1. Introduction. Let (X, , μ) be a probability space, T a linear operator on ℒp(X, , μ), for some p, 1 ≦ p ≦ ∞. Let an be a sequence of complex numbers, n = 0, 1, …, which we shall often refer to as weights. We shall say that the weighted pointwise ergodic theorem holds for T on ℒp, if, for every ƒ in ℒp,

1.1

Let a denote the sequence (an). If (1.1) holds we shall say that a is Birkhoff for T on ℒp, or, more briefly, that (a, T) is Birkhoff.

We are also interested in ergodic theorems for subsequences. Let n(k) be a subsequence. We shall say the pointwise ergodic theorem holds for the subsequence n(k) and the operator T if, for every ƒ in ℒp,

1.2

In 1902, F. R. Moulton (12) gave an early example of a non-Desarguesian plane. Its ‘points” are ordered pairs (x, y) of real numbers. Its “lines” coincide with lines of the real affine plane except that lines of negative slope are “bent” on the x-axis, line {y = b + mx}, for negative m, being replaced by {y = b + mx if y ≤ 0, y = [m/2]. [x + (b/m)] if y > 0}. A certain Desarguesian configuration in the classical plane is shifted just enough to vitiate Desargues’ Theorem for Moulton's geometry. The plane is neither a translation plane (“Veblen-Wedderburn” in the sense of Hall (7), p. 364) nor even the dual of one (Veblen and Wedderburn (17). It is natural to ask if the same construction is feasible when real numbers are replaced by elements from an arbitrary field.

A complex function on the real line is said to be bounded in mean square if it is locally in L2 (i.e. on each finite interval) and satisfies

(1.1)

The set of all such functions clearly forms a linear space over the complex numbers and is a Banach space B under the norm ‖·‖B defined by (1.1). This space, among others, has been discussed by Beurling in [1], where it was shown to be the dual, in the Banach space sense, of a certain Banach (convolution) algebra of functions. We have used Beurling's characterization of B and others of his results throughout this paper, and indeed the essence of one or two of the proofs has been derived from his theorems.

1. Introduction. The elementary portions of the theory of integral representation of numbers or forms by quadratic forms will be somewhat simplified and generalized in this article. This indicates certain directions in which new applications can be made. The applications made here will be largely to the representation of numbers or binary quadratic forms by ternary quadratic forms. Particularly, we shall obtain the correct estimate (Theorem 10) needed to fill a lacuna in certain work of U. V. Linnik [1] on the representation of large numbers by ternary quadratic forms. Since Linnik applied his theorem on ternaries to prove [9] that every large number is a sum of at most seven positive cubes, a lacuna in this proof can now be regarded as filled.

We first prove a nonlinear ergodic theorem for nonexpansive semigroups without convexity in a Hilbert space. Further we prove a fixed point theorem for non-expansive semigroups without convexity which generalizes simultaneously fixed point theorems for left amenable semigroups and left reversible semigroups.

Consider a projective plane with n + 1 points on each line, An oval is a set of n + 1 points, no three of which are collinear.

DEFINITION. (1) A line which contains two points of will be called a secant of .

(2) A line which contains exactly one point of will be called a tangent of or an absolute line.

In (1) Fan showed that if A is a Hermitian matrix with eigenvalues λ1 ≤ … ≤ λn then, for k ≤ n,

,

,

where X1, … , xk run over all sets of k orthonormal (o.n.) vectors in unitary n-space V.

Given a complete $\text{CAT}(0)$ space $X$ endowed with a geometric action of a group $\Gamma $, it is known that if $\Gamma $ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We prove the converse of this statement in the special case where $X$ is a convex subcomplex of the $\text{CAT}(0)$ realization of a Coxeter group $W$, and $\Gamma $ is a subgroup of $W$. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the $\text{CAT}(0)$ realization of arbitrary Tits buildings.

Let H be a finite set; we can suppose H = { 1, 2, …,d } . Consider Hn, the set of length n words over the alphabet H. For every A ∈ Hn define the waiting time for A as the number of experiments needed till A appears as a connected sub-sequence of random elements of H. Formally, let X1, X2,... be i. i. d. random variables, P(X1 = i) = d−1, 1 ≤ i ≤ d then

This article fully characterizes categorically universal varieties of (0, 1)-lattices (that is, lattices with a least element 0 and a greatest element 1 regarded as nullary operations), thereby concluding a series of partial results [3, 5, 8, 10, also 14] which originated with the proof of categorical universality for the variety of all (0, 1)-lattices by Grätzer and Sichler [6].

A category C of algebras of a given type is universal if every category of algebras (and equivalently, according to Hedrlín and Pultr [7 or 14], also the category of all graphs) is isomorphic to a full subcategory of C. The universality of C is thus equivalent to the existence of a full embedding Φ : G→C of the category G of all graphs and their compatible mappings into C. When Φ assigns a finite algebra to every finite graph, we say that C is finite-to-finite universal.

In this paper we use a generalized form of Polya's theorem (1) to obtain generating functions for the number of ordinary graphs with given partition and for the number of bicoloured graphs with given bipartition. Both the points and lines of the graphs are taken as unlabelled. These graph enumeration problems were proposed by Harary in his review article (4). Read (7, 8) solved the problem for unlabelled general graphs and labelled ordinary graphs.

In 1958, G. D. Findlay and J. Lambek defined a relationship between three R-modules, A ≦ B(C), to mean that A ⫅ B and every R-homomorphism from A into C can be uniquely extended to an irreducible partial homomorphism from B into C. If A ≦ B(B), then B is called a rational extension of A and in [5] it is shown that every module has a maximal rational extension in its injective hull which is unique up to isomorphism. A module is called rationally complete provided it has no proper rational extension.

Many functions, F(z), have integral representations of the form

1.1

the so-called Laplace transform of f(t). When f(t) satisfies certain regularity conditions, it is possible to use Cauchy's theorem to deform the contour so that F(z) has the integral representation

1.2

where 7 is a fixed real number, and the path of integration is the straight line joining t = 0 to t = ∞ etr, small indentations in the path of integration being allowed to avoid singularities of f(t) where necessary. When the two functions defined by (1.1) and (1.2) are not the same, (1.2) provides the more general situation for a theoretical discussion of the properties of F(z).

In a paper on the same subject [28] and another coming out at the same time [27], Nagata gave his celebrated Double (treble, really) Sequence Theorem, with which he deduced easily and thus brought together the basic metrization theorems, i.e. theorems in which the conditions for metrizability are given as the availability of bases or subbases of certain descriptions.

In this paper, we solve the isomorphism problem for metacirculant graphs of order $pq$ that are not circulant. To solve this problem, we first extend Babai’s characterization of the $\text{CI}$-property to non-Cayley vertex-transitive hypergraphs. Additionally, we find a simple characterization of metacirculant Cayley graphs of order $pq$, and exactly determine the full isomorphism classes of circulant graphs of order $pq$.

A map p: E → B between metric spaces has the approximate homotopy lifting property with respect to the space X if given a cover Ū of B and maps g: X → E and H: X × [0, 1] → B such that H(x, 0) = pg(x) for all x ϵ X, then there exists a map G: X × [0, 1] → E such that G(x, 0) = g(x) and pGt and Ht are Ū-close for all x ϵ X and t ϵ [0, 1]; i.e. given (x, t) ∊ X × [0, 1], there exists U × Ū such that pG(x, t) and H(x, t) are elements of U.

For a fixed algebraic number α we discuss how closely α can be approximated by a root of a {0, +1, -1} polynomial of given degree. We show that the worst rate of approximation tends to occur for roots of unity, particularly those of small degree. For roots of unity these bounds depend on the order of vanishing, k, of the polynomial at α.

In particular we obtain the following. Let BN denote the set of roots of all {0, +1, -1} polynomials of degree at most N and BN(α k) the roots of those polynomials that have a root of order at most k at α. For a Pisot number α in (1, 2] we show that

and for a root of unity α that

We study in detail the case of α = 1, where, by far, the best approximations are real. We give fairly precise bounds on the closest real root to 1. When k = 0 or 1 we can describe the extremal polynomials explicitly.

Let $\pi :X\,\to \,S$ be a finite type morphism of noetherian schemes. A smooth formal embedding of $X$ (over $S$) is a bijective closed immersion $X\,\subset \,\mathfrak{X}$ , where $\mathfrak{X}$ is a noetherian formal scheme, formally smooth over $S$. An example of such an embedding is the formal completion $\mathfrak{X}={{Y}_{/X}}$ where $X\,\subset \,Y$ is an algebraic embedding. Smooth formal embeddings can be used to calculate algebraic De Rham(co)homology.

Our main application is an explicit construction of the Grothendieck residue complex when $S$ is a regular scheme. By definition the residue complex is the Cousin complex of ${{\pi }^{!}}{{O}_{S}}$, as in $[\text{RD}]$. We start with $\text{I-C}$. Huang's theory of pseudofunctors on modules with 0-dimensional support, which provides a graded sheaf ${{\oplus }_{q}}K_{X/S}^{q}.$ We then use smooth formal embeddings to obtain the coboundary operator $\delta :K_{X/S}^{q}\to K_{X/S}^{q+1}.$ We exhibit a canonical isomorphism between the complex $K_{X/S}^{\cdot },\delta $ and the residue complex of $[\text{RD}]$. When $\pi $ is equidimensional of dimension $n$ and generically smooth we show that ${{\text{H}}^{-n}}K_{X/S}^{\cdot }$ is canonically isomorphic to to the sheaf of regular differentials of Kunz-Waldi $[\text{KW}]$.

Another issue we discuss is Grothendieck Duality on a noetherian formal scheme $\mathfrak{X}$ . Our results on duality are used in the construction of $K_{X/S}^{\cdot }$.

Given a bivariate function defined on some subset of the Cartesian product of two sets, it is natural to ask when that function can be decomposed as the sum of two univariate functions. In particular, is a pointwise limit of such functions itself decomposable? At first glance this might seem obviously true but, as we show, the possibilities are quite subtle. We consider the question of existence and uniqueness of such decompositions for this case and for many generalizations to multivariate functions and to cases where the sets and functions have topological or measure theoretic structure.