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:

We introduce a new type of differentiability, called cofinite Fréchet differentiability. We show that the convex point-of-continuity property of Banach spaces is dual to the cofinite Fréchet differentiability of all equivalent norms. A corresponding result for dual spaces with the weak* convex point-of-continuity property is also established.

It is well known that if A is an n × n complex matrix and AT is its transpose, then there is an invertible n x n complex matrix S such that AT = S-1AS. In this note we wish to point out another simple relation between A and AT.

If A is an n × n complex matrix and AT is its transpose then there are unitary n × n complex matrices U and V such that AT = UAV.

In this paper, we will show that the unitary equivalence of two multiplication operators on the Bergman spaces on polygons depends on the geometry of the polygon.

Set-theoretic solutions to the Yang–Baxter equation can be classified by their universal coverings and their fundamental groupoids. Extending previous results, universal coverings of irreducible involutive solutions are classified in the degenerate case. These solutions are described in terms of a group with a distinguished self-map. The classification in the nondegenerate case is simplified and compared with the description in the degenerate case.

In [2], Ladas and Sficas made two conjectures about the asymptotic behavior of solutions of some neutral differential equations. In this paper we confirm that these conjectures are indeed correct.

We compare, under some assumptions on mass density, the eigenvalues of the Sturm-Liouville problems satisfying homogeneous Dirichlet and Neumann boundary condition.

We show that if the general real method ${{\left( \cdot \,,\,\cdot \right)}_{\Gamma }}$ preserves the Banach-algebra structure, then a bilinear interpolation theorem holds for ${{\left( \cdot \,,\,\cdot \right)}_{\Gamma }}$.

It is shown that the classical Fourier transform can be extended as an algebra isomorphism onto the algebra of all complex-valued functions, which are measurable and finite a.e., under pointwise addition and multiplication. The extended Fourier transform agrees with the distributional Fourier transform on the space of all distributions which have regular transforms. It is defined on an algebra of Mikusiński-type operators in which multiplication is convolution in the subspace of integrable distributions.

Under certain conditions the reduced product space JX of a space X has the same homotopy type as ΩΣX, the loop-space on the suspension of X. Several proofs can be found in the literature. The original proof [6] made unnecessarily strong assumptions. Later, in the last chapter of [3], torn Dieck, Kamps and Puppe gave a proof under much weaker conditions and showed that they could not be further weakened. The question then arose as to whether the reduced product construction could be generalized to provide combinatorial models not only of ΩΣX but also of Ω2Σ2X, Ω3Σ3X and so on. This was answered in the affirmative by May [10], using ideas of Boardman and Vogt [1], and the construction was further developed by Segal [11] and others.

The present note, however, is not concerned with these generalizations. Its purpose is to give a simple proof of the original result, without striving for maximum generality, and to show that the same method can be used to prove an equivariant version of the reduced product theorem and hence a fibrewise version. Fibrewise versions of the reduced product theorem have previously been given by Eggar [5] and, more recently, by one of us [8] but it may be useful to have a relatively simple treatment which is adequate for the majority of applications.

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.

We determine the best constants ${{C}_{p,\infty }}$ and ${{C}_{1,p}},\,1\,<\,p\,<\,\infty $, for which the following holds. If $u,v$ are orthogonal harmonic functions on a Euclidean domain such that $v$ is differentially subordinate to $u$, then

$${{\left\| v \right\|}_{p}}\le {{C}_{p}}{{,}_{\infty }}{{\left\| u \right\|}_{\infty }},\,\,\,\,\,\,\,\,\,\,\,{{\left\| v \right\|}_{1}}\le {{C}_{1,p}}{{\left\| u \right\|}_{p}}.$$

In particular, the inequalities are still sharp for the conjugate harmonic functions on the unit disc of ${{\mathbb{R}}^{2}}$. Sharp probabilistic versions of these estimates are also studied. As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.

The extended completeness theorem of the predicate calculas of the first order. In section 12, we developed a deductive theory of the first order predicate calculus, while in section II we dealt with the semantic theory of that calculus. We now have to consider the connection between these two theories. We recall that a sentence X can be satisfied by a structure M only if X is defined in M. Given a sentence X (a set of sentences K) we shall say that the structure M is a model of X (of K) if X is (all the sentences of K are) satisfied by M.

Topology of positively curved compact Kähler manifolds had been studied by several authors (cf. [6; 2]); these manifolds are simply connected and their second Betti number is one [1]. We will restrict ourselves to the study of some compact homogeneous Kähler manifolds. The aim of this paper is to supplement some results in [9]. We prove, among other results, that a compact, simply connected homogeneous complex manifold whose Euler number is a prime p ≥ 2 is isomorphic to the complex projective space Pp-1 (C); in the p-1 case of surfaces, we prove that a compact, simply connected, homogeneous almost complex surface with Euler-Poincaré characteristic positive, is hermitian symmetric.

It is well known and immediate that in an associative ring a nilpotent one-sided ideal generates a nilpotent two-sided ideal. The corresponding open question for alternative rings was raised by M. Slater [6, p. 476]. Hitherto the question has been answered only in the case of a trivial one-sided ideal J (i.e., in case J2 = 0) [5]. In this note we solve the question in its entirety by showing that a nilpotent one-sided ideal K of an alternative ring generates a nilpotent two-sided ideal. In the process we find an upper bound for the index of nilpotency of the ideal generated. The main theorem provides another proof of the fact that a semiprime alternative ring contains no nilpotent one-sided ideals. Finally we note the analogous result for locally nilpotent one-sided ideals.

Let Un be an n-dimensional subspace of C[0, 1]. We prove that if n ≥ 2, and Un contains a function which is strictly positive on (0, 1), then there exists an f ∈ C[0, 1] which has more than one best one-sided L '-approximation from Un. We also characterize those Un with the property that each f ∈ C[0, 1] has a unique best one-sided L1(w)-approximation from Un with respect to every strictly positive continuous weight function w.

The exact probability of making an error at the first stage of a stepwise regression is found, both when the independent variables are fixed and when they are random. The probabilities differ in the two cases because the random variables involved are pairwise independent but not jointly independent.

The Riemann zeta-function is employed to generate universally overconvergent power series.