On the connectivity of unit distance graphs

For a number fieldK , consider the graphG(K^d), whose vertices are elements ofK ^d, with an edge between any two points at (Euclidean) distance 1. We show thatG(K²) is not connected whileG(K^d) is connected ford 5. We also give necessary and sufficient conditions for the connectedness ofG(K³) andG(K⁴).     

Irreducibility of moduli space of harmonic 2-spheres in S4

In this paper we prove that ford3, the moduli spaces of degreed branched superminimal immersions of the 2-sphere intoS ⁴ has 2 irreducible components. Consequently, the moduli space of degreed harmonic 2-spheres inS ⁴ has 3 irreducible components.     

Feedback numbers of Kautz digraphs

A subset of vertices (resp. arcs) of a graph G is called a feedback vertex (resp. arc) set of G if its removal results in an acyclic subgraph. Let f(d,n) (f_a(d,n)) denote the minimum cardinality over all feedback vertex (resp. arc) sets of the Kautz digraph K(d,n). This paper proves that for any integers d?2 and n?1

     

On a combination method of VDR and patchwork for generating uniform random points on a unit sphere

In this paper, we use a combination of VDR theory and patchwork method to derive an efficient algorithm for generating uniform random points on a unit d-sphere. We first propose an algorithm to generate random vector with uniform distribution on a unit 2-sphere on the plane. Then we use VDR theory to reduce random vector X_d with uniform distribution on a unit d-sphere into , such that the random vector (X_d-1,X_d) is uniformly distributed on a unit 2-sphere and X_d-2 has conditional uniform distribution on a (d-2)-sphere of radius , given V=v with V having the p.d.f. . Finally, we arrive by induction at an algorithm for generating uniform random points on a unit d-sphere.     

On Translational Clouds for a Convex Body

For a d-dimensional convex body K let C(K) denote the minimum size of translational clouds for K. That is, C(K) is the minimum number of mutually non-overlapping translates of K which do not overlap K and block all the light rays emanating from any point of K. In this paper we prove the general upper bound . Furthermore, for an arbitrary centrally symmetric d-dimensional convex body S we show . Finally, for the d-dimensional ball B^d we obtain the bounds .     

Rectangles inscribed in symmetric continua

LetQ be any rectangle and letK ^d (d 2) be a continuum which is either symmetric across a hyperplane or symmetric through a pointz K. We show thatK contains the vertices of a rectangle similar toQ which exhibits the same symmetry as doesK.     

Hadwiger's covering conjecture and low dimensional dual cyclic polytopes

LetK be a convex body in a Euclideand-spaceE ^d withd1. In 1957, H. Hadwiger conjectured thatK can always be covered by 2 ^d smaller homothetic copies ofK. We verify this conjecture in the case thatK is the polar of a cyclicd-polytope andd=3, 4 and 5.     

Factorizations of complete multipartite graphs into generalized cubes

For a positive integer d, the usual d‐dimensional cube Q_d is defined to be the graph (K₂)^d, the Cartesian product of d copies of K₂. We define the generalized cube Q(K_k, d) to be the graph (K_k)^d for positive integers d and k. We investigate the decomposition of the complete multipartite graph K into factors that are vertex‐disjoint unions of generalized cubes Q(K_k, d_i), where k is a power of a prime, n and j are positive integers with j ≤ n, and the d_i may be different in different factors. We also use these results to partially settle a problem of Kotzig on Q_d‐factorizations of K_n. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 144–150, 2000     

Very Ampleness of d-standard Classes on Rational Surfaces

We consider a rational surface X_r, obtained by blowing up ² along a curvilinear zero-dimensional subscheme of length r of the regular locus of a reduced irreducible plane curve of degree d, with d 4; and we give sufficient conditions for d-standard classes to be very ample (resp. base point free or non special) on such a rational surface X_r.     

Lattice points in lattice polytopes

IfK is the underlying point-set of a simplicial complex of dimension at mostd whose vertices are lattice points, and ifG(K) is the number of lattice points inK, then the lattice point enumeratorG(K,t)=1+ _n1 G(nK)t ⁿ takes the formC(K, t)/(1–t) ^d+1, for some polynomialC(K, t). Here,C(K, t) is expressed as a sum of local terms, one for each face ofK. WhenK is a polytope or its boundary, there result inequalities between the numbersG _r (K), whereG(n K)= _r=0 ^d n ^r G _r (K).     

Very Ampleness of <Emphasis Type="Italic">d</Emphasis>-standard Classes on Rational Surfaces

We consider a rational surface X_r, obtained by blowing up ² along a curvilinear zero-dimensional subscheme of length r of the regular locus of a reduced irreducible plane curve of degree d, with d 4; and we give sufficient conditions for d-standard classes to be very ample (resp. base point free or non special) on such a rational surface X_r.Postdoctoral Fellow of the Fund for Scientific Research-Flanders (Belgium).     

A note on the illumination of convex bodies

LetKE ^d be a convex body and letl _r(K) denote the minimum number ofr-dimensional affine subspaces ofE ^d lying outsideK with which it is possible to illuminateK, where 0rd–1. We give a new proof of the theorem thatl _r(K)(d+1)/(r+1) with equality for smoothK.The work was supported by Hung. Nat. Found. for Sci . Research No. 326-0213 and 326-0113.     

<Emphasis Type="Italic">M</Emphasis>-spherical <Emphasis Type="Italic">K</Emphasis>-modules of a rank one semisimple Lie group

Let G =K A N be an Iwasawa decomposition of a connected, noncompact real semisimple Lie group with finite center and let M be the centralizer of A in K . B. Kostant proved that for every irreducible M-spherical K-module V there exists a unique d (the Kostant degree of V) such that V can be realized as a submodule of the space of all -harmonic homogeneous polynomials of degree d on . Here is a Cartan decomposition of the complexification of the Lie algebra of G .In this paper we give an algorithm to obtain a highest weight vector from any M-invariant vector in an irreducible M-spherical K-module. This algorithm allows us to compute a sharp bound for the Kostant degree d(v) of any M-invariant vector v in a locally finite M-spherical K-module V. The method computes d(v) effectively for any V if G is locally isomorphic to SO(n,1) and for if G is locally isomorphic to SU(n,1).Partially supported by Agencia Córdoba Ciencia and CONICET Mathematics Subject Classification (2000):Primary 22E46, Secondary 43A85     

Determining classes of convex bodies by restricted sets of Steiner symmetrizations

In [4] it was shown that a convex body in R ^d (d≥2) is a simplex if and only if each of its Steiner symmetrals has exactly two extreme points outside the corresponding symmetrization space. A natural question arises about restricted sets of symmetrization directions which guarantee this characterization of simplices. Let denote an arbitrary triple of pairwisedistinct great (d-2)-spheres on the unit sphere of R ^d .We shall prove that a convex body K is a simplex if and only if for every direction the corresponding Steiner symmetral of K has the property described above. Weaker conditions characterize additional classes of convex bodies, e.g. (d-2)-fold pyramids over planar, convex 4-gons.     

A characterization of Thalesian orthogonality in Pappian planes of characteristic 2

In an affine plane over a field K, any Thalesian orthogonality relation is equivalent to _d for some dK{0}, where _d denotes the relation with constant of orthogonality d/it (i.e., after suitable coordinatization the slopes m, m*K{0} of orthogonal lines satisfy m·m*=d) (cf. [1], [5]). In the present paper we show that in a Pappian plane of characteristic two any orthogonality relation admitting the same group as ₁ is equivalent to ₁. This gives a characterization of Thalesian orthogonality over perfect fields of characteristic two.     

On polynomial automorphisms of spheres

Let K be an infinite field with characteristic different from two and (K) the n-sphere over K. We show that ambient polynomial automorphisms of (K) preserve the quadratic form x ₀² + ⋯ + x _n ² and the group Aut ((K ⁿ⁺¹, (K)) of such automorphisms of (K) is isomorphic to the (n + 1)-orthogonal group O(n + 1, K) provided K is real. Next, the restriction map Aut (K ³, (K)) → Aut ( (K)) yields a surjection provided K is an algebraically closed field as well. Furthermore, for any such a field K, there is an imbedding

Computing the volume is difficult

For every polynomial time algorithm which gives an upper bound (K) and a lower boundvol(K) for the volume of a convex setKR ^d , the ratio (K)/vol(K) is at least (cd/logd) ^d for some convex setKR ^d .This paper was partly written when both authors were on leave from the Mathematical Institute of the Hungarian Academy of Sciences, 1364 Budapest, P.O. Box 127, Hungary.     

K-theory of blow-ups and vector bundles on the cone over a surface

TheK-theory of the derived categories as well as someK-theoretic invariants associated to the resolution of singularities are applied in order to compute theK ₀-groups of a variety with finitely many singular points. Explicit computations are given in order to determine theK ₀ and the relative CH* groups of the affine cone over a nonsingular surface of degreed in ^d+1 not contained in any hyperplane.     

Forms Represented by Linear Forms

This paper is the third in a series in which the author investigates the question of representation of forms by linear forms. Whereas in the first two treatments the proportion of forms F of degree 3 (resp. degree d) which can be written as a sum of two cubes (resp. d-th powers) of linear forms with algebraic coefficients is determined, the generalization now consists in allowing more general expressions of degree d in two linear forms. The main result is thus to give an asymptotic formula, in terms of their height, for the number or decomposable forms that have a representation

Coincidences of Centers of Edge-Incentric, or Balloon, Simplices

An edge-incentric d-simplex is defined to be a d-simplex S which admits a (d − 1)-sphere that touches all the edges of S internally. The center of such a sphere is called the edge-incenter of S and is denoted by . Equivalently, S is edge-incentric if and only if its vertices are the centers of d + 1 (d − 1)-spheres in mutual external touch, and for this reason one may call such an S a balloon d-simplex. An orthocentric d-simplex is a d-simplex in which the altitudes are concurrent. The point of concurrence is called the orthocenter and is denoted by . The spaces of edge-incentric and of orthocentric d-simplices have the same dimension d in the sense that a d-simplex in either space can be parametrized, up to shape, by d numbers. Edge-incentric and orthocentric tetrahedra are the first two of the four special classes of tetrahedra studied in [1, Chapter IX.B, pp. 294–333]. The degree of regularity implied by the coincidence of two or more centers of a general d-simplex is investigated in [8], where it is shown that the coincidence of the centroid , the circumcenter , and the incenter does not imply much regularity. For an orthocentric d-simplex S, however, it is proved in [9] that if any two of the centers , and coincide, then S is regular. In this paper, the same question is addressed for edge-incentric d-simplices. Among other things, it is proved that if any three of the centers , and of an edge-incentric d-simplex S coincide, then S is regular, and it is also shown that none of the coincidences , and implies regularity (except when d ≤ 3, d ≤ 4, and d ≤ 6, respectively). In contrast with the afore-mentioned results for orthocentric d-simplices, this emphasizes once more the feeling that, regarding many important properties, orthocentric d-simplices are the true generalizations of triangles. Several open questions are posed. Received: June 19, 2006.