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.     

On distance dependent entropy measures of poly propylene imine and zinc porphyrin dendrimers

A key paradigm in contemporary research is the use of graphs to represent physical systems, molecular structures, or particularly metal frameworks. Graphs are increasingly widely used in a variety of fields, including the study of quantum and molecular systems, macromolecules and their interactions, socioeconomic and ecological systems, and technical and infra-structural systems. Understanding how these systems function, are robust, and are stable begins with structural characterization. The use of entropies and entropy-like measurements of graphs/structures of molecules/networks is crucial from both a mathematical and physical standpoint. Several entropy measures of graphs have been defined and studied extensively during the last few decades. The current paper is devoted to investigation of distance dependent entropy measures of Poly Propylene Imine (PPI) dendrimers and Zinc Porphyrin dendrimers. The analytical formulae of distance dependent entropy measures have been developed and their patterns have been presented through graphical tools.     

Asymptotic normality of the size of the giant component via a random walk

In this paper we give a simple new proof of a result of Pittel and Wormald concerning the asymptotic value and (suitably rescaled) limiting distribution of the number of vertices in the giant component of G(n,p) above the scaling window of the phase transition. Nachmias and Peres used martingale arguments to study Karp?s exploration process, obtaining a simple proof of a weak form of this result. We use slightly different martingale arguments to obtain a much sharper result with little extra work.     

Tight products and graph expansion

In this article, we study a new product of graphs called tight product. A graph H is said to be a tight product of two (undirected multi) graphs G₁ and G₂, if V(H) = V(G₁) × V(G₂) and both projection maps V(H)→V(G₁) and V(H)→V(G₂) are covering maps. It is not a priori clear when two given graphs have a tight product (in fact, it is NP‐hard to decide). We investigate the conditions under which this is possible. This perspective yields a new characterization of class‐1 (2k+ 1)‐regular graphs. We also obtain a new model of random d‐regular graphs whose second eigenvalue is almost surely at most O(d^3/4). This construction resembles random graph lifts, but requires fewer random bits. © 2011 Wiley Periodicals, Inc. J Graph Theory     

完全图及完全二部图的点可区别Ⅳ-全染色

设G是简单图,图G的一个k-点可区别Ⅵ-全染色(简记为k-VDIVT染色),f是指一个从V(G)∪E(G)到{1,2,…,k}的映射,满足:()uv,uw∈E(G),v≠w,有,f(uv)≠f(uw);()u,V∈V(G),u≠v,有C(u)≠C(v),其中C(u)={f(u)}∪{f(uv)|uv∈E(G)}.数min{k|G有一个k-VDIVT染色}称为图G的点可区别Ⅵ-全色数,记为x_(vt)~(iv)(G).讨论了完全图K_n及完全二部图K_(m,n)的VDIVT色数.     

(向量)R_n和(向量)G_n的有向优美性

优美图可用在图论中的某些H-分解问题中,很多人研究无向图的优美标号.研究有向优美标号,通过对阶数奇偶性的讨论,给出了n(≥2)阶有向路(向量)P_n和n(≥3)阶有向(向量)C_n圈是有向优美的充分条件.     

K_(2s)×K_(2t)的邻点可区别全染色

基于完全图的邻点可区别全染色,得到了任意偶阶完全图的直积图K_(2s)×K_(2t)的邻点可区别全色数χ_(at)(K_(2s)×K_(2t)=2(s+t)(t、s均为正整数).     

Extremal subgraphs of random graphs

We prove that there is a constant c > 0, such that whenever p ≥ n^‐c, with probability tending to 1 when n goes to infinity, every maximum triangle‐free subgraph of the random graph G_n,p is bipartite. This answers a question of Babai, Simonovits and Spencer (Babai et al., J Graph Theory 14 (1990) 599–622). The proof is based on a tool of independent interest: we show, for instance, that the maximum cut of almost all graphs with M edges, where M ? n and M ≤ /2, is “nearly unique”. More precisely, given a maximum cut C of G_n,M, we can obtain all maximum cuts by moving at most \begin{align*}\mathcal{O}(\sqrt{n^3/M})\end{align*} vertices between the parts of C. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

A necessary and sufficient condition for the total unimodularity of a matrix in terms of graph theory

We present a necessary and sufficient condition for an arbitrary matrix A to be totally unimodular. The matrix A is interpreted as the adjacency matrix of a bipartite graph G(A) The total unimodularity of A corresponds to non-existence of a cycle in G(A) which has an odd column valuation and which is equal to the induced subgraph, Some applications of the results are also discussed.