Analysis of coalescence processes is extensively documented in the literature, but not all of the theoretical frame-works encompass the concept of coalescence itself. We introduce and study Totally Asymmetric Coalescence Process (TACP), a new class of interacting particles in which the environment is functional in the interaction. In particular, the TACP is proved to display a first-order phase transition. Numerical analysis for the TACP is presented as well.

We study a class of one-dimensional probabilistic cellular automata in which each component can be in either state zero or state one. The component interacts with two neighbors: if its neighbors are in an equal state, then the component assumes the same state as its neighbors. If its neighbors are in different states, the following can happen: a one on the right-hand side of a zero, in which case the component becomes one with probability α or zero with probability 1 − α, and conversely, a zero on the right-hand side of a one, in which case the component becomes one with probability β or zero with probability 1 − β. For a set of initial distributions when both neighbors are placed on the right-hand side (respectively, both on the left-hand side) of a component, we prove that the process always converges weakly to the measure concentrated on the configuration where all the components are zeros. When one neighbor is placed on the left-hand side and the other is on the right-hand side, the same convergence happens when β < fN(α), where N is the distance between the neighbors. However, this convergence does not happen for β > 1/2α. Thus, in this case, we get the regimes of ergodicity and non-ergodicity. Moreover, we exhibit another type of phase transition, independent of neighbors’ locations. We also present some numerical studies in which we use mean field approximation and Monte Carlo simulation.

A one-dimensional interacting particle system is revisited. It has discrete time, and its components are located in the set of integers. These components can disappear in the functioning process. Each component assumes two possible states, called plus and minus, and interacts at every time step only with its nearest neighbors. The following two transformations happen: The first one is called flip, under its action, a component in state minus turns into a plus with probability β. The second one is called annihilation, under its action, whenever a component in state plus is a left neighbor of a component in state minus, both components disappear with probability α. Let us consider a set of initial measures to the process. For these measures, we show the upper bound for the mean time of convergence, which is a function of the initial measure. Moreover, we obtain the upper bound to the mean quantity of minuses on the process in each time step. Considering the initial measure concentrated at the configuration whose components are in the state minus, we improved a well-known result that the process is non-ergodic when β < α2/250. Now, we are able to offer non-ergodicity when β < 9α2/1000. We also established new conditions to the ergodicity of the process. Finally, we performed some Monte Carlo simulations for this process.

In dynamical systems, some of the most important questions are related to phase transitions and convergence time. We consider a one-dimensional probabilistic cellular automaton where their components assume two possible states, zero and one, and interact with their two nearest neighbors at each time step. Under the local interaction, if the component is in the same state as its two neighbors, it does not change its state. In the other cases, a component in state zero turns into a one with probability \(\alpha ,\) and a component in state one turns into a zero with probability \(1-\beta \). For certain values of \(\alpha \) and \(\beta \), we show that the process will always converge weakly to \(\delta _{0},\) the measure concentrated on the configuration where all the components are zeros. Moreover, the mean time of this convergence is finite, and we describe an upper bound in this case, which is a linear function of the initial distribution. We also demonstrate an application of our results to the percolation PCA. Finally, we use mean-field approximation and Monte Carlo simulations to show coexistence of three distinct behaviours for some values of parameters \(\alpha \) and \(\beta \).

The aim of this paper is to give some Ulam-Hyers-Rassias stability results for Volterratype stochastic integral equations. The argument makes use of Gronwall lemma and Banach’s fixed point theorem.

This article presents a new example intended to showcase limitations of computer simulations in the study of random processes with local interaction. For this purpose, we examine a new version of the well-known Stavskaya process, which is a discrete-time analog of the well-known contact processes. Like the bulk of random processes studied till now, the Stavskaya process is constant-length, that is, its components do not appear or disappear in the course of its functioning. The process, which we study here and call Variable Stavskaya, VS, is similar to Stavskaya; it is discrete-time; its states are bi-infinite sequences, whose terms take only two values (denoted here as “minus” and “plus”), and the measure concentrated in the configuration “all pluses” is invariant. However, it is a variable length, which means that its components, also called particles, may appear and disappear under its action. The operator VS is a composition of the following two operators. The first operator, called “birth,” depends on a real parameter β; it creates a new component in the state “plus” between every two neighboring components with probability β independently from what happens at other places. The second operator, called “murder,” depends on a real parameter α and acts in the following way: whenever a plus is a left neighbor of a minus, this plus disappears (as if murdered by that minus which is its right neighbor) with probability α independently from what happens to other particles. We prove for any α<1 and any β>0 and any initial measure μ that the sequence μ(𝖵𝖲)t (the result of t iterative applications of VS to μ) tends to the measure δ⊕ (concentrated in “all pluses”) as t→∞. Such a behavior is often called ergodic. However, the Monte Carlo simulations and mean-field approximations, which we performed, behaved as if μ(𝖵𝖲)t tended to δ⊕ much slower for some α,β,μ than for some others. Based on these numerical results, we conjecture that 𝖵𝖲 has phases, but not in that simple sense as the classical Stavskaya process.

Imagine a huge (in theory, infinite) space, whose elements are called components. We say that we have a configuration, if for every component we have specified its state. All components have one and the same finite set \(M\) of possible states and one of the elements of \(M\) is called zero. If all the components of a configuration are zeros, we call it “all zeros”. We are especially interested in those configurations, in which only a bounded set of components are in a state different from zero; such a configuration is called an island. Our time is discrete and we may imagine that, due to the forces of nature, at every time step the whole configuration (landscape) is subject to a deterministic uniform local rule \(D\) such that non-zero components may appear only in the vicinity of already existing non-zero components (like in percolation or contact processes). We say that an operator \(D\) erodes an island \(x\) if there is a natural \(t\) such that \(x D^t =\) “all zeros”, that is \(t\) iterative applications of \(D\) turn \(x\) into “all zeros”. We call \(D\) an eroder if it erodes all islands. We look for an algorithm which decides for any \(D\) whether it is an eroder or not and does it make islands grow or not. In general this problem is algorithmically unsolvable, so we need to restrict our scope; in addition to the afore-mentioned conditions we assume that \(D\) is monotonic. Studying such processes one has to decide whether time and space are discrete or continuous and in this study we choose discrete time and continuous space just because this case is underrepresented in the literature. But especially important is the set of possible states of every single component. Years ago one of us (A. Toom) presented a rule to decide whether \(D\) is an eroder for the case when every component has only two possible states. A. Toom also showed that if \(D\) is an eroder, then it erodes any island in time, which is linear in the diameter of the island. In this work every component has three possible states. It turned out that the difference between two and three states, which may seem trivial, in fact leads to a qualitative difference: in the three-states case an eroder may take non-linear time to reach “all zeros”. Also, unlike two-state case, in the three-state case we have, in addition to eroders, to introduce degraders which reduce any island to an island where states of all components take only two possible values. Our main results are stated as three theorems. Theorem 1 gives sufficient (but regretfully not necessary) conditions for a linear degrader and an eroder. Theorems 2 and 3 concentrate on the case when \(D\) needs a non-linear time to erode an island—a case impossible if states of components take only two values. In addition, Theorem 3 presents a sufficient condition for islands to grow.

We study a non-ergodic one-dimensional probabilistic cellular automata, where each component can assume the states and ­. We obtained the limit distribution for a set of measures on {,­}^. Also, we show that for certain parameters of our process the mean time of convergence can be finite or infinity. When it is finite we have showed that the upper bound is function of the initial distribution.

Given a finite set B (basin) with n>1 elements, which we call points, and a map M:B→B, we call such pairs (B,M) monads. Here we study a class of random monads, where the values of M(⋅) are independently distributed in B as follows: for all a,b∈B the probability of M(a)=a is s and the probability of M(a)=b, where a≠b, is (1−s)/(n−1). Here s is a parameter, 0≤s≤1. We fix a point ⊙∈B and consider the sequence M t (⊙), t=0,1,2,… . A point is called visited if it coincides with at least one term of this sequence. A visited point is called recurrent if it appears in this sequence at least twice; if a visited point appears in this sequence only once, it is called transient. We denote by Vis n , Rec n and Tra n the numbers of visited, recurrent and transient points respectively. We prove that, when n tends to infinity, Vis n and Tra n converge in law to geometric distributions and Rec n converges in law to a distribution concentrated at its lowest value, which is one. Now about moments. The case s=1 is trivial, so let 0≤s<1. For any natural number k there is a number such that the k-th moments of Vis n , Rec n and Tra n do not exceed this number for all n. About Vis n : for any natural k the k-th moment of Vis n is an increasing function of n. So it has a limit when n→∞ and for all n it is less than this limit. About Rec n : for any k the k-th moment of Rec n tends to one when n tends to infinity. About Tra n : for any k the k-th moment of Tra n has a limit when n tends to infinity.

Let us have a finite set B (basin) with n>1 elements, which we call points, and a map M:B→B. Following Vladimir Arnold, we call such pairs (B,M) monads. Here we study a class of random monads, where the values of M(⋅) are independently distributed in B as follows: for all a,b∈B the probability of M(a)=a is s and the probability of M(a)=b, where a≠b, is (1−s)/(n−1). Here s is a parameter in [0,1].

Let us have a non-empty finite set S with n >1 elements which we call points and a map M : S → S . After V.I. Arnold, we call such pairs ( S , M ) monads , but we consider random monads in which all the values of M (⋅) are random, independent and uniformly distributed in S . We fix some ⊙∈ S and consider the infinite sequence M t (⊙), t =0,1,2,… . A point is called visited if it coincides with at least one term of this sequence. A visited point is called recurrent if it appears in this sequence at least twice; if a visited point appears in this sequence only once, it is called transient . We denote by Vis , Rec , Tra the numbers of visited, recurrent and transient points respectively and study their distributions. The distributions of Vis , Rec , Tra are unimodal. The modes of Rec and Tra equal their minimal values, that is 1 and 0 respectively. The mode of Vis is approximated by √(n) , plus-minus a constant. The mathematical expectations: 𝔼(𝑉𝑖𝑠) is approximated by 2 √(π n/8) plus-minus a constant; 𝔼(𝑅𝑒𝑐) and 𝔼(𝑇𝑟𝑎) are approximated by √(π n/8) plus-minus a constant. For the standard deviations σ ( Vis ) and σ ( Rec )= σ ( Tra ) respectively we present the approximations √(4-π/2· n) √(16-3π/24· n), from which they also deviate at most by a constant. We prove that when n tends to infinity, the correlations Corr( Rec , Tra ) and Corr( Rec , Vis )=Corr( Tra , Vis ) converge to 8-3π/16-3π √(12-3π/16-3π).

We present results of Monte Carlo simulation and chaos approximation of a class of Markov processes with a countable or continuous set of states. Each of these states can be written as a finite (finite case) or infinite in both directions (infinite case) sequence of pluses and minuses denoted by circle plus and circle minus. As continuous time goes on, our sequence undergoes the following three types of local transformations: the first one, called flip, changes any minus into plus and any plus into minus with a rate beta; the second, called annihilation, eliminates two neighbor components with a rate a whenever they are in differents states; and the third, called mitosis, doubles any component with a rate gamma. All of them occur at any place of the sequence independently. Our simulations and approximations suggest that with appropriate positive values of alpha, beta and gamma this process has the following two properties. Growth: In the finite case, as the process goes on, the length of the sequence tends to infinity with a probability which tends to 1 when the length of the initial sequence tends to infinity. Nonergodicity: The infinite process is nonergodic and the finite process keeps most of the time at two extremes, occasionally swinging from one to the other.

The flip-annihilation process is a random particle process with one-dimensional local interaction in discrete time, initially presented by one of us, namely Toom in 2004. Its components are enumerated by integer numbers and every component has two states, “minus” and “plus”. At every time step two transformations occur. The first one, called “flip”, independently turns every minus into plus with probability β. The second one, called “annihilation”, acts thus: whenever a plus is a left neighbor of a minus, both disappear with probability α independently from other components. What is interesting about this process is that it is ergodic for β>α/2 and non-ergodic for β<α 2/250. It is natural to conjecture that there is some transition curve, which we call the true curve and denote by \(\beta =\mathsf{true}(\alpha)\), which separates the areas of ergodicity and non-ergodicity of this process from each other. The estimates, mentioned above, albeit rigorous, leave a large gap between them and the present article’s purpose is to obtain some closer, albeit non-rigorous, approximations of the true curve. We do it in two ways, one of which is a chaos approximation and the other is a Monte Carlo simulation. Thus we obtain two curves, which are much closer to each other than the rigorous estimations. Also we fill in, albeit only numerically, another shortcoming of the rigorous estimation β<α 2/250, namely that it leaves us uncertain whether the true curve has a zero or positive slope at the point α=β=0. Both approximate curves have a positive slope at α=0, as we hoped.

Osteoporosis may be characterized by low bone density and its significance is expected to grow as the population of the world both increases and ages. Our purpose here is to model human bone mineral density estimated through dual-energy x-ray absorptiometry, using local volumetric distance spline interpolants. Interpolating the values means the construction of a function F(x,y,z) that mimics the relationship implied by the data (xi,yi,zi;fi), in such a way that F(xi,yi,zi)=fi, i=1,2,…,n, where x,y and z represent, respectively, age, weight and height. This strategy greatly enhances the ability to accurately express the patient's bone density measurements, with the potential to become a framework for bone densitometry in clinical practice. The usefulness of our model is demonstrated in 424 patients and the relevance of our results for diagnosing osteoporosis is discussed.

By means of Monte Carlo simulations performed in the C programming language, an example of scientific programming for the generation of pseudorandom numbers relevant to both teaching and research in the field of biomedicine is presented. The relatively simple algorithm proposed makes possible the statistical analysis of sequences of random numbers. The following three generators of pseudorandom numbers were used: the rand function contained in the stdlib.h library of the C programming language, Marsaglia's generator, and a chaotic function. The statistical properties of the sequences generated were compared, identical parameter values being adopted for this purpose. The properties of two estimators in finite samples of the pseudorandom numbers were also evaluated and, under suitable conditions, both the maximum-likelihood and method of moments proved to be good estimators. The findings demonstrated that the proposed algorithm appears to be suitable for the analysis of data from random experiments, indicating that it has a large variety of possible applications in the clinical practice.

Introduction. Neurodevelopmental outcome studies of school age children who required extracorporeal membrane oxygenation (ECMO) as a newborn due to severe cardiorespiratory failure suggest that these children are at greater risk for learning disabilities and mental retardation than in the general population. In an attempt to further clarify the nature of potential learning disabilities of children who received ECMO, we assessed the neurodevelopmental status of our ECMO survivors at five years of age.

NEURODEVELOPMENTAL OUTCOME AT 3.5 YEARS OF AGE IN ECMO TREATED CHILDREN: RELATIONSHIP TO PRIMARY DIAGNOSIS † 1224

Little is known on the long term neurodevelopmental impact of a cardiopulmonary arrest on neonates with intractable cardiorespiratory failure who are being treated with ECMO. We previously reported on survival and neurodevelopmental outcome in a cohort of ECMO treated neonates matched by diagnosis for arrest (AG) and non-arrest (NAG) status. The current study was undertaken to correlate the cranial CT scans with the neurodevelopmental outcome in the survivors of this cohort. CT scans were performed on the AG at a median of 15 days of age, and on the NAG at a median of 12 days of age. In the AG, CT scans were done at a median of 13 days post arrest. The timing of arrests included 27 prior to ECMO (including 11 at cannulation, and 3 infants with multiple arrests) and 2 post ECMO. At follow up, age 12 to 42 months, the discharge CT scans of 29/30 AG (32±12 months, Mean±SD) and 31/35 NAG (33±11 months) children were reviewed by a neuroradiologist who was blinded to patient arrest and outcome status. Major findings on CT included:*p=0.049 In the AG with CT findings of low perfusion injury (i.e. vascular border zone necrosis), all arrests occurred at cannulation. The one NAG infant with a low perfusion injury had profound intrapartum asphyxia and fetal bradycardia. On neurodevelopmental follow up, 3 infants were abnormal and 1 was suspect. In infants with CT findings of an infarct (i.e. necrosis in a vascular distribution), neurodevelopmental outcome included 1 child whose exam was normal, 2 suspect, and 4 abnormal. In conclusion, neonates meeting the criteria for ECMO, but who have also had an arrest are not at an increased risk for low perfusion injury to the brain over their non-arrest cohorts. The increased incidence of intracranial infarcts in this group deserves continued neurodevelopmental study. All infants with major findings on CT following treatment with ECMO are at very high risk for significant neurodevelopmental problems and require close long term follow up.Table