Narrow Results

UHF flows are the flows obtained as inductive limits of flows on full matrix algebras. We will revisit universal UHF flows and give an explicit construction of such flows on a UHF algebra M_k^∞ for any k and also present a characterization of such flows. Those flows are UHF flows whose cocycle perturbations are almost conjugate to themselves.

We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to the case of biquandles and consider enhancement of the quasi-trivial biquandle cocycle counting invariant by quasi-trivial biquandle cocycles, obtaining invariants of link-homotopy type of links analogous to the quasi-trivial quandle cocycle invariants in Inoue’s paper [A. Inoue, Quasi-triviality of quandles for link-homotopy, J. Knot Theory Ramifications22(6) (2013) 1350026, doi:10.1142/S0218216513500260, MR3070837].

We study the first and second cohomology groups of the ${}^{\ast }$-algebras of the universal unitary and orthogonal quantum groups $U_F^{+}$ and $O_F^{+}$. This provides valuable information for constructing and classifying Lévy processes on these quantum groups, as pointed out by Schürmann. In the case when all eigenvalues of $F^{\ast }F$ are distinct, we show that these ${}^{\ast }$-algebras have the properties (GC), (NC) and (LK) introduced by Schürmann and studied recently by Franz, Gerhold and Thom. In the degenerate case $F=I_d$, we show that they do not have any of these properties. We also compute the second cohomology group of $U_d^{+}$ with trivial coefficients — $H^2(U_d^{+},{}_{𝜖}{ℂ}_{𝜖})\cong {ℂ}^{d^2-1}$ — and construct an explicit basis for the corresponding second cohomology group for $O_d^{+}$ (whose dimension was known earlier, thanks to the work of Collins, Härtel and Thom).

This paper is a survey of recent results on the dynamics of Stochastic Burgers equation (SBE) and two-dimensional Stochastic Navier–Stokes Equations (SNSE) driven by affine linear noise. Both classes of stochastic partial differential equations are commonly used in modeling fluid dynamics phenomena. For both the SBE and the SNSE, we establish the local stable manifold theorem for hyperbolic stationary solutions, the local invariant manifold theorem and the global invariant flag theorem for ergodic stationary solutions. The analysis is based on infinite-dimensional multiplicative ergodic theory techniques developed by D. Ruelle [22] (cf. [20, 21]). The results in this paper are based on joint work of the author with T. S. Zhang and H. Zhao ([17–19]).

Several variants of the classic Fibonacci inflation tiling are considered in an illustrative fashion, in one and in two dimensions, with an eye on changes or robustness of diffraction and dynamical spectra. In one dimension, we consider extension mechanisms of deterministic and of stochastic nature, while we look at direct product variations in a planar extension. For the pure point part, we systematically employ a cocycle approach that is based on the underlying renormalization structure. It allows explicit calculations, particularly in cases where one meets regular model sets with Rauzy fractals as windows.

We provide a rather general perfection result for crude local semi-flows taking values in a Polish space showing that a crude semi-flow has a modification which is a (perfect) local semi-flow which is invariant under a suitable metric dynamical system. Such a (local) semi-flow induces a (local) random dynamical system (RDS). Then we show that this result can be applied to several classes of stochastic differential equations driven by semimartingales with stationary increments such as equations with locally monotone coefficients and equations with singular drift. For these examples it was previously unknown whether they generate a (local) RDS or not.

In this paper we present the property of uniform trichotomy for skew-product semiflows in Banach spaces. We give several characterizations, the obtained theorems and propositions being generalizations of some well-known results on asymptotic behaviors of linear differential equations. There are also presented several examples of semiflows, cocycles and linear skew-product semiflows in Banach spaces.