A priori estimates for the scalar curvature equation on <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

We obtain a priori estimates for solutions to the prescribed scalar curvature equation on S ³. The usual non-degeneracy assumption on the curvature function is replaced by a new condition, which is necessary and sufficient for the existence of a priori estimates, when the curvature function is a positive Morse function.     

Mean–range based distribution-free procedures to minimize “overage” and “underage” costs

We discuss several mean–range based distribution-free decision procedures to minimize several “overage” and “underage” cost functions. For a general cost function, we identify the most favorable distribution and the least favorable distribution associated with the random variable of interest and determine the upper and lower bounds for the cost function. For the quadratic cost function, we recommend the min–max distribution-free decision. For the linear cost function, we identify the range of potential optimal decisions and recommend a hybrid distribution-free decision that has several desirable properties. We provide several numerical examples to demonstrate the robustness of the proposed distribution-free decisions.     

Nonparametric inference for sequential <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> systems

The k-out-of-n model is commonly used in reliability theory. In this model the failure of any component of the system does not influence the components still at work. Sequential k-out-of-n systems have been introduced as an extension of k-out-of-n systems where the failure of some component of the system may influence the remaining ones. We consider nonparametric estimation of the cumulative hazard function, the reliability function and the quantile function of sequential k-out-of-n systems. Furthermore, nonparametric hypothesis testing for sequential k-out-of-n-systems is examined. We make use of counting processes to show strong consistency and weak convergence of the estimators and to derive the asymptotic distribution of the test statistics.     

Singularités topologiques des systoles généralisées

We prove that a generalized systole defined by functions with positive definite Hessians is a topological Morse function such that critical points are eutactic points. We deduce that systole is a topological Morse function on Teichmüller space.     

<Emphasis Type="Italic">W</Emphasis><Superscript>1,<Emphasis Type="Italic">p</Emphasis></Superscript>(R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)-boundedness for maximal functions

In this paper, we get W ^1,p (R ⁿ )-boundedness for tangential maximal function and nontangential maximal function, which improves J.Kinnunen, P.Lindqvist and Tananka’s results. Supported by the key Academic Discipline of Zhejiang Province of China under Grant No.2005 and the Zhejiang Provincial Natural Science Foundation of China.     

Multiple <Emphasis Type="Italic">q</Emphasis>-zeta functions and multiple <Emphasis Type="Italic">q</Emphasis>-polylogarithms

For every positive integer d we define the q-analog of multiple zeta function of depth d and study its properties, generalizing the work of Kaneko et al. who dealt with the case d=1. We first analytically continue it to a meromorphic function on ℂ ^d with explicit poles. In our Main Theorem we show that its limit when q ↑1 is the ordinary multiple zeta function. Then we consider some special values of these functions when d=2. At the end of the paper we also propose the q-analogs of multiple polylogarithms by using Jackson’s q-iterated integrals and then study some of their properties. Our definition is motivated by those of Koornwinder and Schlesinger although theirs are slightly different from ours. Partially supported by NSF grant DMS0139813 and DMS0348258.     

Family of Pearson discrete distributions generated by the univariate hypergeometric function 3F2(α1, α2, α3; γ1, γ2; λ)

In this work we present a study of the Pearson discrete distributions generated by the hypergeometric function ₃F₂(α₁, α₂, α₃;γ₁, γ₂; λ), a univariate extension of the Gaussian hypergeometric function, through a constructive methodology. We start from the polynomial coefficients of the difference equation that lead to such a function as a solution. Immediately after, we obtain the generating probability function and the differential equation that it satisfies, valid for any admissible values of the parameters. We also obtain the differential equations that satisfy the cumulants generating function, moments generating function and characteristic function, From this point on, we obtain a relation in recurrences between the moments about the origin, allowing us to create an equation system for estimating the parameters by the moment method. We also establish a classification of all possible distributions of such type and conclude with a summation theorem that allows us study some distributions belonging to this family. © 1997 by John Wiley & Sons, Ltd.     

On the cubic <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> spline interpolant to the Heaviside function

We prove that the univariate interpolating cubic L ₁ spline to the Heaviside function at three sites to the left of the jump and three sites to the right of the jump entirely agrees with the Heaviside function except in the middle interval where it is the interpolating cubic with zero slopes at the end point. This shows that there is no oscillation near the discontinuous point i.e. no Gibbs’ phenomenon.     

Ball remotality of <Emphasis Type="Italic">M</Emphasis>-ideals in some function spaces and function algebras

In this paper, we show that in many function spaces and function algebras, the unit ball of any M-ideal is densely remotal.     

Pointwise behaviour of <Emphasis Type="Italic">M</Emphasis><Superscript>1,1</Superscript> Sobolev functions

Our main objective is to study Haj?asz type Sobolev functions with the exponent one on metric measure spaces equipped with a doubling measure. We show that a discrete maximal function is bounded in the Haj?asz space with the exponent one. This implies that every such function has Lebesgue points outside a set of capacity zero. We also show that every Haj?asz function coincides with a Hölder continuous Haj?asz function outside a set of small Hausdorff content. Our proofs are based on Sobolev space estimates for maximal functions.     

Some properties of compatibility function based on <Emphasis Type="Italic">T</Emphasis>-operation

In this paper, the concept of compatibility function is proposed for fuzzy value with respect to a kind of the binomial operations which is called t-operation. Besides the continuity of compatibility function on three kinds of metric spaces are discussed, respectively.     

Applications of Feller–Reuter–Riley Transition Functions

A Feller–Reuter–Riley function is a Markov transition function whose corresponding semigroup maps the set of the real-valued continuous functions vanishing at infinity into itself. The aim of this paper is to investigate applications of such functions in the dual problem, Markov branching processes, and the Williams-matrix. The remarkable property of a Feller–Reuter–Riley function is that it is a Feller minimal transition function with a stable q-matrix. By using this property we are able to prove that, in the theory of branching processes, the branching property is equivalent to the requirement that the corresponding transition function satisfies the Kolmogorov forward equations associated with a stable q-matrix. It follows that the probabilistic definition and the analytic definition for Markov branching processes are actually equivalent. Also, by using this property, together with the Resolvent Decomposition Theorem, a simple analytical proof of the Williams' existence theorem with respect to the Williams-matrix is obtained. The close link between the dual problem and the Feller–Reuter–Riley transition functions is revealed. It enables us to prove that a dual transition function must satisfy the Kolmogorov forward equations. A necessary and sufficient condition for a dual transition function satisfying the Kolmogorov backward equations is also provided.     

On set-valued solution of an <Emphasis Type="Italic">n</Emphasis>-order iterative functional equation

A n-order iterative functional equation is considered for set-value function in a nonempty convex compact subset D in Banach space. A result on the existence and uniqueness of its solution is presented without monotonicity. Besides, continuous dependence of the solution upon the given function is obtained.     

ℋ︁‐matrices for the convection‐diffusion equation

Hierarchical matrices (ℋ︁‐matrices) provide a technique for the sparse approximation of large, fully populated matrices. This technique has been shown to be applicable to stiffness matrices arising in boundary element method applications where the kernel function displays certain smoothness properties. The error estimates for an approximation of the kernel function by a separable function can be carried over directly to error estimates for an approximation of the stiffness matrix by an ℋ︁‐matrix, using a certain standard partitioning and admissibility condition for matrix blocks. Similarly, ℋ︁‐matrix techniques can be applied in the finite element context where it is the inverse of the stiffness matrix that is fully populated. Here one needs a separable approximation of Green's function of the underlying boundary value problem in order to prove approximability by matrix blocks of low rank. Unfortunately, Green's function for the convection‐diffusion equation does not satisfy the required smoothness properties, hence prohibiting a straightforward generalization of the separable approximation through Taylor polynomials. We will use Green's function to motivate a modification in the (hierarchical) partitioning of the index set and as a consequence the resulting hierarchy of block partitionings as well as the admissibility condition. We will illustrate the effect of the proposed modifications by numerical results.     

On the Multiplicity of a C^∞-differentiable Function-germ

Risler ＆ Trotman in 1997 proved that the multiplicity of an analytic function germ is left-right lipschitz invariant, which provided a partial answer to Zariski conjecture. In this note, based on the recent work of Comte, Milman ＆ Trotman, we generalize the work of them to prove that the multiplicity of a C^∞ differentiable function germ is also left-right lipschitz invariant.     

The strong <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-<Emphasis Type="Italic">greedy</Emphasis> property of the Walsh system

For any 0 < ? < 1 one can find a measurable set E ? [0, 1] with the measure |E| > 1 ? ? such that for each function f(x) ε L ¹ (0, 1) a function g(x) ε L ¹ (0, 1) exists such that it coincides with f (x) on E, its Fourier—Walsh series converges to it in the metric of L ¹ (0, 1), and all nonzero terms of the sequence of Fourier coefficients of the new function obtained by the Walsh system have the modulo decreasing order; consequently, the greedy algorithm for this function converges to it in the L ¹ (0, 1)-norm.     

Two-Sided Optimal Bounds for Green Functions of Half-Spaces for Relativistic <Emphasis Type="Italic">α</Emphasis>-Stable Process

The purpose of this paper is to find optimal estimates for the Green function of a half-space of the relativistic α -stable process with parameter m on ℝ ^d space. This process has an infinitesimal generator of the form mI–(m ^2/α I–Δ) ^α/2, where 0<α<2, m>0, and reduces to the isotropic α-stable process for m=0. Its potential theory for open bounded sets has been well developed throughout the recent years however almost nothing was known about the behaviour of the process on unbounded sets. The present paper is intended to fill this gap and we provide two-sided sharp estimates for the Green function for a half-space. As a byproduct we obtain some improvements of the estimates known for bounded sets. Our approach combines the recent results obtained in Byczkowski et al. (Bessel Potentials, Hitting Distributions and Green Functions (2006) (preprint). ), where an explicit integral formula for the m-resolvent of a half-space was found, with estimates of the transition densities for the killed process on exiting a half-space. The main result states that the Green function is comparable with the Green function for the Brownian motion if the points are away from the boundary of a half-space and their distance is greater than one. On the other hand for the remaining points the Green function is somehow related the Green function for the isotropic α-stable process. For example, for d≥3, it is comparable with the Green function for the isotropic α-stable process, provided that the points are close enough. Research supported by KBN Grants.     

On the sufficiency of <Emphasis Type="Italic">c</Emphasis>-cyclical monotonicity for optimality of transport plans

We consider a mass transport problem among Polish spaces, and we show that a transport plan concentrated in a c-cyclical monotone set is optimal if the cost function is continuous and possibly + ∞ valued; the same result is proved for a generic l.s.c. cost function in the case when the measures are purely atomic. This generalizes the previously known results.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-consistency of wavelet estimators

This paper deals with the L ^p -consistency of wavelet estimators for a density function based on size-biased random samples. More precisely, we firstly show the L ^p -consistency of wavelet estimators for independent and identically distributed random vectors in R ^d . Then a similar result is obtained for negatively associated samples under the additional assumptions d = 1 and the monotonicity of the weight function.     

Convergence of greedy algorithm in Walsh system in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

Convergence of the greedy algorithm in Walsh system in L _p , p > 1 is studied. It is proved that there exists a function in L _p , 1 < p < 2, with greedy algorithm not converging in measure to that function. A continuous function with divergent in L _p , p > 2, greedy algorithm is constructed and sufficient conditions for convergence of the greedy algorithm in L _p , p > 1 are given.