On Integral Equations the Kernels of Which are Homogeneous Functions of Degree (−1)

The present paper deals with integral equations the kernels of which are homogeneous functions of degree (?1). Factorization approach to such equations is developed. The constructed operator factorization is applied to the equation with a positive symmetric kernel. We prove that in the conservative case, both the homogeneous equation and the corresponding nonhomogeneous equation with a positive free term can possess positive solutions simultaneously.     

When are Fredholm triples operator homotopic?

Fredholm triples are used in the study of Kasparov's -groups, and in Connes's noncommutative geometry. We define an absorption property for Fredholm triples, and give an if and only if condition for a Fredholm triple to be absorbing. We study the interaction of the absorption property with several of the more common equivalence relations for Fredholm triples. In general these relations are coarser than homotopy in the norm topology. We give simple conditions for an equivalence of triples to be implemented by an operator homotopy (i.e. a homotopy with respect to the norm topology). This can be expected to have applications in index theory, as we illustrate by proving two theorems of Pimsner-Popa-Voiculescu type. We show that there is some relationship with the interesting Toms-Winter characterization of -absorbing algebras, recently obtained as part of Elliott's classification program.

     

When are stars the largest cross-intersecting families?

We provide a necessary and sufficient condition for stars to be the largest cross-intersecting families.     

Confidence Intervals of Variance Functions in Generalized Linear Model

In this paper we introduce an appealing nonparametric method for estimating variance and conditional variance functions in generalized linear models （GLMs）, when designs are fixed points and random variables respectively, Bias-corrected confidence bands are proposed for the （conditional） variance by local linear smoothers. Nonparametric techniques are developed in deriving the bias-corrected confidence intervals of the （conditional） variance. The asymptotic distribution of the proposed estimator is established and show that the bias-corrected confidence bands asymptotically have the correct coverage properties. A small simulation is performed when unknown regression parameter is estimated by nonparametric quasi-likelihood. The results are also applicable to nonparamctric autoregressive times series model with heteroscedastic conditional variance.     

The Center Problem for a Linear Center Perturbed by Homogeneous Polynomials

The centers of the polynomial differential systems with homogeneous polynomials have been studied for the degrees s = 2, 3, 4, 5. for s = 2, 3, and partially classified for s = 4, 5. In this paper we recall and we give new centers for s = 6, 7 a linear center perturbed by They are completely classified these results for s = 2, 3, 4, 5,     

When Are All Functions from a Groupoid into Itself Endomorphisms or Translations?

This paper classifies all groupoids that have the property that every map from the groupoid to itself is either an endomorphism or a multiplication map.1991 Mathematics Subject Classification: 20M15     

When are inner mapping groups generated by conjugation maps?

A subloop of a loop Q is said to be normal if it is stabilized by all maps in the inner mapping group of Q. Here we show that in many cases, the inner mapping group of a Moufang loop is actually generated by conjugation maps. This includes any Moufang loop whose cubes generate either the entire loop or a subloop of index three. Such a result can be an extremely useful tool when proving that certain subloops are indeed normal just by showing that they are stabilized by the conjugation maps.     

When are intermediate processes of the same stochastic order?

Let Z_m⁽ⁿ⁾ represent the mth largest order statistic in a random sample of size n. Here we study the process Z_[mt]⁽ⁿ⁾, t > 0, where m(n) is an intermediate sequence such that m → ∞, m/n → 0 as n → ∞.     

When are solutions to nonlinear elliptic boundary value problems convex?

We extend and apply a concavity maximum principle from [10, 9, 7] to some nonlinear elliptic boundary problems and free boundary problems on convex domains Ω?IRⁿ. In particular we extend "convex dead core' results from n = 2 as in [4 ] to arbitrary n. We also show the convexity of the coincidence set in the obstacle problem under suitable assumptions.     

When are small subgraphs of a random graph normally distributed?

Summary LetG be a graph and letX _n count copies ofG in a random graphK(n,p). The random variable is asymptotically normally distributed if and only ifnp ^m andn ² (1-p), wherem=max {e(H)/|H|:HG}. In addition to, and in connection with this main result we investigate the formula for Var (X _n ) and the Poisson convergence ofX _n .Part of the research was done during the author's stay in Division of Mathematics and Science, St. John's University, Staten Island, New York     

Homogeneous nearly K?hler manifolds

The structure of nearly K?hler manifolds was studied by Gray in several articles, mainly in Gray (Math Ann 223:233?C248, 1976). More recently, a relevant progress on the subject has been done by Nagy. Among other results, he proved that a complete strict nearly K?hler manifold is locally a Riemannian product of homogeneous nearly K?hler spaces, twistor spaces over quaternionic K?hler manifolds and six-dimensional (6D) nearly K?hler manifolds, where the homogeneous nearly K?hler factors are also 3-symmetric spaces. In the present article, we show some further properties relative to the structure of nearly K?hler manifolds and, using the lists of 3-symmetric spaces given by Wolf and Gray, we display the exhaustive list of irreducible simply connected homogeneous strict nearly K?hler manifolds. For such manifolds, we give details relative to the intrinsic torsion and the Riemannian curvature.     

Some Analytical Properties of γ-Convex Functions in Normed Linear Spaces

For a fixed positive number γ, a real-valued function f defined on a convex subset D of a normed space X is said to be γ-convex if it satisfies the inequality

Plates made of piezoelectric materials: When are they really piezoelectric?

This paper aims at presenting in a synthetic way mathematical results that have been rigorously and recently derived by the authors. These results deal with the simplified but accurate modeling of linearly piezoelectric thin plates. It is shown how mathematical tools lead to two different situations linked to either sensors or actuators devices. Moreover we enlighten the fact that for some piezoelectric crystal classes, the coupling between the electrical and the mechanical effects disappear in the plate. Finally, we furnish a detailed example of such models in the case of a plate constituted by a 222 symmetry class material.     

Subordination Results for ρ-Valent Meromorphic Functions Associated with a Linear Operator

In this paper, by making use of the Hadamard products, we obtain some subordination results for certain family of meromorphic functions defined by using a new linear operator.     

Homogeneous Schr?dinger Operators on Half-Line

The differential expression L_m=-?_x²+(m²-1/4)x^-2{L_m=-partial_x^2+(m^2-1/4)x^{-2}} defines a self-adjoint operator H _m on L ²(0, ∞) in a natural way when m ² ≥ 1. We study the dependence of H _m on the parameter m show that it has a unique holomorphic extension to the half-plane Re m > −1, and analyze spectral and scattering properties of this family of operators.     

When are static and adjustable robust optimization problems with constraint-wise uncertainty equivalent?

Adjustable robust optimization (ARO) generally produces better worst-case solutions than static robust optimization (RO). However, ARO is computationally more difficult than RO. In this paper, we provide conditions under which the worst-case objective values of ARO and RO problems are equal. We prove that when the uncertainty is constraint-wise, the problem is convex with respect to the adjustable variables and concave with respect to the uncertain parameters, the adjustable variables lie in a convex and compact set and the uncertainty set is convex and compact, then robust solutions are also optimal for the corresponding ARO problem. Furthermore, we prove that if some of the uncertain parameters are constraint-wise and the rest are not, then under a similar set of assumptions there is an optimal decision rule for the ARO problem that does not depend on the constraint-wise uncertain parameters. Also, we show for a class of problems that using affine decision rules that depend on all of the uncertain parameters yields the same optimal objective value as when the rules depend solely on the non-constraint-wise uncertain parameters. Finally, we illustrate the usefulness of these results by applying them to convex quadratic and conic quadratic problems.