Bifurcation theory for finitely smooth planar autonomous differential systems

In this paper we establish bifurcation theory of limit cycles for planar $C^k$ smooth autonomous differential systems, with $k\in \mathbb{N}$. The key point is to study the smoothness of bifurcation functions which are basic and important tool on the study of Hopf bifurcation at a fine focus or a center, and of Poincaré bifurcation in a period annulus. We especially study the smoothness of the first order Melnikov function in degenerate Hopf bifurcation at an elementary center. As we know, the smoothness problem was solved for analytic and $C^{\infty }$ differential systems, but it was not tackled for finitely smooth differential systems. Here, we present their optimal regularity of these bifurcation functions and their asymptotic expressions in the finite smooth case.     

Normal linearization and transition map near a saddle connection with symmetric resonances

We consider a heteroclinic connection in a planar system, between two symmetric hyperbolic saddles of which the eigenvalues are resonant. Starting with a $C^{\infty }$ normal form, defined globally near the connection, we normally linearize the vector field by using finitely smooth tags of logarithmic form. We furthermore define an asymptotic entry–exit relation, and we discuss on two examples how to deal with counting limit cycles near a limit periodic set involving such a connection.     

Regularity of random attractors for fractional stochastic reaction–diffusion equations on Rn

We investigate the regularity of random attractors for the non-autonomous non-local fractional stochastic reaction–diffusion equations in $H^s({\mathbb{R}}^n)$ with $s\in (0,1)$. We prove the existence and uniqueness of the tempered random attractor that is compact in $H^s({\mathbb{R}}^n)$ and attracts all tempered random subsets of $L^2({\mathbb{R}}^n)$ with respect to the norm of $H^s({\mathbb{R}}^n)$. The main difficulty is to show the pullback asymptotic compactness of solutions in $H^s({\mathbb{R}}^n)$ due to the noncompactness of Sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the Wiener process. We establish such compactness by the ideas of uniform tail-estimates and the spectral decomposition of solutions in bounded domains.     

Uniformly recurrent subgroups and simple C⁎-algebras

We study uniformly recurrent subgroups (URS) introduced by Glasner and Weiss [18]. Answering their query we show that any URS Z of a finitely generated group is the stability system of a minimal Z-proper action. We also show that for any sofic URS Z there is a Z-proper action admitting an invariant measure. We prove that for a URS Z all Z-proper actions admits an invariant measure if and only if Z is coamenable. In the second part of the paper we study the separable $C^{?}$-algebras associated to URS's. We prove that if a URS is generic then its $C^{?}$-algebra is simple. We give various examples of generic URS's with exact and nuclear $C^{?}$-algebras and an example of a URS Z for which the associated simple $C^{?}$-algebra is not exact and not even locally reflexive, in particular, it admits both a uniformly amenable trace and a nonuniformly amenable trace.     

Jumps of computably enumerable equivalence relations

We study computably enumerable equivalence relations (or, ceers), under computable reducibility ≤, and the halting jump operation on ceers. We show that every jump is uniform join-irreducible, and thus join-irreducible. Therefore, the uniform join of two incomparable ceers is not equivalent to any jump. On the other hand there exist ceers that are not equivalent to jumps, but are uniform join-irreducible: in fact above any non-universal ceer there is a ceer which is not equivalent to a jump, and is uniform join-irreducible. We also study transfinite iterations of the jump operation. If a is an ordinal notation, and E is a ceer, then let $E^{(a)}$ denote the ceer obtained by transfinitely iterating the jump on E along the path of ordinal notations up to a. In contrast with what happens for the Turing jump and Turing reducibility, where if a set X is an upper bound for the A-arithmetical sets then $X^{(2)}$ computes $A^{(\omega )}$, we show that there is a ceer R such that $R\ge {\mathrm{Id}}^{(n)}$, for every finite ordinal n, but, for all k, $R^{(k)}?{\mathrm{Id}}^{(\omega )}$ (here Id is the identity equivalence relation). We show that if $a,b$ are notations of the same ordinal less than ${\omega }^2$, then $E^{(a)}\equiv E^{(b)}$, but there are notations $a,b$ of ${\omega }^2$ such that ${\mathrm{Id}}^{(a)}$ and ${\mathrm{Id}}^{(b)}$ are incomparable. Moreover, there is no non-universal ceer which is an upper bound for all the ceers of the form ${\mathrm{Id}}^{(a)}$ where a is a notation for ${\omega }^2$.     

Cocycle deformations and Galois objects for semisimple Hopf algebras of dimension p3 and pq2

Let p and q be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension $p^3$ and of dimension $p{q}^2$. We obtain that the $p+1$ non-isomorphic self-dual semisimple Hopf algebras of dimension $p^3$ classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac–Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension $p^3$, established by the third-named author in an appendix.     

Non-differentiability and Hölder properties of self-affine functions

We consider the class of self-affine functions. Firstly, we characterize all nowhere differentiable self-affine continuous functions. Secondly, given a self-affine continuous function $?$, we investigate its Hölder properties. We find its best uniform Hölder exponent and when $?$ is $C^1$, we find the best uniform Hölder exponent of ${?}^{\prime }$. Thirdly, we show that the Hölder cut of $?$ takes the same value almost everywhere for the Lebesgue measure. This last result is a consequence of the Borel strong law of large numbers.     

On the constants for some fractional Gagliardo–Nirenberg and Sobolev inequalities

We consider the inequalities of Gagliardo–Nirenberg and Sobolev in ${\mathbb{R}}^d$, formulated in terms of the Laplacian $\Delta$ and of the fractional powers $D^n?{\sqrt{?\Delta }}^n$ with real $n?0$; we review known facts and present novel, complementary results in this area. After illustrating the equivalence between these two inequalities and the relations between the corresponding sharp constants and maximizers, we focus the attention on the ${?}^2$ case where, for all sufficiently regular $f:{\mathbb{R}}^d\to \mathbb{C}$, the norm $6D^{j}f{6}_{{?}^r}$ is bounded in terms of $6f{6}_{{?}^2}$ and $6D^{n}f{6}_{{?}^2}$, for $1∕r=1∕2?(?n?j)∕d$, and suitable values of $j,n,?$ (with $j,n$ possibly noninteger). In the special cases $?=1$ and $?=j∕n+d∕2n$ (i.e., $r=+\infty$), related to previous results of Lieb and Ilyin, the sharp constants and the maximizers can be found explicitly; we point out that the maximizers can be expressed in terms of hypergeometric, Fox and Meijer functions. For the general ${?}^2$ case, we present two kinds of upper bounds on the sharp constants: the first kind is suggested by the literature, the second one is an alternative proposal of ours, often more precise than the first one. We also derive two kinds of lower bounds. Combining all the available upper and lower bounds, the sharp constants are confined to quite narrow intervals. Several examples are given, including the numerical values of the previously mentioned bounds.