More orthogonal double covers of complete graphs by Hamiltonian paths

An orthogonal double cover (ODC) of the complete graph K_n by a graph G is a collection G of n spanning subgraphs of K_n, all isomorphic to G, such that any two members of G share exactly one edge and every edge of K_n is contained in exactly two members of G. In the 1980s Hering posed the problem to decide the existence of an ODC for the case that G is an almost-Hamiltonian cycle, i.e. a cycle of length n-1. It is known that the existence of an ODC of K_n by a Hamiltonian path implies the existence of ODCs of K_4n and of K_16n, respectively, by almost-Hamiltonian cycles. Horton and Nonay introduced two-colorable ODCs and showed: If there are an ODC of K_n by a Hamiltonian path for some n?3 and a two-colorable ODC of K_q by a Hamiltonian path for some prime power q?5, then there is an ODC of K_qn by a Hamiltonian path. In [U. Leck, A class of 2-colorable orthogonal double covers of complete graphs by hamiltonian paths, Graphs Combin. 18 (2002) 155-167], two-colorable ODCs of K_n and K_2n, respectively, by Hamiltonian paths were constructed for all odd square numbers n?9. Here we continue this work and construct cyclic two-colorable ODCs of K_n and K_2n, respectively, by Hamiltonian paths for all n of the form n=4k²+1 or n=(k²+1)/2 for some integer k.     

Some Results on Cross-Positive Matrices

Let K be a cone in Rⁿ, K1 its dual cone. An n×n matrix A is called cross-positive on K if and only if for all y?K, z?K1 such that (z, y) = 0 we have (z, Ay)?0. In this short note new equivalent conditions for matrices cross-positive on K will be given in terms of the partial ordering in Rⁿ induced by the cone K.     

On indecomposable subcones

Let K₁,K₂ be cones. We say that K₁ is a subcone of K₂ if ExtK₁?ExtK₂. Furthermore, if K₁≠K₂, K₁ is called a proper subcone; if dimK₁=dimK₂, K₁ is called a non-degenerate subcone. We first prove that every n-dimensional indecomposable cone, n?3, contains a non-degenerate indecomposable subcone which has no more than 2n-2 extremals. Then we construct for each n?3 an n-dimensional indecomposable cone with exactly 2n-2 extremals such that each of its proper non-degenerate subcones is decomposable.     

The singular linear preservers of non-singular matrices

Given an arbitrary field K, we reduce the determination of the singular endomorphisms f of M_n(K) such that f(GL_n(K))⊂GL_n(K) to the classification of n-dimensional division algebras over K. Our method, which is based upon Dieudonné’s theorem on singular subspaces of M_n(K), also yields a proof for the classical non-singular case.     

On geometric SDPS-sets of elliptic dual polar spaces

Let n∈N?{0,1} and let K and K^′ be fields such that K^′ is a quadratic Galois extension of K. Let Q⁻(2n+1,K) be a nonsingular quadric of Witt index n in PG(2n+1,K) whose associated quadratic form defines a nonsingular quadric Q⁺(2n+1,K^′) of Witt index n+1 in PG(2n+1,K^′). For even n, we define a class of SDPS-sets of the dual polar space DQ⁻(2n+1,K) associated to Q⁻(2n+1,K), and call its members geometric SDPS-sets. We show that geometric SDPS-sets of DQ⁻(2n+1,K) are unique up to isomorphism and that they all arise from the spin embedding of DQ⁻(2n+1,K). We will use geometric SDPS-sets to describe the structure of the natural embedding of DQ⁻(2n+1,K) into one of the half-spin geometries for Q⁺(2n+1,K^′).     

On Connected 2-K n -Residual Graphs

A graph G is said to be K _n -residual if for every point u in G, the graph obtained by removing the closed neighborhood of u from G is isomorphic to K _n . We inductively define a multiply-K _n -residual graph by saying that G is m-K _n -residual if the removal of the closed neighborhood of any vertex of G results in an (m – 1)-K _n -residual graphs. Erdös, Harary and Klawe [2] determined the minimum order of the m?K _n -residual graphs for all m and n, which are not necessarily connected, the minimum order of connected; K _n -residual graphs, all K _n -residual extremal graphs. They also stated some conjectures regarding the connected case. In this paper, we determine the minimum order of a connected 2-K _n -residual graph and specify the extremal graphs, expect for n = 3. In particular, we determining only one connected 2-K ₄-residual graph of minimal order, and show that there is a connected 2-K ₆-residual graph non isomorphic to K ₈ × K ₃ with minimum order. Finally we present and a revised version of the conjecture in [2].     

Hamiltonian decompositions of complete graphs

It is well known that K_{2n + 1} can be decomposed into n edge-disjoint Hamilton cycles. A novel method for constructing Hamiltonian decompositions of K_{2n + 1} is given and a procedure for obtaining all Hamiltonian decompositions of of K_{2n + 1} is outlined. This method is applied to find a necessary and sufficient condition for a decomposition of the edge set of K_r (r ≤ 2n) into n classes, each class consisting of disjoint paths to be extendible to a Hamiltonian decomposition of K_{2n + 1} so that each of the classes forms part of a Hamilton cycle.     

An Almost Accurate Location of the Maximum Stirling Number(s) of the Second Kind

The Stirling number of the second kind S(n, k) is the number of ways of partitioning a set of n elements into k nonempty subsets. It is well known that the numbers S(n, k) are unimodal in k, and there are at most two consecutive values K _n such that (for fixed n) S(n, K _n ) is maximal. We determine asymptotic bounds for K _n , which are unexpectedly good and improve earlier results. The method used here shows a possible strategy for obtaining numerical bounds such that in almost all cases K _n can be uniquely determined.     

Steiner pentagon systems

A Steiner pentagon system is a pair (K_n, P) where K_n isthe complete undirected graph on n vertices. P is a collection of edge-disjoint pentagons which partition K_n, and such that every part of distinct vertices of K_n is joined by a path of length two in exactly one pentagon of the collection P. The number n is called the order of the system. This paper gives a somplete solution of the existence problem of Steiner pentagon systems. In particular it is shown that the spectrum for Steiner pentagon systems (=the set of all orders for which a Steiner pentagon system exists) is precisely the set of all n ≡ 1 or 5 (mod 10), except 15, for which no such system exists.     

Gelfand pairs on the Heisenberg group and Schwartz functions

Let Hn be the (2n+1)-dimensional Heisenberg group and K a compact group of automorphisms of Hn such that (K?Hn,K) is a Gelfand pair. We prove that the Gelfand transform is a topological isomorphism between the space of K-invariant Schwartz functions on Hn and the space of Schwartz function on a closed subset of Rs homeomorphic to the Gelfand spectrum of the Banach algebra of K-invariant integrable functions on Hn.     

Extremal convex sets

Letf be an extended real valued function on the classK ⁿ of closed convex subsets of euclideann-dimensional space. A setK∈K ⁿ is said to bef-maximal if the conditionsK′∈K ⁿ ,K?K′,K≠K′ implyf(K)<f(K′), andf-minimal ifK′∈K ⁿ,K′∈K,K′≠K impliesf(K′)<f(K). In the cases whenf is the circumradius or inradius allf-maximal andf-minimal sets are determined. Under a certain regularity assumption a corresponding result is obtained for the minimal width. Moreover, a general existence theorem is established and a result concerning the existence of extremal sets with respect to packing and covering densities is proved.     

On a Normal Integral Basis Problem over Cyclotomic Zp-extensions, II

Let p be an odd prime number, K an imaginary abelian field with ζ_p∈K^×, and K_∞/K the cyclotomic Z_p-extension with its nth layer K_n. In the previous paper, we showed that for any n and any unramified cyclic extension L/K_n of degree p, LK_n+1/K_n+1 does have a normal integral basis (NIB) even if L/K_n has no NIB, under the assumption that p does not divide the class number of the maximal real subfield K⁺ (and some additional assumptions on K). In this paper, we show that similar but more delicate phenomena occur for a certain class of tamely ramified extensions of degree p.     

Equivariant collapses and the homotopy type of iterated clique graphs

The clique graph K(G) of a simple graph G is the intersection graph of its maximal complete subgraphs, and we define iterated clique graphs by K⁰(G)=G, Kⁿ⁺¹(G)=K(Kⁿ(G)). We say that two graphs are homotopy equivalent if their simplicial complexes of complete subgraphs are so. From known results, it can be easily inferred that Kⁿ(G) is homotopy equivalent to G for every n if G belongs to the class of clique-Helly graphs or to the class of dismantlable graphs. However, in both of these cases the collection of iterated clique graphs is finite up to isomorphism. In this paper, we show two infinite classes of clique-divergent graphs that satisfy G?Kⁿ(G) for all n, moreover Kⁿ(G) and G are simple-homotopy equivalent. We provide some results on simple-homotopy type that are of independent interest.     

Genera, band sum of knots and Vassiliev invariants

Recently Stoimenow showed that for every knot K and any n∈N and u₀?u(K) there is a prime knot Kn,uo which is n-equivalent to the knot K and has unknotting number u(Kn,uo) equal to u₀. The similar result has been obtained for the 4-ball genus gs of a knot. Stoimenow also proved that any admissible value of the Tristram-Levine signature σξ can be realized by a knot with the given Vassiliev invariants of bounded order. In this paper, we show that for every knot K with genus g(K) and any n∈N and m?g(K) there exists a prime knot L which is n-equivalent to K and has genus g(L) equal to m.     

Covering shadows with a smaller volume

For n?2 a construction is given for convex bodies K and L in Rn such that the orthogonal projection Lu onto the subspace u^⊥ contains a translate of Ku for every direction u, while the volumes of K and L satisfy Vn(K)>Vn(L).A more general construction is then given for n-dimensional convex bodies K and L such that each orthogonal projection Lξ onto a k-dimensional subspace ξ contains a translate of Kξ, while the mth intrinsic volumes of K and L satisfy Vm(K)>Vm(L) for all m>k.For each k=1,…,n, we then define the collection Cn,k to be the closure (under the Hausdorff topology) of all Blaschke combinations of suitably defined cylinder sets (prisms).It is subsequently shown that, if L∈Cn,k, and if the orthogonal projection Lξ contains a translate of Kξ for every k-dimensional subspace ξ of Rn, then Vn(K)?Vn(L).The families Cn,k, called k-cylinder bodies of Rn, form a strictly increasing chain

Cn,1⊂Cn,2⊂?⊂Cn,n−1⊂Cn,n,     

To what extent is a large space of matrices not closed under product?

Let (K) be a field. Given an arbitrary linear subspace V of M_n(K) of codimension less than n-1, a classical result states that V generates the (K)-algebra M_n(K). Here, we strengthen this statement in three ways: we show that M_n(K) is spanned by the products of the form AB with (A,B)∈V²; we prove that every matrix in M_n(K) can be decomposed into a product of matrices of V; finally, when V is a linear perplane of M_n(K) and n>2, we show that every matrix in M_n(K) is a product of two elements of V.     

Spherical functions on Euclidean space

We study spherical functions on Euclidean spaces from the viewpoint of Riemannian symmetric spaces. Here the Euclidean space En=G/K where G is the semidirect product Rn⋅K of the translation group with a closed subgroup K of the orthogonal group O(n). We give exact parameterizations of the space of (G,K)—spherical functions by a certain affine algebraic variety, and of the positive definite ones by a real form of that variety. We give exact formulae for the spherical functions in the case where K is transitive on the unit sphere in En.     

Subspaces of matrices with special rank properties

Let K be a field and let M_m×n(K) denote the space of m×n matrices over K. We investigate properties of a subspace M of M_m×n(K) of dimension n(m-r+1) in which each non-zero element of M has rank at least r and enumerate the number of elements of a given rank in M when K is finite. We also provide an upper bound for the dimension of a constant rank r subspace of M_m×n(K) when K is finite and give non-trivial examples to show that our bound is optimal in some cases. We include a similar a bound for the maximum dimension of a constant rank subspace of skew-symmetric matrices over a finite field.     

Central limit theorems for random polytopes in a smooth convex set

Let K be a smooth convex set with volume one in Rd. Choose n random points in K independently according to the uniform distribution. The convex hull of these points, denoted by Kn, is called a random polytope. We prove that several key functionals of Kn satisfy the central limit theorem as n tends to infinity.     

Almost everywhere convergence of sequences of multiplier operators on local fields

LetK ⁿ be then-dimensional vector space over a local fieldK. Two maximal multiplier theorems onL ^p (K ⁿ ) are proved for certain multiplier operator sequences associated with regularization and dilation respectively. Consequently the a. e. convergence of such multiplier operator sequences is obtained. This sharpens Taibleson’s main result and applies to several important singular integral operators onK ⁿ .