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...

Sia $C$ una curva dello spazio di grado $D$ contenuta in una superficie di grado $r$ e non in una di grado $r-1$. Se $C$ è integra, allora $r\le \sqrt{6D-2}-2$; questo limite superiore, raggiunto in alcuni casi (cfr. [5]), non vale però per curve arbitrarie (cfr. [?, 3 (iii)]). Ogni curva $C$ dello spazio (anche non ridotta o riducibile) può essere ottenuta come schema degli zero di una sezione non nulla di un opportuno fascio riflessivo $F$ di rango 2. Mediante i fasci riflessivi, siamo in grado di estendere alle curve...

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...

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 $q\left(x\right)$.

We make more accessible a neglected simple continued fraction based algorithm due to Lagrange, for deciding the solubility of $a{x}^2+bxy+c{y}^2=N$ in relatively prime integers $x,y$, where $N\ne 0$, gcd$(a,b,c)=\text{gcd}(a,N)=1\text{et}D=b^2-4ac\>0$ is not a perfect square. In the case of solubility, solutions with least positive y, from each equivalence class, are also constructed. Our paper is a generalisation of an earlier paper by the author on the equation $x^2-D{y}^2=N$. As in that paper, we use a lemma on unimodular matrices that gives a much simpler proof than Lagrange’s...

Let $\Omega$ be a bounded, convex planar domain whose boundary has a not too degenerate curvature. In this paper we provide partial answers to an identification question associated with the boundary value problem $$▵u=-cu-d\text{in}\Omega ,u=0\text{on}\partial \Omega .$$ We prove two results: 1) If $\Omega$ is not a ball and if one considers only solutions with $-cu-d\le 0$, then there exist at most finitely many pairs of coefficients $(c,d)$ so that the normal derivative $\frac{\partial u}{\partial \nu }{|}_{\partial \Omega }$ equals a given $\psi \ne 0$. 2) If one imposes no sign condition on the...