A simple example of the weak discontinuity of f↦∫det⁡∇ff\mapsto \int \det \nabla f

In this short note I construct a sequence of conformal diffeomorphisms $f_n$ from the $d$-dimensional unit ball to itself that converges weakly to a constant in $W^{1,d}$. This is a simpler example than the ones that commonly appear in the literature for the discontinuity of the functional $f\mapsto \int \det \nabla f$ in the weak $W^{1,d}$-topology, and shows that this discontinuity is not due to shear or failure of injectivity. It is also a minimizing sequence of the $W^{1,d}$ norm among functions of a given volume of the image

Cooperation under Incomplete Information on the Discount Factors

In the repeated Prisoner's Dilemma, when every player has a different discount factor, the grim-trigger strategy is an equilibrium if and only if the discount factor of each player is higher than some threshold. What happens if the players have incomplete information regarding the discount factors? In this work we look at repeated games in which each player has incomplete information regarding the other player's discount factor, and ask when a pair of grim-trigger strategies is an equilibrium. We provide necessary and sufficient conditions for such strategies to be an equilibrium. We characterize the states of the world in which the strategies are not triggered, i.e., the players cooperate, in such equilibria (or $\epsilon$-equilibria), and ask whether these "cooperation events" are close to those in the complete information case, when the information is "almost" complete, in several senses.Comment: Thesis submitted in partial fulfillment of requirements for the M. Sc. degree in the School of Mathematical Sciences, Tel-Aviv University. The research work for this thesis has been carried out at Tel-Aviv University under the supervision of Prof. Eilon Sola

Limits of elastic models of converging Riemannian manifolds

In non-linear incompatible elasticity, the configurations are maps from a non-Euclidean body manifold into the ambient Euclidean space, $\mathbb{R}^k$. We prove the $\Gamma$-convergence of elastic energies for configurations of a converging sequence, $\mathcal{M}_n\to\mathcal{M}$, of body manifolds. This convergence result has several implications: (i) It can be viewed as a general structural stability property of the elastic model. (ii) It applies to certain classes of bodies with defects, and in particular, to the limit of bodies with increasingly dense edge-dislocations. (iii) It applies to approximation of elastic bodies by piecewise-affine manifolds. In the context of continuously-distributed dislocations, it reveals that the torsion field, which has been used traditionally to quantify the density of dislocations, is immaterial in the limiting elastic model

A Riemannian approach to the membrane limit in non-Euclidean elasticity

Non-Euclidean, or incompatible elasticity is an elastic theory for pre-stressed materials, which is based on a modeling of the elastic body as a Riemannian manifold. In this paper we derive a dimensionally-reduced model of the so-called membrane limit of a thin incompatible body. By generalizing classical dimension reduction techniques to the Riemannian setting, we are able to prove a general theorem that applies to an elastic body of arbitrary dimension, arbitrary slender dimension, and arbitrary metric. The limiting model implies the minimization of an integral functional defined over immersions of a limiting submanifold in Euclidean space. The limiting energy only depends on the first derivative of the immersion, and for frame-indifferent models, only on the resulting pullback metric induced on the submanifold, i.e., there are no bending contributions

Cooperation under Incomplete Information on the Discount Factors

In repeated games, cooperation is possible in equilibrium only if players are sufficiently patient, and long-term gains from cooperation outweigh short-term gains from deviation. What happens if the players have incomplete information regarding each other's discount factors? In this paper we look at repeated games in which each player has incomplete information regarding the other player's discount factor, and ask when full cooperation can arise in equilibrium. We provide necessary and sufficient conditions that allow full cooperation in equilibrium that is composed of grim trigger strategies, and characterize the states of the world in which full cooperation occurs. We then ask whether these "cooperation events" are close to those in the complete information case, when the information on the other player's discount factor is "almost" complete.Comment: appears in International Journal of Game Theory, 201

Variational Convergence of Discrete Geometrically-Incompatible Elastic Models

We derive a continuum model for incompatible elasticity as a variational limit of a family of discrete nearest-neighbor elastic models. The discrete models are based on discretizations of a smooth Riemannian manifold $(M,\mathfrak{g})$, endowed with a flat, symmetric connection $\nabla$. The metric $\mathfrak{g}$ determines local equilibrium distances between neighboring points; the connection $\nabla$ induces a lattice structure shared by all the discrete models. The limit model satisfies a fundamental rigidity property: there are no stress-free configurations, unless $\mathfrak{g}$ is flat, i.e., has zero Riemann curvature. Our analysis focuses on two-dimensional systems, however, all our results readily generalize to higher dimensions.Comment: v3: a more concise version (similar to the published version); proof of Proposition 4.4 corrected, Lemma A.4 adde

Riemannian surfaces with torsion as homogenization limits of locally-Euclidean surfaces with dislocation-type singularities

We reconcile between two classical models of edge-dislocations in solids. The first model, dating from the early 1900s models isolated edge-dislocations as line singularities in locally-Euclidean manifolds. The second model, dating from the 1950s, models continuously-distributed edge-dislocations as smooth manifolds endowed with non-symmetric affine connections (equivalently, endowed with torsion fields). In both models, the solid is modeled as a Weitzenb\"ock manifold. We prove, using a weak notion of convergence [KM15], that the second model can be obtained rigorously as a homogenization limit of the first model, as the density of singular edge-dislocation tends to infinity

On the role of curvature in the elastic energy of non-Euclidean thin bodies

We prove a relation between the scaling $h^\beta$ of the elastic energies of shrinking non-Euclidean bodies $S_h$ of thickness $h\to 0$, and the curvature along their mid-surface $S$. This extends and generalizes similar results for plates [BLS16, LRR] to any dimension and co-dimension. In particular, it proves that the natural scaling for non-Euclidean rods with smooth metric is $h^4$, as claimed in [AAE+12] using a formal asymptotic expansion. The proof involves calculating the $\Gamma$-limit for the elastic energies of small balls $B_h(p)$, scaled by $h^4$, and showing that the limit infimum energy is given by a square of a norm of the curvature at a point $p$. This $\Gamma$-limit proves asymptotics calculated in [AKM+16]

The emergence of torsion in the continuum limit of distributed edge-dislocations

We present a rigorous homogenization theorem for distributed dislocations. We construct a sequence of locally-flat Riemannian manifolds with dislocation-type singularities. We show that this sequence converges, as the dislocations become denser, to a flat non-singular Weitzenb\"ock manifold, i.e. a flat manifold endowed with a metrically-consistent connection with zero curvature and non-zero torsion. In the process, we introduce a new notion of convergence of Weitzenb\"ock manifolds, which is relevant to this class of homogenization problems.Comment: See an erratum regarding Definition 4.1 and Lemma 4.8 in arxiv:1701.08903 (v3 is the same as v2 but for this comment

Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups

We study the geodesic distance induced by right-invariant metrics on the group $\operatorname{Diff}_\text{c}(M)$ of compactly supported diffeomorphisms, for various Sobolev norms $W^{s,p}$. Our main result is that the geodesic distance vanishes identically on every connected component whenever $s<\min\{n/p,1\}$, where $n$ is the dimension of $M$. We also show that previous results imply that whenever $s > n/p$ or $s \ge 1$, the geodesic distance is always positive. In particular, when $n\ge 2$, the geodesic distance vanishes if and only if $s<1$ in the Riemannian case $p=2$, contrary to a conjecture made in Bauer et al. [BBHM13].Comment: Version 2: some changes in presentation of the paper, no mathematical changes. Now includes a link to some related videos in http://www.math.toronto.edu/rjerrard/geo_dist_diffeo/vanishing.htm