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.

Let $G$ be a compactly generated, locally compact group of polynomial growth. Removing a restrictive technical condition from a previous work, we show that the weighted group algebra $L^{1}_{\omega}(G)$ is a symmetric Banach $*$-algebra if and only if the weight function $\omega $ satisfies the GRS-condition. This condition expresses in a precise technical sense that $\omega $ grows subexponentially.

A group $G$ is said to have periodic cohomology with period $q$ after $k$ steps, if the functors $H^{i}(G,-)$ and $H^{i+q}(G,-)$ are naturally equivalent for all $i>k$. Mislin and the author have conjectured that periodicity in cohomology after some steps is the algebraic characterization of those groups $G$ that admit a finite-dimensional, free $G$-CW-complex, homotopy equivalent to a sphere. This conjecture was proved by Adem and Smith under the extra hypothesis that the periodicity isomorphisms are given by the cup product with an element in $H^q (G,\mathbb{Z})$. It is expected that the periodicity isomorphisms will always be given by the cup product with an element in $H^q (G,\mathbb{Z})$; this paper shows that this is the case if and only if the group $G$ admits a complete resolution and its complete cohomology is calculated via complete resolutions. It is also shown that having the periodicity isomorphisms given by the cup product with an element in $H^q (G,\mathbb{Z})$ is equivalent to silp $G$ being finite, where silp $G$ is the supremum of the injective lengths of the projective $\mathbb{Z}G$-modules.

Using reparametrizations of linear flows, we show that there exist area-preserving real analytic maps of the three-dimensional torus that are ‘mixing of all orders’ and do not enjoy the monotone shrinking target property. Prior to that, we give a short proof of a result of Kurzweil from 1955: namely, that a translation $T_\alpha$ of the torus $\mathbb{T}^d$ has the monotone shrinking target property if and only if the vector $\alpha$ is badly approximable (that is, of constant type).

The asymptotic probability theory of conjugacy classes of the finite general groups leads to a probability measure on the set of all partitions of natural numbers. A simple method of understanding these measures in terms of Markov chains is given in this paper, leading to an elementary probabilistic proof of the Rogers–Ramanujan identities. This is compared with work on the uniform measure. The main case of Bailey's lemma is interpreted as finding eigenvectors of the transition matrix of a Markov chain. It is shown that the viewpoint of Markov chains extends to quivers.

Let $f$ be meromorphic of finite order in the plane, such that the second derivative $f^{\prime\prime}$ has finitely many zeros. Then $f$ has finitely many poles. This result was conjectured by the author in 1996, and an example shows that the theorem is sharp.

Let $m,\,g,\,q \in\mathbb{N}$ with $q\,{\geq}\, 2$ and $(m,q-1)=1$. For $n\in\mathbb{N}$, denote by $s_q(n)$ the sum of digits of $n$ in the $q$-ary digital expansion. Given a polynomial $f$ with integer coefficients, degree $d\ge 1$, and such that $f(\mathbb{N})\subset\mathbb{N}$, it is shown that there exists $C=C(f,m,q)>0$ such that for any $g\in\mathbb{Z}$, and all large $N$, \[|\{ 0\,{\leq}\, n\,{\leq}\, N : s_q(f(n))\md gm\}|\,{\geq}\, CN^{\min(1,2/d!)}\]. In the special case $m=q=2$ and $f(n)=n^2$, the value $C=1/20$ is admissible.

The aim of this paper is to exhibit a real Paley–Wiener space sitting inside the Schwartz space, and to give a quick and simple proof of a Paley–Wiener-type theorem. A simple and elementary proof of a theorem postulated by H. H. Bang is also given.

We consider the error term P3(R) = #{n ∈ ℤ3 :|n| [les ] R} − 4|3πR3 which occurs in the counting of lattice points in a sphere of radius R. By considering second and third power moments, we prove that P3(R) = Ω±(R√log R). An upper bound for the gap between the sign changes of P3(R) is also proved.

Define a sequence $(s_{n})$ of two-variable words in variables $x$, $y$ as follows: $s_{0}(x,y)=x$, $s_{n+1}(x,y)=[s_{n}(x,y)^{-y},s_{n}(x,y)]$ for $n\geq0$. It is shown that a finite group $G$ is soluble if and only if $s_{n}$ is a law of $G$ for all but finitely many values of $n$.

A minimal surface of general type with pg(S) = 0 satisfies 1 [les ] K2 [les ] 9, and it is known that the image of the bicanonical map φ is a surface for K2S [ges ] 2, whilst for K2S [ges ] 5, the bicanonical map is always a morphism. In this paper it is shown that φ is birational if K2S = 9, and that the degree of φ is at most 2 if K2S = 7 or K2S = 8.

By presenting two examples of surfaces S with K2S = 7 and 8 and bicanonical map of degree 2, it is also shown that this result is sharp. The example with K2S = 8 is, to our knowledge, a new example of a surface of general type with pg = 0.

The degree of φ is also calculated for two other known surfaces of general type with pg = 0 and K2S = 8. In both cases, the bicanonical map turns out to be birational.

A proof is given that for each non-negative integer $g$, there is an infinite family of knotted surfaces of genus $g$, none of which is ambient isotopic to itself with the orientation reversed.

Here, a classical observation of Siegel is generalized by determining all the linear relations among the initial Fourier coefficients of a modular form on $\SL_2(\ZZ)$. As a consequence, spaces $M_k$ are identified, in which there are universal $p$-divisibility properties for certain $p$-power coefficients. As a corollary, let $f(z)=\sum_{n=1}^{\infty}a_f(n)q^n \in S_k\cap O_{L}[[q]]$ be a normalized Hecke eigenform (note that $q:=e^{2\pi i z}$), and let $k\equiv \delta(k)\pmod{12}$, where $\delta(k)\in \{4, 6, 8, 10, 14\}$. Reproducing earlier results of Hatada and Hida, if $p$ is a prime for which $k\equiv \delta(k)\pmod{p-1}$, and $\mathfrak{p}\subset O_L$ is a prime ideal above $p$, a proof is given to show that $a_f(p)\equiv 0\pmod{\mathfrak{p}}$.

This paper treats the existence of multiple positive solutions of the semi-positone Sturm–Liouville boundary value problem. \begin{eqnarray*} &\lambda (p(t)y'(t))'+g(t)f(t,y(t))=0 \qquad\mbox{almost everywhere on } [R_{0},R_{1}],\\ &\alpha z(R_{0})-\beta p(R_{0})z'(R_{0})=0,\\ &\gamma z(R_{1})+\delta p(R_{1})z'(R_{1})=0, \end{eqnarray*} where $g\in L_{+}^{\infty}[R_{0}, R_{1}]$ and $f$ is allowed to take negative values (that is, $f$ is semi-positone). When $\lambda=1$, new results on the existence of one or two nonzero positive solutions are obtained. These results generalize previous results for positone cases (that is, $f\ge 0$) to the semi-positone cases. We illustrate our results with an explicit example which has two nonzero positive solutions. These results are used to deduce results on intervals of eigenvalues for which there exist one or two nonzero positive eigenfunctions. Applications of these eigenvalue results are provided.

It is proved that if $R$ is an associative ring that is cotorsion as a left module over itself, and $J$ is the Jacobson radical of $R$, then the quotient ring $R/J$ is a left self-injective von Neumann regular ring and idempotents lift modulo $J$. In particular, if $R$ is indecomposable, then it is a local ring.

Solutions are provided to several questions concerning topologically transitive and hypercyclic continuous linear operators on Hausdorff locally convex spaces that are not Fréchet spaces. Among others, the following results are presented. (1) There exist transitive operators on the space $\varphi $ of all finite sequences endowed with the finest locally convex topology (it was already known that there is no hypercyclic operator on $\varphi$). (2) The space of all test functions for distributions, which is also a complete direct sum of Fréchet spaces, admits hypercyclic operators. (3) Every separable infinite-dimensional Fréchet space contains a dense hyperplane that admits no transitive operator.

In iteration theory of rational functions, it is well known that any Fatou component is mapped onto another in an n-to-1 manner, and that periodic components are simply, doubly or infinitely connected.

For meromorphic functions, the situation is much more complicated. Using Ahlfors' theory of covering surfaces, we prove that Fatou components are mapped ‘nearly’ onto others, and that periodic components are again simply, doubly or infinitely connected. Instead of considering meromorphic functions with only one essential singularity, we allow countable sets of singularities and partly even sets of logarithmic capacity zero.

It remains open whether doubly connected periodic components of meromorphic functions with only one singularity are necessarily Herman rings (as holds for rational functions). However, there is a function with two singularities and a doubly connected periodic component which is not an Herman ring.

Let Ω be an open subset of IRd, and let Tp for p∈[1, ∞) be consistent C0-semigroups given by kernels that satisfy an upper heat kernel estimate. Denoting their generators by Ap, we show that the spectrum σ(Ap) is independent of p∈[1, ∞). We also treat the case of weighted Lp-spaces for weights that satisfy a subexponential growth condition. An example shows that independence of the spectrum may fail for an exponential weight.

This paper presents explicit constructions of universal ℝ-trees as certain spaces of functions, and also proves that a 2ℵ0-universal ℝ-tree can be isometrically embedded at infinity into a complete simply connected manifold of negative curvature, or into a non-abelian free group. In contrast to asymptotic cone constructions, asymptotic spaces are built without using the axiom of choice.

A cyclotomic polynomial $\Phi(z)$ is said to be of order 3 if $n=pqr$ for three distinct odd primes $p$, $q$, and $r$. We establish the existence of an infinite family of such polynomials whose coefficients do not exceed 1 in modulus.