The Collatz conjecture is that there exists a positive integer $n$ which satisfies $f^n(m)=1$ for any integer $m\ge 3$, where $f$ is the function on the rational number field defined by $f(m)=m/2$ if the numerator of $m$ is even and $f(m)=(3m+1)/2$ if the numerator of $m$ is odd. Let $m$ be a rational number such that . Then we show that, if $m$ has some simple sequences, then the total number of positive integer $m$ is finite, by estimating $f(m)-m$.

From a viewpoint of the general theory for convex functions it is shown that invariance of closed convex subsets under semigroups of nonlinear operators is characterized in complex Hilbert spaces. Some examples, which generalize positivity of semigroups, are given for semigroups generated by non-linear elliptic operators.

In this paper, we consider the Cauchy problem for the following nonlinear Schrödinger equations

where

${\text{λ}}_{\alpha \beta }\in ℂ,l\ge 2,r\ge 1$ if $N=3,4$, or $r\ge 0$ if $N\ge 5$. We study analyticity of global solutions for (NLS) with small initial data $u_0$. To be precise, we show that global solutions of (NLS) are analytic in space and time for $\left|t\right|\ne 0$ if the norm of the initial data is sufficiently small in analytical space with respect to $x\cdot \nabla$.

The $\ast$-exponential function $e_{\ast }^{ti{e}^{av}\ast u}$ is defined in a transcendently extended Weyl algebra. In the Weyl ordering expression, this is given as the real analytic solution in $t$ of

$${\partial }_t{F}_t(u,v)=i{e}^{av}\{(u+\frac{\hslash ia}{2})F_t(u+\frac{\hslash ia}{2},v)-\frac{\hslash i}{2}{\partial }_v{F}_t(u+\frac{\hslash ia}{2},v)\},F_0(u,v)=1.$$.

For generic initial functions, this equation can be solved for all $t$, but the uniqueness holds only for one direction.