Volume 17, Issue 6

Let M  be a connected compact Kähler manifold equipped with an antiholomorphic involution τ . Let G be a complex reductive group; fix a real structure on G . We consider holomorphic principal G  -bundles over M equipped with a lift of τ   as an antiholomorphic involution of the total space of E G  . We extend the notion of polystability to such bundles with involution and prove that polystability is equivalent to the existence of an Einstein-Hermitian connection compatible with the involution. We also give a criterion for such a bundle over a compact Riemann surface to have a holomorphic connection compatible with the involution.

We estimate exponential sums with the function ρ  ( n  ) defined as the average of the prime divisors of an integer n  ≥ 2 (we also put ρ  (1) = 0). Our bound implies that the fractional parts of the numbers {  ρ  ( n  ) :  n  ≥ 1 } are uniformly distributed over the unit interval. We also estimate the discrepancy of the distribution, and we determine the precise order of the counting function of the set of those positive integers n   such that ρ  ( n  ) is an integer.

We study the horizontal Laplacian Δ H associated to the Hopf fibration S 3  → S 2 with arbitrary Chern number k . We use representation theory to calculate the spectrum, describe the heat kernel and obtain the complete heat trace asymptotics of Δ H . We express the Green functions for associated Poisson semigroups and obtain bounds for their contraction properties and Sobolev inequalities for Δ H . The bounds and inequalities improve as | k  | increases.

An inclusion of a Moufang polygon into another is called algebraic if the algebraic structures which describe them can be chosen in such a way that the one is a substructure of the other. We show that an inclusion of Moufang n  -gons is always algebraic if n  ∈ {3, 6, 8}, but that this is not always true when n  = 4. We classify the algebraic inclusions of Moufang quadrangles in the case where none of the root groups is 2-torsion, which corresponds to the fact that the characteristic of the underlying (skew) field is different from 2. Finally, we show that all full and ideal inclusions of Moufang quadrangles without 2-torsion root groups are algebraic.

In the two parts of this paper we prove that the Reidemeister torsion invariants determine topological equivalence of G  -representations, for G   a finite cyclic group.

We investigate Dirichlet forms in the case that an infinite extension of a local field is given as a state space. Actually, we see the theory of Dirichlet space is applicable to construct Markov processes on the infinite extension. We study non-linear capacities arising from an extensive research into potential theory based on the associated Markov process.

The configuration spaces of arachnoid mechanisms are analyzed in this paper. These mechanisms consist of k   branches each of which has an arbitrary number of links and a fixed initial point, while all branches end at one common end-point. It is shown that generically, the configuration spaces of such mechanisms are manifolds, and the conditions for the exceptional cases are determined. The configuration space of planar arachnoid mechanisms having k   branches, each with two links is analyzed for both the non-singular and the singular cases.