Showing papers in "Ricerche Di Matematica in 2017"

Cross validation of discontinuous Galerkin method and Monte Carlo simulations of charge transport in graphene on substrate

Abstract: The aim of this work is to simulate the charge transport in a monolayer graphene on a substrate. This requires the inclusion of the scatterings of the charge carriers with the impurities and the phonons of the substrate, besides the interaction mechanisms already present in the graphene layer. As physical model, the semiclassical Boltzmann equation will be assumed. Two approaches will be used for the simulations: a numerical scheme based on the Discontinuous Galerkin method for finding deterministic (non stochastic) solutions and a new Direct Monte Carlo Simulation formulated in Romano et al. (J Comput Phys 302:267–284, 2015) in order to deal in the appropriate way with the Pauli exclusion principle for degenerate Fermi gases. A cross validation of the deterministic and stochastic solutions shows the robustness and accuracy of both the approaches.

On kinetic models for polyatomic gases and their hydrodynamic limits

Abstract: Starting from a kinetic Boltzmann or BGK description of a polyatomic gas, on the basis of a discrete structure of internal energy levels, an asymptotic Chapman–Enskog analysis is performed in the continuum limit in order to achieve consistent fluid-dynamic Navier–Stokes equations for the macroscopic fields. Among the various transport coefficients, emphasis is given to the dynamical pressure, which characterizes molecules with non-translational degrees of freedom, and which vanishes in the mono-atomic limit.

On an anisotropic problem with singular nonlinearity having variable exponent

Abstract: We consider the following anisotropic problem, with singular nonlinearity having a variable exponent $$\begin{aligned} \left\{ \begin{array}{ll} -\sum \limits _{i=1}^{N}\partial _{i}\left[ \left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\right] =\frac{f}{u^{\gamma (x) }} &{} \quad in~\Omega , \\ u=0 &{} \quad on~\Omega , \\ u\ge 0 &{} \quad in~\Omega ; \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a bounded regular domain in $${\mathbb {R}}^{N}$$ and $$\gamma (x)>0$$ is a smooth function, having a convenient behavior near $$\partial \Omega .$$ f is assumed to be a non negative function belonging to a suitable Lebesgue space $$L^{m}\left( \Omega \right) .$$ We will also assume without loss of generality that $$2\le p_{1}\le p_{2}\le \cdots \le p_{N}.$$ Using approximation techniques, we obtain existence and regularity of positive solutions to the considered problem.

Shock structure and multiple sub-shocks in binary mixtures of Eulerian fluids

Abstract: The problem of sub-shock formation within a shock structure solution of hyperbolic systems of balance laws is investigated for a binary mixture of multi-temperature Eulerian fluids. The main purpose of this work is the analysis of the ranges of Mach numbers characterizing shock-structure solutions with different features, continuous or not, and to show the existence of ranges, below the maximum unperturbed characteristic velocity, for which each constituent of the mixture may develop a sub-shock within a smooth shock structure profile. The theoretical results are supported by numerical calculations.

Stability of hydromagnetic laminar flows in an inclined heated layer

Abstract: Laminar flows of conducting fluids with an imposed magnetic field play an important role in many applications, for instance in geophysics, astrophysics, e.g. when dealing with solar winds, industry, biology, in metallurgy, in biofilms, etc. Also many engineering applications require heating at the boundaries. The inclination has been examined by some authors mainly in theoretical applications, geophysical studies, and materials processing. In Falsaperla et al. (Laminar hydromagnetic flows in an inclined heated layer, 2016) we have investigated analytical solutions of stationary laminar flows of an inclined layer filled with a hydromagnetic fluid heated from below and subject to the gravity field. In this article we study linear instability and nonlinear stability of some of the above solutions and investigate the critical stability/instability thresholds.

Existence and uniqueness results for a class of singular elliptic problems in perforated domains

Abstract: In this paper we prove some existence, uniqueness and regularity results for a weak solution of a quasilinear elliptic nonlinear problem posed in a domain perforated by one or more holes. The nonlinear term is of the form $$f\zeta (u)$$ , where $$\zeta (s)$$ is singular at $$s=0$$ . On the boundary of the holes we impose a nonlinear Robin condition, while on the exterior boundary we prescribe a homogeneous Dirichlet condition. The difficulty here arises in dealing simultaneously with the quasilinear matrix field, the singular datum and the nonlinear Robin condition. To show the existence of a solution we approximate the problem with a sequence of nonsingular problems for which the existence of a solution is proved via the Schauder fixed-point theorem. The main tool when passing to the limit in the approximate problem is to split the integral of the singular term into the sum of two integrals, one on the set where the solution is very close to the singularity and one where it is far from it. To obtain the uniqueness of the solution we need to require additional assumptions on the quasilinear term and a monotonicity property for the singular one. Under suitable stronger hypotheses on the data, we also prove the boundedness of the solution.

Qualitative analysis of the invasion free boundary problem in biofilms

Abstract: The work presents the qualitative analysis of the free boundary value problem related to the invasion model for multispecies biofilms. This model is based on the continuum approach for biofilm modeling and consists of a system of nonlinear hyperbolic partial differential equations for microbial species growth and spreading, a system of semilinear elliptic partial differential equations describing the substrate trends and a system of semilinear elliptic partial differential equations accounting for the diffusion and reaction of motile species within the biofilm. The free boundary evolution is regulated by a nonlinear ordinary differential equation. Overall, this leads to a free boundary value problem essentially hyperbolic. By using the method of characteristics, the partial differential equations constituting the invasion model are converted to Volterra integral equations. Then, the fixed point theorem is used for the uniqueness and existence result. The work is completed with numerical simulations describing the invasion of nitrite oxidizing bacteria in a biofilm initially constituted by ammonium oxidizing bacteria.

Nonlinear first order partial differential equations reducible to first order homogeneous and autonomous quasilinear ones

Abstract: A theorem providing necessary conditions enabling one to map a nonlinear system of first order partial differential equations to an equivalent first order autonomous and homogeneous quasilinear system is given. The reduction to quasilinear form is performed by constructing the canonical variables associated to the Lie point symmetries admitted by the nonlinear system. Some applications to relevant partial differential equations are given.

Regular solutions to the Navier–Stokes equations with an initial data in L(3,\infty )

Abstract: We consider the Navier–Stokes initial boundary value problem in exterior domains $$\Omega \subset \mathbb {R}^n$$ , $$n\ge 3$$ . We assume that the initial data belongs to $${\mathbb {L}}(n,\infty )$$ suitable subspace of $$L(n,\infty )$$ Lorentz space. We are able to prove on an interval (0, T) the existence of a unique regular solution, global in time for small data. The solution enjoys some new estimates and a new approach to the proof is exhibited.

Lie group analysis of a wave equation with a small nonlinear dissipation

Abstract: In this paper, the problem of finding approximate symmetries of a wave equation with a small nonlinear dissipation is investigated. It is also explored the relationship between the approximate symmetries of our model and other models.

Score functions, generalized relative Fisher information and applications

Abstract: Generalizations of the linear score function, a well-known concept in theoretical statistics, are introduced. As the Gaussian density and the classical Fisher information are closely related to the linear score, nonlinear (respectively fractional) score functions allow to identify generalized Gaussian densities (respectively Levy stable laws) as the (unique) probability densities for which the score of a random variable X is proportional to \(-X\). In all cases, it is shown that the variance of the relative to the generalized Gaussian (respectively Levy) score provides an upper bound for \(L^1\)-distance from the generalized Gaussian density (respectively Levy stable laws). Connections with nonlinear and fractional Fokker–Planck type equations are introduced and discussed.

Note on the existence theory for evolution equations with pseudo-monotone operators

Abstract: In this note we present a framework which allows to prove an abstract existence result for evolution equations with pseudo-monotone operators. The assumptions on the spaces and the operators can be easily verified in concrete examples.

A study on skew Hurwitz series rings

Abstract: In this paper, we continue the study of skew Hurwitz series ring \((HR, \alpha )\), where R is a ring equipped with an endomorphism \(\alpha \). Necessary and sufficient conditions are obtained for \((HR, \alpha )\) to satisfy a certain ring property which is among being local, semilocal, semiperfect, left quasi-duo, clean, exchange, right stable range one, 2-good, projective-free, semiprime, semiregular, I-ring, respectively. Furthermore, we prove that \((HR,\alpha )\) is a domain satisfying the ascending chain condition on principal left (resp. right) ideals if and only if R is a domain satisfying the ascending chain condition on principal left (resp. right) ideals, \(\alpha \) is injective and \(char(R)=0\). Finally, we study the prime radical of skew Hurwitz series ring.

Continuum approach to mathematical modelling of multispecies biofilms

Abstract: The work presents a contribution to the mathematical modelling of formation and growth of multispecies biofilms in the framework of continuum approach, without claiming to be complete. Mathematical models for biofilms often lead to consider free boundary value problems for nonlinear PDEs. The emphasis is on the qualitative analysis, uniqueness and existence of solutions and their main properties. Biofilm life is a complex biological process formed by several phases from the formation, development of colonies, attachment and detachment of microbial mass from (to) biofilm to (from) bulk liquid. Most of these processes are modelled and discussed. Moreover, some problems of interest for engineering and biological applications are considered. Indeed, we discuss the free boundary value problem related to biofilm reactors extensively used in wastewater treatment, and the invasion of new species into an already constituted biofilm with the successive colonizations. The main mathematical methodology used is the method of characteristics. The original differential problem is converted to integral equations. Then, the fixed point theorem is applied.

Exact solutions and optical soliton solutions for the nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity

Abstract: In this article, we apply the soliton ansatz method combined with the Jacobi elliptic equation method which is different from the F-expansion method to obtain several types of Jacobi elliptic function solutions, the optical bright-dark-singular soliton solutions and trigonometric function solutions of the nonlinear Schrodinger equation with fourth-order dispersion and cubic-quintic nonlinearity, self-steeping and self-frequency shift effects which describes the propagation of an optical pulse in optical fibers. Comparison between our results in this article and the well-known results are given.

On the annihilator-ideal graph of commutative rings

Abstract: Let R be a commutative ring with nonzero identity. We denote by AG(R) the annihilator graph of R, whose vertex set consists of the set of nonzero zero-divisors of R, and two distint vertices x and y are adjacent if and only if \(\mathrm {ann}(x) \cup \mathrm {ann}(y) e \mathrm {ann}(xy)\), where for \(t\in R\), we set \(ann(t) := \lbrace r\in R\ \vert \ rt=0\rbrace \). In this paper, we define the annihiator-ideal graph of R, which is denoted by \(A_{I}(R)\), as an undirected graph with vertex set \(A^*(R)\), and two distinct vertices I and J are adjacent if and only if \(\mathrm {ann}(I) \cup \mathrm {ann}(J) e \mathrm {ann}(IJ)\). We study some basic properties of \(A_{I}(R)\) such as connectivity, diameter and girth. Also we investigate the situations under which the graphs AG(R) and \(A_{I}(R)\) are coincide. Moreover, we examin the planarity of the graph \(A_{I}(R)\).

Some asymptotic limits of reaction–diffusion systems appearing in reversible chemistry

Abstract: This paper concerns reaction–diffusion systems consisting of three or four equations, which come out of reversible chemistry. We introduce different scalings for those systems, which make sense in various situations (species with very different concentrations or very different diffusion rates, chemical reactions with very different rates, etc.). We show how recently introduced mathematical tools allow to prove that the formal asymptotics associated to those scalings indeed hold at the rigorous level.

Classification of non-local rings with projective 3-zero-divisor hypergraph

Abstract: Let R be a commutative ring with identity and let Z(R, k) be the set of all k-zero-divisors in R and $$k>2$$ an integer. The k-zero-divisor hypergraph of R, denoted by $$\mathcal {H}_k(R)$$ , is a hypergraph with vertex set Z(R, k), and for distinct elements $$x_1,x_2,\ldots ,x_k$$ in Z(R, k), the set $$\{x_1,x_2, \ldots , x_k\}$$ is an edge of $$\mathcal {H}_k(R)$$ if and only if $$\prod \limits _{i=1}^kx_i =0$$ and the product of any $$(k-1)$$ elements of $$\{x_1,x_2,\ldots ,x_k\}$$ is nonzero. In this paper, we characterize all finite commutative non-local rings R with identity whose $$\mathcal {H}_3(R)$$ has crosscap one.

Uncertainty principles for continuous wavelet transforms related to the Riemann–Liouville operator

Abstract: In this paper we define and study the continuous wavelet transforms associated with the Riemann–Liouville operator, we give nice harmonic analysis results. Next we establish an analogue of Heisenberg’s inequality related to the wavelet transform. Last, we study the wavelet transform on subset of finite measures which deals with time frequency theory.

Existence of renormalized solution of nonlinear elliptic problems with lower order term in Orlicz spaces

Abstract: In this paper, we shall be concerned with the existence of renormalized solution of the following problem, $$\begin{aligned} \left\{ \begin{array}{l} -\text {div}\Big (a(x,u, abla u)\Big )-\text {div}(\Phi (x,u))= f \ \ \mathrm{in}\ \Omega ,\\ u=0 \text { on } \partial \Omega , \end{array} \right. \end{aligned}$$ with the second term f belongs to \(L^1(\Omega )\). The growth and the coercivity conditions on the monotone vector field a are prescribed by a N-function M. We assume any restriction on M, therefore we work with Orlicz-Sobolev spaces which are not necessarily reflexive. The lower order term \(\Phi \) is a Caratheodory function which is not coercive.

Graded semiprime submodules and graded semi-radical of graded submodules in graded modules

Abstract: Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. In this paper, we introduce several results concerning graded semiprime submodules. And we introduce the concept of graded semi-radical of graded submodule in graded module and give a number of its properties.

On attainability of optimal controls in coefficients for system of Hammerstein type with anisotropic p-Laplacian

Abstract: In this paper we consider an optimal control problem (OCP) for the coupled system of a nonlinear monotone Dirichlet problem with anisotropic p-Laplacian and matrix-valued \(L^\infty (\varOmega ,\mathbb {R}^{N\times N})\)-controls in its coefficients and a nonlinear equation of Hammerstein type. Using the direct method in calculus of variations, we prove the existence of an optimal control in considered problem and provide sensitivity analysis for a specific case of considered problem with respect to two-parameter regularization.

Non-equilibrium diffusion temperatures in mixture of gases via Maxwellian iteration

Abstract: This paper studies the difference of non-equilibrium temperatures between constituents, so-called diffusion temperature in the model of mixtures of gases in which each constituent has assigned its own velocity and temperature field. In a previous study (Ruggeri and Simic, Phys. Rev. E 80:026317, 2009), the constitutive equation for the diffusion temperatures, akin to Fick law for the diffusion flux, was derived by means of Maxwellian iteration. In the first order approximation the diffusion temperatures vanish if the constituents have the same ratio of specific heats, i.e. the molecules have the same number of degrees of freedom. This study proceeds to second iteration in the particular case of binary mixture, and provides a second order correction for the diffusion temperature. In this way the classical limit of non-equilibrium temperatures, which covers all the cases, is obtained at the lowest order. A comparison with known results is provided, revealing the corrections and generalizations brought by our results. A special case of constant pressure Fickian diffusion is analyzed and showed that diffusion temperature does not vanish as long as there exist mechanical diffusion between the constituents.

Transitions in a stratified Kolmogorov flow

Abstract: We study the Kolmogorov flow with weak stratification. We consider a stabilizing uniform temperature gradient and examine the transitions leading the flow to chaotic states. By solving the equations numerically we construct the bifurcation diagram describing how the Kolmogorov flow, through a sequence of transitions, passes from its laminar solution toward weakly chaotic states. We consider the case when the Richardson number (measure of the intensity of the temperature gradient) is $$Ri=10^{-5}$$ , and restrict our analysis to the range $$0

A speciality theorem for curves in $${\mathbb {P}}^5$$ contained in Noether–Lefschetz general fourfolds

Abstract: Let \(C\subset {\mathbb {P}}^r\) be an integral projective curve. We define the speciality index e(C) of C as the maximal integer t such that \(h^0(C,\omega _C(-t))>0\), where \(\omega _C\) denotes the dualizing sheaf of C. In the present paper we consider \(C\subset {\mathbb {P}}^5\) an integral degree d curve and we denote by s the minimal degree for which there exists a hypersurface of degree s containing C. We assume that C is contained in two smooth hypersurfaces F and G, with \(deg(F)=n>k=deg (G)\). We assume additionally that F is Noether–Lefschetz general, i.e. that the 2-th Neron–Severi group of F is generated by the linear section class. Our main result is that in this case the speciality index is bounded as \(e(C)\le {\frac{d}{snk}}+s+n+k-6.\) Moreover equality holds if and only if C is a complete intersection of \(T:=F\cap G\) with hypersurfaces of degrees s and \({\frac{d}{snk}}\).

Uniqueness of renormalized solutions for a class of parabolic equations

Abstract: In this paper, we prove uniqueness of renormalized solution for a class of doubly nonlinear parabolic problems. $$\begin{aligned} \left\{ \begin{array}{lll}\displaystyle \frac{\partial e^{\beta u}}{\partial t} - \triangle _{p}u + {\text {div}}(c(x,t)|u|^{\gamma -1}u) + d(x,t)| abla u|^{\delta -1} = f -{\text {div}}(F) &{} {\text { in }}\, Q_{T},\\ u(x,t)=0 &{} {\text { on }} \partial \varOmega \times (0,T),\\ e^{\beta (u(x,0))}=e^{\beta (u_{0}(x))} &{} {\text { in }} \varOmega . \end{array} \right. \end{aligned}$$ (1)

Perfect groups and normal subgroups related to an automorphism

Abstract: In this paper, we introduce the concept of \(\alpha \)-normal subgroup of a finite group G, where \(\alpha \) is an automorphism of G. We also introduce the concept of absolute normal subgroup and investigate all absolute normal subgroups of some groups. Furthermore, we define a group G\(\alpha \)-perfect if the \(\alpha \)-commutator subgroup of G coincides with G. We prove that for every finite abelian group G, there exists a finite abelian group H and \(\alpha \in Aut(H)\) such that \(D_{\alpha }(H)=G\).

Competing risks driven by Mittag-Leffler distributions, under copula and time transformed exponential model

Abstract: We consider a stochastic model for competing risks involving the Mittag-Leffler distribution, inspired by fractional random growth phenomena. We prove the independence between the time to failure and the cause of failure, and investigate some properties of the related hazard rates and ageing notions. We also face the general problem of identifying the underlying distribution of latent failure times when their joint distribution is expressed in terms of copulas and the time transformed exponential model. The special case concerning the Mittag-Leffler distribution is approached by means of numerical treatment. We finally adapt the proposed model to the case of a random number of independent competing risks. This leads to certain mixtures of Mittag-Leffler distributions, whose parameters are estimated through the method of moments for fractional moments.

The numerical solution of nonlinear integral equations of the second kind using thin plate spline discrete collocation method

Abstract: In this article, the thin plate splines are given to obtain the numerical solution of nonlinear Fredholm integral equations of the second kind. The scheme approximates the solution using the discrete collocation method based on the shape functions of thin plate splines constructed on a set of scattered points. The thin plate splines can be seen as a type of the free shape parameter radial basis functions which are used to establish an effective technique for the interpolation of an unknown function. The numerical method developed in the current paper utilizes the non-uniform Gauss–Legendre quadrature rule to compute its integrals. Since the proposed scheme does not require any background mesh for approximations and numerical integrations, it is meshless. This approach can be easily implemented and its algorithm is simple and effective to solve nonlinear integral equations. Moreover the error estimate and the convergence rate of the method are presented. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates.

Symmetric form for the hyperbolic-parabolic system of fourth-gradient fluid model

Abstract: The fourth-gradient model for fluids—associated with an extended molecular mean-field theory of capillarity—is considered. By producing fluctuations of density near the critical point like in computational molecular dynamics, the model is more realistic and richer than van der Waals’ one and other models associated with a second order expansion. The aim of the paper is to prove—with a fourth-gradient internal energy already obtained by the mean field theory—that the quasi-linear system of conservation laws can be written in an Hermitian symmetric form implying the stability of constant solutions. The result extends the symmetric hyperbolicity property of governing-equations’ systems when an equation of energy associated with high order deformation of a continuum medium is taken into account.