Abstract: The cube polynomial of a graph is the counting polynomial for the number of induced k-dimensional hypercubes (k?0). We determine the cube polynomial of Fibonacci cubes and Lucas cubes, as well as the generating functions for the sequences of these cubes. Several explicit formulas for the coefficients of these polynomials are obtained, in particular they can be expressed with convolved Fibonacci numbers. Zeros of the studied cube polynomials are explicitly determined. Consequently, the coefficients sequences of cube polynomials of Fibonacci and Lucas cubes are unimodal.

Abstract: The coupling of surface shape dynamics and surface composition for a material growing by electrodeposition has been found in recent modelling work by the authors, to give rise to a rich morphogenetic scenario. In this paper we concentrate on a systematic description of morphogenesis occurring during metal and alloy electrodeposition. First of all we give a brief presentation of the physical facts relevant to the generation of shapes in electrochemical growth processes, essentially the balance between change-transfer and mass-transport rates. Hence, we review the mathematical modelling work recently published by the authors, based on analytical and numerical studies of a system of reaction-diffusion equations for morphological and adsorption dynamics. Eventually, a selection of examples of experimental validation of the model by means of numerical simulations is proposed.

Abstract: In this paper, we consider the three-dimensional (3D) incompressible Boussinesq equations. We obtain some regularity criteria for the three-dimensional (3D) Boussinesq equations in the Morrey-Campanato space.

Abstract: In this paper we present a hydrodynamical model which, in principle, is able to describe charge transport in a generic compound semiconductor. The model makes use of an analytic approximation for the conduction bands. Energy dispersion relationships in the neighbors (valleys) of the lowest minima are, in fact, taken to be spherical, nonparabolic. The model considers the main scattering mechanisms in polar semiconductors, that is the acoustic, polar optical, intervalley non-polar optical phonon interactions and the ionized impurity scattering. Simulations are shown for the cases of bulk GaN and SiC.

Abstract: In view of solving in a closed form initial and/or boundary value problems of interest in nonlinear hyperbolic and dissipative wave processes it is considered a reduction approach based upon appending differential constraints to quasilinear nonhomogeneous hyperbolic systems of first order PDEs. In this context a governing model of traffic flow is analyzed thoroughly and a classification of possible constraints along with sets of consistent response functions involved therein is worked out whereupon the classes of corresponding exact solutions are determined. To some extent these solutions generalize the classical simple wave solutions and may also incorporate dissipative effects. Furthermore, in order to solve a Riemann Problem, an exact rarefaction wave-like solution is built. Finally an application of the results to the so-called “green traffic light problem” is also illustrated.

Abstract: We prove existence of solutions of a two-compressible (liquid and gas) phase flow model in porous media with two components (water and hydrogen). This model is obtained by writing the mass conservation for each component in each phase. We suppose that the mass exchange between dissolved hydrogen and hydrogen in the gas phase is supposed finite. This mass exchange is modeled by a source term on each mass conservation equations.

Abstract: This paper deals with the well-posedness of the b-family equation in analytic function spaces. Using the Abstract Cauchy-Kowalewski theorem we prove that the b-family equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic and it belongs to H s with s>3/2, and the momentum density u 0−u 0,xx does not change sign, we prove that the solution stays analytic globally in time, for b≥1. Using pseudospectral numerical methods, we study, also, the singularity formation for the b-family equations with the singularity tracking method. This method allows us to follow the process of the singularity formation in the complex plane as the singularity approaches the real axis, estimating the rate of decay of the Fourier spectrum.

Abstract: In this article, we continue our study of a system described by a class of initial boundary value problem (IBVP) of the Korteweg-de Vries (KdV) equation and the KdV Burgers (KdVB) equation posed on a finite interval with nonhomogeneous boundary conditions. While the system is known to be locally well-posed (Kramer et al. arXiv:1012.1057 , [2010]; Rivas et al. in Math. Control Relat. Fields 1:61---81, [2011]) and its small amplitude solutions are known to exist globally, it is not clear whether its large amplitude solutions would blow up in finite time or not. This problem is addressed in this article from control theory point of view: look for some appropriate feedback control laws (with boundary value functions as control inputs) to ensure that the finite time blow-up phenomena would never occur. In this article, a simple, but nonlinear, feedback control law is proposed and the resulting closed-loop system is shown not only to be globally well-posed, but also to be locally exponentially stable for the KdV equation and globally exponentially stable for the KdVB equation.

Abstract: The phenomenon of resonance will be dealt with from the viewpoint of dynamical systems depending on parameters and their bifurcations. Resonance phenomena are associated to open subsets in the parameter space, while their complement corresponds to quasi-periodicity and chaos. The latter phenomena occur for parameter values in fractal sets of positive measure. We describe a universal phenomenon that plays an important role in modelling. This paper gives a summary of the background theory, veined by examples.

Abstract: We study the behaviour of iterates of Mellin-Fejer type operators with respect to pointwise and uniform convergence. We introduce a new method in the construction of linear combinations of Mellin type operators using the iterated kernels. In some cases this provides a better order of approximation.

Abstract: Convection problem in anisotropic and inhomogeneous porous media has been analysed. In particular, the effect of variable permeability and thermal diffusivity with respect to the vertical direction, has been studied. Linear and nonlinear stability analysis of the conduction solution have been performed.

Abstract: The one-dimensional propagation of magnetic domain walls in ferromagnetic nanostrips is investigated in the framework of the extended Landau-Lifshitz-Gilbert equation which includes the effects of spin-polarized currents. The generalized model herein considered explicitly takes also into account two nonlinear mechanisms of dissipation, a rate-dependent viscous-like and a rate-independent dry-like, which are introduced for a better description of the relaxation processes in real samples. By adopting the traveling waves ansatz, we characterize the domain wall motion in two dynamical regimes, steady and precessional. The analytical results are also evaluated numerically in order to elucidate the corresponding physical implications.

Abstract: The paper deals with a third order semilinear equation which characterizes exponentially shaped Josephson junctions in superconductivity. The initial-boundary problem with Dirichlet conditions is analyzed. When the source term F is a linear function, the problem is explicitly solved by means of a Fourier series with properties of rapid convergence. When F is nonlinear, appropriate estimates of this series allow to deduce a priori estimates, continuous dependence and asymptotic behaviour of the solution.

Abstract: In this paper, we are concerned with the existence criteria for positive solutions of the following nonlinear arbitrary order fractional differential equations with deviating argument $$\left \{\begin{array}{l@{\quad}l}D_{0^+}^{\alpha}u(t)+h(t)f(u(\theta(t)))=0, & t\in ( 0,1 ),\ n-1 3 (n??), $D_{0^{+}}^{\alpha}$ is the standard Riemann-Liouville fractional derivative of order ?,f:[0,?)?[0,?), h(t):[0,1]?(0,?) and ?:(0,1)?(0,1] are continuous functions. Some novel sufficient conditions are obtained for the existence of at least one or two positive solutions by using the Krasnosel'skii's fixed point theorem, and some other new sufficient conditions are derived for the existence of at least triple positive solutions by using the fixed point theorems developed by Leggett and Williams etc. In particular, the existence of at least n or 2n?1 distinct positive solutions is established by using the solution intervals and local properties. From the viewpoint of applications, two examples are given to illustrate the effectiveness of our results.

Abstract: The option pricing problem when the asset is driven by a stochastic volatility process and in the presence of transaction costs leads to solving a nonlinear partial differential equation. The nonlinear term in the PDE reflects the presence of transaction costs. Under a particular market completion assumption we derive the nonlinear PDE whose solution may be used to find the price of options. In this paper under suitable conditions, we give an algorithmic scheme to obtain the solution of the problem by an iterative method and provide numerical solutions using the finite difference method.

Abstract: Classes of 2×2 first order quasilinear partial differential equations involving arbitrary continuously differentiable functions that can be mapped into autonomous and homogeneous form through equivalence transformations are considered. Equivalence transformations are point transformations of independent and dependent variables of differential equations involving arbitrary elements. The transformations act on the arbitrary elements as point transformations of an augmented space of independent, dependent variables and additional variables representing values taken by the arbitrary elements. Projecting the admitted symmetries into the space determined by the independent and dependent variables, we determine some finite transformations mapping the system into autonomous and homogeneous form. Some physical applications are considered and a comparison with reduction of quasilinear first order systems to autonomous and homogeneous form through Lie point symmetries is discussed.

Abstract: We briefly present new results on the existence and uniqueness of solutions u of a large class of initial-boundary-value problems characterized by a quasi-linear third order equation on a finite space interval with Dirichlet, Neumann or pseudoperiodic boundary conditions. The class includes equations arising in Superconductor Theory and in the Theory of Viscoelastic Materials.

Abstract: We consider an epidemic model for the dynamics of a vaccine-preventable disease, which incorporates the treatment and an imperfect vaccine given to susceptible individuals. We show that in spite of the simple structure of the model, a backward bifurcation may always occur if the treatment rate is above a threshold value. This occurs regardless of the specific form of the force of infection, which is only required to be infinitesimal of the same order of the size of the infectious compartment I, as I→0. This includes many commonly used functionals, as the linear, the monotone saturating Michaelis-Menten and the non-monotone force of infection used to represent the ‘psychological effect’.

Abstract: The fourth order nonlinear differential equations A with regularly varying coefficient q(t) are studied in the framework of regular variation. It is shown that thorough and complete information can be acquired about the existence of all possible regularly varying solutions of (A) and their accurate asymptotic behavior at infinity.

Abstract: In this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for the second order Hamiltonian system on time scale $\mathbb{T}$ $$\left\{\begin{array}{[email protected]{\quad}l}u^{\Delta^{2}}(t)+A(\sigma(t))u(\sigma(t))+ abla F(\sigma(t),u(\sigma(t)))=0,& \hbox{\ $\Delta$-a.e. $t\in [0,T]_{_{\mathbb{T}}}^{\kappa}$,} \\u(0)-u(T)=0,\qquad u^{\Delta}(0)-u^{\Delta}(T)=0,& \hbox{}\end{array}\right.$$ where u Δ(t) denotes the delta (or Hilger) derivative of u at t, $u^{\Delta^{2}}(t)=(u^{\Delta})^{\Delta}(t)$ , � is the forward jump operator, T is a positive constant, A(t)=[d ij (t)] is a symmetric N�N matrix-valued function defined on $[0,T]_{\mathbb{T}}$ with $d_{ij}\in L^{\infty}([0,T]_{\mathbb{T}},\mathbb{R})$ for all i,j=1,2,�,N, and $F:[0,T]_{\mathbb{T}}\times \mathbb{R}^{N}\rightarrow\mathbb{R}$ . By establishing a proper variational setting, two existence results and two multiplicity results are obtained. Finally, three examples are presented to illustrate the feasibility and effectiveness of our results.

Abstract: Given a class $\mathcal{F(\theta)}$ of differential equations with arbitrary element ?, the problems of symmetry group, nonclassical symmetry and conservation law classifications are to determine for each member $f\in\mathcal{F(\theta)}$ the structure of its Lie symmetry group G f , conditional symmetry Q f and conservation law $\mathop {\rm CL} olimits _{f}$ under some proper equivalence transformations groups. In this paper, an extensive investigation of these three aspects is carried out for the class of variable coefficient (1+1)-dimensional nonlinear telegraph equations with coefficients depending on the space variable f(x)u tt =(g(x)H(u)u x ) x +h(x)K(u)u x . The usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements are first constructed. Then using the technique of variable gauges of arbitrary elements under equivalence transformations, we restrict ourselves to the symmetry group classifications for the equations with two different gauges g=1 and g=h. In order to get the ultimate classification, the method of furcate split is also used and consequently a number of new interesting nonlinear invariant models which have non-trivial invariance algebra are obtained. As an application, exact solutions for some equations which are singled out from the classification results are constructed by the classical method of Lie reduction. The classification of nonclassical symmetries for the classes of differential equations with gauge g=1 is discussed within the framework of singular reduction operator. This enabled to obtain some exact solutions of the nonlinear telegraph equation which are invariant under certain conditional symmetries. Using the direct method, we also carry out two classifications of local conservation laws up to equivalence relations generated by both usual and extended equivalence groups. Equivalence with respect to these groups and correct choice of gauge coefficients of equations play the major role for simple and clear formulation of the final results.

Abstract: In this paper we discuss the existence and global attractivity of k-pseudo almost automorphic sequence solution of a model of bidirectional cellular neural networks. We consider the corresponding difference equation analogue of the model system using suitable discretization method and obtain certain conditions for the existence of solution. The k-pseudo almost automorphic sequence solutions generalize the results of pseudo almost periodic, almost periodic and almost automorphic sequences solutions. Moreover the results proved in this paper are new and compliment the existing one.

Abstract: We study the duality of reconstruction systems, which are g-frames in a finite dimensional setting. These systems allow redundant linear encoding-decoding schemes implemented by the so-called dual reconstruction systems. We are particularly interested in the projective reconstruction systems that are the analogue of fusion frames in this context. Thus, we focus on dual systems of a fixed projective system that are optimal with respect to erasures of the reconstruction system coefficients involved in the decoding process. We consider two different measures of the reconstruction error in a blind reconstruction algorithm. We also study the projective reconstruction system that best approximate an arbitrary reconstruction system, based on some well known results in matrix theory. Finally, we present a family of examples in which the problem of existence of a dual projective system of a reconstruction system of this type is considered.

Abstract: We study the periodic homogenization for a family of functionals defined on Orlicz-Sobolev spaces. One fundamental in this topic is to extend the classical compactness results of the two-scale convergence method to this type of spaces.

Abstract: Formation of energy bands in the system of rotation-vibration quantum states of molecules is described within semi-quantum models under the presence of a symmetry group characterizing the equilibrium molecular configuration. Effective rotation-vibration Hamiltonians are written in two-quantum state models with rotational variables treated as classical ones. Eigen-line bundles associated with eigenvalues of 2×2 Hermitian matrix defined on rotational classical phase space which is a two-dimensional sphere are characterized by the first Chern class. Explicit procedure for the calculation of Chern numbers are suggested and realized for a family of Hamiltonians depending on extra control parameters in the presence of symmetry. Effective Hamiltonians for two vibrational states transforming according to some representations of the cubic symmetry group are studied. Chern numbers are evaluated for respective model Hamiltonians. The iso-Chern diagrams are introduced which split the parameter space into regions with fixed Chern numbers.

Abstract: In this article, we address the question of relating the stability properties of an operator with the stability properties of its associate symmetric operator. The linear-algebra results of Bendixson and Hirsch indicate that the symmetric part of a matrix is always less stable than the matrix itself. We show that in a variety of cases, including infinite dimensional cases associated to systems of PDEs, the same result is valid. We also discuss the applicability to non-autonomous systems, and we show that, in general, this result is not valid. We also review some of the literature that in these years has appeared on the subject.

Abstract: The magnetorotational instability (MRI) triggers turbulence and enables outward transport of angular momentum in hydrodynamically stable rotating shear flows, e.g., in accretion disks. What laws of differential rotation are susceptible to the destabilization by axial, azimuthal, or helical magnetic field? The answer to this question, which is vital for astrophysical and experimental applications, inevitably leads to the study of spectral and geometrical singularities on the instability threshold. The singularities provide a connection between seemingly discontinuous stability criteria and thus explain several paradoxes in the theory of MRI that were poorly understood since the 1950s.

Abstract: The central problem in dynamical systems is the asymptotic behavior or topological structure of the orbits. Nevertheless only orbits of points with certain recurrence and form a set of full measure are truly of importance. Of course, such a set is desired to be as small (in the sense of set inclusion) as possible. In this paper we discuss such two sets: the set of weakly almost periodic points and the set of quasi-weakly almost periodic points. While the two sets are different from each other by definitions, we prove that their closures both coincide with the measure center (or the minimal center of attraction) of the dynamical systems. Generally, a point may have three levels of orbit-structure: the support of an invariant measure generated by the point, its minimal center of attraction and its ?-limit set. We study the three levels of orbit-structure for weakly almost periodic points and quasi-weakly almost periodic points. We prove that quasi-weakly almost periodic points possess especially rich topological orbit-structures. We also present a necessary and sufficient condition for a point to belong to its own minimal center of attraction.

Abstract: In order to achieve prescribed drug release kinetics some authors have been investigating bi-phasic and possibly multi-phasic releases from blends of biodegradable polymers. Recently, experimental data for the release of paclitaxel have been published by Lao et al. (Lao and Venkatraman in J. Control. Release 130:9---14, 2008; Lao et al. in Eur. J. Pharm. Biopharm. 70:796---803, 2008). In Blanchet et al. (SIAM J. Appl. Math. 71(6):2269---2286, 2011) we validated a two-parameter quadratic ordinary differential equation (ODE) model against their experimental data from three representative neat polymers. In this paper we provide a gradient flow interpretation of the ODE model. A three-dimensional partial differential equation (PDE) model for the drug release in their experimental set up is introduced and its parameters are related to the ones of the ODE model. The gradient flow interpretation is extended to the study of the asymptotic concentrations that are solutions of the PDE model to determine the range of parameters that are suitable to simulate complete or partial drug release.