This paper examines the asymptotic behavior of solutions of the three dimensional viscoelastic problem with long-term memory and Tresca friction law in a thin domain ${\Sigma }^{\epsilon }$. We study the asymptotic behavior of this problem when the thickness $\epsilon$ tends to zero and we prove a convergence theorem for the displacement and velocity in appropriate functional spaces. Besides, the limit problem with the limit of Tresca free boundary conditions and a specific Reynolds limit equation is obtained.

Let $f$ and $f^{\prime }$ be non-constant, complex, polynomial function-germs at the origin $\mathbf{0}\in {\mathrm{ℂ}}^n$ such that $f\left(\mathbf{0}\right)=f^{\prime }\left(\mathbf{0}\right)=0$. We suppose that $f$ and $f^{\prime }$ have an isolated singularity at $\mathbf{0}$ and the same Milnor number (respectively, the same Teissier ${\mu }^{*}$-sequence). We show that if $f$ and $f^{\prime }$ can be connected by a continuous path in the $\mu$-constant (respectively, ${\mu }^{*}$-constant) stratum, then they can also be connected by a piecewise complex-analytic path (defined in the relevant natural way) in this stratum.

We give a detailed account of Agol’s theorem and his proof concerning two-meridional-generator subgroups of hyperbolic $2$-bridge link groups, which is included in the slide of his talk at the Bolyai conference 2001. We also give a generalization of the theorem to two-parabolic-generator subgroups of hyperbolic $3$-manifold groups, which gives a refinement of a result due to Boileau-Weidmann.

In this paper, we consider the problem of exponential instability behavior of random dynamical systems described by cocycles in Banach spaces. We prove some continuous and discrete versions of Datko type theorem for the exponential instability in average of cocycles. In addition, two characterizations of the exponential instability in average in terms of Lyapunov functions are given.

We investigate a mean curvature flow obtained via elliptic regularization, and prove that it is not only a Brakke flow, but additionally a generalized BV flow proposed by Stuvard and Tonegawa. In particular, we show that the change in volume of the evolving phase can be expressed in terms of the generalized mean curvature of the Brakke flow.