We prove the existence of a power series having radius of convergence 0, whose partial sums have universal approximation properties on any compact set with connected complement that is contained in a finite union of circles centred at 0 and having rational radii, but do not have such properties on any compact set with nonempty interior. This relates to a theorem of A. I. Seleznev.

The author of this paper has shown previously how a complex representation of a finite group can be split into a virtual sum of representations induced from one-dimensional representations of subgroups in a natural way (sometimes known as explicit Brauer induction). Here the modular case is treated, yielding an analogous result at the level of Brauer characters in general, and in the Green ring for trivial source modules.

Regenerative phenomena were introduced some forty years ago to address problems in the theory of continuous-time Markov processes. The early work in the theory left a number of difficult unsolved problems, in the classification of $p$-functions, oscillation and inequalities, the multiplicative theory, and the theory of unbounded semi-$p$-functions. Recent years have shown progress on all of these fronts, and this paper surveys these results, while drawing attention to significant problems that remain open.

In this paper, we apply Moriwaki's arithmetic height functions to obtain an analogue of Silverman's specialization theorem for families of Abelian varieties over $K$, where $K$ is any field finitely generated over $\mathbb{Q}$.

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.

Given a first order elliptic partial differential equation we construct a solution which solves a given Riemann–Hilbert boundary value problem whose coefficients have singularities of the first kind at a finite number of some prescribed isolated points and are Holder–continuous outside those points while the free term has a finite number of integrable power singularities at some prescribed points. It is shown that the solution belongs to some weighted Sobolev space $W_{1, p}(D; \rho)$, where the weight function $\rho = \rho(z; \partial D)$ is the distance of the variable point $z$ from the boundary $\partial D$ raised to a certain power. The problem is solved by first reducing it to an analogous problem for holomorphic functions. The latter is then solved.

An example is provided of a space that does not have the homotopy type of an H-space, and yet has the same $n$ -type as a double loop space for each natural number $n$ . It is also locally homotopy equivalent to a double loop space at each prime $p$ .

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 $A_p(E)$ .

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.

In this paper, we give an elementary proof of a curious identity of elliptic functions. It is very similar to a beautiful proof given by Coates of a different identity. The result was strongly motivated by Wildeshaus' generalisation of Zagier's conjecture.

The notion of an asymptotic cycle is extended to flows on non-compact spaces, and it is proved that for a suitable class of flows on manifolds it is true that the set of points with asymptotic cycles defined has measure one with respect with every invariant probability measure.

v. paulsen and d. singh gave a characterization of certain hilbert spaces contractively contained in $l^p$, $p \geq 2$. here, the analog of their theorem is proved for $1 \leq p < 2$.

A compact subset of the unit circle is an interpolation set for conformal mappings if and only if it has logarithmic capacity zero.

The local Caratheodory conjecture on the index of an isolated singularity of the principal foliations in surface theory is equivalent to a conjecture of Loewner on the index of the isolated singularities of the Hessian operator of a smooth function in the plane. Here we prove the latter conjecture for a special class of functions.

It is shown that, for λ > ω, λ-orthogonality classes in locally λ-presentable categories coincide with full subcategories that are closed under limits and under λ-directed colimits. This comes as a surprise, since the statement is known to be false for λ = ω.

It is proved that a ring (≠0) is projective-free precisely when it has invariant basis number and every idempotent matrix not equal to I is non-full.

The norm of a group G is the subgroup of elements of G which normalise every subgroup of G. We shall denote it κ(G). An ascending series of subgroups κi(G) in G may be defined recursively by: κ0(G) = 1 and, for i[ges ]0, κi+1(G)/κi(G) = κ(G/κi(G)). For each i, the section κi+1(G)/κi(G) clearly contains the centre of the group G/κi(G). A result of Schenkman [8] gives a very close connection between this norm series and the upper central series: ζi(G)⊆κi(G) ⊆ζ2i(G).

An upper bound on operator norms of compound matrices is presented, and special cases that involve the l1, l2 and l∞ norms are investigated. The results are then used to obtain bounds on products of the largest or smallest eigenvalues of a matrix.

A classical theorem of Valiron states that a function which is holomorphic in the unit disk, unbounded, and bounded on a spiral that accumulates at all points of the unit circle, has asymptotic value $\infty$. This property, and various other properties of such functions, are shown to hold for more general classes of functions which are bounded on a subset of the disk that has a suitably large set of nontangential limit points on the unit circle.