This is a survey of recent advances on zero-sum problems and Snevily’s conjecture concerning finite abelian groups. In particular, we will introduce Reiher’s recent solution to the Kemnitz conjecture and our simplification. 1. On Zero-sum Problems The theory of zero-sums began with the following celebrated theorem. The Erdős-Ginzburg-Ziv Theorem [Bull. Research Council. Israel, 1961]. For any c...

Various algebraic structures have recently appeared in a parallel way in the framework of Hilbert schemes of points on a surface and respectively in the framework of equivariant K-theory [N1, Gr, S2, W], but direct connections are yet to be clarified to explain such a coincidence. We provide several non-trivial steps toward establishing our main conjecture on the isomorphism between the Hilbert...

A permutation graph is an intersection graph of segments lying between two parallel lines. A Seidel complementation of a finite graph at a vertex v consists in complementing the edges between the neighborhood and the non-neighborhood of v. Two graphs are Seidel complement equivalent if one can be obtained from the other by a sequence of Seidel complementations. In this paper we introduce the ne...

The chromatic polynomial P (G,λ) gives the number of λ-colourings of a graph. If P (G,λ) = P (H1, λ)P (H2, λ)/P (Kr , λ), then the graph G is said to have a chromatic factorisation with chromatic factors H1 and H2. It is known that the chromatic polynomial of any clique-separable graph has a chromatic factorisation. In this paper we construct an infinite family of graphs that have chromatic fac...

This paper studies the geometric decay property of the joint queue-length distribution {p(n1, n2)} of a two-node Markovian queueing system in the steady state. For arbitrarily given positive integers c1, c2, d1 and d2, an upper bound η(c1, c2) of the decay rate is derived in the sense exp © lim supn→∞ n −1 log p(c1n+ d1, c2n+ d2) a ≤ η(c1, c2) < 1. It is shown that the upper bound coincides wit...

Since the first proposal that fullerenes are capable of hosting atoms, ions, or clusters by the late Smalley in 1985, tremendous examples of endohedral metallofullerenes (EMFs) have been reported. Breaking the dogma that monometallofullerenes (mono-EMFs) always exist in the form of M@C2n while clusterfullerenes always require multiple (two to four) metal cations to stabilize a cluster that is u...

X iv :m at h/ 95 12 22 4v 1 [ m at h. C O ] 1 9 D ec 1 99 5 An Explicit Formula for the Number of Solutions of X = 0 in Triangular Matrices Over a Finite Field Shalosh B. EKHAD and Doron ZEILBERGER Abstract: We prove an explicit formula for the number of n × n upper triangular matrices, over GF (q), whose square is the zero matrix. This formula was recently conjectured by Sasha Kirillov and Ann...

In this paper, we show how to use stochastic approximation to compute hitting time of a stochastic process, based on the study of the time for a fluid approximation of this process to be at distance 1/N of its fixed point. This approach is developed to study a generalized version of the coupon collector problem. The system is composed by N independent identical Markov chains. At each time step,...

A. Nachawaty,1,2 M. Yang,3 W. Desrat,1 S. Nanot,1 B. Jabakhanji,4 D. Kazazis,5,6 R. Yakimova,7 A. Cresti,8 W. Escoffier,3 and B. Jouault1 1Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier, F-Montpellier, France 2Laboratoire de Physique et Modélisation (LPM), EDST, Lebanese University, Tripoli, Lebanon 3Laboratoire National des Champs Magnétiques Intenses (LNCMI-EMFL), ...