1. The properties of the circulant determinant or the circulant matrix are familiar. The circulant matrix C of order 4 x 4, with elements in the complex field, will serve for illustration.

The four matrix coefficients of c0, c1 c2, c3 form a reducible matrix representation of the cyclic group ℐ4, so that C is a group matrix for this. Let ω be a primitive 4th root of 1. Then Ω as below, its columns being normalized latent vectors of C,

is unitary and symmetric, and reduces Cto diagonal form thus,

where the μr, the latent roots of C, are given by

All of the above extends naturally to the n x n case.

We determine the rings of invariants $S^G$ where $S$ is the symmetric algebra on the dual of a vector space $V$ over ${\mathbb F}_2$ and $G$ is the orthogonal group preserving a non-singular quadratic form on $V$. The invariant ring is shown to have a presentation in which the difference between the number of generators and the number of relations is equal to the minimum possibility, namely $\dim V$, and it is shown to be a complete intersection. In particular, the rings of invariants computed here are all Gorenstein and hence Cohen-Macaulay.

In this paper, groups are investigated in which all subgroups, all normal subgroups, or all characteristic subgroups have a proper supplement. This supplement can be either an arbitrary subgroup, a normal or a characteristic subgroup, resulting in nine classes of groups. Properties of these classes are studied such as containment and closure properties, and characterizations for several of these classes are given.

1991 Mathematics Subject Classification Primary 20E34, Secondary 20E15.

In this paper, we prove some Hardy-type inequalities for the degenerate operators, $L_{p,\alpha}u\,{=}\,{\rm div}_L(|\nabla_Lu|^{p-2}\nabla_Lu)$, where $\nabla_Lu\,{=}\,(\frac{\partial u}{\partial z_1},\ldots,\frac{\partial u}{\partial z_n},|z|^\alpha \frac{\partial u}{\partial t_1},\ldots,|z|^\alpha\frac{\partial u}{\partial t_m})$. These inequalities are established for the whole space, the pseudo-ball and the external domain of the pseudo-ball. We also give a generalization of a result in [8]. Finally, a sharp inequality for $L_{\alpha}\,{=}\,L_{2,\alpha}$ is obtained.

In § 2 a number of infinite series of E-functions are summed by expressing the E-functions as Barnes integrals and interchanging the order of summation and integration.

The Barnes integral employed is

where | amp z | < π and the integral is taken up the η;-axis, with loops, if necessary, to ensure that the origin lies to the left of the contour and the points α1, α2,… αp to the right of the contour. Zero and negative integral values of the α's and p's are excluded, and the α's must not differ by integral values. When p < q + 1 the contour is bent to the left at each end.

Let X be an infinite, locally finite, almost transitive graph with polynomial growth. We show that such a graph X is the inverse limit of an infinite sequence of finite graphs satisfying growth conditions which are closely related to growth properties of the infinite graph X.

1991 Mathematics Subject Classification. Primary 05C25, Secondary 20F8.

The nth order polylogarithm Lin(z) is defined for |z| ≦ 1 by

([4, p. 169], cf. [2, §1. 11 (14) and § 1. 11. 1]). The definition can be extended to all values of zin the z-plane cut along the real axis from 1 to ∝ by the formula

[2, §1. 11(3)]. Then Lin(z) is regular in the cut plane, and there is a differential recurrence relation [4, p. 169]

It is convenient to extend the sequence Lin(z) backwards in the manner suggested by (2) and define

Then Li1(z)= – log(l–z), and Lin(z) is a rational function of z for n= 0, – 1, – 2,…. Formula (2) now holds for all integers n.

In Section 33 of [2], Bonsall and Duncan define an element t of a Banach algebra to act compactly on if the map a → tat is a compact operator on . In this paper, the arguments and technique of [1] are used to study this question for C*-algebras (see also [10]). We determine the elements b of a C*-algebra for which the maps a→ba, a→ab, a→ab + ba, a→bab are compact (respectively weakly compact), determine the C*-algebras which are compact in the sense of Definition 9, of [2, p. 177] and give a characterization of the *-automorphisms of which are weakly compact perturbations of the identity.

A hereditarily indecomposable asymptotic $\ell_2$ Banach space is constructed. The existence of such a space answers a question of B. Maurey and verifies a conjecture of W. T. Gowers.

In present paper we define a new kind of quantized enveloping algebra of (2). We denote this algebra by Ur,t, where r, t are two non-negative integers. It is a non-commutative and non-cocommutative Hopf algebra. If r = 0, then the algebra Ur,t is isomorphic to a tensor product of the algebra of infinite cyclic group and the usual quantum enveloping algebra of (2) as Hopf algebras. The representation of this algebra is studied.

A module M over a commutative ring R has an almost trivial dual if there is no homomorphism from M onto a free R-module of countable infinite rank. Using a new combinatorial principle (the ℵn-Black Box), which is provable in ordinary set theory, we show that for every natural number n, there exist arbitrarily large ℵn-free R-modules with almost trivial duals, when R is a complete discrete valuation domain. A corresponding result for torsion modules is also obtained.

Meta-centralizers of non-locally compact group algebras are studied. Theorems about their representations with the help of families of generalized measures are proved. Isomorphisms of group algebras are investigated in relation with meta-centralizers.

We show, under some natural restrictions, that orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime p. We also show that for all but finitely many initial points either the multiplicative order of this point or the length of the orbit it generates (both modulo a large prime p) is large. The approach is based on the results of Dvornicich and Zannier (Duke Math. J.139 (2007), 527–554) and Ostafe (2017) on roots of unity in polynomial orbits over the algebraic closure of the field of rational numbers.

In this paper, we develop the theory of stratifying systems in the context of exact categories as a generalisation of the notion of stratifying systems in module categories, which have been studied by different authors. We prove that attached to a stratifying system in an exact category $(\mathcal{A},\mathcal{E})$ there is an standardly stratified algebra B such that the category $\mathscr{F}$F(Θ), of F-filtered objects in the exact category $(\mathcal{A},\mathcal{E})$ is equivalent to the category $\mathscr{F}$(Δ) of Δ-good modules associated to B. The theory we develop in exact categories, give us a way to produce standardly stratified algebras from module categories by just changing the exact structure on it. In this way, we can construct exact categories whose bounded derived category is equivalent to the bounded derived category of an standardly stratified algebra. Finally, applying the relative homological algebra developed by Auslander–Solberg, we can construct examples of stratifying systems that are not a stratifying system in the classical sense, so our approach really produces new stratifying systems.

Except for blocks with a cyclic or Klein four defect group, it is not known in general whether the Morita equivalence class of a block algebra over a field of prime characteristic determines that of the corresponding block algebra over a p-adic ring. We prove this to be the case when the defect group is quaternion of order 8 and the block algebra over an algebraically closed field k of characteristic 2 is Morita equivalent to kÃ4. The main ingredients are Erdmann's classification of tame blocks [6] and work of Cabanes and Picaronny [4, 5] on perfect isometries between tame blocks.

A Banach space E is said to have Property (w) if every operator from E into E' is weakly compact. This property was introduced by E. and P. Saab in [9]. They observe that for Banach lattices, Property (w) is equivalent to Property (V*), which in turn is equivalent to the Banach lattice having a weakly sequentially complete dual. Thus the following question was raised in [9].

Does every Banach space with Property (w) have a weakly sequentially complete dual, or even Property (V*)?

In this paper, we give two examples, both of which answer the question in the negative. Both examples are James type spaces considered in [1]. They both possess properties stronger than Property (w). The first example has the property that every operator from the space into the dual is compact. In the second example, both the space and its dual have Property (w). In the last section we establish some partial results concerning the problem (also raised in [9]) of whether (w) passes from a Banach space E to C(K, E).

As in [4], a specification is a list (r, s, t1 h0, hmc (l), …, c(ho)) such that

(i) each of r, s, t1 h0, h∞ is a non-negative integer,

(ii) for each i, c(i) is a positive integer,

(iii) if h∞ = 0 then ho = ∞,

(iv) if h∞ = 1 and t1 + h0 is finite then t1 is even,

(v) r + s + t1 + h0 + h∞ = ∞.

In [7], Z. Tang and H. Zakeri introduced the concept of co-Cohen-Macaulay Artinian module over a quasi-local commutative ring R (with identity): a non-zero Artinian R-module A is said to be a co-Cohen-Macaulay module if and only if codepth A = dim A, where codepth A is the length of a maximalA-cosequence and dimA is the Krull dimension of A as defined by R. N. Roberts in [2]. Tang and Zakeriobtained several properties of co-Cohen-Macaulay Artinian R-modules, including a characterization of such modules by means of the modules of generalized fractions introduced by Zakeri and the present second author in [6]; this characterization is explained as follows.

We study the duality problem for order weakly compact operators by giving sufficient and necessary conditions under which the order weak compactness of an operator implies the order weak compactness of its adjoint and conversely.

Let $G(o)$ and $G(*)$ be two groups of finite order $n$, and suppose that each of the sets $\{u\in G;\ uo v=u*v$ for all $v\in G\}$ and $\{v\in G;\ uo v=u*v$ for all $u\in G\}$ has $n/2$ elements. Then $G(*)$ can be obtained from $G(o)$ by one of the two general constructions that are discussed in the paper.