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     

Self‐Intersection Local Time for 𝒮′(ℝd)‐Ornstein‐Uhlenbeck Processes Arising from Immigration Systems

Fluctuation limits of an immigration branching particle system and an immigration branching measure‐valued process yield different types of 𝒮′(ℝ^d)‐valued Ornstein‐Uhlenbeck processes whose covariances are given in terms of an excessive measure for the underlying motion in Rd, which is taken to be a symmetric α‐stable process. In this paper we prove existence and path continuity results for the self‐intersection local time of these Ornstein‐Uhlenbeck processes. The results depend on relationships between the dimension d and the parameter α.     

Contextual regularity and complexity of neuronal activity: From stand‐alone cultures to task‐performing animals

Precursors of the superior information processing capabilities of our cortex can most probably be traced back to simple invertebrate systems. Using a unique set of newly developed neuronal preparations and state‐of‐the‐art analysis tools, we show that insect neurons have the ability to self‐regulate the information capacity of their electrical activity. We characterize the activity of a distinct population of neurons under progressive levels of structural and functional constraints: self‐formed networks of neuron clusters in vitro; isolated ex vivo ganglions; in vivo task‐free, and in vivotask‐forced neuronal activity in the intact animal. We show common motifs and identify trends of increasing self‐regulated complexity. This important principle may have played a key role in the gradual transition from simple neuronal motor control to complex information processing. © 2004 Wiley Periodicals, Inc. Complexity 9: 25–32, 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.     

Local time and a particle picture for Euclidean field theory

We study intersection properties of Wiener processes in the plane. For each positive integer k we show that k independent Wiener processes intersect almost surely in a set of Hausdorff dimension two, and that the set of points a single process visits at least k distinct times also has dimension two. We construct a functional on configurations of k independent Wiener processes that measures the extent to which the trajectories of the k processes intersect. We prove certain L^p estimates for this functional and show that it is a local time for a certain vector-valued multiparameter stochastic process.     

Projective models of Enriques surfaces in scrolls

We study Cossec's ? ‐function, which is defined for divisors with positive self‐intersection on an Enriques surface. In this paper we study the existence of pairs (C ², ? (C )) with C an irreducible curve. The ? ‐function gives in a natural way scrolls containing Enriques surfaces. We compute scroll types to some of these scrolls. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Large deviations ordering of point processes in some queueing networks

Given a stochastic ordering between point processes, say that a p.p. N is smooth if it is less than the Poisson process with the same average intensity for this ordering. In this article we investigate whether initially smooth processes retain their smoothness as they cross a network of FIFO ·/D/1 queues along fixed routes. For the so-called strong variability ordering we show that point processes remain smooth as they proceed through a tandem of quasi-saturated (i.e., loaded to 1) M+·/D/1 queues. We then introduce the Large Deviations ordering, which involves comparison of the rate functions associated with Large Deviations Principles satisfied by the point processes. For this ordering, we show that smoothness is retained when the processes cross a feed-forward network of unsaturated ·/D/1 queues. We also examine the LD characteristics of a deterministic p.p. at the output of an M+·/D/1 queue. This revised version was published online in June 2006 with corrections to the Cover Date.     

POSSIBILITIES AND LIMITATIONS OF USING HISTORIC PROVENANCE TESTS TO INFER FOREST SPECIES GROWTH RESPONSES TO CLIMATE CHANGE

Abstract. Under projected changes in global climate, the growth and survival of existing forests will depend on their ability to adjust physiologically in response to environmental change. Quantifying their capacity to adjust and whether the response is species‐ or population‐specific is important to guide forest management strategies. New analyses of historic provenance tests data are yielding relevant insights about these responses. Yet, differences between the objectives used to design the experiments and current objectives impose limitations to what can be learned from them. Our objectives are (i) to discuss the possibilities and limitations of using such data to quantify growth responses to changes in climate and (ii) to present a modeling approach that creates a species‐ and population‐specific model. We illustrate the modeling approach for Larix occidentalis Nutt. We conclude that the reanalysis of historic provenance tests data can lead to the identification of species that have population‐specific growth responses to changes in climate, provide estimates of optimum transfer distance for populations and species, and provide estimates of growth changes under different climate change scenarios. Using mixed‐effects modeling techniques is a sound statistical approach to overcome some of the limitations of the data.     

Random processes in Sobolev-Orlicz spaces

We establish conditions under which the trajectories of random processes from Orlicz spaces of random variables belong with probability one to Sobolev-Orlicz functional spaces, in particular to the classical Sobolev spaces defined on the entire real axis. This enables us to estimate the rate of convergence of wavelet expansions of random processes from the spaces L _p (Ω) and L ₂ (Ω) in the norm of the space L _q (ℝ). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1340–1356, October, 2006.     

Discovery Through Gossip

We study randomized gossip‐based processes in dynamic networks that are motivated by information discovery in large‐scale distributed networks such as peer‐to‐peer and social networks. A well‐studied problem in peer‐to‐peer networks is resource discovery, where the goal for nodes (hosts with IP addresses) is to discover the IP addresses of all other hosts. Also, some of the recent work on self‐stabilization algorithms for P2P/overlay networks proceed via discovery of the complete network. In social networks, nodes (people) discover new nodes through exchanging contacts with their neighbors (friends). In both cases the discovery of new nodes changes the underlying network — new edges are added to the network — and the process continues in the changed network. Rigorously analyzing such dynamic (stochastic) processes in a continuously changing topology remains a challenging problem with obvious applications. This paper studies and analyzes two natural gossip‐based discovery processes. In the push discovery or triangulation process, each node repeatedly chooses two random neighbors and connects them (i.e., “pushes” their mutual information to each other). In the pull discovery process or the two‐hop walk, each node repeatedly requests or “pulls” a random contact from a random neighbor and connects itself to this two‐hop neighbor. Both processes are lightweight in the sense that the amortized work done per node is constant per round, local, and naturally robust due to the inherent randomized nature of gossip. Our main result is an almost‐tight analysis of the time taken for these two randomized processes to converge. We show that in any undirected n‐node graph both processes take rounds to connect every node to all other nodes with high probability, whereas is a lower bound. We also study the two‐hop walk in directed graphs, and show that it takes time with high probability, and that the worst‐case bound is tight for arbitrary directed graphs, whereas Ω(n²) is a lower bound for strongly connected directed graphs. A key technical challenge that we overcome in our work is the analysis of a randomized process that itself results in a constantly changing network leading to complicated dependencies in every round. We discuss implications of our results and their analysis to discovery problems in P2P networks as well as to evolution in social networks. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 48, 565–587, 2016     

Extreme self‐adjoint extensions of a semibounded ‐difference operator

For a certain q‐difference operator introduced and studied in a series of articles by the same authors, we investigate its extreme self‐adjoint extensions, i.e., the so‐called Friedrichs and Kre?n extensions. We show that for the interval of parameters under consideration, the Friedrichs extension and the Kre?n extension are distinct and give values of the parameter in the von Neumann formulas that correspond to those extensions and describe their resolvent operators. A crucial rôle in our investigation plays the fact that both the Friedrichs and the Kre?n extensions are scale invariant.     

The Tax Identity For Markov Additive Risk Processes

Taxed risk processes, i.e. processes which change their drift when reaching new maxima, represent a certain type of generalizations of Lévy and of Markov additive processes (MAP), since the times at which their Markovian mechanism changes are allowed to depend on the current position. In this paper we study generalizations of the tax identity of Albrecher and Hipp (2007) from the classical risk model to more general risk processes driven by spectrally-negative MAPs. We use the Sparre Andersen risk processes with phase-type interarrivals to illustrate the ideas in their simplest form.     

Small ball probabilities for smooth Gaussian fields and tensor products of compact operators

We find the logarithmic L₂‐small ball asymptotics for a class of zero mean Gaussian fields with covariances having the structure of “tensor product”. The main condition imposed on marginal covariances is slow growth at the origin of counting functions of their eigenvalues. That is valid for Gaussian functions with smooth covariances. Another type of marginal functions considered as well are classical Wiener process, Brownian bridge, Ornstein–Uhlenbeck process, etc., in the case of special self‐similar measure of integration. Our results are based on a new theorem on spectral asymptotics for the tensor products of compact self‐adjoint operators in Hilbert space which is of independent interest. Thus, we continue to develop the approach proposed in the paper 6 , where the regular behavior at infinity of marginal eigenvalues was assumed.     

Long time approximations for solutions of wave equations associated with the Steklov spectral homogenization problems

The interest in the use of quasimodes, or almost frequencies and almost eigenfunctions, to describe asymptotics for low‐frequency and high‐frequency vibrations in certain singularly perturbed spectral problems, which depend on a small parameter ε, has been recently highlighted in many papers. In this paper we deal with the low frequencies for a Steklov‐type eigenvalue homogenization problem: we consider harmonic functions in a bounded domain of ?², and strongly alternating boundary conditions of the Dirichlet and Steklov type on a part of the boundary. The spectral parameter appears in the boundary condition on small segments T^ε of size O(ε) periodically distributed along the boundary; ε also measures the periodicity of the structure. We consider associated second‐order evolution problems on spaces of traces that depend on ε, and we provide estimates for the time t in which standing waves, constructed from quasimodes, approach their solutions u^ε(t) as ε→0. Copyright © 2009 John Wiley & Sons, Ltd.     

Iterated line graphs are maximally ordered

A graph G is k‐ordered if for every ordered sequence of k vertices, there is a cycle in G that encounters the vertices of the sequence in the given order. We prove that if G is a connected graph distinct from a path, then there is a number t_G such that for every t ≥ t_G the t‐iterated line graph of G, L^t (G), is (δ(L^t (G)) + 1)‐ordered. Since there is no graph H which is (δ(H)+2)‐ordered, the result is best possible. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 171–180, 2006     

Atomic Latin squares of order eleven

A Latin square is pan‐Hamiltonian if the permutation which defines row i relative to row j consists of a single cycle for every i ≠ j. A Latin square is atomic if all of its conjugates are pan‐Hamiltonian. We give a complete enumeration of atomic squares for order 11, the smallest order for which there are examples distinct from the cyclic group. We find that there are seven main classes, including the three that were previously known. A perfect 1‐factorization of a graph is a decomposition of that graph into matchings such that the union of any two matchings is a Hamiltonian cycle. Each pan‐Hamiltonian Latin square of order n describes a perfect 1‐factorization of K_n,n, and vice versa. Perfect 1‐factorizations of K_n,n can be constructed from a perfect 1‐factorization of K_n+1. Six of the seven main classes of atomic squares of order 11 can be obtained in this way. For each atomic square of order 11, we find the largest set of Mutually Orthogonal Latin Squares (MOLS) involving that square. We discuss algorithms for counting orthogonal mates, and discover the number of orthogonal mates possessed by the cyclic squares of orders up to 11 and by Parker's famous turn‐square. We find that the number of atomic orthogonal mates possessed by a Latin square is not a main class invariant. We also define a new sort of Latin square, called a pairing square, which is mapped to its transpose by an involution acting on the symbols. We show that pairing squares are often orthogonal mates for symmetric Latin squares. Finally, we discover connections between our atomic squares and Franklin's diagonally cyclic self‐orthogonal squares, and we correct a theorem of Longyear which uses tactical representations to identify self‐orthogonal Latin squares in the same main class as a given Latin square. © 2003 Wiley Periodicals, Inc.     

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.     

Structure relations in special <Emphasis Type="Italic">A</Emphasis><Subscript>∞</Subscript>-bialgebras

We compute the structure relations in special A _∞-bialgebras whose operations are limited to those defining the underlying A _∞-(co)algebra substructure. Such bialgebras appear as the homology of certain loop spaces. Whereas structure relations in general A _∞-bialgebras depend upon the combinatorics of permutahedra, only Stasheff’s associahedra are required here. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 43, Topology and Its Applications, 2006.     

Parameter estimation in systems of differential equations,based on three-way data arrays and random time changes

Consider a system of n₁ × n₂ differential equations depending on a vector θ of unknown parameters. We suggest an iterative estimation procedure for θ, based on a three-way array of observations. The method involves random time changes driven by multidimensional integrated Ornstein-Uhlenbeck processes.     

Periodic Billiard Trajectories in Smooth Convex Bodies

We consider billiard trajectories in a smooth convex body in \mathbbR^d{\mathbb{R}^{d}} and estimate the number of distinct periodic trajectories that make exactly p reflections per period at the boundary of the body. In the case of prime p we obtain the lower bound (d – 2)(p – 1) + 2, which is much better than the previous estimates.