Let $G$ be a group that has a presentation with $n$ generators and $m$ relators, where $m\,{<}\,n$, and let $S$ be an arbitrary generating set for $G$. Then some subset of $n-m$ elements of $S$ freely generates a free subgroup of $G$. This leads to the proof of a conjecture of Gromov on growth of groups.

The values taken by $\Gamma_0(n)$-modular functions at elliptic points of order 2 for the Fricke group $\Gamma_0(n)^\dagger$ that lie outside $\Gamma_0(n)$ are studied. In the case of a principal modulus (‘Hauptmodul’) for $\Gamma_0(n)$ or $\Gamma_0(n)^\dagger$, the class fields generated by these values are determined.

The paper introduces a new grading on the preprojective algebra of an arbitrary locally finite quiver. Viewing the algebra as a left module over the path algebra, the author uses the grading to give an explicit geometric construction of a canonical collection of exact sequences of its submodules. If a vertex of the quiver is a source, the above submodules behave nicely with respect to the corresponding reflection functor. It follows that when the quiver is finite and without oriented cycles, the canonical exact sequences are the almost split sequences with preprojective terms, and the indecomposable direct summands of the submodules are the non-isomorphic indecomposable preprojective modules. The proof extends that given by Gelfand and Ponomarev in the case when the finite quiver is a tree.

Let $L/K$ be a finite Galois extension of fields whose Galois group is a nonabelian simple group. It is shown that $L/K$ admits exactly two Hopf–Galois structures.

This paper proves bounds for the commutator width of a wreath product of two groups. As a corollary, it is shown that the commutator width of finite perfect linear groups of dimension 15 is unbounded. It follows that the covering number of these groups is unbounded. On the other hand, the commutator width of iterated wreath products of nonabelian finite simple groups is bounded by an absolute constant.

The authors solve the following problem: for which connected Lie groups is the group of homotopy classes of self maps commutative?

The class $P_n(I)$ of matrix monotone functions of order $n$, defined on an open interval $I$, is considered in this paper. Explicit examples are given to prove that if $I$ is different from the whole real line, then $P_{n+1}(I)$ is a proper subset of $P_n(I)$ for each natural number $n$. The subhomogeneous C*-algebras are then characterized in terms of matrix monotone functions.

The semilinear elliptic eigenvalue problem with superlinear pure power nonlinearity is considered. This problem is treated from the standpoint of $L^2$-theory and the precise asymptotic formula for the eigenvalue parameter $\lambda \,{=}\, \lambda(\alpha)$ as $\alpha \,{\to}\, \infty$ is established, where $\alpha$ is the $L^2$-norm of the solution $u$ associated with $\lambda$.

A well-known formula of Bendixson states that solutions of first-order differential equations, as functions of their initial conditions, satisfy a certain partial differential equation. A consequence is Alekseev's nonlinear variation of parameters formula. In this paper, corresponding results are proved for difference equations. To achieve this, use is made of the recently introduced concept of alpha derivatives, rather than of differences or of the usual derivatives. This technique allows the results to be generalized to alpha dynamic equations, which include among others ordinary differential and difference equations.

The omega limit sets of a nonlinear operator $T$ which is defined on a positive cone and satisfies certain ray-contractive type conditions are discussed. Under the assumption that the restriction of $T$ to a compact subset is surjective, the following alternatives are proved: the omega limit set of a point in the cone either consists of a fixed point or forms a 2-cycle. In addition, new proofs and extensions to relevant results are given.

Under very general conditions, a proof is given, via Morse theory, of the existence of solutions for asymptotically ‘linear’ $p$-Laplacian equations, where the asymptotic limit may be greater than the second eigenvalue. The existence of nonzero solutions is also considered.

Cet article présente des minorations universelles de la première valeur propre non nulle du laplacien des formes différentielles sur les domaines euclidiens ou hémisphériques convexes ou les hypersurfaces euclidiennes convexes.

In terms of the upper bounds of a second-order elliptic operator acting on specific Lyapunov-type functions with compact level sets, sufficient conditions are presented for the corresponding Dirichlet form to satisfy the Poincaré and the super-Poincaré inequalities. Here, the elliptic operator is assumed to be symmetric on $L^2(\mu)$ with some probability measure $\mu$. As applications, proofs are given for a class of (non-symmetric) diffusion operators generating $C_0$-semigroups on $L^1(\mu)$: that their $L^p(\mu)$-essential spectrum is empty for $p>1$. This follows since it is proved that their $C_0$-semigroups are compact.Supported in part by the NNSFC for Distinguished Young Scholars (10025105), the 973-Project and TRAPOYT in China.

For $f$ meromorphic in the complex plane and $\varphi$ meromorphic in the unit disc, sharp upper bounds are obtained for $$m\left(r,\frac{f^{(k)}}{f^{(j)}}\right)=\frac{1}{2\pi}\int_0^{2\pi} \log^{+}\left|\frac{f^{(k)}(re^{i\theta})}{f^{(j)}(re^{i\theta})}\right |\, d\theta,\qquad r<\infty,$$ and $$m\left(r,\frac{\vp^{(k)}}{\vp^{(j)}}\right)=\frac{1}{2\pi}\int_0^{2\pi} \log^{+}\left|\frac{\vp^{(k)}(re^{i\theta})}{\vp^{(j)}(re^{i\theta})}\right |\, d\theta,\qquad r<1,$$ where $k$ and $j$ are integers satisfying $k>j\geq 0$. The results generalize the logarithmic derivative estimate due to Gol'dberg and Grinshtein to derivatives of higher order.

A boundary point of a domain $D$ in $\Bbb{R}^n$ is said to be broadly accessible if it ‘almost lies’ on the boundary of a round ball contained in $D$. If $f$ is a quasiconformal mapping of the unit ball $B^n$ onto $D$, then it is shown that broadly accessible boundary points on $\partial D$ correspond under $f$ to a set of full measure on $\partial B^n$.This research was carried out while R. Näkki was visiting The University of Texas at Austin in 1985–86.