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In the paper we introduce a new method how to use only an orthonormality relation of coefficients of Dirichlet series defining given L-functions from the Selberg class to prove joint universality.

Singular hyperbolicity is a weakened form of hyperbolicity that has been introduced for vector fields in order to allow non-isolated singularities inside the non-wandering set. A typical example of a singular hyperbolic set is the Lorenz attractor. However, in contrast to uniform hyperbolicity, singular hyperbolicity does not immediately imply robust topological properties, such as the transitivity. In this paper, we prove that on an open and dense subset of the space of $$C^1$$ vector fields of a compact manifold, any singular hyperbolic attractors is robustly transitive.

In this paper, we investigate the essential approximate point spectrum and the essential defect spectrum of a 2 × 2 block operator matrix on a Banach space. Furthermore, we apply the obtained results to two-group transport operators in the Banach space L p ([−a, a] × [−1, 1]) × L p ([−a, a] × [−1, 1]), a > 0, p ≥ 1.

Peter Schmid Mathematisches Institut der Universit~it, Auf der Morgenstelle 10, D-7400-Tiibingen1, Federal Republic of Germany To Helmut Wielandt, on his seventieth birthday, 19 December, 1980 Let A be a Q-module where A and Q are finite p-groups. By a theorem of Gaschtitz and Uchida A is cohomologically trivial provided the (Tare) cohomology H"(Q, A)=0 for just one integer n (cf. [-1], p. 110). This has been used in order to produce noninner p-automorphisms for p-groups. In fact, when N is a normal subgroup of a finite p-group G and Q = G/N, it often happens that the Q-mod- ule A =Z(N), the centre of N, has nontrivial cohomology. The object of this paper is to record the following result. Theorem. Let G be a regular p-group and N a nontrivial normal subgroup of G. If Q = G/N is not cyclic, H"(Q, Z(N)) =t= 0 for all n. If A is a cyclic p-group, p an odd prime, and Q a p-subgroup of Aut(A), the semidirect product G=Q. A is regular but Hn(Q,A)=O. Thus the hypothesis in the theorem cannot be omitted. ' One might ask when extensions of regular p-groups are aga!n regular. Some necessary condition may be taken from Proposition 2 below.' Note, however, that even direct products of regular p-groups need not be regular; see Wielandt's example in [-2], p. 323. 1. Cohomological Triviality and Fixed Points We begin with some basic observations concerning cohomologically trivial mod- ules for p-groups. If A is a (right) Q-module, A o denotes the submodule of fixed points under Q and [A,Q] the commutator submodule. The trace map a ~-+ a ~, x of A is written z = zQ, its image A t and its kernel A,. In dealing with

For applications of angle-bounded mappings and operators satisfying condition (,) to nonlinear equations of Hammerstein type in Banach spaces we refer to [1-3]. It is clear by definition that the adjoint of an angle-bounded mapping is again angle-bounded. The purpose of this paper is to show that the class of linear operators satisfying condition (,) is invariant under passage to the adjoint operator.

In this paper, we use the Approximation Formula for the Fourier transform of the solution set of lattice points on k-spheres and methods of Bourgain and Ionescu to refine the \(\ell ^p(\mathbb {Z}^d)\)-boundedness results for discrete k-spherical maximal functions to a restricted weak-type result at the endpoint. We introduce a density-parameter, which may be viewed as a discrete version of Minkowski dimension used in related works on the continuous analgoue, in order to exploit recent progress of Wooley and Bourgain–Demeter–Guth on the Vinogradov mean value conjectures via a novel Approximation Formula for a single average, and obtain improved bounds for lacunary discrete k-spherical maximal functions when \(k \ge 3\).