Synchronization versus stability of the invariant distribution for a class of globally coupled maps

We study a class of globally coupled maps in the continuum limit, where the individual maps are expanding maps of the circle. The circle maps in question are such that the uncoupled system admits a unique absolutely continuous invariant measure (acim), which is furthermore mixing. Interaction arises in the form of diffusive coupling, which involves a function that is discontinuous on the circle. We show that for sufficiently small coupling strength the coupled map system admits a unique absolutely continuous invariant distribution, which depends on the coupling strength $\varepsilon$. Furthermore, the invariant density exponentially attracts all initial distributions considered in our framework. We also show that the dependence of the invariant density on the coupling strength $\varepsilon$ is Lipschitz continuous in the BV norm. When the coupling is sufficiently strong, the limit behavior of the system is more complex. We prove that a wide class of initial measures approach a point mass with support moving chaotically on the circle. This can be interpreted as synchronization in a chaotic state

A note on a maximal Bernstein inequality

We show somewhat unexpectedly that whenever a general Bernstein-type maximal inequality holds for partial sums of a sequence of random variables, a maximal form of the inequality is also valid.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ304 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

The limit distribution of ratios of jumps and sums of jumps of subordinators

Let $V_{t}$ be a driftless subordinator, and let denote $m_{t}^{(1)} \geq m_{t}^{(2)} \geq\ldots$ its jump sequence on interval $[0,t]$. Put $V_{t}^{(k)} = V_{t} - m_{t}^{(1)} - \ldots- m_{t}^{(k)}$ for the $k$-trimmed subordinator. In this note we characterize under what conditions the limiting distribution of the ratios $V_{t}^{(k)} / m_{t}^{(k+1)}$ and $m_{t}^{(k+1)} / m_{t}^{(k)}$ exist, as $t \downarrow0$ or $t \to\infty$.Comment: 14 page

Couplings and Strong Approximations to Time Dependent Empirical Processes Based on I.I.D. Fractional Brownian Motions

We define a time dependent empirical process based on $n$ i.i.d.~fractional Brownian motions and establish Gaussian couplings and strong approximations to it by Gaussian processes. They lead to functional laws of the iterated logarithm for this process.Comment: To appear in the Journal of Theoretical Probability. 37 pages. Corrected version. The results on quantile processes are taken out and it will appear elsewher

Oscillating spin-orbit interaction in two-dimensional superlattices: sharp transmission resonances and time-dependent spin polarized currents

We consider ballistic transport through a lateral, two-dimensional superlattice with experimentally realizable, sinusoidally oscillating Rashba-type spin-orbit interaction. The periodic structure of the rectangular lattice produces a spin-dependent miniband structure for static SOI. Using Floquet theory, transmission peaks are shown to appear in the mini-bandgaps as a consequence of the additional, time-dependent SOI. A detailed analysis shows that this effect is due to the generation of harmonics of the driving frequency, via which e.g., resonances that cannot be excited in the case of static SOI become available. Additionally, the transmitted current shows space and time-dependent partial spin-polarization, in other words, polarization waves propagate through the superlattice.Comment: 8 pages, 6 figure

Two-colorings with many monochromatic cliques in both colors

Color the edges of the n-vertex complete graph in red and blue, and suppose that red k-cliques are fewer than blue k-cliques. We show that the number of red k-cliques is always less than cknk, where ck∈(0, 1) is the unique root of the equation zk=(1-z)k+kz(1-z)k-1. On the other hand, we construct a coloring in which there are at least cknk-O(nk-1) red k-cliques and at least the same number of blue k-cliques. © 2013 Elsevier Inc

Local unitary invariants for multipartite quantum systems

A method is presented to obtain local unitary invariants for multipartite quantum systems consisting of fermions or distinguishable particles. The invariants are organized into infinite families, in particular, the generalization to higher dimensional single particle Hilbert spaces is straightforward. Many well-known invariants and their generalizations are also included.Comment: 13 page

The twistor geometry of three-qubit entanglement

A geometrical description of three qubit entanglement is given. A part of the transformations corresponding to stochastic local operations and classical communication on the qubits is regarded as a gauge degree of freedom. Entangled states can be represented by the points of the Klein quadric ${\cal Q}$ a space known from twistor theory. It is shown that three-qubit invariants are vanishing on special subspaces of ${\cal Q}$. An invariant vanishing for the $GHZ$ class is proposed. A geometric interpretation of the canonical decomposition and the inequality for distributed entanglement is also given.Comment: 4 pages RevTeX

Explaining the elongated shape of 'Oumuamua by the Eikonal abrasion model

The photometry of the minor body with extrasolar origin (1I/2017 U1) 'Oumuamua revealed an unprecedented shape: Meech et al. (2017) reported a shape elongation b/a close to 1/10, which calls for theoretical explanation. Here we show that the abrasion of a primordial asteroid by a huge number of tiny particles ultimately leads to such elongated shape. The model (called the Eikonal equation) predicting this outcome was already suggested in Domokos et al. (2009) to play an important role in the evolution of asteroid shapes.Comment: Accepted by the Research Notes of the AA