Voronoi Diagrams of Lines in 3-Space Under Polyhedral Convex Distance Functions

The combinatorial complexity of the Voronoi diagram ofnlines in three dimensions under a convex distance function induced by a polytope with a constant number of edges is shown to beO(n²α(n)log n), where α(n) is a slowly growing inverse of the Ackermann function. There are arrangements ofnlines where this complexity can be as large as Ω(n²α(n)).     

Incremental and Decremental Maintenance of Planar Width

We present an algorithm for maintaining the width of a planar point set dynamically, as points are inserted or deleted. Our algorithm takes time O(kn^ε) per update, where k is the amount of change the update causes in the convex hull, n is the number of points in the set, and ε > 0 is any arbitrarily small constant. For incremental or decremental update sequences, the amortized time per update is O(n^ε).     

A logarithmic Gauss curvature flow and the Minkowski problem

Let X₀ be a smooth uniformly convex hypersurface and f a postive smooth function in Sⁿ. We study the motion of convex hypersurfaces X(·,t) with initial X(·,0)=θX₀ along its inner normal at a rate equal to log(K/f) where K is the Gauss curvature of X(·,t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ^*>0 such that if θ<θ^*, they shrink to a point in finite time and, if θ>θ^*, they expand to an asymptotic sphere. Finally, when θ=θ^*, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f(x).     

Ray shooting and intersection searching amidst fat convex polyhedra in 3-space

We present a data structure for ray-shooting queries in a set of convex fat polyhedra of total complexity n in . The data structure uses O(n^2+ε) storage and preprocessing time, and queries can be answered in O(log²n) time. A trade-off between storage and query time is also possible: for any m with n<m<n², we can construct a structure that uses O(m^1+ε) storage and preprocessing time such that queries take time.We also describe a data structure for simplex intersection queries in a set of n convex fat constant-complexity polyhedra in . For any m with n<m<n³, we can construct a structure that uses O(m^1+ε) storage and preprocessing time such that all polyhedra intersecting a query simplex can be reported in O((n/m^1/3)logn+k) time, where k is the number of answers.     

Graphs satisfying inequality θ(G)θ(G)

In this paper, we study the edge clique cover number of squares of graphs. More specifically, we study the inequality θ(G²)θ(G) where θ(G) is the edge clique cover number of a graph G. We show that any graph G with at most θ(G) vertices satisfies the inequality. Among the graphs with more than θ(G) vertices, we find some graphs violating the inequality and show that dually chordal graphs and power-chordal graphs satisfy the inequality. Especially, we give an exact formula computing θ(T²) for a tree T.     

Gray code enumeration of families of integer partitions

In this paper we show that the elements of certain families of integer partitions can be listed in a minimal change, or Gray code, order. In particular, we construct Gray code listings for the classes P_δ(n, k) and D(n, k) of partitions of n into parts of size at most k in which, for P_δ(n, k), the parts are congruent to one modulo δ and, for D(n, k), the parts are distinct. It is shown that the elements of these classes can be listed so that the only change between successive partitions is the increase of one part by δ (or the addition of δ ones) and the decrease of one part by δ (or the removal of δ ones), where, in the case of D(n, k), δ = 1.     

Some Mean Values Related to Average Multiplicative Orders of Elements in Finite Fields

For any positive integer n let α(n) denote the average order of elements in the cyclic group Z_n. In this note, we investigate the functions α(n)/n and α(n)/φ(n) when n ranges through numbers of the form p−1 with p prime, and when n ranges through numbers of the form 2^m−1 with m a positive integer. In particular, we show that such functions have limiting distributions, and we compute their average values, and their minimal and maximal orders.To Jean-Louis Nicolas at his 60th birthday2000 Mathematics Subject Classification: Primary—11N45; Secondary—05A16, 11N37This work was supported in part by Grant SEP-CONACYT 37259-E.     

On the adjacencies of triangular meshes based on skeleton-regular partitions

For any 2D triangulation τ, the 1-skeleton mesh of τ is the wireframe mesh defined by the edges of τ, while that for any 3D triangulation τ, the 1-skeleton and the 2-skeleton meshes, respectively, correspond to the wireframe mesh formed by the edges of τ and the “surface” mesh defined by the triangular faces of τ. A skeleton-regular partition of a triangle or a tetrahedra, is a partition that globally applied over each element of a conforming mesh (where the intersection of adjacent elements is a vertex or a common face, or a common edge) produce both a refined conforming mesh and refined and conforming skeleton meshes. Such a partition divides all the edges (and all the faces) of an individual element in the same number of edges (faces). We prove that sequences of meshes constructed by applying a skeleton-regular partition over each element of the preceding mesh have an associated set of difference equations which relate the number of elements, faces, edges and vertices of the nth and (n−1)th meshes. By using these constitutive difference equations we prove that asymptotically the average number of adjacencies over these meshes (number of triangles by node and number of tetrahedra by vertex) is constant when n goes to infinity. We relate these results with the non-degeneracy properties of longest-edge based partitions in 2D and include empirical results which support the conjecture that analogous results hold in 3D.     

Minimum degree of minimal defect -extendable bipartite graphs

A near perfect matching is a matching saturating all but one vertex in a graph. If G is a connected graph and any n independent edges in G are contained in a near perfect matching, then G is said to be defect n-extendable. If for any edge e in a defect n-extendable graph G, G−e is not defect n-extendable, then G is minimal defect n-extendable. The minimum degree and the connectivity of a graph G are denoted by δ(G) and κ(G) respectively. In this paper, we study the minimum degree of minimal defect n-extendable bipartite graphs. We prove that a minimal defect 1-extendable bipartite graph G has δ(G)=1. Consider a minimal defect n-extendable bipartite graph G with n≥2, we show that if κ(G)=1, then δ(G)≤n+1 and if κ(G)≥2, then 2≤δ(G)=κ(G)≤n+1. In addition, graphs are also constructed showing that, in all cases but one, there exist graphs with minimum degree that satisfies the established bounds.     

Pseudoprime values of the Fibonacci sequence, polynomials and the Euler function

We show that if α > 1 is any fixed integer, then for a sufficiently large x > 1, the nth Fibonacci number F_n is a base α pseudopfime only for at most (4 + o(1))π(x) of posifive integers n x. The same result holds for Mersenne numbers 2ⁿ — 1 and for one more general class of Lucas sequences. A slight modification of our method also leads to similar results for polynomial sequences f(n), where f ∊ [X]. Finally, we use a different technique to get a much sharper upper bound on the counting fimction of the positive integers n such that φ(n) is a base α pseudoprime, where φ is the Euler function.     

New text indexing functionalities of the compressed suffix arrays

New text indexing functionalities of the compressed suffix arrays are proposed. The compressed suffix array proposed by Grossi and Vitter is a space-efficient data structure for text indexing. It occupies only O(n) bits for a text of length n; however it also uses the text itself that occupies bits for the alphabet . In this paper we modify the data structure so that pattern matching can be done without any access to the text. In addition to the original functions of the compressed suffix array, we add new operations search, decompress and inverse to the compressed suffix arrays. We show that the new index can find occ occurrences of any substring P of the text in O(|P|logn+occlog^εn) time for any fixed 1ε>0 without access to the text. The index also can decompress a part of the text of length m in O(m+log^εn) time. For a text of length n on an alphabet such that , our new index occupies only bits where is the order-0 entropy of the text. Especially for ε=1 the size is bits. Therefore the index will be smaller than the text, which means we can perform fast queries from compressed texts.     

Blumenthal''s Theorem for Laurent Orthogonal Polynomials

We investigate polynomials satisfying a three-term recurrence relation of the form B_n(x)=(x−β_n)B_n−1(x)−α_nxB_n−2(x), with positive recurrence coefficients α_n+1,β_n (n=1,2,…). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. We analyse in more detail the case where α_n→α and β_n→β and show that the zeros of B_n are dense on an interval and that the support of the Laurent orthogonality measure is equal to this interval and a set which is at most denumerable with accumulation points (if any) at the endpoints of the interval. This result is the Laurent version of Blumenthal's theorem for orthogonal polynomials.     

Non-microstates free entropy dimension for groups

We show that for any discrete finitely-generated group G and any self-adjoint n-tuple X₁,...,X_n of generators of the group algebra Voiculescu’s non-microstates free entropy dimension δ^*(X₁,...,X_n) is exactly equal to β₁(G) − β₀(G) + 1 where β_i are the ℓ²-Betti numbers of G.Received: January 2004 Revision: October 2004 Accepted: January 2005     

On the Erd s-diameter of sets

Let δ(n) denote the minimum diameter of a set of n points in the plane in which any two positive distances, if they are different, differ by at least one. Erd s conjectured that for n sufficiently big we have δ(n) = n − 1, the extremal configuration being n equidistant points on a line. In this note we prove an asymptotic version of this conjecture for the special case of sets which lie in a parallel half-strip.     

Bounds on margin distributions in learning problems

Let be a probability space and let P_n be the empirical measure based on i.i.d. sample (X₁,…,X_n) from P. Let be a class of measurable real valued functions on For define F_f(t):=P{ft} and F_n,f(t):=P_n{ft}. Given γ(0,1], define _n,γ(δ):=1/(n^1−γ/2δ^γ). We show that if the L₂(P_n)-entropy of the class grows as ^−α for some α(0,2), then, for all and all δ(0,Δ_n), Δ_n=O(n^1/2),

and

where and c(σ)↓1 as σ↓0 (the above inequalities hold for any fixed σ(0,1] with a high probability). Also, define

Then for all

uniformly in and with probability 1 (for the above ratio is bounded away from 0 and from ∞). The results are motivated by recent developments in machine learning, where they are used to bound the generalization error of learning algorithms. We also prove some more general results of similar nature, show the sharpness of the conditions and discuss the applications in learning theory.     

Some Infinite Families of Locally Partial Geometries Generalizing the Classical Laguerre Planes

Starting with a subgeometry Ω embedded in a β-dimensional projective space PG(β, q), β 1, we construct inductively a series of rank n residually connected geometries Γ^{(n, β, Ω)}, n β, by putting Γ^{(β, β, Ω)} = Ω and extending Γ^{(n - 1, β, Ω)} with a partial geometry.     

Linear Optimization Queries

Let Γ₀ be a set of n halfspaces in E^d (where the dimension d is fixed) and let m be a parameter, n ≤ m ≤ n^d/2. We show that Γ₀ can be preprocessed in time and space O(m^1+δ) (for any fixed δ > 0) so that given a vector c E^d and another set Γ_q of additional halfspaces, the function c · x can be optimized over the intersection of the halfspaces of Γ₀ Γ_q in time O((n/m^1/d/2 + |Γ_q|)log^4d+3n). The algorithm uses a multidimensional version of Megiddo′s parametric search technique and recent results on halfspace range reporting. Applications include an improved algorithm for computing the extreme points of an n-point set P in E^d, improved output-sensitive computation of convex hulls and Voronoi diagrams, and a Monte-Carlo algorithm for estimating the volume of a convex polyhedron given by the set of its vertices (in a fixed dimension).     

Bounds on the number of maximal sum-free sets

We show that the number of maximal sum-free subsets of {1,2,…,n} is at most 2^3n/8+o(n). We also show that 2^0.406n+o(n) is an upper bound on the number of maximal product-free subsets of any group of order n.     

Onα-Symmetric Multivariate Characteristic Functions

Ann-dimensional random vector is said to have anα-symmetric distribution,α>0, if its characteristic function is of the form((|u₁|^α+…+|u_n|^α)^1/α). We study the classesΦ_n(α) of all admissible functions: [0, ∞)→ . It is known that members ofΦ_n(2) andΦ_n(1) are scale mixtures of certain primitivesΩ_nandω_n, respectively, and we show thatω_nis obtained fromΩ_2n−1byn−1 successive integrations. Consequently, curious relations between 1- and 2- (or spherically) symmetric distributions arise. An analogue of Askey's criterion gives a partial solution to a question of D. St. P. Richards: If(0)=1,is continuous, lim_t→∞ (t)=0, and⁽²ⁿ⁻²⁾(t) is convex, thenΦ_n(1). The paper closes with various criteria for the unimodality of anα-symmetric distribution.     

An Algorithm for the Modular Decomposition of Hypergraphs

We propose an O(n⁴) algorithm to build the modular decomposition tree of hypergraphs of dimension three and show how this algorithm can be generalized to compute in O(n^{3k − 5}) time the decomposition of hypergraphs of any fixed dimension k.