high fraction of the bijections generate M₀(G), then G has few involutions (compared to the order of G). Received: 10 January 2005     

A Class of Operators Similar to the Shift on H 2(G)

For a compact subset K in the complex plane, let A(K) denote the algebra of all functions continuous on K and analytic on K° and let R(K) denote the uniform closure of the rational functions with poles off K. Let G is a bounded open subset whose complement in the plane has a finite number of components. Suppose that and every function in H^∞(G) is the pointwise limit of a bounded sequence of functions in . The purpose of this paper is to characterize all subnormal operators similar to M_z, the operator of multiplication by the independent variable z on the Hardy space H²(G). We also characterize all bounded linear operators that are unitarily equivalent to M_z in the case when each of the components of G is simply connected. In particular, our similarity result extends a well-known result of W. Clary on the unit disk to multiply connected domains.     

Best-possible bounds on sets of bivariate distribution functions

The fundamental best-possible bounds inequality for bivariate distribution functions with given margins is the Fréchet–Hoeffding inequality: If H denotes the joint distribution function of random variables X and Y whose margins are F and G, respectively, then max(0,F(x)+G(y)−1)H(x,y)min(F(x),G(y)) for all x,y in [−∞,∞]. In this paper we employ copulas and quasi-copulas to find similar best-possible bounds on arbitrary sets of bivariate distribution functions with given margins. As an application, we discuss bounds for a bivariate distribution function H with given margins F and G when the values of H are known at quartiles of X and Y.     

Slice regular functions on real alternative algebras

In this paper we develop a theory of slice regular functions on a real alternative algebra A. Our approach is based on a well-known Fueter's construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in A and we prove a Cauchy integral formula for slice functions of class C¹.     

Birkhoff-functions that are bounded on prescribed sets

We prove the existence of entire functions which are universal under translations and bounded on certain prescribed sets. It is also shown that the family of all these universal functions is a dense but not a G_δ-subset in the space of entire functions provided with a natural metric. Received: 24 November 2004; revised: 12 April 2005     

Asymptotic Solution of a Differential System Related to the Generalized Hypergeometric Equation

We introduce a first‐order differential system Y′(x) =A(x)Y(x) on [a, ∞) particular cases of which are equivalent to standard forms of the generalized hypergeometric equation. Our purpose is to obtain the asymptotic solution of the system as x → ∞ by defining suitable transformations of the solution vector Y and using ideas from a unified asymptotic theory of differential systems. Thus our methods place the system within the scope of this unified theory, and they are independent of specialized properties of the Meijer G‐function solutions of generalized hypergeometric equations. As such, our methods are also capable of extension to other situations not covered by these special functions.     

Minimization of extended quadratic functions

Summary A method is given for constructing algorithms which generate conjugate search directions when applied to functions of the formF o q whereq is a positive-definite quadratic andF is a non-negative strictly increasing function of a single variable with a non-vanishing derivative. Perfect line searches are not assumed.     

Differentiable functions of quaternion variables

We investigate differentiability of functions defined on regions of the real quaternion field and obtain a noncommutative version of the Cauchy-Riemann conditions. Then we study the noncommutative analog of the Cauchy integral as well as criteria for functions of a quternion variable to be analytic. In particular, the quaternionic exponential and logarithmic functions are being considered. Main results include quaternion versions of Hurwitz' theorem, Mittag-Leffler's theorem and Weierstrass' theorem.     

Transitive partial actions of groups

J. Kellendonk and M. V. Lawson established that each partial action of a group G on a set Y can be extended to a global action of G on a set Y _G containing a copy of Y. In this paper we classify transitive partial group actions. When G is a topological group acting on a topological space Y partially and transitively we give a condition for having a Hausdorff topology on Y _G such that the global group action of G on Y _G is continuous and the injection Y into Y _G is an open dense equivariant embedding.      

Topological nearrings whose additive groups are Euclidean

Let (G,+) be a group with a locally compact Hausdorff topology for which the binary operation + is continuous. Those, binary operation * onG for which (G, +, *) is a topological nearring are described. In the case whereG is abelian, those binary operations * for which (G, +, *) is a topological ring are also described. Versions of these results are then obtained in the special case where the group is the topological Euclideann-group,R ⁿ. A family of binary operations * for which (R ⁿ, +, *)_is a topological nearring is then investigated in some detail. Most of these nearrings turn out to be planar. Their ideals are completely determined and we characterize those nearrings which are simple. The multiplicative semi-groups (R ⁿ, *) of these nearrings are then investigated. Green's relations are completely determined and it is shown that a number of familiar properties of semigroups are equivalent for these particular semigroups. Finally, all those binary operations * for which (R, +, *) is a topological nearring are completely described. It is determined when any two of these nearrings are isomorphic and for each of these nearrings, its automorphism group, is completely determined.     

Uniform Approximation by Entire Functions That Are All Bounded on a Given Set

For two closed sets F and G in the complex plane C, G C , we solve the following problem Under what conditions on F and G can every function f , continuous on F and analytic in its interior, be uniformly approximated by entire functions, each of which is bounded on G ? February 7, 1995. Date revised: October 31, 1995.     

Analysis of a Discrete-Time Finite-Capacity Single-Server Queue with Markovian-Arrival Process and Random-Bulk-Service: D-MAP/G Y /1/M

In this article, we consider a single-server, finite-capacity queue with random bulk service rule where customers arrive according to a discrete-time Markovian arrival process (D-MAP). The model is denoted by D-MAP/G ^Y /1/M where server capacity (bulk size for service) is determined by a random variable Y at the starting point of services. A simple analysis of this model is given using the embedded Markov chain technique and the concept of the mean sojourn time of the phase of underlying Markov chain of D-MAP. A complete solution to the distribution of the number of customers in the D-MAP/G ^Y /1/M queue, some computational results, and performance measures such as the average number of customers in the queue and the loss probability are presented.     

Gorenstein Projective,Injective, and Flat Complexes

Abstract

We study the classification of those finite groups G having a non-inner class preserving automorphism. Criteria for these automorphisms to be inner are established. Let G be a nilpotent-by-nilpotent group and S?∈?Sy l ₂(G). If S is abelian, generalized quaternion or S is dihedral, and in this case G is also metabelian, then Out _c (G)?=?1. If S is generalized quaternion, 𝒵(S)???𝒵(G) and S ₄ is not a homomorphic image of G, then Out _c (G)?=?1. As a consequence, it follows that the normalizer problem of group rings has a positive answer for these groups.     

The quantum McKay correspondence for polyhedral singularities

Let G be a polyhedral group, namely a finite subgroup of SO(3). Nakamura’s G-Hilbert scheme provides a preferred Calabi-Yau resolution Y of the polyhedral singularity ℂ³/G. The classical McKay correspondence describes the classical geometry of Y in terms of the representation theory of G. In this paper we describe the quantum geometry of Y in terms of R, an ADE root system associated to G. Namely, we give an explicit formula for the Gromov-Witten partition function of Y as a product over the positive roots of R. In terms of counts of BPS states (Gopakumar-Vafa invariants), our result can be stated as a correspondence: each positive root of R corresponds to one half of a genus zero BPS state. As an application, we use the Crepant Resolution Conjecture to provide a full prediction for the orbifold Gromov-Witten invariants of [ℂ³/G].     

Class-Preserving Coleman Automorphisms of Finite Groups

 Let G be a finite group whose Sylow 2-subgroups are either cyclic, dihedral, or generalized quaternion. It is shown that a class-preserving automorphism of G of order a power of 2 whose restriction to any Sylow subgroup of G equals the restriction of some inner automorphism of G is necessarily an inner automorphism. Interest in such automorphisms arose from the study of the isomorphism problem for integral group rings, see [6, 7, 13, 14].