A limited resource model of fault-tolerant capability against cascading failure of complex network

We propose a novel capacity model for complex networks against cascading failure. In this model, vertices with both higher loads and larger degrees should be paid more extra capacities, i.e. the allocation of extra capacity on vertex i will be proportional to k_i ^γ , where k_i is the degree of vertex i and γ > 0 is a free parameter. We have applied this model on Barabási-Albert network as well as two real transportation networks, and found that under the same amount of available resource, this model can achieve better network robustness than previous models.     

Abstract Commensurators of Groups Acting on Rooted Trees

We show that the abstract commensurator of a nearly level transitive weakly branch group H coincides with the relative commensurator of H in the homeomorphism group of the boundary of the tree on which H acts. It is also shown that the commensurator of an infinite group which is commensurable with its own nth direct power contains a Higman–Thompson group as a subgroup. Applying these results to the Grigorchuk 2-group G we show that the commensurator of G is a finitely presented infinite simple group.     

Maximally Symmetric Trees

We characterize the best model geometries for the class of virtually free groups, and we show that there is a countable infinity of distinct best model geometries in an appropriate sense – these are the maximally symmetric trees. The first theorem gives several equivalent conditions on a bounded valence, cocompact tree T without valence 1 vertices saying that T is maximally symmetric. The second theorem gives general constructions for maximally symmetric trees, showing for instance that every virtually free group has a maximally symmetric tree for a model geometry.     

A composition law for Runge-Kutta methods applied to index-2 differential-algebraic equations

This paper presents a new composition law for Runge-Kutta methods when applied to index-2 differential-algebraic systems. Applications of this result to the study of the order of composite methods and of symmetric methods are given.     

On the irregularity strength of trees

For any graph G, let n_i be the number of vertices of degree i, and . This is a general lower bound on the irregularity strength of graph G. All known facts suggest that for connected graphs, this is the actual irregularity strength up to an additive constant. In fact, this was conjectured to be the truth for regular graphs and for trees. Here we find an infinite sequence of trees with λ(T) = n₁ but strength converging to . © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 241–254, 2004     

Cluster Analysis of Gene Expression Data

The expression levels of many thousands of genes can be measured simultaneously by DNA microarrays (chips). This novel experimental tool has revolutionized research in molecular biology and generated considerable excitement. A typical experiment uses a few tens of such chips, each dedicated to a single sample—such as tissue extracted from a particular tumor. The results of such an experiment contain several hundred thousand numbers, that come in the form of a table, of several thousand rows (one for each gene) and 50–100 columns (one for each sample). We developed a clustering methodology to mine such data. In this review I provide a very basic introduction to the subject, aimed at a physics audience with no prior knowledge of either gene expression or clustering methods. I explain what genes are, what is gene expression and how it is measured by DNA chips. Next I explain what is meant by clustering and how we analyze the massive amounts of data from such experiments, and present results obtained from analysis of data from colon cancer, brain tumors and breast cancer.     

Minimal Length Elements of Thompson's Group F

Elements of the group are represented by pairs of binary trees and the structure of the trees gives insight into the properties of the elements of the group. The review section presents this representation and reviews the known relationship between elements of F and binary trees. In the main section we give a method of determining the minimal lengths of elements of Thompson's group F in the two generator presentation

<Emphasis Type="Italic">X</Emphasis>-Trees and Weighted Quartet Systems

In this note, we consider a finite set X and maps W from the set $ \mathcal{S}_{2|2} (X) $ of all 2, 2- splits of X into $ \mathbb{R}_{\geq 0} $. We show that such a map W is induced, in a canonical way, by a binary X-tree for which a positive length $ \mathcal{l} (e) $ is associated to every inner edge e if and only if (i) exactly two of the three numbers W(ab|cd),W(ac|bd), and W(ad|cb) vanish, for any four distinct elements a, b, c, d in X, (ii) $ a \neq d \quad\mathrm{and}\quad W (ab|xc) + W(ax|cd) = W(ab|cd) $ holds for all a, b, c, d, x in X with #{a, b, c, x} = #{b, c, d, x} = 4 and $ W(ab|cx),W(ax|cd) $ > 0, and (iii) $ W (ab|uv) \geq \quad \mathrm{min} (W(ab|uw), W(ab|vw)) $ holds for any five distinct elements a, b, u, v, w in X. Possible generalizations regarding arbitrary $ \mathbb{R} $-trees and applications regarding tree-reconstruction algorithms are indicated.AMS Subject Classification: 05C05, 92D15, 92B05.     

On the Combinatorics of Rooted Binary Phylogenetic Trees

We study subtree-prune-and-regraft (SPR) operations on leaf-labelled rooted binary trees, also known as rooted binary phylogenetic trees. This study is motivated by the problem of graphically representing evolutionary histories of biological sequences subject to recombination. We investigate some basic properties of the induced SPR-metric on the space of leaf-labelled rooted binary trees with n leaves. In contrast to the case of unrooted trees, the number |U(T)| of trees in which are one SPR operation away from a given tree depends on the topology of T. In this paper, we construct recursion relations which allow one to determine the unit-neighbourhood size |U(T)| efficiently for any tree topology. In fact, using the recursion relations we are able to derive a simple closed-form formula for the unit-neighbourhood size. As a corollary, we construct sharp upper and lower bounds on the size of unit-neighbourhoods and investigate the diameter of . Lastly, we consider an enumeration problem relevant to population genetics.AMS Subject Classification: 05C05, 92D15.     

The Approximation Numbers of Hardy-Type Operators on Trees

The Hardy operator T_a on a tree is defined by Properties of T_a as a map from L^p() into itselfare established for 1 p . The main result is that, with appropriateassumptions on u and v, the approximation numbers a_n(T_a) ofT_a satisfy for a specified constant _p and 1 p < . This extends results of Naimark, Newmanand Solomyak for p = 2. Hitherto, for p 2, (*) was unknowneven when is an interval. Also, upper and lower estimates forthe l^q and weak-l^q norms of a_n(T_a) are determined. 2000 MathematicalSubject Classification: 47G10, 47B10.