We give a combinatorial interpretation for the positive moments of the values at the edge of the critical strip of the $L$-functions of modular forms of $GL\left(2\right)$ and $GL\left(3\right)$. We deduce some results about the asymptotics of these moments. We extend this interpretation to the moments twisted by the eigenvalues of Hecke operators.

In this paper we determine, or give lower and upper bounds on, the 2-dipath and oriented L(2, 1)-span of the family of planar graphs, planar graphs with girth 5, 11, 16, partial k-trees, outerplanar graphs and cacti.

Floor diagrams are a class of weighted oriented graphs introduced by E. Brugallé and the second author. Tropical geometry arguments lead to combinatorial descriptions of (ordinary and relative) Gromov–Witten invariants of projective spaces in terms of floor diagrams and their generalizations. In a number of cases, these descriptions can be used to obtain explicit (direct or recursive) formulas for the corresponding enumerative invariants. In particular, we use this approach to enumerate rational...

This paper gives a shortest path algorithm for CFG (context free grammar) labeled and weighted digraphs where edge weights may be positive or negative, but negative-weight cycles are not allowed in the underlying unlabeled graph. These results build directly on an algorithm of Barrett et al. [SIAM J. Comput.30 (2000) 809–837]. In addition to many other results, they gave a shortest path algorithm for CFG labeled and weighted digraphs where all edges are nonnegative. Our algorithm is based closely...

A graph G of size q is graceful if there exists an injective function f:V(G)→ 0,1,...,q such that each edge uv of G is labeled |f(u)-f(v)| and the resulting edge labels are distinct. Also, a (p,q) graph G with q ≥ p is harmonious if there exists an injective function $f:V\left(G\right)\to Z_q$ such that each edge uv of G is labeled f(u) + f(v) mod q and the resulting edge labels are distinct, whereas G is felicitous if there exists an injective function $f:V\left(G\right)\to Z_{q+1}$ such that each edge uv of G is labeled f(u) + f(v) mod q and the...

The geometry of complex networks is closely related with their structure and function. In this paper, we investigate the Gromov-hyperbolicity of the Newman-Watts model of small-world networks. It is known that asymptotic Erdős-Rényi random graphs are not hyperbolic. We show that the Newman-Watts ones built on top of them by adding lattice-induced clustering are not hyperbolic as the network size goes to infinity. Numerical simulations are provided to illustrate the effects of various parameters...

The memoir presented by Lagrange, which this paper examines, is usually considered as an elegant, but scarcely practicable, contribution to numerical analysis. The purpose of this study is to show the significance of the novel mathematical ideas it contains, and in particular to look at this essay from the perspective of generating function theory, for which the theoretical foundations would be laid some little time later by Laplace. This excursus of Lagrange’s does indeed proffer an abundance of...

Let ${𝔖}_n$ denote the symmetric group with $n$ letters, and $g\left(n\right)$ the maximal order of an element of ${𝔖}_n$. If the standard factorization of $M$ into primes is $M=q_1^{{\alpha }_1}{q}_2^{{\alpha }_2}...q_k^{{\alpha }_k}$, we define $\ell \left(M\right)$ to be $q_1^{{\alpha }_1}+q_2^{{\alpha }_2}+...+q_k^{{\alpha }_k}$; one century ago, E. Landau proved that $g\left(n\right)={max}_{\ell \left(M\right)\le n}M$ and that, when $n$ goes to infinity, $logg\left(n\right)\sim \sqrt{nlog\left(n\right)}$.There exists a basic algorithm to compute $g\left(n\right)$ for $1\le n\le N$; its running time is $𝒪\left(N^{3/2}/\sqrt{logN}\right)$ and the needed memory is $𝒪\left(N\right)$; it allows computing $g\left(n\right)$ up to, say, one million. We describe an algorithm to calculate $g\left(n\right)$ for $n$ up to ${10}^{15}$. The main idea is to use the so-called $\ell$-superchampion...