Sums of the form where f(n) is a multiplicative arithmetical function and denotes summation over those values of n for which f(n)>0 and f(n) ≠1, were studied by De Koninck [2], De Koninck and Galambos [3], Brinitzer [1] and Ivič [5]. The aim of this note is to give an asymptotic formula for a certain class of multiplicative, positive, primeindependent functions (an arithmetical function is prime-independent if f(pv) = g(v) for all primes p and v = 1, 2, …). This class of functions includes, among others, the functions a(n) and τ(e)(n), which represent the number of nonisomorphic abelian groups of order n and the number of exponential divisors of n respectively, and none of the estimates of the above-mentioned papers may be applied to this class of functions. We prove the following.

Given a minimal action $\alpha $ of a countable group on the Cantor set, we show that the alternating full group $\mathsf {A}(\alpha )$ is non-amenable if and only if the topological full group $\mathsf {F}(\alpha )$ is $C^*$ -simple. This implies, for instance, that the Elek–Monod example of non-amenable topological full group coming from a Cantor minimal $\mathbb {Z}^2$ -system is $C^*$ -simple.

Let D be a PID with infinitely many maximal ideals. J. W. Brewer has asked whether some simple ring extension D[α] of D must have nontrivial Picard group. We show that this question has a negative answer.

We study the boundedness and compactness of weighted composition operators acting on weighted Bergman spaces and weighted Dirichlet spaces by using the corresponding Carleson measures. We give an estimate for the norm and the essential norm of weighted composition operators between weighted Bergman spaces as well as the composition operators between weighted Hilbert spaces.

Using the method of non-well-founded generic ultra-powers, we shall prove a generalization of a theorem of Taylor that certain values of completeness and saturatedness of a uniform ideal rule out certain sizes of the underlying index set.

An upper bound is given for the order of the kernel of the map on Sideal class groups that is induced by For some special types of number fields F the connection between the size of the above kernel for and the units and norms in are examined. Let K2(O) denote the Milnor K-group of the ring of integers of a number field. In some cases a formula by Conner, Hurrelbrink and Kolster is extended to show how closely the 4-rank of is related to the 4-rank of the S-ideal class group of

The problem to be considered in this note, in its most concrete form, is the determination of all quartets f1, f2, g1, g2 of functions analytic on some domain and satisfying

*

where p > 0. When p = 2 the question can be reformulated in terms of finding a necessary and sufficient condition for (two-dimensional) Hilbert space valued analytic functions to have equal pointwise norms, and the answer (Theorem 1) justifies this point of view. If p ≠ 2, the problem is solved by reducing to the case p = 2, and the reformulation in terms of the norm equality of lp valued analytic functions gives no clue to the answer.

The purpose of this note is to obtain an improved lower bound for the multidimensional Bohr radius introduced by L. Aizenberg (2000, Proceedings of the American Mathematical Society 128, 1147–1155), by means of a rather simple argument.

Kaplansky's Lemma [3] states: the number of k-combinations of 1, 2,…, n with no two consecutive integers in any selection is Using this, Abramson [l; lemma 3] solves the problem: find the, number of k-combinations so that no two integers i and i+2 appear in any selection. (This is generalized by Abramson in [2].) An interesting solution, also using Kaplansky's lemma, is obtained as follows.

If n = 2m, we choose s from among the m even integer s, no two consecutive, and k- s from among the m odd integers, no two consecutive.

The space wp of strongly Cesàro summable sequences of index p > 0 has been investigated by several authors. In [2], Kuttner proved that no Toeplitz matrix could sum all sequences in wP, a result which was extended to coregular matrices by Maddox [5]. In [1], Borwein considered the continuous dual space of wp. The more general space w(p) has also been considered [3, 4], where p = (pk) is a strictly positive sequence. The r-convexity of the spaces w∞(p) and w0(p) was dealt with in a partial way in [8]. In the present note we establish criteria for the rconvexity of some general classes of [A, p]0 and [A, p]∞ spaces (see [6] and [7] for definitions), and in particular we give the necessary and sufficient conditions for the r-convexity of w∞(p) and w0(p). For most of the relevant definitions and notation we refer to [8].

Soit M une variété réelle de classe C∞ (connexe, paracompacte) et ω une un-forme complètement intégrable sur M (ω ∧ dω = 0). On admet qu’en certains points, dits singuliers, la un-forme peut s’annuler. Désignons par Jω(A) la matrice jacobienne de ω en un tel point A. Un point singulier A de ω est non dégénéré si le rang de Jω(A) est maximum. II est clair que ces singularités sont toujours isolées dans M.

Nous proposons de construire des polynômes à coefficients p-adiques, de degré D, sans terme constant, ayant plus que 2D + 1 variables. Ces polynômes n'ont dans Zp que le zéro trivial modulo p2. Le but que nous cherchons à atteindre, à partir des polynômes ainsi construit, est de minorer les valeurs de p pour lesquelles l'équation F(x1,...,X2D+1) ≡ 0 (mod p2) admet une solution primitive, F est un élément de Zp[X1,... ,X2D+1] de degré D sans term constant.

With the aid of a recent result obtained by the first author, an expression is derived which unifies the well-known Jensen and Gould formulas.

Suppose that $\gamma \in {{C}^{2}}\left( \left[ 0,\infty \right]) \right)$ is a real-valued function such that $\gamma \left( 0 \right)\,=\,{\gamma }'\left( 0 \right)\,=\,0$, and ${\gamma }''\left( t \right)\,\approx \,{{t}^{m-2}}$, for some integer $m\,\ge \,2$. Let $\Gamma \left( t \right)\,=\,\left( t,\,\gamma \left( t \right) \right),\,t\,>\,0$, be a curve in the plane, and let $d\text{ }\!\!\lambda\!\!\text{ }\,\text{=}\,dt$ be a measure on this curve. For a function $f$ on ${{\mathbf{R}}^{2}}$, let

$$Tf\left( x \right)\,=\,\left( \text{ }\lambda \text{ }*f \right)\left( x \right)=\int_{0}^{\infty }{f\left( x-\Gamma \left( t \right) \right)dt,\,\,x\in {{\mathbf{R}}^{2}}}.$$

An elementary proof is given for the optimal ${{L}^{p}}-{{L}^{q}}$ mapping properties of $T$.

Mr. President, ladies and gentlemen, I have had great pleasure in accepting the invitation to deliver this tenth Jeffery-Williams lecture. As one of that generation who were strongly influenced by Ralph Jeffery and Lloyd Williams, I also appreciate the challenge of maintaining the high standard that these lectures named in their honour have established. May I also say that the lecture earlier this morning by Professor Dieudonnè was certainly fortunate for us.

In this paper, we investigate the holomorphic sections of holomorphic Finsler bundles over both compact and non-compact complete complex manifolds. We also inquire into the holomorphic vector fields on compact and non-compact complete complex Finsler manifolds. We get vanishing theorems in each case according to different certain curvature conditions. This work can be considered as generalizations of the classical results on Kähler manifolds and hermitian bundles.

In this paper we prove the following result: let R be a prime ring with no non-zero nil left ideals whose characteristic is different from 2 and let U be a non central Lie ideal of R.

If d ≠ 0 is a derivation of R such that d(u) is invertible or nilpotent for all u ∈ U, then either R is a division ring or R is the 2 X 2 matrices over a division ring. Moreover in the last case if the division ring is non commutative, then d is an inner derivation of R.

The consideration of tensor products of 0-Hecke algebramodules leads to natural analogs of the Bessel $J$-functions in the algebra of noncommutative symmetric functions. This provides a simple explanation of various combinatorial properties of Bessel functions.

A new tautological relation of ${{\overline{M}}_{3,\,1}}$ in codimension 3 is derived and proved, using an invariance constraint from our previous work.

Ziegler’s Indecomposability Criterion is used to prove that a totally transcendental, i.e., $\sum $-pure injective, indecomposable left module over a left noetherian ring is a directed union of finitely generated indecomposable modules. The same criterion is also used to give a sufficient condition for a pure injective indecomposable module $_{R}U$ to have an indecomposable local dual $U_{R}^{\#}.$