Vortex filament equation in the Euclidean space has a long time solution for any closed initial data, because it can be converted into a standard nonlinear Schrödinger equation. While the Riemannian version of vortex filament equation is not integrable at all, we prove that it has a long time solution for any closed initial data.

In this paper we continue systematic study of the dimension estimate of the global attractor for the chemotaxis-growth system. Using nonnegativity of solu-tions we manage significantly to improve dimension estimates with respect to the chemotactic parameter.

We study the action of the mapping class group $\mathcal{M}(F)$ on the complex of curves of a non-orientable surface $F$. Following the outline of [1] we obtain, using the result of [4], a presentation for $\mathcal{M}(F)$ defined in terms of the mapping class groups of the complementary surfaces of collections of curves, provided that $F$ is not sporadic, i.e. the complex of curves of $F$ is simply connected. We also compute a finite presentation for the mapping class group of each sporadic surface.

We study traveling waves for a two-dimensional lattice dynamical system with monostable nonlinearity. We prove that there is a minimal speed such that a traveling wave exists if and only if its speed is above this minimal speed. Then we show the uniqueness (up to translations) of wave profile for each given speed. Moreover, any wave profile is strictly monotone.

For $\mathbf{r}=(r_{1},\ldots,r_{d}) \in\mathbb{R}^{d}$ the map $\tau_{\mathbf{r}}\colon \mathbb{Z}^{d} \to \mathbb{Z}^{d}$ given by \[ \tau_{\mathbf{r}}(a_{1},\ldots,a_{d}) =(a_{2},\ldots,a_{d},-\lfloor r_{1}a_{1}+ \cdots + r_{d}a_{d} \rfloor) \] is called a shift radix system if for each $\mathbf{a} \in \mathbb{Z}^{d}$ there exists an integer $k>0$ with $\tau_{\mathbf{r}}^{k}(\mathbf{a})=0$. As shown in the first two parts of this series of papers shift radix systems are intimately related to certain well-known notions of number systems like $\beta$-expansions and canonical number systems. In the present paper further structural relationships between shift radix systems and canonical number systems are investigated. Among other results we show that canonical number systems related to polynomials \[ \sum_{i=0}^{d} p_{i} X^{i} \in \mathbb{Z}[X] \] of degree $d$ with a large but fixed constant term $p_{0}$ approximate the set of ($d-1$)-dimensional shift radix systems. The proofs make extensive use of the following tools: Firstly, vectors $\mathbf{r} \in\mathbb{R}^{d}$ which define shift radix systems are strongly connected to monic real polynomials all of whose roots lie inside the unit circle. Secondly, geometric considerations which were established in Part I of this series of papers are exploited. The main results establish two conjectures mentioned in Part II of this series of papers.

We construct a family of cyclic extensions of number fields, in which every finite place is unramified, from an elliptic curve with a rational torsion point. As an application, we obtain such polynomials $F(X)$ of rational coefficients that have the following property: For a rational number $\xi$ chosen at random, the class number of the field generated by the square root of $F(\xi)$ is ``often'' divisible by 3, 5 or by 7.

We investigate crosscap numbers by essential tangle decompositions. We show that each of the crosscap numbers of the Kinoshita-Terasaka knot and the Conway knot is four.

For certain $(1,1)$-knots in lens spaces with a longitudinal surgery yielding the 3-sphere, we determine a non-negative integer derived from its $(1,1)$-splitting. The value will be an invariant for such knots. Roughly, it corresponds to a `minimal' self-intersection number when one consider projections of a knot on a Heegaard torus. As an application, we give a necessary and sufficient condition for such knots to be hyperbolic.

By using excursions from the maximum, we get asymptotic information on the hitting law of a fluctuating Brownian functional. This extends a result of Isozaki and Kotani who considered the case when the underlying Lévy process is stable.

The $*$-spread of an ideal is defined as the minimal number of generators of an ideal which is minimal with respect to having the same tight closure as the original ideal. We prove an asymptotic length formula for the $*$-spread.

In a recent paper ([2]), Nicolas Bouleau provides a new tool, based on the language of Dirichlet forms, to study the propagation of errors and reinforce the historical approach of Gauss. In the same way that the practical use of the normal distribution in statistics may be explained by the central limit theorem, the aim of this paper is to underline the importance of a family of error structures by asymptotic arguments.

We prove that for $2$-bridge knots, the diameter, $D$, of the set of boundary slopes is twice the crossing number, $c$. This constitutes partial verification of a conjecture that, for all knots in $S^{3}$, $D \leq 2 c$. In addition, we characterize the $2$-bridge knots with four or fewer boundary slopes and show that they each have a boundary slope of genus two or less.

The aim of this paper is to derive $L^{p}$-$L^{q}$ estimates for strictly hyperbolic equations with time-dependent coefficients which are of Lipschitz class. Further-more, $L^{p}$-$L^{q}$ estimates for Kirchhoff equation can be obtained by applying the Schauder-Tychonoff fixed point theorem.

We determine the cut locus of an arbitrary non-simply connected, compact and irreducible Riemannian symmetric space explicitly, and compute injectivity radius and diameter for every type of them.

In this paper, we reconstruct the link invariant of framed links introduced in [1] by the universal $R$-matrix of $\mathcal{U}_{q}(\sl_{2})$ and name it the colored Alexander invariant. We check that the optimistic limit $\mathop{\mathrm{o-lim}}$ of this invariant is determined by the volume of the knot and link cone-manifold for figure eight knot, Whitehead link and Borromean rings. We also propose the $A$-polynomials of these examples obtained from the colored Alexander invariant.