Dans un récent article [2, p. 379], R. M. Robinson a émis une conjecture intéressante sur le diamètre transfini des ensembles compacts et symétriques du plan. La conjecture est la suivante:

Soit E un ensemble compact de points du plan, symétrique par rapport á l'axe réel R. Pour n ≥ 2, choisissons n points z1, …, zn de E et posons xi = zi,   pour i=1, 2,…, n.

Soient:

Let ${\mathcal{A}}$ be a star-shaped polygon in the plane, with rational vertices, containing the origin. The number of primitive lattice points in the dilate $t{\mathcal{A}}$ is asymptotically $\frac{6}{\unicode[STIX]{x1D70B}^{2}}\text{Area}(t{\mathcal{A}})$ as $t\rightarrow \infty$. We show that the error term is both $\unicode[STIX]{x1D6FA}_{\pm }(t\sqrt{\log \log t})$ and $O(t(\log t)^{2/3}(\log \log t)^{4/3})$. Both bounds extend (to the above class of polygons) known results for the isosceles right triangle, which appear in the literature as bounds for the error term in the summatory function for Euler’s $\unicode[STIX]{x1D719}(n)$.

If $A$ is a set of $M$ positive integers, let $G(A)$ be the maximum of ${{a}_{i}}/\,\gcd ({{a}_{i}},\,\,{{a}_{j}}\,)$ over ${{a}_{i}},\,{{a}_{j}}\,\,\in \,\,A.$ We show that if $G(A)$ is not too much larger than $M$, then $A$ must have a special structure.

In [2: p. 415], P. Gabriel proves that if R is a ring with 1 and S is a non-empty multiplicative set such that 0∉S, then S-1R exists if and only if for every pair (a, s)∈R×S, there is a pair (b, t)∈R×S such that at=sb and if s1a=0 for some s1 in S then as2=0 for some s2 in S. The purpose of this note is to give a self contained elementary proof of Gabriel’s result.

Statistics in the 20th century has been enlivened by a passionate, occasionally bitter, and still vibrant debate on the foundations of statistics and in particular on Bayesian vs. frequentist approaches to inference. In 1975 D. V. Lindley predicted a Bayesian 21st century for statistics. This prediction has often been discussed since, but there is still no consensus on the probability of its correctness. Recent developments in the asymptotic theory of statistics are, surprisingly, shedding new light on this debate, and may have the potential to provide a common middle ground.

In this paper, we consider the behavior of the commutators of convolution operators on the Triebel–Lizorkin spaces ${{\dot{F}}_{p}}^{s,q}$.

This note concerns elements (called trivectors) of the third Grassmann product of a complex vector space U. Usually there are many ways to write a given trivector as the sum of simple or decomposable trivectors, and it is an interesting problem to find those representations which contain the smallest possible number of decomposables. This number we shall call the length of the trivector. Let N(n) denote the length of the longest trivector in ∧3U where U has dimension n. In this note we give upper bounds for N(n) when n ≤ 8.

We classify real hypersurfaces in complex projective space whose structure Jacobi operator satisfies two conditions at the same time.

In this paper we prove the sharp inequality

$$\left| P_{n}^{\left( s \right)}\left( x \right) \right|\le P_{n}^{\left( s \right)}\left( 1 \right)\left( {{\left| x \right|}^{n}}+\frac{n-1}{2s+1}\left( 1-{{\left| x \right|}^{n}} \right) \right)$$

where $P_{n}^{\left( s \right)}\left( x \right)$ is the classical ultraspherical polynomial of degree $n$ and order $s\ge n\frac{1+\sqrt{5}}{4}$. This inequality can be refined in $\left[ 0,z_{n}^{s} \right]$ and $\left[ z_{n}^{s},1 \right]$, where $z_{n}^{s}$ denotes the largest zero of $P_{n}^{\left( s \right)}\left( x \right)$.

Let

1

be the simple continued fraction (SCF) of an irrational number x. The partial quotients ai which we shall sometimes refer to as the "terms" of the SCF are integers and, for i ≥ 2, they are positive. If x is a quadratic irrationality then, by Lagrange's Theorem, the right side of (1) becomes periodic from some point on.

Let f be an additive arithmetical function having a distribution F. For any sequence let

In this note, we determine the slowest growing function b so that Qn{b, f) tends weakly to F, for various f.

La portée et les buts de cet article sont signalés au début des différents paragraphes et dans les remarques qui suivent le théorème 3, le lemme 5 et le théorème 21. Limitons-nous à mentionner ici que le sujet traité est à la fois en relation avec la théorie des systèmes d'inéquations linéaires (théorie des polyèdres coniques), avec le concept de point Λ-extrême, avec les diverses notions d'efficacité par rapport à un système d'objectifs linéaires et avec la programmation à objectifs linéaires multiples.

The article introduces and studies Hausdorff–Berezin operators on the unit ball in a complex space. These operators are a natural generalization of the Berezin transform. In addition, the class of such operators contains, for example, the invariant Green potential, and some other operators of complex analysis. Sufficient and necessary conditions for boundedness in the space of p – integrable functions with Haar measure (invariant with respect to involutive automorphisms of the unit ball) are given. We also provide results on compactness of Hausdorff–Berezin operators in Lebesgue spaces on the unit ball. Such operators have previously been introduced and studied in the context of the unit disc in the complex plane. Present work is a natural continuation of these studies.

We generalize a result of E. E. Floyd on unicoherent Peano continua with involution to unicoherent locally connected regular hereditarily Lindelöf spaces. The result has an application in the theory of connectivity functions.

Let Zm denote the ring of integers modulo an integer m > 1 and let G(Zm) be the group under composition of all Zm-automorphisms of Zm[X]. In this paper we determine Zm[X]G(zm) the subring of Zm[X] left fixed by elements of G(Zm).

We characterize two important notions of amenability and compactness of a locally compact quantum group $\mathbb{G}$ in terms of certain homological properties. For this, we show that $\mathbb{G}$ is character amenable if and only if it is both amenable and co-amenable. We finally apply our results to Arens regularity problems of the quantum group algebra ${{L}^{1}}\left( \mathbb{G} \right)$. In particular, we improve an interesting result by Hu, Neufang, and Ruan.

A famous theorem of Krull's is that the integral closure of an integral domain D is the intersection of the valuation domains that contain D. An example is given to show that the same result need not hold for the integral closure of a ring with zero divisors.

In this paper we consider how much we can say about an irreducible symmetric space M which admits a hypersurface N with at most two distinct principal curvatures. Then we will obtain that (1) if N is locally symmetric, then M must be a sphere, a real projective space and their noncompact duals (2) if N is Einstein, then M must be rank 1.

It is shown that the ergodic Hilbert transform exists for a class of bounded symmetric admissible processes relative to invertible measure preserving transformations. This generalizes the well-known result on the existence of the ergodic Hilbert transform.