Showing papers in "Ricerche Di Matematica in 2016"

Solving Volterra’s population growth model of arbitrary order using the generalized fractional order of the Chebyshev functions

Abstract: Volterra’s model for population growth in a closed system includes an integral term to indicate accumulated toxicity in addition to the usual terms of the logistic equation, that occurs in ecology. In this paper, a new numerical approximation is introduced for solving this model of arbitrary (integer or fractional) order. The proposed numerical approach is based on the generalized fractional order Chebyshev orthogonal functions of the first kind and the collocation method. Accordingly, we employ a collocation approach, by computing through Volterra’s population model in the integro-differential form. This method reduces the solution of a problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show that the new method is efficient and applicable.

Super-critical and sub-critical bifurcations in a reaction-diffusion Schnakenberg model with linear cross-diffusion

Abstract: In this paper the Turing pattern formation mechanism of a two components reaction-diffusion system modeling the Schnakenberg chemical reaction is considered. In Ref. (Madzavamuse et al., J Math Biol 70(4):709–743, 2015) it was shown how the presence of linear cross-diffusion terms favors the destabilization of the constant steady state. We perform the weakly nonlinear multiple scales analysis to derive the equations for the amplitude of the Turing patterns and to show how the cross-diffusion coefficients influence the occurrence of super-critical or sub-critical bifurcations. We present a numerical exploration of far from equilibrium regimes and prove the existence of multistable stationary solutions.

Weighted fully measurable grand Lebesgue spaces and the maximal theorem

Abstract: Anatriello and Fiorenza (J Math Anal Appl 422:783–797, 2015) introduced the fully measurable grand Lebesgue spaces on the interval $$(0,1)\subset \mathbb R$$ , which contain some known Banach spaces of functions, among which there are the classical and the grand Lebesgue spaces, and the $$EXP_\alpha $$ spaces $$(\alpha >0)$$ . In this paper we introduce the weighted fully measurable grand Lebesgue spaces and we prove the boundedness of the Hardy–Littlewood maximal function. Namely, let $$\begin{aligned} \Vert f\Vert _ {p[\cdot ],\delta (\cdot ), w}={{\mathrm{ess\,sup}}}_{x\in (0,1)} \left( \int _0^1 (\delta (x)f(t))^{p(x)} w(t)\mathrm{dt}\right) ^{\frac{1}{p(x)}}, \end{aligned}$$ where w is a weight, $$0<\delta (\cdot )\le 1\le p(\cdot )<\infty $$ , we show that if $$\displaystyle {p^+}=\Vert p\Vert _\infty <+\infty $$ , the inequality $$\begin{aligned}\Vert Mf\Vert _{p[\cdot ],\delta (\cdot ),w} \le c\Vert f\Vert _{p[\cdot ],\delta (\cdot ),w} \end{aligned}$$ holds with some constant c independent of f if and only if the weight w belongs to the Muckenhoupt class $$A_{p^+}$$ .

An operatorial model for long-term survival of bacterial populations

Abstract: This paper deals with the application of the operatorial techniques of quantum physics to model a closed ecosystem in a two-dimensional region. In particular, we consider a model with four compartments, represented by fermionic operators, distributed on a finite square grid. The dynamics is governed by a self-adjoint Hamiltonian containing the interactions among the compartments (nutrients, bacteria and two different kinds of garbage) as well as some diffusive phenomena along the cells.

Porous MHD convection: stabilizing effect of magnetic field and bifurcation analysis

Abstract: MHD convection in a horizontal porous-layer filled by a plasma, imbedded in a transverse magnetic field and heated from below, is investigated. The critical Rayleigh number is found and, in simple algebraic closed forms, conditions necessary and sufficient for the onset of steady and oscillatory convection are obtained. It is shown that the stabilizing effect of the magnetic field grows with $$Q^2$$ , Q being the Chandrasekhar number. The linearization principle (Rionero, Rend Lincei Mat Appl 25:368, 2014): “Decay of linear energy for any initial data implies decay of nonlinear energy at any instant” continues to hold also in the case at stake and allows to recover for the global nonlinear stability the conditions of linear stability.

Recent results on nonlinear extended thermodynamics of real gases with six fields Part I: general theory

Abstract: We review the recently developed theory of extended thermodynamics (ET) of real gases with six independent fields, i.e., the mass density, the velocity, the temperature and the dynamic pressure, without adopting near-equilibrium approximation. We discuss the polytropic and non-polytropic cases of rarefied polyatomic gases in detail, including the closure via nonlinear molecular ET.

Band gap engineering of graphene through quantum confinement and edge distortions

Abstract: Based on the density functional theory approach we explore the chances endured by energy gap (EG) of semiconducting (armchair) graphene nanoribbons (AGNRs) when Stone-Wales (SW) defects are placed inside their lattices. Our results show that the AGNRs, which belong to the $$3\hbox {m} + 2$$ family experience an increase in their EG value. On the other hand, those belonging to 3m and $$3\hbox {m} + 1$$ families experience decrease in their EG. The maximum observed EG for pristine and distorted ribbons were $$\sim $$ 2.6 and $$\sim $$ 1.6 eV, respectively. Our results can be useful to understand the semiconducting properties of wider graphene nanoribbons which are already available experimentally.

Generalized Riemann problems and exact solutions for p-systems with relaxation

Abstract: In this paper a p-system with relaxation is considered. Within the theoretical framework of the differential constraints method, we determine two possible classes of exact solutions of the governing model both parameterized by one arbitrary function. This allows to solve classes of initial value problems of interest in nonlinear wave propagation. In fact a generalized Riemann problem is solved by determining a smooth solution which plays the role of the well known rarefaction wave of the homogeneous case.

Approximation method for solving fixed point problem of Bregman strongly nonexpansive mappings in reflexive Banach spaces

Abstract: In this paper, we introduce a new iteration process for solving fixed point problem of Bregman strongly nonexpansive mappings, and then we study a strong convergence theorem for a common fixed point of Bregman strongly nonexpansive mappings in the framework of reflexive Banach spaces.

Modelling multiple relapses in drug epidemics

Abstract: Drug dependence is a ‘chronic disease’ treatable through rehabilitation. Many drug addicts progress through a series of rehabilitation and relapsing episodes. In this paper, we formulate a mathematical model with n-alternate stages of rehabilitation and relapsing. The dynamics of drug abuse are treated as an infectious disease that spreads through a population. The model analysis shows that the model has two equilibria, the drug free equilibrium and the drug persistent equilibrium, that are both globally stable when the threshold $${\mathcal {R}}_{0}<1$$ and $${\mathcal {R}}_{0}>1$$ respectively. The model is fitted to data on individuals under repeated rehabilitation and parameter values that give the best fit chosen. The projections carried out the long term trends of proportions for repeated rehabilitants. The relative impact for each subgroup is determined to find out which population subgroup is responsible for a disproportionate number of initiations. The results have huge implications to designing policies aligned to rehabilitation processes.

The fractional Fisher information and the central limit theorem for stable laws

Abstract: A new information-theoretic approach to the central limit theorem for stable laws is presented. The main novelty is the concept of relative fractional Fisher information, which shares most of the properties of the classical one, included Blachman-Stam type inequalities. These inequalities relate the fractional Fisher information of the sum of n independent random variables to the information contained in sums over subsets containing \(n-1\) of the random variables. As a consequence, a simple proof of the monotonicity of the relative fractional Fisher information in central limit theorems for stable law is obtained, together with an explicit decay rate.

Extended generalized \((Zakh\frac{G^{\prime }}{G})\)-expansion method for solving the nonlinear quantum Zakharov–Kuznetsov equation

Abstract: In this article, we apply the extended generalized \((\frac{G^{\prime }}{G})\)-expansion method combined with the Jacobi elliptic equation to find new exact solutions of the nonlinear quantum Zakharov–Kuznetsov (QZK) equation with the aid of computer algebraic system Maple. Soliton solutions, periodic solutions, rational functions solutions and Jacobi elliptic functions solutions are obtained. Based on reductive perturbation technique and a series of transformation, the nonlinear QZK had been derived by many authors which can be reduced to a nonlinear ordinary differential equation (ODE) using the wave transformation. The extended generalized \((\frac{G^{\prime }}{G})\)-expansion method is straightforward and concise, and it can also be applied to other nonlinear PDEs in mathematical physics.

An improved model of alliances between political parties

Abstract: We consider an operatorial model of alliances between three political parties which interact with their electors, with the undecided voters, and with the electors of the other parties. This extends what was done in a previous paper, where this last type of interactions was not considered. Of course, taking them into account makes the system closer to real life. To produce an exactly solvable model, we restrict here to quadratic Hamiltonians, so that the equations of motion turn out to be linear. The dynamics of the so-called decision functions are deduced, and some explicit situations are considered in details.

Importance of Darcy or Brinkman laws upon resonance in thermal convection

Abstract: Thermal convection is analysed in a layer of Brinkman or Darcy porous material when the buoyancy force is quadratic in the temperature field and there is also present a constant heat source. In this situation resonance may occur in the sense that convective motion may commence simultaneously in two separate layers in the porous medium. It is shown that whether resonance occurs or not depends crucially on whether a Brinkman or Darcy law holds and this indicates how important it is to understand the model for flow in porous media.

Modelling the effects of malaria infection on mosquito biting behaviour and attractiveness of humans

Abstract: We develop and analyse a deterministic population-based ordinary differential equation of malaria transmission to consider the impact of three common assumptions of malaria models: (1) malaria infection does not change the attractiveness of humans to mosquitoes; (2) exposed mosquitoes (infected with malaria but not yet infectious to humans) have the same biting rate as susceptible mosquitoes; and (3) mosquitoes infectious to humans have the same biting rate as susceptible mosquitoes. We calculate the basic reproductive number, \(R_0\), for this model and show the existence of a transcritical bifurcation at \(R_0=1\), in common with most epidemiological models. We further show that for some sets of parameter values, this bifurcation can be backward (subcritical). We show with numerical simulations that increasing the relative attractiveness of infectious humans, increases \(R_0\) but reduces the equilibrium prevalence of infectious humans; decreasing the biting rate of exposed mosquitoes increases \(R_0\) and the equilibrium prevalence of infectious humans and mosquitoes; and increasing the biting rate of infectious mosquitoes has no impact on \(R_0\) or the equilibrium prevalence of infectious humans, but decreases the infectious prevalence of infectious mosquitoes. These analyses of a simple malaria model show that common assumptions around the relative attractiveness of infectious humans and the relative biting rates of exposed and infectious mosquitoes can have substantial and counter-intuitive effects on malaria transmission dynamics.

The non-linear energy stability of Brinkman thermosolutal convection with reaction

Abstract: We use the energy method to obtain the non-linear stability threshold for thermosolutal convection porous media of Brinkman type with reaction. The obtained non-linear boundaries for different values of the reaction terms are compared with the relevant linear instability boundaries obtained by Wang and Tan (Phys Lett A 373:776–780, 2009). Using the energy theory we obtain the non-linear stability threshold below which the solution is globally stable. The compound matrix numerical technique is implemented to solve the associated system of equations with the corresponding boundary conditions. Two systems are investigated, the heated below salted above case and the heated below salted below case. The effect of the reaction terms and Brinkman term on the Rayleigh number is discussed and presented graphically.

On a class of moving boundary problems for the potential mkdV equation: conjugation of Bäcklund and reciprocal transformations

Abstract: A class of moving boundary problems for the solitonic potential mkdV equation is set down which is reciprocal to generalised Stefan-type problems that admit exact solution in terms of a Painleve II symmetry reduction. The latter allows the construction of the exact solution of the moving boundary problems via the iterated action of a Backlund transformation. This results in solution representation in terms of Yablonski–Vorob’ev polynomials. It is indicated how the results may be extended, via a novel reciprocal link, to moving boundary problems for a canonical solitonic equation generated as a member of the WKI inverse scattering system.

Population dispersal and Allee effect

Abstract: This paper studies influences of population dispersal on the dynamics of populations that live in patches and grow under Allee effect. Analytical conditions for the global stability of the model in the case of weak Allee effect are established by using the theory of monotonic dynamical systems. Numerical simulations are provided for the case of two patches and strong Allee effect, which reveal that a moderate migration to the better patch is beneficial to overall population, whereas a larger one is harmful.

Nonlinear parabolic inequalities in Lebesgue-Sobolev spaces with variable exponent

Abstract: We give an existence result of the obstacle parabolic problem: $$\begin{aligned} \left\{ \begin{array}{ll} u\ge \psi &{}\ \ \text{ a.e. } \text{ in } \ \Omega \times (0,T)\\ \displaystyle \frac{\partial b(x,u)}{\partial t} -div(a(x,t,u, abla u))+div(\phi (x,t,u)) =f &{}\ \ \text{ in }\ \Omega \times (0,T) .\\ \end{array} \right. \end{aligned}$$ The main contribution of our work is to prove the existence of an entropy solution without the coercivity condition on \(\phi \). The second term f belongs to \(L^{1}(\Omega \times (0,T))\) and \(b(.,u_0)\in L^1(\Omega )\). The proof is based on the penalization methods.

Recent results on nonlinear extended thermodynamics of real gases with six fields Part II: shock wave structure

Abstract: On the basis of the nonlinear extended thermodynamics theory discussed in Part I, the shock wave structure in a rarefied non-polytropic gas is analyzed. It is found that the effect of nonlinearity in the constitutive equation on the shock wave structure becomes significant only when the Mach number is large. The deviation from the exponential decay in the relaxation profile of the mass density for large Mach numbers is also discussed.

Kernel estimates for Schrödinger type operators with unbounded coefficients and critical exponents

Abstract: We consider the Schrodinger type operator $$(1+|x|^\alpha )\Delta +c|x|^{\alpha -2}$$ , for $$\alpha > 2$$ , $$c<0$$ and $$N>2$$ . Heat kernel estimates of the associated semigroup are obtained using the equivalence between weighted Nash inequalities and “weighted” ultracontractivity of a symmetric Markov semigroup. Moreover we give estimates of the eigenfunctions of the operator for large values of |x|.

Wavefront invasion for a chemotaxis model of Multiple Sclerosis

Abstract: In this work we study wavefront propagation for a chemotaxis reaction-diffusion system describing the demyelination in Multiple Sclerosis. Through a weakly non linear analysis, we obtain the Ginzburg–Landau equation governing the evolution of the amplitude of the pattern. We validate the analytical findings through numerical simulations. We show the existence of traveling wavefronts connecting two different steady solutions of the equations. The proposed model reproduces the progression of the disease as a wave: for values of the chemotactic parameter below threshold, the wave leaves behind a homogeneous plaque of apoptotic oligodendrocytes. For values of the chemotactic coefficient above threshold, the model reproduces the formation of propagating concentric rings of demyelinated zones, typical of Balo’s sclerosis.

Convection in multi-component rotating fluid layers via the auxiliary system method

Abstract: The onset of thermal convection in a uniformly rotating horizontal layer filled by a Navier–Stokes multi-component fluid mixture, heated from below and salted partly from above and partly from below, is investigated via the new approach named auxiliary system method Rionero (Rend Lincei Mat Appl 25:1–44, 2014). In the free–free case, via the generalization of the Rionero Linearization Principle: “Decay of linear energy for any initial data implies decay of nonlinear energy at any instant” [given in Rionero (Rend Lincei Mat Appl 25:1–44, 2014) in the absence of rotation], it is shown that conditions guaranteeing linear stability of thermal conduction solution guarantee also absence of subcritical instabilities and global exponential asymptotic nonlinear stability. The classical Benard problem is investigated via a procedure different from the celebrated one given in Chandrasekhar (Hydrodynamic and hydromagnetic stability, 1981).

Complex singularities in KdV solutions

Abstract: In the small dispersion regime, the KdV solution exhibits rapid oscillations in its spatio-temporal dependence. We show that these oscillations are caused by the presence of complex singularities that approach the real axis. We give a numerical estimate of the asymptotic dynamics of the poles.

On very short and intense laser–plasma interactions

Abstract: We briefly report on some results regarding the impact of very short and intense laser pulses on a cold, low-density plasma initially at rest, and the consequent acceleration of plasma electrons to relativistic energies. Locally and for short times the pulse can be described by a transverse plane electromagnetic travelling-wave and the motion of the electrons by a purely Magneto-Fluido-Dynamical model with a very simple dependence on the transverse electromagnetic potential, while the ions can be regarded as at rest; the Lorentz–Maxwell and continuity equations are reduced to the Hamilton equations of a Hamiltonian system with 1 degree of freedom, in the case of a plasma with constant initial density, or a collection of such systems otherwise. We can thus describe both the well-known wakefield behind the pulse and the recently predicted slingshot effect, i.e. the backward expulsion of high energy electrons just after the laser pulse has hit the surface of the plasma.

Extended thermodynamics of dense gases with at least 24 moments

Abstract: This article belongs to the research trend on dense and polyatomic gases which began with a well known article by Profs. Arima, Taniguchi, Ruggeri, and Sugiyama following the idea of considering two blocks of balance equations. Since a model with more moments better fits the experimental data, this case has been considered by Profs. Arima, Mentrelli and Ruggeri in the framework of the kinetic approach. The present article pursues a similar end in the framework of the macroscopic approach which is more general than the kinetic one. The general and explicit closure for the 24 moments model is found up to whatever order with respect to equilibrium. It gets to first base on which to build also the model with an arbitrary number of moments; this more general model heavily needs the results of the present article.

Groups satisfying the double chain condition on subnormal subgroups

Abstract: If $$\theta $$ is a subgroup property, a group G is said to satisfy the double chain condition on $$\theta $$ -subgroups if it admits no infinite double sequences $$\begin{aligned} \cdots

Onset of linear instability driven by electric currents in magnetic systems: a Lagrangian approach

Abstract: A Lagrangian formalism is used to compute the onset of linear instability in magnetic heterostructures subject to two competing dissipative phenomena: the intrinsic damping and the current-induced spin-transfer-torque. The small-amplitude precessional dynamics undergone by the magnetization vector at the excitation threshold is described in terms of linearized Lagrange equations which are recast as a complex generalized non-Hermitian eigenvalue problem. The numerical solution of such a problem allows to characterize those magnetic normal modes which become unstable when the “negative” losses induced by the electric current fully compensate the intrinsic “positive” ones. An illustrative example is also carried out in order to test the capability of the proposed method to determine accurately such an instability threshold when geometric or material properties are varied.

Cofiniteness and Artinianness of certain local cohomology modules

Abstract: Let R be a commutative Noetherian ring, I, J be two ideals of R, M be an R-module and \({\mathcal {S}}\) be a Serre class of R-modules. A positive answer to the Huneke’s conjecture is given for a Noetherian ring R and minimax R-module M of Krull dimension less than 3, with respect to \({\mathcal {S}}\). There are some results on cofiniteness and Artinianness of local cohomology modules with respect to a pair of ideals. For a ZD-module M of finite Krull dimension and an integer \(n\in {\mathbb {N}}\), if \({{\mathrm{\mathrm{H}}}}^{i}_{I,J}(M)\in {\mathcal {S}}\) for all \(i>n\), then \({{\mathrm{\mathrm{H}}}}^{i}_{I,J}(M)/{\mathfrak {a}}^{j}{{\mathrm{\mathrm{H}}}}^{i}_{I,J}(M)\in {\mathcal {S}}\) for any \({\mathfrak {a}}\in \tilde{W}(I,J)\), all \(i\ge n\), and all \(j\ge 0\). By introducing the concept of Serre cohomological dimension of M with respect to (I, J), for an integer \(r\in {\mathbb {N}}_0\), \({{\mathrm{\mathrm{H}}}}^{j}_{I,J}(R)\in {\mathcal {S}}\) for all \(j>r\) iff \({{\mathrm{\mathrm{H}}}}^{j}_{I,J}(M)\in {\mathcal {S}}\) for all \(j>r\) and any finite R-module M.

Infinitely many solutions for a nonlinear difference equation with oscillatory nonlinearity

Abstract: In this paper, we study a discrete nonlinear boundary value problem that involves a nonlinear term oscillating near the origin and a power-type nonlinearity $$u^p$$ . By using variational methods, we establish the existence of a sequence of non-negative weak solutions that converges to 0 if $$p\ge 1$$ . In the sublinear case, we prove that for all n positive integer, the problem has at least n weak solutions if the parameter lies in a certain range.