Space-efficient geometric divide-and-conquer algorithms

We develop a number of space-efficient tools including an approach to simulate divide-and-conquer space-efficiently, stably selecting and unselecting a subset from a sorted set, and computing the kth smallest element in one dimension from a multi-dimensional set that is sorted in another dimension. We then apply these tools to solve several geometric problems that have solutions using some form of divide-and-conquer. Specifically, we present a deterministic algorithm running in time using extra memory given inputs of size n for the closest pair problem and a randomized solution running in expected time and using extra space for the bichromatic closest pair problem. For the orthogonal line segment intersection problem, we solve the problem in time using extra space where n is the number of horizontal and vertical line segments and k is the number of intersections.     

Minimum weight pseudo-triangulations

We consider the problem of computing a minimum weight pseudo-triangulation of a set of n points in the plane. We first present an -time algorithm that produces a pseudo-triangulation of weight which is shown to be asymptotically worst-case optimal, i.e., there exists a point set for which every pseudo-triangulation has weight , where is the weight of a minimum weight spanning tree of . We also present a constant factor approximation algorithm running in cubic time. In the process we give an algorithm that produces a minimum weight pseudo-triangulation of a simple polygon.     

The minimum-area spanning tree problem

Motivated by optimization problems in sensor coverage, we formulate and study the Minimum-Area Spanning Tree (mast) problem: Given a set of n points in the plane, find a spanning tree of of minimum “area”, where the area of a spanning tree is the area of the union of the n−1 disks whose diameters are the edges in . We prove that the Euclidean minimum spanning tree of is a constant-factor approximation for mast. We then apply this result to obtain constant-factor approximations for the Minimum-Area Range Assignment (mara) problem, for the Minimum-Area Connected Disk Graph (macdg) problem, and for the Minimum-Area Tour (mat) problem. The first problem is a variant of the power assignment problem in radio networks, the second problem is a related natural problem, and the third problem is a variant of the traveling salesman problem.     

A result about the density of iterated line intersections in the plane

Let S be a finite set of points in the plane and let be the set of intersection points between pairs of lines passing through any two points in S. We characterize all configurations of points S such that iteration of the above operation produces a dense set. We also discuss partial results on the characterization of those finite point-sets with rational coordinates that generate all of through iteration of .     

Line-segment intersection made in-place

We present a space-efficient algorithm for reporting all k intersections induced by a set of n line segments in the plane. Our algorithm is an in-place variant of Balaban's algorithm and, in the worst case, runs in time using extra words of memory in addition to the space used for the input to the algorithm.     

A quadratic distance bound on sliding between crossing-free spanning trees

Let S be a set of n points in the plane and let be the set of all crossing-free spanning trees of S. We show that it is possible to transform any two trees in into each other by O(n²) local and constant-size edge slide operations. Previously no polynomial upper bound for this task was known, but in [O. Aichholzer, F. Aurenhammer, F. Hurtado, Sequences of spanning trees and a fixed tree theorem, Comput. Geom.: Theory Appl. 21 (1–2) (2002) 3–20] a bound of O(n²logn) operations was conjectured.     

The number of translates of a closed nowhere dense set required to cover a Polish group

For a Polish group let be the minimal number of translates of a fixed closed nowhere dense subset of required to cover . For many locally compact this cardinal is known to be consistently larger than which is the smallest cardinality of a covering of the real line by meagre sets. It is shown that for several non-locally compact groups . For example the equality holds for the group of permutations of the integers, the additive group of a separable Banach space with an unconditional basis and the group of homeomorphisms of various compact spaces.     

Nonlinear maps preserving Lie products on factor von Neumann algebras

In this paper, we prove that every bijective map preserving Lie products from a factor von Neumann algebra into another factor von Neumann algebra is of the form A→ψ(A)+ξ(A), where is an additive isomorphism or the negative of an additive anti-isomorphism and is a map with ξ(AB-BA)=0 for all .     

Linear maps preserving idempotency of products or triple jordan products of operators

Let be the algebra of all bounded linear operators on a complex Banach space . We give the concrete forms of linear surjective maps on which preserve the nonzero idempotency of either products of two operators or triple Jordan products of two operators.     

Computing homotopic shortest paths efficiently

We give deterministic and randomized algorithms to find shortest paths homotopic to a given collection Π of disjoint paths that wind amongst n point obstacles in the plane. Our deterministic algorithm runs in time , and the randomized algorithm runs in expected time O(k_out+k_inlogn+n(logn)^1+ε). Here k_in is the number of edges in all the paths of Π, and k_out is the number of edges in the output paths.     

A note on 2-factors with two components

In this note, we consider a minimum degree condition for a hamiltonian graph to have a 2-factor with two components. Let G be a graph of order n3. Dirac's theorem says that if the minimum degree of G is at least , then G has a hamiltonian cycle. Furthermore, Brandt et al. [J. Graph Theory 24 (1997) 165–173] proved that if n8, then G has a 2-factor with two components. Both theorems are sharp and there are infinitely many graphs G of odd order and minimum degree which have no 2-factor. However, if hamiltonicity is assumed, we can relax the minimum degree condition for the existence of a 2-factor with two components. We prove in this note that a hamiltonian graph of order n6 and minimum degree at least has a 2-factor with two components.     

Uniform asymptotic expansions of the Pollaczek polynomials

Two uniform asymptotic expansions are obtained for the Pollaczek polynomials P_n(cosθ;a,b). One is for , , in terms of elementary functions and in descending powers of . The other is for , in terms of a special function closely related to the modified parabolic cylinder functions, in descending powers of n. This interval contains a turning point and all possible zeros of P_n(cosθ) in θ(0,π/2].     

Sparse geometric graphs with small dilation

Given a set S of n points in , and an integer k such that 0k<n, we show that a geometric graph with vertex set S, at most n−1+k edges, maximum degree five, and dilation O(n/(k+1)) can be computed in time O(nlogn). For any k, we also construct planar n-point sets for which any geometric graph with n−1+k edges has dilation Ω(n/(k+1)); a slightly weaker statement holds if the points of S are required to be in convex position.     

Graph drawings with few slopes

The slope-number of a graph G is the minimum number of distinct edge slopes in a straight-line drawing of G in the plane. We prove that for Δ5 and all large n, there is a Δ-regular n-vertex graph with slope-number at least . This is the best known lower bound on the slope-number of a graph with bounded degree. We prove upper and lower bounds on the slope-number of complete bipartite graphs. We prove a general upper bound on the slope-number of an arbitrary graph in terms of its bandwidth. It follows that the slope-number of interval graphs, cocomparability graphs, and AT-free graphs is at most a function of the maximum degree. We prove that graphs of bounded degree and bounded treewidth have slope-number at most . Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.     

Bivariate quartic spline spaces and quasi-interpolation operators

In this paper, we study two bivariate quartic spline spaces and , and present two classes of quasi-interpolation operators in the two spaces, respectively. Some results on the operators are given.     

On quarks confinement and asymptotic freedom

We review briefly previous works on quarks confinement and asymptotic freedom. Subsequently we introduce the magnetic monopole pair inverse coupling in analogy to the Cooper pairs electromagnetic inverse fine structure constant .

Finally, we show that confinement is absolute at certain energy scale limit where the Planck energy is M_p = (10)¹⁹ Gev and the expectation value of a number of generation equal to 16 + k plays a crucial role. It is important to note that is de facto replaced by and . The phenomena discussed here have a classical analogy in the engineering theory of plasticity called strain hardening or negative super spring.     

Sign balance for finite groups of Lie type

A product formula for the parity generating function of the number of 1’s in invertible matrices over is given. The computation is based on algebraic tools such as the Bruhat decomposition. It is somewhat surprising that the number of such matrices with odd number of 1’s is greater than the number of those with even number of 1’s. The same technique can be used to obtain a parity generating function also for symplectic matrices over . We present also a generating function for the sum of entries of matrices over an arbitrary finite field calculated in . The Mahonian distribution appears in these formulas.     

Limit behavior of global attractors for the complex Ginzburg–Landau equation on infinite lattices

In this work, the authors first show the existence of global attractors for the following lattice complex Ginzburg–Landau equation:

and for the following lattice Schrödinger equation:

Then they prove that the solutions of the lattice complex Ginzburg–Landau equation converge to that of the lattice Schrödinger equation as ε→0+. Also they prove the upper semicontinuity of as ε→0+ in the sense that .     

Identifying phylogenetic trees

A central problem that arises in evolutionary biology is that of displaying partitions of subsets of a finite set X on a tree whose vertices are partially labelled with the elements of X. Such a tree is called an X-tree and, for a collection of partitions of subsets of X, characterisations for the existence and uniqueness of an X-tree that displays have been previously given in terms of chordal graphs. In this paper, we obtain two closely related characterisations also in terms of chordal graphs. The first describes when identifies an X-tree, and the second describes when a compatible subset of is of maximum size.     

Maximal realcompact spaces and measurable cardinals

Comfort and Hager investigate the notion of a maximal realcompact space and ask about the relationship to the first measurable cardinal . A space is said to be a space if the intersection of fewer than open sets is again open. They ask if each realcompact space is maximal realcompact. We establish that this question is undecidable.