On sets having finitely many points of local nonconvexity and propertyP m

If a pointq ofS has the property that each neighborhood ofq contains pointsx andy such that the segmentxy is not contained byS, q is called a point of local nonconvexity ofS. LetQ denote the set of points of local nonconvexity ofS. Tietze’s well known theorem that a closed connected setS in a linear topological space is convex ifQ=φ is generalized in the result:If S is a closed set in a linear topological space such that S ∼ Q is connected and |Q|=n<∞,then S is the union of n+1or fewer closed convex sets. Letk be the minimal number of convex sets needed in a convex covering ofS. Bounds fork in terms ofm andn are obtained for sets having propertyP _m and |Q|=n.     

A new generalization of Moulton affine planes

We consider a variation on W. A. Pierce's construction of Moulton planes. For any pseudo-ordered fieldF, the pairs of elements ofF are taken as points, and straight lines are given by the equationsx=c,y=mx+n withm≥0 andg(y=mf(x)+n withm < 0, wheref andg are mappings ofF into itself which have to satisfy a number of conditions.     

On the intersection of maximalm-convex subsets

A subsetS of a real linear spaceE is said to bem-convex providedm≧2, there exist more thanm points inS, and for eachm distinct points ofS at least one of the ( ₂ ^m ) segments between thesem points is included inS. InE, letxy denote the segment between two pointsx andy. For any pointx inSυE, letS _x ={y: xyυS}. The kernel of a setS is then defined as {xεS: S _x=S}. It is shown that the kernel of a setS is always a subset of the intersection of all maximalm-convex subsets ofS. A sufficient condition is given for the intersection of all the maximalm-convex subsets of a setS to be the kernel ofS.     

A new approach to the dynamic maintenance of maximal points in a plane

A pointp _i=(x _i, y_i) in thex–y plane ismaximal if there is no pointp _j=(x _j, y_j) such thatx _j>x_i andy _j>y_i. We present a simple data structure, a dynamic contour search tree, which contains all the points in the plane and maintains an embedded linked list of maximal points so thatm maximal points are accessible inO(m) time. Our data structure dynamically maintains the set of points so that insertions takeO(logn) time, a speedup ofO(logn) over previous results, and deletions takeO((logn)²) time.The research of the first author was partially supported by the National Science Foundation under Grant No. DCR-8320214 and by the Office of Naval Research on Contract No. N 00014-86-K-0689. The research of the second author was partially supported by the Office of Naval Research on Contract No. N 00014-86-K-0689.     

A sparse graph almost as good as the complete graph on points inK dimensions

A setV ofn points ink-dimensional space induces a complete weighted undirected graph as follows. The points are the vertices of this graph and the weight of an edge between any two points is the distance between the points under someL _p metric. Let ε≤1 be an error parameter and letk be fixed. We show how to extract inO(n logn+ε ^−k log(1/ε)n) time a sparse subgraphG=(V, E) of the complete graph onV such that: (a) for any two pointsx, y inV, the length of the shortest path inG betweenx andy is at most (1+∈) times the distance betweenx andy, and (b)|E|=O(ε^−k n).     

Note on the solution of a certain boundary-value problem

A simple algorithm is described for inverting the operatorD _x D _y (D _x andD _y here and subsequently denote partial differentiation with respect tox andy respectively) which occurs in the iterative solution of the equationD _x D _y f (x, y)=g (x, y, f, D _x f, D _x ² f,D _x D _y f, D _y ² f) when boundary values off(x,y) are given along the sides of the rectangle in thexy-plane whose corners are at the points (a,b); (a+(n+1)k,b); (a+(n+1)k,b+(n+1)h); (a,b+(n+1)h).Communication M. R. 43 of the Computation Department of the Mathematical Centre, Amsterdam.     

AnO(n logn) algorithm for computing the link center of a simple polygon

We present an algorithm that determines the link center of a simplen-vertex polygonP inO(n logn) time. The link center of a simple polygon is the set of pointsx insideP at which the maximal link-distance fromx to any other point inP is minimized. The link distance between two pointsx andy insideP is defined to be the smallest number of straight edges in a polygonal path insideP connectingx andy. Using our algorithm we also obtain anO(n logn)-time solution to the problem of determining the link radius ofP. The link radius ofP is the maximum link distance from a point in the link center to any vertex ofP. Both results are improvements over theO(n ²) bounds previously established for these problems. The research of J.-R. Sack was supported by the Natural Sciences and Engineering Research Council of Canada.     

Protecting convex sets

A point-setS is protecting a collection F =T ₁,T ₂,..., _n ofn mutually disjoint compact sets if each one of the setsT _i is visible from at least one point inS; thus, for every setT _i ∈F there are points xS andy T _i such that the line segment joining x to y does not intersect any element inF other thanT _i . In this paper we prove that [2(n-2)/3] points are always sufficient and occasionally necessary to protect any family F ofn mutually disjoint compact convex sets. For an isothetic family F, consisting ofn mutually disjoint rectangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary to protect it. IfF is a family of triangles, [4n/7] points are always sufficient. To protect families ofn homothetic triangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary.     

The 2-circle and 2-disk problems on trees

Our purpose here is to consider on a homogeneous tree two Pompeiutype problems which classically have been studied on the plane and on other geometric manifolds. We obtain results which have remarkably the same flavor as classical theorems. Given a homogeneous tree, letd(x, y) be the distance between verticesx andy, and letf be a function on the vertices. For each vertexx and nonnegative integern let Σ _n f(x) be the sum Σ _{d(x, y)=n} f(y) and letB _n f(x)=Σ _{d(x, y)≦n} f(y). The purpose is to study to what extent Σ _n f andB _n f determinef. Since these operators are linear, this is really the study of their kernels. It is easy to find nonzero examples for which Σ _n f orB _n f vanish for one value ofn. What we do here is to study the problem for two values ofn, the 2-circle and the 2-disk problems (in the cases of Σ _n andB _n respectively). We show for which pairs of values there can exist non-zero examples and we classify these examples. We employ the combinatorial techniques useful for studying trees and free groups together with some number theory.     

Quasi-symmetric 2, 3, 4-designs

Quasi-symmetric designs are block designs with two block intersection numbersx andy It is shown that with the exception of (x, y)=(0, 1), for a fixed value of the block sizek, there are finitely many such designs. Some finiteness results on block graphs are derived. For a quasi-symmetric 3-design with positivex andy, the intersection numbers are shown to be roots of a quadratic whose coefficients are polynomial functions ofv, k and λ. Using this quadratic, various characterizations of the Witt—Lüneburg design on 23 points are obtained. It is shown that ifx=1, then a fixed value of λ determines at most finitely many such designs.     

The road-colouring problem

LetG be a finite directed graph which is irreducible and aperiodic. Assume each vertex ofG leads to at least two other vertices, and assumeG has a cycle of prime length which is a proper subset ofG. Then there exist two functionsr:G →G andb:G →G such that ifr(x)=y andb(x)=z thenx →y andx →z inG andy ≠z and such that some composition ofr’s andb’s is a constant function. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. I am grateful to Cornell University whose kind hospitality I enjoyed while working on this problem.     

A Helmholtz-Lie type characterization of ellipsoids,II

A closed convex surfaceS in is an ellipsoid if and only if for anyx, y εS there is an affinity mappingx ontoy and a neighborhood ofx inS onto a neighborhood ofy inS.     

Semigroup actions on homogeneous spaces

LetG be a connected semi-simple Lie group with finite center andS⊄G a subsemigroup with interior points. LetG/L be a homogeneous space. There is a natural action ofS onG/L. The relationx≤y ify ∈Sx, x, y ∈G/L, is transitive but not reflexive nor symmetric. Roughly, a control set is a subsetD ⊄G/L, inside of which reflexivity and symmetry for ≤ hold. Control sets are studied inG/L whenL is the minimal parabolic subgroup. They are characterized by means of the Weyl chambers inG meeting intS. Thus, for eachw ∈W, the Weyl group ofG, there is a control setD _w .D ₁ is the only invariant control set, and the subsetW(S)={w:D _w =D ₁} turns out to be a subgroup. The control sets are determined byW(S)/W. The following consequences are derived: i)S=G ifS is transitive onG/H, i.e.Sx=G/H for allx ∈G/H. HereH is a non discrete closed subgroup different fromG andG is simple. ii)S is neither left nor right reversible ifS #G iii)S is maximal only if it is the semigroup of compressions of a subset of some minimal flag manifold. Research partially supported by CNPq grant n^o 50.13.73/91-8     

A NEW PROPERTY OF BINARY UNDIRECTED de BRUIJN GRAPHS

1.IntroductionThen-dimensionalbinarydirecteddeBruijngraph,denotedbyB(n),hasthevertex-set{xlxz'xu:xiE{0,1}}.TherearetwoarcsfromavertexxlxZ'xutothevenicesxzxa'x.-lx.0andxZx3'x.-l-c.1.Then-dimensionalbinaryundirecteddeBruijngraph,denotedbyUB(n),isobtainedfromB(n)bydeletingtheorientationofthearcsandthenomittingmultipleedgesandloops.ItiswellknownthatUB(n)hasdiameternandconnectivitytwo.ThedeBruijngraphshavebeenwidelyusedincodingtheory[6].Theyhavebeenalsoproposedasapossiblegoodcomputeri…     

Paths in graphs,reducing the edge-connectivity only by two

For every pair of verticesx andy in a connected, finite, undirected graphG, there is a pathP joiningx andy such that deleting the edges ofP fromG, for every pair of vertices ofG, the local edge-connectivity decreases by at most two.     

On the lattice property of the plane and some problems of Dirac,Motzkin and Erdős in combinatorial geometry

LetS be a set ofn non-collinear points in the Euclidean plane. It will be shown here that for some point ofS the number ofconnecting lines through it exceedsc · n. This gives a partial solution to an old problem of Dirac and Motzkin. We also prove the following conjecture of Erdős: If any straight line contains at mostn−x points ofS, then the number of connecting lines determined byS is greater thanc · x · n. Dedicated to Paul Erdős on his seventieth birthday     

On (a, b)-consecutive petersen graphs

The generalized Petersen graphsP(n,k), n≥3, 1≤k<n/2, consist of an outern-cyclex _o x ₁ x ₂...x _n−1 , a set ofn spokesx _i y _i (0≤i≤n−1), andn inner edgesy _i y _i +k with indices taken modulon. This paper deals with (a,b)-consecutive labelings of generalized Petersen graphP(n,k).     

Bitableaux Bases for the Diagonally Invariant Polynomial Quotient Rings

LetR=Q[x₁, x₂, …, x_n,y₁, y₂, …, y_n,z₁, …, z_n,w₁, …, w_n], letR^S_n={P∈R:σP=P∀σ∈S_n} and letμandνbe hook shape partitions ofn. WithΔ_μ(X, Y) andΔ_ν(Z, W) being appropriately defined determinants, ∂_xibeing the partial derivative operator with respect tox_iandP(∂)=P(∂_x1, …, ∂_xn, ∂_y1, …, ∂_wn), define _{μ, ν}={P∈R^S_n:P(∂)Δ_μ(X, Y)Δ_ν(Z, W)=0}. A basis is constructed for the polynomial quotient ringR^S_n/_{μ, ν}that is indexed by pairs of standard tableaux. The Hilbert series ofR^S_n/_{μ, ν}is related to the Macdonaldq, t-Kostka coefficients.     

Homomorphisms of extensions of a semigroup within a variety

LetV be a semigroup variety containing all commutative semigroups such that the law of exponents (xy) ⁿ =xⁿyⁿ fails inV for everyn > 1. For every semigroupS V such that the reflection of the semigroup obtained fromS by an adding unity has only one idempotent there exists a semigroupT V extendingS without non-trivial endomorphisms. In more general, the full subcategory ofV formed by all extensions ofS withinV is universal.Presented by B. M. Schein.     

A helmholtz-lie type characterization of ellipsoids,I

A closed convex surfaceS in withd odd, is an ellipsoid if and only if it has the following property: for any pair of pointsx, y inS there is an affine transformation which mapsx ontoy and a suitable neighborhood ofx inS onto a neighborhood ofy inS.