Related papers: Drawing butterflies from the almost Mathieu operat…

In the following we are interested in the spectral gaps of discrete quasiperiodic Schr\"odinger operators when the frequency is Diophantine, the potential is analytic, and in the subcritical regime. The gap-labelling theorem asserts in this…

It is known that the spectral type of the almost Mathieu operator depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency. We study the competition between those factors and…

A bipartite graph extensively models relationships between real-world entities of two different types, such as user-product data in e-commerce. Such graph data are inherently becoming more and more streaming, entailing continuous insertions…

With the idea of an eventual classification of 3-bridge links,\ we define a very nice class of 3-balls (called butterflies) with faces identified by pairs, such that the identification space is $S^{3},$ and the image of a prefered set of…

We prove almost Lipshitz continuity of spectra of singular quasiperiodic Jacobi matrices and obtain a representation of the critical almost Mathieu family that has a singularity. This allows us to prove that the Hausdorff dimension of its…

Bipartite graphs serve as a natural model for representing relationships between two different types of entities. When analyzing bipartite graphs, butterfly counting is a fundamental research problem that aims to count the number of…

We report on a study of topological properties of Fibonacci quasicrystals. Chern numbers which label the dense set of spectral gaps, are shown to be related to the underlying palindromic symmetry. Topological and spectral features are…

We introduce self-similar versions of the one-dimensional almost Mathieu operators. Our definition is based on a class of self-similar Laplacians instead of the standard discrete Laplacian, and includes the classical almost Mathieu…

For non-critical almost Mathieu operators with Diophantine frequency, we establish exponential asymptotics on the size of spectral gaps, and show that the spectrum is homogeneous. We also prove the homogeneity of the spectrum for…

This paper is concerned with the fast computation of Fourier integral operators of the general form $\int_{\R^d} e^{2\pi\i \Phi(x,k)} f(k) d k$, where $k$ is a frequency variable, $\Phi(x,k)$ is a phase function obeying a standard…

We present families of large undirected and directed Cayley graphs whose construction is related to butterfly networks. One approach yields, for every large $k$ and for values of $d$ taken from a large interval, the largest known Cayley…

In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, \[ (H_{b,\phi} x)_n= x_{n+1} +x_{n-1} + b \cos(2 \pi n \omega + \phi)x_n \] on $l^2(\mathbb{Z})$ and its associated eigenvalue equation…

Recent advances in realizing artificial gauge fields on optical lattices promise experimental detection of topologically non-trivial energy spectra. Self-similar fractal energy structures generally known as Hofstadter butterflies depend…

The quantitative information on the spectral gaps for the linearized Boltzmann operator is of primary importance on justifying the Boltzmann model and study of relaxation to equilibrium. This work, for the first time, provides numerical…

Some aspects of analysis on disconnected open subsets of the plane with connected fractal boundary are discussed.

For three classes of elliptic pseudodifferential operators on a compact manifold with boundary which have `geometric K-theory', namely the `transmission algebra' introduced by Boutet de Monvel, the `zero algebra' introduced by Mazzeo and…

Spectral and numerical properties of classes of random orthogonal butterfly matrices, as introduced by Parker (1995), are discussed, including the uniformity of eigenvalue distributions. These matrices are important because the…

We present a general and useful method to predict the existence, frequency, and spatial properties of gap states in photonic (and other) structures with a gapped spectrum. This method is established using the scattering approach. It offers…

Large-scale coarse-grained molecular dynamics simulations of inhomogeneous gel networks were performed to investigate abnormal butterfly patterns in two-dimensional scattering. The networks were diamond lattice-based with distributions in…

Kicked Harper operators and on-resonance double kicked rotor operators model quantum systems whose semiclassical limits exhibit chaotic dynamics. Recent computational studies indicate a striking resemblance between the spectrums of these…