A pair of bodies rolling on each other is an interesting example of nonholonomic systems in control theory. There is a geometric condition equivalent to the rolling constraint which enables us to generalize the rolling motions for any two-dimensional Riemannian manifolds. This system has a five-dimensional phase space. In order to study the controllability of the rolling surfaces, we lift the system to a six-dimensional space and show that the lifted system is controllable unless the two surfaces have isometric universal covering spaces. In the non-controllable case there are some three-dimensional orbits each of which corresponds to an isometry of the universal covering spaces.

This paper concerns closed hypersurfaces of dimension $n{\geq}2$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature evolving in direction of its normal vector, where the speed equals a power ${\beta}{\geq}1$ of the mean curvature. The main result is that if the initial closed, weakly h-convex hypersurface satisfies that the ratio of the biggest and smallest principal curvature at everywhere is close enough to 1, depending only on n and ${\beta}$, then under the flow this is maintained, there exists a unique, smooth solution of the flow which converges to a single point in ${\mathbb{H}}_{\kappa}^{n+1}$ in a maximal finite time, and when rescaling appropriately, the evolving hypersurfaces exponential convergence to a unit geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$.

In this paper, we first prove some Liouville type theorems for elliptic inequalities on weighted manifolds which support a weighted Sobolev-type inequality. Secondly, applying the Liouville type theorems to self-shrinkers, we obtain some global rigidity theorems.

The solutions of equations are usually found using iterative methods whose convergence order is determined by Taylor expansions. In particular, the local convergence of the method we study in this paper is shown under hypotheses reaching the third derivative of the operator involved. These hypotheses limit the applicability of the method. In our study we show convergence of the method using only the first derivative. This way we expand the applicability of the method. Numerical examples show the applicability of our results in cases earlier results cannot.

The main purpose of this paper is to propose a scheme of a proof of the nonsimpleness of the group $Homeo^{\Omega}$ ($D^2$, ${\partial}D^2$) of area preserving homeomorphisms of the 2-disc $D^2$. We first establish the existence of Alexander isotopy in the category of Hamiltonian homeomorphisms. This reduces the question of extendability of the well-known Calabi homomorphism Cal : $Diff^{\Omega}$ ($D^1$, ${\partial}D^2$)${\rightarrow}{\mathbb{R}}$ to a homomorphism ${\bar{Cal}}$ : Hameo($D^2$, ${\partial}D^2$)${\rightarrow}{\mathbb{R}}$ to that of the vanishing of the basic phase function $f_{\underline{F}}$, a Floer theoretic graph selector constructed in [9], that is associated to the graph of the topological Hamiltonian loop and its normalized Hamiltonian ${\underline{F}}$ on $S^2$ that is obtained via the natural embedding $D^2{\hookrightarrow}S^2$. Here Hameo($D^2$, ${\partial}D^2$) is the group of Hamiltonian homeomorphisms introduced by $M{\ddot{u}}ller$ and the author [18]. We then provide an evidence of this vanishing conjecture by proving the conjecture for the special class of weakly graphical topological Hamiltonian loops on $D^2$ via a study of the associated Hamiton-Jacobi equation.

In this paper, we study entire solutions of some nonlinear difference equations and transcendental meromorphic solutons of some nonlinear differential equations. Our results generalize the results due to [11], [17].

Let M be an invariant subspace of $H^2$ over the bidisk. Associated with M, we have the fringe operator $F^M_z$ on $M{\ominus}{\omega}M$. It is studied the Fredholmness of $F^M_z$ for (generalized) zero based invariant subspaces M. Also ker $F^M_z$ and ker $(F^M_z)^*$ are described.

We construct the extension of a hyperelliptic K3 surface to a Fano 6-fold with extraordinary properties in moduli. This leads us to a family of surfaces of general type with $p_g=1$, q = 0, $K^2=2$ and hyperelliptic canonical curve, each of which is a weighted complete inter-section inside a Fano 6-fold. Finally, we use these hyperelliptic surfaces to determine an 8-parameter family of Godeaux surfaces with ${\pi}_1={\mathbb{Z}}/2$.

Firstly, we establish spectral mapping theorems for normal eigenvalues (due to Browder) of a $C_0$-semigroup and its generator. Secondly, we discuss relationships between normal eigenvalues of the compact monodromy operator and the generator of the evolution semigroup on $P_{\tau}(X)$ associated with the ${\tau}$-periodic evolutionary process on a Banach space X, where $P_{\tau}(X)$ stands for the space of all ${\tau}$-periodic continuous functions mapping ${\mathbb{R}}$ to X.

Chen and Ishikawa studied the surfaces of revolution of the polynomial and the rational kind of finite type in Euclidean 3-space $E^3$ [10]. Here, the pedal of revolution surfaces of the polynomial and the rational kind are discussed. Also, as a special case of general revolution surfaces, the sphere and catenoid are studied for the kind of finite type.

Let p(t, x) be the fundamental solution to the problem $${\partial}^{\alpha}_tu=-(-{\Delta})^{\beta}u,\;{\alpha}{\in}(0,2),\;{\beta}{\in}(0,{\infty})$$. If ${\alpha},{\beta}{\in}(0,1)$, then the kernel p(t, x) becomes the transition density of a Levy process delayed by an inverse subordinator. In this paper we provide the asymptotic behaviors and sharp upper bounds of p(t, x) and its space and time fractional derivatives $$D^n_x(-{\Delta}_x)^{\gamma}D^{\sigma}_tI^{\delta}_tp(t,x),\;{\forall}n{\in}{\mathbb{Z}}_+,\;{\gamma}{\in}[0,{\beta}],\;{\sigma},{\delta}{\in}[0,{\infty})$$, where $D^n_x$ x is a partial derivative of order n with respect to x, $(-{\Delta}_x)^{\gamma}$ is a fractional Laplace operator and $D^{\sigma}_t$ and $I^{\delta}_t$ are Riemann-Liouville fractional derivative and integral respectively.