Showing papers in "Ricerche Di Matematica in 2013"

Finite groups with $$\mathbb{P }$$ -subnormal subgroups

Abstract: A subgroup $$H$$ of a group $$G$$ is called $$\mathbb{P }$$ -subnormal in $$G$$ whenever either $$H=G$$ or there is a chain of subgroups $$H=H_0\subset H_1\subset \cdots \subset H_n=G$$ such that $$|H_i:H_{i-1}|$$ is a prime for all $$i$$ . In this paper we study groups with $$\mathbb{P }$$ -subnormal 2-maximal subgroups, and groups with $$\mathbb{P }$$ -subnormal primary cyclic subgroups.

Modelling alcohol problems: total recovery

Abstract: Binge drinking in the UK is an increasing problem, resulting in negative health, social and economic effects. Mathematical modelling allows for future predictions to be made and may provide valuable information regarding how to approach solving the problem of binge drinking in the UK. We develop a 3-equation model for alcohol problems, specifically binge drinking, which allows for total recovery. Individuals are split into those that are susceptible to developing an alcohol problem, those with an alcohol problem and those in treatment. We find that the model has two equilibrium points: one without alcohol problems and one where alcohol problems are endemic in the population. We compare our results with those of an existing model that does not allow for total recovery. We show that without total recovery, the threshold for alcohol problems to become endemic in the population is lowered. The endemic equilibrium solution is also affected, with an increased proportion of the population in the treatment class and a decreased proportion in the susceptible class. Including totally recovery does not determine whether the proportion of individuals with alcohol problems increases or decreases, however it does effect the size of the change. Parameter estimates are made from information regarding binge drinking where we find an increase in the recovery rate decreases the proportion of binge drinkers in the population.

Soret effects on the onset of convection in rotating porous layers via the “auxiliary system method”

Abstract: The onset of convection in a ternary horizontal porous fluid layer, heated from below and salted from above and below, in the presence of Soret thermo-diffusive effects (in the Darcy–Boussinesq scheme) is investigated. Via a new approach (“auxiliary system method”), it is shown that do not exist subcritical instabilities and that the global nonlinear stability is guaranteed by the linear stability. The Soret stabilizing-destabilizing effects, via algebraic conditions in closed forms, are obtained also discovering symmetries and skew-symmetries hidden in the Darcy–Boussinesq equations.

On the nonlinear stability of an epidemic SEIR reaction-diffusion model

Abstract: This paper deals with a reaction-diffusion SEIR model for infections. The longtime behaviour of the solutions is analyzed and, in particular, absorbing sets in the phase space are determined. By using a peculiar Lyapunov function, the nonlinear asymptotic stability of endemic equilibrium is investigated.

Local higher integrability for parabolic quasiminimizers in metric spaces

Abstract: Using purely variational methods, we prove in metric measure spaces local higher integrability for minimal p-weak upper gradients of parabolic quasiminimizers related to the heat equation. We assume the measure to be doubling and the underlying space to be such that a weak Poincare inequality is supported. We define parabolic quasiminimizers in the general metric measure space context, and prove an energy type estimate. Using the energy estimate and properties of the underlying metric measure space, we prove a reverse Holder inequality type estimate for minimal \(p\)-weak upper gradients of parabolic quasiminimizers. Local higher integrability is then established based on the reverse Holder inequality, by using a modification of Gehring’s lemma.

On the stability-instability of vertical throughflows in double diffusive mixtures saturating rotating porous layers with large pores

Abstract: The long-time behaviour of the solutions of the Darcy–Oberbeck– Boussinesq system modeling fluid motion in horizontal porous layers, is investigated. The layer is supposed to be uniformly heated and salted from below, rotating around the vertical axis, showing large pores. Necessary and sufficient conditions guaranteeing the stability of a vertical constant throughflow are obtained. The non-linear, global, asymptotic \(L^2-\)stability of the throughflow solution, is investigated.

Parabolic oblique derivative problem with discontinuous coefficients in generalized Morrey spaces

Abstract: We study the global Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with \(VMO\) coefficients.

CR immersions and Lorentzian geometry

Abstract: Using tools from Lorentzian geometry (arising from the presence of the Fefferman metric) we prove a Takahashi type theorem (for a class of pseudohermitian immersions covered by connection-preserving equivariant immersions among the total spaces of the canonical circle bundles) thus relating the geometry of a pseudohermitian immersion from a strictly pseudoconvex CR manifold $$M$$ into an odd dimensional sphere, to the spectrum of the sublaplacian on $$M$$ .

Hausdorff measures of subgroups of \mathbb{R }/\mathbb{Z } and \mathbb{R }

Abstract: For a sequence \(\underline{u}=(u_n)_{n\in \mathbb{N }}\) of integers, let \(t_{\underline{u}}(\mathbb{T })\) be the group of all topologically \(\underline{u}\)-torsion elements of the circle group \(\mathbb{T }:=\mathbb{R }/\mathbb{Z }\). We show that for any \(s\in ]0,1[\) and \(m\in \{0,+\infty \}\) there exists \(\underline{u}\) such that \(t_{\underline{u}}(\mathbb{T })\) has Hausdorff dimension \(s\) and \(s\)-dimensional Hausdorff measure equal to \(m\) (no other values for \(m\) are possible). More generally, for dimension functions \(f,g\) with \(f(t)\prec g(t), f(t)\prec \!\!\!\prec t\) and \(g(t)\prec \!\!\!\prec t\) we find \(\underline{u}\) such that \(t_{\underline{u}}(\mathbb{T })\) has at the same time infinite \(f\)-measure and null \(g\)-measure.

On minimal non-M_rC-groups

Abstract: A group \(G\) is said to be a minimax group if it has a finite series whose factors satisfy either the minimal or the maximal condition. Let \(D(G)\) denotes the subgroup of \(G\) generated by all the Chernikov divisible normal subgroups of \(G\). If \(G\) is a soluble-by-finite minimax group and if \(D(G)=1\) , then \(G\) is said to be a reduced minimax group. Also \(G\) is said to be an \( M_{r}C\)-group (respectively, \(PC\)-group), if \(G/C_{G} \left(x^{G}\right)\) is a reduced minimax (respectively, polycyclic-by-finite) group for all \(x\in G\) . These are generalisations of the familiar property of being an \(FC\)-group. Finally, if \(\mathfrak X \) is a class of groups, then \(G\) is said to be a minimal non-\(\mathfrak X \)-group if it is not an \(\mathfrak X \)-group but all of whose proper subgroups are \(\mathfrak X \)-groups. Belyaev and Sesekin characterized minimal non-\(FC\)-groups when they have a non-trivial finite or abelian factor group. Here we prove that if \(G\) is a group that has a proper subgroup of finite index, then \(G\) is a minimal non-\(M_{r}C\)-group (respectively, non-\(PC\)-group) if, and only if, \(G\) is a minimal non-\(FC\)-group.

Minimal subgroups and the structure of finite groups

Abstract: A subgroup \(H\) of a finite group \(G\) is weakly-supplemented in \(G\) if there exists a proper subgroup \(K\) of \(G\) such that \(G=HK\). In this paper we prove that a finite group \(G\) is \(p\)-nilpotent if every minimal subgroup of \(P\bigcap G^{N}\) is weakly-supplemented in \(G\), and when \(p=2\) either every cyclic subgroup of \(P\bigcap G^{N}\) with order 4 is weakly-supplemented in \(G\) or \(P\) is quaternion-free, where \(p\) is the smallest prime number dividing the order of \(G\), \(P\) a sylow \(p\)-subgroup of \(G\).

A representation of the chamber set Hecke algebra of an affine building

Abstract: We define a representation of the chamber set Hecke algebra of an affine building. This representation allows to extend to every affine buildings of rank 3 the Kato’s condition for the bijectivity of the Poisson transform.

Implicit mapping theorem for extended metric regularity in metric spaces

Abstract: In this paper we prove an extension of the contraction mapping principle for set-valued mappings stated by A.L. Dontchev and W.W. Hager dealing with more general assumptions containing modulus instead of pseudo-contractive multifunctions. Using this result we show that the update Graves theorem, in company with the stability of metric regularity under perturbations can be extended to a much broader framework of set-valued mappings acting in abstract spaces.

On a class of metahamiltonian groups

Abstract: A group is metahamiltonian if all its non-abelian subgroups are normal. It is known that any infinite (generalized) soluble group whose proper subgroups are metahamiltonian is itself metahamiltonian. Moreover, it turns out that the study of soluble groups whose infinite proper subgroups are metahamiltonian can be reduced to the case of a finite extension of a central subgroup of type \(p^\infty \) for some prime \(p\). A classification of metahamiltonian groups in this latter class is given.

An anisotropic geometrical approach for non-relativistic extended dynamics

Abstract: In this paper we present the distinguished (d-) Riemannian geometry (in the sense of nonlinear connection, Cartan canonical linear connection, together with its d-torsions and d-curvatures) for a possible Lagrangian inspired by optics in non-uniform media. The corresponding equations of motion are also exposed, and some particular solutions are given. For instance, we obtain as geodesic trajectories some circular helices (depending on an angular velocity $$\omega $$ ), certain circles situated in some planes (ones are parallel with $$xOy $$ , and other ones are orthogonal on $$xOy$$ ), or some straight lines which are parallel with the axis $$Oz$$ . All these geometrical geodesics are very specific because they are completely determined by the non-constant index of refraction $$n(x)$$ .

Waves, particles and fields: an explicitly covariant approach

Abstract: The idea of associating particle trajectories with wave propagation rays exploited in a previous paper in the context of general relativity with a synchronous gauge, here is examined with no assumptions on co-ordinate choice (no synchronous gauge condition on the metric). Identification of particle Hamilton–Jacobi equation with wave-sheet equation in a space–time with more than 4 dimensions, is performed in an explicitly covariant formulation, leading to a Kaluza–Klein type theory involving Klein–Gordon equation arising from dilaton field equations. De Broglie and Einstein-Planck quantum relations are also deduced in a natural way. Adding suitable Yang–Mills fields provides unification of gravitational, electromagnetic, weak and strong interactions into a \(16\) dimensional space–time geometry. The electron mass gap is also avoided compactifying extra dimensional co-ordinates on fractalized closed paths.

Mean oscillations of the logarithmic function

Abstract: The main property of functions with bounded mean oscillations-viz. the exponential decay of the distribution function, is considered. This property is represented by the John–Nirenberg inequality but the exact constant in the exponent is known only for the functions defined on those intervals which are at least one-sided bounded. In the present paper, an estimate of this constant for the functions having bounded mean oscillation on the whole real line is given. Moreover, an estimate of mean oscillations for the even extension of the functions defined on a semi-axis is established.

On generalizing transitivity, persistence, and sensitivity

Abstract: A subgroup property \(\alpha \) is transitive in a group \(G\) if \(U \alpha V\) and \(V \alpha G\) imply that \(U \alpha G\) whenever \(U \le V \le G\), and \(\alpha \) is persistent in \(G\) if \(U \alpha G\) implies that \(U \alpha V\) whenever \(U \le V \le G\). Even though a subgroup property \(\alpha \) may be neither transitive nor persistent, a given subgroup \(U\) may have the property that each \(\alpha \)-subgroup of \(U\) is an \(\alpha \)-subgroup of \(G\), or that each \(\alpha \)-subgroup of \(G\) in \(U\) is an \(\alpha \)-subgroup of \(U\). We call these subgroup properties \(\alpha \)-transitivity and \(\alpha \)-persistence, respectively. We introduce and develop the notions of \(\alpha \)-transitivity and \(\alpha \)-persistence, and we establish how the former property is related to \(\alpha \)-sensitivity. In order to demonstrate how these concepts can be used, we apply the results to the cases in which \(\alpha \) is replaced with “normal” and the “cover-avoidance property.” We also suggest ways in which the theory can be developed further.

A model for the problem of the cooperation/competition between infinite continuous species

Abstract: In this paper we consider a particular type of differential equation that we can consider as a simple model for the problem of the cooperation/competition of infinite species. In this model each of the species meets each of the other species with a degree of competition or cooperation and their arrangements affect the evolution of the species. A first result of the existence of a unique, local-in-time, solution is given.

Nonexistence and multiplicity of nontrivial solutions for some nonuniformly nonlinear systems

Abstract: The goal of this paper is to study the nonexistence and multiplicity of nontrivial weak solutions for a class of nonlinear systems. The results are proved by Minimum principle and the Mountain pass theorem

Finite planar spaces with a spread of short lines

Abstract: In this paper a class of finite planar spaces with a spread of short lines is characterized.

Hyper-(rank one) groups

Abstract: A group \(G\) is called a \(\mathcal{P }_1\)-group if it has a normal series of finite length whose factors have rank \(1\), while \(G\) is an \(\mathcal{H }_1\)-group if it has an ascending normal series of the same type. This paper investigates properties of \(\mathcal{P }_1\)-groups and \(\mathcal{H }_1\)-groups which correspond to known properties of nilpotent and supersoluble groups.

$$F$$-harmonic maps as global maxima

Abstract: In this note, we show that some $$F$$ -harmonic maps into spheres are global maxima of the variations of their energy functional on the conformal group of the sphere. Our result extends partially those obtained in El Soufi and Lejune [C.R.A.S. 315(Serie I):1189–1192, 1992] and El Soufi [Compositio Math 95:343–362,1995] for harmonic and $$p$$ -harmonic maps.