Let ϕ0 be an Anzai transformation on the 2-torus T2 defined by ϕ0(x,y) = (e2πiθx,xy) and ϕy a Furstenberg transformation on T2 defined by ϕf(x,y) = (e2πiθx,e2πif(x)xy) where θ is an irrational number and f is a real valued continuous function on the 1-torus T. In the present note we will show that ϕf has topologically quasi-discrete spectrum if and only if ϕf is topologically conjugate to ϕ0. Furthermore we will show that for any irrational number θ there is a real valued continuous function f on T such that ϕf does not have topologically quasi-discrete spectrum but is uniquely ergodic.

A Banach space has the complete continuity property (CCP) if each bounded linear operator from L1 into is completely continuous (i.e., maps weakly convergent sequences to norm convergent sequences). The main theorem shows that a Banach space failing the CCP has a subspace with a finite dimensional decomposition which fails the CCP. If furthermore the space has some nice local structure (such as fails cotype or is a lattice), then the decomposition may be strengthened to a basis.

Results are given for a class of square {0,1}-matrices which provide information about the 4-rank of the ideal class group of certain quadratic number fields.

This paper deals with the relative trace formula in the case of base change. Two truncations of the kernel are introduced, both based on the ideas of Arthur, and their integrals are shown to be asymptotic to each other. We also consider products of the kernel with automorphic forms, as these appear when comparing trace formulae (see [5]).

Let R be a ring and P(R) the sum of all periodic ideals of R. We prove that P(R) is the intersection of all prime ideals Pα such that contains no nontrivial periodic ideals. We also prove that P(R) = 0 if and only if Rs is a subdirect product of prime rings Rα with P(Rα) = 0.

In [1], Biggs stated that the Coxeter graph can be embedded in an orientable surface of genus 3. The purpose of this note is to point out that Biggs' embedding is in fact into a non-orientable surface. Further, it is shown that the orientable genus is 3 and the non-orientable genus is 6.

Let E be one of the spaces C(K) and L1, F be an arbitrary Banach space, p > 1, and (X, σ) be a space with a finite measure. We prove that E is isometric to a subspace of the Lebesgue-Bochner space LP(X; F) only if E is isometric to a subspace of F. Moreover, every isometry T from E into Lp(X; F) has the form Te(x) = h(x)U(x)e, e ∊ E, where h: X —> R is a measurable function and, for every x ∊ X, U(x) is an isometry from E to F

We prove that given an operator space structure on a dual Banach space Y*, it is not necessarily the dual one of some operator space structure on Y. This allows us to show that Sakai's theorem providing the identification between C*-algebras having a predual and von Neumann algebras does not extend to the category of operator spaces. We also include a related result about completely bounded operators from B(ℓ2)* into the operator Hilbert space OH.

A conjecture of Brown and Zassenhaus (see [2]) states that the first log/? primes generate a primitive root (mod p) for almost all primes p. As a consequence of a Theorem of Burgess and Elliott (see [3]) it is easy to see that the first log2p log log4+∊p primes generate a primitive root (mod p) for almost all primes p. We improve this showing that the first log2p/ log log p primes generate a primitive root (mod p) for almost all primes p.

We give a simple and explicit example of a complex Banach space which is not isomorphic to its complex conjugate, and hence of two real-isomorphic spaces which are not complex-isomorphic.

Our aim is to give an arithmetical expression of the class number formula of real quadratic fields. Starting from the classical Dirichlet class number formula, our proof goes along arithmetical lines not depending on any analytical method such as an estimate for

By using the principle of mathematical induction a simple algebraic formula is derived for an m-dimensional Schlömilch series. The result yields a countably infinite number of representations for null-functions on increasingly larger open intervals.

A linearly ordered set A is said to shuffle into another linearly ordered set B if there is an order preserving surjection A —> B such that the preimage of each member of a cofinite subset of B has an arbitrary pre-defined finite cardinality. We show that every countable linearly ordered set shuffles into itself. This leads to consequences on transformations of subsets of the real numbers by order preserving maps.

For the trace map on an irreducible semigroup of n × n matrices over a field, I. N. Herstein presented a theorem in [3] which enables us to limit the nature of matrix groups of a certain kind. However, this is incorrect in general. For the theorem, we shall present a counter example, a revision, and some generalizations to non-irreducible semigroups.

We describe two methods to combine sets of lines of different 2-dimensional projective planes into line sets of new 2-dimensional projective planes. Using these methods we describe several ways in which sets of circles of different 2-dimensional circle planes can be combined into circle sets of new 2-dimensional circle planes.

Let be a Lie algebra homomorphism from the Lie algebra of G to the Lie algebra of H in the following cases: (i) G and H are irreducible algebraic groups over an algebraically closed field of characteristic 0, or (ii) G and H are linear complex analytic groups. In this paper, we present some equivalent conditions for ϕ to be a differential in the above two cases. That is, ϕ is the differential of a morphism of algebraic groups or analytic groups as appropriate.

In the algebraic case, for example, it is shown that ϕ is a differential if and only if ϕ preserves nilpotency, semisimplicity, and integrality of elements. In the analytic case, ϕ is a differential if and only if ϕ maps every integral semisimple element of into an integral semisimple element of , where G0 and H0 are the universal algebraic subgroups of G and H. Via rational elements, we also present some equivalent conditions for ϕ to be a differential up to coverings of G in the algebraic case, and for ϕ to be a differential up to finite coverings of G in the analytic case.

We prove that every invertible operator in a properly infinite von Neumann algebra, in particular in L(H) for infinite dimensional H, is a product of 7 positive invertible operators. This improves a result of Wu, who proved that every invertible operator in L(H) is a product of 17 positive invertible operators.

For finite-dimensional subspaces of ℓ∞ over a non-archimedean valued base field K we study orthocomplementation as related to the Hahn-Banach property and strictness. As a corollary we obtain that, if K is not spherically complete, a closed hyperplane in c0 having the Hahn-Banach property is orthocomplemented (Theorem 2.1, Remark 2).

Let A be a commutative k-algebra with 1. We present a characterization of α-derivations, for α: A →> A a morphism of algebras, using α-Taylor series. When S = C[x,x-1,ξ] and α(x) = qx, α(ξ) = qξ, we compare the q-de Rham cohomology of the C-algebra S with the Hochschild homology of Dq, the algebra of q-difference operators on C[x,x-1], for q ∊ C, q ≠ 0,1.

The unit group of the modular group algebra of a finite p-group in characteristic p is nilpotent. The p-groups for which it is of nilpotency class 3 were determined in work of Coleman, Passman, Shalev and Mann when p ≥ 3. We resolve the p = 2 case here which completes the classification.