## Conference Publications

2009

Select all articles

Export/Reference:

*+*[Abstract](814)

*+*[PDF](172.9KB)

**Abstract:**

The existence of energy solutions to the Cauchy-Neumann problem for the porous medium equation of the form $v_t - \Delta (|v|^{m-2}v) = \alpha v$ with $m \geq 2$ and $\alpha \in \mathbb{R}$ is proved, by reducing the equation to an evolution equation involving two subdifferential operators and exploiting subdifferential calculus recently developed by the author.

*+*[Abstract](816)

*+*[PDF](197.0KB)

**Abstract:**

In this paper we consider the global existence and the asymptotic behavior of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and damping $$ \left\{\begin{array}{l} \psi_t = -(1-\alpha) \psi - \theta_x + \alpha \psi_{x x} + \psi\psi_x, (E)\\ \theta_t = -(1-\alpha)\theta + \nu \psi_x + 2\psi\theta_x + \alpha \theta_{x x}, \end{array} \right. $$ with initial data converging to different constant states at infinity $$(\psi,\theta)(x,0)=(\psi_0(x), \theta_0(x)) \rightarrow (\psi_{\pm}, \theta_{\pm}) \ \ {as} \ \ x \rightarrow \pm \infty, (I) $$ where $\alpha$ and $\nu$ are positive constants such that $\alpha <1$, $\nu <4\alpha(1-\alpha)$. Under the assumption that $|\psi_+ - \psi_- |+| \theta_+ - \theta_-|$ is sufficiently small, we show that if the initial data is a small perturbation of the convection-diffusion waves defined by (11) which are obtained by the parabolic system (9), solutions to Cauchy problem (E) and (I) tend asymptotically to those convection-diffusion waves with exponential rates. We mainly propose a better asymptotic profile than that in the previous work by [13,3], and derive its decay rates by weighted energy method instead of considering the linearized structure as in [3].

*+*[Abstract](769)

*+*[PDF](169.3KB)

**Abstract:**

This paper characterizes the existence of coexistence states for a class of sublinear elliptic cooperative systems with a linear equation and the other non-linear, but yet linear on a subdomain of the underlying domain. The analysis of this problem is imperative for ascertaining the dynamics of wider general classes of cooperative systems with spatially heterogeneous nonlinearities, like those introduced in López-Gómez & Molina-Meyer [10] for weakly coupled systems. Our characterization relies upon a

*spectral bound*associated to a certain non-local second order differential operator, which has not been previously documented in the literature.

*+*[Abstract](933)

*+*[PDF](223.5KB)

**Abstract:**

This work deals with the relationship between a continuous dynamical system and numerical methods for its computer simulations, viewed as discrete dynamical systems. The term 'dynamic consistency' of a numerical scheme with the associated continuous system is usually loosely defined, meaning that the numerical solutions replicate some of the properties of the solutions of the continuous system. Here, this concept is replaced with

*topological dynamic consistency*, which is defined in precise terms through the topological equivalence of maps. This ensures that

*all*the topological properties (e.g., fixed points and their stability, periodic solutions, invariant sets, etc.) are preserved. Two examples are provided which demonstrate that numerical schemes satisfying this strong notion of dynamic consistency can be constructed using the nonstandard finite difference method.

*+*[Abstract](711)

*+*[PDF](159.7KB)

**Abstract:**

We develop and implement an efficient algorithm to estimate the 5 parameters of Heston's model from arbitrary given series of joint observations for the stock price and volatility. We consider the time interval T separating two observations to be unknown and estimate it from the data, thereby estimating 6 parameters with a clear gain in fit accuracy. We compare the maximum likelihood parameter estimates based on an Euler discretization scheme to analogous estimates derived from the more accurate Milstein discretization scheme; we derive explicit conditions under which the two set of estimates are asymptotically equivalent, and we compute the asymptotic distribution of the difference of the two set of estimates. We show that parameter estimates derived from the Euler scheme by constrained optimization of the approximate maximum likelihood are consistent, and we compute their asymptotic variances. Numerically, our estimation algorithms are easy to implement,and require only very moderate amounts of CPU. We have performed extensive simulations which show that for standard range of the process parameters, the empirical variances of our parameter estimates are correctly approximated by their theoretical asymptotic variances.

*+*[Abstract](599)

*+*[PDF](133.3KB)

**Abstract:**

In the paper a system of differential equations $y_i^' = f_i(t, y_1,..., y_{n-1}) g_i(y_n)$, $i=1,..., n$ is studied. Sufficient (necessary) conditions for the existence of a solution $y$ fulfilling $lim_{t\to \tau_-} y_i(t)= C_i$, $i=1,2,..., n-1$, $lim_{t\to\tau_-}|y_n(t)|=\infty $ are derived where $\tau<\infty $ and $C_i \in \mathbb{R}$ are given.

*+*[Abstract](820)

*+*[PDF](255.9KB)

**Abstract:**

In this paper we show existence of finite energy solutions for the Cauchy problem associated with a semilinear wave equation with interior damping and supercritical source terms. The main contribution consists in dealing with super-supercritical source terms (terms of the order of $|u|^p$ with $p\geq 5$ in $n=3$ dimensions), an open and highly recognized problem in the literature on nonlinear wave equations.

*+*[Abstract](788)

*+*[PDF](160.2KB)

**Abstract:**

It is proved the existence of infinitely many solutions to a superquadratic Dirac-type boundary value problem of the form $\tau z = \nabla_z F(t,z)$, $y(0) = y(\pi) = 0$ ($z=(x,y)\in \mathbb{R}^2 $). Solutions are distinguished by using the concept of rotation number. The proof is performed by a global bifurcation technique.

*+*[Abstract](726)

*+*[PDF](165.1KB)

**Abstract:**

In this paper we investigate a class of parabolic inclusions with a nonlocal condition of integral type. We provide sufficient conditions that guarantee the existence of at least one solution. Our technique is based on Green's function for linear parabolic partial differential equations and fixed point theorems for multivalued maps.

*+*[Abstract](668)

*+*[PDF](162.6KB)

**Abstract:**

We consider a predator-prey system with logistic-type growth and linear diffusion for the prey, Holling type II functional response and the nonlinear diffusion $\nabla \left( \sigma n b \nabla b)$ for the predator, where $n$ is the prey (nutrient) and $b$ is the predator (bacteria) density, respectively. This corresponds to a collective-type behavior for predators: they spread faster when numerous enough at a front line. We present the complete linear stability analysis for this case, discuss some results of numerical simulations: the asymptotic behavior of the model (with the zero Neumann boundary conditions in a 2-D domain) was similar to the relevant Lotka-Volterra system of ordinary differential equations.

*+*[Abstract](779)

*+*[PDF](235.4KB)

**Abstract:**

In this article, we present a new continuous model for tumor growth. This model describes the evolution of three components: sane tissue, cancer cells and extracellular medium. In order to render correctly the cellular division, this model uses a discrete description of the cell cycle (the set of steps a cell has to undergo in order to divide). To account for cellular adhesion and the mechanics which may influence the growth, we assume a viscoelastic mechanical behavior. This model extends the one presented in [18] with a more realistic description of the forces that drive the movement.

*+*[Abstract](778)

*+*[PDF](426.7KB)

**Abstract:**

The one-dimensional Swift-Hohenberg equation is known to exhibit a variety of localized states within the so-called pinning or snaking region. Single-pulse states consist of single localized structures within the spatial domain, and are organized into a snakes-and-ladders structure within the pinning region. Multipulse states consist of two or more localized structures within the domain, but their detailed organization within the pinning region is not known. In this paper we consider multipulse solutions of the one-dimensional Swift-Hohenberg equation on large but periodic domains, and show that while these are also confined to the pinning region the details of their organization depend on whether the pulses are equidistant or not. For large domains the required branch-following becomes delicate and may lead to erroneous results unless performed with great care.

*+*[Abstract](701)

*+*[PDF](115.3KB)

**Abstract:**

We prove the solvability of the following boundary value problem on the real line

$\Phi(u'(t))'=f(t,u(t),u'(t))$ on $\mathbb{R}$,

$u(-\infty)=-1,$ $u(+\infty)=1,$

with a singular $\Phi$-Laplacian operator.

We assume $f$ to
be a continuous function that satisfies suitable symmetry
conditions. Moreover some growth conditions in a neighborhood of
zero are imposed.

*+*[Abstract](755)

*+*[PDF](244.9KB)

**Abstract:**

The modeling of the microphysics of a population of atmospheric particles interacting through a common medium leads to the solution of a large system of weakly coupled differential-algebraic equations. An implicit time discretization of the system of differential-algebraic equations is solved with a Newton method at each time step. The structure of the global system and the sparsity of the Newton matrix allow the efficient use of a Schur complement approach for the decoupling of the various subsystems at the discrete level. A numerical approach for the decomposition of the population into fast and slow subsystems is proposed. Numerical results are presented for organic atmospheric particles to illustrate the properties of the method.

*+*[Abstract](708)

*+*[PDF](174.4KB)

**Abstract:**

Since Palais' pioneer paper in 1963, Condition $(C)$ in both the Palais--Smale version and Cerami's variant has been widely used in order to prove minimax existence theorems for $C^1$ functionals in Banach spaces. Here, we introduce a weaker version of these conditions so that a Deformation Lemma still holds and some critical points theorems can be stated. Such abstract results apply to $p$--Laplacian type elliptic problems.

*+*[Abstract](828)

*+*[PDF](151.7KB)

**Abstract:**

We show that the heat equation modelled as $y'=Ay+u$ in $L^1(\Omega)$ is null controllable by controls in $L^{\infty}(0,T;L^p(\Omega))$ with $1< p<\infty$. Moreover, the corresponding minimal time function is Hölder continuous.

*+*[Abstract](914)

*+*[PDF](4502.6KB)

**Abstract:**

The aim of this paper is to numerically investigate multiple solutions of semilinear elliptic systems with zero Dirichlet boundary conditions

-$\Delta u=F_u(x;u,v),$ $x\in\Omega,

-$\Delta v=F_v(x;u,v),$ $x\in\Omega,

where $\Omega \subset \mathbb{R}^{N}$ ($N\ge 1$) is a bounded domain. A strongly coupled case where the potential $F(x;u,v)$ takes the form $|u|^{\alpha_1}|v|^{\alpha_2}$ with $\alpha_1, \alpha_2>1$ is specially studied. By using a local min-orthogonal method, both positive and sign-changing solutions are found and displayed.

*+*[Abstract](717)

*+*[PDF](178.2KB)

**Abstract:**

In this paper we define $u_\infty$-quasisimilarity in order to unify $t_\infty$-quasisimilarity and $n_\infty$-quasisimilarity and then study the stability for solutions of linear dynamic equations on time scales by using the concept of $u_\infty$-quasisimilarity and dynamic inequality.

*+*[Abstract](841)

*+*[PDF](214.7KB)

**Abstract:**

We study the dynamics of the sine-Gordon equation's kink soliton solutions under the coarse-grain description via two "collective variables": the position of the "center" of a soliton and its characteristic width ("size"). Integral expressions for the interaction potential and the quasi-particles' cross-masses are derived. However, these cannot be evaluated in closed form when the solitons have varying widths, so we develop a perturbation approach with the velocity of the faster soliton as the small parameter. This enables us to derive a system of four coupled second-order ODEs, one for each collective variable. The resulting initial-value problem is very stiff and numerical instabilities make it difficult to solve accurately, so a semi-empirical iterative approach to its solution is proposed. Then, we demonstrate that, even though it appears the solitons pass through each other, the quasi-particles actually "exchange" their pseudomasses during a collision.

*+*[Abstract](570)

*+*[PDF](253.3KB)

**Abstract:**

We consider $n$ tubes exiting a junction and filled with a non viscous isentropic or isothermal fluid. In each tube a copy of the $p$-system in Euler coordinates is considered. The aim of the presentation is to compare three different notions of solutions at the junctions: p-solutions, Q-solutions and P-solutions.

*+*[Abstract](661)

*+*[PDF](135.7KB)

**Abstract:**

We present various models in classical mechanics which exhibit 'exotic' orbits. We give an example of a smooth $|\mathbf{r}|$-independent potential

$V$ in dimension three, which exhibits an orbit that spirals as time goes to infinity. This kind of orbits cannot occur for this class of potentials in dimension two [4] or, see below, if ${Cr}=\{\omega\in S^{n-1}:\nabla V(\omega)=0\}$, $n\geq 3$, is totally disconnected. In addition, for each $\mu>2$ we give an example of a potential of the form $V(r,\theta)=O(r^{-\mu})$, in two dimensions, which is not radially symmetric and has a zero-energy orbit that escapes towards infinity in spirals. Zero energy orbits escaping towards infinity in spirals cannot occur for radial potentials with the same rate of decay.

*+*[Abstract](686)

*+*[PDF](178.6KB)

**Abstract:**

We consider a variation of the Kuramoto-Sivashinsky Equation in space dimension two. We show that under some assumptions the equation is globally well-posed and posseses a global attractor in the periodic case. The analysis is based on the Lyapunov function approach, point dissipativeness and asymptotic compactness.

*+*[Abstract](788)

*+*[PDF](225.9KB)

**Abstract:**

In the framework of bidifferential graded algebras, we present universal solution generating techniques for a wide class of integrable systems.

*+*[Abstract](854)

*+*[PDF](177.9KB)

**Abstract:**

Here we discuss a geometric integrator for nonholonomic mechanical systems that preserves the nonholonomic constraints, the discrete nonholonomic momentum map, and is also energy-preserving in some important cases. This method does not require a predefined discretization of the nonholonomic constraints. In Euclidean space, it yields a generalization of the classical SHAKE and RATTLE algorithms to the nonholonomic setting. This article shows that the method is second order convergent.

*+*[Abstract](642)

*+*[PDF](161.8KB)

**Abstract:**

The onset of a typical bacterial growth curve shows a period of very slow increase in population counts. This is a period of physiological adaptation to new environmental conditions. While in mathematical biology much progress was made in recent years to describe physiologically structured populations, these models typically have too many degrees of freedom to easily allow a model identification against experimental data. Therefore, and for all practical purposes, microbiologists have proposed simpler models of physiological adaptation in the past, usually in connection with standard growth curves. In this paper we compare the performance of four such lag-time models, each of which described by a scalar differential equation, when combined with a model of a siderophore producing bacterial population under iron limitation. In each case this yields a system of five nonlinear ordinary differential equations that we compare against experimental data, by solving the associated vector optimization problem. Our main finding is that a big step in accuracy is made already by including a simple lag-time model that only introduces one additional degree of freedom in the parameter identification problem (the initial state of health of the population), and that this can be reliably improved if a further degree of freedom, describing the dynamics of the physiological recovery process, is included. The vector optimization problem is solved by scalarizing it with a linear functional and solving the resulting scalar optimization problem. The growth parameters that are identified in this procedure are found to be robust with respect to the scalarization coefficient.

*+*[Abstract](766)

*+*[PDF](173.2KB)

**Abstract:**

Obstacle problems, mathematical models of some nonlinear phenomena accompanying a free boundary, have been well studied. In this paper, the existence and uniqueness of a system between the obstacle problem and the Navier-Stokes equations is considered. The abstract theory for evolution equations governed by a subdifferential of the indicator functional on a time-dependent, closed, and convex set is applied to show the main theorem. $L^\infty $-estimate is an important lemma to prove the existence theorem.

*+*[Abstract](901)

*+*[PDF](147.9KB)

**Abstract:**

We present an iterative scheme for solving inclusions of the form $f(x) + F(x) \ni 0$ where $f$ is a Lipschitz continuous function admitting a first order divided difference while $F$ stands for a set-valued mapping, both of them acting between Banach spaces. We prove the convergence of our method under several regularity properties for $F$ and without any differentiability assumption on $f$. We investigate, subsequently, the case when the mapping $F$ is metrically regular, strongly metrically regular and strongly metrically subregular.

*+*[Abstract](667)

*+*[PDF](220.5KB)

**Abstract:**

The basin of attraction of an equilibrium or periodic orbit can be determined by sublevel sets of a Lyapunov function. A Lyapunov function is a function with negative orbital derivative, which is defined by $LV(t,x)= {\nabla_x V(t,x),f(t,x)} + \partial_t V(t,x)$. We construct a Lyapunov function by approximately solving a Cauchy problem with a linear PDE for its orbital derivative and boundary conditions on a non-characteristic hypersurface. For the approximation we use meshless collocation. We describe the general approximate reconstruction of multivariate functions, which are periodic in one variable, from discrete data sets and derive error estimates. This method has already been applied to autonomous dynamical systems. In this paper, however, we consider a time-periodic ODE $\dot{x}=f(t,x)$, $x\in \mathbb R^n$, and study the basin of attraction of an exponentially asymptotically stable periodic orbit.

*+*[Abstract](871)

*+*[PDF](124.9KB)

**Abstract:**

We consider a three-point boundary value problem for the beam equation. Some

*a priori*estimates to the positive solutions for the boundary value problem are obtained. Sufficient conditions for the existence and nonexistence of positive solutions for the boundary value problem are established. The results are illustrated with an example.

*+*[Abstract](695)

*+*[PDF](146.3KB)

**Abstract:**

The authors study a higher order three point boundary value problem. Estimates for positive solutions are given; these estimates improve some recent results in the literature. Using these estimates, new sufficient conditions for the existence and nonexistence of positive solutions of the problem are obtained. An example illustrating the results is included.

*+*[Abstract](782)

*+*[PDF](224.0KB)

**Abstract:**

We start off this paper with a brief introduction to modeling Human Immunodeficiency Virus (HIV) and Acquired Immune Deficiency Syndrome (AIDS) amongst sharing, injecting drug users (IDUs). Then we describe the mathematical model which we shall use which extends an existing model of the spread of HIV and AIDS amongst IDUs by incorporating loss of HIV infectivity over time. This is followed by the derivation of a key epidemiological parameter, the basic reproduction number $sf(R)_0$. Next we give some analytical equilibrium, local and global stability results. We show that if $sf(R)_0 \le 1$ then the disease will always die out. For $sf(R)_0 > 1$ there is the disease-free equilibrium (DFE) and a unique endemic equilibrium. The DFE is unstable. An approximation argument shows that we expect the endemic equilibrium to be locally stable. We next discuss a more realistic version of the model, relaxing the assumption that the number of addicts remains constant and obtain some results for this model. The subsequent section gives simulations for both models confirming that if $sf(R)_0 \le 1$ then the disease will die out and if $sf(R)_0 > 1$ then if it is initially present the disease will tend to the unique endemic equilibrium. The simulation results are compared with the original model with no loss of HIV infectivity. Next the implications of these results for control strategies are considered. A brief summary concludes the paper.

*+*[Abstract](724)

*+*[PDF](205.7KB)

**Abstract:**

A two-dimensional microeconomic model with three bounded controls is created and investigated. The model describes a manufacturer producing a consumer good and a retailer that buys this product in order to resell it for a profit. Two types of differential hierarchical games will be applied in order to model the interactions between the manufacturer and retailer. We will consider the difficult case in which the maximum of the objective functions can be reached only on the boundary of the admissible set. Optimal strategies for manufacturer and retailer in both games will be found. The object of our interest is the investigation of the vertical integration of retail and industrial groups. We will determine the conditions of interaction that produce a stable and maximally effective structure over given planning periods.

*+*[Abstract](735)

*+*[PDF](117.1KB)

**Abstract:**

In order to obtain the existence of periodic and almost periodic solutions of Volterra difference equation: $ x(n+1)=f(n,x(n))+\sum_{s=-\infty}^{n}F(n,s,x(n+s),x(n)) $, we consider certain two stability properties, which are referred to as (K, $ \rho $)-weakly uniformly asymptotically stable and (K, $ \rho $)-uniformly asymptotically stable.

*+*[Abstract](1016)

*+*[PDF](162.8KB)

**Abstract:**

In this paper we are concerned with the convergence analysis of splitting methods for nonautonomous abstract evolution equations. We introduce a framework that allows us to analyze the popular Lie, Peaceman--Rachford and Strang splittings for time dependent operators. Our framework is in particular suited for analyzing dimension splittings. The influence of boundary conditions is discussed.

*+*[Abstract](639)

*+*[PDF](138.0KB)

**Abstract:**

The following elliptic equation with nonlinear boundary condition is considered: $-\Delta u+bu=f(x)$ in $\Omega$, $-\frac{\partial u}{\partial n}=\beta(u)-g(u)$ on $\partial\Omega$, where $b\geq0$, $f\in L^2(\Omega)$, $\beta(u)$ is a monotone increasing function on $\mathbb (R)^1$ and $g(u)$ is its small perturbation. It is shown that this problem admits a solution $u$ belonging to $H^2(\Omega)$ under suitable conditions on $\beta$ and $g$. The method of our proof relies on some approximation procedures and the classical but new arguments for $H^2$-estimates near the boundary which can work under (non-monotone) nonlinear boundary conditions.

*+*[Abstract](745)

*+*[PDF](149.8KB)

**Abstract:**

Nonoscillatory solutions of a general class of second order functional neutral differential equations of the form

$(r(t)(x(t)+p(t)x(t-\tau))')'+f(t,x(\sigma_{1}(t)),x(\sigma_{2}(t)),...,x(\sigma_{n}(t)))=0$

have been classified in accordance with their asymptotic behavior.

*+*[Abstract](820)

*+*[PDF](163.1KB)

**Abstract:**

In this paper, we are concerned with the following quasilinear elliptic equations:

-div${a(x)|\nabla u|^{p-2}\nabla u$ $=b(x)|u|^(q-2)u $ in $\Omega$

$u(x)$ $= 0$
on $\partial\Omega$

where $\Omega$ is a domain in $\mathbf R^N$ $(N \ge 1)$ with smooth
boundary.

When $a$ and $b$ are positive constants, there are many results on the
nonexistence of nontrivial solutions for the equation (E).
The main purpose of
this paper is to discuss the nonexistence results for
(E) with a class of weak solutions under some assumptions on $a$ and $b$.

*+*[Abstract](650)

*+*[PDF](146.1KB)

**Abstract:**

We consider a coupled system of Hamiltonian partial differential equations introduced by Popowicz, which has the appearance of a two-field coupling between the Camassa-Holm and Degasperis-Procesi equations. The latter equations are both known to be integrable, and admit peaked soliton (peakon) solutions with discontinuous derivatives at the peaks. A combination of a reciprocal transformation with Painlevé analysis provides strong evidence that the Popowicz system is non-integrable. Nevertheless, we are able to construct exact travelling wave solutions in terms of an elliptic integral, together with a degenerate travelling wave corresponding to a single peakon. We also describe the dynamics of $N$-peakon solutions, which is given in terms of an Hamiltonian system on a phase space of dimension $3N$.

*+*[Abstract](1014)

*+*[PDF](183.2KB)

**Abstract:**

We consider a delayed neural network model with diffusion. By analyzing the distributions of the eigenvalues of the system and applying the center manifold theory and normal form computation, we show that, regarding the connection coefficients as the perturbation parameter, the system, with the different boundary conditions, undergoes some bifurcations including transcritical bifurcation, Hopf bifurcation and Hopf-zero bifurcation. The normal forms are given to determine the stabilities of the bifurcated solutions.

*+*[Abstract](715)

*+*[PDF](150.4KB)

**Abstract:**

In this note we discuss the existence of positive solutions for some nonlocal boundary value problem where the boundary conditions involve linear functionals on the space $C[0,1]$ and the involved nonlinearity might be singular. Our main ingredient is the theory of fixed point index.

*+*[Abstract](711)

*+*[PDF](163.2KB)

**Abstract:**

Existence, stability, and shape of periodic solutions are derived for the differential-difference equation $\varepsilon\dot x(t)+x(t)=f(x([t-1])), 0<\varepsilon\<\<1,$ where $[\cdot]$ is the integer part function. The equation can be viewed as a special discretization (discrete version) of the singularly perturbed differential delay equation $\varepsilon\dot x(t)+x(t)=f(x(t-1))$. The principal analysis is based on reduction to the two-dimensional map $F: (u,v)\to (v, f(u)+ [v-f(u)]e^{-1/\varepsilon}),$ many relevant properties of which follow from those of the one-dimensional map $f$.

*+*[Abstract](846)

*+*[PDF](157.0KB)

**Abstract:**

In this paper, we present a framework of 4D variational data assimilation (4D-Var) in Hilbert spaces and discuss Marchuk-Strang operator splitting methods for 4D-Var. Convergence analysis of the operator splitting methods is made.

*+*[Abstract](915)

*+*[PDF](828.4KB)

**Abstract:**

'No man is an island' [John Donne]. Human and technological networks play a vital part in our lives, and their failures have often caused severe adverse consequences. In this paper we address this crucial issue by presenting a model to prevent not only network failures but also their propagation to the remaining network elements. Our model forecasts the number of packets each node is able to service without becoming overloaded, by determining the transition probabilities assigned to each link. Thus, our model ensures that nodes receive as many packets as their network resources prescribe. The model is portable to any type of topology and is based on Ordinary Differential Equations (ODEs), which are numerically solved as a multivariable, coupled system, over a variety of topologies. Our numerical algorithm is based on the classic Runge--Kutta 4th order, which is adjusted to integrate graph principles.

*+*[Abstract](875)

*+*[PDF](137.9KB)

**Abstract:**

We give sufficient conditions on the value $\tau \in (0, T]$ such that the nonlinear fractional boundary value problem

$\D_0^\alpha + u(t) + f(t, u(t)) = 0,$ $t \in (0, \tau),$

$I^\gamma u(0^+) = 0,$ $I^\beta u(\tau) = 0,$

where $1 - \alpha < \gamma \leq 2 - \alpha,$ $2 - \alpha < \beta < 0$, $\D_(0+)^\alpha$ is the Riemann-Liouville differential operator of order $\alpha $, and $f \in C([0,T] \times \mathbb{R})$ is nonnegative, has a positive solution. We also present a nonexistence result.

*+*[Abstract](918)

*+*[PDF](131.6KB)

**Abstract:**

In this work we consider the dynamical response of a non-linear beam with viscous damping, perturbed in both the transverse and axial directions interacting with a potential flow. In particular we show that for a class of boundary conditions (clamped beam) and given inlet velocity flow for the fluid, there exists appropriate energy norm for the parameters of the beam (displacements) and flow (potential) bounded by the inlet boundary condition for the fluid flow.

*+*[Abstract](816)

*+*[PDF](407.8KB)

**Abstract:**

To describe sampling - reconstruction procedure (SRP) of Markov processes the conditional mean rule is used. There are two types of stochastic differential equations under consideration: 1) linear with varying in time coefficients; 2) non linear coefficients. In the first Gaussian case it is sufficiently to obtain the expression for conditional covariance function and then to calculate the reconstruction function and the error reconstruction function. In the case 2 it is necessary to obtain the solution of the corresponding Fokker - Plank - Kolmogorov equation for the conditional probability density functions (pdf). We obtain the required conditional pdf with two fixed samples and then determine the reconstruction function and the error reconstruction function. The jitter effect is described by random variable with the beta-distribution. Some examples are given.

*+*[Abstract](1011)

*+*[PDF](168.5KB)

**Abstract:**

This paper extends the dual-Petrov-Galerkin method proposed by Shen [16] and further developed by Yuan, Shen and Wu [23] to several integrable and non-integrable fifth-order KdV type equations. These fifth-order equations arise in modeling different wave phenomena and involve various nonlinear terms. The method is implemented to compute the solitary wave solutions of these equations and the numerical results imply that this scheme is capable of capturing, with very high accuracy, the details of these solutions with modest computational costs. It is also shown that the scheme is stable under a very mild stability constraint, and is second-order accurate in time and spectrally accurate in space.

*+*[Abstract](625)

*+*[PDF](115.5KB)

**Abstract:**

We reduce the question of local nonsolvability of the Darboux equation, and hence of the isometric embedding problem for surfaces, to the local nonsolvability of a simple linear equation whose type is explicitly determined by the Gaussian curvature.

*+*[Abstract](688)

*+*[PDF](163.3KB)

**Abstract:**

We study the nonlinear boundary value problem consisting of the equation $y^{''}+ w(t)f(y)=0$ on $[a,b]$ and a multi-point boundary condition. By relating it to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes of the existence of different types of nodal solutions as the problem changes.

*+*[Abstract](601)

*+*[PDF](205.8KB)

**Abstract:**

The paper is devoted to the state estimation problem in control theory under uncertainty. The approach for estimating the reachable sets of the linear impulsive differential systems is presented. The reachable sets are approximated by the ones for special discrete time systems. The degree of convergence is established. The families of external and internal polyhedral (parallelepiped-valued and parallelotope-valued) estimates of the reachable sets of the auxiliary systems are introduced. Evolution of estimates is determined by systems of recurrence relations. The families which ensure the exact representations of the reachable sets of the auxiliary systems as well as the families of the touching and tight estimates are found. This technique gives the possibility to construct the guaranteed estimates (including $\epsilon$-touching and $\epsilon$-tight ones) for the reachable sets of the primary systems. The results of numerical simulations are presented.

*+*[Abstract](874)

*+*[PDF](139.3KB)

**Abstract:**

We study a non-isothermal phase separation model of the Penrose-Fife type. We introduce the notion of a generalized solution and prove its unique existence.

*+*[Abstract](690)

*+*[PDF](375.4KB)

**Abstract:**

We consider a vectorial nonlinear diffusion equation with inhomogeneous terms in one-dimensional space. In this paper we study approximating problems of singular diffusion equations with a piecewise constant initial data. Also we consider the relationship between the singular diffusion problem and its approximating ones. Moreover we give some numerical experiments for the approximating equation with inhomogeneous terms and a piecewise constant initial data.

*+*[Abstract](786)

*+*[PDF](297.6KB)

**Abstract:**

In this paper, we consider a convection-diffusion boundary value problem with singular perturbation. A finite element method (FEM) is proposed based on discontinuous Galerkin (DG) discretization of least-squares variational formulation. Numerical tests on representative problems reveal that the method is robust and efficient.

*+*[Abstract](618)

*+*[PDF](168.7KB)

**Abstract:**

Wronskian determinants are used to construct exact solution to integrable equations. The crucial steps are to apply Hirota's bilinear forms and explore linear conditions to guarantee the Plücker relations. Upon solving the linear conditions, the resulting Wronskian formulations bring solution formulas, which can yield solitons, negatons, positions and complexitons. The solution process is illustrated by the Korteweg-de Vries equation and applied to the Boussinesq equation.

*+*[Abstract](928)

*+*[PDF](213.7KB)

**Abstract:**

In this paper we consider a boundary value problem for the following semilinear elliptic equation :$-\varepsilon^{2}\Delta u=h(|x|)^2(u-a(|x|))(1-u^2)$ in $B_1(0)$ with homogeneous Neumann boundary condition. The function $a$ is a $C^1$ function satisfying $|a(r)|< 1$ for $r\in [0,1]$ and $a^'(0)=0$. The function $h$ is a positive $C^1$ function satisfying $h^'(0)=0$. The nonlinear function in the equation is a typical example of the so-called {\it bistable} nonlinearity. Functions $a$ and $h$ in the nonlinearity represent spatial inhomogeneity. In particular, we consider the case where $a(r)=0$ on some interval $I\subset (0,1)$. When $\varepsilon>0$ is very small, there exist stationary solutions with sharp transition layers. We investigate asymptotic locations of transition layers of a global minimizer corresponding to an energy functional as $\varepsilon\to 0$. We use the variational procedure used in [4] with a few modifications prompted by the presence of the function $h$.

*+*[Abstract](802)

*+*[PDF](202.0KB)

**Abstract:**

This paper is concerned with time-delayed reaction-diffusion equations. For all traveling wavefronts, they are proved to be stable time-asymptotically by the technical weighted energy method with the comparison principle together, which extends the wave stability results obtained in [7,8]. Some numerical simulations are also carried out, which confirm our theoretical results.

*+*[Abstract](853)

*+*[PDF](162.0KB)

**Abstract:**

A one dimensional version of the dynamic Marguerre-Vlasov system in the presence of thermal effects is considered. The system depends on a parameter $\epsilon>0$ in a singular way as $\epsilon\to0$. Our interest is twofold: 1) To find the limit system as $\epsilon\to0$ and 2) To study the asymptotic behavior as $t\to+\infty$ of the total energy $E_{\epsilon}(t)$ and compare it with the total energy of the limit system.

*+*[Abstract](729)

*+*[PDF](155.6KB)

**Abstract:**

In the paper we consider a stochastic integral inclusion with discontinuous multivalued right hand side, driven by a continuous semimartingale. Using selection properties and lower and upper solutions methods we demonstrate the existence of strong solutions for such inclusions. We extend some recent results both for deterministic differential inclusions and for stochastic differential equations for increasing operators.

*+*[Abstract](652)

*+*[PDF](97.0KB)

**Abstract:**

We derive a nonstandard finite difference scheme for the coupled, nonlinear PDE's modeling laser generated electrons and holes in a semiconductor. Our scheme has the essential feature of giving numerical solutions for which the charge densities are non-negative. Many of the standard schemes do not have this physically required property.

*+*[Abstract](643)

*+*[PDF](145.9KB)

**Abstract:**

In this paper it is considered a fourth order problem composed of a fully nonlinear differential equation and functional boundary conditions satisfying some monotone conditions.This functional dependence on $u,u^' $ and $u^{''}$and generalizes several types of boundary conditions such as Sturm-Liouville, multipoint, maximum and/or minimum arguments, or nonlocal. The main theorem is an existence and location result as it provides not only the existence, but also some qualitative information about the solution.

*+*[Abstract](835)

*+*[PDF](167.9KB)

**Abstract:**

We present a Sobolev space approach for semilinear heat equations $u_t=\Delta u + F(u(t,x))$ for $t>0$ on a bounded domain $\Omega\subset\mathbf{R}^n$. By proving that there exists a solution in the anisotropic Sobolev space $W^{1,2}_p( \R_+\times\Omega)$, we can deduce more than just global existence in time. For example, both the solution and its time derivative are of class $L^p$, and the solution tends to zero in $L^\infty(\Omega)$ as $t\to\infty$. The main result shows that the existence of a solution in $W^{1,2}_p$ depends primarily on the existence of an appropriate

*a priori*estimate on the $L^\infty$ norm of solutions as the initial data is deformed to zero.

*+*[Abstract](739)

*+*[PDF](142.8KB)

**Abstract:**

It is the main objective of this paper to discuss applications of the abstract results recently evolved in [16,17] to the solvability of variational inequalities with constraints depending on the unknown functions, which are called quasi-variational inequalities.

*+*[Abstract](785)

*+*[PDF](167.1KB)

**Abstract:**

In this paper we study the Cauchy problem for the weakly coupled system of damped wave equations. Recently Sun and Wang [12] have shown the existence and nonexistence of the Cauchy problem for the weakly coupled system of damped wave equations, provided that the space dimension $n=1, 3$. In this paper we will generalize their existence result to the case where $n=1,2.3$, and we improve time decay estimates when $n=3$. Moreover, the Cauchy problem with slow decaying initial data is treated.

*+*[Abstract](975)

*+*[PDF](173.6KB)

**Abstract:**

Compressible, stationary Navier-Stokes (N-S) equations are considered. The shape sensitivity analysis is performed in the case of small perturbations of the so-called

*it approximate solutions*. The proposed method of shape sensitivity analysis is general, and can be used to establish the well-posedness for distributed and boundary control problems as well as for inverse problems in the case of the state equations in the form of compressible N-S equations.

*+*[Abstract](867)

*+*[PDF](185.5KB)

**Abstract:**

We show convergence of solutions to equilibria for quasilinear and fully nonlinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional $C^1$-manifold which is normally stable.

*+*[Abstract](670)

*+*[PDF](128.7KB)

**Abstract:**

In this paper, we demonstrate new methods to prove existence of travelling front solutions and better estimates of minimum travelling speed to reaction diffusion systems modelling cubic Auto-Catalysis chemical reactions A + 2B $\rightarrow$ 3B involving two chemical species, a reactant A and an auto-catalyst B. Furthermore, we show the development of interface in the form of travelling fronts for quadratic Auto-Catalysis chemical reactions A + B $\rightarrow$ 2B when initial values are set up similar to an experiment involving Auto-Catalysis as a key step.

*+*[Abstract](708)

*+*[PDF](169.5KB)

**Abstract:**

This paper considers modeling the initiation of atherosclerosis, as an inflammatory instability. Motivated by the disease paradigm articulated by Russell Ross, atherogenesis is viewed as an inflammatory spiral with positive feedback loop involving key cellular and chemical species interacting and reacting within the intimal layer of muscular arteries. The inflammation is modeled through a system of nonlinear reaction/diffusion/convection partial differential equations. The inflammatory spiral is initiated as an instability from a healthy state which is defined to be an equilib- rium state devoid of certain key inflammatory markers. Disease initiation is studied through a linear, asymptotic stability analysis of a healthy equilibrium state. Conditions on system parameters guaranteeing stability of the health state and conditions on system parameters leading to instability are given. Among the questions addressed in the analysis is the possible mitigating effect of anti-oxidants upon transition to the inflammatory spiral.

*+*[Abstract](724)

*+*[PDF](193.9KB)

**Abstract:**

Consider nonlinear stochastic difference equations

$X(n+1) = X(n)+hf(X(n))+\sqrthg(X(n))\xi_{n+1},$ $n \in \N,$ $X(0) =$ ς $\in \mathbb{R},$ (1)

where $\{\xi_n\}_{n\in \N}$ are independent $fr{N} (0,1)$-distributed random variables, $h>0$, can be viewed as a discretization of Itô stochastic differential equations (SDEs).

We discuss the following. If, for all $t\ge 0$, the solution $Y(t)$ of the corresponding SDE is positive, or $Y(t) \in [0,K]$ for some $K>0$, does the solution $X(n)$ of related discretization (1) possess the same properties with large probability? In general, the answer is no. However in many cases we are able to discretize the SDE related to (1) over a compact interval $[0,T]$ in such a way that an adequate qualitative behavior is observed with an arbitrarily high probability.

*+*[Abstract](710)

*+*[PDF](144.8KB)

**Abstract:**

Sufficient conditions are given such that the discrete time competition models constructed by applying nonstandard finite difference (NSFD) schemes for the Lotka-Volterra competition models are dynamically consistent. The derived discrete models preserve the positivity of solutions, local stability conditions, boundedness, and the monotonicity of the continuous Lotka-Volterra system. In other words, we are able to construct discrete-time competition models that behave just like the continuous-time Lotka-Volterra competition models.

*+*[Abstract](737)

*+*[PDF](173.5KB)

**Abstract:**

This paper discusses the quasilinearization process, that is, reduction of a given boundary value problem to a quasi-linear one. Quasi-linear BVP has a solution of definite type which is induced by oscillatory properties of the linear part of a system. Conditions are given for a solution of a modified (quasi-linear) BVP to satisfy also the original problem. It is shown that multiple quasilinearizations with different linear parts if possible lead to multiple solutions.

*+*[Abstract](672)

*+*[PDF](154.5KB)

**Abstract:**

We study an asymptotically linear Schrödinger equation in the whole Euclidean space $\mathbb{R}^N$. By using a suitable linking theorem we prove that the problem admits not only the trivial solution but also either a sign--changing solution or one positive and one negative solution.

*+*[Abstract](676)

*+*[PDF](186.5KB)

**Abstract:**

The main result is a very general solution formula for the noncommutative AKNS system, extending work by Bauhardt and Pöppe. As an application, we construct for the matrix sine-Gordon equation $N$-soliton solutions analogous to the multisoliton solutions for the KdV equation due to Goncharenko.

*+*[Abstract](610)

*+*[PDF](141.5KB)

**Abstract:**

In this note we discuss blow-up at space infinity for quasilinear parabolic equation $u_t = \Delta u^m + u^{p}$. It is known that if initial data is not a constant and takes its maximum at space infinity in a certain sense, the solution blows up only at space infinity at minimal blow-up time. We show that if $m \ge 1$ and a solution blows up at minimal blow-up time, then it blows up completely at the blow-up time.

*+*[Abstract](809)

*+*[PDF](303.4KB)

**Abstract:**

This paper is devoted to the stability analysis for two dimensional interfaces in solid-liquid phase transitions, represented by some types of Allen-Cahn equations. Each Allen-Cahn equation is derived from a free energy, associated with a two dimensional Finsler norm, under the so-called crystalline type setting, and then the Wulff shape of the Finsler norm is supposed to correspond to the basic structural unit of masses of pure phases (crystals). Consequently, special piecewise smooth Jordan curves, based on Wulff shapes, will be exemplified in the main theorems, as the geometric representations of the stability condition.

*+*[Abstract](751)

*+*[PDF](307.2KB)

**Abstract:**

The system of Coupled Nonlinear Schrödinger's Equations (CNLSE) is solved numerically by means of a conservative difference scheme. A new kind of repelling collision is discovered for negative values of the cross-modulation coupling parameter, $\alpha_2$. The results show that as the latter becomes increasingly negative, the behavior of the solitons during interaction change drastically. While for $\alpha_2 >0$, the solitons pass through each other, a negative threshold value $\alpha^*_2 < 0$ is found below which the solitons repell each other. This is a novel result for this kind of models and the conservation of momentum for the system of quasi-particles (QPs) is thoroughly investigated.

*+*[Abstract](805)

*+*[PDF](173.7KB)

**Abstract:**

In modeling of populations and in many other applications parameters are either measured directly or determined by fitting parameters to a mathematical model. These parameters have variability depending on experimental error, the actual population used and many other factors. In this paper we consider that those parameters are random variables with given distributions. We write and solve random differential equations that model Monod growth kinetics. This type of kinetics is useful, for example, in modeling biofilm growth.

*+*[Abstract](783)

*+*[PDF](189.0KB)

**Abstract:**

We consider the Kuramoto-Sivashinsky (KS) equation in finite domains of the form $[-L,L]$. Our main result provides effective new estimates for higher Sobolev norms of the solutions in terms of powers of $L$ for the one-dimentional differentiated KS. We illustrate our method on a simpler model, namely the regularized Burger's equation. The underlying idea in this result is that

*a priori*control of the $L^2$ norm is enough in order to conclude higher order regularity and in fact, it allows one to get good estimates on the high-frequency tails of the solution.

*+*[Abstract](680)

*+*[PDF](106.6KB)

**Abstract:**

In this paper, we will develop a comparison principle with Razumikhin condition relative to stability theory of impulsive functional differential system with infinite delay in terms of two different measures.

*+*[Abstract](765)

*+*[PDF](180.5KB)

**Abstract:**

We consider an abstract second order semilinear evolution equation with a bounded dissipation. We establish an equivalence between the stabilization of this system and the observability of the corresponding undamped system. Our technique of proof relies on an appropriate decomposition of the solution, and the energy method. Our result generalizes an earlier one by Haraux [5] who studied the same type of problem for linear systems. Some applications of our result are provided, and the paper ends with a few open problems.

*+*[Abstract](826)

*+*[PDF](128.1KB)

**Abstract:**

A mathematical model of MAPK and BMP-Smad1 signaling pathways in the embryonic development of

*Xenopus laevis*is constructed. The model consists of a system of 4 coupled, nonlinear ordinary differential equations. Numerical computations characterize the biological result that a 4 to 6-fold increase in MAPK activity inhibits Smad1 activity and triggers the neural fate of the embryo's ectodermal cells. Bifurcation analysis of the model shows that this biological result can be explained via transcritical bifurcations involving steady-state MAPK and Smad1 activity levels.

*+*[Abstract](647)

*+*[PDF](455.2KB)

**Abstract:**

In this paper, modified Korteweg-de Vries (mKdV) and Harry Dym (HD) surfaces are considered which are arisen from using soliton surface technique and a variational principle. Some of these surfaces belong to Willmore-like and Weingarten surfaces, and surfaces that solve the generalized shape equation classes. Moreover, parameterized form of these surfaces are found for given solutions of the mKdV and HD equations.

*+*[Abstract](764)

*+*[PDF](131.7KB)

**Abstract:**

A simplified mathematical model of solid tumor regrowth is analyzed. When the model system is disturbed by radiation and chemotherapy, which are given by discontinuous functions, the system loses its smoothness. For the purpose of comparison and verification of therapy efficacy, a weak solution is constructed. Some suggestions about effective combination of treatments are also given.

*+*[Abstract](791)

*+*[PDF](813.7KB)

**Abstract:**

The system of Coupled Nonlinear Schrödinger Equations (CNLSEs) is solved by a conservative difference scheme in complex arithmetic developed in earlier author's work. The initial condition represents a superposition of two one-soliton solutions of different circular polarizations. The interaction (collision) of the solitons and their quasi-particle (QP) behavior is examined for different configurations of the initial system of QPs. We found that the polarization angle of a QP can change after a collision with another QP depending on the configuration of the initial phases. The effects found in the present work seem to be novel and enrich the knowledge about the intimate mechanisms of interaction of polarized QPs of CNLSEs.

*+*[Abstract](693)

*+*[PDF](226.4KB)

**Abstract:**

This paper presents a continuum - traffic flow like - model for the flow of products through complex production networks, based on statistical information obtained from extensive observations of the system. The resulting model consists of a system of hyperbolic conservation laws, which, in a relaxation limit, exhibit the correct diffusive properties given by the variance of the observed data.

*+*[Abstract](763)

*+*[PDF](172.3KB)

**Abstract:**

The asymptotic behavior of stochastic wave equations on $\mathbb{R}^n$ is studied. The existence of a random attractor for the corresponding random dynamical system in $H^1(\mathbb{R}^n) \times L^2(\mathbb{R}^n)$ is established, where the nonlinearity has an arbitrary growth order for $n \le 2$ and is subcritical for $n=3$.

*+*[Abstract](695)

*+*[PDF](126.3KB)

**Abstract:**

The paper deals with the existence and nonexistence of positive radial solutions for the weakly coupled quasilinear system div$( | \nabla u|^{p-2}\nabla u ) + \lambda f(v)=0$, div $( | \nabla v|^{p-2}\nabla v ) + \lambda g(u)=0$ in $\B$, and $\u =v=0$ on $\partial B,$ where $p>1$, $B$ is a finite ball, $f$ and $g$ are continuous and nonnegative functions. We prove that there is a positive radial solution for the problem for various intevals of $\lambda$ in sublinear cases. In addition, a nonexistence result is given. We shall use fixed point theorems in a cone.

*+*[Abstract](689)

*+*[PDF](151.5KB)

**Abstract:**

We consider the system of integral equations

$u_i(t)=int_0^Tg_i(t,s)[a_i(s,u_1(s),u_2(s),...,u_n(s))+b_i(s,u_1(s),u_2(s),...,u_n(s))]ds,$ $t \in [0,T],$ $1<=i<=n,$

where $T>0$ is fixed
and the nonlinearities $a_i(t,u_1,u_2,\cdots,u_n)$ can be *
singular* at $t=0$ and $u_j=0$ where $j\in\{1,2,\cdots,n\}.$
Criteria are established for the existence of * fixed-sign
solutions* $(u_1^*,u_2^*,\cdots,u_n^*)$ to the above system, i.e.,
$\theta_iu_i^*(t)\geq 0$ for $t\in [0,T]$ and $1\leq i\leq n,$ where
$\theta_i\in \{1,-1\}$ is fixed. We also include an example to
illustrate the usefulness of the results obtained.

*+*[Abstract](871)

*+*[PDF](184.6KB)

**Abstract:**

In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem

$-\Delta u=\lambda u-b(x)uf(u)$ in $ \Omega,$

$u =+\infty$ on $\partial \Omega,$

*+*[Abstract](815)

*+*[PDF](133.6KB)

**Abstract:**

In a flow of two immiscible incompressible viscous fluids, jump discontinuities of flow quantities appear at the two-fluid interface. The immersed interface method can accurately and efficiently simulate the flow without smearing the sharp interface by incorporating necessary jump conditions into a numerical scheme. In this paper, we systematically derive the principal jump conditions for the velocity, the pressure, and their normal derivatives.

*+*[Abstract](932)

*+*[PDF](191.3KB)

**Abstract:**

This paper is concerned with the Belousov-Zhabotinskii reaction model. We consider the reaction-diffusion model due to Keener-Tyson. After constructing a dynamical system, we will construct exponential attractors and will estimate the attractor dimension from below. In particular, it will be shown that, as the excitability $\epsilon > 0$ tends to zero, the attractor dimension tends to infinity, although the exponential attractor can depend on the excitability continuously.

*+*[Abstract](684)

*+*[PDF](179.0KB)

**Abstract:**

The existence of a global attractor in the $L^2$ product phase space for the solution semiflow of the modified Schnackenberg equations with the Dirichlet boundary condition on a bounded domain of space dimension $n\le 3$ is proved. This reaction-diffusion system features two pairs of oppositely-signed nonlinear terms so that the dissipative sign-condition is not satisfied. The proof features two types of rescaling and grouping estimation in showing the absorbing property and the uniform smallness in proving the asymptotical compactness by the approach of a new decomposition.

*+*[Abstract](516)

*+*[PDF](176.0KB)

**Abstract:**

We consider symbolic flows over finite alphabets and study certain kinds of repetitions in these sequences. Positive and negative results for the existence of such repetitions are given for codings of interval exchange transformations and codings of quadratic polynomials.

## Readers

## Email Alert

Add your name and e-mail address to receive news of forthcoming issues of this journal:

[Back to Top]