Showing papers in "Le Matematiche in 2007"

Biharmonic Green functions

Abstract: The harmonic Green and Neumann function and a particular Robin function are used to construct bi-harmonic Green, Neumann and particular Robin functions. Moreover hybrid bi-harmonic Green functions are given. They all are constructed via a convolution of the mentioned harmonic particular fundamental solutions. In case of the unit disc they are explicitly expressed. Besides these 9 bi-harmonic Green functions there is another bi-harmonic Green function in explicit form for the unit disc not defined by convolution. Related boundary value problems are not all well posed. In case they are, the unique solutions are given. For the other cases solvability conditions are determined and the unique solutions found. There are all together 12 Dirichlet kind boundary value problems for the inhomogeneous bi-harmonic equation treated. The investigation is restricted to the two dimensional case and complex notation is used.

Some remarks on the Cantor pairing function

Abstract: In this paper, some results and generalizations about the Cantor pairing function are given. In particular, it is investigated a very compact expression for the n -degree generalized Cantor pairing function (g.C.p.f., for short), that permits to obtain n −tupling functions which have the characteristics to be n -degree polynomials with rational coefficients. A recursive formula for the n -degree g.C.p.f. is also provided.

On the maximum principle for viscosity solutions of fully nonlinear elliptic equations in general domain

Abstract: We analyze the validity of the Maximum Principle for viscosity solutions of fully nonlinear second order elliptic equations in general unbounded domains under suitable structure conditions on the equation allowing notably quadratic growth in the gradient terms.

Nonlinear elliptic equations and Gauss measure

Abstract: We prove existence and regularity results for weak solutions to nonlinear elliptic equations.

Zeros of bessel functions: monotonicity, concavity, inequalities

Abstract: We present a survey of the most important inequalities and monotonicity, concavity (convexity) results of the zeros of Bessel functions. The results refer to the definition Jνκ of the zeros of Cν (x) = Jν (x) cosα −Yν (x) sinα, formulated in [6], where κ is a continuous variable. Sometimes, also the Sturm comparison theorem is an important tool of our results.

An analysis of two dimensional integral equations of the second kind

Abstract: In this article, A numerical method is used to solve the two dimensional Fredholm integral equation of the second kind with weak singular kernel using the Toeplitz matrix and product Nystrom method. The numerical results given in this paper are computed using maple 8. The error, in each case, is computed.

Wiener criterion at the boundary related to p-homogeneous strongly local dirichlet forms

Abstract: We state a Wiener criterion at the boundary related to p -homogeneous strongly local Riemannian type Dirichlet forms.

Oscillation estimates relative to p-homogeneous forms and Kato measures data

Abstract: We state pointwise estimate for the positive subsolutions associated to a p -homogeneous form and nonnegative Radon measures data. As a by-product we establish an oscillation’s estimate for the solutions relative to Kato measures data.

Harnack inequality for harmonic functions relative to a nonlinear p-homogeneous Riemannian Dirichlet form

Abstract: We consider a measure valued map α(u) defined on D where D is a subspace of L ^p(X,m) with X a locally compact Hausdorff topological space with a distance under which it is a space of homogeneous type. Under assumptions of convexity, Gateaux differentiability and other assumptions on α which generalize the properties of the energy measure of a Dirichlet form, we prove the Holder continuity of the local solution u of the problem ∫ X µ(u,v)(dx) = 0 for each v belonging to a suitable space of test functions, where µ(u,v) = .

Clean rings of skew Hurwitz series

Abstract: In this paper we study the transfer of some algebraic properties from the ring R to the ring of skew Hurwitz series T = (HR,σ) where σ is an automorphism of R and vice versa. We show that T = (HR,σ) is a clean(strongly clean) ring if and only if R is clean (strongly clean). Different properties of skew Hurwitz series are studied such as simplicity, primeness and semiprime.

Regularity of solutions of the Neumann problem for the Laplace equation

Abstract: Let u be a solution of the Neumann problem for the Laplace equation in G with the boundary condition g . It is shown that u ∈ L q (∂ G ) (equivalently, u ∈ B q,2 1/q ( G ) for 1 , u ∈ L q 1/q (G ) for 2 ≤ q ) if and only if the single layer potential corresponding to the boundary condition g is in L q (∂ G ) . As a consequence we give a regularity result for some nonlinear boundary value problem.

Comparison results for solutions of parabolic equations with a singular potential

Abstract: We consider the solution u of the Cauchy-Dirichlet problem for a class of linear parabolic equations in which the coefficient of the zero order term could have a singularity at the origin of the type 1/|x|^2. We prove that u can be compared “in the sense of rearrangements” with the solution v of a problem whose data are radially symmetric with respect to the space variable.

On generalized derivatives and formal powers for pseudoanalytic functions

Abstract: We consider pseudoanalytic functions depending on two or three real variables. They are characterized by the corresponding Bers-Vekua equations. In the case of two dimensions we use the complex notation whereas for the case of three variables the concept of complex quaternions serves for our investigations. In a particular plane case we give an explicit representation of formal powers with which a complete system of solutions of the corresponding Bers-Vekua equation can be given. By an example we show how the concept of formal powers may also be applied to the case of three variables.

On a theorem of Faltings on formal functions

Abstract: In 1980, Faltings proved, by deep local algebra methods, a local result regarding formal functions which has the following global geometric fact as a consequence Theorem − Let k be an algebraically closed field (of any characteristic) Let Y be a closed subvariety of a projective irreducible variety X defined over k Assume that X C P^n, dim(X) = d > 2 and Y is the intersection of X with r hyperplanes of P^n, with r ≤ d − 1 Then, every formal rational function on X along Y can be (uniquely) extended to a rational function on X Due to its importance, the aim of this paper is to provide two elementary global geometric proofs of this theorem

Stability and free energies in linear visco-elasticity

Abstract: Dedicated to the memory of Gaetano Fichera In this paper we show that several free energies can be related to a material with fading memory with different domains of definition and topologies. As Fichera has proved, the study of stability can be affected by the space and then by the topology which we chose, while the material must be independent by this one. In the last part of the work by the semi-group theory, we use the new notion of minimum state to study the asymptotic behavior of the differential system.

On the boundary behavior of the holomorphic sectional curvature of the Bergman metric

Abstract: We obtain a conceptually new differential geometric proof of P. F. Klembeck’s result (cf. [9]) that the holomorphic sectional curvature k g (z ) of the Bergman metric of a strictly pseudoconvex domain Ω ⊂ ℂ n approaches −4/(n + 1) (the constant sectional curvature of the Bergman metric of the unit ball) as z → ∂ Ω .

Analog of Dini-Riemann theorem for non-absolutely convergent integrals

Abstract: An analogue of classical Dini-Riemann theorem related to non-absolutely convergent series of real number is proved for the Lebesgue improper integral

Existence of continuous solutions to evolutionary quasi-variational inequalities with applications

Abstract: The author presents dynamic elastic traffic equilibrium problems with data depending explicitly on time and studies under which assumptions the continuity of solutions with respect to the time can be ensured. In particular, regularity results for solutions to time-dependent quasi-variational inequalities associated to a general class of closed lower semicontinuous multifunctions will be showed. These results will be obtained making use of the property of the Mosco’s convergence. At last, it will be applied an example of the dynamic elastic traffic equilibrium problem.

Completeness theorems: Fichera's fundamental results and some new contributions

Abstract: After recalling Fichera’s fundamental results in the study of the problem of the completeness of particular solutions of a partial differential equation, we give some new completeness theorem. They concern the Dirichlet problem for a general elliptic operator of higher order with real constant coefficients in any number of variables.

On reducibility of linear difference systems

Abstract: We study problems related to the existence of a nondegenerate substitution (of the Lyapunov type) that reduces a system of linear difference equations to a system with constant coefficients

Discrete mathematics, discrete physics and numerical methods

Abstract: Discrete mathematics has been neglected for a long time. It has been put in the shade by the striking success of continuous mathematics in the last two centuries, mainly because continuous models in physics proved very reliable, but also because of the greater difficulty in dealing with it. This perspective has been rapidly changing in the last years owing to the needs of the numerical analysis and, more recently, of the so called discrete physics. In this paper, starting from some sentences of Fichera about discrete and continuous world, we shall present some considerations about discrete phenomena which arise when designing numerical methods or discrete models for some classical physical problems.

On bernoulli boundary value problem

Abstract: We give a constructive proof of the existence and uniqueness of the solution, under certain conditions, by Picard’s iteration Moreover Newton’s iteration method is considered for the numerical computation of the solution

Groups with finitely many normalizers of non-subnormal subgroups

Abstract: It is proved that a group G has finitely many normalizers of non-subnormal subgroups if and only if each subgroup of G either is subnormal or has finitely many conjugates; groups with this latter property have been completely described in [8]. Moreover, groups with finitely many normalizers of infinite non-subnormal subgroups are described.

Multi-variable gould-hopper and laguerre polynomials

Abstract: The monomiality principle was introduced by G. Dattoli, in order to derive the properties of special or generalized polynomials starting from the corresponding ones of monomials. In this article we show a general technique to extend themonomiality approach tomulti-index polynomials in several variables. Application to the case of Hermite, Laguerre-type and mixed-type (i.e. between Laguerre and Hermite) are derived.

An order drop theorem

Abstract: A drop theorem on ordered metric spaces is established from the (pre) order version of Ekeland’s variational principle in Turinici [An St UAIC Ia (Math), 36 (1990), 329-352]. The logical equivalence between these results is also discussed.

Application of the residue theorem to bilateral hypergeometric series

Abstract: The application of the residue theorem to bilateral hypergeometric series identities is systematically reviewed by exemplifying three classes of summation theorems due to Dougall (1907), Jackson (1949, 1952) and Slater-Lakin (1953).

Diffraction of electromagnetic wave on the system of metallic strips in the stratified medium

Abstract: In this paper, we study diffraction of electromagnetic wave on the system of metallic strips in the stratified medium. The integral equation which represent this problem is solved by Galerkin’s method. To this equation the plane diffraction problem for TE-polarizable electromagnetic wave on the system of metallic strips in the stratified medium was reduced.

n-blocks collections on Fano manifolds and sheaves with regularity −∞

Abstract: Let X be a smooth Fano manifold equipped with a “ nice ” n -blocks collection in the sense of [3] and F a coherent sheaf on X. Assume that X is Fano and that all blocks are coherent sheaves. Here we prove that F has regularity −∞ in the sense of [3] if Supp(F) is finite, the converse being true under mild assumptions. The corresponding result is also true when X has a geometric collection in the sense of [2].

Parabolic transmission problems across irregular layers

Abstract: We study second order transmission problems across either a fractal surface or across the corresponding pre-fractal surface. Existence uniqueness and regularity results for the strict solution in both cases are proved. The asymptotic behaviour of the solutions of the approximating problems is also investigated.

Approximate approximations on nonuniform grids

Abstract: We present an extension of approximate quasi-interpolation on uniformly distributed nodes, to functions given on a set of nodes close to an uniform, not necessarily cubic, grid.