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...

Let $Q$ denote the field of rational numbers. Let $K$ be a cyclic quartic extension of $Q$. It is known that there are unique integers $A$, $B$, $C$, $D$ such that $$K=Q\left(\sqrt{A(D+B\sqrt{D})}\right),$$ where $$A\text{is}\text{squarefree}\text{and}\text{odd},D=B^2+C^2\text{is}\text{squarefree},B>0,C>0,GCD(A,D)=1.$$ The conductor $f\left(K\right)$ of $K$ is $f\left(K\right)=2^l\left|A\right|D$, where $$l=\left\{\begin{array}{c}3,\text{if}D\equiv 2(mod4)\text{or}D\equiv 1(mod4),B\equiv 1(mod2),\hfill \\ 2,\text{if}D\equiv 1(mod4),B\equiv 0(mod2),A+B\equiv 3(mod4),\hfill \\ 0,\text{if}D\equiv 1(mod4),B\equiv 0(mod2),A+B\equiv 1(mod4).\hfill \end{array}\right.$$ A simple proof of this formula for $f\left(K\right)$ is given, which uses the basic properties of quartic Gauss sums.

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 $m$.

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 $q\left(x\right)$.

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...

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...