We study the moderate growth generalized Whittaker functions, associated to a unitary character $\psi $ of a unipotent subgroup, for the non-tempered cohomological representation of $G\,=\,\text{Sp}\left( 2,\,\mathbb{R} \right)$. Through an explicit calculation of a holonomic system which characterizes these functions we observe that their existence is determined by the including relation between the real nilpotent coadjoint $G$-orbit of $\psi $ in $\mathfrak{g}_{\mathbb{R}}^{*}$ and the asymptotic support of the cohomological representation.

Let R be a 2-torsion free simple artinian ring with involution*. The element u of R is said to be unitary if u is invertible with inverse u*. In this paper we shall be concerned with the subalgebras W of R over its centre Z such that uWu* ⊆ W, for all unitaries u of R. We prove that if R has rank superior to 1 over a division ring D containing more than 5 elements and if R is not 4-dimensional then any such subalgebra W must be one of the trivial subalgebras 0, Z or R, under one of the following extra finiteness assumptions: W contains inverses, W satisfies a polynomial identity, the ground division ring D is algebraic, the involution is a conjugate-transpose involution such that D equipped with the induced involution is generated by unitaries.

In this paper we note a simple connection between the (ƒ, dn) method of summability (defined by Smith (8)) and a composition of [F, dn] (defined by Jakimovski (4)) and Sonnenschein methods (9; 10). This connection is then used to supply some sufficient conditions for the regularity of (ƒ, dn) methods by using known regularity conditions for various [F, dn] and Sonnenschein methods. Finally, the connection is further exploited to obtain information about the Lebesgue constants for a certain class of [F, dn] methods by investigating related (ƒ, dn) methods.

2. Definitions and the regularity theorem. The (ƒ, dn) method of summability is defined by Smith (8) essentially as follows. Letƒ(z) be a nonconstant entire function satisfying ƒ (l) = 1 and let {dn} be a sequence of complex numbers satisfying dt ≠ – 1, di ≠ – ƒ(0) (i ≧ 1).

On associe à toute structure de proto-bigèbre de Lie sur un espace vectoriel $F$ de dimension finie des structures d’algèbre de Lie d’homotopie définies respectivement sur la suspension de l’algèbre extérieure de $F$ et celle de son dual ${{F}^{*}}$. Dans ces algèbres, tous les crochets $n$-aires sont nuls pour $n\,\ge \,4$ du fait qu’ils proviennent d’une structure de proto-bigèbre de Lie. Plus généralement, on associe à un élément de degré impair de l’algèbre extérieure de la somme directe de $F$ et ${{F}^{*}}$, une collection d’applications multilinéaires antisymétriques sur l’algèbre extérieure de $F$ (resp. ${{F}^{*}}$), qui vérifient les identités de Jacobi généralisées, définissant les algèbres de Lie d’homotopie, si l’élément donné est de carré nul pour le grand crochet de l’algèbre extérieure de la somme directe de $F$ et de ${{F}^{*}}$.

The usual development of the Lebesgue integral starts with a measure that may have been derived from some simpler set function, its associated class of measurable sets, the corresponding set of measurable functions, and operations which ultimately define the integral of any given function of this class, except for certain ones which are unbounded above and below. Here we propose to define a process of integration with respect to a set function more general than a measure. This process allows us to integrate virtually all functions real-valued on our space. The integrals thus obtained are all completely additive on a certain completely additive class of sets. Under rather mild hypotheses, we are able to delineate this class of sets explicitly.

In this paper, we study the Dirichlet problem of Hessian quotient equations of the form $S_k(D^2u)/S_l(D^2u)=g(x)$ in exterior domains. For $g\equiv \mbox {const.}$, we obtain the necessary and sufficient conditions on the existence of radially symmetric solutions. For g being a perturbation of a generalized symmetric function at infinity, we obtain the existence of viscosity solutions by Perron’s method. The key technique we develop is the construction of sub- and supersolutions to deal with the non-constant right-hand side g.

A basic tool in the usual presentation of the Morita theorems is the correspondence theorem for projective modules. Let RM be a left R-module and B = HomR(M, M). When M is a progenerator, there is a close connection (in fact a lattice isomorphism) between left R-submodules of M and left ideals of B, which can be applied to the solution of problems such as characterizing when the endomorphism ring of a finitely generated projective faithful module is simple or right Noetherian. More generally, Faith proved that this connection can be retained in suitably modified form when M is just a generator in R-mod ([4], [2], [3]). In this form the correspondence theorem can be applied to show, e.g., that, when RM is a generator, then (a): RM is finite-dimensional if and only if B is a left finite-dimensional ring and in this case d(RM) = d(BB), and (b): If RM is nonsingular then B is a left nonsingular ring ([6]).

Given an affine Coxeter group W, the corresponding Shi arrangement is a refinement of the corresponding Coxeter hyperplane arrangements that was introduced by Shi to study Kazhdan–Lusztig cells for W. Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in W. Low elements in W were introduced to study the word problem of the corresponding Artin–Tits (braid) group and turns out to produce automata to study the combinatorics of reduced words in W. In this article, we show, in the case of an affine Coxeter group, that the set of minimal length elements of the regions in the Shi arrangement is precisely the set of low elements, settling a conjecture of Dyer and the second author in this case. As a by-product of our proof, we show that the descent walls – the walls that separate a region from the fundamental alcove – of any region in the Shi arrangement are precisely the descent walls of the alcove of its corresponding low element.

Indices of normal spaces with countable basis for equivariant mappings have been investigated by Bourgin [4; 6] and by Wu [11; 12] in the case where the transformation groups are of prime order p. One of us has extended the concept to the case where the transformation group is a cyclic group of order pt and discussed its applications to the Kakutani Theorem (see [10]). In this paper we will define the Jp-index of a normal space with countable basis in the case where the transformation group is a cyclic group of order n, where n is divisible by p. We will decide, by means of the spectral sequence technique of Borel [1; 2], the Jp-index of SO(n) where n is an odd integer divisible by p. The method used in this paper can be applied to find the Jp-index of a classical group G whose cohomology ring over Jp has a system of universally transgressive generators of odd degrees.

Let S be a subset of some linear topological space. The set S is said to be m-convex, m ≧ 2, if and only if for every m-member subset of line segments determined by these points lies in S. A point x in S is called a point of local convexity of S if and only if there is some neighborhood N of x such that if y, z ∈ N⋂ S, then [y, z] ⊆ S. If S fails to be locally convex at some point q in S, then q is called a point of local nonconvexity (lnc point) of S.

There is a body of results for lattices known as “Decomposition Theory” which is aimed at proving certain existence and uniqueness theorems concerning irredundant representations of elements of a compactly generated lattice. The motivation for these results is certainly the quest for sufficient conditions on congruence lattices to insure irredundant subdirect representations of algebras. These theorems usually include some kind of modularity or distribut i v e hypothesis (for uniqueness) and some atomicity hypothesis (for existence); the precise details can be found in [3]. The atomicity condition is usually the hypothesis that the lattice in question is strongly atomic or at least atomic. Now, it is well-known that every algebra has a weakly atomic congruence lattice.

We consider threefolds that admit a fibration by $\text{K3}$ surfaces over a nonsingular curve, equipped with a divisorial sheaf that defines a polarization of degree two on the general fibre. Under certain assumptions on the threefold we show that its relative log canonical model exists and can be explicitly reconstructed from a small set of data determined by the original fibration. Finally, we prove a converse to this statement: under certain assumptions, any such set of data determines a threefold that arises as the relative log canonical model of a threefold admitting a fibration by $\text{K3}$ surfaces of degree two.

Bruck and Kleinfeld [3] proved that any alternative ring with characteristic prime to 2 must satisfy the identity

where the associator (x,y,z) is defined by (x, y, z) = (xy)z – x(yz) and . Linearization of the identity (x2, y, z) = 2x · (x, y, z) yields for characteristic prime to 2 an equivalent identity

(1)

Using the right alternative law (x, y, z) = –(y, x, z) and the flexible law (x, y, z) = –(z, y, x) which is satisfied in any alternative ring we obtain

(2)

and

(3)

Suppose $K$ is an imaginary quadratic field and $E$ is an elliptic curve over a number field $F$ with complex multiplication by the ring of integers in $K$. Let $p$ be a rational prime that splits as ${{\mathfrak{p}}_{1}}{{\mathfrak{p}}_{2}}$ in $K$. Let ${{E}_{{{p}^{n}}}}$ denote the ${{p}^{n}}$-division points on $E$. Assume that $F\left( {{E}_{{{p}^{n}}}} \right)$ is abelian over $K\,\text{for}\,\text{all}\,n\,\ge \,0$. This paper proves that the Pontrjagin dual of the $\mathfrak{p}_{1}^{\infty }$-Selmer group of $E$ over $F\left( {{E}_{{{p}^{\infty }}}} \right)$ is a finitely generated free $\wedge $-module, where $\wedge $ is the Iwasawa algebra of $\text{Gal}\left( F\left( {{E}_{{{p}^{\infty }}}} \right)/F\left( E\mathfrak{p}_{1}^{\infty }{{\mathfrak{p}}_{2}} \right) \right)$ . It also gives a simple formula for the rank of the Pontrjagin dual as a $\wedge $-module.

The chief purpose of this paper is to find all pairs (G, θ) where G is a finite special p-group, and θ is an automorphism of G acting trivially on the Frattini subgroup and irreducibly on the Frattini quotient. This problem arises in the context of describing finite groups having an abelian maximal subgroup. In fact, we solve a more general problem for a wider class of p-groups, which we call special F-groups, where F is a finite field of characteristic p. We point out that if p is odd, then an F-group has exponent p. On the other hand, every special 2-group is also a special GF(2)-group.

Let G be a compact connected Lie group with H a connected subgroup of maximal rank. Suppose there exists a compact connected Lie subgroup K with H ⊂ K ⊂ G. Then there exists a smooth fiber bundle G/H → G/K with K/H as the fiber. (See for example [13].) This can be incorporated into a diagram involving the classifying spaces as follows:

(1)

Here π, ϕ, ϕ1 and ϕ2 denote fibrations. We also know that the homogeneous spaces and the Lie groups, which are homotopy equivalent to the loop spaces of their respective classifying spaces, are homotopy equivalent to connected finite complexes.

Now suppose H is a maximal subgroup. Can there still exist spaces, which we will call BK, K/H, and G/K, and fibrations so that diagram (1) is still valid? This paper will show that in many cases either G/K or K/H will be homo topically trivial.

If K is a compact subset of Cn, will denote the polynomial hull of K:

arises in the study of uniform algebras as the maximal ideal space of the algebra P(K) of uniform limits on K of polynomials (see [3]). The condition (K is polynomially convex) is a necessary one for uniform approximation on K of continuous functions by polynomials (P(K) = C(K)). If K is not polynomially convex, the question of existence of analytic structure in is of particular interest. For n = 1, is the union of K and the bounded components of C\K. The determination of in dimensions greater than one is a more difficult problem. Among the special classes of compact sets K whose polynomial hulls have been determined are those invariant under certain group actions on Cn.

A set in a metric space is called a Čebyšev set if it has a unique “nearest neighbour” to each point of the space. In this paper we generalize this notion, defining a set to be Čebyšev relative to another set if every point in the second set has a unique “nearest neighbour” in the first. We are interested in Čebyšev sets in some hyperspaces over ${{\text{R}}^{n}}$, endowed with the Hausdorff metric, mainly the hyperspaces of compact sets, compact convex sets, and strictly convex compact sets.

We present some new classes of Čebyšev and relatively Čebyšev sets in various hyperspaces. In particular, we show that certain nested families of sets are Čebyšev. As these families are characterized purely in terms of containment,without reference to the semi-linear structure of the underlyingmetric space, their properties differ markedly from those of known Čebyšev sets.

1. Let G be a compact group, let I b e a subset of its dual object 𝚪, which, without loss of generality, will be assumed to be a countable subset. Let Di, i ∈ I , be irreducible representations of G of degree di. The Fourier series of a function F in L1(G) is denned by

where

All topological spaces under discussion are assumed to be Tychonoff.

For any topological space X let τ(X) denote the topology of X. If X ᑕ Y then a function κ : τ(X) ⟶ τ(Y) is called an extender provided that κ(U) ∩ X = U for all U ∊ τ(X). In addition, X is said to be Kn-embedded in Y (cf. [3]) provided there is an extender κ : τ(X) ⟶ τ(Y) such that

The extender κ is called a Kn-function (cf. [3]).

Eric van Douwen has asked whether there is a space X with a subspace Z which is Ki-embedded but not K0-embedded. The aim of this note is to answer this question.