Nonlinear dynamics and intermittency in a long-term copepod time series

We consider the nonlinear dynamics of a long-term copepod (small crustaceans) time series sampled weekly in the Mediterranean sea from 1967 to 1992. Such population dynamics display a high variability that we consider here in an interdisciplinary study, using tools borrowed from the field of statistical physics. We analyse the extreme events of male and female abundances, and of the total population, and show that they both have heavy tailed probability density functions (pdf). We provide hyperbolic fits of the form p(x) ∼ 1/x^μ+1, and estimate the value of μ using Hill’s estimator. We then study the ratio of male to female abundances, compared to the female abundances. Using conditional probability density functions and conditional averages, we show that this ratio is independent of the female density, when the latter is larger than a given threshold. This property is very useful for modelization. We also consider the product of male to female abundances, which can be ecologically related to the encounters. We show that this product is extremely intermittent, and link its pdf to the female pdf.     

On a muted family of bivariate distributions

This paper introduces ‘muted bivariate distribution (MBD)’ as a new family of bivariate distributions possessing the property (P) that the concomitant of nth record value on one variable arising from any distribution of this family is distributed identically as the largest order statistic of a sample of size n arising from the marginal distribution of the same variable. Also if the property P is observed on any distribution, then we illustrate a process of impounding the probability density function (pdf) of the nth record value on one variable resulting due to P with the known form of the marginal pdf on the other variable to build up an appropriate MBD of the parent population. Some properties of the newly defined MBD are further derived. We identify the forms of the pdf of the concomitant of generalized upper record value and the pdf of the concomitant of the smallest order statistic of a sample of size n in order that each of the above pdf’s again can be used to generate MBD of the parent population, alternatively. We further observe MBD as a suitable model for describing an astronomical bivariate dataset relating to the galaxy luminosity measurements made under two conditions on globular clusters. For illustrative purposes, a real-life dataset on acacia trees obtained via concomitant record ranked set sampling methodology is considered as an application of this new family of bivariate distributions.     

Non-differentiability of alpha function at the boundary of flat

With the variational method introduced by Mather,we construct a mechanical Hamiltonian system whose α function has a flat F and is non-differentiable at the boundary F.In the case of two degrees of freedom,we prove that this phenomenon is stable under perturbations of Maé's.     

An application of Taylor series in the approximation of solutions to stochastic differential equations with time-dependent delay

The subject of this paper is the analytic approximation method for solving stochastic differential equations with time-dependent delay. Approximate equations are defined on equidistant partitions of the time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. It will be shown, without making any restrictive assumption for the delay function, that the approximate solutions converge in L^p-norm and with probability 1 to the solution of the initial equation. Also, the rate of the L^p convergence increases when the degrees in the Taylor approximations increase, analogously to what is found in real analysis. At the end, a procedure will be presented which allows the application of this method, with the assumption of continuity of the delay function.     

Anisotropic mesh refinement for finite element methods based on error reduction

In this paper, we propose an anisotropic adaptive refinement algorithm based on the finite element methods for the numerical solution of partial differential equations. In 2-D, for a given triangular grid and finite element approximating space V, we obtain information on location and direction of refinement by estimating the reduction of the error if a single degree of freedom is added to V. For our model problem the algorithm fits highly stretched triangles along an interior layer, reducing the number of degrees of freedom that a standard h-type isotropic refinement algorithm would use.     

Probability density functions of some skew tent maps

We consider a family of chaotic skew tent maps. The skew tent map is a two-parameter, piecewise-linear, weakly-unimodal, map of the interval F_a,b. We show that F_a,b is Markov for a dense set of parameters in the chaotic region, and we exactly find the probability density function (pdf), for any of these maps. It is well known (Boyarsky A, Góra P. Laws of chaos: invariant measures and dynamical systems in one dimension. Boston: Birkhauser, 1997), that when a sequence of transformations has a uniform limit F, and the corresponding sequence of invariant pdfs has a weak limit, then that invariant pdf must be F invariant. However, we show in the case of a family of skew tent maps that not only does a suitable sequence of convergent sequence exist, but they can be constructed entirely within the family of skew tent maps. Furthermore, such a sequence can be found amongst the set of Markov transformations, for which pdfs are easily and exactly calculated. We then apply these results to exactly integrate Lyapunov exponents.     

Simulated annealing with a potential function with discontinuous gradient on Rd

In this paper, we have proven that the simulated annealing processdX _t = ?β(t) ?U (X _t ) + √2dW _tt with a potential function on R^d, of which the gradient is discontinuous, converges in probability to a neighborhood of the global minima of the potential function.     

Estimating the parameters of a truncated normal distribution

Let ø(x) be a truncated normal pdf over the interval [a,b], that is, assume ø(x)=exp[-(x–μ)²/2σ²]/∝^b_aexp[-(x–μ)²/2σ²]dx for - ∞<a?x?b?< + ∞ and zero elsewhere. Suppose that X₁,X₂,…,X_n is a random sample of size n from this truncated distribution. Using known properties of exponential families of distributions and the system of Legendre polynomials over the interval [-1,1], we examine the maximum likelihood estimation of the parameters μ and σ².     

A family of minimax estimators of a multivariate normal mean

Let X be a p-dimensional normal random vector with unknown mean vector θ and covariance σ²I. Let S/σ², independent of X, be chi-square with n degrees of freedom. Relative to the squared error loss, James and Stein (1961) have obtained an estimator which dominates the usual estimator X. Baranchik (1970) has extended James and Stein's results. We obtain a theorem which can provide a different family of minimax estimators containing James-Stein's estimator. Two interesting minimax estimators are presented in this paper.     

A One-Dimensional Optimization Algorithm and Its Convergence Rate under the Wiener Measure

In this paper we describe an adaptive algorithm for approximating the global minimum of a continuous function on the unit interval, motivated by viewing the function as a sample path of a Wiener process. It operates by choosing the next observation point to maximize the probability that the objective function has a value at that point lower than an adaptively chosen threshold. The error converges to zero for any continuous function. Under the Wiener measure, the error converges to zero at rate e^−nδ_n, where {δ_n} (a parameter of the algorithm) is a positive sequence converging to zero at an arbitrarily slow rate.     

Further improved recursions for a class of compound Poisson distributions

In the present paper we develop more efficient recursive formulae for the evaluation of the t-order cumulative function Γ^th(x) and the t-order tail probability Λ^th(x) of the class of compound Poisson distributions in the case where the derivative of the probability generating function of the claim amounts can be written as a ratio of two polynomials. These efficient recursions can be applied for the exact evaluation of the probability function (given by De Pril [De Pril, N., 1986a. Improved recursions for some compound Poisson distributions. Insurance Math. Econom. 5, 129-132]), distribution function, tail probability, stop-loss premiums and t-order moments of stop-loss transforms of compound Poisson distributions. Also, efficient recursive algorithms are given for the evaluation of higher-order moments and r-order factorial moments about any point for this class of compound Poisson distributions. Finally, several examples of discrete claim size distributions belonging to this class are also given.     

Lattices on simplicial partitions

In this paper, (d+1)-pencil lattices on simplicial partitions in R^d are studied. The barycentric approach naturally extends the lattice from a simplex to a simplicial partition, providing a continuous piecewise polynomial interpolant over the extended lattice. The number of degrees of freedom is equal to the number of vertices of the simplicial partition. The constructive proof of this fact leads to an efficient computer algorithm for the design of a lattice.     

Approximation with the output of linear control systems

In this paper we give a new proof that for controllable and observable linear systems every L²[0,T] function can be approximated in the L²[0,T] sense with an output function generated by an L²[0,T] input function. We also give a new characterization of how continuous functions on [0,T] are uniformly approximated by an output generated by a continuous input function. The relative degree of the transfer function of the system determines those functions that can be approximated. We further show that if the initial data is allowed to vary then every continuous function is uniformly approximated by outputs generated by continuous functions.     

The possible number of islands on the sea

Let a height function f be a real valued function on R². A connected subset of R² is called an island, if there is a water level such that H is an island in the classical sense. We show that an island system is always laminar. Among others, in this paper we prove that the cardinality of a maximal laminar system is either countable or continuum.     

Implicit Sampling, with Application to Data Assimilation

There are many computational tasks, in which it is necessary to sample a given probability density function (or pdf for short), i.e., to use a computer to construct a sequence of independent random vectors x i (i = 1, 2, ··· ), whose histogram converges to the given pdf. This can be difficult because the sample space can be huge, and more importantly, because the portion of the space, where the density is significant, can be very small, so that one may miss it by an ill-designed sampling scheme. Indeed, Markovchain Monte Carlo, the most widely used sampling scheme, can be thought of as a search algorithm, where one starts at an arbitrary point and one advances step-by-step towards the high probability region of the space. This can be expensive, in particular because one is typically interested in independent samples, while the chain has a memory. The authors present an alternative, in which samples are found by solving an algebraic equation with a random right-hand side rather than by following a chain; each sample is independent of the previous samples. The construction in the context of numerical integration is explained, and then it is applied to data assimilation.     

Subsonic irrotational flows in a two-dimensional finitely long curved nozzle

This is a continuation of our previous paper Du et al. (http://www.ims.cuhk.edu.hk/publications/reports/2012-06.pdf), where we have characterized a set of physical boundary conditions that ensures the existence and uniqueness of subsonic irrotational flow in a flat nozzle. In this paper, we will investigate the influence of the incoming flow angle and the geometry structure of the nozzle walls on subsonic flows in a finitely long curved nozzle. It turns out to be interesting that the incoming flow angle and the angle of inclination of nozzle walls play the same role as the end pressure for the stabilization of subsonic flows. In other words, the L ² and L ^∞ bounds of the derivative of these two quantities cannot be too large, similar as we have indicated in Du et al. (http://www.ims.cuhk.edu.hk/publications/reports/2012-06.pdf) for the end pressure. The curvatures of the nozzle walls will also play an important role in the stability of the subsonic flow.     

Viability with Probabilistic Knowledge of Initial Condition, Application to Optimal Control

In this paper we provide an extension of the Viability and Invariance Theorems in the Wasserstein metric space of probability measures with finite quadratic moments in ? ^d for controlled systems of which the dynamic is bounded and Lipschitz. Then we characterize the viability and invariance kernels as the largest viability (resp. invariance) domains. As application of our result we consider an optimal control problem of Mayer type with lower semicontinuous cost function for the same controlled system with uncertainty on the initial state modeled by a probability measure. Following Frankowska, we prove using the epigraphical viability approach that the value function is the unique lower semicontinuous proximal episolution of a suitable Hamilton Jacobi equation.     

Probability that the commutator of two group elements is equal to a given element

In this paper we study the probability that the commutator of two randomly chosen elements in a finite group is equal to a given element of that group. Explicit computations are obtained for groups G which |G^′| is prime and G^′≤Z(G) as well as for groups G which |G^′| is prime and G^′∩Z(G)=1. This paper extends results of Rusin [see D.J. Rusin, What is the probability that two elements of a finite group commute? Pacific J. Math. 82 (1) (1979) 237-247].     

Uniformly convergent H(div)-conforming rectangular elements for Darcy-Stokes problem

We consider a singular perturbation problem which describes 2D Darcy-Stokes flow.An H(div)-conforming rectangular element,DS-R14,is proposed and analyzed frst.This element has 14 degrees of freedom for velocity and is proved to be uniformly convergent with respect to perturbation constant.We then simplify this element to get another H(div)-conforming rectangular element,DS-R12,which has 12 degrees of freedom for velocity.The uniform convergence is also obtained for this element.Finally,we construct a de Rham complex corresponding to DS-R12 element.