In this paper we consider degenerate parabolic equations, and obtain an interior and a boundary Harnack inequalities for nonnegative solutions to the degenerate parabolic equations. Furthermore we obtain boundedness and continuity of the solutions.

Let $\mathcal{A}$ be a simple Abelian variety of dimension $g$ over $\mathbb{Q}$, and let $\ell$ be the rank of the Mordell-Weil group $\mathcal{A}(\mathbb {Q})$. Assume $\ell \geq 1$. A conjecture of Mazur asserts that the closure of $\mathcal{A}(\mathbb {Q})$ into $\mathcal{A}(\mathbb{R})$ for the real topology contains the neutral component $\mathcal{A} (\mathbb{R})^{0}$ of the origin. This is known only under the extra hypothesis $\ell \geq g^{2}-g+1$. We investigate here a quantitative refinement of this question: for each given positive $h$, the set of points in $\mathcal{A}(\mathbb {Q})$ of Néron-Tate height $\leq h$ is finite, and we study how these points are distributed into the connected component $\mathcal{A}(\mathbb {R})^{0}$. More generally we consider an Abelian variety $\mathcal{A}$ over a number field $K$ embedded in $\mathbb{R}$, and a subgroup $\Gamma$ of $\mathcal{A}(K)$ of sufficiently large rank. The effective result of density we obtain relies on an estimate of Diophantine approximation, namely a lower bound for linear combinations of determinants involving Abelian logarithms.

In this paper we prove the local existence and uniqueness of $C^{1+\gamma}$ solutions of the Boussinesq equations with initial data $v_0$, $\theta_0 \in C^{1+\gamma}$, $\omega_0, \Delta\theta_0\in L^q$ for $0 < \gamma < 1$ and $1 < q < 2$. We also obtain a blow-up criterion for this local solutions. More precisely we show that the gradient of the passive scalar $\theta$ controls the breakdown of $C^{1+\gamma}$ solutions of the Boussinesq equations.

We consider a problem whether Kodaira's $\partial \overline{\partial}$-Lemma holds on weakly $1$-complete Kähler manifolds or not. This problem was proposed by S. Nakano. We prove that the Lemma holds for some class of complex quasi-tori $\mathbb{C}^{n}/\Gamma$, and it does not hold for the other class of them. Every complex quasi-tori is weakly $1$-complete and complete Käahler. Then we get a negative answer for the above problem.

We study the asymptotic behavior at low energy of scattering amplitudes in two dimensional magnetic fields with compact support. The obtained result depends on the total flux of magnetic fields. It should be noted that magnetic potentials do not necessarily fall off rapidly at infinity. The main body of argument is occupied by the resolvent analysis at low energy for magnetic Schrödinger operators with perturbations of lang-range class. We can show that the dimension of resonance spaces at zero energy does not exceed two. As a simple application, we also discuss the scattering by magnetic field with small support and the convergence to the scattering amplitude by $\delta$-like magnetic field.

It is shown that a Siegel-Hecke eigenform of integral weight $k$ and genus 2 is uniquely determined by its Fourier coefficients indexed by $nT$ where $T$ runs over all half-integral positive definite primitive matrices of size 2 and $n$ over all squarefree positive integers. The proof uses analytic arguments involving Koecher-Maaß series and spinor zeta functions.

Let $H_{1}, H_{2}, \ldots, H_{q}$ be hyperplanes in $P^{N}(\mathbb{C})$ in general position. Previously, the author proved that, in the case where $q \geq 2N+3$, the condition $\nu(f, H_{j}) = \nu(g, H_{j})$ imply $f = g$ for algebraically nondegenerate meromorphic maps $f, g : \mathbb{C}^{n} \to P^{N}(\mathbb{C})$, where $\nu(f, H_{j})$ denote the pull-backs of $H_{j}$ through $f$ considered as divisors. In this connection, it is shown that, for $q \geq 2N+2$, there is some integer $\ell_{0}$ such that, for any two nondegenerate meromorphic maps $f, g : \mathbb{C}^{n} \to P^{N}(\mathbb {C})$ with $\min(\nu(f, H_{j}), \ell_{0}) = \min(\nu(g, H_{j}), \ell_{0})$ the map $f \times g$ into $P^{N}(\mathbb{C}) \times P^{N}(\mathbb{C})$ is algebraically degenerate. He also shows that, for $N = 2$ and $q = 7$, there is some $\ell_{0}$ such that the conditions $\min(\nu(f, H_{j}), \ell_{0}) = \min(\nu(g, H_{j}), \ell_{0})$ imply $f = g$ for any two nondegenerate meromorphic maps $f, g$ into $P^{2}(\mathbb{C})$ and seven generic hyperplanes $H_{j}$'s.

Let $(M, {\cal V}, \theta)$ be a real analytic $(2n+1)$-dimensional pseudo-hermitian manifold with nondegenerate Levi form and $F$ be a pseudo-hermitian embedding into $\mathbb{C}^{n+1}$. We show under certain generic conditions that $F$ satisfies a complete system of finite order. We use a method of prolongation of the tangential Cauchy-Riemann equations and pseudo-hermitian embedding equation. Thus if $F \in C^{k}(M)$ for sufficiently large $k$, $F$ is real analytic. As a corollary, if $M$ is a real hypersurface in $\mathbb{C}^{n+1}$, then $F$ extends holomorphically to a neighborhood of $M$ provided that $F$ is sufficiently smooth.