Archivio della ricerca di Triestehttps://arts.units.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Thu, 21 Jan 2021 22:28:46 GMT2021-01-21T22:28:46Z10201Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equationhttp://hdl.handle.net/11368/2507945Titolo: Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation
Abstract: We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation
\begin{equation*}
-\Big( u'/{ \sqrt{1-{u'}^2}}\Big)'
= f(t,u).
\end{equation*}
Depending on the behaviour of $f=f(t,s)$ near $s=0$, we prove the existence of either
one, or two, or three, or infinitely many positive solutions. In general,
the positivity of $f$ is not required.
All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/11368/25079452012-01-01T00:00:00ZExtending Nearly-Linear Modelshttp://hdl.handle.net/11368/2945506Titolo: Extending Nearly-Linear Models
Abstract: Nearly-Linear Models are a family of neighbourhood models, obtaining lower/upper probabilities from a given probability by a linear affine transformation with barriers. They include a number of known models as special cases, among them the Pari-Mutuel Model, the ε-contamination model, the Total Variation Model and the vacuous lower/upper probabilities. We classified Nearly-Linear models, investigating their consistency properties, in previous work. Here we focus on how to extend those Nearly-Linear Models that are coherent or at least avoid sure loss. We derive formulae for their natural extensions, interpret a specific model as a natural extension itself of a certain class of lower probabilities, and supply a risk measurement interpretation for one of the natural extensions we compute.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11368/29455062019-01-01T00:00:00ZThe Dirichlet problem for a prescribed anisotropic mean curvature equation: existence, uniqueness and regularity of solutionshttp://hdl.handle.net/11368/2848504Titolo: The Dirichlet problem for a prescribed anisotropic mean curvature equation: existence, uniqueness and regularity of solutions
Abstract: We discuss existence, uniqueness, regularity and boundary behaviour of solutions of the Dirichlet problem for the prescribed anisotropic mean curvature equation
\begin{equation*}
{\rm -div}\left({\nabla u}/{\sqrt{1 + |\nabla u|^2}}\right) = -au + {b}/{\sqrt{1 + |\nabla u|^2}},
\end{equation*}
where $a,b>0$ are given parameters and $\Omega$ is a bounded Lipschitz domain in $\RR^N$.
This equation appears in the modeling theory of capillarity phenomena for compressible fluids and in the description of the geometry of the human cornea.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/11368/28485042016-01-01T00:00:00ZInference with Nearly-Linear uncertainty modelshttp://hdl.handle.net/11368/2965326.2Titolo: Inference with Nearly-Linear uncertainty models
Abstract: Several simplified uncertainty models are derived from a given probability of which they are a perturbation. Among these, we introduced in previous work Nearly-Linear (NL) models. They perform a linear affine transformation of with barriers, obtaining a couple of conjugate lower/upper probabilities, and generalise several well known neighbourhood models. We classified NL models, partitioning them into three subfamilies, and established their basic consistency properties in [5]. In this paper we investigate how to extend NL models that avoid sure loss by means of their natural extension, a basic, although operationally not always simple, inferential procedure in Imprecise Probability Theory. We obtain formulae for computing directly the natural extension in a number of cases, supplying a risk measurement interpretation for one of them. The results in the paper also broaden our knowledge of NL models: we characterise when they avoid sure loss, express some of them as linear (or even convex) combinations of simpler models, and explore relationships with interval probabilities.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11368/2965326.22020-01-01T00:00:00ZEssential mathematics for economicshttp://hdl.handle.net/11368/2929245Titolo: Essential mathematics for economics
Abstract: This short book is aimed at undergraduate students with a very standard knowledge of algebra learnt at the high school, and it appears as "self contained". According to the long teaching experience of the authors, it was written in order to furnish nearly all the material both for a 90-hour course of "Mathematics for Economics", and for a 45-hour course of "Financial Mathematics", the first part being covered by the chapters from 1 to 10, and the second by the chapters from 11 to 14.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11368/29292452018-01-01T00:00:00ZRadial solutions of the Dirichlet problem for a class of quasilinear elliptic equations arising in optometryhttp://hdl.handle.net/11368/2932001Titolo: Radial solutions of the Dirichlet problem for a class of quasilinear elliptic equations arising in optometry
Abstract: This paper deals with the quasilinear elliptic problem egin{align*} { m -div} left({ abla u}/{sqrt{1 + | abla u|^2}} ight)+a(x) u &= b(x)/sqrt{1 + | abla u|^2} ext { in } B, ;; u=0 , ext{ on } partial B, end{align*} where $B$ is an open ball in $RR^N$, with $Nge 2$, and $a,b in C^1(overline B) $ are given radially symmetric functions, with $a(x) ge 0$ in $B$. This class of anisotropic prescribed mean curvature equations appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids. Unlike all previous works published on these subjects, existence and uniqueness of solutions of the above problem are here analyzed in the case where the coefficients $a, b$ are not necessarily constant and no sign condition is assumed on $b$.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11368/29320012019-01-01T00:00:00ZA prescribed anisotropic mean curvature equation modeling the corneal shape: a paradigm of nonlinear analysishttp://hdl.handle.net/11368/2915510Titolo: A prescribed anisotropic mean curvature equation modeling the corneal shape: a paradigm of nonlinear analysis
Abstract: In this paper we survey, complete and refine some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation egin{equation*} { m -div}left({ abla u}/{sqrt{1 + | abla u|^2}} ight) = -au + {b}/{sqrt{1 + | abla u|^2}}, end{equation*} in a bounded Lipschitz domain $Omega subset RR^N$, with $a,b>0$ parameters. This equation appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids. Here we show how various techniques of nonlinear functional analysis can successfully be applied to derive a complete picture of the solvability patterns of the problem.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11368/29155102018-01-01T00:00:00ZThe Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz-Minkowski spacehttp://hdl.handle.net/11368/2888466Titolo: The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz-Minkowski space
Abstract: We discuss existence, multiplicity, localisation and stability properties of solutions of the Dirichlet problem associated with the gradient dependent prescribed mean curvature equation in the Lorentz-Minkowski space {−div(∇u/√1−|∇u|²)=f(x,u,∇u) in Ω, u=0 on ∂Ω . The obtained results display various peculiarities, which are due to the special features of the involved differential operator and have no counterpart for elliptic problems driven by other quasilinear differential operators. This research is also motivated by some recent achievements in the study of prescribed mean curvature graphs in certain Friedmann–Lemaître–Robertson–Walker, as well as Schwarzschild–Reissner–Nordström, spacetimes.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/11368/28884662017-01-01T00:00:00ZWeak Dutch Books versus strict consistency with lower previsionshttp://hdl.handle.net/11368/2903912Titolo: Weak Dutch Books versus strict consistency with lower previsions
Abstract: Several consistency notions for lower previsions (coherence, convexity, others) require that the suprema of certain gambles, having the meaning of gains, are non-negative. The limit situation that a gain supremum is zero is termed Weak Dutch Book (WDB). In the literature, the special case of WDBs with precise probabilities has mostly been analysed, and strict coherence has been proposed as a radical alternative. In this paper the focus is on WDBs and generalised strict coherence, termed strict consistency, with imprecise previsions. We discuss properties of lower previsions incurring WDBs and conditions for strict consistency, showing in both cases how they are differentiated by the degree of consistency of the given uncertainty assessment.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/11368/29039122017-01-01T00:00:00ZWeak Dutch Books with Imprecise Previsionshttp://hdl.handle.net/11368/2903910Titolo: Weak Dutch Books with Imprecise Previsions
Abstract: Uncertainty assessments for imprecise previsions based on coherence and related concepts require that the suprema of certain random numbers (interpreted as gains) are non-negative. The extreme situation that a supremum is zero represents what is called a Weak Dutch Book (WDB) in a betting interpretation language. While most of the previous dedicated literature focused on WDBs for de Finetti's coherence with precise probabilities, in this paper we analyse the properties of WDBs with imprecise previsions, notably for conditional (Williams') coherent lower previsions. We show that WDB assessments ensure a certain `local precision' property and imply, in the agent's evaluation, some kind of `protection' against real losses. Further, these properties vary with the consistency notion we adopt, tending to vanish with weaker ones. A generalisation of the classical strict coherence and other alternative approaches to WDBs are also discussed.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/11368/29039102017-01-01T00:00:00Z