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

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

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

Selective disassembly planning considering process capability and component quality utilizing reinforcement learning

Disassembly is a crucial process for achieving circular products, enabling function recovery, material reuse, and recycling. Disassembly planning is complex due to epistemic uncertainty associated with each unique product’s conditions, i.e., quality and aleatoric uncertainty about the capabilities of available resources and processes, and the cost benefits of associated operations impede planning. Therefore, the disassembly is intended to result in keeping the maximum value for the disassembled units of the product. In selective disassembly, the specification of the units of the product to be disassembled is acquired, leaving the rest of the product intact. The benefit of selective disassembly is to minimize waste during dismantling and maximize the reuse of the disassembled components for economic and ecological sustainability. The challenges in disassembly sequence planning include product complexity, operational and technological process capabilities, and the lack of information regarding the product architecture. For this complex planning task, limited studies have been performed on incorporating process capabilities with respect to the operations resources for selective disassembly planning. In this paper, an approach for optimal sequence planning of the selective disassembly process is put forward, taking into account multiple constraints, i.e., quality, time, and process capability. The intelligent planning approach takes advantage of a reinforcement learning model to handle the complexity of the planning problem. The approach has been implemented and tested on an industrial reference assembly. The result shows that the complex task of selective disassembly planning can be efficiently performed utilizing the proposed approach

Ios tidal response precludes a shallow magma ocean.

Io experiences tidal deformation as a result of its eccentric orbit around Jupiter, which provides a primary energy source for Ios continuing volcanic activity and infrared emission1. The amount of tidal energy dissipated within Io is enormous and has been suggested to support the large-scale melting of its interior and the formation of a global subsurface magma ocean. If Io has a shallow global magma ocean, its tidal deformation would be much larger than in the case of a more rigid, mostly solid interior2. Here we report the measurement of Ios tidal deformation, quantified by the gravitational tidal Love number k2, enabled by two recent flybys of the Juno spacecraft. By combining Juno3,4 and Galileo5-7 Doppler data from the NASA Deep Space Network and astrometric observations, we recover Re(k2) of 0.125 ± 0.047 (1σ) and the tidal dissipation parameter Q of 11.4 ± 3.6 (1σ). These measurements confirm that a shallow global magma ocean in Io does not exist and are consistent with Io having a mostly solid mantle2. Our results indicate that tidal forces do not universally create global magma oceans, which may be prevented from forming owing to rapid melt ascent, intrusion and eruption8,9, so even strong tidal heating-such as that expected on several known exoplanets and super-Earths10-may not guarantee the formation of magma oceans on moons or planetary bodies