The existence of time local $C^{\infty }$-solutions is shown for Cauchy problem of the porous medium equations. Our arguments rely on the "$L^{\infty }$-energy method" developed in our previous paper [16] and a new method based on the theory of evolution equations in the $L^2$-framework which enables us to handle with perturbations which can be decomposed into monotone parts and small parts in Sobolev spaces of higher order.

In this paper we consider Jørgensen's inequality for classical Schottky groups of real types, that is, the third, sixth and eighth types. The infimum of Jørgensen's numbers for the groups of the third, sixth and eighth types are 4, 16 and 16, respectively.

To a Hodge structure $V$ of weight $k$ with CM by a field $K$ we associate Hodge structures $V_{-n/2}$ of weight $k+n$ for $n$ positive and, under certain circumstances, also for $n$ negative. We show that these 'half twists' come up naturally in the KugaSatake varieties of weight two Hodge structures with CM by an imaginary quadratic field.

In this paper, we discuss factorization of the solutions of some linear ordinary differential equations with transcendental entire coefficients. We give a condition which shows that the solutions for some differential equations are prime, for some are factorizable.

We study the non-linearlizability conjecture (NLC) for polynomials at non-Brjuno irrationally indifferent fixed points. A polynomial is $n$-subhyperbolic if it has exactly $n$ recurrent critical points corresponding to irrationally indifferent cycles, other ones in the Julia set are preperiodic and no critical orbit in the Fatou set accumulates to the Julia set. In this article, we show that NLC and, more generally, the cycle-version of NLC are true in a subclass of $n$-subhyperbolic polynomials. As a corollary, we prove the cycle-version of the Yoccoz Theorem for quadratic polynomials.

We also study several specific examples of $n$-subhyperbolic polynomials. Here we also show the scaling invariance of the Brjuno condition: if an irrational number $\alpha$ satisfies the Brjuno condition, then so do $m\alpha$ for every positive integer $m$.

We give a new a priori estimate for a classical solution of the inhomogeneous wave equation in $R^n\times R$, where $n=2,3$. As an application of the estimate, we study the asymptotic behavior as $t\to \pm \infty$ of solutions $u(x,t)$ and $v(x,t)$ to a system of semilinear wave equations: ${\partial }_t^{2}u-\Delta u=|v{|}^p$, ${\partial }_t^{2}v-\Delta v=|u{|}^q$ in $R^n\times R$, where $(n+1)/(n$ -1 $)$ with $n=2$ or $n=3$. More precisely, it is known that there exists a critical curve $\Gamma =\Gamma (p,q,n)=0$ on the p-q plane such that, when $\Gamma >0$, the Cauchy problem for the system has a global solution with small initial data and that, when $\Gamma \le 0$, a solution of the problem generically blows up in finite time even if the initial data are small. In this paper, when $\Gamma >0$, we construct a global solution $\left(u\right(x,t),$$v(x,t))$ of the system which is asymptotic to a pair of solutions to the homogeneous wave equation with small initial data given, as $t\to -\infty$, in the sense of both the energy norm and the pointwise convergence. We also show that the scattering operator exists on a dense set of a neighborhood of 0 in the energy space.

There is proved that the Hochschild cohomology algebras of finitedimensional self-injective $K$-algebras over a field $K$ are invariants of stable equivalences of Morita type.

Let $KO\left(C{P}^m\right)$ be the $KO$-ring of the complex projective space $C{P}^m$. By means of methods of rational $D$-series [4], a formula for the $J$-orders of elements of $KO\left(C{P}^m\right)$ is given. Explicit formulas are given for computing the $J$-orders of the canonical generators of $KO\left(C{P}^m\right)$ and the $J$-order of any complex line bundle over $C{P}^m$.

In this article we study nonsingular rational open surfaces of logarithmic Kodaira dimension zero with connected boundaries at infinity defined over an algebraically closed field of arbitrary characteristic. We establish a classification theory of nonsingular alfine surfaces of logarithmic Kodaira dimension zero and give a characterization of $A_{*}^1\times A_{*}^1$ in arbitrary characteristic.

This paper is devoted to the study of rational approximations of the ratio $\eta \left(\lambda \right)/\omega \left(\lambda \right)$, where $\omega \left(\lambda \right)$ and $\eta \left(\lambda \right)$ are the real period and real quasi-period, respectively, of the elliptic curve $y^2=x(x-1)(x-\lambda )$. Using monodromy principle for hypergeometric function in the logarithm case we obtain rational approximations of $(\eta /\omega )\left(\lambda \right)$ with $\lambda ∊Q$ and we shall find new measures of irrationality, both in the archimedean and non archimedean case.

Let $G$ be a finite group. A gap $G$-module $V$ is a finite dimensional real $G$-representation space satisfying the following two conditions:

(1) The following strong gap condition holds: $\dim V^P>2\dim V^H$ for all $H\le G$ such that $P$ is of prime power order, which is a sufficient condition to define a Gsurgery obstruction group and a $G$-surgery obstruction.

(2)$V$ has only one $H$-fixed point 0 for all large subgroups $H$, namely $H∊\mathcal{L}\left(G\right)$. A finite group $G$ not of prime power order is called a gap group if there exists a gap Gmodule. We discuss the question when the direct product $K\times L$ is a gap group for two finite groups $K$ and $L$. According to [5], if $K$ and $K\times C_2$ are gap groups, so is $K\times L$. In this paper, we prove that if $K$ is a gap group, so is $K\times C_2$. Using [5], this allows us to show that if a finite group $G$ has a quotient group which is a gap group, then $G$ itself is a gap group. Also, we prove the converse: if $K$ is not a gap group, then $K\times D_{2n}$ is not a gap group. To show this we define a condition, called NGC, which is equivalent to the non-existence of gap modules.