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Vengono descritte le gare a squadre delle Olimpiadi della Matematica, una competizione annuale molto seguita dagli studenti delle scuole superiori in Italia. Le squadre, composte da sette membri, si allenano durante tutto l'anno e si sfidano con spirito agonistico. Le gare a squadre stimolano la collaborazione alla risoluzione dei problemi e l'aspetto ludico aiuta a rendere la matematica meno temuta., The paper describes the team competitions of the Mathematical Olympiad, an annual competition much awaited by students in Italy. The teams are composed of seven members and they train throughout the year and challenge each other with a competitive spirit. Mathematical team competitions stimulate collaboration in problemsolving and the playful aspect helps to fight the fear of mathematics.

We study the continuous version of a hyperbolic rescaling of a discrete game, called open mancala. The resulting PDE turns out to be a singular transport equation, with a forcing term taking values in , and discontinuous in the solution itself. We prove existence and uniqueness of a certain formulation of the problem, based on a nonlocal equation satisfied by the free boundary dividing the region where the forcing is one (active region) and the region where there is no forcing (tail region). Several examples, most notably the Riemann problem, are provided, related to singularity formation. Interestingly, the solution can be obtained by a suitable vertical rearrangement of a multi-function. Furthermore, the PDE admits a Lyapunov functional.

A handlebody link is a union of handlebodies of positive genus embedded in 3-space, which generalises the notion of links in classical knot theory. In this paper, we consider handlebody links with a genus two handlebody and n-1 solid tori, n > 1. Our main result is the classification of such handlebody links with six crossings or less, up to ambient isotopy.

X-ray crystallography is a powerful method that has significantly contributed to our understanding of the biological function of proteins and other molecules. This method relies on the production of crystals that, however, are usually a bottleneck in the process. For some molecules, no crystallization has been achieved or insufficient crystals were obtained. Some other systems do not crystallize at all, such as nanoparticles which, because of their dimensions, cannot be treated by the usual crystallographic methods. To solve this, whole pair distribution function has been proposed to bridge the gap between Bragg and Debye scattering theories. To execute a fitting, the spectra of several different constructs, composed of millions of particles each, should be computed using a particle–pair or particle–particle (pp) distance algorithm. Using this computation as a test bench for current field-programmable gate array (FPGA) technology, we evaluate how the parallel computation capability of FPGAs can be exploited to reduce the computation time. We present two different solutions to the problem using two state-of-the-art FPGA technologies. In the first one, the main C program uses OmpSs (a high-level programming model developed at the Barcelona Supercomputing Center, that enables task offload to different high-performance computing devices) for task invocation, and kernels are built with OpenCL using reduced data sizes to save transmission time. The second approach uses task and data parallelism to operate on data locally and update data globally in a decoupled task. Benchmarks have been evaluated over an Intel D5005 Programmable Acceleration Card, computing a model of 2 million particles in 81.57 s – 24.5 billion atom pairs per second (bapps)– and over a ZU102 in 115.31 s. In our last test, over an up-to-date Alveo U200 board, the computation lasted for 34.68 s (57.67 bapps). In this study, we analyze the results in relation to the classic terms of speed-up and efficiency, This work was partially supported by the Spanish Government (AEI) and European Union (FEDER) (projects CTQ2017-88948-P, PID2021-124848OB-I00, RTI2018-095232-B-C21, SEV-2015-0493 and TIN2015-65316-P, grant BES-2016-078046) and by the Generalitat de Catalunya (contracts 2017-SGR-1742, 2017-SGR-1414, 2017-SGR-670 and 2017-SGR-1328)., Peer Reviewed, Postprint (published version)

A handlebody link is a union of handlebodies of positive genus embedded in 3-space, which generalises the notion of links in classical knot theory. In this paper, we consider handlebody links with a genus two handlebody and n - 1 solid tori, n > 1. Our main result is the classification of such handlebody links with six crossings or less, up to ambient isotopy.

X-ray crystallography is a powerful method that has significantly contributed to our understanding of the biological function of proteins and other molecules. This method relies on the production of crystals that, however, are usually a bottleneck in the process. For some molecules, no crystallization has been achieved or insufficient crystals were obtained. Some other systems do not crystallize at all, such as nanoparticles which, because of their dimensions, cannot be treated by the usual crystallographic methods. To solve this, whole pair distribution function has been proposed to bridge the gap between Bragg and Debye scattering theories. To execute a fitting, the spectra of several different constructs, composed of millions of particles each, should be computed using a particle-pair or particle-particle (pp) distance algorithm. Using this computation as a test bench for current field-programmable gate array (FPGA) technology, we evaluate how the parallel computation capability of FPGAs can be exploited to reduce the computation time. We present two different solutions to the problem using two state-of-the-art FPGA technologies. In the first one, the main C program uses OmpSs (a high-level programming model developed at the Barcelona Supercomputing Center, that enables task offload to different high-performance computing devices) for task invocation, and kernels are built with OpenCL using reduced data sizes to save transmission time. The second approach uses task and data parallelism to operate on data locally and update data globally in a decoupled task. Benchmarks have been evaluated over an Intel D5005 Programmable Acceleration Card, computing a model of 2 million particles in 81.57 s - 24.5 billion atom pairs per second (bapps)- and over a ZU102 in 115.31 s. In our last test, over an up-to-date Alveo U200 board, the computation lasted for 34.68 s (57.67 bapps). In this study, we analyze the results in relation to the classic terms of speed-up and efficiency

Associated to an embedded surface in the three-sphere, we construct a diagram of fundamental groups, and prove that it is a complete invariant, whereform we deduce complete invariants of handlebody links, tunnels of handlebody links, and spatial graphs. The main ingredients in the proof of the completeness include a generalization of the Kneser conjecture for three-manifolds with boundary proved here, and extensions of Waldhausen's theorem by Evans, Tucker and Swarup. Computable invariants of handlebody links derived therefrom are calculated.

In this paper we first review the covering space method with constrained BV functions for solving the classical Plateau's problem. Next, we carefully analyze some interesting examples of soap films compatible with the covering space method: in particular, the case of a soap film only partially wetting a space curve, a soap film spanning a cubical frame but having a large tunnel, aa soap film that retracts onto its boundary, hence not modelable with the Reifenberg method, and various soap films spanning an octahedral frame.

In this paper, we study the diffusion of a water release into an important real water basin, Lake Garda in northern Italy, coupling a variety of models from hydrodynamics (full Navier-Stokes for the motion of water) to statistics (for the modelling of rain inflow and the time-periodic wind motion on the surface). Using a recent digital bathymetry, we are able to present some novel forecasts of water dynamics, in the presence of an uncommon water release by river Adige into Lake Garda, necessary to prevent flood situations in case of heavy rainfall, and taking into account common inflows, wind speed on the surface and friction on the bottom. The output shows the trajectories of the released particles, which are typically much colder than the existing ones, and may help to individuate critical sites and shores.

We construct a complete invariant of oriented connected closed surfaces in 3, which generalizes the notion of peripheral system of a knot group. As an application, we define two computable invariants to investigate handlebody knots and bi-knotted surfaces with homeomorphic complements. In particular, we obtain an alternative proof of inequivalence of Ishii, Kishimoto, Moriuchi and Suzuki's handlebody knots 51 and 64.

Per mantenere vivo il ricordo di Martin Gardner il gruppo G4G (Gathering for Gardner) si riunisce periodicamente contribuendo con nuovi rompicapo, puzzle, oggetti ingegnosi.

By using two different invariants for the Rubik’s Magic puzzle, one of metric type, the other of topological type, we can dramatically reduce the universe of constructible configurations of the puzzle. Finding the set of actually constructible shapes remains however a challenging task, that we tackle by first reducing the target shapes to specific configurations: the octominoid 3D shapes, with all tiles parallel to one coordinate plane; and the planar “face-up” shapes, with all tiles (considered of infinitesimal width) lying in a common plane and without superposed consecutive tiles. There are still plenty of interesting configurations that do not belong to either of these two collections. The set of constructible configurations (those that can be obtained by manipulation of the undecorated puzzle from the starting situation) is a subset of the set of configurations with vanishing invariants. We were able to actually construct all octominoid shapes with vanishing invariants and most of the planar “face-up” configurations. Particularly important is the topological invariant, of which we recently found mention in a paper by Tom Verhoeff.

Si tratta di una illusione descritta da Martin Gardner in uno dei suoi libri. Nella presente nota si descrive un metodo semplice ed intuitivo per costruire questa catena di anelli

In this paper, we estimate from above the area of the graph of a singular map u taking a disk to three vectors, the vertices of a triangle, and jumping along three C2-embedded curves that meet transversely at only one point of the disk. We show that the singular part of the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to “fill the hole” in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces, placed vertically over the jump of u, coupled together via a triple point in the target triangle. Such a construction depends on the choice of a target triple point, and on a connection passing through it, which dictate the boundary condition for the three minimal surfaces. We show that the singular part of the relaxed area of u cannot be larger than what we obtain by minimizing over all possible target triple points and all corresponding connections.

Using a suitable triple covering space it is possible to model the construction of a non-simply connected minimal surface spanning all six edges of an elongated tetrahedron, working in the space of BV functions and interpreting the film as the boundary of a Caccioppoli set in the covering space. The possibility of using covering spaces for minimal surfaces was first proposed by Brakke. After a question raised by R. Hardt in the late 1980's, it seems common opinion that an area-minimizing surface of this sort does not exist for a regular tetrahedron, although a proof of this fact is still missing. In this paper we show that there exists a non-simply connected surface spanning the boundary of an elongated tetrahedron and having area strictly less than the area of the minimal contractible surface.

We prove the optimal bound for the maximal number of moves required to reach a periodic configuration of open mancala (also called open owari or with other names), inspired by a popular african game. The proof relies on the notion of s-monotonicity, which is a stronger version of the mere monotonicity.

Motivated by a variational model concerning the depth of the objects in a picture and the problem of illusory contours, this book investigates one of the central problems of computer vision: the topological and algorithmical reconstruction of a smooth three dimensional scene starting from the visible part of an apparent contour. The authors focus their attention on the manipulation of apparent contours using a finite set of elementary moves, which correspond to consider diffeomorphic three dimensional scenes. A large part of the book is devoted to the algorithmic part, with implementations, experiments, and computed examples. The book is intended also as a user's guide to the software code appcontour, written for the manipulation of apparent contours and their invariants.

We study the asymptotic analysis of a singularly perturbed weakly parabolic system of m equations of anisotropic reaction-diffusion type. Our main result formally shows that solutions to the system approximate a geometric motion of a hypersurface by anisotropic mean curvature. The anisotropy, supposed to be uniformly convex, is explicit and turns out to be the dual of the star-shaped combination of the m original anisotropies.

In this paper we are concerned with labelled apparent contours, namely with apparent contours of generic orthogonal projections of embedded surfaces in R3, endowed with a suitable depth information. We show that there exists a finite set of elementary moves (i.e. local topological changes) on labelled apparent contours such that the following theorem holds: two generic embeddings of a closed surface S in R3 are isotopic if and only if their apparent contours can be connected using only smooth isotopies and a finite sequence of moves.

In this paper we study the nonconvex anisotropic mean curvature flow of a hypersurface. This corresponds to an anisotropic mean curvature flow where the anisotropy has a nonconvex Frank diagram. The geometric evolution law is therefore forward-backward parabolic in character, hence ill-posed in general. We study a particular regularization of this geometric evolution, obtained with a nonlinear version of the so-called bidomain model. This is described by a degenerate system of two uniformly parabolic equations of reaction-diffusion type, scaled with a positive parameter $\epsilon$. We analyze some properties of the formal limit of solutions of this system as $\epsilon \to 0$, and show its connection with nonconvex mean curvature flow. Several numerical experiments substantiating the formal asymptotic analysis are presented.

We prove that the semidiscrete schemes of a Perona-Malik type equation converge, in a long time scale, to a suitable system of ordinary differential equations defined on piecewise constant functions. The proof is based on a formal asymptotic expansion argument, and on a careful construction of discrete sub and supersolutions. Despite the equation has a region where it is backward parabolic, we prove a discrete comparison principle, which is the key tool for the convergence result.

We improve an estimate given by Acerbi and Dal Maso in 1994, concerning the area of the graph of a singular map from the disk of R2 into R2, taking only three values, and jumping on three half-lines meeting at the origin in a triple junction.

We give a complete classification of the stability properties of the equilibria for the semi-discrete one-dimensional Perona-Malik equation, with Dirichlet boundary conditions. We also give the $\Gamma$-expansion of the corresponding discretized functionals up to the order two, as the discretization parameter goes to zero.

We derive an explicit formula for the computation of a Vassiliev-type invariant of an apparent contour introduced recently by Ohmoto and Aicardi using appropriate linking numbers of its Legendrian lift. Our formula does not require the construction of the Legendrian lift and only takes into account the nodes, the cusps, the extremal points and the orientation of the apparent contour. In this way the computation can be implemented into a computer program; we demonstrate its use with some examples.

We show that the completion problem of reconstructing the hidden arcs of the contours of an image, given only the visible ones, has a solution. More precisely we prove that, given an oriented plane graph K having as vertices only T-junctions and nonexterior terminal points, there exists an apparent contour G such that K is the visible part of G. This result is sharp, since the converse statement is easily seen to be satisfied. As a consequence, from K we can reconstruct a solid shape E in three-dimensional space such that K coincides with the visible part of the apparent contour of E. The main tools to prove our result are a Morse description of K and the Huffman labelling for apparent contours.

We introduce and study a two-dimensional model for the reconstruction of a smooth generic three-dimensional scene E, which may handle the self-occlusions and that can be considered as an improvement of the 2.1D sketch of Nitzberg and Mumford. We characterize from the topological viewpoint the apparent contour of E, namely, we characterize those planar graphs G that are apparent contours of some scene E. Moreover, we show that if E and F are two of these scenes, then E and F differ by a global homeomorphism which is strictly increasing on each fiber along the direction of the eye of the observer. These two topological theorems allow to find the domain of the functional F describing the model. Compactness, semicontinuity and relaxation properties of F are then studied, as well as connections of our model with the problem of completion of hidden contours.

We prove the convergence, up to a subsequence, of the spatial semidiscrete scheme for the one-dimensional Perona-Malik equation $u_t = (\phi'(u_x))_x$, $\phi(p) := \log(1+p^2)/2$, when the initial datum $\bar u$ is 1-Lipschitz out of a finite number of jump points, and we characterize the problem satisfied by the limit solution. In the more difficult case when $\bar u$ has a whole interval where $\phi''(\bar u_x)$ is negative, we construct a solution by a careful inspection of the behaviour of the approximating solutions in a space-time neighbourhood of the jump points. The limit solution u we obtain is the same as the one obtained by replacing $\phi(\cdot)$ with the truncated function $\min(\phi(\cdot),1)$, and it turns out that $u$ solves a free boundary problem. The free boundary consists of the points dividing the region where $|u_x| > 1$ from the region where $|u_x| \leq 1$. Finally, we consider the full space-time discretization (implicit in time) of the Perona-Malik equation, and we show that, if the time step is small with respect to the spatial grid h, then the limit is the same as the one obtained with the spatial semidiscrete scheme. On the other hand, if the time step is large with respect to h, then the limit solution equals $\bar u$, i.e., the standing solution of the convexified problem.

In this extended abstract we describe a software code that is able to simbolically manipulate an apparent contour described in terms of its "region description" or "morse description"

We discuss some examples of nonuniqueness for the crystalline curvatur e flow, when the Wulff shape is a square not centered at the origin

In this extended abstract we investigate some relations between existence conditions of a particular capillarity problem with tangential contact angle and evolution by mixed crystalline curvature with a cylindrical Wulff shape. Bending/breaking of faces for crystalline evolution indeed correspond to cross section for the capillary container for which no solution of the capillarity problem exists in the form of a subgraph.

Un software di geometria interattiva non puo' sostituire una dimostrazione rigorosa, come viene esemplificato tramite una particolare costruzione geometrica che mette in evidenza gli errori di arrotondamento del calcolatore.

We provide the complete argument for the proof of step 2 in the demonstration of Lemma 4.2 in the paper "Some results on minimal barriers in the sense of De Giorgi applied to driven motion by mean curvature", 1995 of the same authors.