The Kowalevski gyrostat in two constant fields is known as the unique example of an integrable rigid body problem described by the Hamiltonian system with three degrees of freedom not reducible to a family of systems in fewer dimensions. The practical explicit integration of this system can hardly be obtained by the existing techniques. Then the challenging problem becomes to fulfill the qualitative investigation based on the study of the Liouville foliation of the phase space. As the first approach to topological analysis of this system we find the stratified critical set of the momentum map; this set is represented as the union of manifolds with induced almost Hamiltonian systems having less than three degrees of freedom. We obtain the equations of the bifurcation diagram in three-dimensional space. These equations have the form convenient for the classification of the bifurcation sets arising on 5-dimensional iso-energetic levels.

Let $\cM_{g,n}$ and $\cH_{g,n}$, for $2g-2+n>0$, be, respectively, the moduli stack of $n$-pointed, genus $g$ smooth curves and its closed substack consisting of hyperelliptic curves. Their topological fundamental groups can be identified, respectively, with $\GG_{g,n}$ and $H_{g,n}$, the so called Teichmüller modular group and hyperelliptic modular group. A choice of base point on $\cH_{g,n}$ defines a monomorphism $H_{g,n}\hookra\GG_{g,n}$.

Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichmüller group $\GG_{g,n}$ is the group of isotopy classes of diffeomorphisms of the surface $S_{g,n}$ which preserve the orientation and a given order of the punctures. As a subgroup of $\GG_{g,n}$, the hyperelliptic modular group then admits a natural faithful representation $H_{g,n}\hookra\out(\pi_1(S_{g,n}))$.

The congruence subgroup problem for $H_{g,n}$ asks whether, for any given finite index subgroup $H^\ld$ of $H_{g,n}$, there exists a finite index characteristic subgroup $K$ of $\pi_1(S_{g,n})$ such that the kernel of the induced representation $H_{g,n}\ra\out(\pi_1(S_{g,n})/K)$ is contained in $H^\ld$. The main result of the paper is an affirmative answer to this question for $n\geq 1$.

We consider the initial value problem for a system of parabolic partial differential equations modeling chemotaxis in $\R^n (n\ge 1)$, and give the asymptotic profiles for a specific class of solutions by space-time higher-order asymptotic expansions.

We clarified the variational meaning of the special values $\zeta(2M)\ (M=1,2,3,\cdots)$ of Riemann zeta function $\zeta(s)$. These are essentially the best constant of Sobolev inequality. In the background we consider Dirichlet-Neumann boundary value problem for a differential operator $(-1)^M(d/dx)^{2M}$. Its Green function is found and expressed in terms of the well-known Bernoulli polynomial. The supremum of the diagonal value of Green function is equal to the best constant for corresponding Sobolev inequality. Discrete version of the corresponding Sobolev inequality is also presented.

After the works of Kauffman-Banchoff and Yamasaki, it is known that a local move called the pass move is strongly related to the Arf invariant, which is equivalent to the parity of the coefficient of the degree two term in the Conway polynomial. Our main result is the following: There exists a pair of knots such that their Conway polynomials coincide, and that the sets of Conway polynomials of knots obtained from them by a single pass move do not coincide.

The induced transformation on a properly chosen interval of a 2-interval exchange transformation is a 2-interval exchange transformation again. This induction is explicitly integrated by using the simple continued fraction algorithm. Therefore, we can say that the simple continued fraction algorithm acts well as a ‘‘multiplicative Rauzy induction’’ on the family of 2-interval exchange transformations. Now, we have the following question; On what kind of family of 3-interval exchange transformations does the negative slope algorithm act well as a multiplicative Rauzy induction? The purpose of this paper is to give the answer to this question.