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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 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.

We investigate the breaking and bending phenomena of a facet of a three dimensional crystal which evolves under crystalline mean curvature flow. We give necessary and sufficient conditions for a facet to be calibrable, i.e. not to break or bend under the evolution process. We also give a criterion which allows to predict exactly where a subdivision of a not calibrable facet takes place in the evolution process.

Let p be a given positively one-homogeneous convex function, and let W be the corresponding unit ball. Motivated by motion by crystalline mean curvature in three dimensions, we introduce and study a class of smooth boundaries in the relative geometry induced by p. One can realize that, even when W is a polytope, such smooth class cannot be reduced to the class of polyhedral boundaries (locally resembling the boundary of W). Curved portions must be necessarily included and this fact (as well as the nonsmoothness of W) is source of several technical difficulties, related to the geometry of Lipschitz manifolds. Given a boundary ∂E in our class, we study a variational problem on vector fields defined on ∂E. It turns out that the minimizers have a uniquely determined (intrinsic) tangential divergence on ∂E. We define such a divergence to be the p-mean curvature kp of ∂E; the function kp is expected to be the initial velocity of ∂E, whenever ∂E is considered as the initial datum for the corresponding (nonsmooth) anisotropic mean curvature flow. We prove that kp is bounded on ∂E, and that it is of bounded variation on facets F of ∂E corresponding to facets of the boundary of W (if any). Finally, further regularity properties of kp and of its jump set are inspected: in particular, in three space dimensions, we relate the sublevel sets of kp on F with the geometry of the boundary of W. PART 2.

Anisotropic curvature flow equations with singular interfacial energy are important for good understanding of motion of phase-boundaries. If the energy and the interfacial surface were smooth, then the speed of the interface would be equal to the gradient of the energy. However, this is not so simple in the case of non-smooth crystalline energy. But it's well-known that a unique gradient characterization of the velocity is possible if the interface is a curve in the two-dimensional space. In this paper we propose a notion of solution in the three-dimensional space by introducing geometric subdifferentials and characterizing the speed. We also give a counterexample to a problem concerning the Cahn-Hoffman vector field on a facet, a flat portion of the interface.

Let p be a given positively one-homogeneous convex function, and let W be the corresponding unit ball. Motivated by motion by crystalline mean curvature in three dimensions, we introduce and study a class of smooth boundaries in the relative geometry induced by p. One can realize that, even when W is a polytope, such smooth class cannot be reduced to the class of polyhedral boundaries (locally resembling the boundary of W). Curved portions must be necessarily included and this fact (as well as the nonsmoothness of W) is source of several technical difficulties, related to the geometry of Lipschitz manifolds. Given a boundary ∂E in our class, we study a variational problem on vector fields defined on ∂E. It turns out that the minimizers have a uniquely determined (intrinsic) tangential divergence on ∂E. We define such a divergence to be the p-mean curvature kp of ∂E; the function kp is expected to be the initial velocity of ∂E, whenever ∂E is considered as the initial datum for the corresponding (nonsmooth) anisotropic mean curvature flow. We prove that kp is bounded on ∂E, and that it is of bounded variation on facets F of ∂E corresponding to facets of the boundary of W (if any). Finally, further regularity properties of kp and of its jump set are inspected: in particular, in three space dimensions, we relate the sublevel sets of kp on F with the geometry of the boundary of W. PART 1.

We consider a discrete approximation of the Mumford-Shah functional defined on finite element spaces. For some meaningful model case we study global and local minima, then, from their behavior, we can propose two minimization methods, based on the quasi-Newton algorithm, which can find the absolute minimum of the functional.

Evolution by mean curvature is recently attracting large attention especially when the underlying anisotropic structure degenerates to become crystalline. In such situation new phenomena must be taken into account: the evolution law becomes nonlocal and "hyperbolic" across facets; moreover events like face breaking or bending have to be considered especially in three dimensions. For this reason the ODE approach suggested by J. Taylor long ago cannot be used directly and the required modifications to the algorithm are not clear at the moment. The well known diffused interface approximation for the classical mean curvature flow, which leads to the Allen-Cahn equation, can be applied in this contex, resulting in a bistable reaction-diffusion equation with good convergence properties to the sharp interface evolution. This equation can then be discretized using finite elements in space and forward differences in time. Numerical simulations with the resulting scheme seem to recover the face breaking and face bending phenomena.

We show two examples of facet-breaking for three-dimensional polyhedral surfaces evolving by crystalline mean curvature. The analysis shows that creation of new facets during the evolution is a common phenomenon. The first example is completely rigorous, and the evolution after the subdivision of one facet is explicitly computed for short times. Moreover, the constructed evolution is unique among the crystalline flows with the given initial datum. The second example suggests that curved portions of the boundary may appear even starting from a polyhedral set close to the Wulff shape.

In this paper we consider the so called prescribed curvature problem approximated by a singularly perturbed double obstacle variational inequality. We extend [Pao97] with the introduction of the same nonregular potential used for the evolution problem [NPV94c] and we prove an optimal O(e2) error estimate for nondegenerate minimizers (where e stands for the perturbation parameter). Following [Pao97] the result relies on the construction of precise barriers suggested by formal asymptotics combined with the use of the maximum principle. Key ingredients are the construction of a sub(super)solution containing appropriate shape corrections and the use of a modified distance function based on the principal eigenfunction of the second variation of the prescribed curvature functional. This analysis is next extended to a piecewise linear finite element discretization of the elliptic PDE of bistable type to prove the same error extimate for discrete minima using the Rannacher-Scott L\infty-estimates and under appropriated restrictions on the mesh size (h2=O(es) with s > 5/2).

We prove that two linked circles in R3 evolving by curvature develop three dimensional fattening at finite time. The same result holds for two planar convex closed linked curves.

In this paper we present a few numerical simulations of a nonsymmetric anisotropic evolution by mean curvature which leads to the so-called fattening of the interface. The numerical simulations are based on a diffused interface approximation via a bistable reaction-diffusion equation which is then discretized by means of finite elements in space and forward differences in time. An adaptive strategy, together with the use of a dynamic mesh, is used to take advantage of the local behaviour of the equation at hand. However, this particular choice of anisotropy, with a large ratio between the maximal and minimal surface energy, seems to be highly critical for the type of approximation presented here, leading to very expensive computations in terms of CPU time.

In this paper we present and discuss the results of some numerical simulations in order to investigate the mean curvature flow problem in presence of a nonconvex anisotropy. Mathematically, nonconvexity of the anisotropy leads to the ill-posedness of the evolution problem, which becomes forward-backward parabolic. Simulations presented here refer to two different settings: curvature driven vertical motion of graphs (nonparametric setting) and motion in the normal direction by anisotropic mean curvature of surfaces (parametric setting). In the latter case we first relax the problem via an Allen-Cahn type reaction-diffusion equation, in the context of Finsler geometry (diffused interface approximation). Our results suggest three main points. A nonconvex anisotropy and its convexification give rise, for both settings and the discretizations considered, to different evolutions. Wrinkled regions seem to appear only in correspondence to locally concave parts of the anisotropy. Moreover, locally convex regions (interior to the convexification of the so-called Frank diagram) seem to play an important role.

We study the convergence of the singularly perturbed anisotropic, nonhomogeneous reaction-diffusion equation $\epsilon \partial_t u - \epsilon^2\text{div} T^o(x, \nabla u) + f(u) - \epsilon \frac{c_1}{c_0} g = 0$ where f is the derivative of a bistable quartic-like potential with unequal wells, $T^o (x, \cdot)$ is a nonlinear monotone operator homogeneous of degree one and g is a given forcing term. More precisely we prove that an appropriate level set of the solution satisfies an $O (\epsilon^3 |\log\epsilon|^2)$ error bound (in the Hausdorff distance) with respect to a hypersurface moving with the geometric law $V = (c - \epsilon \kappa_\phi) n_\phi +$ g-dependent terms, where $n_\phi$ is the so-called Cahn-Hoffmann vector and $\kappa_\phi$ denotes the anisotropic mean curvature of the hypersurface. We also discuss the connection between the anisotropic reaction-diffusion equation and the bidomain model, which is described by a system of equations modeling the propagation of an electric stimulus in the cardiac tissue.

A new finite element method is discussed for approximating evolving interfaces in $\Rn$ whose normal velocity equals mean curvature plus a forcing function. The method is insensitive to singularity formation and retains the local structure of the limit problem, and thus exhibits a computational complexity typical of $\R^{n-1}$ without having the drawbacks of front-tracking strategies. A graded dynamic mesh around the propagating front is the sole partition present at any time step and is significantly smaller than a full mesh. Time stepping is explicit, but stability constraints force small time steps only when singularities develop, whereas relatively large time steps are allowed before or past singularities, when the evolution is smooth. The explicit marching scheme also guarantees that at most one layer of elements has to be added or deleted per time step, thereby making mesh updating simple, and thus practical. Performance and potentials are fully documented via a number of numerical simulations in 2D, 3D, 4D, and 8D, with axial symmetries. They include tori and cones for the mean curvature flow, minimal and prescribed mean curvature surfaces with given boundary, fattening for smooth driving force, and volume constraint.

In a recent paper, Fischer and Finn have proposed a procedure to improve the accuracy in the measurament of capillary contact angles, based on the use of vessels with canonical cross-sections. We simulate numerically the behaviour of such shapes for a number of cross-sections and fluid contact angles. Our approximation consists of the minimization of a suitable convex functional discretized by finite elements.

We study the anisotropic motion of a hypersurface in the context of the geometry of Finsler spaces. This amounts in considering the evolution in relative geometry, where all quantities are referred to the given Finsler metric $\phi$ representing the anisotropy, which we allow to be a function of space. Assuming that the anisotropy is strictly convex and smooth, we prove that the natural evolution law is of the form "velocity = $H_\phi$", where $H_\phi$ is the relative mean curvature vector of the hypersurface. We derive this evolution law using different approches, such as the variational method of Almgren-Taylor-Wang, the Hamilton-Jacobi equation, and the approximation by means of a reaction-diffusion equation.

In this paper, we prove that solutions of the anisotropic Allen-Cahn equation in double-obstacle form with kinetic term \begin{displaymath} \varepsilon \beta(\nabla \varphi) \partial_t \varphi - \varepsilon \nabla A'(\nabla \varphi) - \frac{1}{\varepsilon} \varphi = \frac{\pi}{4} u \quad \mbox{ in } [|\varphi| < 1], \end{displaymath} where $\ A\ $ is a convex function, homogeneous of degree two, and $\ \beta\ $ depends only on the direction of $\ \nabla \varphi\ $, converge to an anisotropic mean-curvature flow \begin{displaymath} \beta(N) V_N = - \mbox{tr}(B(N) D^2 B(N) R) - B(N) u. \end{displaymath} Here $\ V_N \mbox{ and } R\ $ respectively denote the normal velocity and the second fundamental form of the interface, and $\ B := \sqrt{2A}\ $. We prove this in the case when the above flow admits a smooth solution, and we establish that the Hausdorff-distance between the zero-level set of $\ \varphi\ $ and the interface of the flow is of order $\ O(\varepsilon^2)\ $.

We study some relations between the concepts of perimeter, Hausdorff measure, and Minkowsky content, when Rn is endowed with a convex Finsler metric depending in a continuous way on the position. We show some connections with the theory of Gamma-convergence and with the anisotropic motion of a smooth hypersurface by mean curvature.

In this paper we describe some numerical simulations in the context of mean curvature flow. We recover a few different approaches in modelling the evolution of an interface $\Sigma$ which evolves according to the law: $V = \kappa + g$ where $V$ is the velocity in the inward normal direction, $\kappa$ is the sum of the principal curvatures and $g$ is a given forcing term. We will discuss about the phenomenon of fattening or nonuniqueness of the solution, recovering what is known about this subject. Finally we show some interesting numerical simulations that suggests evidence of fattening starting from different initial interfaces. Of particular interest is the result obtained for a torus in $\bold R^4$ which would be a first example of a regular and compact surface showing evidence of fattening in the case of pure motion by mean curvature (no forcing term).

In this paper we present some numerical simulations of motion by mean curvature associated to an underlying anisotropy of crystalline type. Such anisotropies are characterized by a Frank diagram (and consequently a Wulff shape) of polygonal type. Our simulations are based on an Allen-Cahn type regularization, performed in the context of a Finsler geometry. The choice of a nonregular double well potential leads to a double obstacle formulation that allows us to exploit the dynamic mesh algorithm, thus reducing considerably the computational cost. A number of simulations show the robustness of our discretization process, which is capable to deal with situations (presence of a generic forcing term) that are critical for other known numerical techniques.

A number of numerical simulations of surfaces evolving by mean curvature in anisotropic materials are presented and discussed. We approximate an evolution law based on an appropriate definition of anisotropic mean curvature. This is first regularized by means of a relaxation procedure that produces a reaction-diffusion equation similar to the model proposed by Allen and Cahn for the evolution of antiphase boundaries. The successive discretization process is based on linear finite elements in space and forward differences in time. The final algorithm also exploits the localized behaviour of the solution using a dynamically maintained mesh that spans the sole transition region of the relaxed solution, corresponding to a thin strip around the moving interface.