Analytics and numerics of drug release tracking

The study of the dependence of fluxes, concentrations and response times, on the characteristic properties of drug delivery polymeric devices, plays an important role in the design of drug release platforms. The aim of this paper is to develop mathematical tools for an in-depth understanding of drug release tracking. The mathematical model presented takes into account the viscoelastic properties of the polymer and the state of the dispersed drug: free or chemically bound to the matrix. For nonlinear chemical bounds the process is described by a nonlinear integro-differential system and the drug release tracking is treated numerically. For linear chemical bounds closed formulas for the fluxes and response times are established in terms of the parameters that characterize the drug and the platform. These formulas provide a set of a priori estimations for the variables of the model. Numerical examples which show the effectiveness of the approach are included.     

Stability of solutions of chemotaxis equations in reinforced random walks

In this paper we consider a nonlinear system of differential equations consisting of one parabolic equation and one ordinary differential equation. The system arises in chemotaxis, a process whereby living organisms respond to chemical substance by moving toward higher, or lower, concentrations of the chemical substance, or by aggregating or dispersing. We prove that stationary solutions of the system are asymptotically stable.     

A path-transformation for random walks and the Robinson-Schensted correspondence

The author and Marc Yor recently introduced a path-transformation with the property that, for belonging to a certain class of random walks on , the transformed walk has the same law as the original walk conditioned never to exit the Weyl chamber . In this paper, we show that is closely related to the Robinson-Schensted algorithm, and use this connection to give a new proof of the above representation theorem. The new proof is valid for a larger class of random walks and yields additional information about the joint law of and . The corresponding results for the Brownian model are recovered by Donsker's theorem. These are connected with Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix theory. The connection we make between the path-transformation and the Robinson-Schensted algorithm also provides a new formula and interpretation for the latter. This can be used to study properties of the Robinson-Schensted algorithm and, moreover, extends easily to a continuous setting.

     

Multiparameter models for random walks in disordered lattice systems

Multidimensional asymptotically exactly solvable models are suggested for random walks in a stationary random lattice environment. These models differ from the well-known ones in that they involve arbitrarily many independent local random parameters per lattice site and allow for slowly decreasing intensities with increasing intersite distance. In particular, the suggested models describe the first nontrivial exactly solvable multidimensional systems with symmetrical interaction. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 2 pp. 332–344, May 1999.     

On the uniform asymptotic stability for a linear integro-differential equation of Volterra type

In this paper, we give a necessary and sufficient condition on the uniform asymptotic stability of the zero solution of a linear integro-differential equation of Volterra type where the ordinary part is ax(t). We put emphasis on the case a>0. The proofs of our results are carried out by using the root analysis of the characteristic equation. In Section 5 we give some conjectures.     

Reaction–diffusion in viscoelastic materials

In this paper we study initial boundary value problems that describe reaction–diffusion phenomena in viscoelastic materials. The mathematical model, represented by an integro-differential equation coupled with an ordinary differential equation, is analyzed from theoretical and numerical viewpoints.     

Wiener's test for space-time random walks and its applications

This paper establishes a criterion for whether a -dimensional random walk on the integer lattice visits a space-time subset infinitely often or not. It is a precise analogue of Wiener's test for regularity of a boundary point with respect to the classical Dirichlet problem. The test obtained is applied to strengthen the harder half of Kolmogorov's test for the random walk.

     

THE EFFECT OF MEMORY TERMS IN DIFFUSION PHENOMENA

In this paper the effect of integral memory terms in the behavior of diffusion phenomenais studied.The energy functional associated with different models is analyzed and stabilityinequalities are established.Approximation methods for the computation of the solutionof the integro-differentiaI equations are constructed.Numerical results are included.     

Continuous random dynamical systems

We study a dynamical system generalizing continuous iterated function systems and stochastic differential equations disturbed by Poisson noise. The main results provide us with sufficient conditions for the existence and uniqueness of an invariant measure for the considered system. Since the dynamical system is defined on an arbitrary Banach space (possibly infinite dimensional), to prove the existence of an invariant measure and its stability we make use of the lower bound technique developed by Lasota and Yorke and extended recently to infinite-dimensional spaces by Szarek. Finally, it is shown that many systems appearing in models of cell division or gene expressions are exactly as those we study. Hence we obtain their stability as well.     

Linear stability of general linear methods for systems of neutral delay differential equations

This paper is concerned with the numerical solution of delay differential equations (DDEs). We focus on the stability of general linear methods for systems of neutral DDEs with multiple delays. A type of interpolation procedure is considered for general linear methods. Linear stability properties of general linear methods with this interpolation procedure are investigated. Many extant results are unified.     

Range reliability in random walks

This paper introduces a definition of reliability based on a process range. Thus, process failure is defined when the range of a process first reaches a given and unacceptable level. The Mean Time To Failure (MTTF) which is denned as the mean of the first time for a range to attain a given amplitude is then calculated for an asymmetric random walk process. The probability distribution of the range is then given and the process reliability over long periods of system operations are then calculated. Applications such as the control of wings movements, stock price and exchange rates volatility (defined in terms of reliability) are also used to motivate the usefulness of range processes in reliability studies. Finally, we point out that there is necessarily a relationship between the range reliability and the propensity of a series to become chaotic.     

一类函数方程的稳定性及微商配置解

本文对一类重要的函数方程建立解稳定性定理,提出微商配置解,证明了收敛性.     

Scaling exponents of random walks in random sceneries

We compute the exact asymptotic normalizations of random walks in random sceneries, for various null recurrent random walks to the nearest neighbours, and for i.i.d., centered and square integrable random sceneries. In each case, the standard deviation grows like n with . Here, the value of the exponent is determined by the sole geometry of the underlying graph, as opposed to previous examples, where this value reflected mainly the integrability properties of the steps of the walk, or of the scenery. For discrete Bessel processes of dimension d[0;2[, the exponent is . For the simple walk on some specific graphs, whose volume grows like n^d for d[1;2[, the exponent is =1−d/4. We build a null recurrent walk, for which without logarithmic correction. Last, for the simple walk on a critical Galton–Watson tree, conditioned by its nonextinction, the annealed exponent is . In that setting and when the scenery is i.i.d. by levels, the same result holds with .     

Localization for branching random walks in random environment

We consider branching random walks in d$d$-dimensional integer lattice with time–space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If d≥3$d\ge 3$ and the environment is “not too random”, then, the total population grows as fast as its expectation with strictly positive probability. If, on the other hand, d≤2$d\le 2$, or the environment is “random enough”, then the total population grows strictly slower than its expectation almost surely. We show the equivalence between the slow population growth and a natural localization property in terms of “replica overlap”. We also prove a certain stronger localization property, whenever the total population grows strictly slower than its expectation almost surely.     

Branching random walks with random environments in time

We consider a branching random walk on N with a random environment in time （denoted by ξ）. Let Zn be the counting measure of particles of generation n, and let Zn（t） be its Laplace transform. We show the convergence of the free energy n-llog Zn（t）, large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn（t）/E[Zn（t）ξ].     

Asymptotic direction for random walks in random environments

In this paper we study the existence of an asymptotic direction for random walks in random i.i.d. environments (RWRE). We prove that if the set of directions where the walk is transient contains a non-empty open set, the walk admits an asymptotic direction. The main tool to obtain this result is the construction of a renewal structure with cones. We also prove that RWRE admits at most two opposite asymptotic directions.     

Completions of nonlinear DAE flows based on index reduction techniques and their stabilization

Differential algebraic equations (DAEs) define a differential equation on a manifold. A number of ways have been developed to numerically solve some classes of DAEs. Motivated by problems in control theory, numerical simulation, and the use of general purpose modeling environments, recent research has considered the embedding of the DAE solutions of a general DAE into the solutions of an ODE where the added dynamics have special properties. This paper both provides new results on the linear time-varying case and considers the important nonlinear case.     

Large deviations for random walks in a mixing random environment and other (non‐Markov) random walks

We extend a recent work by S. R. S. Varadhan [8] on large deviations for random walks in a product random environment to include more general random walks on the lattice. In particular, some reinforced random walks and several classes of random walks in Gibbs fields are included. © 2004 Wiley Periodicals, Inc.