Let S be a ring, and let (ei) be an orthogonal system of a finite number of idempotents. Then e = Σei has the following properties:

(i) Se Σ Sei and eS = Σ ei S.

(ii) The mappings v: Se → Π Sei and w: eS → Π ei S defined by v(x) = [xei] and w(x) = [ei x] respectively are isomorphisms.

Next assume that (ei)i∈I is a set of idempotents indexed by a totally ordered set I such that ei ej = 0 for every i < j. If I is finite, it is evident that

has the above two properties.

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.

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.

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.

There are three lattice-theoretic properties that are generally used to open a discussion on sublattices of a free lattice:

(W) for all a, b, c, d, a Λ b ≦ c V d implies a A b ≦ c, a Λ b g rf, a ≦cVd,orb ≦ c V d;

(SDv) for all a, b, c, a V b = a V c implies a V b = a V (b Λ c ) ;

(SDΛ) for all a, b, c, a Λ b = a Λ c implies a Λ b = a Λ (b V c).

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.

In 1955, J. Surányi and P. Turán (8) initiated the problem of existence and uniqueness of interpolatory polynomials of degrees less than or equal to 2n — 1 when their values and second derivatives are prescribed on n given nodes. This kind of interpolation was termed (0, 2)-interpolation. Later, Balázs and Turán (1) gave the explicit representation of the interpolatory polynomials for the case when the n given nodes (n even) are taken to be the zeros of πn(x) = (1 — x2)Pn′(x), where Pn–i(x) is the Legendre polynomial of degree n — 1. In this case the explicit representation of interpolatory polynomials turns out to be simple and elegant.

Balázs and Turán (2) proved the convergence of these polynomials when f(x) has a continuous first derivative satisfying certain conditions of modulus of continuity. They noted (1) that a significant application of lacunary interpolation could possibly be given in the theory of a differential equation of the form y′ + A(x)y= 0.

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.

W. Casselman defined a basis ${{f}_{u}}$ of Iwahori fixed vectors of a spherical representation $(\pi ,\,V)$ of a split semisimple $p$-adic group $G$ over a nonarchimedean local field $F$ by the condition that it be dual to the intertwining operators, indexed by elements $u$ of the Weyl group $W$. On the other hand, there is a natural basis ${{\psi }_{u}}$, and one seeks to find the transition matrices between the two bases. Thus, let ${{f}_{u}}\,=\,{{\sum }_{v}}\overset{\tilde{\ }}{\mathop{m}}\,(u,\,v){{\psi }_{v}}$ and ${{\psi }_{u}}\,=\,{{\sum }_{v}}m(u,\,v){{f}_{v}}$. Using the Iwahori–Hecke algebra we prove that if a combinatorial condition is satisfied, then $m(u,\,v)\,=\,{{\Pi }_{\alpha }}\,\frac{1-{{q}^{-1}}\,{{z}^{\alpha }}}{1-{{z}^{\alpha }}}$ , where $\mathbf{z}$ are the Langlands parameters for the representation and $\alpha $ runs through the set $S(u,\,v)$ of positive coroots $\alpha \,\in \,\hat{\Phi }$ (the dual root systemof $G$) such that $u\,\le \,v{{r}_{\alpha }}\,<\,v$ with ${{r}_{\alpha }}$ the reflection corresponding to $\alpha $. The condition is conjecturally always satisfied if $G$ is simply-laced and the Kazhdan–Lusztig polynomial ${{P}_{{{w}_{0}}v,\,{{w}_{0}}u}}\,=\,1$ with ${{w}_{0}}$ the long Weyl group element. There is a similar formula for $\tilde{m}$ conjecturally satisfied if ${{P}_{u,\,v}}\,=\,1$. This leads to various combinatorial conjectures.

The aim here is to define connections on a parabolic principal bundle. Some applications are given.

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.

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.

We consider the analogue of the $\wedge (p)$―problem for subsets of the Legendre polynomials or more general ultraspherical polynomials. We obtain the “best possible” result that if $2\,<\,p\,<\,4$ then a random subset of $N$ Legendre polynomials of size ${{N}^{4/p-1}}$ spans an Hilbertian subspace. We also answer a question of König concerning the structure of the space of polynomials of degree $n$ in various weighted ${{L}_{p}}$-spaces.

1. Introduction. As is well known, the spectral theorem plays an important part in mathematics because of its many applications. Unfortunately, the theorem is valid for normal operators only. In view of this, attempts have been made by several mathematicians to obtain a theorem about a more general class of operators, which will reduce to the ordinary spectral theorem if the operator is normal. Brown (1) has developed a unitary equivalence theory for a certain class of operators.

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.

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

The purpose of this paper is to prove the following theorem:

Theorem. Given a convex curve C, of perimeter length I and three points M, N, and P which divide the perimeter of C into three parts of equal length, the perimeter length of the triangle MNP is never less than ½l. Equality holds only in the case where C is an equilateral triangle and M, N, and P are the mid-points of the three sides.

All spaces in this paper are completely regular Hausdorff and all maps are continuous onto, unless otherwise stated. The purpose of this paper is to investigate the realcompactness of a space X which contains a Lindelöf space L such that every zero-set Z (in X) disjoint from L is realcompact. We show in § 2 that such a space X is very close to being realcompact (Theorems I, II and III). But in general such a space fails to be realcompact. Indeed, in §§ 3 and 4 the following questions of Mrówka [18, 19] are answered, both in the negative:

(Q. 1) If X = L ∪ G where L is Lindelöf closed and G is E-compact, then is X E-compact?

(Q. 2) Suppose f:X → Y is a perfect map such that the set M(f) = {y ∊ Y| |f−l(y)| > 1} of multiple points of f is Lindelöf (especially, countable) closed. If X is E-compact, is Y also E-compact?

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].

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.