For T in our main Theorem 5, T* is called Lamperti in [11], whose terminology and notation we shall follow in the sequel. To avoid longish expressions, we shall also say that T* here is disjunctive and, dually, T = (T*)* is codisjunctive. The present work grows out of an attempt to establish a DEE for the general power bounded positive operator on Lp, in view of the success in the contraction case [1, 11], and forms a continuation of [11]. (In passing, we note that Calderon's technique [2] mentioned in [11] was anticipated in 1938 by M. Fukamiya [7], though in a variant form and for a more classical case, namely that of a positive Lp isometry induced by an invertible, measure preserving transformation on a totally finite measure space. Calderon's case does not assume invertibility nor total finiteness.)

A classical question in algebraic geometry is whether a given projection of a projective space induces an isomorphism on a given closed subvariety. To answer it, one investigates secant lines to the subvariety. There has been a lot of recent activity in this field ([12], [14],[18], [21], [23]): see [14] and [12] for references).

An obvious generalization of the secant lines is provided by the secant r-planes, which intersect a given closed subvariety in r + 1 linearly independent points. The closure of the set of these secant r-planes is the secant variety, and the aim of this paper is to determine its rational equivalence class in the case of curves. There is an extensive classical literature about this problem.

Proximity spaces and contiguity spaces, and more recently nearness spaces, have been studied not just because they provide various approaches to uniform structure. Possibly of greater importance is that they can be used as a means of introducing compactifications and more general extensions of the topological spaces on which they are defined. Riesz [20] was probably the first to recognize this connection. Since then the idea was used by Freudenthal [9], Alexandroff [1], Smirnov [21], Leader [17] and Ivanov and Ivanova [13, 14, 15] among others.

Recently Reed [19] using work of Bentley [2, 4] and Herrlich [11, 12] studied the 1 – 1 correspondence between the class of all cluster generated nearness spaces and the class all principal T 1-extensions of a given T 1-space. She succeeded in showing that the mapping induces a 1 – 1 correspondence between the contingual nearness spaces in and the compactifications in .

In this paper we study the inverse limit cohomology h*(K(Z, 3)) of an Eilenberg-MacLane object K(Z, 3) for certain cohomology theories h. Our main result gives a complete description of all non-trivial differentials of the Atiyah-Hirzebruch spectral sequence (AHSS) H*(X;h*(pt)) ⇒ h*(X) for X = K(Z, 3) and h either of the complex K-theories K*( ;Z/p) and K*( ;Z(p) ). This is achieved inductively using the finite symmetric product spaces SPkS 3, k = pr. Identification of cycles and boundaries of each non-trivial differential leads to an explicit description of BP<1>*(K(Z, 3); Z/p) and some information about BP<1>*(K(Z, 3)).

Let R 1 and R 2 be commutative rings with identities, G a group and R 1 G and R 2 G the group ring of G over R 1 and R 2 respectively. The problem that motivates this work is to determine what relations exist between R 1 and R 2 if R 1 G and R 2 G are isomorphic. For example, is the coefficient ring R 1 an invariant of R 1 G? This is not true in general as the following example shows. Let H be a group and

If R 1 is a commutative ring with identity and R 2 = R 1 H, then

but R 1 needn't be isomorphic to R 2.

Several authors have investigated the problem when G = <x>, the infinite cyclic group, partly because of its closeness to R[x], the ring of polynomials over R.

Functions on R (or on R/Z, or Q/Z, or the interval (0,1)) which satisfy the identity

1.1

for positive integers m and fixed complex s, appear in several branches of mathematics (see [8], p. 65-68). They have recently been studied systematically by Kubert [6] and Milnor [12]. Milnor showed that for each complex s there is a one-dimensional space of even functions and a one-dimensional space of odd functions which satisfy (1.1). These functions can be expressed in terms of either the Hurwitz partial zeta-function or the polylogarithm functions.

My purpose is to prove an analogous theorem for p-adic functions. The p-adic analog is slightly more general; it allows for a Dirichlet character χ 0(m) in front of m s–l in (1.1). The functions satisfying (1.1) turn out to be p-adic “partial Dirichlet L-functions”, functions of two p-adic variables (x, s) and one character variable χ 0 which specialize to partial zeta-functions when χ 0 is trivial and to Kubota-Leopoldt L-functions when x = 0.

The generalized Euler numbers may be defined by

Since is zero unless m divides n, we shall write for . Leeming and MacLeod [12] recently gave some congruences for these numbers. They found congruences (mod 16) for where m = 3, 6, 8, 12, and 16. Thus for m = 3, their congruence is

They also proved that , and , and they made several conjectures which may be stated as follows:

C1

C2

C3

C4

Let A be a C*-algebra, let p be a polynomial over C, and let a in M(A) (the multiplier algebra of A) be such that p(ad a) = 0. In this paper we study the following problem: when does there exist λ in Z(M(A)) (the centre of M(A)) such that p(a – λ) = 0? The first result of this type known to us is due to I. N. Herstein [7], who showed that for a simple ring with identity, such a λ always exists when p is of the form p(x) = xk for some positive integer k. Later, in [8], C. R. Miers showed that the result is true for any primitive unital C*-algebra and any polynomial whatever. It was also shown in [8] that if A is a unital C*-algebra acting on H and p is any polynomial, then such a λ exists in the larger algebra Z(A″). In particular, the strict result holds for any von Neumann algebra, A.

The “combinatorial line conjecture” states that for all q ≧ 2 and there exists such that if , X is a q-element set, and A is any subset of X n (= cartesian product of n copies of X) with more than elements (that is, A has density greater than ), then A contains a combinatorial line. (For a definition of combinatorial line, together with statements and proofs of many results related to the combinatorial line conjecture, including all those results mentioned below, see [5]. Since we are not directly concerned with combinatorial lines in this paper, we do not reproduce the definition here.)

This conjecture (which is a strengthened version of a conjecture of Moser [7]), if true, would bear the same relation to the Hales-Jewett theorem that Szemerédi's theorem bears to van der Waerden's theorem.

Let σ be a finite positive singular Borel measure defined on Euclidean space R N . For w ∈ R N and y > 0, its Poisson integral is defined by the formula

where CN is chosen so that

Since σ is singular, almost everywhere with respect to Lebesgue measure on R N . On the other hand, almost everywhere dσ. It follows that for all sufficiently small y,

is a non-empty open subset of R N . If σ has compact support then |E y | → 0 as y → 0, where |E y | denotes the Lebesgue measure of E y . In this paper we give a lower bound on the rate at which |E y | may go to zero. The lower bound depends on the smoothness of the measure; the smoother the measure, the more slowly |E y | may approach 0.

The subject of this paper arises from the familiar process whereby an automorphism of a field generates new representations from old. One may think of that process spatially, as a change of vector space structure in the representation space by means of the automorphism. The operators of the representation acting in the “new“ space then constitute the new representation. This point of view makes visible an algebraic structure we call a scalar action. A scalar action f of a ring R (with unity) in an abelian group Kis a ring homomorphism f:R → End(V) taking the unity element of R to the identity operator in End(V). If f is a scalar action of a field F and ϕ is an automorphism of F then f ∘ ϕ is another scalar action of F, and it is this construction which is used to define the “new” representation space mentioned above. But the variety of scalar actions goes rather beyond that construction.