Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies

Let K \subset R^N be a convex body containing the origin. A measurable set G \subset R^N with positive Lebesgue measure is said to be uniformly K-dense if, for any fixed r > 0, the measure of G \cap (x + rK) is constant when x varies on the boundary of G (here, x + rK denotes a translation of a dilation of K). We first prove that G must always be strictly convex and at least C1,1-regular; also, if K is centrally symmetric, K must be strictly convex, C1,1-regular and such that K = G - G up to homotheties; this implies in turn that G must be C2,1- regular. Then for N = 2, we prove that G is uniformly K-dense if and only if K and G are homothetic to the same ellipse. This result was already proven by Amar, Berrone and Gianni in [3]. However, our proof removes their regularity assumptions on K and G and, more importantly, it is susceptible to be generalized to higher dimension since, by the use of Minkowski's inequality and an affine inequality, avoids the delicate computations of the higher-order terms in the Taylor expansion near r = 0 for the measure of G\cap(x+rK) (needed in [3])

The heart of a convex body

We investigate some basic properties of the {\it heart} $\heartsuit(\mathcal{K})$ of a convex set $\mathcal{K}.$ It is a subset of $\mathcal{K},$ whose definition is based on mirror reflections of euclidean space, and is a non-local object. The main motivation of our interest for $\heartsuit(\mathcal{K})$ is that this gives an estimate of the location of the hot spot in a convex heat conductor with boundary temperature grounded at zero. Here, we investigate on the relation between $\heartsuit(\mathcal{K})$ and the mirror symmetries of $\mathcal{K};$ we show that $\heartsuit(\mathcal{K})$ contains many (geometrically and phisically) relevant points of $\mathcal{K};$ we prove a simple geometrical lower estimate for the diameter of $\heartsuit(\mathcal{K});$ we also prove an upper estimate for the area of $\heartsuit(\mathcal{K}),$ when $\mathcal{K}$ is a triangle.Comment: 15 pages, 3 figures. appears as "Geometric Properties for Parabolic and Elliptic PDE's", Springer INdAM Series Volume 2, 2013, pp 49-6

Using the Process Digital Twin as a tool for companies to evaluate the Return on Investment of manufacturing automation

The fourth industrial revolution is gaining momentum, but still lacks full realization. Several studies suggest that many companies around the world have begun the digital transformation undertaking, but most are still far from full adoption and yet fail to see the full economic potential, being stuck in what has been called "pilot purgatory”. Digitalization is largely recognized as an accelerator and enabler for full automation in manufacturing, but companies are still struggling to assess the return on investment and the impact on operational performance indicators. Therefore, companies, especially SMEs characterized by dynamic, high-value, high-mix, and low-volume contexts, are reluctant to invest further. By incorporating simulation, data analytics and behavioral models, digital twins may also be used to support automation solutions ramp-up, demonstrate their impact evaluation, usage scenarios, eliminating the need for physical prototypes, reducing development time, and improving quality. Few forward-thinking companies are pursuing the digital transformation path, while the majority are clipping the wings of a transformation that is essential for a sustainable manufacturing. This paper describes a theoretical approach to exploit the digital twin technology to gather insights towards a realistic economical assessment of full automation solutions, to back and encourage investments to realize the potential of the digital manufacturing transformation. The approach is being tested under the European Union’s Horizon 2020 research and innovation program under grant agreement No. 958363, which provides an opportunity to assess how the various components of the method are constructed, how complex they are, and what level of effort is required, using a practical example

Solutions of elliptic equations with a level surface parallel to the boundary: stability of the radial configuration

Positive solutions of homogeneous Dirichlet boundary value problems or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of their level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. In fact, we show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is almost radially symmetric, in the sense that is contained in and contains two concentric balls Bre and Bri, with the difference re 12ri (linearly) controlled by a suitable norm of the deviation of the solution from a constant. The proof relies on and enhances arguments developed in a paper by Aftalion, Busca and Reichel

Recent advances in modelling and simulation of surface integrity in machining - A review

Machining is one of the final steps in the manufacturing value chain, where the dimensional tolerances are fine-tuned, and the functional surfaces are generated. Many factors such as the process type, cutting parameters, tool geometry and wear can influence the surface integrity (SI) in machining. Being able to predict and monitor the influence of different parameters on surface integrity provides an opportunity to produce surfaces with predetermined properties. This paper presents an overview of the recent advances in computational and artificial intelligence methods for modelling and simulation of surface integrity in machining and the future research and development trends are highlighted

Harmonic fields on the extended projective disc and a problem in optics

The Hodge equations for 1-forms are studied on Beltrami's projective disc model for hyperbolic space. Ideal points lying beyond projective infinity arise naturally in both the geometric and analytic arguments. An existence theorem for weakly harmonic 1-fields, changing type on the unit circle, is derived under Dirichlet conditions imposed on the non-characteristic portion of the boundary. A similar system arises in the analysis of wave motion near a caustic. A class of elliptic-hyperbolic boundary-value problems is formulated for those equations as well. For both classes of boundary-value problems, an arbitrarily small lower-order perturbation of the equations is shown to yield solutions which are strong in the sense of Friedrichs.Comment: 30 pages; Section 3.3 has been revise

Long-term proactive management of psoriasis with calcipotriol and betamethasone dipropionate foam: an Italian consensus through a combined nominal group technique and Delphi approach

Background: Although long-term management of psoriasis is paramount, this approach is challenging in clinical practice. In the recent PSO-LONG trial, a fixed-dose combination of betamethasone dipropionate (BD) and calcipotriol (Cal) foam applied twice a week on non-consecutive days for 52 weeks (proactive treatment) reduced the risk of relapse. However, the role of Cal/BD foam in the long-term management of psoriasis needs further clarifications. The ProActive Management (PAM) program, a nationwide Italian project, aims at reaching a consensus on the role of proactive management of psoriasis. Methods: A steering committee generated some statements through the nominal group technique (NGT). The statements were voted by an expert panel in an adapted Delphi voting process. Results: Eighteen statements were proposed, and the majority of them (14/18) reached a consensus during the Delphi voting. The need to provide long-term proactive topical treatment to reduce the risk of relapse for the treatment of challenging diseases sites or in patients where phototherapy or systemic therapies are contraindicated/ineffective was widely recognized. A consensus was reached about the possibility to associate the proactive treatment with systemic and biological therapies, without the need for dose intensification, thus favoring a prolonged remission. Moreover, the proactive treatment was recognized as more effective than weekend therapy in increasing time free from relapses. Approaches to improve adherence, on the other hand, need further investigation. Conclusions: The inclusion in guidelines of a proactive strategy among the effective treatment options will be a fundamental step in the evolution of a mild-moderate psoriasis therapeutic approach