Let $G$ be a finite group. The strong symmetric genus$\sigma ^{0}(G)$ is the minimum genus of any Riemann surface on which $G$ acts, preserving orientation. For any non-negative integer $g$, there is at least one group of strong symmetric genus $g$. For $g \neq 2$, one such group has the form $Z_{k} \times D_{n}$ for some $k$ and $n$.

Analogues of the Funk–Hecke formula for spherical harmonics are proved for Dunkl's h-harmonics associated to the reflection groups, and for orthogonal polynomials related to h-harmonics on the unit ball. In particular, an analogue and its application are discussed for the weight function (1−[mid ]x[mid ]2)μ−1/2 on the unit ball in ℝd.

Answering a question of Gromov [7], we shall present an example of a finitely generated group Γ and two non-principal ultrafilters [Ascr ], [Bscr ] such that the asymptotic cones Con[Ascr ] Γ and Con[Bscr ] Γ are not homeomorphic.

let $r$ be a ring and $t$ a 1-tilting right $r$-module. then $t$ is of countable type. moreover, $t$ is of finite type in the case where $r$ is a prüfer domain.

One of the basic facts of group theory is that each finite group contains a Sylow p-subgroup for each prime p which divides the order of the group. In this note we show that each vertex-transitive self- complementary graph has an analogous property. As a consequence of this fact, we obtain that each prime divisor p of the order of a vertex-transitive self-complementary graph satisfies the congruence pm ≡ 1(mod 4), where pm is the highest power of p which divides the order of the graph.

We give a simple direct proof of the interpolation inequality $ \|\nabla f\|^2_{L^{2p}} \le C \|f\|_{\rm BMO} \|f\|_{W^{2,p}}$, where $ 1<p<\infty$. For $p=2$ this inequality was obtained by Meyer and Rivière via a different method, and it was applied to prove a regularity theorem for a class of Yang–Mills fields. We also extend the result to higher derivatives, sharpening all those cases of classical Gagliardo–Nirenberg inequalities where the norm of the function is taken in $L^\infty$ and other norms are in $L^q$ for appropriate $q>1$.

We consider the uniqueness of the positive solution to a semilinear elliptic equation with Dirichlet boundary condition and the nonlinearity satisfying $f(0)<0$ and having asymptotic sublinear growth rate. A similar idea is also applied to the nonexistence of a positive solution to a superlinear problem.

It is proved here that if $f : {\rm C\!\!\!I} \rightarrow \bar{{\rm C\!\!\!I}}$ is an elliptic function and $q$ is the maximal multiplicity of all poles of $f$, then the Hausdorff dimension of the Julia set of $f$ is greater than $2q/(q+1)$, and the Hausdorff dimension of the set of points that escape to infinity is less than or equal to $2q/(q+1)$. In particular, the area of this latter set is equal to 0.

We show that the conjugacy problem is solvable in [finitely generated free]-by-cyclic groups, by using a result of O. Maslakova that one can algorithmically find generating sets for the fixed subgroups of free group automorphisms, and one of P. Brinkmann that one can determine whether two cyclic words in a free group are mapped to each other by some power of a given automorphism. We also solve the power conjugacy problem, and give an algorithm to recognize whether two given elements of a finitely generated free group are twisted conjugated to each other with respect to a given automorphism.

The theory of isometric group actions on R-trees is extended to actions by homeomorphisms with the following non-nesting property: no group element maps an arc properly into itself. A finitely presented group acting freely by homeomorphisms on an R-tree is free abelian or splits over a (possibly trivial) cyclic group.

Let K be a compact subset of ℝn, 0[les ]s[les ]n. Let Ps0, [Pscr ]s denote s-dimensional packing premeasure and measure, respectively. We discuss in this paper the relation between Ps0 and [Pscr ]s. We prove: if Ps0(K)<∞, then [Pscr ]s(K) = Ps0(K); and if Ps0(K) = ∞, then for any ε>0, there exists a compact subset F of K such that [Pscr ]s(F) = Ps0(F) and [Pscr ]s(F)[ges ] [Pscr ]s(K)−ε.

We provide a criterion for the energy measures of regular Dirichlet forms on post-critically finite self-similar sets to be singular with respect to every self-similar measure. A probabilistic interpretation of the singularity is also discussed.

We consider the Abel equation

formula here

where p and q are polynomials, and a is a fixed constant. We denote the solution of (1) by y(x, c), where y(a, c) = c. Standard existence theorems ensure that y(x, c) is well-defined and analytic in both its arguments, for c sufficiently small. If y(b, c) = c, then we call y(x, c) a periodic solution. Likewise, if y(b, c) ≡ c for all c close to 0, then we say that the system has a centre between a and b. The numbers a and b are not important; by a simple transformation, we can always choose a = 0 and b = 1, and we shall usually do so from now on.

Abel equations arise in several circumstances, but perhaps the main reason for their recent study is connected to the family of systems

formula here

where M and N are homogeneous polynomials of the same degree n. A transformation due to Cerkas allows us to bring these systems to the form (1), where p and q are now trigonometric polynomials. It is not hard to show that, setting a = 0 and b = 2π, the definitions of periodic solution and centre for (1) coincide with their usual definitions in the planar system (2). There are also transformations to Abel-type equations for more general systems; see [8, 10].

This trigonometric Abel equation has been used in a large number of works in order to estimate the number of limit cycles or obtain centre conditions, as well as in more general investigations relating the derivatives of the return map with iterated integrals. However, studying system (2) in whatever form is by no means easy, and a natural question is to ask whether we can still capture the essence of this problem if we take p and q to be polynomials, in the hope that the calculations will become easier. The recent series of investigations by Briskin, Françoise and Yomdin [3, 4, 5] seems to indicate that this could be the case.

Our interest here is to see what conditions the existence of a centre in (1) imposes on the defining equations. For ease of reference, we shall always denote the antiderivative of the polynomials p and q as P and Q; that is,

formula here

Let K be a cone in ℝn. Then K is a uniqueness set for the spherical Radon transform if and only if it is not contained in the zero set of any (nontrivial) homogeneous harmonic polynomial. A local version of this result is also proved.

Twenty-five years ago, Conway and Norton published in this journal their remarkable paper ‘Monstrous Moonshine’, proposing a completely unexpected relationship between finite simple groups and modular functions. We review the progress made in broadening and understanding that relationship.

This article is an extended version of a lecture given in Oxford on 12 May 1995 at the invitation of the London Mathematical Society and the British Society for the History of Mathematics.

Contents

1. A few figures

2. Taylor series before 1900. A strange statement of Borel

3. Fourier series before 1900. A strange field

4. Brownian motion around 1900. A rising subject

5. Fourier and Taylor series after 1900. A revival

6. Lacunarity and randomness

7. The appearance of random series of functions

8. The Wiener theory of Brownian motion

9. The merging of Brownian motion and random Fourier series

10. The non-differentiability and local behaviour of Brownian motion

11. Three ways to figure out the Brownian motion

12. The plane Brownian motion

13. Applications of Brownian motion to Taylor series and analytic functions

14. Applications of Brownian motion to Fourier series and harmonic analysis

Let $\Gamma$ be a non-Euclidean crystallographic group. $\Gamma$ is said to be non-maximal if there exists a non-Euclidean crystallographic group $\Gamma'$ such that $\Gamma \le \Gamma'$ and the dimension of the Teichmüller space of $\Gamma$ equals the dimension of the Teichmüller space of $\Gamma'$. The full list of such pairs of groups is computed in the case when $\Gamma$ is non-normal in $\Gamma'$. The corresponding problem for Fuchsian groups was solved by Singerman.

Several inequalities for trace norms of sums of $n$ operators with roots of unity coefficients are proved in this paper. When $n=2$, these reduce to the classical Clarkson inequalities and their non-commutative analogues.

Let Ω⊆ℝd be an open set of measure 1. An open set D⊆ℝd is called a ‘tight orthogonal packing region’ for Ω if D−D does not intersect the zeros of the Fourier transform of the indicator function of Ω, and D has measure 1. Suppose that Λ is a discrete subset of ℝd. The main contribution of this paper is a new way of proving the following result: D tiles ℝd when translated at the locations Λ if and only if the set of exponentials EΛ = {exp 2πi〈λ, x〉 [ratio ] λ ∈ Λ} is an orthonormal basis for L2(Ω). (This result has been proved by different methods by Lagarias, Reeds and Wang [9] and, in the case of Ω being the cube, by Iosevich and Pedersen [3]. When Ω is the unit cube in ℝd, it is a tight orthogonal packing region of itself.) In our approach, orthogonality of EΛ is viewed as a statement about ‘packing’ ℝd with translates of a certain non-negative function and, additionally, we have completeness of EΛ in L2(Ω) if and only if the above-mentioned packing is in fact a tiling. We then formulate the tiling condition in Fourier analytic language, and use this to prove our result.

A real-valued function $f$ defined on an open, convex set $D$ of a real normed space is called $(\varepsilon,\delta)$-midconvex if it satisfies $$f\left(\frac{x+y}{2}\right) \le \frac{f(x)+f(y)}{2} + \varepsilon|x-y| + \delta, \quad\hbox{for } x,y\in D.$$ The main result of the paper states that if $f$ is locally bounded from above at a point of $D$ and is $(\varepsilon,\delta)$-midconvex, then it satisfies the convexity-type inequality $$f(\lambda x+(1-\lambda) y) \leq \lambda f(x)+(1-\lambda)f(y)+2\delta +2\varepsilon \varphi(\lambda)|x-y| \quad\hbox{for } x,y\in D, \, \lambda\in[0,1],$$ where $\varphi:[0,1]\to{\mathbb R}$ is a continuous function satisfying $$\max(-\lambda\log_2\lambda,\,-(1-\lambda)\log_2(1-\lambda)) \le\varphi(\lambda)\le 1.4\max(-\lambda\log_2\lambda,\,-(1-\lambda)\log_2(1-\lambda))$$. The particular case $\varepsilon=0$ of this result is due to Ng and Nikodem (Proc. Amer. Math. Soc. 118 (1993) 103–108), while the specialization $\varepsilon=\delta=0$ yields the theorem of Bernstein and Doetsch (Math. Ann. 76 (1915) 514–526).