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.     

Limit theorems for random walks

We consider a random walk $S_{\tau }$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner’s subordination. We prove that under certain conditions on the subordinator $\tau$ appropriately scaled random walk $S_{\tau }$ converges in the Skorohod space to the symmetric $\alpha$-stable process $B^{\alpha }$. We also prove asymptotic formula for the transition function of $S_{\tau }$ similar to the Pólya’s asymptotic formula for $B^{\alpha }$.     

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.     

Eigenvalues of rank one perturbations of unstructured matrices

Let A be a fixed complex matrix and let $u\text{,}v$ be two vectors. The eigenvalues of matrices $A+\tau {\mathit{uv}}^{?}$ $(\tau \in \mathbb{R})$ form a system of intersecting curves. The dependence of the intersections on the vectors $u\text{,}v$ is studied.     

Multivariate polynomial interpolation and sampling in Paley–Wiener spaces

In this paper, an equivalence between existence of particular exponential Riesz bases for spaces of multivariate bandlimited functions and existence of certain polynomial interpolants for functions in these spaces is given. Namely, polynomials are constructed which, in the limiting case, interpolate ${\left\{({\tau }_n,f\left({\tau }_n\right))\right\}}_n$ for certain classes of unequally spaced data nodes ${\left\{{\tau }_n\right\}}_n$ and corresponding ${?}_2$ sampled data ${\left\{f\left({\tau }_n\right)\right\}}_n$. Existence of these polynomials allows one to construct a simple sequence of approximants for an arbitrary multivariate bandlimited function $f$ which demonstrates $L_2$ and uniform convergence on ${\mathbb{R}}^d$ to $f$. A simpler computational version of this recovery formula is also given at the cost of replacing $L_2$ and uniform convergence on ${\mathbb{R}}^d$ with $L_2$ and uniform convergence on increasingly large subsets of ${\mathbb{R}}^d$. As a special case, the polynomial interpolants of given ${?}_2$ data converge in the same fashion to the multivariate bandlimited interpolant of that same data. Concrete examples of pertinent Riesz bases and unequally spaced data nodes are also given.     

On the lower central series of the IA-automorphism group of a free group

In this paper, we study the Johnson homomorphisms ${\tau }_k^{\prime }$ of the automorphism group of a free group of rank $n$, which are defined on the graded quotients of the lower central series of the IA-automorphism group. In particular, we determine the cokernel of ${\tau }_k^{\prime }$ for any $k\ge 2$ and $n\ge k+2$.     

Bifurcation of peakons and periodic cusp waves for the generalization of the Camassa–Holm equation

In this paper, we investigate the generalization of the Camassa–Holm equation $u_t+K{\left(u^m\right)}_x?{\left(u^n\right)}_{xxt}={[\frac{{\left({\left(u^n\right)}_x\right)}^2}{2}+u^n{\left(u^n\right)}_{xx}]}_x\text{,}$ where $K$ is a positive constant and $m,n\in \mathbb{N}$. The bifurcation and some explicit expressions of peakons and periodic cusp wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. Further, in the process of obtaining the bifurcation of phase portraits, we show that $K=\frac{m+n}{1+n}{c}^{\frac{n?m+1}{n}}$ is the peakon bifurcation parameter value for the equation. From the bifurcation theory, in general, the peakons can be obtained by taking the limit of the corresponding periodic cusp waves. However, we find that in the cases of $n\ge 2,m=n+1$, when $K$ tends to the corresponding bifurcation parameter value, the periodic cusp waves will no longer converge to the peakons, instead, they will still be the periodic cusp waves. To the best of our knowledge, up until now, this phenomenon has not appeared in any other literature. By further studying the cause of this phenomenon, we show that this planar system has some different characters from the previous Camassa–Holm systems. What is more, we obtain some periodic cusp wave solutions in the form of polynomial functions, which are different from those in the form of exponential functions. Some previous results are extended.