Structural and functional growth in self‐reproducing cellular automata

In this article, we analyze the dynamics of change in two‐dimensional self‐reproducers, identifying the processes that drive their evolution. We show that changes in self‐reproducers structure and behavior depend on their genetic memory. This consists of distinct yet interlinked components determining their form and function. In some cases these components degrade gracefully, changing only slightly; in others the changes destroy the original structure and function of the self‐reproducer. We sketch these processes at the genotype and the phenotype level—showing that they follow distinct trajectories within mutation space and quantifying the degree of change produced by different trajectories. We show that changes in structure and behavior depend on the interplay between the genotype and the phenotype. This determines universal structures, from which it is possible to construct a great number of self‐reproducing systems, as we observe in biology. Creative processes of change produce divergent and/or convergent methods for the generation of self‐reproducers. Divergence involves the creation of completely new information convergence involves local change and specialization of the structures concerned. © 2006 Wiley Periodicals, Inc. Complexity 11: 12–29, 2006     

Claw‐free 3‐connected P11‐free graphs are hamiltonian

We show that every 3‐connected claw‐free graph which contains no induced copy of P₁₁ is hamiltonian. Since there exist non‐hamiltonian 3‐connected claw‐free graphs without induced copies of P₁₂ this result is, in a way, best possible. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 111–121, 2004     

Pseudo‐random properties of self‐complementary symmetric graphs

There are some results in the literature showing that Paley graphs behave in many ways like random graphs G(n, 1/2). In this paper, we extend these results to the other family of self‐complementary symmetric graphs. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 310–316, 2004     

Life‐like self‐reproducers

In order to understand the interplay among information, genetic instructions, and phenotypic variations, self‐reproducers discovered in two‐dimensional cellular automata are considered as proto‐organisms, which undergo to mutations as they were in a real environmental situation. We realized a computational model through which we have been able to discover the genetic map of the self‐reproducers and the networks they use. Identifying in these maps sets of different functional genes, we found that mutations in the genetic sequences could affect both external shapes and behavior of the self‐reproducers, thus realizing different life‐like strategies in the evolution process. The results highlight that some strategies evolution uses in selecting organisms that are fitting with changing environmental situations maintain the self‐reproducing function, whereas other variations create new self‐reproducers. These self‐reproducers in turn realize different genetic networks, which can be very different from the basic ancestors pools. The mutations that are disruptive bring self‐reproducers to disappear, while other proto‐organisms are generated. © 2004 Wiley Periodicals, Inc. Complexity 9: 38–55, 2004     

Convex‐round graphs are circular‐perfect

We show a connection between two concepts that have hitherto been investigated separately, namely convex‐round graphs and circular cliques. The connections are twofold. We prove that the circular cliques are precisely the cores of convex‐round graphs; this implies that convex‐round graphs are circular‐perfect, a concept introduced recently by Zhu [10]. Secondly, we characterize maximal K_r‐free convex‐round graphs and show that they can be obtained from certain circular cliques in a simple fashion. Our proofs rely on several structural properties of convex‐round graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 182–194, 2002     

Self‐dual and self‐petrie‐dual regular maps

Regular maps are cellular decompositions of surfaces with the “highest level of symmetry”, not necessarily orientation‐preserving. Such maps can be identified with three‐generator presentations of groups G of the form G = 〈a, b, c|a² = b² = c² = (ab)^k = (bc)^m = (ca)² = … = 1〉; the positive integers k and m are the face length and the vertex degree of the map. A regular map (G;a, b, c) is self‐dual if the assignment b?b, c?a and a?c extends to an automorphism of G, and self‐Petrie‐dual if G admits an automorphism fixing b and c and interchanging a with ca. In this note we show that for infinitely many numbers k there exist finite, self‐dual and self‐Petrie‐dual regular maps of vertex degree and face length equal to k. We also prove that no such map with odd vertex degree is a normal Cayley map. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 69:152‐159, 2012     

The shape of large Galton‐Watson trees with possibly infinite variance

Let T be a critical or subcritical Galton‐Watson family tree with possibly infinite variance. We are interested in the shape of T conditioned to have a large total number of vertices. For this purpose we study random trees whose conditional distribution given their size is the same as the respective conditional distribution of T. These random family trees have a simple probabilistic structure if decomposed along the lines of descent of a number of distinguished vertices chosen uniformly at random. The shape of the subtrees spanned by the selected vertices and the root depends essentially on the tail of the offspring distribution: While in the finite variance case the subtrees are asymptotically binary, other shapes do persist in the limit if the variance is infinite. In fact, we show that these subtrees are Galton‐Watson trees conditioned on their total number of leaves. The rescaled total size of the trees is shown to have a gamma limit law. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

TOWARDS A STANDARD FOR THE INDIVIDUAL‐BASED MODELING OF PLANT POPULATIONS: SELF‐THINNING AND THE FIELD‐OF‐NEIGHBORHOOD APPROACH

ABSTRACT. In classical theoretical ecology there are numerous standard models which are simple, generally applicable, and have well‐known properties. These standard models are widely used as building blocks for all kinds of theoretical and applied models. In contrast, there is a total lack of standard individual‐based models (IBM's), even though they are badly needed if the advantages of the individual‐based approach are to be exploited more efficiently. We discuss the recently developed ‘field‐of‐neighborhood’ approach as a possible standard for modeling plant populations. In this approach, a plant is characterized by a circular zone of influence that grows with the plant, and a field of neighborhood that for each point within the zone of influence describes the strength of competition, i.e., growth reduction, on neighboring plants. Local competition is thus described phenomenologically. We show that a model of mangrove forest dynamics, KiWi, which is based on the FON approach, is capable of reproducing self‐thinning trajectories in an almost textbook‐like manner. In addition, we show that the entire biomass‐density trajectory (bdt) can be divided into four sections which are related to the skewness of the stem diameter distributions of the cohort. The skewness shows two zero crossings during the complete development of the population. These zero crossings indicate the beginning and the end of the self‐thinning process. A characteristic decay of the positive skewness accompanies the occurrence of a linear bdt section, the well‐known self‐thinning line. Although the slope of this line is not fixed, it is confined in two directions, with morphological constraints determining the lower limit and the strength of neighborhood competition exerted by the individuals marking the upper limit.     

H‐self‐adjoint matrices. Application to radiosity

In computer graphics, in the radiosity context, a linear system Φx=b must be solved and there exists a diagonal positive matrix H such that H Φ is symmetric. In this article, we extend this property to complex matrices: we are interested in matrices which lead to Hermitian matrices under premultiplication by a Hermitian positive‐definite matrix H. We shall prove that these matrices are self‐adjoint with respect to a particular innerproduct defined on ?ⁿ. As a result, like Hermitian matrices, they have real eigenvalues and they are diagonalizable. We shall also show how to extend the Courant–Fisher theorem to this class of matrices. Finally, we shall give a new preconditioning matrix which really improves the convergence speed of the conjugate gradient method used for solving the radiosity problem. Copyright © 2002 John Wiley & Sons, Ltd.     

Forbidden minors for wye‐delta‐wye reducibility

A graph is Y Δ Y reducible if it can be reduced to a single vertex by a sequence of series‐parallel reductions and Y Δ Y transformations. The class of Y Δ Y reducible graphs is minor closed. We found a large number of minor minimal Y Δ Y irreducible graphs: a family of 57578 31‐edge graphs and another 40‐edge graph. It is still an open problem to characterize Y Δ Y reducible graphs in terms of a finite set of forbidden minors. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 317–321, 2004     

Approach to self‐similarity in Smoluchowski's coagulation equations

We consider the approach to self‐similarity (or dynamical scaling) in Smoluchowski's equations of coagulation for the solvable kernels K(x, y) = 2, x + y and xy. In addition to the known self‐similar solutions with exponential tails, there are one‐parameter families of solutions with algebraic decay, whose form is related to heavy‐tailed distributions well‐known in probability theory. For K = 2 the size distribution is Mittag‐Leffler, and for K = x + y and K = xy it is a power‐law rescaling of a maximally skewed α‐stable Lévy distribution. We characterize completely the domains of attraction of all self‐similar solutions under weak convergence of measures. Our results are analogous to the classical characterization of stable distributions in probability theory. The proofs are simple, relying on the Laplace transform and a fundamental rigidity lemma for scaling limits. © 2003 Wiley Periodicals, Inc.     

Non‐Cayley‐isomorphic self‐complementary circulant graphs

Non‐CI self‐complementary circulant graphs of prime‐squared order are constructed and enumerated. It is shown that for prime p, there exists a self‐complementary circulant graph of order p² not Cayley isomorphic to its complement if and only if p ≡ 1 (mod 8). Such graphs are also enumerated. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 128–141, 2000     

Semi‐transitive graphs

In this paper, we first consider graphs allowing symmetry groups which act transitively on edges but not on darts (directed edges). We see that there are two ways in which this can happen and we introduce the terms bi‐transitive and semi‐transitive to describe them. We examine the elementary implications of each condition and consider families of examples; primary among these are the semi‐transitive spider‐graphs PS(k,N;r) and MPS(k,N;r). We show how a product operation can be used to produce larger graphs of each type from smaller ones. We introduce the alternet of a directed graph. This links the two conditions, for each alternet of a semi‐transitive graph (if it has more than one) is a bi‐transitive graph. We show how the alternets can be used to understand the structure of a semi‐transitive graph, and that the action of the group on the set of alternets can be an interesting structure in its own right. We use alternets to define the attachment number of the graph, and the important special cases of tightly attached and loosely attached graphs. In the case of tightly attached graphs, we show an addressing scheme to describe the graph with coordinates. Finally, we use the addressing scheme to complete the classification of tightly attached semi‐transitive graphs of degree 4 begun by Marus?ic? and Praeger. This classification shows that nearly all such graphs are spider‐graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 1–27, 2004     

First and second moments for self‐similar couplings and Wasserstein distances

We study aspects of the Wasserstein distance in the context of self‐similar measures. Computing this distance between two measures involves minimising certain moment integrals over the space of couplings, which are measures on the product space with the original measures as prescribed marginals. We focus our attention on self‐similar measures associated to equicontractive iterated function systems consisting of two maps on the unit interval and satisfying the open set condition. We are particularly interested in understanding the restricted family of self‐similar couplings and our main achievement is the explicit computation of the 1st and 2nd moment integrals for such couplings. We show that this family is enough to yield an explicit formula for the 1st Wasserstein distance and provide non‐trivial upper and lower bounds for the 2nd Wasserstein distance for these self‐similar measures.     

Spanning k‐arc‐strong subdigraphs with few arcs in k‐arc‐strong tournaments

Given a k‐arc‐strong tournament T, we estimate the minimum number of arcs possible in a k‐arc‐strong spanning subdigraph of T. We give a construction which shows that for each k ≥ 2, there are tournaments T on n vertices such that every k‐arc‐strong spanning subdigraph of T contains at least arcs. In fact, the tournaments in our construction have the property that every spanning subdigraph with minimum in‐ and out‐degree at least k has arcs. This is best possible since it can be shown that every k‐arc‐strong tournament contains a spanning subdigraph with minimum in‐ and out‐degree at least k and no more than arcs. As our main result we prove that every k‐arc‐strong tournament contains a spanning k‐arc‐strong subdigraph with no more than arcs. We conjecture that for every k‐arc‐strong tournament T, the minimum number of arcs in a k‐arc‐strong spanning subdigraph of T is equal to the minimum number of arcs in a spanning subdigraph of T with the property that every vertex has in‐ and out‐degree at least k. We also discuss the implications of our results on related problems and conjectures. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 265–284, 2004     

Constructing regular self‐complementary uniform hypergraphs

In this article, we examine the possible orders of t‐subset‐regular self‐complementary k‐uniform hypergraphs, which form examples of large sets of two isomorphic t‐designs. We reformulate Khosrovshahi and Tayfeh–Rezaie's necessary conditions on the order of these structures in terms of the binary representation of the rank k, and these conditions simplify to a more transparent relation between the order n and rank k in the case where k is a sum of consecutive powers of 2. Moreover, we present new constructions for 1‐subset‐regular self‐complementary uniform hypergraphs, and prove that these necessary conditions are sufficient for all k, in the case where t = 1. © 2011 Wiley Periodicals, Inc. J Combin Designs 19: 439‐454, 2011     

Highly edge‐connected detachments of graphs and digraphs

Let G = (V,E) be a graph or digraph and r : V → Z₊. An r‐detachment of G is a graph H obtained by ‘splitting’ each vertex ν ∈ V into r(ν) vertices. The vertices ν₁,…,ν_r(ν) obtained by splitting ν are called the pieces of ν in H. Every edge uν ∈ E corresponds to an edge of H connecting some piece of u to some piece of ν. Crispin Nash‐Williams 9 gave necessary and sufficient conditions for a graph to have a k‐edge‐connected r‐detachment. He also solved the version where the degrees of all the pieces are specified. In this paper, we solve the same problems for directed graphs. We also give a simple and self‐contained new proof for the undirected result. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 67–77, 2003     

K‐linked graphs with girth condition

Recently, Mader [ 7 ] proved that every 2k‐connected graph with girth g(G) sufficiently large is k‐linked. We show here that g(G ≥ 11 will do unless k = 4,5. If k = 4,5, then g(G) ≥ 19 will do. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 48–50, 2004     

On the Lipschitz equivalence of self‐affine sets

Recently Lipschitz equivalence of self‐similar sets on has been studied extensively in the literature. However for self‐affine sets the problem is more awkward and there are very few results. In this paper, we introduce a w‐Lipschitz equivalence by repacing the Euclidean norm with a pseudo‐norm w. Under the open set condition, we prove that any two totally disconnected integral self‐affine sets with a common matrix are w‐Lipschitz equivalent if and only if their digit sets have equal cardinality. The main methods used are the technique of pseudo‐norm and Gromov hyperbolic graph theory on iterated function systems.     

Point‐wise decay estimate for the global classical solutions to quasilinear hyperbolic systems

In this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19 :1263–1317; Nonlinear Anal. 1997; 28 :1299–1322; Chin. Ann. Math. 2004; 25B :37–56). We give a new, very simple proof of this result and also give a sharp point‐wise decay estimate of the solution. Then, we consider the mixed initial‐boundary‐value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12 (1):59–78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point‐wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd.