We consider a Cauchy problem $$y^{\prime }\left(x\right)+y^2\left(x\right)=q\left(x\right),{\left.y\left(x\right)\right|}_{x=x_0}=y_0$$ where $x_0$ , $y_0\in \mathbb{R}$ and $q\left(x\right)\in L_1^{\text{loc}}\left(R\right)$ is a non-negative function satisfying the condition: $${\int }_{-\mathrm{\infty }}^{x}q\left(t\right)dt>0,{\int }_x^{\mathrm{\infty }}q\left(t\right)dt>0\text{ for }x\in \mathbb{R}.$$ We obtain the conditions under which $y\left(x\right)$ can be continued to all of $\mathbb{R}$. This depends on $x_0$ , $y_0$ and the properties of $q\left(x\right)$.

Let G be a group generated by r elements $g_1,\dots ,g_r$. Among the reduced words in $g_1,\dots ,g_r$ of length n some, say ${\gamma }_n$, represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of ${\gamma }_{2n}$ has a limit, called the cogrowth exponent with respect to the generators $g_1,\dots ,g_r$. We show by analytic methods that the numbers ${\gamma }_n$ vary regularly, i.e. the ratio ${\gamma }_{2n+2}/{\gamma }_{2n}$ is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function...

We obtain a sharp estimate for the best constant $C>0$ in the Wirtinger type inequality $${\int }_0^{2\pi }{\gamma }^p{\omega }^2\le C{\int }_0^{2\pi }{\gamma }^q\omega ^{\prime }{}^2$$ where $\gamma$ is bounded above and below away from zero, $w$ is $2\pi$-periodic and such that ${\int }_0^{2\pi }{\gamma }^p\omega =0$, and $p+q\ge 0$. Our result generalizes an inequality of Piccinini and Spagnolo.

In this paper, we study the relationship between the long time behavior of a solution $u(t,x)$ of the nonlinear heat equation $u_t-\Delta u+{\left|u\right|}^{\alpha }u=0$ on ${ℝ}^N$ (where $\alpha \>0$) and the asymptotic behavior as $\left|x\right|\to \infty$ of its initial value $u_0$. In particular, we show that if the sequence of dilations ${\lambda }_n^{2/\alpha }{u}_0({\lambda }_n·)$ converges weakly to $z(·)$ as ${\lambda }_n\to \infty$, then the rescaled solution $t^{1/\alpha }u(t,·\sqrt{t})$ converges uniformly on ${ℝ}^N$ to $𝒰\left(1\right)z$ along the subsequence $t_n={\lambda }_n^2$, where $𝒰\left(t\right)$ is an appropriate flow. Moreover, we show there exists an initial value $U_0$ such that the set of all possible $z$ attainable...

Let $q$ be a complex number, $m$ be a positive rational integer and $l_m\left(q\right)=inf\{\left|P\left(q\right)\right|,P\in {\mathbb{Z}}_m\left[X\right],P\left(q\right)\ne 0\}$, where ${\mathbb{Z}}_m\left[X\right]$ denotes the set of polynomials with rational integer coefficients of absolute value $\le m$. We determine in this note the maximum of the quantities $l_m\left(q\right)$ when $q$ runs through the interval $]m,m+1[$. We also show that if $q$ is a non-real number of modulus $\>1$, then $q$ is a complex Pisot number if and only if $l_m\left(q\right)\>0$ for all $m$.