arXiv subjects

Sergio Venturini

Publications and source records attributed to Sergio Venturini.

11 recordsLinked to original sources

arXiv2024

Lipshitzian Vector Fields, Upper Gradients And Distributional Derivatives

We prove that given a locally integrable function $f$ on an open set of an Euclidean space the distributional derivative $Xf$ with respect to a locally Lipshitzian vector field $X$ is locally integrable if, and only if, the function $f$ admits a locally integrable upper gradient along the vector field $X$; in this case $Xf$ coincides with the Lie derivative $L_X f$ and $|Xf|$ is the least upper gradient of the function $f$. Applications to systems of locally Lipshitzian vector fields are given.

math.MG↗

arXiv2020

If $L(χ,1)=0$ then $ζ(1/2+it)\neq0$

Let $L(s)=\sum_{n=1}^{+\infty}\dfrac{a(n)}{n^s}$ be a Dirichlet series were $a(n)$ is a bounded completely multiplicative function. We prove that if $L(s)$ extends to a holomorphic function on the open half space $\Re s >1-δ$, $δ>0$ and $L(1)=0$ then such a half space is a zero free region of the Riemann zeta function $ζ(s)$. Similar results is proven for completely multiplicative functions defined on the space of the ideals of the ring of the algebraic integers of a number field of finite degree.

math.NT↗

arXiv2015

Generalized Quantile Treatment Effect: A Flexible Bayesian Approach Using Quantile Ratio Smoothing

We propose a new general approach for estimating the effect of a binary treatment on a continuous and potentially highly skewed response variable, the generalized quantile treatment effect (GQTE). The GQTE is defined as the difference between a function of the quantiles under the two treatment conditions. As such, it represents a generalization over the standard approaches typically used for estimating a treatment effect (i.e., the average treatment effect and the quantile treatment effect) because it allows the comparison of any arbitrary characteristic of the outcome's distribution under the two treatments. Following Dominici et al. (2005), we assume that a pre-specified transformation of the two quantiles is modeled as a smooth function of the percentiles. This assumption allows us to link the two quantile functions and thus to borrow information from one distribution to the other. The main theoretical contribution we provide is the analytical derivation of a closed form expression for the likelihood of the model. Exploiting this result we propose a novel Bayesian inferential methodology for the GQTE. We show some finite sample properties of our approach through a simulation study which confirms that in some cases it performs better than other nonparametric methods. As an illustration we finally apply our methodology to the 1987 National Medicare Expenditure Survey data to estimate the difference in the single hospitalization medical cost distributions between cases (i.e., subjects affected by smoking attributable diseases) and controls.

math.ST↗

arXiv2013

On the Orbits of not Expansive Mappings in Metric Spaces

Let $\MSpace$ be a locally compact metric space and let $\pMap:\MSpace\to\MSpace$ be a not expansive map. We prove that for each $\ppa_0\in\MSpace$ the sequence $\ppa_0,\pMap(\ppa_0),\pMap^2(\ppa_0),\ldots$ is either relatively compact in $\MSpace$ or compactly divergent in $\MSpace$. As applications we study the structure of the functions which are limits of the iterates of the map $\pMap$ and we prove the analyticity of the set of $\pMap$-recurrent points when $\pMap:\MSpace\to\MSpace$ is a holomorphic and $\MSpace$ is a complex hyperbolic spaces in the sense of Kobayashi.

math.CV↗

arXiv2012

On Cimmino Integrals as Residues of Zeta Functions

The following paper is a variation on a theme of Gianfranco Cimmino on some integral representation formulas for the solution of a linear equations system. Cimmino was probably motivated for giving a representation formula suitable not only for theoretical investigations but also for applied computation. In this paper we will prove that the Cimmino integrals are strictly related to the residues of some zeta-like functions associated to the linear system.

math.CV↗

arXiv2011

Complex Gradient Systems

Let $M$ be a complex manifold of complex dimension $n+k$. We say that the functions $u_1,...s,u_k$ and the vector fields $ξ_1,...,ξ_k$ on $M$ form a \emph{complex gradient system} if $ξ_1,...,ξ_k,Jξ_1,...,Jξ_k$ are linearly independent at each point $p\in M$ and generate an integrable distribution of $TM$ of dimension $2k$ and $du_α(ξ_β)=0$, $\d^c\u_α(ξ_β)=δ_{αβ}$ for $α,β=1,...,k$. We prove a Cauchy theorem for such complex gradient systems with initial data along a $\CR-$submanifold of type $(\CRdim,\CRcodim)$. We also give a complete local characterization for the complex gradient systems which are \emph{holomorphic} and \emph{abelian}, which means that the vector fields $ξ_α^c=ξ_α-Jξ_β$, $α=1,...,k$ are holomorphic and satisfy $[ξ_alpha^c,\bar{ξ_β^c}]=0$ for each $α,β=1,...,k$.

math.CV↗

arXiv2010

Adapted complex tubes on the symplectization of pseudo-Hermitian manifolds

Let $(M,ω)$ be a pseudo-Hermitian space of real dimension $2n+1$, that is $\RManBase$ is a $\CR-$manifold of dimension $2n+1$ and $ω$ is a contact form on $M$ giving the Levi distribution $HT(M)\subset TM$. Let $M^ω\subset T^*M$ be the canonical symplectization of $(M,ω)$ and $M$ be identified with the zero section of $M^ω$. Then $M^ω$ is a manifold of real dimension $2(n+1)$ which admit a canonical foliation by surfaces parametrized by $\mathbb{C}\ni t+iσ\mapsto ϕ_p(t+iσ)=σω_{g_t(p)}$, where $p\inM$ is arbitrary and $g_t$ is the flow generated by the Reeb vector field associated to the contact form $ω$. Let $J$ be an (integrable) complex structure defined in a neighbourhood $U$ of $M$ in $M^ω$. We say that the pair $(U,J)$ is an {adapted complex tube} on $M^ω$ if all the parametrizations $ϕ_p(t+iσ)$ defined above are holomorphic on $ϕ_p^{-1}(U)$. In this paper we prove that if $(U,J)$ is an adapted complex tube on $M^ω$, then the real function $E$ on $M^ω\subset T^*M$ defined by the condition $α=E(α)ω_{π(α)}$, for each $α\in M^ω$, is a canonical equation for $M$ which satisfies the homogeneous Monge-Ampère equation $(dd^c E)^{n+1}=0$. We also prove that if $M$ and $ω$ are real analytic then the symplectization $M^ω$ admits an unique maximal adapted complex tube.

math.CV↗

arXiv2009

Contact geometry of one dimensional holomorphic foliations

Let V be a real hypersurface of class C^k, k>=3, in a complex manifold M of complex dimension n+1, HT(V) the holomorphic tangent bundle to V giving the induced CR structure on V. Let θbe a contact form for (V,HT(V)), ξ_0 the Reeb vector field determined by θand assume that ξ_0 is of class C^k. In this paper we prove the following theorem (cf. Theorem 4.1): if the integral curves of ξ_0 are real analytic then there exist an open neighbourhood N\subset M of V and a solution u\in C^k(N) of the complex Monge-Ampère equation (dd^c u)^(n+1)=0 on N which is a defining equation for V. Moreover, the Monge-Ampère foliation associated to u induces on V that one associated to the Reeb vector field. The converse is also true. The result is obtained solving a Cauchy problem for infinitesimal symmetries of CR distributions of codimension one which is of independent interest (cf. Theorem 3.1).

math.CV↗

arXiv2008

Volumes, Traces and Zeta Functions

Let $Q(x)$ be a quadratic form over $\mathbb{R}^n$. The Epstein zeta function associated to $Q(x)$ is a well known function in number theory. We generalize the construction of the Epstein zeta function to a class of function $ϕ(x)$ defined in $\mathbb{R}^n$ that we call $A-$homogeneous, where $A$ is a real aquare matrix of order $n$ having each eigenvalue in the left hal space $\Reλ>0$. Such a class includes all the homogeneous polynomials (positive outside the origin) and all the norms on $\mathbb{R}^n$ which are smooth outside the origin. As in the classical (i.e. quadratic) case we prove that such zeta functions are obtained from the Mellin transforms of theta function of Jacobi type associated to the $A-$homogeneous function $ϕ(x)$. We prove that the zeta function associated to a $A-$homogeneous function $ϕ(x)$ which is positive and smooth outside the origin is an entire meromorphic function having a unique simple pole at $s=α$ the trace of the matrix $A$ with residue given by the product of the trace $α$ and the Lebesgue volume of the unit ball associated to $ϕ(x)$, that is the volume of the set $x\in\R^n$ satisfying $ϕ(x)<1$. We also prove that the theta funtion associated to $ϕ(x)$ has an asymptotic expansion near the origin. We find that the coefficients of such expansion depend on the values that the zeta function associated to $ϕ(x)$ assumes at the negative integers.

math.CV↗

arXiv2008

Gamma shape mixtures for heavy-tailed distributions

An important question in health services research is the estimation of the proportion of medical expenditures that exceed a given threshold. Typically, medical expenditures present highly skewed, heavy tailed distributions, for which (a) simple variable transformations are insufficient to achieve a tractable low-dimensional parametric form and (b) nonparametric methods are not efficient in estimating exceedance probabilities for large thresholds. Motivated by this context, in this paper we propose a general Bayesian approach for the estimation of tail probabilities of heavy-tailed distributions, based on a mixture of gamma distributions in which the mixing occurs over the shape parameter. This family provides a flexible and novel approach for modeling heavy-tailed distributions, it is computationally efficient, and it only requires to specify a prior distribution for a single parameter. By carrying out simulation studies, we compare our approach with commonly used methods, such as the log-normal model and nonparametric alternatives. We found that the mixture-gamma model significantly improves predictive performance in estimating tail probabilities, compared to these alternatives. We also applied our method to the Medical Current Beneficiary Survey (MCBS), for which we estimate the probability of exceeding a given hospitalization cost for smoking attributable diseases. We have implemented the method in the open source GSM package, available from the Comprehensive R Archive Network.

stat.AP↗

arXiv2008

Maximal plurisubharmonic models

An analytic pair of dimension n and center V is a pair (V, M) where M is a complex manifold of (complex) dimension n and V is a closed totally real analytic submanifold of dimension n. To an analytic pair (V, M) we associate the class of the functions u from M to a positive bounded interval which are plurisubharmonic in M and such that u(p) = 0 for each p in V. If the class admits a maximal function u, the triple (V, M, u) is said to be a maximal plurisubharmonic model. After defining a pseudo-metric E(V,M) on the center V of an analytic pair (V, M) we prove (see Theorem 4.1, Theorem 5.1) that maximal plurisubharmonic models provide a natural generalization of the Monge-Ampere models introduced by Lempert and Szoke in [16].

math.CV↗