Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)
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A new characterization and a Rodrigues formula for generalized Hermite orthogonal polynomials
Full text linkIn this paper, we consider the raising operator Rξ = ξ Tμ + xI, ξ ̸= 0, where Tμ and I are the Dunkl operator and the identity operator respec- tively. Our purpose is to determine all monic orthogonal polynomials se- quences {Pn(x)}n≥0 such that the sequence of polynomials {(Rξ Pn)(x)}n≥0 is also orthogonal. We prove that the only sequence of polynomials sat- isfying this condition is, up to a dilation, the generalized Hermite poly- nomial sequence. Then, we explore our result to deduce a Rodrigues formula for the generalized Hermite polynomials sequence
On the complements of union of open balls of fixed radius in the Euclidean space
Full text linkLet an R-body be the complement of the union of open balls of radius R in Ed . The R-hulloid of a closed not empty set A, the minimal R-body containing A, is investigated; if A is the set of the vertices of a simplex, the R-hulloid of A is completely described (if d = 2) and if d > 2 special examples are studied. The class of R-bodies is compact in the Hausdorff metric if d = 2, but not compact if d > 2.
 
Singular quasilinear problems with quadratic growth in the gradient
Full text linkIn this paper we consider the problem
where Ω is an open bounded set of RN (N ≥ 3), A(x) is a coercive matrix with coefficients in L∞(Ω), H(x,s,ξ) is a Carathe ́odory function which satisfies for a given γ > 0 and some c0 ≥ 0
The nonnegative term a0 belongs to LN/2(Ω), χ{u̸=0} is caracteristic function, f belongs to LN/2(Ω) and 0 < θ < 1. For f and a0 sufficiently small (and more precisely when f and a0 satisfy the smallness condition (2.11)), we prove the existence of at least one solution u such that eδ |u| − 1 belongs to H01(Ω) for some δ ≥ γ. Some a priori estimates are obtained
Further applications of two minimax theorems
Full text linkIn this paper, we deal with new applications of two minimax theorems of B. Ricceri ([5],[9]). Here is a particular case of one of the results that we obtain: Let (T,F,μ) be a non-atomic measure space, with μ(T) < +∞, (E , ∥ · ∥) a real Banach space, I ⊆ E an unbounded set whose closure does not contain 0. Moreover, let p, q, r, s be four numbers such that 0 < s < q ≤ p, p ≥ 1, r > 1. Set X := {f ∈ Lp(T,E) : f(T) ⊆ I}. Then, one ha
Boundary value problems in general relativity
Full text linkCertain theorems of existence, non-existence and uniqueness for boundary value problems modeling axial symmetric problems in general relativity are presented using the Weyl\u27s metric. A solution related to the classical Poiseuille solution of non-relativistic fluid mechanics is also presented
Computing totally real hyperplane sections and linear series on algebraic curves
Full text linkGiven a real algebraic curve, embedded in projective space, we study the computational problem of deciding whether there exists a hyperplane meeting the curve in real points only. More generally, given any divisor on such a curve, we may ask whether the corresponding linear series contains an effective divisor with totally real support. This translates into a particular type of parametrized real root counting problem that we wish to solve exactly. On the other hand, it is known that for a given genus and number of real connected components, any linear series of sufficiently large degree contains a totally real effective divisor. Using the algorithms described in this paper, we solve a number of examples, which we can compare to the best known bounds for the required degree.
 
Idempotent factorization of matrices over a Prüfer domain of rational functions
Full text linkWe consider the smallest subring D of R(X) containing every element of the form 1/(1+x2), with x ϵ R(X). D is a Prüfer domain called the minimal Dress ring of R(X). In this paper, addressing a general open problem for Prüfer non Bézout domains, we investigate whether 2x2 singular matrices over D can be decomposed as products of idempotent matrices. We show some conditions that guarantee the idempotent factorization in M2(D).
 
The geometry of discotopes
Full text linkWe study a class of semialgebraic convex bodies called discotopes. These are instances of zonoids, objects of interest in real algebraic geometry and random geometry. We focus on the face structure and on the boundary hypersurface of discotopes, highlighting interesting birational properties which may be investigated using tools from algebraic geometry. When a discotope is the Minkowski sum of two-dimensional discs, the Zariski closure of its set of extreme points is an irreducible hypersurface. In this case, we provide an upper bound for the degree of the hypersurface, drawing connections to the theory of classical determinantal varieties
On characterizations of Dunkl-semiclassical orthogonal polynomials
Full text linkIn this paper, the Dunkl-semiclassical orthogonal polynomials will be studied as a generalization ofthe Dunkl-classical ones. We obtain some characterizations for such polynomials. Moreover, an example of non-symmetric Dunkl-semiclassical orthogonal polynomials is given
Dynamics of a polymerization model on a graph
Full text linkThis work is concerned with the dynamics of a polymerization process coupled with mass transfer and monomers injection, modeled by means of an infinite-dimensional system of Smoluchowski\u27s equations in a finite graph. Under suitable assumptions on the system\u27s aggregation coefficients, we show that, as a consequence of the injection mechanism, a sizable depletion of the pool of available reacting substances occurs at some finite time, that can be estimated in terms of the parameters of the problem. By analogy with well-known results in chemical engineering, we interpret that result as the onset of a sol-gel phase transition. We suggest that this property might have some interest in the mathematical modeling of neurodegenerative processes, where the polymerization of some soluble proteins and their eventual aggregation into insoluble plaques play a remarkable role, which is not well understood as yet