Approximate range searching using binary space partitions

We show how any BSP tree for the endpoints of a set of n disjoint segments in the plane can be used to obtain a BSP tree of size for the segments themselves, such that the range-searching efficiency remains almost the same. We apply this technique to obtain a BSP tree of size O(nlogn) such that -approximate range searching queries with any constant-complexity convex query range can be answered in O(min_>0{(1/)+k}logn) time, where k is the number of segments intersecting the -extended range. The same result can be obtained for disjoint constant-complexity curves, if we allow the BSP to use splitting curves along the given curves.We also describe how to construct a linear-size BSP tree for low-density scenes consisting of n objects in such that -approximate range searching with any constant-complexity convex query range can be done in O(logn+min_>0{(1/^d−1)+k}) time.     

A polynomial time approximation scheme for the two-source minimum routing cost spanning trees

Let G be an undirected graph with nonnegative edge lengths. Given two vertices as sources and all vertices as destinations, we investigated the problem how to construct a spanning tree of G such that the sum of distances from sources to destinations is minimum. In the paper, we show the NP-hardness of the problem and present a polynomial time approximation scheme. For any >0, the approximation scheme finds a (1+)-approximation solution in O(n^1/+1) time. We also generalize the approximation algorithm to the weighted case for distances that form a metric space.     

Theorems about the attractor for incompressible non-Newtonian flow driven by external forces that are rapidly oscillating in time but have a smooth average

This paper discusses the incompressible non-Newtonian fluid with rapidly oscillating external forces g(x,t)=g(x,t,t/) possessing the average g⁰(x,t) as →0⁺, where 0<₀<1. Firstly, with assumptions (A₁)–(A₅) on the functions g(x,t,ξ) and g⁰(x,t), we prove that the Hausdorff distance between the uniform attractors and in space H, corresponding to the oscillating equations and the averaged equation, respectively, is less than O() as →0⁺. Then we establish that the Hausdorff distance between the uniform attractors and in space V is also less than O() as →0⁺. Finally, we show for each [0,₀].     

Maximal Inequalities and Lebesgue's Differentiation Theorem for Best Approximant by Constant over Balls

For fL_p( ⁿ), with 1p<∞, >0 and x ⁿ we denote by T(f)(x) the set of every best constant approximant to f in the ball B(x, ). In this paper we extend the operators T_p to the space L_p−1( ⁿ)+L_∞( ⁿ), where L₀ is the set of every measurable functions finite almost everywhere. Moreover we consider the maximal operators associated to the operators T_p and we prove maximal inequalities for them. As a consequence of these inequalities we obtain a generalization of Lebesgue's Differentiation Theorem.     

Uniform Kazhdan constant for some families of linear groups

Let R be a ring generated by l elements with stable range r. Assume that the group EL_d(R) has Kazhdan constant ₀>0 for some dr+1. We prove that there exist (₀,l)>0 and , s.t. for every nd, EL_n(R) has a generating set of order k and a Kazhdan constant larger than . As a consequence, we obtain for where n3, a Kazhdan constant which is independent of n w.r.t. generating set of a fixed size.     

Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization

The following reaction-diffusion system in spatially non-homogeneous almost-periodic media is considered in a bounded domain : (1)∂_tu=Au−f(u)+g, u|_∂Ω=0. Here u=(u¹,…,u^k) is an unknown vector-valued function, f is a given nonlinear interaction function and the second order elliptic operator A has the following structure: where a_ij^l(y) are given almost-periodic functions. We prove that, under natural assumptions on the nonlinear term f(u), the longtime behavior of solutions of (1) can be described in terms of the global attractor of the associated dynamical system and that the attractors  , 0<<₀1, converge to the attractor of the homogenized problem (1) as →0. Moreover, in the particular case of periodic media, we give explicit estimates for the distance between the non-homogenized and the homogenized attractors in terms of the parameter .     

The maximum asymptotic bias of S-estimates for regression over the neighborhoods defined by certain special capacities

The maximum asymptotic bias of an S-estimate for regression in the linear model is evaluated over the neighborhoods (called (c,γ)-neighborhoods) defined by certain special capacities, and its lower and upper bounds are derived. As special cases, the (c,γ)-neighborhoods include those in terms of -contamination, total variation distance and Rieder's (,δ)-contamination. It is shown that when the model distribution is normal and the (,δ)-contamination neighborhood is adopted, the lower and upper bounds of an S-estimate (including the LMS-estimate) based on a jump function coincide with the maximum asymptotic bias. The tables of the maximum asymptotic bias of the LMS-estimate are given. These results are an extension of the corresponding ones due to Martin et al. (Ann. Statist. 17 (1989) 1608), who used -contamination neighborhoods.     

A note on perturbation of non-symmetric Dirichlet forms by signed smooth measures

This article discusses the perturbation of a non-symmetric Dirichlet form,(ε, D(ε)), by a signed smooth measure μ, whereμ=μ1 -μ2 with μ1 and μ2 being smooth measures. It gives a sufficient condition for the perturbed form (εμ, D(εμ)) (for some αo ≥ 0) to be a coercive closed form.     

A (2 + )-Approximation Scheme for Minimum Domination on Circle Graphs

The main result of this paper is a (2 + )-approximation scheme for the minimum dominating set problem on circle graphs. We first present an O(n²) time 8-approximation algorithm for this problem and then extend it to an time (2 + )-approximation scheme for this problem. Here n and m are the number of vertices and the number of edges of the circle graph. We then present simple modifications to this algorithm that yield (3 + )-approximation schemes for the minimum connected and the minimum total dominating set problems on circle graphs. Keil (1993, Discrete Appl. Math.42, 51–63) shows that these problems are NP-complete for circle graphs and leaves open the problem of devising approximation algorithms for them. These are the first O(1)-approximation algorithms for domination problems on circle graphs.     

On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point

This work focuses on the reducibility of the following real nonlinear analytical quasiperiodic system:

where A is a real 2×2 constant matrix, and f(t,0,)=O() and ∂_xf(t,0,)=O() as →0. With some non-resonant conditions of the frequencies with the eigenvalues of A and without any nondegeneracy condition with respect to , by an affine analytic quasiperiodic transformation we change the system to a suitable norm form at the zero equilibrium for most of the sufficiently small perturbation parameter .     

Inscribing an axially symmetric polygon and other approximation algorithms for planar convex sets

Given a planar convex set C, we give sublinear approximation algorithms to determine approximations of the largest axially symmetric convex set S contained in C, and the smallest such set S^′ that contains C. More precisely, for any >0, we find an axially symmetric convex polygon QC with area |Q|>(1−)|S| and we find an axially symmetric convex polygon Q^′ containing C with area |Q^′|<(1+)|S^′|. We assume that C is given in a data structure that allows to answer the following two types of query in time T_C: given a direction u, find an extreme point of C in direction u, and given a line ℓ, find C∩ℓ. For instance, if C is a convex n-gon and its vertices are given in a sorted array, then T_C=O(logn). Then we can find Q and Q^′ in time O(^−1/2T_C+^−3/2). Using these techniques, we can also find approximations to the perimeter, area, diameter, width, smallest enclosing rectangle and smallest enclosing circle of C in time O(^−1/2T_C).     

Turán type reverse Markov inequalities for compact convex sets

For a compact convex set the well-known general Markov inequality holds asserting that a polynomial p of degree n must have p^′c(K)n²p. On the other hand for polynomials in general, p^′ can be arbitrarily small as compared to p.The situation changes when we assume that the polynomials in question have all their zeroes in the convex set K. This was first investigated by Turán, who showed the lower bounds p^′(n/2)p for the unit disk D and for the unit interval I[-1,1]. Although partial results provided general lower estimates of order , as well as certain classes of domains with lower bounds of order n, it was not clear what order of magnitude the general convex domains may admit here.Here we show that for all bounded and convex domains K with nonempty interior and polynomials p with all their zeroes lying in K p^′c(K)np holds true, while p^′C(K)np occurs for any K. Actually, we determine c(K) and C(K) within a factor of absolute numerical constant.     

A note on asymptotic behavior for negative drift random walk with dependent heavy-tailed steps and its application to risk theory

In this article, the dependent steps of a negative drift random walk are modelled as a two-sided linear process Xn =-μ ∞∑j=-∞ψn-jεj, where {ε, εn; -∞＜ n ＜ ∞}is a sequence of independent, identically distributed random variables with zero mean, μ＞0 is a constant and the coefficients {ψi;-∞＜ i ＜∞} satisfy 0 ＜∞∑j=-∞|jψj| ＜∞. Under the conditions that the distribution function of |ε| has dominated variation and ε satisfies certain tail balance conditions, the asymptotic behavior of P{supn≥0(-nμ ∞∑j=-∞εjβnj) ＞ x}is discussed. Then the result is applied to ultimate ruin probability.     

Existence of HSOLSSOMs of type

In this paper we investigate the existence of holey self-orthogonal Latin squares with a symmetric orthogonal mate of type 2ⁿu¹ (HSOLSSOM(2ⁿu¹)). For u2, necessary conditions for existence of such an HSOLSSOM are that u must be even and n3u/2+1. Xu Yunqing and Hu Yuwang have shown that these HSOLSSOMs exist whenever either (1) n9 and n3u/2+1 or (2) n263 and n2(u-2). In this paper we show that in (1) the condition n9 can be extended to n30 and that in (2), the condition n263 can be improved to n4, except possibly for 19 pairs (n,u), the largest of which is (53,28).     

Small noise asymptotics for invariant densities for a class of diffusions: A control theoretic view

We consider multi-dimensional nondegenerate diffusions with invariant densities, with the diffusion matrix scaled by a small >0. The o.d.e. limit corresponding to =0 is assumed to have the origin as its unique globally asymptotically stable equilibrium. Using control theoretic methods, we show that in the ↓0 limit, the invariant density has the form ≈exp(−W(x)/²), where the W is characterized as the optimal cost of a deterministic control problem. This generalizes an earlier work of Sheu. Extension to multiple equilibria is also given.     

Interlacing and spacing properties of zeros of polynomials, in particular of orthogonal and -minimal polynomials,

Let be a sequence of polynomials with real coefficients such that uniformly for [α-δ,β+δ] with G(eⁱ)≠0 on [α,β], where 0α<βπ and δ>0. First it is shown that the zeros of are dense in [α,β], have spacing of precise order π/n and are interlacing with the zeros of p_n+1(cos) on [α,β] for every nn₀. Let be another sequence of real polynomials with uniformly on [α-δ,β+δ] and on [α,β]. It is demonstrated that for all sufficiently large n the zeros of p_n(cos) and strictly interlace on [α,β] if on [α,β]. If the last expression is zero then a weaker kind of interlacing holds. These interlacing properties of the zeros are new for orthogonal polynomials also. For instance, for large n a simple criteria for interlacing of zeros of Jacobi polynomials on [-1+,1-], >0, is obtained. Finally it is shown that the results hold for wide classes of weighted L_q-minimal polynomials, q[1,∞], linear combinations and products of orthogonal polynomials, etc.     

Succinct data structures for flexible text retrieval systems

We propose succinct data structures for text retrieval systems supporting document listing queries and ranking queries based on the tf*idf (term frequency times inverse document frequency) scores of documents. Traditional data structures for these problems support queries only for some predetermined keywords. Recently Muthukrishnan proposed a data structure for document listing queries for arbitrary patterns at the cost of data structure size. For computing the tf*idf scores there has been no efficient data structures for arbitrary patterns.Our new data structures support these queries using small space. The space is only 2/ times the size of compressed documents plus 10n bits for a document collection of length n, for any 0<1. This is much smaller than the previous O(nlogn) bit data structures. Query time is O(m+qlogn) for listing and computing tf*idf scores for all q documents containing a given pattern of length m. Our data structures are flexible in a sense that they support queries for arbitrary patterns.     

Entropy of C(K)-Valued Operators

We investigate how the entropy numbers (e_n(T)) of an arbitrary Hölder-continuous operator T: E→C(K) are influenced by the entropy numbers (_n(K)) of the underlying compact metric space K and the geometry of E. We derive diverse universal inequalities relating finitely many _n(K)'s with finitely many e_n(T)'s which yield statements about the asymptotically optimal behaviour of the sequence (e_n(T)) in terms of the sequence (_n(K)). As an application we present new methods for estimating the entropy numbers of a precompact and convex subset in a Banach space E, provided that the entropy numbers of its extremal points are known.     

Homology of Newtonian Coalgebras

Given a Newtonian coalgebra we associate to it a chain complex. The homology groups of this Newtonian chain complex are computed for two important Newtonian coalgebras arising in the study of flag vectors of polytopes:R a, b and Rc, d. The homology of Ra, b corresponds to the homology of the boundary of then -crosspolytope. In contrast, the homology of Rc, d depends on the characteristic of the underlying ring R. In the case the ring has characteristic 2, the homology is computed via cubical complexes arising from distributive lattices. This paper ends with a characterization of the integer homology ofZ c, d.     

Preliminary test estimators and phi-divergence measures in generalized linear models with binary data

We consider the problem of estimation of the parameters in Generalized Linear Models (GLM) with binary data when it is suspected that the parameter vector obeys some exact linear restrictions which are linearly independent with some degree of uncertainty. Based on minimum -divergence estimation (ME), we consider some estimators for the parameters of the GLM: Unrestricted ME, restricted ME, Preliminary ME, Shrinkage ME, Shrinkage preliminary ME, James–Stein ME, Positive-part of Stein-Rule ME and Modified preliminary ME. Asymptotic bias as well as risk with a quadratic loss function are studied under contiguous alternative hypotheses. Some discussion about dominance among the estimators studied is presented. Finally, a simulation study is carried out.