Anisotropic functions : a genericity result with crystallographic implications

Victor J. Mizel; Alexander J. Zaslavski

Displaying similar documents to “Anisotropic functions : a genericity result with crystallographic implications”

A proof of the crossing number of $K_{3, n}$ in a surface

Pak Tung Ho (2007)

Discussiones Mathematicae Graph Theory

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In this note we give a simple proof of a result of Richter and Siran by basic counting method, which says that the crossing number of $K_{3, n}$ in a surface with Euler genus ε is ⎣n/(2ε+2)⎦ n - (ε+1)(1+⎣n/(2ε+2)⎦).

Gradient theory for plasticity via homogenization of discrete dislocations

Adriana Garroni, Giovanni Leoni, Marcello Ponsiglione (2010)

Journal of the European Mathematical Society

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We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations. We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so that the mathematical formulation will involve a two-dimensional variational problem. The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so-called core region. We show that the $Γ$ -limit of this energy (suitably...

Symmetry of minimizers with a level surface parallel to the boundary

Giulio Ciraolo, Rolando Magnanini, Shigeru Sakaguchi (2015)

Journal of the European Mathematical Society

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We consider the functional $ℐ_{Ω} (v) = \int_{Ω} [f (| D v |) - v] d x,$ where $Ω$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$ , Crasta [Cr1] has shown that if $ℐ_{Ω}$ admits a minimizer in $W_{0}^{1, 1} (Ω)$ depending only on the distance from the boundary of $Ω$ , then $Ω$ must be a ball. With some restrictions on $f$ , we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss...

Flexibility of surface groups in classical simple Lie groups

Inkang Kim, Pierre Pansu (2015)

Journal of the European Mathematical Society

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We show that a surface group of high genus contained in a classical simple Lie group can be deformed to become Zariski dense, unless the Lie group is $S U (p, q)$ (resp. $S O^{*} (2 n)$ , $n$ odd) and the surface group is maximal in some $S (U (p, p) \times U (q - p)) \subset S U (p, q)$ (resp. $S O^{*} (2 n - 2) \times S O (2) \subset S O^{*} (2 n)$ ). This is a converse, for classical groups, to a rigidity result of S. Bradlow, O. García-Prada and P. Gothen.

Bridgeland-stable moduli spaces for $K$ -trivial surfaces

Daniele Arcara, Aaron Bertram (2013)

Journal of the European Mathematical Society

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We give a one-parameter family of Bridgeland stability conditions on the derived category of a smooth projective complex surface $S$ and describe “wall-crossing behavior” for objects with the same invariants as $𝒪_{C} (H)$ when $H$ generates Pic $(S)$ and $C \in |H|$ . If, in addition, $S$ is a $K 3$ or Abelian surface, we use this description to construct a sequence of fine moduli spaces of Bridgeland-stable objects via Mukai flops and generalized elementary modifications of the universal coherent sheaf. We also discover...

Scaling limit and cube-root fluctuations in SOS surfaces above a wall

Pietro Caputo, Eyal Lubetzky, Fabio Martinelli, Allan Sly, Fabio Lucio Toninelli (2016)

Journal of the European Mathematical Society

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Consider the classical $(2 + 1)$ -dimensional Solid-On-Solid model above a hard wall on an $L \times L$ box of $ℤ^{2}$ . The model describes a crystal surface by assigning a non-negative integer height $η_{x}$ to each site $x$ in the box and 0 heights to its boundary. The probability of a surface configuration $η$ is proportional to $\exp (- β ℋ (η))$ , where $β$ is the inverse-temperature and $ℋ (η)$ sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough...

Symplectic critical surfaces in Kähler surfaces

Xiaoli Han, Jiayu Li (2010)

Journal of the European Mathematical Society

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Let $M$ be a Kähler surface and $Σ$ be a closed symplectic surface which is smoothly immersed in $M$ . Let $α$ be the Kähler angle of $Σ$ in $M$ . We first deduce the Euler-Lagrange equation of the functional $L = \int_{Σ} \frac{1}{cos α} d μ$ in the class of symplectic surfaces. It is ${cos}^{3} α H = {(J {(J \nabla cos α)}^{⊤})}^{⊥}$ , where $H$ is the mean curvature vector of $Σ$ in $M$ , $J$ is the complex structure compatible with the Kähler form $ω$ in $M$ , which is an elliptic equation. We call such a surface a symplectic critical surface. We show that, if $M$ is a Kähler-Einstein surface...

Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces

Mikhail Karpukhin, Gerasim Kokarev, Iosif Polterovich (2014)

Annales de l’institut Fourier

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We prove two explicit bounds for the multiplicities of Steklov eigenvalues $σ_{k}$ on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index $k$ of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given Riemannian surface with smooth boundary the multiplicities of Steklov eigenvalues $σ_{k}$ are uniformly bounded in $k$ .

A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types

Lucas Döring, Radu Ignat, Felix Otto (2014)

Journal of the European Mathematical Society

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We study the Landau-Lifshitz model for the energy of multi-scale transition layers – called “domain walls” – in soft ferromagnetic films. Domain walls separate domains of constant magnetization vectors $m_{α}^{\pm} \in 𝕊^{2}$ that differ by an angle $2 α$ . Assuming translation invariance tangential to the wall, our main result is the rigorous derivation of a reduced model for the energy of the optimal transition layer, which in a certain parameter regime confirms the experimental, numerical and physical predictions:...

Dimension vs. genus: A surface realization of the little k-cubes and an $E_{\infty}$ operad

Ralph M. Kaufmann (2009)

Banach Center Publications

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We define a new $E_{\infty}$ operad based on surfaces with foliations which contains $E_{k}$ suboperads. We construct CW models for these operads and provide applications of these models by giving actions on Hochschild complexes (thus making contact with string topology), by giving explicit cell representatives for the Dyer-Lashof-Cohen operations for the 2-cubes and by constructing new Ω spectra. The underlying novel principle is that we can trade genus in the surface representation vs. the dimension...

Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II

Fabrizio Catanese, Frédéric Mangolte (2009)

Annales scientifiques de l'École Normale Supérieure

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Let $W \to X$ be a real smooth projective 3-fold fibred by rational curves such that $W (ℝ)$ is orientable. J. Kollár proved that a connected component $N$ of $W (ℝ)$ is essentially either Seifert fibred or a connected sum of lens spaces. Answering three questions of Kollár, we give sharp estimates on the number and the multiplicities of the Seifert fibres (resp. the number and the torsions of the lens spaces) when $X$ is a geometrically rational surface. When $N$ is Seifert fibred over a base orbifold $F$ , our...

Generalised functions of bounded deformation

Gianni Dal Maso (2013)

Journal of the European Mathematical Society

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We introduce the space $G B D$ of generalized functions of bounded deformation and study the structure properties of these functions: the rectiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for $G B D$ , which leads to a compactness result for the space $G S B D$ of generalized special functions of bounded deformation. The latter is connected to the existence of solutions to a weak formulation...

On Verlinde sheaves and strange duality over elliptic Noether-Lefschetz divisors

Alina Marian, Dragos Oprea (2014)

Annales de l’institut Fourier

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We extend results on generic strange duality for $K 3$ surfaces by showing that the proposed isomorphism holds over an entire Noether-Lefschetz divisor in the moduli space of quasipolarized $K 3$ s. We interpret the statement globally as an isomorphism of sheaves over this divisor, and also describe the global construction over the space of polarized $K 3 s$ .

A continuous Helson surface in $𝐑^{3}$

Detlef Müller (1984)

Annales de l'institut Fourier

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For some time it has been known that there exist continuous Helson curves in $R^{2}$ . This result, which is related to Lusin’s rearrangement problem, had been proved first by Kahane in 1968 with the aid of Baire category arguments. Later McGehee and Woodward extended this result, giving a concrete construction of a Helson $k$ -manifold in $R^{n k}$ for $n \geq k + 1$ . We present a construction of a Helson 2-manifold in $R^{3}$ . With modification, our method should even suffice to prove that there are Helson hypersurfaces...

On a bifurcation problem arising in cholesteric liquid crystal theory

Carlo Greco (2017)

Commentationes Mathematicae Universitatis Carolinae

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In a cholesteric liquid crystal the director field $n (x, y, z)$ tends to form a right-angle helicoid around a twist axis in order to minimize the internal energy; however, a fixed alignment of the director field at the boundary (strong anchoring) can give rise to distorted configurations of the director field, as oblique helicoid, in order to save energy. The transition to this distorted configurations depend on the boundary conditions and on the geometry of the liquid crystal, and it is known...

Vortex collisions and energy-dissipation rates in the Ginzburg–Landau heat flow. Part I: Study of the perturbed Ginzburg–Landau equation

Sylvia Serfaty (2007)

Journal of the European Mathematical Society

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We study vortices for solutions of the perturbed Ginzburg–Landau equations $Δ u + (u / ε^{2}) {(1 - | u |}^{2}) = f_{ε}$ where $f_{ε}$ is estimated in $L^{2}$ . We prove upper bounds for the Ginzburg–Landau energy in terms of $∥ f_{ε} ∥_{L^{2}}$ , and obtain lower bounds for $∥ f_{ε} ∥_{L^{2}}$ in terms of the vortices when these form “unbalanced clusters” where $\sum_{i} d_{i}^{2} \neq {(\sum_{i} d_{i})}^{2}$ . These results will serve in Part II of this paper to provide estimates on the energy-dissipation rates for solutions of the Ginzburg–Landau heat flow, which allow one to study various phenomena occurring in this flow,...