ISSN:

1531-3492

eISSN:

1553-524X

All Issues

## Discrete & Continuous Dynamical Systems - B

September 2012 , Volume 17 , Issue 6

Special issue

dedicated to professor Avner Friedman on his 80th birthday

Select all articles

Export/Reference:

2012, 17(6): i-iii
doi: 10.3934/dcdsb.2012.17.6i

*+*[Abstract](1027)*+*[PDF](3306.6KB)**Abstract:**

It is our great privilege to serve as Guest Editors for this special issue of Discrete and Continuous Dynamical Systems, Series B honoring Professor Avner Friedman on his 80th birthday.

For more information please click the "Full Text" above.

2012, 17(6): 1605-1638
doi: 10.3934/dcdsb.2012.17.1605

*+*[Abstract](1158)*+*[PDF](921.4KB)**Abstract:**

We consider a class of excitable system whose dynamics is described by Fitzhugh-Nagumo (FN) equations. We provide a description for rigidly rotating spirals based on the fact that one of the unknowns develops abrupt jumps in some regions of the space. The core of the spiral is delimited by these regions. The description of the spiral is made using a mixture of asymptotic and rigorous arguments. Several open problems whose rigorous solution would provide insight in the problem are formulated.

2012, 17(6): 1639-1649
doi: 10.3934/dcdsb.2012.17.1639

*+*[Abstract](976)*+*[PDF](365.1KB)**Abstract:**

Formal asymptotic expansions have long been used to study the singularly perturbed Allen-Cahn type equations and reaction-diffusion systems, including in particular the FitzHugh-Nagumo system. Despite their successful role, it has been largely unclear whether or not such expansions really represent the actual profile of solutions with rather general initial data. By combining our earlier result and known properties of eternal solutions of the Allen-Cahn equation, we prove validity of the principal term of the formal expansions for a large class of solutions.

2012, 17(6): 1651-1672
doi: 10.3934/dcdsb.2012.17.1651

*+*[Abstract](1167)*+*[PDF](1286.5KB)**Abstract:**

Radiotherapy is an important clinical tool to fight malignancies. To do so, a key point consists in selecting a suitable radiation dose that could achieve tumour control without inducing significant damage to surrounding healthy tissues. In spite of recent significant advances, any radiotherapy planning in use relies principally on experience-based decisions made by clinicians among several possible choices.

In this work we consider a mathematical problem related to that decision-making process. More precisely, we assume that a well-defined target region, called planning target volume (PTV), is given. We then consider the question of determining which radiation distribution is able to achieve a maximum impact on tumour cells and a minimum one in healthy ones. Such dose distribution is defined as the solution of a multi-parameter minimization problem over the PTV and healthy tissues, subject to a number of constraints arising from clinical and technical requirements. For any choice of parameters, sufficient conditions for the existence of a unique solution of that problem are derived. Such solution is then approximated by means of a suitable numerical algorithm. Finally, some examples are considered, on which the dependence on model parameters of different clinical efficiency indexes is discussed.

2012, 17(6): 1673-1684
doi: 10.3934/dcdsb.2012.17.1673

*+*[Abstract](1156)*+*[PDF](332.7KB)**Abstract:**

A two-component reaction-diffusion system modelling a class of spatially structured epidemic systems is considered. More specifically, the system describes the spread of malaria mediated by a population of infected mosquitoes. A relevant problem, related to the possible eradication of the epidemic, is the so called zero-stabilization. We prove that it is possible to diminish exponentially the epidemic process, in the whole habitat, just by acting on the segregation rate between the population of infected mosquitoes and the susceptible human population in a nonempty and sufficiently large subset of the spatial domain.

2012, 17(6): 1685-1691
doi: 10.3934/dcdsb.2012.17.1685

*+*[Abstract](1145)*+*[PDF](265.2KB)**Abstract:**

We consider a porous medium flow in which the material is initially distributed in the exterior of an empty region (a hole) and study the final stage of the hole-filling process as well as the initial stage of the post filling regime. It is known that in axially symmetric flow the hole-filling is asymptotically described by a self-similar solution which depends on a constant determined by the initial distribution. The post filling accumulation process is also locally described by a self-similar solution which in turn is characterized by a constant. In general, these constants must be found either experimentally or numerically. Here we present an example of a one-dimensional flow where the constants are obtained explicitly.

2012, 17(6): 1693-1706
doi: 10.3934/dcdsb.2012.17.1693

*+*[Abstract](982)*+*[PDF](339.5KB)**Abstract:**

Necessary and sufficient conditions for quasiconvexity, also called level-set convexity, of a function are given in terms of first-order partial differential equations. Solutions to the equations are understood in the viscosity sense and the conditions apply to nonsmooth and semicontinuous functions. A comparison principle, implying uniqueness of solutions, is shown for a related partial differential equation. This equation is then used in an iterative construction of the quasiconvex envelope of a function. The results are then extended to robustly quasiconvex functions, that is, functions which are quasiconvex under small linear perturbations.

2012, 17(6): 1707-1728
doi: 10.3934/dcdsb.2012.17.1707

*+*[Abstract](858)*+*[PDF](545.1KB)**Abstract:**

In this paper we analyze a free-boundary model for free-standing fibers made from smectic layers of kinked (bent-core) liquid crystal molecules. In [1] a radial model was proposed to explain how fibers form (assuming radially symmetric configurations) based on the distinctive packing and ferroelectric properties of bent--core molecules. We develop this model further to include smectic energy terms so as to allow for non--circular cross--sections with non--radial configurations and fields. We show that the relative size of the energy's elasticity constants can be used to determine the stability (instability) of radially symmetric fibers with respect to non--radial perturbations.

2012, 17(6): 1729-1750
doi: 10.3934/dcdsb.2012.17.1729

*+*[Abstract](1308)*+*[PDF](497.6KB)**Abstract:**

We deal with nonlinear elliptic and parabolic systems that are the Bellman systems associated to stochastic differential games as a main motivation. We establish the existence of weak solutions in any dimension for an arbitrary number of equations ("players"). The method is based on using a

*renormalized sub- and super-solution technique*. The main novelty consists in the new structure conditions on the critical growth terms with allow us to show weak solvability for Bellman systems to certain classes of stochastic differential games.

2012, 17(6): 1751-1759
doi: 10.3934/dcdsb.2012.17.1751

*+*[Abstract](826)*+*[PDF](368.6KB)**Abstract:**

We show the free boundary of the American put option with dividend payment is $C^{\infty}$.

2012, 17(6): 1761-1774
doi: 10.3934/dcdsb.2012.17.1761

*+*[Abstract](873)*+*[PDF](370.7KB)**Abstract:**

We study the dead-core rate for the solution of the porous medium equation with a strong absorption. It is known that solutions with certain class of initial data develop a dead-core in finite time. We prove that, unlike the cases of semilinear heat equation and fast diffusion equation, there are solutions with the self-similar dead-core rate. This result is based on the construction of a Lyapunov functional, some a priori estimates, and a delicate analysis of the associated re-scaled ordinary differential equation.

2012, 17(6): 1775-1794
doi: 10.3934/dcdsb.2012.17.1775

*+*[Abstract](1361)*+*[PDF](1273.0KB)**Abstract:**

Keizer's paradox refers to the observation that deterministic and stochastic descriptions of chemical reactions can predict vastly different long term outcomes. In this paper, we use slow manifold analysis to help resolve this paradox for four variants of a simple autocatalytic reaction. We also provide rigorous estimates of the spectral gap of important linear operators, which establishes parameter ranges in which the slow manifold analysis is appropriate.

2012, 17(6): 1795-1807
doi: 10.3934/dcdsb.2012.17.1795

*+*[Abstract](917)*+*[PDF](491.1KB)**Abstract:**

We combine a simple quasi-geostrophic flow model with the Zabusky-McWilliams theory of atmospheric vortex dynamics to address a hurricane-tracking problem of interest to the insurance industry. This enables us to make predictions about the "follow-my-leader" phenomenon.

2012, 17(6): 1809-1829
doi: 10.3934/dcdsb.2012.17.1809

*+*[Abstract](916)*+*[PDF](433.5KB)**Abstract:**

Employing and generalizing the (continuous time, incomplete market) equity valuation (and hedging) theory introduced recently by one of the authors, the effect of stock dilution and buy-back on the equity value is quantified. Both, neutral and indifference pricing methodologies are considered, and results of different levels of complexity are provided. Hedging results are provided as well. Both pricing and hedging results are obtained as special cases of the general methodology of pricing and hedging in incomplete markets recently developed by one of the authors.

2012, 17(6): 1831-1840
doi: 10.3934/dcdsb.2012.17.1831

*+*[Abstract](1011)*+*[PDF](319.3KB)**Abstract:**

We extend the deterministic mathematical malaria model framework of Dembele

*et al.*and use it to study the impact of protecting humans from mosquito bites and mass killing of mosquito vectors on malaria incidence in Missira, a village in Mali. As a case study, we fit our model to Missira malaria incidence data. Using the fitted model, we compute the optimal proportion of protected human population from infected mosquito bites and optimal proportion of killed moquitoes that would lead to the eradication of malaria in Missira.

2012, 17(6): 1841-1858
doi: 10.3934/dcdsb.2012.17.1841

*+*[Abstract](981)*+*[PDF](444.6KB)**Abstract:**

The local positivity of solutions to logarithmically singular diffusion equations is investigated in some open space-time domain $E\times(0,T]$. It is shown that if at some time level $t_o\in(0,T]$ and some point $x_o\in E$ the solution $u(\cdot,t_o)$ is not identically zero in a neighborhood of $x_o$, in a measure-theoretical sense, then it is strictly positive in a neighborhood of $(x_o, t_o)$. The precise form of this statement is by an intrinsic Harnack-type inequality, which also determines the size of such a neighborhood.

2012, 17(6): 1859-1887
doi: 10.3934/dcdsb.2012.17.1859

*+*[Abstract](1115)*+*[PDF](1391.6KB)**Abstract:**

Two types of aggregation systems with Fisher-KPP growth are proposed. One is described by a normal reaction-diffusion system, and the other is described by a cross-diffusion system. If the growth effect is dominant, a spatially constant equilibrium solution is stable. When the growth effect becomes weaker and the aggregation effect become dominant, the solution is destabilized so that spatially non-constant equilibrium solutions, which exhibit Turing's patterns, appear. When the growth effect weakens further, the spatially non-constant equilibrium solutions are destabilized through Hopf bifurcation, so that oscillatory Turing's patterns appear. Finally, when the growth effect is extremely weak, there appear spatio-temporal periodic solutions exhibiting infinite dimensional relaxation oscillation.

2012, 17(6): 1889-1902
doi: 10.3934/dcdsb.2012.17.1889

*+*[Abstract](1160)*+*[PDF](357.7KB)**Abstract:**

We analyze a variational method for reconstructing a bar code signal from a blurry and noisy measurement. The bar code is modeled as a binary function with a finite number of transitions and a parameter controlling minimal feature size. The measured signal is the convolution of this binary function with a Gaussian kernel. In this work, we assume that the blur kernel is known and establish conditions (involving noise level and variance of the convolution kernel) under which the variational method considered recovers essentially the correct bar code.

2012, 17(6): 1903-1937
doi: 10.3934/dcdsb.2012.17.1903

*+*[Abstract](1012)*+*[PDF](742.6KB)**Abstract:**

In hollow fibres dialyzers blood flows in the fibres channel and plasma filtrates through their permeable wall to feed the flow of the permeate (dialyzate), which takes place among the fibres in the opposite direction. We investigate this fluid dynamical problem exploiting the existence of two separate scales: the one in the fibres direction ($\sim \,20\,cm$), and the one along the fibre radius ($\sim \,0.1\,mm$). We formulate a mathematical model based on a two-scale approach providing a full description of the flows of the blood, of the dialyzate and the of the plasma through the membrane, as well as of the progressive increase of the hematocrit. The problem is characterized by various rather unusual features like the slip condition of blood on the membrane and the feedback loop of boundary data for the hematocrit. Blood rheology is assumed to be of shear-thinning type, with hematocrit dependent coefficients. Under some simplifications explicit solutions are found. We show how the necessity of respecting several constraints and of reaching some specific targets influences the selection of the geometrical and physical parameters of the system. Once the fluid dynamical problem has been solved, the removal from blood of chemicals like urea, etc. has been studied.

2012, 17(6): 1939-1967
doi: 10.3934/dcdsb.2012.17.1939

*+*[Abstract](1050)*+*[PDF](1027.3KB)**Abstract:**

We study the solutions to Poisson-Boltzmann equation for electrolytic solutions in a domain $\Omega$, surrounded by an uncharged dielectric medium. We establish existence, uniqueness and regularity of solutions and study in detail their asymptotic behaviour close to $\partial\Omega$ when a characteristic length, called the Debye length, is sufficiently small. This is a double layer with a thickness that changes from point to point along $\partial\Omega$ depending on the normal derivative of a harmonic function outside $\Omega$ and the mean curvature of $\partial\Omega$. We also provide numerical evidence of our results based on a finite elements approximation of the problem.

2012, 17(6): 1969-1990
doi: 10.3934/dcdsb.2012.17.1969

*+*[Abstract](918)*+*[PDF](778.5KB)**Abstract:**

We extend the development of homogenized models for excitable tissues coupled through "doughball" gap junctions. The analysis admits nonlinear Fickian fluxes in rather general ways and includes, in particular, calcium-gated conductance. The theory is motivated by an attempt to understand wave propagation and failure observed in the pancreatic islets of Langerhans. We reexamine, numerically, the role that gap junctional strength is generally thought to play in pattern formation in continuum models of islets.

2012, 17(6): 1991-1999
doi: 10.3934/dcdsb.2012.17.1991

*+*[Abstract](1210)*+*[PDF](318.7KB)**Abstract:**

We study a fully nonlinear equation of complex Monge-Ampère type on Hermitian manifolds. We establish the

*a priori*estimates for solutions of the equation up to the second order derivatives with the help of a subsolution.

2012, 17(6): 2001-2016
doi: 10.3934/dcdsb.2012.17.2001

*+*[Abstract](950)*+*[PDF](388.0KB)**Abstract:**

In this study, we establish a financial credit derivative pricing model for a contract which is subject to counterparty risks. The model leads to a fully nonlinear partial differential equation problem. We study this PDE problem and obtained a solution as the limit of a sequence of semi-linear PDE problems which also arise from financial models. Moreover, the problems and methods build a bridge between two main risk frameworks: structure and intensity models. We obtain the uniqueness, regularities and some properties of the solution of this problem.

2012, 17(6): 2017-2046
doi: 10.3934/dcdsb.2012.17.2017

*+*[Abstract](983)*+*[PDF](3312.4KB)**Abstract:**

In this paper, we propose a new continuous time model to recover the volatility of underlying asset from observed market European option price. The model is a couple of fully nonlinear parabolic partial differential equations (see (34), (36)). As an inverse problem, the model is deduced from a Tikhonov regularization framework. Based on our method, the recovering procedure is stable and accurate. It is justified not only in theoretical proofs, but also in the numerical experiments.

2012, 17(6): 2047-2072
doi: 10.3934/dcdsb.2012.17.2047

*+*[Abstract](1071)*+*[PDF](378.0KB)**Abstract:**

Random dispersal describes the movement of organisms between adjacent spatial locations. However, the movement of some organisms such as seeds of plants can occur between non-adjacent spatial locations and is thus non-local. We propose to study a mixed dispersal strategy, which is a combination of random dispersal and non-local dispersal. More specifically, we assume that a fraction of individuals in the population adopt random dispersal, while the remaining fraction assumes non-local dispersal. We investigate how such mixed dispersal affects the invasion of a single species and also how mixed dispersal strategy will evolve in spatially heterogeneous but temporally constant environment.

2012, 17(6): 2073-2090
doi: 10.3934/dcdsb.2012.17.2073

*+*[Abstract](1053)*+*[PDF](425.5KB)**Abstract:**

We consider linear elliptic and parabolic equations with measurable coefficients and prove two types of $L_{p}$-estimates for their solutions, which were recently used in the theory of fully nonlinear elliptic and parabolic second order equations in [1]. The first type is an estimate of the $\gamma$th norm of the second-order derivatives, where $\gamma\in(0,1)$, and the second type deals with estimates of the resolvent operators in $L_{p}$ when the first-order coefficients are summable to an appropriate power.

2012, 17(6): 2091-2151
doi: 10.3934/dcdsb.2012.17.2091

*+*[Abstract](1072)*+*[PDF](4710.2KB)**Abstract:**

SELEX (Systematic Evolution of Ligands by EXponential Enrichment) is an iterative separation process by which a pool of nucleic acids that bind with varying specificities to a fixed target molecule or a fixed mixture of target molecules, i.e., single or multiple targets, can be separated into one or more pools of pure nucleic acids. In its simplest form, as introduced in [6], the initial pool is combined with the target and the products separated from the mixture of bound and unbound nucleic acids. The nucleic acids bound to the products are then separated from the target. The resulting pool of nucleic acids is expanded using PCR (polymerase chain reaction) to bring the pool size back up to the concentration of the initial pool and the process is then repeated. At each stage the pool is richer in nucleic acids that bind best to the target. In the case that the target has multiple components, one obtains a mixture of nucleic acids that bind best to at least one of the components. A further refinement of multiple target SELEX, known as alternate SELEX, is described below. This process permits one to specify which nucleic acids bind best to each component of the target.

These processes give rise to discrete dynamical systems based on consideration of statistical averages (the law of mass action) at each step. A number of interesting questions arise in the mathematical analysis of these dynamical systems. In particular, one of the most important questions one can ask about the limiting pool of nucleic acids is the following: Under what conditions on the individual affinities of each nucleic acid for each target component does the dynamical system have a global attractor consisting of a single point? That is, when is the concentration distribution of the limiting pool of nucleic acids independent of the concentrations of the individual nucleic acids in the initial pool, assuming that all nucleic acids are initially present in the initial pool? The paper constitutes a summary of our theoretical and numerical work on these questions, carried out in some detail in [9], [11], [13].

2012, 17(6): 2153-2170
doi: 10.3934/dcdsb.2012.17.2153

*+*[Abstract](1214)*+*[PDF](501.3KB)**Abstract:**

Dynamical system models of complex biochemical reaction networks are high-dimensional, nonlinear, and contain many unknown parameters. The capacity for multiple equilibria in such systems plays a key role in important biochemical processes. Examples show that there is a very delicate relationship between the structure of a reaction network and its capacity to give rise to several positive equilibria. In this paper we focus on networks of reactions governed by mass-action kinetics. As is almost always the case in practice, we assume that no reaction involves the collision of three or more molecules at the same place and time, which implies that the associated mass-action differential equations contain only linear and quadratic terms. We describe a general injectivity criterion for quadratic functions of several variables, and relate this criterion to a network's capacity for multiple equilibria. In order to take advantage of this criterion we look for explicit general conditions that imply non-vanishing of polynomial functions on the positive orthant. In particular, we investigate in detail the case of polynomials with only one negative monomial, and we fully characterize the case of affinely independent exponents. We describe several examples, including an example that shows how these methods may be used for designing multistable chemical systems in synthetic biology.

2012, 17(6): 2171-2184
doi: 10.3934/dcdsb.2012.17.2171

*+*[Abstract](986)*+*[PDF](777.4KB)**Abstract:**

In this paper we present a mathematical analysis of a model involving target-mediated drug disposition involving two targets, developed to fit a series of data sets. The two targets are receptors with very different characteristics: one has high affinity to the drug, a small capacity and a short half-life, whilst the other receptor has low affinity to the drug, high capacity and its half-life is large. The analysis of this model yields a qualitative and quantitative understanding of the dynamics of this two-receptor model and in particular identifies different time scales over which the amounts of free drug and drug-receptor complexes vary. Thus it yields analytical tools to make long-term predictions on the basis of medium term data sets.

2012, 17(6): 2185-2200
doi: 10.3934/dcdsb.2012.17.2185

*+*[Abstract](941)*+*[PDF](495.8KB)**Abstract:**

In this paper we analyze and generalize the dynamical system introduced by Lemaire and co-workers [14] as a model of cell signaling in bone remodeling. We show that for large classes of parameter values, including the physically-realistic baseline values, the system has a unique physically relevant equilibrium which is a global attractor. We generalize that model, minimally, to incorporate a mechanism by which parathyroid hormone (PTH) retards osteoblast apoptosis. We show that with this mechanism, which is a simplified version of that proposed by Bellido and co-workers [4], the model exhibits a well-known phenomenon that has puzzled researchers: the system responds catabolically to the continuous administration of PTH and ally to appropriately pulsed administration of PTH.

2012, 17(6): 2201-2223
doi: 10.3934/dcdsb.2012.17.2201

*+*[Abstract](1006)*+*[PDF](507.6KB)**Abstract:**

In this paper, we use the method of characteristics to study singularities in the flow of a parameterized family of extremals for an optimal control problem. By means of the Lyapunov--Schmidt reduction a characterization of

*fold*and

*cusp*points is given. Examples illustrate the local behaviors of the flow near these singular points. Singularities of fold type correspond to the typical conjugate points as they arise for the classical problem of minimum surfaces of revolution in the calculus of variations and local optimality of trajectories ceases at fold points. Simple cusp points, on the other hand, generate a cut-locus that limits the optimality of close-by trajectories globally to times prior to the conjugate points.

2012, 17(6): 2225-2242
doi: 10.3934/dcdsb.2012.17.2225

*+*[Abstract](791)*+*[PDF](425.4KB)**Abstract:**

We show that for a very general and natural class of curvature functions (for example the curvature quotients $(\sigma_n/\sigma_l)^{\frac{1}{n-l}}$) the problem of finding a complete spacelike strictly convex hypersurface in de Sitter space satisfying $f(\kappa)=\sigma \in (1,\infty)$ with a prescribed compact future asymptotic boundary $\Gamma$ at infinity has at least one smooth solution (if $l=1$ or $l=2$ there is uniqueness). This is the exact analogue of the asymptotic plateau problem in Hyperbolic space and is in fact a precise dual problem. By using this duality we obtain for free the existence of strictly convex solutions to the asymptotic Plateau problem for $\sigma_l=\sigma,\,1 \leq l < n$ in both de Sitter and Hyperbolic space.

2012, 17(6): 2243-2266
doi: 10.3934/dcdsb.2012.17.2243

*+*[Abstract](1634)*+*[PDF](500.8KB)**Abstract:**

The study of spatially explicit integro-difference systems when the local population dynamics are given in terms of discrete-time generations models has gained considerable attention over the past two decades. These nonlinear systems arise naturally in the study of the spatial dispersal of organisms. The brunt of the

*mathematical*research on these systems, particularly, when dealing with

*cooperative*systems, has focused on the study of the existence of traveling wave solutions and the characterization of their spreading speed. Here, we characterize the minimum propagation (spreading) speed, via the convergence of initial data to wave solutions, for a large class of

*non cooperative*nonlinear systems of integro-difference equations. The spreading speed turns out to be the slowest speed from a family of non-constant traveling wave solutions. The applicability of these theoretical results is illustrated through the explicit study of an integro-difference system with

*local*population dynamics governed by Hassell and Comins' non-cooperative competition model (1976). The corresponding integro-difference nonlinear systems that results from the redistribution of individuals via a dispersal kernel is shown to satisfy conditions that guarantee the existence of minimum speeds and traveling waves. This paper is dedicated to Avner Friedman as we celebrate his immense contributions to the fields of partial differential equations, integral equations, mathematical biology, industrial mathematics and applied mathematics in general. His leadership in the mathematical sciences and his mentorship of students and friends over several decades has made a huge difference in the personal and professional lives of many, including both of us.

2012, 17(6): 2267-2280
doi: 10.3934/dcdsb.2012.17.2267

*+*[Abstract](1142)*+*[PDF](367.9KB)**Abstract:**

It is shown how to construct criteria of the form $f(u)\le f'(0)K(u)$ which guarantee that the spreading speed $c^*$ of a reaction-diffusion equation with the reaction term $f(u)$ is linearly determinate in the sense that $c^*=2\sqrt{f'(0)}$. Some of these criteria improve the classical condition $f(u)\le f'(0)u$, and permit the presence of sharp Allee effects. Inequalities which guarantee the failure of linear determinacy are also presented.

2012, 17(6): 2281-2298
doi: 10.3934/dcdsb.2012.17.2281

*+*[Abstract](1207)*+*[PDF](679.9KB)**Abstract:**

Integrodifference equations are discrete in time and continuous in space, and are used to model the spread of populations that are growing in discrete generations, or at discrete times, and dispersing spatially. We investigate optimal harvesting strategies, in order to maximize the profit and minimize the cost of harvesting. Theoretical results on the existence, uniqueness and characterization, as well as numerical results of optimized harvesting rates are obtained. The order of how the three events, growth, dispersal and harvesting, are arranged affects the harvesting behavior.

2017 Impact Factor: 0.972

## Readers

## Authors

## Editors

## Referees

## Librarians

## More

## Email Alert

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

[Back to Top]