Irrational Directions Research Articles

Controlling and Monitoring Crack Propagation in Monolayer Graphene Single Crystals

AbstractThe evolution and propagation of cracks in exfoliated monolayer graphene single crystals and their associated stress fields are studied using Raman spectroscopy as a strain sensor. Such stress fields are found to extend over several microns, and their analysis leads to the fracture behavior of graphene being interpreted using linear elastic fracture mechanics. It is shown that propagation of a sharp crack occurs at a critical stress intensity factor Kc ≈ 4.0 MPa m1/2, similar to the earlier findings of theoretical simulations. In contrast, a blunt crack lying along an irrational direction of the crystal is found to have a less localized stress field at the crack tip and propagate at a value of Kc > 9 MPa m1/2. It is also shown that once crack propagation takes place, the cracks rotate to become aligned approximately perpendicular to the tensile axis, regardless of the crystallographic orientation of the graphene. Based upon this, a simple strain engineering method is proposed to control the propagation of cracks. This has important implications for the use of graphene in practical applications and suggests a simple method to control the formation of cracks in graphene through the application of strain.

Finsler perturbation with nondense geodesics with irrational directions

We show that given any Liouville direction and flat Finsler torus, one can make a $C^{\infty}$-small perturbation on an arbitrarily small disc to get a nondense geodesic in the given direction.

Homogenization of the boundary value for the Dirichlet problem

In this paper, we give a mathematically rigorous proof of the averaging behavior of oscillatory surface integrals. Based on ergodic theory, we find a general geometric condition which we call irrational direction dense condition, abbreviated as IDDC, under which the averaging takes place. It should be stressed that IDDC does not imply any control on the curvature of the given surface. As an application, we prove homogenization for elliptic systems with Dirichlet boundary data, in \begin{document}$ C^1 $\end{document} -domains.

The stable norm on the 2-torus at irrational directions

We study properties of the stable norm on the first homology group of the 2-torus with respect to Riemannian or Finsler metrics, focusing on points with irrational slope. Our results show that the stable norm detects KAM-tori and hyperbolicity in the geodesic flow. Along the way, we shall prove new inequalities for the stable norm near rational directions. Moreover, we study the stable norm in some natural examples reflecting the new results in this paper.

On self-similar sets with overlaps and inverse theorems for entropy

We study the Hausdorff dimension of self-similar sets and measures on the line. We show that if the dimension is smaller than the minimum of 1 and the similarity dimension, then at small scales there are super-exponentially close cylinders. This is a step towards the folklore conjecture that such a drop in dimension is explained only by exact overlaps, and confirms the conjecture in cases where the contraction parameters are algebraic. It also gives an affirmative answer to a conjecture of Furstenberg, showing that the projections of the "1-dimensional Sierpinski gasket" in irrational directions are all of dimension 1. As another consequence, if a family of self-similar sets or measures is parametrized in a real-analytic manner, then, under an extremely mild non-degeneracy condition, the set of "exceptional" parameters has Hausdorff dimension 0. Thus, for example, there is at most a zero-dimensional set of parameters 1/2<r<1 such that the corresponding Bernoulli convolution has dimension <1, and similarly for Sinai's problem on iterated function systems that contract on average. A central ingredient of the proof is an inverse theorem for the growth of Shannon entropy of convolutions of probability measures. For the dyadic partition D_n of the line into intervals of length 1/2^n, we show that if H(nu*mu,D_n)/n < H(mu,D_n)/n + delta for small delta and large n, then, when restricted to random element of a partition D_i, 0<i<n, either mu is close to uniform or nu is close to atomic. This should be compared to results in additive combinatorics that give the global structure of measures satisfying H(nu*mu,D_n)/n < H(mu,D_n)/n + O(1/n).

Directional recurrence for infinite measure preserving actions

We define directional recurrence for infinite measure preserving $\mathbb{Z}^{d}$ actions both intrinsically and via the unit suspension flow and prove that the two definitions are equivalent. We study the structure of the set of recurrent directions and show it is always a $G_{{\it\delta}}$ set. We construct an example of a recurrent action with no recurrent directions, answering a question posed in a 2007 paper of Daniel J. Rudolph. We also show by example that it is possible for a recurrent action to not be recurrent in an irrational direction even if all its sub-actions are recurrent.

Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data

In this article we investigate averaging properties of fully nonlinear PDEs in bounded domains with oscillatory Neumann boundary data. The oscillation is periodic and is present both in the operator and in the Neumann data. Our main result states that, when the domain does not have flat boundary parts and when the homogenized operator is rotation invariant, the solutions uniformly converge to the homogenized solution solving a Neumann boundary problem. Furthermore we show that the homogenized Neumann data is continuous with respect to the normal direction of the boundary. Our result is the nonlinear version of the classical result in [3] for divergence-form operators with co-normal boundary data. The main ingredients in our analysis are the estimate on the oscillation on the solutions in half-spaces (Theorem 3.1), and the estimate on the mode of convergence of the solutions as the normal of the half-space varies over irrational directions (Theorem 4.1).

Wannier-Stark states and Bloch oscillations in the honeycomb lattice

We study a quantum particle in a tilted honeycomb lattice in the tight-binding approximation. First we discuss the particle eigenstates, i.e., the stationary Wannier-Stark states. These states are proved to be extended states for the rational directions of the static field and localized states for the irrational directions. We find energy bands of the extended states and analyze the localized states. It is shown, in particular, that the localized honeycomb Wannier-Stark states are chaotic states with irregular dependence of the localization length on the static field magnitude. Second we discuss Bloch oscillations of the quantum particle. Irregular Bloch oscillations for irrational directions are observed.

Plane-Like Minimizers and Differentiability of the Stable Norm

In this paper we investigate the strict convexity and the differentiability properties of the stable norm, which corresponds to the homogenized surface tension for a periodic perimeter homogenization problem (in a regular and uniformly elliptic case). We prove that it is always differentiable in totally irrational directions, while in other directions, it is differentiable if and only if the corresponding plane-like minimizers satisfying a strong Birkhoff property foliate the torus. We also discuss the issue of the uniqueness of the correctors for the corresponding homogenization problem.

Cyclotron-Bloch dynamics of a quantum particle in a two-dimensional lattice. II. Arbitrary electric field directions

We study the quantum dynamics of a charged particle in a two-dimensional lattice, subject to constant and homogeneous electric and magnetic fields. We find that different regimes characterize these motions, depending on a combination of conditions, corresponding to weak and strong electric field intensities, rational or irrational directions of the electric field with respect to the lattice, and small or large values of the magnetic (Peierls) phase.

Future from the Past

The author demonstrates that the bulk of futurological forecasts do not come true as well as predictions contained in scientific fiction. In his view the systematic mis-takes of such forecasts are due not to the shortcomings of predictions, but to the fact that the development of civilization is unfolding in irrational direction. It is connected with the substitution of “paradigm of development” by the “paradigm of entertainment” – the distinguishingly manifested process which has psycho-logical roots. The author states that underestimation of the changes in human psychology is still alive and well in futurological forecasts what diminishes their exactness.

Ergodic theorems for surfaces with minimal random weights

The purpose of this paper is to formulate a higher dimensional analogue of the asymptotic shape theorem of first-passage percolation. The moment condition required depends on the curvature of the boundary: p > 3 2 for the disk and p > 2 for the square. In first-passage percolation, there is a.s. convergence to the time constant even in an irrational direction if the passage times are integrable random variables.

Projecting the one-dimensional Sierpinski gasket

LetS⊂ℝ2 be the Cantor set consisting of points (x,y) which have an expansion in negative powers of 3 using digits {(0,0), (1,0), (0,1)}. We show that the projection ofS in any irrational direction has Lebesgue measure 0. The projection in a rational directionp/q has Hausdorff dimension less than 1 unlessp+q ≡ 0 mod 3, in which case the projection has nonempty interior and measure 1/q. We compute bounds on the dimension of the projection for certain sequences of rational directions, and exhibit a residual set of directions for which the projection has dimension 1.

Expansive Subdynamics

This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let α \alpha be a continuous action of Z d \mathbb {Z}^d on an infinite compact metric space. For each subspace V V of R d \mathbb {R}^d we introduce a notion of expansiveness for α \alpha along V V , and show that there are nonexpansive subspaces in every dimension ≤ d − 1 \le d-1 . For each k ≤ d k\le d the set E k ( α ) \mathbb {E}_k(\alpha ) of expansive k k -dimensional subspaces is open in the Grassmann manifold of all k k -dimensional subspaces of R d \mathbb {R}^d . Various dynamical properties of α \alpha are constant, or vary nicely, within a connected component of E k ( α ) \mathbb {E}_k(\alpha ) , but change abruptly when passing from one expansive component to another. We give several examples of this sort of “phase transition,” including the topological and measure-theoretic directional entropies studied by Milnor, zeta functions, and dimension groups. For d = 2 d=2 we show that, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an E 1 ( α ) \mathbb {E}_1(\alpha ) . The unresolved case is that of the complement of a single irrational direction. Algebraic examples using commuting automorphisms of compact abelian groups are an important source of phenomena, and we study several instances in detail. We conclude with a set of problems and research directions suggested by our analysis.

A study of atomic V-steps formed by cleavage on NaCl crystal surfaces

Abstract Among atomic steps formed by cleavage on the surface of ionic crystals, V-patterns, frequently linked to each other forming zig-zag patterns, are especially typical. In the present work, several features of these patterns (i.e. step orientation, step height and sign, revealed by their splitting due to slow evaporation, formation of multiple V-steps) are studied on decorated replicas of NaCl crystals, and their relation with the dynamics of screw dislocations during crack propagation is discussed. We show that, though both V-sides frequently have irrational directions, one of them sometimes changes abruptly to [100] and [010] directions. This is interpreted in terms of glide and cross-glide motions of dislocations. Therefore a mechanism of zig-zag pattern formation is proposed according to which small dislocation loops, created by the crack, would move short distances, tightly wrapped around the crack tip and would disappear quickly due to an effect of interaction with cleavage steps. In agreement with previous results, these dislocations could not be observed after cleavage and the energy absorbed by their motion would be small with respect to that absorbed by larger loops, which are revealed after cleavage by X-ray topography and chemical etching methods.