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.     

External cofinalities and the antichain condition in partial orders

Does every partial order of singular cofinality λ have an antichain of size ? This is the Singular Cofinality Conjecture. M. Pouzet proved [M. Pouzet, Parties cofinales des ordres partiels ne contenant pas d’antichaines infinies, 1980, preprint] that there must be an infinite antichain. When is uncountable, the positive answer is only consistently true, but unknown in ZFC. In this note we investigate this question from the purely set-theoretic point of view. On the way, we answer a question of Milner and Pouzet from [E.C. Milner, M. Pouzet, Posets with singular cofinality, 1997, preprint].     

A dual open coloring axiom

We discuss a dual of the Open Coloring Axiom introduced by Abraham et al. [U. Abraham, M. Rubin, S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of ₁-dense real order types, Ann. Pure Appl. Logic 29 (2) (1985) 123–206] and show that it follows from a statement about continuous colorings on Polish spaces that is known to be consistent. We mention some consequences of the new axiom and show that implies that all cardinal invariants in Cichoń’s diagram are at least ₂.     

Augmenting the connectivity of geometric graphs

Let G be a connected plane geometric graph with n vertices. In this paper, we study bounds on the number of edges required to be added to G to obtain 2-vertex or 2-edge connected plane geometric graphs. In particular, we show that for G to become 2-edge connected, additional edges are required in some cases and that additional edges are always sufficient. For the special case of plane geometric trees, these bounds decrease to and , respectively.     

Discrete subspaces of countably tight compacta

Our main result is that the following cardinal arithmetic assumption, which is a slight weakening of GCH, “2^κ is a finite successor of κ for every cardinal κ”, implies that in any countably tight compactum X there is a discrete subspace D with . This yields a (consistent) confirmation of Alan Dow’s Conjecture 2 from [A. Dow, Closures of discrete sets in compact spaces, Studia Math. Sci. Hung. 42 (2005) 227–234].     

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.     

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.     

On the consistency strength of the Milner–Sauer conjecture

In their paper from 1981, Milner and Sauer conjectured that for any poset P,≤, if , then P must contain an antichain of cardinality κ. The conjecture is consistent and known to follow from GCH-type assumptions.

We prove that the conjecture has large cardinals consistency strength in the sense that its negation implies, for example, the existence of a measurable cardinal in an inner model. We also prove that the conjecture follows from Martin’s Maximum and holds for all singular λ above the first strongly compact cardinal.     

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].     

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 .     

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.     

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.     

A non-normal topology generated by a two-point selection

We construct a two-point selection , where is the set of the irrational numbers, such that the space is not normal and it is not collectionwise Hausdorff either. Here, τ_f denotes the topology generated by the two-point selection f. This example answers a question posed by V. Gutev and T. Nogura. We also show that if is a two-point selection such that the topology τ_f has countable pseudocharacter, then τ_f is a Tychonoff topology.     

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.     

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.     

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.     

An atlas for tridiagonal isospectral manifolds

Let be the compact manifold of real symmetric tridiagonal matrices conjugate to a given diagonal matrix Λ with simple spectrum. We introduce bidiagonal coordinates, charts defined on open dense domains forming an explicit atlas for . In contrast to the standard inverse variables, consisting of eigenvalues and norming constants, every matrix in now lies in the interior of some chart domain. We provide examples of the convenience of these new coordinates for the study of asymptotics of isospectral dynamics, both for continuous and discrete time.     

Real hypersurfaces in complex projective space with recurrent structure Jacobi operator

We classify real hypersurfaces of complex projective space , m3, with -recurrent structure Jacobi operator and apply this result to prove the non-existence of such hypersurfaces with recurrent structure Jacobi operator.