Let τ (H) be the cover number and ν(H) be the matching number of a hypergraph H. Ryser conjectured that every r-partite hypergraph H satisfies the inequality τ (H) ≤ (r − 1)ν(H). This conjecture is open for all r ≥ 4. For intersecting hypergraphs, namely those with ν(H) = 1, Ryser's conjecture reduces to τ (H) ≤ r − 1. Even this conjecture is extremely difficult and is open for all r ≥ 6. For infinitely many r there are examples of intersecting r-partite hypergraphs with τ (H) = r − 1, demonstrating the tightness of the conjecture for such r. However, all previously known constructions are not optimal as they use far too many edges. How sparse can an intersecting r-partite hypergraph be, given that its cover number is as large as possible, namely τ (H) ≥ r − 1? In this paper we solve this question for r ≤ 5, give an almost optimal construction for r = 6, prove that any r-partite intersecting hypergraph with τ (H) ≥ r − 1 must have at least (3− 1 √ 18 )r(1−o(1)) ≈ 2.764r(1−o(1)) edges, and conjecture that there exist constructions with Θ(r) edges.

Abstract. In this paper we give explicitly a family of probability densities, the moments of which are Fuss-Catalan numbers. The densities appear naturally in random matrices, free probability and other contexts.In this paper we study a family of probability densities π s , s ∈ N, which are uniquely determined by the moment sequencesare known as Fuss-Catalan numbers in free probability theory [7]. The densities π s belong to the class of free Bessel laws [3] and are known to appear in several different contexts, for instance, random matrices [1,3,7], random quantum states [4], free probability and quantum groups [3,6]. More precisely, π s is the limit spectral distribution of random matrices in the forms such as X Indeed, the following relation holds [3]:The distribution π 1,t , the famous Marchenko-Pastur law of parameter t, also called free Poisson law, has an explicit formula: (0.4) π 1,t = max(1 − t, 0)δ 0 + 4t − (x − 1 − t) 2 2πx .

Birkhoff equations on time scales and Noether theorem for Birkhoffian system on time scales are studied. First, some necessary knowledge of calculus on time scales are reviewed. Second, Birkhoff equations on time scales are obtained. Third, the conditions for invariance of Pfaff action and conserved quantities are presented under the special infinitesimal transformations and general infinitesimal transformations, respectively. Fourth, some special cases are given. And finally, an example is given to illustrate the method and results.

The first and second Zagreb indices were first introduced by Gutman and Trinajstić 1972 . It is reported that these indices are useful in the study of anti-inflammatory activities of certain chemical instances, and in elsewhere. Recently, the first and second Zagreb coindices, a new pair of invariants, were introduced in Došlić 2008 . In this paper we introduce the a and a, b -analogs of the above Zagreb indices and coindices and investigate the relationship between the enhanced versions to get a unified theory.

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While the standard Catalan and Schröder theories both have been extensively studied, people have only begun to investigate higher dimensional versions of the Catalan number (see, say, the 1991 paper of Hilton and Pedersen, and the 1996 paper of Garsia and Haiman). In this paper, we study a yet more general case, the higher dimensional Schröder theory. We define $m$-Schröder paths, find the number of such paths from $(0,0)$ to $(mn, n)$, and obtain some other results on the $m$-Schröder paths and $m$-Schröder words. Hoping to generalize classical $q$-analogue results of the ordinary Catalan and Schröder numbers, such as in the works of Fürlinger and Hofbauer, Cigler, and Bonin, Shapiro and Simion, we derive a $q$-identity which would welcome a combinatorial interpretation. Finally, we introduce the ($q, t$)-$m$-Schröder polynomial through "$m$-parking functions" and relate it to the $m$-Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel and Ulyanov.

Noether theorem is an important aspect to study in dynamical systems. Noether symmetry and conserved quantity for the fractional Birkhoffian system are investigated. Firstly, fractional Pfaff actions and fractional Birkhoff equations in terms of combined Riemann-Liouville derivative, Riesz-Riemann-Liouville derivative, combined Caputo derivative and Riesz-Caputo derivative are reviewed. Secondly, the criteria of Noether symmetry within combined Riemann-Liouville derivative, Riesz-Riemann-Liouville derivative, combined Caputo derivative and Riesz-Caputo derivative are presented for the fractional Birkhoffian system, respectively. Thirdly, four corresponding conserved quantities are obtained. The classical Noether identity and conserved quantity are special cases of this paper. Finally, four fractional models, such as the fractional Whittaker model, the fractional Lotka biochemical oscillator model, the fractional Hénon-Heiles model and the fractional Hojman-Urrutia model are discussed as examples to illustrate the results.

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