Simply connected orthogonal polygons as unions of two orthogonally starshaped sets

Let S be a simply connected orthogonal polygon in the plane. The set S is a union of two sets which are starshaped via staircase paths (i.e., orthogonally starshaped) if and only if for every three points of S, at least two of these points see (via staircase paths) a common point of S. Moreover, the simple connectedness condition cannot be deleted.     

Intersection of maximal orthogonally starshaped polygons

Let S be a simply connected orthogonal polygon in and let P(S) denote the intersection of all maximal starshaped via staircase paths orthogonal subpolygons in S. Our result: if , then there exists a maximal starshaped via staircase paths orthogonal polygon , such that . As a corollary, P(S) is a starshaped (via staircase paths) orthogonal polygon or empty. The results fail without the requirement that the set S is simply connected. Received 1 March 1999.     

Dimensions of staircase kernels in orthogonal polygons

LetS be a simply connected orthogonal polygon in the plane. If every four points ofS see (via staircase paths inS) a common segment, then the staircase kernel ofS contains a segment as well. A parallel result holds when every four points ofS see (via staircase paths inS) a common 2-dimensional set. In each case, the number 4 is best possible. However, both results fail without the requirement that setS be simply connected.An example shows that no Helly-type analogue is possible for intersections of orthogonally convex sets in the plane.     

Intersections and Unions of Orthogonal Polygons Starshaped Via Staircase n-Paths

For n ≥ 1, define p (n) to be the smallest natural number r for which the following is true: For any finite family of simply connected orthogonal polygons in the plane and points x and y in , if every r (not necessarily distinct) members of contain a common staircase n-path from x to y, then contains such a path. We show that p(1) = 1 and p(n) = 2 (n − 1) for n ≥ 2. The numbers p(n) yield an improved Helly theorem for intersections of sets starshaped via staircase n-paths. Moreover, we establish the following dual result for unions of these sets: Let be any finite family of orthogonal polygons in the plane, with simply connected. If every three (not necessarily distinct) members of have a union which is starshaped via staircase n-paths, then T is starshaped via staircase (n + 1)-paths. The number n + 1 in the theorem is best for every n ≥ 2.     

On staircase starshapedness in rectilinear spaces

Let P ⁿ be a union of a finite number of boxes whose intersection graph is a tree. If every two boundary points of P are visible via staircase paths from a common point of P, then P is starshaped via staircase paths. The same result holds true when P is a cubical polyhedron of ⁿ , which is the geometric realization of some median graph.This generalizes the recent result of M. Breen, J. Geometry, 51 (1994), established for simple rectilinear polygons.Research for this paper was done while the author was visiting the Mathematisches Seminar der Universität Hamburg, on leave from the Universitatea de Stat din Moldova. The author gratefully acknowledges financial support by the Alexander von Humboldt Stifting.     

Z-cyclic triplewhist tournaments when the number of players involves primes of the form 8u + 5

When the number of players, v, in a whist tournament, Wh(v), is ≡ 1 (mod 4) the only instances of a Z-cyclic triplewhist tournament, TWh(v), that appear in the literature are for v = 21,29,37. In this study we present Z-cyclic TWh(v) for all v ∈ T = {v = 8u + 5: v is prime, 3 ≤ u ≤ 249}. Additionally, we establish (1) for all v ∈ T there exists a Z-cyclic TWh(vⁿ) for all n ≥ 1, and (2) if v_i ∈ T, i = 1,…,n, there exists a Z-cyclic TWh(v… v) for all ?_i ≥ 1. It is believed that these are the first instances of infinite classes of Z-cyclic TWh(v), v ≡ 1 (mod 4). © 1994 John Wiley & Sons, Inc.     

Exhaustive weakly wandering sequences

A sequence of integers {n_i : i = 0, 1…} is an exhaustive weakly wandering sequence for a transformation T if for some measurable set W, X=^∞_i=0TⁿⁱW(disj. We introduce a hereditary Property (H) for a sequence of integers associated with an infinite ergodic transformation T, and show that it is a sufficient condition for the sequence to be an exhaustive weakly wandering sequence for T. We then show that every infinite ergodic transformation admits sequences that possess Property (H), and observe that Property (H) is inherited by all subsequences of a sequence that possess it. As a corollary, we obtain an application to tiling the set of integers with infinite subsets.     

On Intersection of Maximal Orthogonally k-Starshaped Polygons

Let F be a simply connected orthogonal polygon in R ² and let P denote the intersection of all maximal orthogonally k-starshaped polygons in F for any fixed integer k,k2. If P and for every x,y P which are joined in F by a staircase path having two segments there is a similar staircase path from x to y in P, then there exists a maximal orthogonally k-starshaped polygon Q in F such that the staircase k-kernel of Q is a subset of the staircase k-kernel of P. In particular, F is either an orthogonally k-starshaped simply connected polygon in F or empty.     

Temperely-Lieb Algebras and the Four-Color Theorem

The Temperley–Lieb algebra T_n with parameter 2 is the associative algebra over Q generated by 1,e₀,e₁, . . .,e_n, where the generators satisfy the relations if |i–j|=1 and e_ie_j=e_je_i if |i–j|2. We use the Four Color Theorem to give a necessary and sufficient condition for certain elements of T_n to be nonzero. It turns out that the characterization is, in fact, equivalent to the Four Color Theorem.* Partially supported by NSF under Grant DMS-9802859 and by NSA under grant MDA904-97-1-0015. Partially supported by NSF under Grant DMS-9623031 and by NSA under Grant MDA904-98-1-0517.     

Packing random items of three colors

Consider three colors 1,2,3, and forj3, considern items (X _i,j)in of colorj. We want to pack these items inn bins of equal capacity (the bin size is not fixed, and is to be determined once all the objects are known), subject to the condition that each bin must contain exactly one item of each color, and that the total item sizes attributed to any given bin does not exceed the bin capacity. Consider the stochastic model where the random variables (X _i,jj)in,j3 are independent uniformly distributed over [0,1]. We show that there is a polynomial-time algorithm that produces a packing which has a wasted spaceK logn with overwhelming probability.Work partially supported by an N.S.F. grant.     

Characterizing compact unions of two starshaped sets inR d

SetS inR ^d has propertyK ₂ if and only ifS is a finite union ofd-polytopes and for every finite setF in bdryS there exist points c₁,c₂ (depending onF) such that each point ofF is clearly visible viaS from at least one c_i,i = 1,2. The following characterization theorem is established: Let , d2. SetS is a compact union of two starshaped sets if and only if there is a sequence {S _j } converging toS (relative to the Hausdorff metric) such that each setS _j satisfies propertyK ₂. For , the sufficiency of the condition above still holds, although the necessity fails.     

Reconstructing dimensions of intersections of convex sets

Denoting by dimA the dimension of the affine hull of the setA, we prove that if {K _i:i T} and {K _i ^j :i T} are two finite families of convex sets inR ⁿ and if dim {K _i :i S} = dim {K _i ^j :i S}for eachS T such that|S| n + 1 then dim {K _i :i T} = dim {K _i : {i T}}.     

Optimal pairs of score vectors for positional scoring rules

Letw=(w ₁,,w _m ) andv=(v ₁,,v _m-1 ) be nonincreasing real vectors withw ₁>w _m andv ₁>v _m-1 . With respect to a lista ₁,,a _n of linear orders on a setA ofm3 elements, thew-score ofaA is the sum overi from 1 tom ofw _i times the number of orders in the list that ranka inith place; thev-score ofaA{b} is defined in a similar manner after a designated elementb is removed from everya _j .We are concerned with pairs (w, v) which maximize the probability that anaA with the greatestw-score also has the greatestv-score inA{b} whenb is randomly selected fromA{a}. Our model assumes that linear ordersa _j onA are independently selected according to the uniform distribution over them linear orders onA. It considers the limit probabilityP _m (w, v) forn that the element inA with the greatestw-score also has the greatestv-score inA{b}.It is shown thatP _m (m,v) takes on its maximum value if and only if bothw andv are linear, so thatw _i –w _i+1=w _i+1–w _i+2 forim–2, andv _i –v _i+1 =v _i+1 –v _i+2 forim–3. This general result for allm3 supplements related results for linear score vectors obtained previously form{3,4}.     

Samuel multiplicity and Fredholm theory

In this note we combine methods from commutative algebra and complex analytic geometry to calculate the generic values of the cohomology dimensions of a commuting multioperator on its Fredholm domain. More precisely, we prove that, for a given Fredholm tuple T = (T ₁, ..., T _n ) of commuting bounded operators on a complex Banach space X, the limits exist and calculate the generic dimension of the cohomology groups H ^p (z − T, X) of the Koszul complex of T near z = 0. To deduce this result we show that the above limits coincide with the Samuel multiplicities of the stalks of the cohomology sheaves of the associated complex of analytic sheaves at z = 0.     

Illuminating high-dimensional convex sets

A setL of points in thed-spaceE ^d is said toilluminate a familyF={S ₁, ...,S _n } ofn disjoint compact sets inE ^d if for every setS _i inF and every pointx in the boundary ofS _i there is a pointv inL such thatv illuminatesx, i.e. the line segment joiningv tox intersects the union of the elements ofF in exactly {x}.The problem we treat is the size of a setS needed to illuminate a familyF={S ₁, ...,S _n } ofn disjoint compact sets inE ^d . We also treat the problem of putting these convex sets in mutually disjoint convex polytopes, each one having at most a certain number of facets.     

Staircase kernels for orthogonald-polytopes

LetS be a finite union of boxes inR ^d . Forx inS, defineA _x ={yx is clearly visible fromy via staircase paths inS}, and let KerS denote the staircase kernel ofS. Then KerS={A _x x is a point of local nonconvexity ofS}. A similar result holds with clearly visible replaced by visible and points of local nonconvexity ofS replaced by boundary points ofS.Supported in part by NSF grant DMS-9207019.     

Can visibility graphs Be represented compactly?

We consider the problem of representing the visibility graph of line segments as a union of cliques and bipartite cliques. Given a graphG, a familyG={G ₁,G ₂,...,G _k } is called aclique cover ofG if (i) eachG _i is a clique or a bipartite clique, and (ii) the union ofG _i isG. The size of the clique coverG is defined as ∑ _i=1 ^k n _i , wheren _i is the number of vertices inG _i . Our main result is that there are visibility graphs ofn nonintersecting line segments in the plane whose smallest clique cover has size Ω(n ²/log² n). An upper bound ofO(n ²/logn) on the clique cover follows from a well-known result in extremal graph theory. On the other hand, we show that the visibility graph of a simple polygon always admits a clique cover of sizeO(nlog³ n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n logn). The work by the first author was supported by National Science Foundation Grant CCR-91-06514. The work by the second author was supported by a USA-Israeli BSF grant. The work by the third author was supported by National Science Foundation Grant CCR-92-11541.     

Tail bounds for the height and width of a random tree with a given degree sequence

Fix a sequence c = (c₁,…,c_n) of non‐negative integers with sum n ? 1. We say a rooted tree T has child sequence c if it is possible to order the nodes of T as v₁,…,v_n so that for each 1 ≤ i ≤ n, v_i has exactly c_i children. Let ${\mathcal T}Fix a sequence c = (c₁,…,c_n) of non‐negative integers with sum n ? 1. We say a rooted tree T has child sequence c if it is possible to order the nodes of T as v₁,…,v_n so that for each 1 ≤ i ≤ n, v_i has exactly c_i children. Let ${\mathcal T}$ be a plane tree drawn uniformly at random from among all plane trees with child sequence c . In this note we prove sub‐Gaussian tail bounds on the height (greatest depth of any node) and width (greatest number of nodes at any single depth) of ${\mathcal T}$. These bounds are optimal up to the constant in the exponent when c satisfies $\sum_{i=1}^n c_i^2=O(n)$; the latter can be viewed as a “finite variance” condition for the child sequence. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Circuits of Each Length in Tournaments

A tournament is a directed graph whose underlying graph is a complete graph. A circuit is an alternating sequence of vertices and arcs of the form v ₁, a ₁, v ₂, a ₂, v ₃, . . . , v _n-1, a _n-1, v _n in which vertex v _n  = v ₁, arc a _i  = v _i v _i+1 for i = 1, 2, . . . , n?1, and \({a_i \neq a_j}\) if \({i \neq j}\) . In this paper, we shall show that every tournament T _n in a subclass of tournaments has a circuit of each length k for \({3 \leqslant k \leqslant \theta(T_n)}\) , where \({\theta(T_n) = \frac{n(n-1)}{2}-3}\) if n is odd and \({\theta(T_n) = \frac{n(n-1)}{2}-\frac{n}{2}}\) otherwise. Note that a graph having θ(G) > n can be used as a host graph on embedding cycles with lengths larger than n to it if congestions are allowed only on vertices.