Volume 19, Issue 1

We introduce a new method to solve abstract Cauchy problems in reflexive spaces even if no right half line is in the resolvent set of the corresponding operator and apply our result to PDE with constant coefficients.

We study the chain transitive sets and Morse decompositions of flows on fiber bundles whose fibers are compact homogeneous spaces of Lie groups. The emphasis is put on generalized flag manifolds of semi-simple (and reductive) Lie groups. In this case an algebraic description of the chain transitive sets is given. Our approach consists in shadowing the flow by semigroups of homeomorphisms to take advantage of the good properties of the semigroup actions on flag manifolds. The description of the chain components in the flag bundles generalizes a theorem of Selgrade for projective bundles with an independent proof.

Let g be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus, χ a primitive character of conductor q , and s a point on the critical line ℜ s = ½. It is proved that , where ε > 0 is arbitrary and θ = is the current known approximation towards the Ramanujan–Petersson conjecture (which would allow θ = 0); moreover, the dependence on s and all the parameters of g is polynomial. This result is an analog of Burgess' classical subconvex bound for Dirichlet L -functions. In Appendix 2 the above result is combined with a theorem of Waldspurger and the adelic calculations of Baruch–Mao to yield an improved uniform upper bound for the Fourier coefficients of holomorphic half-integral weight cusp forms.

For an action of a finitely generated group G on a compact space X we define recurrence at a point and then show that, when X is zero dimensional, the conditions (i) pointwise recurrence, (ii) X is a union of minimal sets and (iii) the orbit closure relation is closed in X × X , are equivalent. As a corollary we get that for such flows distality is the same as equicontinuity. In the last part of the paper we describe an example of a ℤ-flow where all points are positively recurrent, but there are points which are not negatively recurrent.

This work is devoted to the study of the elliptic equation Δ u = f ( x, u ) in an exterior non-smooth domain. Applying the method of upper and lower solutions and a diagonal argument, we prove the existence of solutions under various boundary conditions.

In this paper, all spaces are localized at the prime two. Let T (1) be the Ravenel spectrum characterized by the BP ∗ -homology as BP ∗ [ t 1 ], T (1)/( v 1 ) be the cofiber of the self map v 1 : Σ 2 T (1) → T (1) and L 2 denote the Bousfield localization functor with respect to v 2 −1 BP ∗ . In this paper, we compute the homotopy groups π ∗ ( L 2 T (1)/( v 1 )) by determining the E ∞ -term of its Adams-Novikov spectral sequence (ANSS). From the E 2 -term of the ANSS for π ∗ ( L 2 T (1)/( v 1 )), we determine a subgroup of the E 2 -term for π ∗ ( L 2 T (1)). We also show that the E 4 -term for π ∗ ( L 2 T (1)) has horizontal vanishing line.

Let D denote the open unit disc and f : D → be meromorphic and injective in D . We further assume that f has a simple pole at the point p ∈ (0, 1) and an expansion . In particular, we consider functions f that map D onto a domain whose complement with respect to is convex. Because of the shape of f ( D ) these functions will be called concave univalent functions with pole p and the family of these functions is denoted by C o ( p ). It is proved that for fixed p ∈ (0, 1) the domain of variability of the coefficient a n ( f ), n ⩾ 2, f ∈ C o ( p ), is determined by the inequality . This settles two conjectures published by A. E. Livingston in 1994 and by Ch. Pommerenke and the authors of the present article in 2004.

Given uniform probability on words of length M = Np + k , from an alphabet of size p , consider the probability that a word (i) contains a subsequence of letters p , p − 1, …, 1 in that order and (ii) that the maximal length of the disjoint union of p − 1 increasing subsequences of the word is ⩽ M − N . A generating function for this probability has the form of an integral over the Grassmannian of p -planes in ℂ n . The present paper shows that the asymptotics of this probability, when N → ∞, is related to the k th moment of the χ 2 -distribution of parameter 2 p 2 . This is related to the behavior of the integral over the Grassmannian Gr( p , ℂ n ) of p -planes in ℂ n , when the dimension of the ambient space ℂ n becomes very large. A dierent scaling limit for the Poissonized probability is related to a new matrix integral, itself a solution of the Painlevé IV equation. This is part of a more general set-up related to the Painlevé V equation.