Covariance information based on the <Emphasis Type="Italic">t</Emphasis> prior

The exact distribution of the covariance between two populations is derived. It is assumed that one of the population means has a Student's t prior while the other is taken to come from one of normal, Student's t, Laplace, logistic or Bessel families (the five well-known symmetric distributions). The exact distribution is given in terms of the characteristic function. The calculations involve several special functions.     

Existence of the best <Emphasis Type="Italic">n</Emphasis>-term approximants for structured dictionaries

We extend the Pizzetti formulas, i.e., expansions of the solid and spherical means of a function in terms of the radius of the ball or sphere, to the case of real analytic functions and to functions of Laplacian growth. We also give characterizations of these functions. As an application we give a characterization of solutions analytic in time of the initial value problem for the heat equation ∂ _t u = Δu in terms of holomorphic properties of the solid and/or spherical means of the initial data.     

Local means,wavelet bases and wavelet isomorphisms in Besov‐Morrey and Triebel‐Lizorkin‐Morrey spaces

We consider local means with bounded smoothness for Besov‐Morrey and Triebel‐Lizorkin‐Morrey spaces. Based on those we derive characterizations of these spaces in terms of Daubechies, Meyer, Bernstein (spline) and more general r‐regular (father) wavelets, finally in terms of (biorthogonal) wavelets which can serve as molecules and local means, respectively. Hereby both, local means and wavelet decompositions satisfy natural conditions concerning smoothness and cancellation (moment conditions). Moreover, the given representations by wavelets are unique and yield isomorphisms between the considered function spaces and appropriate sequence spaces of wavelet coefficients. These wavelet representations lead to wavelet bases if, and only if, the function spaces coincide with certain classical Besov‐Triebel‐Lizorkin spaces.     

Approximation by Nörlund means of double Fourier series for Lipschitz functions

We study the rate of uniform approximation by Nörlund means of the rectangular partial sums of the double Fourier series of a function |(x, y) belonging to the class Lip α, 0 < α 1, on the two-dimensional torus −π < x, y π. As a special case we obtain the rate of uniform approximation by double Cesàro means.     

On <Emphasis Type="Italic">l</Emphasis><Subscript>2,<Emphasis Type="Italic">p</Emphasis></Subscript>-circle numbers

Circle numbers are defined to reflect the Euclidean area-content and, for p ≠ 2, suitably defined non-Euclidean circumference properties of the l _2,p -circles, p ∈ [1, ∞]. The resulting function is continuous, increasing, and takes all values from [2, 4]. The actually chosen dual l _2,p -geometry for measuring the arc-length is closely connected with a generalization of the method of indivisibles of Cavalieri and Torricelli in the sense that integrating such arc-lengths means measuring area content. Moreover, this approach enables one to look in a new way into the co-area formula of measure theory which says that integrating Euclidean arc-lengths does not yield area content except for p = 2. The new circle numbers play a natural role, e.g., as norming constants in geometric measure representation formulae for p-generalized uniform probability distributions on l _2,p -circles.     

Curves on a Kummer Surface in ℙ3, I

In this paper the asymptotic behaviour of the solutions of x' = A(t)x + h(t,x) under the assumptions of instability is studied, A(t) and h(t,x) being a square matrix and a vector function, respectively. The conditions for the existence of bounded solutions or solutions tending to the origin as t → ∞ are obtained. The method: the system is recasted to an equation with complex conjugate coordinates and this equation is studied by means of a suitable Lyapunov function and by virtue of the Wazevski topological method. Applications to a nonlinear differential equation of the second order are given.     

Approximation of functions by a new construction of Bernstein‐Chlodowsky operators: Theory and applications

The main motivation of this paper is to provide a generalization of Bernstein‐Chlodowsky type operators which depend on function τ by means of two sequences of functions. The newly defined operators fix the test function set {1, τ, τ²} . Then we present the approximation properties of newly defined operators, such as weighted approximation, degree of approximation and Voronovskaya type theorems. Finally, we present a series of numerical examples demonstrating the effectiveness of this newly defined Bernstein‐Chlodowsky operators for computing function approximation.     

Some difference sequence spaces defined by a sequence of <Emphasis Type="Italic">ϕ</Emphasis> -functions

Abstract  In this paper, we introduce new difference sequence spaces combining with de la Vallee-Poussin mean using by a sequence of modulus functions and ϕ -functions. We also studied connections between statistically convergence related with this space. Keywords Difference sequence, Modulus function, ϕ -function, De la Vallee-Poussin means, Statistical convergence Mathematics Subject Classification (2000) 46A45, 40F05, 46A80     

Quasi-<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> norm orthogonal Galerkin expansions in sums of Jacobi polynomials

In the study of differential equations on [ − 1,1] subject to linear homogeneous boundary conditions of finite order, it is often expedient to represent the solution in a Galerkin expansion, that is, as a sum of basis functions, each of which satisfies the given boundary conditions. In order that the functions be maximally distinct, one can use the Gram-Schmidt method to generate a set orthogonal with respect to a particular weight function. Here we consider all such sets associated with the Jacobi weight function, w(x) = (1 − x) ^α (1 + x) ^β . However, this procedure is not only cumbersome for sets of large degree, but does not provide any intrinsic means to characterize the functions that result. We show here that each basis function can be written as the sum of a small number of Jacobi polynomials, whose coefficients are found by imposing the boundary conditions and orthogonality to the first few basis functions only. That orthogonality of the entire set follows—a property we term “auto-orthogonality”—is remarkable. Additionally, these basis functions are shown to behave asymptotically like individual Jacobi polynomials and share many of the latter’s useful properties. Of particular note is that these basis sets retain the exponential convergence characteristic of Jacobi expansions for expansion of an arbitrary function satisfying the boundary conditions imposed. Further, the associated error is asymptotically minimized in an L ^p(α) norm given the appropriate choice of α = β. The rich algebraic structure underlying these properties remains partially obscured by the rather difficult form of the non-standard weighted integrals of Jacobi polynomials upon which our analysis rests. Nevertheless, we are able to prove most of these results in specific cases and certain of the results in the general case. However a proof that such expansions can satisfy linear boundary conditions of arbitrary order and form appears extremely difficult.     

Synchronization and antisynchronization of N‐coupled fractional‐order complex chaotic systems with ring connection

This paper is devoted to investigate synchronization and antisynchronization of N‐coupled general fractional‐order complex chaotic systems described by a unified mathematical expression with ring connection. By means of the direct design method, the appropriate controllers are designed to transform the fractional‐order error dynamical system into a nonlinear system with antisymmetric structure. Thus, by using the recently established result for the Caputo fractional derivative of a quadratic function and a fractional‐order extension of the Lyapunov direct method, several stability criteria are derived to ensure the occurrence of synchronization and antisynchronization among N‐coupled fractional‐order complex chaotic systems. Moreover, numerical simulations are performed to illustrate the effectiveness of the proposed design.     

On the Edgeworth expansion and the <Emphasis Type="Italic">M</Emphasis> out of <Emphasis Type="Italic">N</Emphasis> bootstrap accuracy for a Studentized trimmed mean

We investigate the second order accuracy of the M out of N bootstrap for a Studentized trimmed mean using the Edgeworth expansion derived in a previous paper. Some simulations, which support our theoretical results, are also given. The effect of extrapolation in conjunction with the M out of N bootstrap for Studentized trimmed means is briefly discussed. As an auxiliary result we obtain a Bahadur’s type representation for an M out of N bootstrap quantile. Our results supplement previous work on (Studentized) trimmed means by Hall and Padmanabhan [13], Bickel and Sakov [7], and Gribkova and Helmers [11].      

On noncommutative weighted local ergodic theorems on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces

In the present paper we consider a von Neumann algebra M with a faithful normal semi-finite trace τ, and {α _t }, a strongly continuous extension to L ^p (M, τ) of a semigroup of absolute contractions on L ¹(M, τ). By means of a non-commutative Banach Principle we prove for a Besicovitch function b and x ∊ L ^p (M, τ), that the averages 1/T ∫₀ ^T b(t)α _t (x)dt converge bilateral almost uniformly in L ^p (M, τ) as T → 0. Communicated by Dénes Petz     

Quantitative estimations of bivariate summation‐integral–type operators

In this paper, we study the approximation properties of bivariate summation‐integral–type operators with two parameters . The present work deals within the polynomial weight space. The rate of convergence is obtained while the function belonging to the set of all continuous and bounded function defined on ([0],∞)(×[0],∞) and function belonging to the polynomial weight space with two parameters, also convergence properties, are studied. To know the asymptotic behavior of the proposed bivariate operators, we prove the Voronovskaya type theorem and show the graphical representation for the convergence of the bivariate operators, which is illustrated by graphics using Mathematica. Also with the help of Mathematica, we discuss the comparison by means of the convergence of the proposed bivariate summation‐integral–type operators and Szász‐Mirakjan‐Kantorovich operators for function of two variables with two parameters to the function. In the same direction, we compute the absolute numerical error for the bivariate operators by using Mathematica and is illustrated by tables and also the comparison takes place of the proposed bivariate operators with the bivariate Szász‐Mirakjan operators in the sense of absolute error, which is represented by table. At last, we study the simultaneous approximation for the first‐order partial derivative of the function.     

Unified analysis of <Emphasis Type="Italic">BMAP</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1 cyclic polling models

In this paper we present a unified analysis of the BMAP/G/1 cyclic polling model and its application to the gated and exhaustive service disciplines as examples. The applied methodology is based on the separation of the analysis into service discipline independent and dependent parts. New expressions are derived for the vector-generating function of the stationary number of customers and for its mean in terms of vector quantities depending on the service discipline. They are valid for a broad class of service disciplines and both for zero- and nonzero-switchover-times polling models. We present the service discipline specific solution for the nonzero-switchover-times model with gated and exhaustive service disciplines. We set up the governing equations of the system by using Kronecker product notation. They can be numerically solved by means of a system of linear equations. The resulting vectors are used to compute the service discipline specific vector quantities.     

On the two-dimensional Marcinkiewicz means with respect to Walsh–Kaczmarz system

In this paper we prove that the maximal operator of the Marcinkiewicz means of two-dimensional integrable functions with respect to the Walsh–Kaczmarz system is of weak type (1,1). Moreover, the Marcinkiewicz means converge to f almost everywhere, for any integrable function f.     

Approximation by Nörlund means of Walsh-Fourier series

We study the rate of approximation by Nörlund means for Walsh-Fourier series of a function in L^p and, in particular, in Lip(α, p) over the unit interval [0, 1), where α > 0 and 1 p ∞. In case p = ∞, by L^p we mean C_W, the collection of the uniformly W-continuous functions over [0, 1). As special cases, we obtain the earlier results by Yano, Jastrebova, and Skvorcov on the rate of approximation by Cesàro means. Our basic observation is that the Nörlund kernel is quasi-positive, under fairly general assumptions. This is a consequence of a Sidon type inequality. At the end, we raise two problems.     

（0,1;0）-INTERPOLATION ON INFINITE INTERVAL （-∞, ＋∞）

In this paper, we study the explicit representation and convergence of （0, 1;0）-interpolation on infisite interval, which means to determine a polynomial of degree ≤ 3n - 2 when the function values areprescribed at two set of points namely the zeros of Hn（x） and H′n （x） and the first derivatives at the zerosof H′n（x）.     

Approximations of pseudo-Boolean functions; applications to game theory

This paper studies the approximation of pseudo-Boolean functions by linear functions and more generally by functions of (at most) a specified degree. Here a pseudo-Boolean function means a real valued function defined on {0,1} ⁿ , and its degree is that of the unique multilinear polynomial that expresses it; linear functions are those of degree at most one. The approximation consists in choosing among all linear functions the one which is closest to a given function, where distance is measured by the Euclidean metric onR ²ⁿ . A characterization of the best linear approximation is obtained in terms of the average value of the function and its first derivatives. This leads to an explicit formula for computing the approximation from the polynomial expression of the given function. These results are later generalized to handle approximations of higher degrees, and further results are obtained regarding the interaction of approximations of different degrees. For the linear case, a certain constrained version of the approximation problem is also studied. Special attention is given to some important properties of pseudo-Boolean functions and the extent to which they are preserved in the approximation. A separate section points out the relevance of linear approximations to game theory and shows that the well known Banzhaf power index and Shapley value are obtained as best linear approximations of the game (each in a suitably defined sense).Supported by the Air Force Office of Scientific Research (under grant number AFOSR 89-0512 and AFOSR 90-0008 to Rutgers University), as well as the National Science Foundation (under grant number DMS 89-06870).