Point sets with uniformity properties and orthogonal hypercubes

The theory of (t, m, s)-nets is useful in the study of sets of points in the unit cube with small discrepancy. It is known that the existence of a (0, 2,s)-net in baseb is equivalent to the existence ofs–2 mutually orthogonal latin squares of orderb. In this paper we generalize this equivalence by showing that fort0 the existence of a (t, t+2,s)-net in baseb is equivalent to the existence ofs mutually orthogonal hypercubes of dimensiont+2 and orderb. Using the theory of hypercubes we obtain upper bounds ons for the existence of such nets. Forb a prime power these bounds are best possible. We also state several open problems.This author would like to thank the Mathematics Department of the University of Tasmania for its hospitality during his sabbatical when this paper was written. The same author would also like to thank the NSA for partial support under grant agreement # MDA904-87-H-2023.This author's research was supported by a grant from the Commonwealth of Australia through the Australian Research Council.     

On the distribution of Lachlan nonsplitting bases

 We say that a computably enumerable (c.e.) degree b is a Lachlan nonsplitting base (LNB), if there is a computably enumerable degree a such that a > b, and for any c.e. degrees w,v ≤ a, if a ≤ w or; v or; b then either a ≤ w or; b or a ≤ v or; b. In this paper we investigate the relationship between bounding and nonbounding of Lachlan nonsplitting bases and the high /low hierarchy. We prove that there is a non-Low₂ c.e. degree which bounds no Lachlan nonsplitting base. Received: 11 October 1999 / Published online: 7 May 2002     

Scrambling Sobol' and Niederreiter–Xing Points

Hybrids of equidistribution and Monte Carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter. In particular, randomized (0, m, s)-nets in basebproduce unbiased estimates of the integral, have a variance that tends to zero faster than 1/nfor any square integrable integrand and have a variance that for finitenis never more thane?2.718 times as large as the Monte Carlo variance. Lower bounds thaneare known for special cases. Some very important (t, m, s)-nets havet>0. The widely used Sobol' sequences are of this form, as are some recent and very promising nets due to Niederreiter and Xing. Much less is known about randomized versions of these nets, especially ins>1 dimensions. This paper shows that scrambled (t, m, s)-nets enjoy the same properties as scrambled (0, m, s)-nets, except the sampling variance is guaranteed only to be belowb^t[(b+1)/(b−1)]^stimes the Monte Carlo variance for a least-favorable integrand and finiten.     

Intersecting random translates of invariant Cantor sets

Summary Given two Cantor setsX andY in [0, 1), invariant under the mapxb x mod 1, the Hausdorff dimension of (X+t)Y is constant almost everywhere. WhenX,Y are defined by admissible digits in baseb, and more generally by sofic systems, we compute this dimension in terms of the largest Lyapunov exponent of a random product of matrices. The results are extended to higher dimensions and multiple intersections.Oblatum 17-VIII-1990Support was provided by an IBM fellowship.Partially supported by a grant from the Landau Centre for Mathematical Analysis     

Una fuzione qualunque come densità di una funzione di intervallo

It is proved that for a functionf defined inR ^k, and for every derivation baseD, there exists an additive interval function wich is a primitive off onD. This generalizes a result of Sierpinski. Gruppo di ricerca diretto dal Prof. V. Aversa con fondo MPI.     

Symmetric iterative interpolation processes

Using a baseb and an even number of knots, we define a symmetric iterative interpolation process. The main properties of this process come from an associated functionF. The basic functional equation forF is thatF(t/b)=σ_n F(n/b)F(t-n). We prove thatF is a continuous positive definite function. We find almost precisely in which Lipschitz classes derivatives ofF belong. If a functiony is defined only on integers, this process extendsy continuously to the real axis asy(t=∑ _n y(n)F(t?n). Error bounds for this iterative interpolation are given.     

The Denjoy extension of the Riemann and Mcshane integrals

In this paper we study the Denjoy-Riemann and Denjoy-McShane integrals of functions mapping an interval [a, b] into a Banach space X. It is shown that a Denjoy-Bochner integrable function on [a, b] is Denjoy-Riemann integrable on [a, b], that a Denjoy-Riemann integrable function on [a, b] is Denjoy-McShane integrable on [a, b] and that a Denjoy-McShane integrable function on [a, b] is Denjoy-Pettis integrable on [a, b]. In addition, it is shown that for spaces that do not contain a copy of c ₀, a measurable Denjoy-McShane integrable function on [a, b] is McShane integrable on some subinterval of [a, b]. Some examples of functions that are integrable in one sense but not another are included.     

Self-tuning of fuzzy belief rule bases for engineering system safety analysis

A framework for modelling the safety of an engineering system using a fuzzy rule-based evidential reasoning (FURBER) approach has been recently proposed, where a fuzzy rule-base designed on the basis of a belief structure (called a belief rule base) forms a basis in the inference mechanism of FURBER. However, it is difficult to accurately determine the parameters of a fuzzy belief rule base (FBRB) entirely subjectively, in particular for complex systems. As such, there is a need to develop a supporting mechanism that can be used to train in a locally optimal way a FBRB initially built using expert knowledge. In this paper, the methods for self-tuning a FBRB for engineering system safety analysis are investigated on the basis of a previous study. The method consists of a number of single and multiple objective nonlinear optimization models. The above framework is applied to model the system safety of a marine engineering system and the case study is used to demonstrate how the methods can be implemented.     

Stability number and [a,b]‐factors in graphs

A spanning subgraph whose vertices have degrees belonging to the interval [a,b], where a and b are positive integers, such that a ≤ b, is called an [a,b]‐factor. In this paper, we prove sufficient conditions for existence of an [a,b]‐factor, a connected [a,b]‐factor, and a 2‐connected [a,b]‐factor. The conditions involve the minimum degree, the stability number, and the connectivity of a graph. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 254–264, 2004     

Fibres multiples sur les surfaces: aspects geométriques, hyperboliques et arithmétiques

We show the hyperbolic and complex function field versions of Lang’s conjecture for smooth projective surfaces S having a fibration f:S→C with orbifold base an orbifold curve of general type. The fibre multiplicities defining the orbifold base are non-classical, and simply-connected surfaces S can occur. Also, it is not possible in general to reduce to the easy case where C itself is a curve of general type by a finite étale cover c:S′→S. We leave open the arithmetic version of Lang’s conjecture, which rests on an orbifold version of Mordell’s Conjecture, consequence of the ABC-conjecture, but apparently not of the proofs of Falting’s result.     

Maximum Likelihood for Generalized Linear Models with Nested Random Effects via High-Order,Multivariate Laplace Approximation

Abstract

Nested random effects models are often used to represent similar processes occurring in each of many clusters. Suppose that, given cluster-specific random effects b, the data y are distributed according to f(y|b, Θ), while b follows a density p(b|Θ). Likelihood inference requires maximization of ∫ f(y|b, Θ)p(b|Θdb with respect to Θ. Evaluation of this integral often proves difficult, making likelihood inference difficult to obtain. We propose a multivariate Taylor series approximation of the log of the integrand that can be made as accurate as desired if the integrand and all its partial derivatives with respect to b are continuous in the neighborhood of the posterior mode of b|Θ,y. We then apply a Laplace approximation to the integral and maximize the approximate integrated likelihood via Fisher scoring. We develop computational formulas that implement this approach for two-level generalized linear models with canonical link and multivariate normal random effects. A comparison with approximations based on penalized quasi-likelihood, Gauss—Hermite quadrature, and adaptive Gauss-Hermite quadrature reveals that, for the hierarchical logistic regression model under the simulated conditions, the sixth-order Laplace approach is remarkably accurate and computationally fast.     

[a,b]-factorizations of graphs

Let a and b be integers with b ? a ? 0. A graph G is called an [a,b]-graph if a ? d_G(v) ? b for each vertex v ∈ V(G), and an [a,b]-factor of a graph G is a spanning [a,b]-subgraph of G. A graph is [a,b]-factorable if its edges can be decomposed into [a,b]-factors. The purpose of this paper is to prove the following three theorems: (i) if 1 ? b ? 2a, every [(12a + 2)m + 2an,(12b + 4)m + 2bn]-graph is [2a, 2b + 1]-factorable; (ii) if b ? 2a ?1, every [(12a ?4)m + 2an, (12b ?2)m + 2bn]-graph is [2a ?1,2b]-factorable; and (iii) if b ? 2a ?1, every [(6a ?2)m + 2an, (6b + 2)m + 2bn]-graph is [2a ?1,2b + 1]-factorable, where m and n are nonnegative integers. They generalize some [a,b]-factorization results of Akiyama and Kano [3], Kano [6], and Era [5].     

The use of finite-segmentp-adic arithmetic for exact computation

A finite number system for doing exact computer arithmetic, due to Krishnamurthy, Rao, and Subramanian, is described. For each rational numbera/b, with |a| and |b| suitably bounded, the firstr digits of the (infinite)p-adic expansion ofa/b are used as a coded representation fora/b (the Hensel code). Arithmetic operations on the Hensel codes produce Hensel codes for the exact results of the arithmetic operations.     

Infinite families of surface symmetries

Consider a sequenceN _a,b :g ↦ag+b, g=2,3,… wherea, b are rational numbers anda≥0. We say thatN _a,b is admissible if for infinitely manyg’s there exists a finite groupG of orientation-preserving homeomorphisms of a compact orientable surface of genusg such that |G|=N _a,b (g). In this case we shall also say thatG belongs toN _a,b . The main result of the paper is that ifa+b≠0, then any group belonging toN _a,b contains a cyclic subgroup of bounded index, where the bound depends only ona,b. Moreovera+b is positive, and 2b/a is an integer. This implies, for instance, that there are only finitely many possibilities for perfect subgroups or quotient groups of groups belonging toN _a,b . Partially supported by an NFS grant, and a PSC-CUNY award. Dedicated to A. M. Macbeath with much respect     

Cauchy’s residue theorem for a class of real valued functions

Let [a, b] be an interval in ℝ Rand let F be a real valued function defined at the endpoints of [a, b] and with a certain number of discontinuities within [a, b]. Assuming F to be differentiable on a set [a, b] | E to the derivative f, where E is a subset of [a, b] at whose points F can take values ±∞ or not be defined at all, we adopt the convention that F and f are equal to 0 at all points of E and show that KH-vt ∝ _a ^b f = F(b) − F(a), where KH-vt denotes the total value of the Kurzweil-Henstock integral. The paper ends with a few examples that illustrate the theory.     

Finite Rank Hankel Operators over the Complex Wiener Space

This work studies finite rank Hankel operators H _b on a Hilbert space of holomorphic, square integrable Wiener functionals. The main tool to investigate these operators is their unitary equivalent representation on the Hilbert space of skeletons. The finite rank property is characterized in terms of a functional equation for the symbol b, which generalizes the well known equation b(z+w)=b(z)b(w). Also finite rank symbols of polynomial type are characterized in terms of their chaos expansions.     

On the decimal numbers base n

This paper focuses on the representation of a proper fraction a/b by a decimal number base n where n is any integer greater than 1. The scope is narrowed to look at only fractions where a, b are positive integers with a<b and b not equal to 0 nor equal to 1. Some relationships were found between b and n, which determine whether the representation will become either finite decimal, pure recurring decimal or mixed decimal base n. Three theorems have been proven to indicate the deciding factors and the relationships. In addition, the length of the finite decimal numbers base n was further explored.     

On the Distribution of the Sums of Binomial Coefficients Modulo a Prime

 Let be the binomial coefficient modulo b (b prime), with if l is greater than c, and let be the sum of binomial coefficients modulo b, that is (mod b). We prove the following property: the for which the couples c, l verify and are uniformly distributed in the residue classes modulo b as n tends to infinity. The method, using the Perron-Frobenius theory, applies also to and gives a new proof of the well known result for the non-zero binomial coefficients modulo b. (Received 21 June 1999; in revised form 13 July 2000)     

Hardy inequalities for weighted Dirac operator

An inequality of Hardy type is established for quadratic forms involving Dirac operator and a weight r ^−b for functions in \mathbbRⁿ{\mathbb{R}^n}. The exact Hardy constant c _b  = c _b (n) is found and generalized minimizers are given. The constant c _b vanishes on a countable set of b, which extends the known case n = 2, b = 0 which corresponds to the trivial Hardy inequality in \mathbbR²{\mathbb{R}^2}. Analogous inequalities are proved in the case c _b  = 0 under constraints and, with error terms, for a bounded domain.