Given a class of topological spaces and a class of maps of topological spaces, our interest is in characterization of the class () of topological spaces whose every -image lies in . The class () is referred to as the -resolvent of and is the largest class of spaces smaller than closed under -images (provided is closed under composition and includes identity maps).

In the present paper, will be either the class of paracompact spaces or the class of regular spaces, and the conditions determining will always include separation axioms on the range, some of which can be dispensed with now by agreeing that all spaces are Hausdorff.

Soit M un facteur de type IIIλ 0 ≦ λ< 1, agissant sur un espace hilbertien à base dénombrable. Le propos de ce travail est l'étude des états normaux d'un tel facteur à équivalence unitaire près. L'espace quotient n'est pas séparé; l'espace séparé associé est obtenu comme quotient de l'espace des états normaux par la relation d'équivalence R dont les classes sont les fermetures des orbites précédentes.

Pour étudier cet espace, on définit un calcul fonctionnel sur M+* (espace des formes linéaires positives normales sur M) à valeurs dans M. Ce calcul fonctionnel est déterminé par la donnée d'une C* algèbre d'applications de M+ * dans M “continues” en un certain sens, notée C*(M).

Suppose that $Y$ is a cyclic cover of projective space branched over a hyperplane arrangement $D$ and that $U$ is the complement of the ramification locus in $Y$. The first theorem in this paper implies that the Beilinson–Hodge conjecture holds for $U$ if certain multiplicities of $D$ are coprime to the degree of the cover. For instance, this applies when $D$ is reduced with normal crossings. The second theorem shows that when $D$ has normal crossings and the degree of the cover is a prime number, the generalized Hodge conjecture holds for any toroidal resolution of $Y$. The last section contains some partial extensions to more general nonabelian covers.

The following well known inequality was first proved by Bernstein [2].

THEOREM A. If pn(x) is a polynomial of degree n, such that |pn(x)| ≦ 1 for –1 ≦ x = +1, then

1

The dominant n(1 – x2)–;1/2 is best possible only at the zeros of the Tchebychev polynomial

but the bound is precise at every interior point as far as the exponent of n is concerned.

Theorem A was extended to the case of higher derivatives by Duffin and Schaeffer in [4]. In that paper they make extensive use of the oscillation property of the polynomial Tn(x) and of the related function

A regular T1 space is said to be R-closed if there is no regular T1 space in which it can be embedded as a nonclosed subspace. A regular T1 space is said to be minimal regular if no regular T1 topology on the underlying set is strictly weaker than the given topology. It is known (see [1, Theorem 4, p. 455]) that every minimal regular space is R-closed. An R-closed space, however, need not be minimal regular [3, Example 2, p. 288].

In this work, we obtain the pointwise almost everywhere convergence for two families of multilinear operators: (a) the doubly truncated homogeneous singular integral operators associated with $L^q$ functions on the sphere and (b) lacunary multiplier operators of limited smoothness. The a.e. convergence is deduced from the $L^2\times \cdots \times L^2\to L^{2/m}$ boundedness of the associated maximal multilinear operators.

When a set of axioms is laid down as the basis of any mathematical doctrine, it must be proved that this set never leads to a contradiction. In this note we turn the question around. A set of axioms is given and we wish to adjoin an axiom of a specified type. How far does the demand of non-contradiction limit the choice of the new axiom?

1. The Theorem of Le Roux. Let U(x, y, α) be any solution of the linear hyperbolic differential equation

1

containing a parameter α. J. Le Roux (5) has shown that the function u(x, y) defined by

2 α0 = const.

We consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible isomorphism types for these ideals. Following up on that work, we now analyze the possible degrees of the elements in such regular sequences. For each case of our classification, we provide some criteria guaranteeing the existence of regular sequences in certain degrees.

We shall say that a plane set D has the Kakeya property if a unit segment can be turned continuously in D through 360° back to its original position. The famous solution of this problem by A. S. Besicovitch (1; 2; 4; 5; 6), to the effect that there are sets of arbitrarily small area having the Kayeka property, leaves open the problem obtained by adding the new condition that the set D be also simply connected. Since we do not know whether there is an attainable minimum, we define the Kakeya constant K to be the greatest lower bound of areas of simply connected sets having the Kakeya property. We shall refer to such sets as Kakeya sets.

Let (X, T, π) denote a flow, where X is a compact topological space metrizable by d, and T is a closed non-trivial subgroup of the reals under addition. T is recurrent if and only if for each and s > 0, there exists t > s such that x ∈ X implies . If T is almost-periodic, then T is both recurrent and distal. In §§ 4 and 5, it is shown that, under more stringent hypotheses, the recurrence of T is neither a necessary nor a sufficient condition for T to be distal. Let S be a closed non-trivial subgroup of T. It is shown in § 3 that T is recurrent if and only if S is recurrent. From this result, we obtain a solution to a problem posed by Nemyckiĭ (16, p. 492, Problem 6).

The purpose of this paper is to examine the extent to which a homomorphism of a ring with involution is determined by its action on the symmetric elements of the ring. Assuming that the ring is "suitably free" of 2 × 2 matrix rings, we show that any homomorphism is uniquely determined if its image is semi-prime without nonzero central ideals. To obtain this result we first investigate automorphisms of quotients of rings with involution.

In (5),I considered two-term conditions in π-exact couples, of which the exact couple of Federer (7) is an example. Let M(X, Y)be the space of all maps from X to Y with the compact-open topology. Our aim in this paper is to construct a π-exact couple , where Xis a finite-dimensional (in the sense of Lebesgue) metric space and , a certain (rather large) class of spaces. Specifically, is the class of all topological spaces Xwhich possess the following property (P).

(P) Let Y be a (possibly infinite) simplicial complex. There exists x0 ∈ X and y0 ∊ Y such that [X, x0]≃ [Y, y0].

In § 5 it will be seen that contains all CW complexes and all metric absolute neighbourhood retracts (ANR)s.

In this paper we represent certain linear operators in a space with indefinite metric. Such a space may be a pair (H, B), where H is a separable Hilbert space, B is a bilinear functional on H given by B(x, y) = [Jx, y], [, ] is the Hilbert inner product in H, and J is a bounded linear operator such that J = J* and J2 = I. If T is a linear operator in H, then ‖T‖ is the usual operator norm. The operator J above has two eigenspaces corresponding to the eigenvalues + 1 and –1.

In case the eigenspace in which J induces a positive operator has finite dimension k, a general spectral theory is known and has been developed principally by Pontrjagin [25], Iohvidov and Kreĭn [13], Naĭmark [20], and others.

A sequence, a1 < a2 < a3 < …, of positive integers is called lacunary if the difference sequence dn = an+l — an tends to infinity as n → ∞.

In several recent papers we have made use of these sequences in analysis and combinatorics. In [6] we show that the class of all sets which are either finite or the range of a lacunary sequence is “full” in the sense that if (tk) is a real sequence and for each then (tk) is an l1 sequence, that is,

In [3] the class of all finite unions of sets of is shown to consist of exactly those sets of integers, A, whose characteristic sequence, χA, is in the well known summability space bs + c0. More recently, in [1], we study lacunary sequences in connection with the conjecture of P.

Let $\phi :\,X\,\to \,M$ be a morphism from a smooth irreducible complex quasi-projective variety $X$ to a Grassmannian variety $M$ such that the image is of dimension ≥ 2. Let $D$ be a reduced hypersurface in $M$, and $\gamma $ a general linear automorphism of $M$. We show that, under a certain differential-geometric condition on $\phi (X)$ and $D$, the fundamental group ${{\text{ }\!\!\pi\!\!\text{ }}_{1}}\left( {{\left( \gamma \,o\,\phi \right)}^{-1}}\,\left( M\,\backslash \,D \right) \right)$ is isomorphic to a central extension of ${{\pi }_{1}}\left( M\,\backslash \,D \right)\,\,\times \,{{\pi }_{1}}\left( X \right)$ by the cokernel of ${{\pi }_{2}}\left( \phi \right)\,:\,{{\pi }_{2}}\left( X \right)\,\to {{\pi }_{2}}\left( M \right)$.

A remarkable theorem of Herstein [1, Theorem 2] of which we have made several uses states: If R is a semiprime ring of characteristic different from 2 and if U is both a Lie ideal and a subring of R then either U ⊂ Z (the centre of R) or U contains a nonzero ideal of R. In a recent paper [3] Herstein extends the above mentioned result significantly and has proved that if R is a semiprime ring of characteristic different from 2 and V is an additive subgroup of R such that [V, U] ⊂ V, where U is a Lie ideal of R, then either [V, U] = 0 or V ⊃ [M, R] ≠ 0 where M is an ideal of R. In this paper our object is to prove the following.

To understand the p-adic étale cohomology of a proper smooth variety over a p-adic field, Faltings compared it to the cohomology of his ringed topos, by the so-called Faltings’ main p-adic comparison theorem, and then deduced various comparisons with p-adic cohomologies originating from differential forms. In this article, we generalize the former to any proper and finitely presented morphism of coherent schemes over an absolute integral closure of $\mathbb {Z}_p$ (without any smoothness assumption) for torsion abelian étale sheaves (not necessarily finite locally constant). Our proof relies on our cohomological descent for Faltings’ ringed topos, using a variant of de Jong’s alteration theorem for morphisms of schemes due to Gabber–Illusie–Temkin to reduce to the relative case of proper log-smooth morphisms of log-smooth schemes over a complete discrete valuation ring proved by Abbes–Gros. A by-product of our cohomological descent is a new construction of Faltings’ comparison morphism, which does not use Achinger’s results on $K(\pi ,1)$-schemes.

We study convergence in weighted convolution algebras ${{L}^{1}}\left( \omega \right)$ on ${{R}^{+}}$, with the weights chosen such that the corresponding weighted space $M\left( \omega \right)$ of measures is also a Banach algebra and is the dual space of a natural related space of continuous functions. We determine convergence factor $\eta $ for which weak$*$-convergence of $\left\{ {{\text{ }\!\!\lambda\!\!\text{ }}_{n}} \right\}$ to $\text{ }\!\!\lambda\!\!\text{ }$ in $M\left( \omega \right)$ implies norm convergence of ${{\text{ }\!\!\lambda\!\!\text{ }}_{n}}*f$ to $\text{ }\!\!\lambda\!\!\text{ *}f$ in ${{L}^{1}}\left( \omega \eta \right)$. We find necessary and sufficent conditions which depend on $\omega$ and $f$ and also find necessary and sufficent conditions for $\eta$ to be a convergence factor for all ${{L}^{1}}\left( \omega \right)$ and all $f$ in ${{L}^{1}}\left( \omega \right)$. We also give some applications to the structure of weighted convolution algebras. As a preliminary result we observe that $\eta$ is a convergence factor for $\omega$ and $f$ if and only if convolution by $f$ is a compact operator from $M\left( \omega \right)$ (or ${{L}^{1}}\left( \omega \right)$) to ${{L}^{1}}\left( \omega \eta \right)$.

1. In this paper, X will always denote a locally compact Hausdorff space, C0(X) the algebra of all complex-valued continuous functions vanishing at infinity on X and B(X) the algebra of all bounded continuous complex-valued functions defined on X. If X is compact, C0(X) is identical to B (X) and all the results of this paper are obvious. Therefore, we will assume at the outset that X is not compact. If A represents an algebra of functions, AR will denote the algebra of all real-valued functions in A.