For a locally compact group $G$ and $1\,<\,p\,<\,\infty $, let ${{A}_{p}}\left( G \right)$ be the Herz-Figà-Talamanca algebra and let $P{{M}_{p}}\left( G \right)$ be its dual Banach space. For a Banach ${{A}_{p}}\left( G \right)$-module $X$ of $P{{M}_{p}}\left( G \right)$, we prove that the multiplier space $\text{M}\left( {{A}_{p}}\left( G \right),{{X}^{*}} \right)$ is the dual Banach space of ${{Q}_{X}}$, where ${{Q}_{X}}$ is the norm closure of the linear span ${{A}_{p}}\left( G \right)X\,\text{of}\,u\,f\,\text{for}\,u\,\in \,{{A}_{p}}\left( G \right)\,\text{and}\,f\,\in \,X$ in the dual of $\text{M}\left( {{A}_{p}}\left( G \right),{{X}^{*}} \right)$. If $p\,=\,2$ and $P{{F}_{p}}\left( G \right)\subseteq X$, then ${{A}_{p}}\left( G \right)X$ is closed in $X$ if and only if $G$ is amenable. In particular, we prove that the multiplier algebra $M{{A}_{p}}\left( G \right)\,\text{of}\,{{A}_{p}}\left( G \right)$ is the dual of $Q$, where $Q$ is the completion of ${{L}^{1}}\left( G \right)$ in the $||\cdot |{{|}_{M}}$-norm. $Q$ is characterized by the following: $f\,\in \,Q$ if an only if there are ${{u}_{i}}\,\in \,{{A}_{p}}\left( G \right)$ and ${{f}_{i}}\in P{{F}_{p}}\left( G \right)\left( i=1,2,... \right)$ with $\sum\nolimits_{i=1}^{\infty }{||}\,{{u}_{i}}\,|{{|}_{{{A}_{p}}\left( G \right)}}||fi|{{|}_{P{{F}_{p}}\left( G \right)}}\,<\,\infty $ such that $f=\sum{_{i=1}^{\infty }\,{{u}_{i}}{{f}_{i}}}$ on $M{{A}_{p}}\left( G \right)$. It is also proved that if ${{A}_{p}}\left( G \right)$ is dense in $M{{A}_{p}}\left( G \right)$ in the associated ${{w}^{*}}$-topology, then the multiplier norm and $||\cdot |{{|}_{{{A}_{p}}\left( G \right)}}$-norm are equivalent on ${{A}_{p}}\left( G \right))$ if and only if $G$ is amenable.

In this paper we study second order linear differential systems. We examine the relationship between oscillation of n-dimensional systems and certain associated n-dimensional systems, where m ≧ n. Several theorems are presented which unify and encompass in the linear case a number of results from the literature. In particular, we present a transformation which extends an oscillation theorem due to Allegretto and Erbe [1], and a comparison theorem due to Kreith [9], and explains some work of Howard [7].

We compute the loops of units in the integral alternative loop rings of six Moufang loops. Four of these are subloops of the loop of matrices of determinant one in Zorn's vector matrix algebra over a ring of integers while the remaining two are closely related to this interesting algebra. This paper thus serves, in part, to highlight a Moufang analogue of SL(2, Z) which the author suggests is worthy of further study.

Let F denote the Galois field GF(pr) with pr elements, where p is an odd prime and r is a positive integer. Suppose further that m and n are arbitrary elements of F and that αi, βi pt (i = 1, …, s) are nonzero elements of F. The purpose of this paper is to evaluate the function Ns(m, n), defined, for an arbitrary positive integer s, to be the number of simultaneous solutions in F of the equations

1.1.

In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form $H$ and extend the result to $n$-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimensions.

We prove an integration formula involving Jack polynomials conjectured by I. P. Goulden and D.M. Jackson in connection with enumeration of maps in surfaces.

Although the literature on splines has grown vastly during the last decade [11], the study of polynomial splines on the circle seems to have suffered neglect. The first to study the subject in depth seem to be Ahlberg, Nilson and Walsh [1]. Almost at the same time I. J. Schoenberg [8] studied the problem of interpolation at the roots of unity by splines and its relation to quadrature on the circle. For discrete polynomial splines on the circle we refer to [5]. M. Golomb [3] also considers interpolation by a class of “spline” functions in the complex plane but his point of view is based on minimum norm properties of spline functions. Perhaps the reason for this neglect may be attributed to the fact that one can pass from the circle to the line by means of the transformation z → exp 2-πix. This changes the problem on the circle into periodic interpolation on the line with the difference that instead of interpolation by piecewise polynomial, we now consider piecewise exponential polynomials with complex exponents.

The topology of abstract 2-metric (area-metric) spaces has been the object of study in recent papers of Gähler (1) and Froda (2). The geometric properties of such spaces, however, have remained largely untouched since the initial work of Menger (3). As in ordinary metric spaces, a notion of 2-betweenness, or interiorness, can be easily defined in 2-metric spaces. In abstract metric spaces the betweenness relation is characterized among all relations defined on each triple of points of every metric space by six natural properties (4, pp. 33-40; 5). The purpose of this paper is to prove a similar theorem characterizing the relation of 2-betweenness in 2-metric spaces.

Affine and projective Hjelmslev planes are generalizations of ordinary affine and projective planes where two points (lines) may be joined by (may intersect in) more than one line (point). The elements involved in multiple joinings or intersections are neighbours, and the neighbour relations on points respectively lines are equivalence relations whose quotient spaces define an ordinary affine or projective plane called the canonical image of the Hjelmslev plane. An affine or projective Hjelmslev plane is a topological plane (briefly a TH-plane and specifically a TAH-plane respectively a TPH-plane) if its point and line sets are topological spaces so that the joining of non-neighbouring points, the intersection of non-neighbouring lines and (in the affine case) parallelism are continuous maps, and the neighbour relations are closed.

In this paper we continue our investigation of such planes initiated by the author in [38] and [39].

The main theorems concern the relation between the - compact spaces and the -regular spaces, and their analogues in uniform spaces. In either of the categories of Tychonoff spaces or uniform spaces, let be a class of spaces, let be the epi-reflective hull of se (closed subspaces of products of members of ), let be the “onto-reflective” hull of (all subspaces of products of members of ), and let r and o be the associated functors. Let be the class of spaces which admit a perfect map into a member of . Then, is epi-reflective (and in Tych, = but in Unif, the equality fails); call the functor p.

Notation,p and q are generic symbols for prime numbers. N(H, p) denotes the number of primes q, not exceeding Hy which are primitive roots (mod p).

g(p) denotes the least positive primitive root (mod p).

g*(p) is the least prime primitive root (mod p).

v(m) denotes the number of distinct prime divisors of the integer m.

τk(m) is the number of ways of representing the integer m as the product of k integers, order being important.

π(x, k, r) is the number of primes p, not exceeding x, which satisfy p ≡ r(mod k); while π(x) denotes the total number of p ≦ x.

logm x denotes the mth iterated logarithmic function which is defined, when possible, by

We study skeleton C*-subalgebras of a given C*-algebra. We show that if A is a unital (non-unital but σ-unital) simple C*-algebra, ℳ is any unital (nonunital) matroid C* -algebra, then A contains a skeleton C*-subalgebra B with a quotient which is isomorphic to ℳ. Other results for skeleton C*-subalgebras are also obtained. Applications of these results to the structure of quasi-multipliers and perturbations of C*-algebras are given.

In this paper, we will prove, as a consequence of the main theorem,

THEOREM A. (See Corollary 2.6). The group of an alternating knot, for which the leading coefficient of the knot polynomial is a prime power, is residually finite and solvable.

The maximal ideals in a commutative Banach algebra with identity have been elegantly characterized [5; 6] as those subspaces of codimension one which do not contain invertible elements. Also, see [1]. For a function algebra A, a closed separating subalgebra with constants of the algebra of complex-valued continuous functions on the spectrum of A, a compact Hausdorff space, this characterization can be restated: Let F be a linear functional on A with the property:

(*) For each ƒ in A there is a point s, which may depend on f, for which F(f) = f(s).

We determine and completely describe all pure realizations of the finite regular toroidal polyhedra of types {3, 6} and {6, 3}.

We develop the analog of crystalline Dieudonné theory for $p$-divisible groups in the arithmetic of function fields. In our theory $p$-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson’s abelian $t$-modules and $t$-motives, and Drinfeld shtukas. Moreover, we discuss the existence of a Verschiebung map and apply it to deformations of local shtukas and divisible local Anderson modules. As a tool we use Faltings’s and Abrashkin’s theories of strict modules, which we review briefly.

In a number of recent papers, especially by Wilansky (4; 6), Zeller (8), and Peyerimhoff (3), the sequence-to-sequence transformation

A : (n = 0, 1, …)

has been studied under certain conditions, designated by FAK, PMI, etc. (see §3). The purpose of this note is to point out some relations among these conditions, and to show that some theorems previously obtained hold under weaker assumptions.

We show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives an example where the partial transpose produces freeness at the operator level. Finally, we investigate the case of real Wishart matrices.

We give a geometrical construction of the canonical automorphic factor for the Jacobi group and construct new vector valued modular forms from Jacobi forms by differentiating them with respect to toroidal variables and then evaluating at zero.

1. Introduction. The ternary orthogonal group of projectivities over a finite field leaves a non-singular conic ✗ invariant, but the geometry consequent thereupon does not appear to have been investigated. The group is isomorphic to a binary group of fractional substitutions over the same field and this fact may, since these binary groups and their subgroups are so well known, have inhibited projects to embark on a detailed description of the geometry of the ternary group.