Showing papers in "Ricerche Di Matematica in 2015"

Some mathematical properties of the ROC curve and their applications

Abstract: In this paper we present ROC methodology and analyze the ROC curve. We describe first the historical background and its relation with signal detection theory. Some mathematical properties of this curve are given, and in particular the relation with stochastic orders and statistical hypotheses testing are described. We present also a medical application of the Neymann–Pearson lemma.

Piecewise Hermite interpolation via barycentric coordinates

Abstract: Piecewise interpolation methods, as spline or Hermite cubic interpolation methods, define the interpolating function by means of polynomial pieces and ensure that some regularity conditions hold at the break-points. In this paper, starting from a previous work, where we proposed a class of piecewise interpolating functions whose expression depends on the barycentric coordinates, we extend that approach to obtain an interpolant for which Hermite constraints are assigned. The underlying idea is to define the interpolant on each subinterval, by weighting two Taylor polynomials, centered at the endpoints, of a function that generated the Hermite conditions. The weights depend on a suitable function of the barycentric coordinates of the evaluation point. We show that the interpolating function inherits the properties of regularity from such a weight function. Moreover, we study the interpolation error and provide bounds in terms of both the degree of the Taylor polynomials and properties of the weight function. Numerical experiments confirm the theoretical results and show the reliability of the bounds provided for the interpolation error.

Closed-form solutions for the first-passage-time problem and neuronal modeling

Abstract: The Gauss–Diffusion processes are here considered and some relations between their infinitesimal moments and mean and covariance functions are remarked. The corresponding linear stochastic differential equations are re-written specifying the coefficient functions and highlighting their meanings in theoretical and application contexts. We resort the Doob-transformation of a Gauss–Markov process as a transformed Wiener process and we represent some time-inhomogeneous processes as transformed Ornstein–Uhlenbeck process. The first passage time problem is considered in order to discuss some neuronal models based on Gauss–Diffusion processes. We recall some different approaches to solve the first passage time problem specifying when a closed-form result exists and numerical evaluations are required when the latter is not available. In the contest of neuronal modeling, relations between firing threshold, mean behavior of the neuronal membrane voltage and input currents are given for the existence of a closed-form result useful to describe the firing activity. Finally, we collect in an unified way some models and the corresponding Gauss–Diffusion processes already considered by us in some previous papers.

Extended thermodynamics of dense gases in the presence of dynamic pressure

Abstract: Extended thermodynamics (ET) developed up to now fails when a gas is very dense and is composed of molecules with small internal degrees of freedom because the condition of convexity (stability) is violated. The aim of this paper is to explore a possible approach to construct an ET theory that is valid for any dense gas with the condition that it reduces to the usual ET theory when a gas is sufficiently rarefied. We restrict our study, for simplicity, within the simplest case in which the dissipation is only due to the dynamic pressure. Therefore the basic system of equations is the simplest variant of the Euler system, that is, the system composed of the equations for the conservation laws and an equation for the dynamic pressure (6-field theory).

On solving two higher-order nonlinear PDEs describing the propagation of optical pulses in optic fibers using the $$\left( \frac{G^{\prime }}{G},\frac{1}{G}\right) $$-expansion method

Abstract: The propagation of the optical solitons is usually governed by the nonlinear Schrodinger equations. In this article, the two variable $$(\frac{ G^{\prime }}{G},\frac{1}{G})$$ -expansion method is employed to construct the exact traveling wave solutions with parameters of two nonlinear PDEs namely, the (2 $$+$$ 1)-dimensional nonlinear cubic–quintic Ginzburg–Landau equation and the (1 $$+$$ 1)-dimensional resonant nonlinear Schrodinger’s equation with dual-power law nonlinearity which describe the propagation of optical pulses in optic fibers. When the parameters are replaced by special values, the well-known solitary wave solutions of these equations rediscovered from the traveling waves. This method can be thought of as the generalization of well-known original $$(\frac{G^{\prime }}{G})$$ -expansion method proposed by M. Wang et al. It is shown that the two variable $$(\frac{G^{\prime }}{G}, \frac{1}{G})$$ -expansion method provides a more powerful mathematical tool for solving many other nonlinear PDEs in mathematical physics.

L 2 -energy decay of convective nonlinear PDEs reaction-diffusion systems via auxiliary ODEs systems

Abstract: The $$L^2$$ -energy longtime behaviour of convective binary P.D.Es. reaction–diffusion systems under the action of: self and cross diffusion; nonlinear reaction terms and Robin boundary conditions, is investigated. An auxiliary ODEs system, depending on a positive parameter $$\mu $$ , is introduced. Via the energy decay of the auxiliary system, for general classes of nonlinear reaction terms, the absence of subcritical instabilities and the $$L^2$$ -energy asymptotic decay, are obtained. Estimates of the attraction basin are furnished. Instead of Sobolev type inequalities, only algebraic inequalities are requested. Applications to a celebrated model are furnished.

Numerical analysis of a Cahn–Hilliard type equation with dynamic boundary conditions

Abstract: We study space and time discretizations of a Cahn–Hilliard type equation with dynamic boundary conditions. We first study a semi-discrete version of the equation and we prove optimal error estimates in energy norms and weaker norms. Then, we study the stability of the fully discrete scheme obtained by applying the Euler backward scheme to the space semi-discrete problem. In particular, we show that this fully discrete problem is unconditionally stable. Some numerical results in two space dimensions conclude the paper.

Analysis of a mosquito–borne disease transmission model with vector stages and nonlinear forces of infection

Abstract: We study a mosquito-borne epidemic model where the vector population is distinct in aquatic and adult stages and a saturating effect of disease transmission is assumed to occur when the population of infectious carriers becomes large enough. A qualitative analysis, including centre manifold analysis, has been performed to determine the existence of stability–instability thresholds.

Global nonlinear stability and “cold convection instability” of non-constant porous throughflows, 2D in vertical planes

Abstract: Porous horizontal layers are considered. A class of non-constant porous throughflows, 2D in vertical planes, is obtained. The global stability conditions and the “cold convection instability” conditions are investigated.

Volterra integral equations on time scales: stability under constant perturbations via Liapunov direct method

Abstract: In this paper we consider Volterra integral equations on time scales and describe our study about the long time behavior of their solutions. We provide sufficient conditions for the stability under constant perturbations by using the direct Lyapunov method and we present some examples of application.

Some properties of the Riesz potentials in Dunkl analysis

Abstract: In the present paper, we study first the behavior at infinity of the Riesz potential for Dunkl transform of a non compactly supported function. Second, we give for \(1 < p \le q < +\infty \), weighted \(Lp\rightarrow Lq\) boundedness of the Riesz potentials with sufficient conditions. As application, we prove a weighted generalized Sobolev inequality.

A simple pointview for Kadec-1/4 theorem in the complex case

Abstract: This paper contains a simple and different pointview from the literature, in the best of my knowledge, to generalize well known results by Kadec (in R) and Duffin and Eachus (in C), concerning Riesz bases. Main goal of the present work is to overcome, atleastpartially,thelimitationsexhibited inthepaper ofDuffinandEachus and in the book of Young for the Riesz bases. A consequence of the main theorem and its corollary is that the constant log2 π can be replaced by 1/4 (for complex λn).

On some martingale inequalities for mean oscillations in weak spaces

Abstract: Given a Banach function space \(X\) over a nonatomic probability space, we define, in a natural way, a quasi-Banach function space which is denoted by \(\mathrm {w}\text {-}{X}\). In this paper, we consider some inequalities in \(\mathrm {w}\text {-}{X}\) involving the mean oscillation and the maximal function of a martingale. The main results give necessary and sufficient conditions for the inequalities to be valid.

Energy estimates for the Cauchy problem associated to a class of hyperbolic operators with double characteristics in presence of transition

Abstract: The aim of the paper is to establish energy estimates for solutions to the Cauchy problem for the class of hyperbolic second order operators with double char- acteristics in presence of transition P = D 20 − D 21 − (x0 + λ − α(x1)) 2 D 2 x2 .

Some questions concerning proper subrings

Abstract: Given a ring extension \(R\subset S\) of integral domains, it is shown that if each proper subring of \(S\) containing \(R\) is integrally closed, then \(S\) is integrally closed As an application, we show that if each proper subring of \(S\) containing \(R\) is a valuation (resp, Prufer, resp Principal ideal) domain, then so is \(S\)

Relativistic extended thermodynamics from the Lagrangian view-point

Abstract: The salient points of relativistic extended thermodynamics (R.E.T.) are here revised according to the Lagrangian view-point: attention is focused to each material particle and to its physical properties, during all the motion of the same particle. The results for the non relativistic case are already present in literature. Here a similar procedure is followed for R.E.T. with an arbitrary number of moments. It is also shown how the Einstenian Relativity Principle and some symmetry condition, which are present in the Eulerian view-point, can be “translated” in the Lagrangian view-point, where they are no more so self-evident.

Sharp interactions among A_\infty -weights on the real line

Abstract: We prove that any weight $$v\in L^1_{loc}({{\mathbb {R}}})$$ $$v:{{\mathbb {R}}}\rightarrow [0,+\infty )$$ of the Muckenhoupt class $$A_\infty $$ has the form $$v=h^\prime $$ where $$h:{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$$ is a bi-Sobolev map. As an application we improve the known results on exact continuation and reciprocal imbeddings for Gehring $$G_q$$ and Muckenhoupt $$A_p$$ classes, providing exact bounds in all cases. The method relies on a duality formula due to Johnson and Neugebauer [Rev Mat Iberoam 3(2)249–273, (1987)] $$\begin{aligned} A_p((h^{-1} )^\prime ) =G_q(h^\prime ), \end{aligned}$$ $$\frac{1}{p}+\frac{1}{q}=1$$ .

Maximal non-treed subring of its quotient field

Abstract: An integral domain R is called maximal non-treed subring of its quotient field, if R is not treed, and every proper overring of R is treed. We do prove that R is a maximal non-treed subring of its quotient field if and only if R is non-treed, local with maximal ideal m and its integral closure \(\overline{R}\) is a semi-local Prufer domain with two maximal ideals M, N such that \(M\cap N=m\), and there is no field lying properly between R / m and \(\overline{R}/M\times \overline{R}/N\). Additional characterizations are settled in terms of pullback rings or minimal overrings.

Equimeasurable rearrangements of functions satisfying the reverse Hölder or the reverse Jensen inequality

Abstract: Equimeasurable rearrangements of functions $$f$$ satisfying the reverse Holder or the reverse Jensen inequality are studied. It is shown that the equimeasurable rearrangements belong to the same class as the function $$f$$ , and a sharp estimate of the rearrangement norms via the norm of the function $$f$$ is obtained.

nD-pD Dimensional reduction of micromagnetic structures

Abstract: Starting from a \(nD, n\ge 2\), non-convex and nonlocal micromagnetic energy, we determine, via an asymptotic analysis, the free energy of a \(pD\) ferromagnetic domain, \(1\le p

Some special classes of finite soluble PST-groups

Abstract: In this survey paper several classes of finite soluble PST-groups are introduced and studied. We also determine how these classes are related.

Homogenization of an elastic medium having three phases

Abstract: We study an elastostatic problem in an $${\varepsilon }$$ -periodic medium having three phases: matrix, fibers, and fiber coatings. The rigidity is of order one along the fibers and is scaled by $${{\varepsilon }^2}$$ (the so-called double porosity scaling) in both the transverse directions and the fiber coatings. Using the homogenization process, we show that both the effective transverse traction and the longitudinal stress in the fibers are mainly influenced by the elastic properties of the fiber coatings.

Primary abelian almost n\text {-}\Sigma -groups

Abstract: We define the class of \(p\)-primary abelian almost \(n\)-\(\Sigma \)-groups and study their basic properties. One of the main results is that almost \(\Sigma \)-groups are almost \(p^{\omega +1}\)-projective exactly when they are almost dsc groups of length less than or equal to \(\omega +1\). Some other characterizations of this new group class are also established.

A note on autocentral automorphisms of groups

Abstract: The absolute centre L(G) of a group G is the subgroup of all elements fixed by every automorphism of G, and an automorphism of G is autocentral if it acts trivially on the factor group G / L(G) Autocentral automorphisms have been introduced by Moghaddam and Safa (Ricerche Mat 59:257–264, 2010) The aim of this paper is to obtain new informations on the behaviour of autocentral automorphisms of a group We also consider the relations between the group of autocentral automorphisms and that of class preserving automorphisms of a group

A note on groups with a modularity condition on infinite subsets

Abstract: Finitely generated groups in which every infinite set of cyclic subgroups satisfies a suitable modularity condition are studied.

Coherent states and Berezin quantization for non-scalar holomorphic representations

Abstract: Let \(G\) be a quasi-Hermitian Lie group and let \(K\) be a maximal compactly embedded subgroup of \(G\). Let \(\pi \) be a unitary representation of \(G\) which is holomorphically induced from a unitary representation \(\rho \) of \(K\). Then \(\pi \) is usually realized in a Hilbert space of vector-valued holomorphic functions. Here we introduce a realization of \(\pi \) in a reproducing kernel Hilbert space of complex-valued functions and we compute the (scalar-valued) coherent states. We study the corresponding Berezin quantization map and, in particular, we show that this map is equivalent to that introduced by Cahen (Rend Istit Mat Univ Trieste 46:157–180, 2014).

Groups in which every finite subnormal subgroup is normal

Abstract: A group G is called a T-group if normality in G is a transitive relation, i.e. all subnormal subgroups of G are normal. The structure of soluble T-groups is well-known, and several authors have investigated groups in which the normality condition is imposed only to a relevant system of subnormal subgroups. We consider here (generalized) soluble groups in which all finite subnormal subgroups are normal.

On F-normal subgroups of finite groups

Abstract: A subgroup \(H\) of a finite group \(G\) is called F-normal in \(G\) if there exists a proper normal subgroup \(K\) of \(G\) such that \(G=HK\) and \(H\cap K \le \Phi (H)\), where \(\Phi (H)\) is the intersection of all maximal subgroups of \(H\). We investigate the influence of F-normal subgroups on the structure of \(G\).

Nonexistence of positive solutions for a class of semilinar elliptic systems in a ball

Abstract: We prove the nonexistence of positive solutions to the system $$\begin{aligned} \left\{ \begin{array}{l} -\Delta u=\lambda f(v)\text { in }B, \\ -\Delta v=\mu g(u) \text { in }B, \\ u=v=0\text { on }\partial B, \end{array} \right. \end{aligned}$$ where $$B$$ is the open unit ball in $$\mathbb {R}^{N}$$ , $$\ N>1,\ \lambda ,\mu $$ are positive constants bounded away from $$0$$ with $$\lambda \mu $$ large, $$f,g$$ are smooth functions with $$f(0),g(0)<0,\ f\circ (cg)$$ and $$g\circ (cf)$$ growing at least linearly at $$\infty $$ .