One investigates the existence of an $\omega$-periodic solution of the problem $u_t=u_{xx}+cu+g(t,x)+ϵf(t,x,u,u_x,ϵ),u(t,0)=h_0\left(t\right)+ϵ{\chi }_0(t,u(t,0),u(t,\pi )),u(t,\pi )=h_1\left(t\right)+ϵ{\chi }_1(t,u(t,0),u(t,\pi ))$, provided the functions $g,f,h_0,h_1,{\chi }_0,{\chi }_1$ are sufficiently smooth and $\omega$-periodic in $t$. If $c\ne k^2$, $k$ natural, such a solution always exists for sufficiently small $ϵ>0$. On the other hand, if $c=l^2$, $l$ natural, some additional conditions have to be satisfied.

In the paper the conditions for the existence of a $2\pi$-periodic solution in $t$ of the system $u_{tt}-u_{rr}-(2/r)u_r=ϵf(t,r,u,u_t,u_r)$, $\left|u(t,0)\right|<+\infty ,u(t,\pi )=0$ are investigated provided that $f$ is sufficiently smooth and $2\pi$-periodic in $t$.

Let $E$ be a modular elliptic curve over $\mathbb{Q}$ $L^{\left(n\right)}(s,E)$ denote the $n$-th derivative of its Hasse-Weil $L$-series. We estimate the number of twisted elliptic curves $E_d,d\le D$ such that $L^{\left(n\right)}(1,E_d)\ne 0$.

In this paper, the existence of an $\omega$-periodic weak solution of a parabolic equation (1.1) with the boundary conditions (1.2) and (1.3) is proved. The real functions $f(t,r),h\left(t\right),a\left(t\right)$ are assumed to be $\omega$-periodic in $t,f\in L_2(S,H),a,h$ such that $a^{\text{'}}\in L_{\infty }\left(R\right),h^{\text{'}}\in L_{\infty }\left(R\right)$ and they fulfil (3). The solution $u$ belongs to the space $L_2(S,V)\cap L_{\infty }(S,H)$, has the derivative $u^{\text{'}}\in L_2(S,H)$ and satisfies the equations (4.1) and (4.2). In the proof the Faedo-Galerkin method is employed.

Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in ${ℂ}^n$ of codimension 1, $vo{l}_{2n-2}\left(D^{n-1}\right)\le vo{l}_{2n-2}\left(H\cap D^n\right)\le 2vo{l}_{2n-2}\left(D^{n-1}\right)$. The lower bound is attained if and only if H is orthogonal to the versor $e_j$ of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector $e_j+\sigma e_k$ for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify ${ℂ}^n$ with ${ℝ}^{2n}$; by $vo{l}_k\left(·\right)$ we denote the usual k-dimensional volume in ${ℝ}^{2n}$. The result is a complex counterpart of Ball’s [B1]...