Ideal einstein,conformally flat and semi-symmetric immersions

Recently, B. Y. Chen introduced a new intrinsic invariant of a manifold, and proved that everyn-dimensional submanifold of real space formsR ^m (ε) of constant sectional curvature ε satisfies a basic inequality δ(n ₁,…,n _k )≤c(n ₁,…,n _k )H ²+b(n ₁,…,n _k )ε, whereH is the mean curvature of the immersion, andc(n ₁,…,n _k ) andb(n ₁,…,n _k ) are constants depending only onn ₁,…,n _k ,n andk. The immersion is calledideal if it satisfies the equality case of the above inequality identically for somek-tuple (n ₁,…,n _k ). In this paper, we first prove that every ideal Einstein immersion satisfyingn≥n ₁+…+n _k +1 is totally geodesic, and that every ideal conformally flat immersion satisfyingn≥n ₁+…+n _k +2 andk≥2 is also totally geodesic. Secondly we completely classify all ideal semi-symmetric hypersurfaces in real space forms. The author was supported by the NSFC and RFDP.     

Asymptotic results on infinite tandem queueing networks

We consider an infinite tandem queueing network consisting of ·/GI/1/∞ stations with i.i.d. service times. We investigate the asymptotic behavior of t(n, k), the inter-arrival times between customers n and n + 1 at station k, and that of w(n, k), the waiting time of customer n at station k. We establish a duality property by which w(n, k) and the “idle times”y(n, k) play symmetrical roles. This duality structure, interesting by itself, is also instrumental in proving some of the ergodic results. We consider two versions of the model: the quadrant and the half-plane. In the quadrant version, the sequences of boundary conditions {w(0,k), k∈ℕ} and {t(n, 0), n∈ℕ}, are given. In the half-plane version, the sequence {t(n, 0), n∈ℕ} is given. Under appropriate assumptions on the boundary conditions and on the services, we obtain ergodic results for both versions of the model. For the quadrant version, we prove the existence of temporally ergodic evolutions and of spatially ergodic ones. Furthermore, the process {t(n, k), n∈ℕ} converges weakly with k to a limiting distribution, which is invariant for the queueing operator. In the more difficult half plane problem, the aim is to obtain evolutions which are both temporally and spatially ergodic. We prove that 1/n∑ _k=1 ⁿ w(0, k) converges almost surely and in L ₁ to a finite constant. This constitutes a first step in trying to prove that {t(n,k), n∈ℤ} converges weakly with k to an invariant limiting distribution. Received: 23 March 1999 / Revised version: 5 January 2000 / Published online: 12 October 2000     

Almost Symmetric Cycles in Large Digraphs

We prove that every digraph D with n≥7, n≥+6 vertices and at least (n−k−1)(n−1)+k(k+1) arcs contains all symmetric cycles of length at most n−k−2, an almost symmetric cycle of length n−k−1, and with some exceptions, also an almost symmetric cycle of length n−k. Consequently, D contains all orientations of cycles of length at most n−k, unless D is an exception. The research was partially supported by the AGH University of Science and Technology grant No 11 420 04     

Distributions of Points in <Emphasis Type="Italic">d</Emphasis> Dimensions and Large <Emphasis Type="Italic">k</Emphasis>-Point Simplices

We consider a variant of Heilbronn’s triangle problem by investigating for a fixed dimension d≥2 and for integers k≥2 with k≤d distributions of n points in the d-dimensional unit cube [0,1] ^d , such that the minimum volume of the simplices, which are determined by (k+1) of these n points is as large as possible. Denoting by Δ _k,d (n), the supremum of this minimum volume over all distributions of n points in [0,1] ^d , we show that c _k,d ⋅(log n)^1/(d−k+1)/n ^k/(d−k+1)≤Δ _k,d (n)≤c _k,d ′/n ^k/d for fixed 2≤k≤d, and, moreover, for odd integers k≥1, we show the upper bound Δ _k,d (n)≤c _k,d ″/n ^{k/d+(k−1)/(2d(d−1))}, where c _k,d ,c _k,d ′,c _k,d ″>0 are constants. A preliminary version of this paper appeared in COCOON ’05.     

An additive divisor problem with a growing number of factors

Let τk(n) be the number of representations ofn as the product ofk positive factors, τ(n)=τ(n). The asymptotics of Σ _n≤x τ _k (n)τ(n+1) for 80k ¹⁰ (lnlnx)³≤lnx is shown to be uniform with respect tok. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 391–406, March, 1997. Translated by N. K. Kulman     

Eigenvalues and suspension structure of compact Riemannian orbifolds with positive Ricci curvature

Let M be a compact n-dimensional Riemannian orbifold of Ricci curvature ≥n−1. We prove that for 1 ≤k≤n, the k ^th nonzero eigenvalue of the Laplacian on M is equal to the dimension n if and only if M is isometric to the k-times spherical suspension over the quotient S ^{n − k} }Γ of the unit (n−k)-sphere by a finite group Γ⊂O(n−k+1) acting isometrically on S ^{n − k} ⊂ℝ ^{n − k +}. Received: 21 September 1998 / Revised version: 23 February 1999     

Sum Complexes—a New Family of Hypertrees

A k-dimensional hypertree X is a k-dimensional complex on n vertices with a full (k−1)-dimensional skeleton and \binomn-1k\binom{n-1}{k} facets such that H _k (X;ℚ)=0. Here we introduce the following family of simplicial complexes. Let n,k be integers with k+1 and n relatively prime, and let A be a (k+1)-element subset of the cyclic group ℤ _n . The sum complex X _A is the pure k-dimensional complex on the vertex set ℤ _n whose facets are σ⊂ℤ _n such that |σ|=k+1 and ∑ _x∈σ x∈A. It is shown that if n is prime, then the complex X _A is a k-hypertree for every choice of A. On the other hand, for n prime, X _A is k-collapsible iff A is an arithmetic progression in ℤ _n .     

On the number of affine equivalence classes of spherical tube hypersurfaces

We consider Levi non-degenerate tube hypersurfaces in \mathbbCⁿ⁺¹{\mathbb{C}^{n+1}} that are (k, n − k)-spherical, i.e. locally CR-equivalent to the hyperquadric with Levi form of signature (k, n − k), with n ≤ 2k. We show that the number of affine equivalence classes of such hypersurfaces is infinite (in fact, uncountable) in the following cases: (i) k = n − 2, n ≥ 7; (ii) k = n − 3, n ≥ 7; (iii) k ≤ n − 4. For all other values of k and n, except for k = 3, n = 6, the number of affine classes was known to be finite. The exceptional case k = 3, n = 6 has been recently resolved by Fels and Kaup who gave an example of a family of (3, 3)-spherical tube hypersurfaces that contains uncountably many pairwise affinely non-equivalent elements. In this paper we deal with the Fels–Kaup example by different methods. We give a direct proof of the sphericity of the hypersurfaces in the Fels–Kaup family, and use the j-invariant to show that this family indeed contains an uncountable subfamily of pairwise affinely non-equivalent hypersurfaces.     

Orders on Multisets and Discrete Cones

We study additive representability of orders on multisets (of size k drawn from a set of size n) which satisfy the condition of independence of equal submultisets (IES) introduced by Sertel and Slinko (Ranking committees, words or multisets. Nota di Laboro 50.2002. Center of Operation Research and Economics. The Fundazione Eni Enrico Mattei, Milan, 2002, Econ. Theory 30(2):265–287, 2007). Here we take a geometric view of those orders, and relate them to certain combinatorial objects which we call discrete cones. Following Fishburn (J. Math. Psychol., 40:64–77, 1996) and Conder and Slinko (J. Math. Psychol., 48(6):425–431, 2004), we define functions f(n,k) and g(n,k) which measure the maximal possible deviation of an arbitrary order satisfying the IES and an arbitrary almost representable order satisfying the IES, respectively, from a representable order. We prove that g(n,k) = n − 1 whenever n ≥ 3 and (n, k) ≠ (5, 2). In the exceptional case, g(5,2) = 3. We also prove that g(n,k) ≤ f(n,k) ≤ n and establish that for small n and k the functions g(n,k) and f(n,k) coincide.      

The structure of digraphs associated with the congruence <Emphasis Type="Italic">x</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript> ≡ <Emphasis Type="Italic">y</Emphasis> (mod <Emphasis Type="Italic">n</Emphasis>)

We assign to each pair of positive integers n and k ⩾ 2 a digraph G(n, k) whose set of vertices is H = {0, 1, ..., n − 1} and for which there is a directed edge from a ∈ H to b ∈ H if a ^k ≡ b (mod n). We investigate the structure of G(n, k). In particular, upper bounds are given for the longest cycle in G(n, k). We find subdigraphs of G(n, k), called fundamental constituents of G(n, k), for which all trees attached to cycle vertices are isomorphic.     

On K 1,k -factorization of bipartite multigraphs

A K1,k-factorization of λKm,n is a set of edge-disjoint K1,k-factors of λKm,n, which partition the set of edges of λKm,n. In this paper, it is proved that a sufficient condition for the existence of K1,k-factorization of λKm,n, whenever k is any positive integer, is that （1） m ≤ kn, （2） n ≤ km, （3） km-n = kn-m ≡ 0 （mod （k^2- 1）） and （4） λ（km-n）（kn-m） ≡ 0 （mod k（k- 1）（k^2 - 1）（m ＋ n））.     

Estimates on binomial sums of partition functions

Let p(n) denote the partition function and define where p(0)= 1. We prove that p(n,k) is unimodal and satisfies for fixed n≥ 1 and all 1≤k≤n. This result has an interesting application: the minimal dimension of a faithful module for a k-step nilpotent Lie algebra of dimension n is bounded by p(n,k) and hence by , independently of k. So far only the bound n ^{n −1} was known. We will also prove that for n≥ 1 and . Received: 17 December 1999     

Hypergraphs with independent neighborhoods

For each k ≥ 2, let ρ _k ∈ (0, 1) be the largest number such that there exist k-uniform hypergraphs on n vertices with independent neighborhoods and (ρ _k + o(1))( _k ⁿ ) edges as n → ∞. We prove that ρ _k = 1 − 2logk/k + Θ(log log k/k) as k → ∞. This disproves a conjecture of Füredi and the last two authors.     

Factoriality of complete intersections in ℙ<Superscript>5</Superscript>

Let X be a complete intersection of two hypersurfaces F _n and F _k in ℙ⁵ of degree n and k, respectively, with n ≥ k, such that the singularities of X are nodal and F _k is smooth. We prove that if the threefold X has at most (n + k − 2)(n − 1) − 1 singular points, then it is factorial.     

Edge Maximal Non-Bipartite Graphs Without Odd Cycles of Prescribed Lengths

 Let ?(n;3,5,…,2k+1) denote the class of non-bipartite graphs on n vertices having no odd cycle of length ≤2k+1. We prove that for every G∈?(n;3,5,…,2k+1) and characterize the extremal graphs. We also study the subclass ℋ(n;3,5,…,2k+1) consisting of the hamiltonian members of ?(n;3,5,…, 2k+1). For this subclass the above upper bound holds for odd n. For even n we establish the following sharp upper bound:

A combinatorial identity arising from symplectic geometry

In this note, we give a generalization of the famous combinational identity （-1）^nn! = Σk=1^n （nk）（-1）^kk^n arising from symplectic geometry.     

Resolvable Coverings of 2-Paths by Cycles

 Let K _n be the complete graph on n vertices. A C(n,k,λ) design is a multiset of k-cycles in K _n in which each 2-path (path of length 2) of K _n occurs exactly λ times. A C(lk,k,1) design is resolvable if its k-cycles can be partitioned into classes so that every vertex appears exactly once in each class. A C(n,n,1) design gives a solution of Dudeney's round table problem. It is known that there exists a C(n,n,1) design when n is even and there exists a C(n,n,2) design when n is odd. In general the problem of constructing a C(n,n,1) design is still open when n is odd. Necessary and sufficient conditions for the existence of C(n,k,λ) designs and resolvable C(lk,k,1) designs are known when k=3,4. In this paper, we construct a resolvable C(n,k,1) design when n=p ^e +1 ( p is a prime number and e≥1) and k is any divisor of n with k≠1,2. Received: October, 2001 Final version received: September 4, 2002 RID="*" ID="*" This research was supported in part by Grant-in-Aid for Scientific Research (C) Japan     

Diagnostic tests for local coalescences of variables in Boolean functions

The article studies diagnostic tests for local k -fold coalescences of variables in Boolean functions f( [(x)\tilde]ⁿ )( 1 ￡ k ￡ n,  1 ￡ t ￡ 2^2k ) f\left( {{{\tilde{x}}^n}} \right)\left( {1 \leq k \leq n,\;1 \leq t \leq {2^{{2^k}}}} \right) . Upper and lower bounds are proved for the Shannon function of the length of the diagnostic test for local k -fold coalescences generated by the system of functions F_t^k \Phi_t^k . The Shannon function of the length of a complete diagnostic test for local k -fold coalescences behaves asymptotically as 2 ^k (n − k + 1) for n → ∞, k → ∞.     

The largest projections of regular polytopes

The projections of the regularn-dimensional simplex and crosspolytope intoR ^k with the largestk-volume are determined here for the casesk=2,n≧2 andk=3, 4≦n≦6. The proofs involve a combination of exterior algebra and computer gradient methods.     

Super-Magic Complete k-Partite Hypergraphs

 We deal with complete k-partite hypergraphs and we show that for all k≥2 and n≠2,6 its hyperedges can be labeled by consecutive integers 1,2,…,n ^k such that the sum of labels of the hyperedges incident to (k−1) particular vertices is the same for all (k−1)-tuples of vertices from (k−1) independent sets. Received: December 8, 1997 Final version received: July 26, 1999