Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in De Sitter space

In this paper we give a partially affirmative answer to the following question posed by Haizhong Li: is a complete spacelike hypersurface in De Sitter space , n?3, with constant normalized scalar curvature R satisfying totally umbilical?     

Arithmetical properties of wendt's determinant

Wendt's determinant of order n is the circulant determinant W_n whose (i,j)-th entry is the binomial coefficient , for 1?i,j?n, where n is a positive integer. We establish some congruence relations satisfied by these rational integers. Thus, if p is a prime number and k a positive integer, then and . If q is another prime, distinct from p, and h any positive integer, then . Furthermore, if p is odd, then . In particular, if p?5, then . Also, if m and n are relatively prime positive integers, then W_mW_n divides W_mn.     

Composition operators on the polydisc induced by smooth symbols

We study composition operators CΦ on the Hardy spaces Hp and weighted Bergman spaces of the polydisc Dn in Cn. When Φ is of class C² on , we show that CΦ is bounded on Hp or if and only if the Jacobian of Φ does not vanish on those points ζ on the distinguished boundary Tn such that Φ(ζ)∈Tn. Moreover, we show that if ε>0 and if , then CΦ is bounded on .     

Bound states and the Szeg? condition for Jacobi matrices and Schrödinger operators

For Jacobi matrices with a_n=1+(−1)ⁿαn^−γ, b_n=(−1)ⁿβn^−γ, we study bound states and the Szeg? condition. We provide a new proof of Nevai's result that if , the Szeg? condition holds, which works also if one replaces (−1)ⁿ by . We show that if α=0, β≠0, and , the Szeg? condition fails. We also show that if γ=1, α and β are small enough ( will do), then the Jacobi matrix has finitely many bound states (for α=0, β large, it has infinitely many).     

On the b-chromatic number of Kneser graphs

In this note, we prove that for any integer n≥3 the b-chromatic number of the Kneser graph KG(m,n) is greater than or equal to . This gives an affirmative answer to a conjecture of [6].     

Spectral radius of graphs with given matching number

In this paper, we show that of all graphs of order n with matching number β, the graphs with maximal spectral radius are K_n if n = 2β or 2β + 1; if 2β + 2 ? n < 3β + 2; or if n = 3β + 2; if n > 3β + 2, where is the empty graph on t vertices.     

Covering numbers of different points in Dvoretzky covering

Consider the Dvoretzky random covering on the circle T with a decreasing length sequence {?n}_n?1 such that . We study, for a given β?0, the set Fβ of points which are asymptotically covered by a number βLn of the first n randomly placed intervals where . Three typical situations arise, delimited by two “phase transitions”, according to is zero, positive-finite or infinite, where . More precisely, if ?n tends to zero rapidly enough so that then, with probability one, dimHFβ=1 for all β?0; if ?n is moderate so that then, with probability one, we have for and Fβ=∅ for where and is the interval consisting of β's such that ; eventually, if ?n is so slow that then, with probability one, F₁=T. This solves a problem raised by L. Carleson in a rather satisfactory fashion.Analogous results are obtained for the Poisson covering of the line, which is studied as a tool.     

Haj?asz-Sobolev imbedding and extension

The author establishes some geometric criteria for a Haj?asz-Sobolev -extension (resp. -imbedding) domain of Rn with n?2, s∈(0,1] and p∈[n/s,∞] (resp. p∈(n/s,∞]). In particular, the author proves that a bounded finitely connected planar domain Ω is a weak α-cigar domain with α∈(0,1) if and only if for some/all s∈[α,1) and p=(2−α)/(s−α), where denotes the restriction of the Triebel-Lizorkin space on Ω.     

Large monochromatic components in colorings of complete 3-uniform hypergraphs

Let f(n,r) be the largest integer m with the following property: if the edges of the complete 3-uniform hypergraph are colored with r colors then there is a monochromatic component with at least m vertices. Here we show that and . Both results are sharp under suitable divisibility conditions (namely if n is divisible by 7, or by 6 respectively).     

On the log-Sobolev constant for the simple random walk on the n-cycle: the even cases

Consider the simple random walk on the n-cycle . For this example, Diaconis and Saloff-Coste (Ann. Appl. Probab. 6 (1996) 695) have shown that the log-Sobolev constant α is of the same order as the spectral gap λ. However the exact value of α is not known for n>4. (For n=2, it is a well known result of Gross (Amer. J. Math. 97 (1975) 1061) that α is . For n=3, Diaconis and Saloff-Coste (Ann. Appl. Probab. 6 (1996) 695) showed that . For n=4, the fact that follows from n=2 by tensorization.) Based on an idea that goes back to Rothaus (J. Funct. Anal. 39 (1980) 42; 42 (1981) 110), we prove that if n?4 is even, then the log-Sobolev constant and the spectral gap satisfy . This implies that when n is even and n?4.     

Further results on the least eigenvalue of connected graphs

In this paper, we identify within connected graphs of order n and size n+k (with and ) the graphs whose least eigenvalue is minimal. It is also observed that the same graphs have the largest spectral spread if n is large enough.     

Packing 1×2×4 bricks in a cubical box

Given a cubical box C_2n+1 of side 2n+1 and a supply of 1×2×4 bricks, it is proved that if n≥2, then

(A1)

one can pack bricks for n odd, and bricks for n even,

(A2)

the capacity of C_2n+1 is , and if n≡1 or 2 (mod4), this upper bound for the capacity can be reduced by 1.

     

Directed cyclic Hamiltonian cycle systems of the complete symmetric digraph

In this paper, we prove that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph, , exist if and only if n is odd with n≠15 and n≠p^α for p an odd prime and α≥2 or with n≠2p^α for p an odd prime and α≥1. We also show that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph minus a set of n/2 vertex-independent digons, (K_n−I)^∗, exist if and only if .     

On complete subsets of the cyclic group

A subset X of an abelian G is said to be complete if every element of G can be expressed as a nonempty sum of distinct elements from X.Let A⊂Zn be such that all the elements of A are coprime with n. Solving a conjecture of Erd?s and Heilbronn, Olson proved that A is complete if n is a prime and if . Recently Vu proved that there is an absolute constant c, such that for an arbitrary large n, A is complete if , and conjectured that 2 is essentially the right value of c.We show that A is complete if , thus proving the last conjecture.     

Reflexivity and hyperreflexivity of operator spaces

In this paper, we show that if is an n-dimensional subspace of L(V) such that every nonzero transformation of has rank greater than or equal to 2n−1 then is algebraically reflexive. If is an n-dimensional subspace of B(H) such that every nonzero transformation of  has rank greater than or equal to 2n−1 then is hyperreflexive. We also consider how to construct some new hyperreflexive subspaces.     

The Dirichlet Markov Ensemble

We equip the polytope of n×n Markov matrices with the normalized trace of the Lebesgue measure of Rⁿ². This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean (1/n,…,1/n). We show that if is such a random matrix, then the empirical distribution built from the singular values of tends as n→∞ to a Wigner quarter-circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of tends as n→∞ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of is of order when n is large.     

Steiner almost self-complementary graphs and halving near-Steiner triple systems

We show that for every admissible order v≡0 or there exists a near-Steiner triple system of order v that can be halved. As a corollary we obtain that a Steiner almost self-complementary graph with n vertices exists if and only if n≡0 or .     

Markov interlacing property for exponential polynomials

Let Un be an extended Tchebycheff system on the real line. Given a point , where x₁<?<xn, we denote by the polynomial from Un, which has zeros x₁,…,xn. (It is uniquely determined up to multiplication by a constant.) The system Un has the Markov interlacing property (M) if the assumption that and interlace implies that the zeros of and interlace strictly, unless . We formulate a general condition which ensures the validity of the property (M) for polynomials from Un. We also prove that the condition is satisfied for some known systems, including exponential polynomials and . As a corollary we obtain that property (M) holds true for Müntz polynomials , too.