Related papers: Average Length of Cycles in Rectangular Lattice
Cyclic lattices and ideal lattices were introduced by Micciancio in \cite{D2}, Lyubashevsky and Micciancio in \cite{L1} respectively, which play an efficient role in Ajtai's construction of a collision resistant Hash function (see \cite{M1}…
We consider the generating function of the algebraic area of lattice walks, evaluated at a root of unity, and its relation to the Hofstadter model. In particular, we obtain an expression for the generating function of the n-th moments of…
In the past 20 years, the enumeration of plane lattice walks confined to a convex cone -- normalized into the first quadrant -- has received a lot of attention, stimulated the development of several original approaches, and led to a rich…
We introduce in this paper two dimensional lattice models whose continuum limit belongs to the $N=2$ series. The first kind of model is integrable and obtained through a geometrical reformulation, generalizing results known in the $k=1$…
Enzyme kinetics are cyclic. A more realistic reversible three-step mechanism of the Michaelis-Menten kinetics is investigated in detail, and three kinds of waiting cycle times $T$, $T_{+}$, $T_{-}$ are defined. It is shown that the mean…
Excitations of three-dimensional spin glasses are computed numerically. We find that one can flip a finite fraction of an LxLxL lattice with an O(1) energy cost, confirming the mean field picture of a non-trivial spin overlap distribution…
A $3$-uniform loose cycle, denoted by $C_t$, is a $3$-graph on $t$ vertices whose vertices can be arranged cyclically so that each hyperedge consists of three consecutive vertices, and any two consecutive hyperedges share exactly one…
The Collatz problem is generalized into $3n + 3^k$ problem. It is shown that as long as the Collatz function iterates converge to the cycle passing through the number 1, the $3n + 3^k$ sequence converges to the cycle passing through the…
We investigate the typical cycle lengths, the total number of cycles, and the number of finite cycles in random permutations whose probability involves cycle weights. Typical cycle lengths and total number of cycles depend strongly on the…
A mutation cycle is a cycle in a graph whose vertices are labeled by the quivers in a given mutation class and whose edges correspond to single mutations. For any fixed $n\ge 4$, we describe arbitrarily long mutation cycles involving…
A Schr\"oder path is a lattice path from $(0,0)$ to $(2n,0)$ with steps $(1,1)$, $(1,-1)$ and $(2,0)$ that never goes below the $x-$axis. A small Schr\"{o}der path is a Schr\"{o}der path with no $(2,0)$ steps on the $x-$axis. In this paper,…
In a graph, $k$ cycles are {\em admissible} if their lengths form an arithmetic progression with common difference one or two. Let $G$ be a 2-connected graph with minimum degree at least $k\geqslant 4$. We prove that \begin{itemize} \item…
We prove the following theorem. Let $r\ge 4$ be an integer, and $G$ be a $K_{1,r}$-free $r$-edge-connected $r$-regular graph. Then, for every set $W$ of even number of vertices of $G$ such that the distance between any two vertices of $W$…
Trying to enumerate all of the walks in a 2D lattice is a fun combinatorial problem and there are numerous applications, from polymers to sports. Computers provide a wonderful tool for analyzing these walks; we provide a Maple package for…
Random sequential adsorption of binary mixtures of extended objects on a two-dimensional triangular lattice is studied numerically by means of Monte Carlo simulations. The depositing objects are formed by self-avoiding random walks on the…
We investigate a functional equation which resembles the functional equation for the generating function of a lattice walk model for the quarter plane. The interesting feature of this equation is that its orbit sum is zero while its…
A coding lattice $\Lambda_c$ and a shaping lattice $\Lambda_s$ forms a nested lattice code $\mathcal{C}$ if $\Lambda_s \subseteq \Lambda_c$. Under some conditions, $\mathcal{C}$ is a finite cyclic group formed by rectangular encoding. This…
There is a well-known asymptotic formula, due to W. M. Schmidt (1968) for the number of full-rank integer lattices of index at most $V$ in $\mathbb{Z}^n$. This set of lattices $L$ can naturally be partitioned with respect to the factor…
The $I=2$ $\pi\pi$ elastic $s$-wave scattering phase shift is measured by lattice QCD with $N_f=3$ flavors of the Asqtad-improved staggered fermions. The lattice-calculated energy-eigenvalues of $\pi\pi$ systems at one center of mass frame…
The quantum diffusion of a particle in an initially localized state on a cyclic lattice with N sites is studied. Diffusion and reconstruction time are calculated. Strong differences are found for even or odd number of sites and the limit…