Universal Upper Bound for the Growth of Artin Monoids

In this paper we study the growth rates of Artin monoids, and we show that 4 is a universal upper bound. We also show that the generating functions of the associated right-angled Artin monoids are given by families of Chebyshev polynomials. Applications to Artin groups and positive braids are given.     

EXCEPTIONAL SEQUENCES FOR QUIVERS OF DYNKIN TYPE

Let kQ be the path algebra of a quiver Q without oriented cycles with n vertices. An indecomposable kQ-module without self-extensions is called exceptional. The braid group B _n with n ? 1 generators acts naturally on the set of complete exceptional sequences. Crawley-Boevey (Proceedings of ICRA VI, Carleton-Ottawa, 1992) and Ringel (Contemp. Math. 1994, 171, 339–352) have pointed out that this action is transitive. The number of complete exceptional sequences for kQ representation finite will be computed here and it is shown to be independent of the orientation of the arrows of the quiver Q. The factor group of the braid group which acts freely on the set of complete exceptional sequences can be regarded as a subgroup of the symmetric group S _{? n} , where ? _n is the number of complete exceptional sequences of the algebra kQ. This group is known for certain special types of quivers. Some other interesting relations of the acting group will be given.     

Retarded regular graphs are regular or semiregular

A graph G is said to be retarded regular if there is a positive integral number s such that the number of walks of length s starting at vertices of G is a constant function. Regular and semiregular graphs are retarded regular with s?=?1 and s\!≤ \!2, respectively. We prove that any retarded regular connected graph is either regular or semiregular.     

Sparse random graphs: Eigenvalues and eigenvectors

In this paper, we prove the semi‐circular law for the eigenvalues of regular random graph G_n,d in the case d →∞, complementing a previous result of McKay for fixed d. We also obtain a upper bound on the infinity norm of eigenvectors of Erd?s–Rényi random graph G(n,p), answering a question raised by Dekel–Lee–Linial. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

The number of graphs and a random graph with a given degree sequence

We consider the set of all graphs on n labeled vertices with prescribed degrees D = (d₁,…,d_n). For a wide class of tame degree sequences D we obtain a computationally efficient asymptotic formula approximating the number of graphs within a relative error which approaches 0 as n grows. As a corollary, we prove that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D. We also establish an asymptotic formula for the number of bipartite graphs with prescribed degrees of vertices, or, equivalently, for the number of 0‐1 matrices with prescribed row and column sums. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Critical behavior in inhomogeneous random graphs

We study the critical behavior of inhomogeneous random graphs in the so‐called rank‐1 case, where edges are present independently but with unequal edge occupation probabilities. The edge occupation probabilities are moderated by vertex weights, and are such that the degree of vertex i is close in distribution to a Poisson random variable with parameter w_i, where w_i denotes the weight of vertex i. We choose the weights such that the weight of a uniformly chosen vertex converges in distribution to a limiting random variable W. In this case, the proportion of vertices with degree k is close to the probability that a Poisson random variable with random parameter W takes the value k. We pay special attention to the power‐law case, i.e., the case where \begin{align*}{\mathbb{P}}(W\geq k)\end{align*} is proportional to k^‐(τ‐1) for some power‐law exponent τ > 3, a property which is then inherited by the asymptotic degree distribution. We show that the critical behavior depends sensitively on the properties of the asymptotic degree distribution moderated by the asymptotic weight distribution W. Indeed, when \begin{align*}{\mathbb{P}}(W > k) \leq ck^{-(\tau-1)}\end{align*} for all k ≥ 1 and some τ > 4 and c > 0, the largest critical connected component in a graph of size n is of order n^2/3, as it is for the critical Erd?s‐Rényi random graph. When, instead, \begin{align*}{\mathbb{P}}(W > k)=ck^{-(\tau-1)}(1+o(1))\end{align*} for k large and some τ∈(3,4) and c > 0, the largest critical connected component is of the much smaller order n^{(τ‐2)/(τ‐1)}. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 480–508, 2013     

A Reduced Order Model for Control of a Single Reactor Activated Sludge Process

The IAWQ activated sludge model no 1, slightly modified by adding a nitrogen limiting growth function for heterotrophic microorganisms, has been validated on a sequencing single sludge wastewater reactor doing nitrification and denitrification. However, this model is too complex to be used for control design. The main purpose of this paper is to show how it is possible to simplify the full model from biological and mathematical considerations. Properties of observability, controllability and theoretical identifiability have been analyzed. The parameter identification has been carried out by using the simplex method of Nelder & Mead. Simulation results performed over a range of six hours (two aerobic / anoxic cycles), show that there exists a good fit between the simulated solution and the actual behavior of a lab scale pilot plant.