We give a normal form theorem for arithmetical derivations. It is proved by induction up to $\epsilon_{1}$ and implies the $\omega-$ consistency of arithmetic.

Let $A$ be a finite dimension algebra over an algebraically closed field such that all its idempotent ideals are projective. We show that if $A$ is representation-infinite and not hereditary, then there exist infinitely many nonisomorphic indecomposable $A$-modules of infinite projective dimension.

Let $X$ be a complex manifold, $M$ a real analytic submanifold of $X^{R}$, $T^{*}X$ the cotangent bundle to $X$, $T_{M}^{*}X$ the conormal bundle to $M$ in $X$. Assume that $T_{M}^{*}X$ is regular and $CR$ in $T^{*}X$. We then show that $T_{M^{*}}X$ is locally defined as the zero-set of the real and/or imaginary part of holomorphic symplectic coordinates of $T^{*}X$. It is well known that the similar description of $M$ in local complex coordinates of $X$ is true only if $M$ is Levi flat. As an application we obtain a generalization of the celebrated edge of the wedge Theorem.

In this paper we investigate the life span of the Cauchy Problem for nonlinear systems of the form $(*)$ $\left\{\begin{array}{l}\partial_{t}u=f(t,x,u,\partial_{1}u,\ldots,\partial_{n}u)\\u(0,x)=\epsilon\phi(x).\end{array}\right.$ Assuming that $(*)$ is weakly hyperbolic and has the solution $u\equiv 0$ with $\phi\equiv 0$, we prove that i) lifespan $T_{\epsilon}\rightarrow\in\infty$ as $\epsilon\rightarrow 0$. $T_{\epsilon}$ admits the asymptotic estimate $T_{\epsilon}\geq\psi^{-1}(\mu\log\log(1/\epsilon)$, where $\psi(t)=\int_{0^{t}}|f(\tau)|d\tau,\mu \gt 0$. ii) $u=0$ is a stable solution. In order to get this fact, we first consider the case of linear systems and then apply to nonlinear systems.

Given a triple covering $X$ of genus $g$ of a general (in the sense of Brill-Noether) curve $C$ of genus $h$, we show the existence of base-point-free pencils of degree $d$ which are not composed with the triple covering for any $d\geq g-[(3h+1)/2]-1$ by utilizing some enumerative methods and computations. We also discuss about the sharpness of our main result and the so-called Castelnuovo-Severi bound by exhibiting some examples.