Let $D$ be a bounded domain in ${\bb R}^n$ . For a function $f$ on the boundary $\partial D$ , the Dirichlet solution of $f$ over $D$ is denoted by $H_D f$ , provided that such a solution exists. Conditions on $D$ for $H_D$ to transform a Hölder continuous function on $\partial D$ to a Hölder continuous function on $D$ with the same Hölder exponent are studied. In particular, it is demonstrated here that there is no bounded domain that preserves the Hölder continuity with exponent 1. It is also also proved that a bounded regular domain $D$ preserves the Hölder continuity with some exponent $\alpha$ , $0 < \alpha < 1$ , if and only if $\partial D$ satisfies the capacity density condition, which is equivalent to the uniform perfectness of $\partial D$ if $n = 2$ .

A proof is given that for primes r, s, not both 2, and for finite simple classical groups G of sufficiently large rank, the probability that two randomly chosen elements in G of orders r and s generate G tends to 1 as |G| → ∞.

For the general one-dimensional Schrödinger operator −d2/dx2+q(x) with real q ∈ L1(ℝ), this paper presents a new series representation of the Jost solution which, in turn, implies a new asymptotic representation of the Weyl m-function for locally summable q. This representation is then applied to smooth potentials q to obtain Weyl m-function power asymptotics. The condition q(N) ∈ L1(x0, x0+δ), for N ∈ ℕ0, allows one to derive the (N+1) term for almost all x ∈ [x0, x0+δ), thereby refining a relevant result by Danielyan, Levitan and Simon.

A surprising relationship is established in this paper, between the behaviour modulo a prime $p$ of the number $s_n({\cal G})$ of index $n$ subgroups in a group ${\cal G}$ , and that of the corresponding subgroup numbers for a normal subgroup in ${\cal G}$ with cyclic quotient of $p$ –power order. The proof relies, among other things, on a twisted version due to Philip Hall of Frobenius' theorem concerning the equation $x^m=1$ in finite groups. One of the applications of this result, presented here, concerns the explicit determination modulo $p$ of $s_n({\cal G})$ in the case when ${\cal G}$ is the fundamental group of a tree of groups all of whose vertex groups are cyclic of $p$ –power order. Furthermore, a criterion is established (by a different technique) for the function $S_n({\cal G})$ to be periodic modulo $p$ .

Let $K$ and $L$ be two convex bodies in ${\bb R}^n$ . The volume ratio ${\rm vr}(K, L)$ of $K$ and $L$ is defined by ${\rm vr}(K, L) = \inf(\vert K\vert/\vert T(L)\vert)^{1/n}$ , where the infimum is over all affine transformations $T$ of ${\bb R}^n$ for which $T(L) \subseteq K$ . It is shown in this paper that ${\rm vr}(K, L) \leqslant c \sqrt{n} \log n$ , where $c > 0$ is an absolute constant. This is optimal up to the logarithmic term.

Using Chern's method of transgression, the Euler class of a compact orientable Riemann–Finsler space is represented by polynomials in the connection and curvature matrices of a torsion-free connection. When using the Chern connection (and hence the Christoffel–Levi–Civita connection in the Riemannian case), this result extends the Gauss–Bonnet formula of Bao and Chern to Finsler spaces whose indicatrices need not have constant volume.

A proof is given that the full symmetric group over any infinite set is the product of finitely many Abelian subgroups. In fact, 289 subgroups suffice. Sharp bounds are also obtained on the minimal number $k$ , such that the finite symmetric group $S_n$ is the product of $k$ Abelian subgroups. Using this, $S_n$ is proved to be the product of $72n^{1/2}(\log n)^{3/2}$ cyclic subgroups.

Let (Pn) be a sequence of polynomials which converges with a geometric rate on some arc in the complex plane to an analytic function. It is shown that if the sequence has restricted growth on a closed plane set E which is non-thin at ∞, then the limit function has a maximal domain of existence, and (Pn) converges with a locally geometric rate on this domain. If (snk) is a sequence of partial sums of a power series, a similar growth restriction on E forces the power series to have Ostrowski gaps. Moreover, the requirement of non-thinness of E at ∞ is necessary for these conclusions.

In the paper, three lower bounds are given for the Morse index of a constant mean curvature torus in Euclidean $3$ -space, in terms of its spectral genus $g$ . The first two lower bounds grow linearly in $g$ and are stronger for smaller values of $g$ , while the third grows quadratically in $g$ but is weaker for smaller values of $g$ .

Several results are proved, related to an old problem posed by G. R. MacLane, namely whether functions in the class [Ascr ] that are locally univalent can have arc tracts. In particular, a proof is given of an assertion of MacLane that if f ∈ [Ascr ] is locally univalent and has no arc tracts, then f′ ∈ [Ascr ].

If the subset $E$ of ${\bb T}$ supports a synthesizable $p$ -pseudofunction $T$ , then the restriction $A_p(E)$ to $E$ of the Herz algebra on ${\bb T}$ is not Arens regular. The proof is direct, by brute force, and does not show the existence of Day points for $A_p(E)$ .

Some generalizations of the classical Hurewicz formula are obtained for the extension dimension and $C$ -spaces. A characterization is also given of the class of metrizable spaces that are absolute neighborhood extensors for all metrizable $C$ -spaces.

Let Pf(x) =−if′(x) and Qf(x) = xf(x) be the canonical operators acting on an appropriate common dense domain in L2(ℝ). The derivations DP(A) = i(PA−AP) and DQ(A) = i(QA−AQ) act on the *-algebra [Ascr ] of all integral operators having smooth kernels of compact support, for example, and one may consider the noncommutative ‘Laplacian’, L = D2P+D2Q, as a linear mapping of [Ascr ] into itself.

L generates a semigroup of normal completely positive linear maps on [Bscr ](L2(ℝ)), and this paper establishes some basic properties of this semigroup and its minimal dilation to an E0-semigroup. In particular, the author shows that its minimal dilation is pure and has no normal invariant states, and he discusses the significance of those facts for the interaction theory introduced in a previous paper.

There are similar results for the canonical commutation relations with n degrees of freedom, where 1 [les ] n < 1.

Hyperbolic invariant sets Λ of C1+γ diffeomorphisms where either the stable or unstable leaves are 1-dimensional are considered in this paper. Under the assumption that the Λ has local product structure, the authors prove that the holonomies between the 1-dimensional leaves are C1+α for some 0 < α < 1.

The main theorem of this paper generalizes a classic Aumann's characterization of compact convex sets, via the acyclicity of their hypersections, to arbitrary weakly closed subsets of a locally convex linear space.

Based on computing spherical lengths of polygonal curves and spherical areas of domains bounded by polygonal curves on the complex plane, this paper proves a property of holomorphic functions and, as an application, gives a very brief proof of a famous inequality obtained by L. V. Ahlfors which easily implies Picard's theorem. Taken together with the arguments given by Alhfors, therefore, this paper in fact gives an elementary, brief and geometrical proof of Picard's theorem.

Let Γ be a discrete group acting on a compact manifold X, let V be a Γ-equivalent Hermitian vector bundle over X, and let D be a first-order elliptic self-adjoint Γ-equivalent differential operator acting on sections of V. This data is used to define Toeplitz operators with symbols in the transformation group C*-algebra C(X)[rtimes ]Γ, and it is shown that if the symbol of such a Toeplitz operator is invertible, then the operator is Fredholm. In the case where Γ is finite and acts freely on X, a geometric-topological formula for the index is stated that involves an explicitly constructed differential form associated to the symbol.

An example is given of a random time $\rho$ associated with Brownian motion such that $\rho$ is not a stopping time but ${\bf E}M_{\rho}={\bf E}M_0$ for every uniformly integrable martingale $M$ .

The main purpose of this paper is to use the properties of Gaussian sums and the analytic method to study the problem of calculating one kind of character sums modulo $q$ , and to give an interesting identity.

In this paper, a growth result is obtained for normalized $k$ -fold symmetric convex biholomorphic mappings on the unit ball of a complex Banach space.