Subregion-Adaptive Integration of Functions Having a Dominant Peak

Abstract

Many statistical multiple integration problems involve integrands that have a dominant peak. In applying numerical methods to solve these problems, statisticians have paid relatively little attention to existing quadrature methods and available software developed in the numerical analysis literature. One reason these methods have been largely overlooked, even though they are known to be more efficient than Monte Carlo for well-behaved problems of low dimensionality, may be that when applied naively they are poorly suited for peaked-integrand problems. In this article we use transformations based on “split t” distributions to allow the integrals to be efficiently computed using a subregion-adaptive numerical integration algorithm. Our split t distributions are modifications of those suggested by Geweke and may also be used to define Monte Carlo importance functions. We then compare our approach to Monte Carlo. In the several examples we examine here, we find subregion-adaptive integration to be substantially more efficient than importance sampling.     

Jackson-type Theorems on a Finite Interval with Weights Having Non-symmetric Inner Singularities

We consider Jackson-type theorems for weighted polynomial approximation when the weight has certain type of zeros inside and/or at the end-points of the interval. The approximating polynomials are constructed from truncated convolution integrals.     

On a Two-Stage Procedure Having Second-Order Properties with Applications

Under a fairly general setup, we first modify the Stein-type two-stage methodology in order to incorporate some partial information in the form of a known and positive lower bound for the otherwise unknown nuisance parameter, 0(> 0). This revised methodology is then shown to enjoy various customary second-order properties and expansions for functions of the associated stopping variable, under appropriate conditions. Such general machineries are later applied in different types of estimation as well as selection and ranking problems, giving a sense of a very broad spectrum of possibilities. This constitutes natural extensions of these authors' earlier paper (Mukhopadhyay and Duggan (1997a, Sankhya Ser. A, 59, 435 448)) on the fixed-width confidence interval estimation problem exclusively for the mean of a normal distribution having an unknown variance.     

Optimal M/G/1 Server Location on a Network Having a Fixed Facility

This paper considers the problem of locating a single mobile service unit on a network G where the servicing of a demand includes travel time to a permanent facility which is located at a predetermined point on G. Demands for service, which occur solely on the nodes of the network, arrive in a homogeneous Poisson manner. The server, when free, can be immediately dispatched to a demand: the service unit travels to the demand, performs some on-scene service, continues to the permanent facility, where off-scene service is rendered, and then it returns to its ‘home’ location, where possibly additional off-scene service is given. Previous research has examined the same problem, however without the presence of a permanent facility. The paper discusses methods of solving two cases when the server is unable to be immediately dispatched to service a demand: (1) the zero-capacity queueing system; (2) the infinite-capacity queueing system. For the first case we prove that the optimal location is included in a small set of points in the network, and we show how to find this set. For the second case, we present an 0(n³) algorithm (n is the number of nodes) to obtain the optimal location.     

Complex Polynomial Vector Fields Having a Finitely Curved Orbit

The author proves that a non-singular polynomial vector field without invariant lines and having an entire finitely curved transcendent orbit on C^2 must be equivalent to a trivial vector field by a holomorphic change of coordinates. Other classification results are obtained for polynomial vector fields having a finitely curved orbit on C^2.     

W-Surfaces Having Some Properties

We investigate some characteristic properties of specific Weingarten surfaces in the Euclidean space E³ using the nets of the lines of curvature resp. the asymptotic lines on both central surfaces of them.     

Higher-Order Differential Equations Having a Singularity in an Interior Point

Ordinary differential equations of an arbitrary order having a non-integrable singularity inside the interval are considered under additional matching conditions for solutions at the singular point. We construct special fundamental systems of solutions for this class of differential equations, study their asymptotical, analytical and structural properties and the behavior of the corresponding Stokes multipliers. These fundamental systems of solutions are used in spectral analysis of differential operators with singularities.     

Kant, Spinoza, and the Metaphysics of the Ontological Proof

This paper provides an interpretation and evaluation of Spinoza’s highly original version of the ontological proof in terms of the concept of substance instead of the concept of perfection in the first book of his Ethics. Taking the lead from Kant’s critique of ontological arguments in the Critique of Pure Reason, the paper explores the underlying ontological and epistemological presuppositions of Spinoza’s proof. The main topics of consideration are the nature of Spinoza’s definitions, the way he conceives of the relation between a substance and its essence, and his conception of existence. Once clarity is shed upon these fundamental issues, it becomes possible to address the proof in its own terms. It is then easy to see that Kant’s objections miss their target and that the same is true of those advanced by another of the ontological argument’s most famous critics, Bertrand Russell. Finally, several interpretations of Spinoza’s proof are proposed and critically evaluated; on all of them, the argument turns out to be either invalid or question-begging.     

On Rings Having McCoy-Like Conditions

ABSTRACT

In this paper, we study the behavior of the couniform (or dual Goldie) dimension of a module under various polynomial extensions. For a ring automorphism σ ∈ Aut(R), we use the notion of a σ-compatible module M _R to obtain results on the couniform dimension of the polynomial modules M[x], M[x ^?1], and M[x, x ^?1] over suitable skew extension rings.     

Perdurance, Endurance, and ‘Having a Property Atemporally’

In this paper, I argue that both perdurance theory and the ‘relations-to-times’ endurantist view rely on an atemporal notion of property instantiation and relation bearing. I distinguish two possible meanings of ‘atemporal’ which result in two different understandings of what it is for an object to have a property or to bear a relation atemporally. I show that standard presentations of the theories considered are indeterminate as to which of these two understandings is the intended one. I claim that even if both understandings are admissible, one of them is more attractive and has more to recommend than the other.     

Second-Order Elliptic Equation of Divergence Form Having a Compactly Supported Solution

An equation of the form -div (gu) = u has a solution u of class C ₀ , where g is a real positive definite matrix-valued function belonging to the Holder classes with exponent less than 1. From the spectral point of view, this means that there exists a Schrodinger operator with periodic metric and the spectrum of this operator contains an eigenvalue of infinite multiplicity. Bibliography: 9 titles.     

A Model Nonlinear Problem Having a Continuous Locus of Singular Points

The asymptotic behavior of solutions of a singularly perturbed boundary value problem whose reduced equation has continuous bran,ches of singular points is investigated. In particular, unexpected limiting values of solutions for certain choices of the boundary conditions are found by means of a priori estimates.     

具有$2$为无符号拉普拉斯特征值的完美匹配单圈图

本文给出了$2$为完美匹配单圈图的无符号拉普拉斯特征值的充分必要条件.     

Real Hypersurfaces Having Many Pseudo-hyperplanes

Abstract

Let nand dbe natural integers satisfying n ≥ 3 and d ≥ 10. Let Xbe an irreducible real hypersurface Xin ? ⁿ of degree dhaving many pseudo-hyperplanes. Suppose that Xis not a projective cone. We show that the arrangement ? of all d ? 2 pseudo-hyperplanes of Xis trivial, i.e., there is a real projective linear subspace Lof ? ⁿ (?) of dimension n ? 2 such that L ? Hfor all H ∈ ?. As a consequence, the normalization of Xis fibered over ?¹in quadrics. Both statements are in sharp contrast with the case n = 2; the first statement also shows that there is no Brusotti-type result for hypersurfaces in ? ⁿ , for n ≥ 3.