We prove here that the p - 1 first derivatives of the fundamental period of the Carliz module are algebraically independent. For that purpose we will show to use Mahler's method in this situtaion.

Extending previous results of the first author, some new estimates are obtained for maximal operators of Schrödinger type with a complex parameter.

We prove that there is a correspondence between Ramanujan-type formulas for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1/\pi $ and formulas for Dirichlet $L$-values. Our method also allows us to reduce certain values of the Epstein zeta function to rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker.

An equidistant permutation array (EPA) is a ν × r array defined on an r-set, R, such that (i) each row is a permutation of the elements of R and (ii) any two distinct rows agree in λ positions (that is, the Hamming distance is (r−λ)).

Such an array is said to have order ν. In this paper we give several recursive constructions for EPA's.

The first construction uses a resolvable regular pairwise balanced design of order v to construct an EPA of order ν. The second construction is a generalization of the direct product construction for Room squares.

We also give a construction for intersection permutation arrays, which arrays are a generalization of EPA's.

This paper discusses the relationship between two equivalence relations on the class of finite nilpotent groups. Two finite groups are conformal if they have the same number of elements of all orders. (Notation: G ≈ H.) This relation is discussed in [4] pp 107–109 where it is shown that conformality does not necessarily imply isomorphism, even if one of the groups is abelian. However, if both groups are abelian the position is much simpler. Finite conformal abelian groups are isomorphic.

We investigate the number of symmetric matrices of nonnegative integers with zero diagonal such that each row sum is the same. Equivalently, these are zero-diagonal symmetric contingency tables with uniform margins, or loop-free regular multigraphs. We determine the asymptotic value of this number as the size of the matrix tends to infinity, provided the row sum is large enough. We conjecture that one form of our answer is valid for all row sums. An example appears in Figure 1.

In a recent issue of this Journal, Pu [3] has given an interesting construction of a nonmeasurable subset A of R such that for all intervals I in R. [Throughout this note, the symbol λ denotes Lebesgue outer measure on R or Haar outer measure on a general locally compact group.] This solves a problem stated in [2], p. 295, Exercise (18.30).

Let k be an infinite cardinal, F a field, and let GL(k, F) be the group of all non-singular linear transformations on a ki-dimensional vector space V over F. Various examples are given of maximal subgroups of GL(k, F). These include (i) stabilizers of families of subspaces of V which are like filters or ideals on a set, (ii) almost stabilizers of certain subspaces of V, (iii) almost stabilizers of a direct decomposition of V into two k-dimensional subspaces.

It is also noted that GL(k, F) is not the union of any chain of length k of proper subgroups.

We exhibit a canonical geometric pairing of the simple closed curves of the degree three cover of the modular surface, Γ3\ℋ, with the proper single self-intersecting geodesics of Crisp and Moran. This leads to a pairing of fundamental domains for Γ3 with Markoff triples.

The routes of the simple closed geodesics are directly related to the above. We give two parametrizations of these. Combining with work of Cohn, we achieve a listing of all simple closed geodesics of length within any bounded interval. Our method is direct, avoiding the determination of geodesic lengths below the chosen lower bound.

John W. Moon has discovered several computational errors in our article above (J. Austral. Math. Soc. (Series A) 20 (1975), 483–503). The five constants reported for identity trees below Table 1 at the bottom of page 502 are all wrong. The correct values are

We shall introduce the notions of strong Morita equivalence for unital inclusions of unital $C^{\ast }$-algebras and conditional expectations from an equivalence bimodule onto its closed subspace with respect to conditional expectations from unital $C^{\ast }$-algebras onto their unital $C^{\ast }$-subalgebras. Also, we shall study their basic properties.

The spherical functions of triangle buildings can be described in terms of certain two-dimensional orthogonal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynimial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups. We present strong laws of large numbers, a central limit theorem, and a local limit theorem; all these results are well-known for homogeneous trees. Proofs are based on moment functions on hypergroups and on explicit expansions of the hypergroup characters in terms of certain two-dimensional Tchebychev polynimials.

The aim of this note is to evaluate an integral involving the product of two H-functions.

Let Β1, Β2 be a pair of Banach spaces and T be a vector valued martingale transform (with respect to general filtration) which maps Β1-valued martingales into Β2-valued martingales. Then, the following statements are equivalent: T is bounded from into for some p (or equivalently for every p) in the range 1 < p < ∞; T is bounded from into BMOB2; T is bounded from BMOB1 into BMOB2; T is bounded from into . Applications to UMD and martingale cotype properties are given. We also prove that the Hardy space defined in the case of a general filtration has nice dense sets and nice atomic decompositions if and only if Β has the Radon-Nikodým property.

The finite Fourier transform of a family of orthogonal polynomials is the usual transform of these polynomials extended by $0$ outside their natural domain of orthogonality. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families.

In an earlier paper we showed that the set ψ+ (X, Y) of super Tauberian transformations between two Banach spaces X and Y forms an open subset of B(X, Y) which is closed under perturbation by super weakly compact transformations. In this note we characterize a class dual to ψ+ (X, Y) which we denote by ψ-(X, Y). We show that T∈ψ+(X, Y) if and only if T′ ∈ ψ-(Y′, X′) and that T′∈ψ+(Y′, X′) if and only if T ∈ ψ-(X, Y) and provide standard and nonstandard characterizations of elements of ψ-(X, Y). These two classes thus play in some ways analogous roles to the sets of semi-Fredholm transforms ϕ+ (X, Y) and ϕ-(X, Y).

Moreover en forms an open subset of B(X, Y) closed under the taking of adjoints, under the taking of nonstandard hull extensions, and under perturbation by super weakly compact transformations.

Quasi-dual-continuous modules, which generalize the concept of dual-continuous modules, are studied Mohamed, Müller and Singh had obtained some decomposition theorems and their partial converses for dual-continuous modules. It is shown that these results can be extended to quasi-dual-continuous modules. Further, a short proof of a decomposition theorem for quasi-dual-continuous modules established recently by Oshiro is given. Some more structure theorems for such modules are established. Finally, quasi-dual-continuous covers are studied, and duals for results of Müller and Rizvi are derived.

We solve the following problem which was posed by Barnes in 1962. For which abelian groups G and H of the same prime power order is it possible to embed the subgroup lattice of G in that of H? It follows from Barnes' results and a theorem of Herrmann and Huhn that if there exists such an embedding and G contains three independent elements of order p2, then G and H are isomorphic. This reduces the problem to the case that G is the direct product of cyclic p-groups only two of which have order larger than p. We determine all groups H for which the desired embedding exists.

In this paper we follow the approach of Bertrand–Beukers (and of Bertrand’s later work), based on differential Galois theory, to prove a very general version of Shidlovsky’s lemma that applies to Padé-approximation problems at several points, both at functional and numerical levels (that is, before and after evaluating at a specific point). This allows us to obtain a new proof of the Ball–Rivoal theorem on irrationality of infinitely many values of the Riemann zeta function at odd integers, inspired by the proof of the Siegel–Shidlovsky theorem on values of $E$-functions: Shidlovsky’s lemma is used to replace Nesterenko’s linear independence criterion with Siegel’s, so that no lower bound is needed on the linear forms in zeta values. The same strategy provides a new proof, and a refinement, of Nishimoto’s theorem on values of $L$-functions of Dirichlet characters.

For a point set in the multidimensional unit torus we introduce an Lk-measure of uniformity of distribution, which for k=2 reduces to diaphony (and thus in this case essentially coincides with Weyl L2-discrepancy). For k ∈ [1, 2] we establish a sharp asymptotic for this new measure as the number of points of the set tends to infinity. Upper and lower-bound estimates are given also for k >2.