Variational Approximation of Functionals Defined on 1-dimensional Connected Sets: The Planar Case

In this paper we consider variational problems involving 1-dimensional connected sets in the Euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert--Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica--Mortola type energies proving a $Gamma$-convergence result. We also introduce a suitable convex relaxation and develop the corresponding numerical implementations. The proposed methods are quite general and the results we obtain can be extended to $n$-dimensional Euclidean space or to more general manifold ambients, as shown in the companion paper [M. Bonafini, G. Orlandi, and E. Oudet, Variational Approximation of Functionals Defined on 1-Dimensional Connected Sets in $mathbb{R}^n$, preprint, 2018]

Landmarking-based unsupervised clustering of human faces manifesting labio-schisis dysmorphisms

Ultrasound scans, Computed Axial Tomography, Magnetic Resonance Imaging are only few examples of medical imaging tools boosting physicians in diagnosing a wide range of pathologies. Anyway, no standard methodology has been dened yet to extensively exploit them and current diagnoses procedures are still carried out mainly relying on physician's experience. Although the human contribution is always fundamental, it is self-evident that an automatic procedure for image analysis would allow a more rapid and eective identication of dysmorphisms. Moving toward this purpose, in this work we address the problem of feature extraction devoted to the detection of specic dis- eases involving facial dysmorphisms. In particular, a bounded Depth Minimum Steiner Trees (D-MST) clustering algorithm is presented for discriminating groups of individu- als relying on the manifestation/absence of the labio-schisis pathology, commonly called cleft lip. The analysis of three-dimensional facial surfaces via Dierential Geometry is adopted to extract landmarks. The extracted geometrical information is furthermore elaborated to feed the unsupervised clustering algorithm and produce the classication. The clustering returns the probability of being aected by the pathology, allowing physi- cians to focus their attention on risky individuals for further analysis

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology

Introduction to the R package TDA

We present a short tutorial and introduction to using the R package TDA, which provides some tools for Topological Data Analysis. In particular, it includes implementations of functions that, given some data, provide topological information about the underlying space, such as the distance function, the distance to a measure, the kNN density estimator, the kernel density estimator, and the kernel distance. The salient topological features of the sublevel sets (or superlevel sets) of these functions can be quantified with persistent homology. We provide an R interface for the efficient algorithms of the C++ libraries GUDHI, Dionysus and PHAT, including a function for the persistent homology of the Rips filtration, and one for the persistent homology of sublevel sets (or superlevel sets) of arbitrary functions evaluated over a grid of points. The significance of the features in the resulting persistence diagrams can be analyzed with functions that implement recently developed statistical methods. The R package TDA also includes the implementation of an algorithm for density clustering, which allows us to identify the spatial organization of the probability mass associated to a density function and visualize it by means of a dendrogram, the cluster tree

The Fermat-Torricelli problem in the case of three-point sets in normed planes

In the paper the Fermat-Torricelli problem is considered. The problem asks a point minimizing the sum of distances to arbitrarily given points in d-dimensional real normed spaces. Various generalizations of this problem are outlined, current methods of solving and some recent results in this area are presented. The aim of the article is to find an answer to the following question: in what norms on the plane is the solution of the Fermat-Torricelli problem unique for any three points. The uniqueness criterion is formulated and proved in the work, in addition, the application of the criterion on the norms set by regular polygons, the so-called lambda planes, is shown.Comment: 13 pages, 9 figure

Locally minimal uniformly oriented shortest networks

AbstractThe Steiner problem in a λ-plane is the problem of constructing a minimum length network interconnecting a given set of nodes (called terminals), with the constraint that all line segments in the network have slopes chosen from λ uniform orientations in the plane. This network is referred to as a minimum λ-tree. The problem is a generalization of the classical Euclidean and rectilinear Steiner tree problems, with important applications to VLSI wiring design.A λ-tree is said to be locally minimal if its length cannot be reduced by small perturbations of its Steiner points. In this paper we prove that a λ-tree is locally minimal if and only if the length of each path in the tree cannot be reduced under a special parallel perturbation on paths known as a shift. This proves a conjecture on necessary and sufficient conditions for locally minimal λ-trees raised in [M. Brazil, D.A. Thomas, J.F. Weng, Forbidden subpaths for Steiner minimum networks in uniform orientation metrics, Networks 39 (2002) 186–222]. For any path P in a λ-tree T, we then find a simple condition, based on the sum of all angles on one side of P, to determine whether a shift on P reduces, preserves, or increases the length of T. This result improves on our previous forbidden paths results in [M. Brazil, D.A. Thomas, J.F. Weng, Forbidden subpaths for Steiner minimum networks in uniform orientation metrics, Networks 39 (2002) 186–222]

A weak acceleration effect due to residual gravity in a multiply connected universe

Could cosmic topology imply dark energy? We use a weak field (Newtonian) approximation of gravity and consider the gravitational effect from distant, multiple copies of a large, collapsed (virialised) object today (i.e. a massive galaxy cluster), taking into account the finite propagation speed of gravity, in a flat, multiply connected universe, and assume that due to a prior epoch of fast expansion (e.g. inflation), the gravitational effect of the distant copies is felt locally, from beyond the naively calculated horizon. We find that for a universe with a $T^1xR^2$ spatial section, the residual Newtonian gravitational force (to first order) provides an anisotropic effect that repels test particles from the cluster in the compact direction, in a way algebraically similar to that of dark energy. For a typical test object at comoving distance $\chi$ from the nearest dense nodes of the cosmic web of density perturbations, the pressure-to-density ratio $w$ of the equation of state in an FLRW universe, is w \sim - (\chi/L)^3, where $L$ is the size of the fundamental domain, i.e. of the universe. Clearly, |w|<<1. For a T^3 spatial section of exactly equal fundamental lengths, the effect cancels to zero. For a T^3 spatial section of unequal fundamental lengths, the acceleration effect is anisotropic in the sense that it will *tend to equalise the three fundamental lengths*. Provided that at least a modest amount of inflation occurred in the early Universe, and given some other conditions, multiple connectedness does generate an effect similar to that of dark energy, but the amplitude of the effect at the present epoch is too small to explain the observed dark energy density and its anisotropy makes it an unrealistic candidate for the observed dark energy.Comment: 12 pages, 8 figures, accepted by Astronomy & Astrophysics; v2 includes 3D calculation and result; v3 includes analysis of numerical simulation, matches accepted versio