Volume 17, Issue 4

We define Hecke operators U m that sift out every m -th Taylor series coefficient of a rational function in one variable, defined over the reals. We prove several structure theorems concerning the eigenfunctions of these Hecke operators, including the pleasing fact that the point spectrum of the operator U m is simply the set {± m k | k ∈ ℕ} ∪ {0}. It turns out that the simultaneous eigenfunctions of all of the Hecke operators involve Dirichlet characters mod L , giving rise to the result that any arithmetic function of m that is completely multiplicative and also satisfies a linear recurrence must be a Dirichlet character times a power of m . We also define the notions of level and weight for rational eigenfunctions, by analogy with modular forms, and we show the existence of some interesting finite-dimensional subspaces of rational eigenfunctions (of fixed weight and level), whose union gives all of the rational functions whose coefficients are quasi-polynomials.

When Γ is a ring extension of R , conditions are found to insure that the notion of n -tilting module and n -cotilting module pass from a left R -module V to the induced left Γ -modules Tor i R ( Γ, V ) and Ext R i ( Γ, V ), 0 ≤ i ≤ n .

We consider the variational problem for curves in real projective plane defined by the projective arclength functional and discuss the integrability of its stationary curves in a geometric setting. We show how methods from the subject of exterior differential systems and the reduction procedure for Hamiltonian systems with symmetries lead to the integration by quadratures of the extrema. A scheme of integration is illustrated.

We construct a space of boundary values of the minimal symmetric singular Dirac operator acting in the Hilbert space L A 2 ([ a, b ); ℂ 2 ) (−∞ < a < b ≤ ∞), and in Weyl’s limit-circle case. A description of all maximal dissipative, maximal accretive, selfadjoint, and other extensions of such a symmetric Dirac operator is given in terms of boundary conditions. We investigate two classes of maximal dissipative operators with separated boundary conditions, called ‘dissipative at a ’ and ‘dissipative at b ’. In each of these cases we construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of the Titchmarsh-Weyl function of a selfadjoint operator. Finally, we prove the theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative Dirac operators.

General formulas giving numerical constraints for projective invariants of embedded, complex, projective manifolds are explicitly worked out in the framework of adjunction theory.

The purpose of this paper is to continue the study of the so-called holomorphic Q classes via two conformal deformations of the fractional image area of a holomorphic function on the open unit disk. Results include fractional order Carleson measures, embeddings of Choquet spaces with respect to Hausdorff capacities, preduals, strong isoperimetric inequalities, boundary values, harmonic extensions, Brownian motions, function modules, value distributions, and harmonic majorants of solutions to Euler-Lagrange differential equation subject to finite conformally invariant energy with fractional order.

In this paper we extend Gaschütz’ theorem to profinite groups (cf. Thm. A) and use this result to prove two theorems on the minimal number of generators of a pro(finite-soluble) group (cf. Thm. B, Thm. C). In section 3 we give a parameterization of the set of isomorphism classes of irreducible discrete ℱ p 〚 G 〛-modules, where G is either a finitely generated free pro(finite-abelian) group or a tame ramification group T q , q = p f (cf. Prop. 3.3, Prop. 3.6). This parameterization together with Theorem A and B allows us to calculate the probabilistic ζ -function for finitely generated free pro(finite-metabelian) groups and the absolute Galois groups of l -adic number fields explicitly (cf. Thm. D, Thm. E).

Let (Ω, ≤) be any doubly homogeneous chain and Aut(Ω) its group of order-automorphisms. We show that if J is a set of generators of Aut(Ω), then there is a positive integer n such that every element of Aut(Ω) is a product of at most n members of J  ∪ J −1 . Also, Aut(Ω) cannot be written as the union of a countable chain of proper subgroups of Aut(Ω).