Let G be a locally compact topological group. A number of characterizations are given of the class of compact groups in terms of the geometric properties such as Radon-Nikodym property, Dunford-Pettis property and Schur property of Ap(G), and the properties of the multiplication operator on PFp(G). We extend and improve several results of Lau and Ülger [17] to Ap(G) and Bp(G) for arbitrary p.

Let A be a finite dimensional associative K-algebra with an identity over an algebraically closed field K, d a natural number, and modA(d) the affine variety of d-dimensional A-modules. The general linear group Gld(K) acts on modA(d) by conjugation, and the orbits correspond to the isomorphism classes of d-dimensional modules. For M and N in modA(d), N is called a degeneration of M, if TV belongs to the closure of the orbit of M. This defines a partial order ≤deg on modA(d). There has been a work [1], [10], [11], [21] connecting ≤deg with other partial orders ≤ext and ≤deg on modA(d) defined in terms of extensions and homomorphisms. In particular, it is known that these partial orders coincide in the case A is representation-finite and its Auslander-Reiten quiver is directed. We study degenerations of modules from the additive categories given by connected components of the Auslander-Reiten quiver of A having oriented cycles. We show that the partial orders ≤ext, ≤deg and < coincide on modules from the additive categories of quasi-tubes [24], and describe minimal degenerations of such modules. Moreover, we show that M ≤degN does not imply M ≤ext N for some indecomposable modules M and N lying in coils in the sense of [4].

The purpose of this paper is to characterize the radical ideals of principal ideal domains and Dedekind domains. We show that if T is a radical class and R is a PID, then T(R) is an intersection of prime ideals of R. More specifically, if

then T(R) = (p1p2 … pk), where p1, p2, … , pk are distinct primes, and where (p1p2 … Pk) denotes the principal ideal of R generated by p1p2 … pk. We also characterize the radical ideals of commutative principal ideal rings. For radical ideals of Dedekind domains we obtain a characterization similar to the one given for PID's.

A subset $X$ of a Polish group $G$ is called Haar null if there exist a Borel set $B\,\supset \,X$ and Borel probability measure $\mu$ on $G$ such that $\mu \left( g\,Bh \right)\,=\,0$ for every $g,\,h\,\in \,G$. We prove that there exist a set $X\,\subset \,\text{R}$ that is not Lebesgue null and a Borel probability measure $\mu$ such that $\mu \left( X\,+\,t \right)\,=\,0$ for every $t\,\in \,\text{R}$. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.)

This result motivates the following Baire category analogue. It is consistent with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor set $C\,\subset \,G$ such that for every non-meagre set $X\,\subset \,\text{G}$ there exists a $t\in \text{G}$ such that $C\,\cap \,\left( X\,+\,t \right)$ is relatively non-meagre in $C$. This essentially generalizes results of Bartoszyński and Burke–Miller.

Let $G$ be an affine Kac-Moody group, ${{\pi }_{0}},...,{{\pi }_{r}},{{\pi }_{\delta }}$ its fundamental irreducible representations and ${{\chi }_{0}},...,{{\chi }_{r}},{{\chi }_{\delta }}$ their characters. We determine the set of all group elements $x$ such that all ${{\pi }_{i}}(x)$ act as trace class operators, i.e., such that ${{\chi }_{i}}(x)$ exists, then prove that the ${{\chi }_{i}}$ are class functions. Thus, ($\chi \,:=\,({{\chi }_{0}},...,{{\chi }_{r}},\,{{\chi }_{\delta }})$)factors to an adjoint quotient $\bar{\chi }$ for $G$. In a second part, following Steinberg, we define a cross-section $C$ for the potential regular classes in $G$. We prove that the restriction $\chi \text{ }\!\!|\!\!\text{ }c$ behaves well algebraically. Moreover, we obtain an action of ${{\mathbb{C}}^{\times }}$ on $C$, which leads to a functional identity for $\text{ }\chi |\text{ c}$ which shows that $\text{ }\chi |\text{ c}$ is quasi-homogeneous.

Let L be a line bundle on a connected, smooth, algebraic, projective surface X. In this paper we have studied the following questions:

1) Under which conditions is L spanned by global sections? I.e., if ɸL : X →PN denotes the map associated to the space Г(L) of the sections of L, when is ɸL a morphism?

2) Under which conditions is L very ample? I.e., when does ɸL give an embedding?

These problems arise naturally in the study, and in particular in the classification, of algebraic surfaces (see [8], [3], [5]).

Horadam (11) has established the existence of a locus L in [8] (projective 8-space) having order 45 and dimension 4, which is invariant under a group of order 51840 X 81 (the Clifford similarity transform group CT). Associated with CT are two other groups, the Clifford collineation group CG of order 81, and the Clifford substitution group CS of order 51840. Furthermore, CS may be regarded as either a subgroup of CT, or a symplectic group of index matrices of size 4. Among the matrices of size 9 which perform the operations of CT, there is a set of 81 involutory, symmetric, orthogonal matrices JW. As collineation matrices in [8], these produce 81 pairs of invariant spaces Σ, Π of dimensions 3 and 4 respectively. These [4]'s give rise to a configuration C invariant under the operations of CT, consisting of 360 points, 1080 lines, 120 Jacobian planes, and 81 [4]'s, and their various inter-relationships.

All our notation is as denned in [2] with the restriction to n = 1. However, for our purposes, we introduce a sequence of norms by

in It is not difficult to see that turns out to be a fundamental space.

It is a well-known fact that the Watson transform and the Mellin transform are connected by the fact that

and

if and only if K(s)K(l — s) = 1, where K(s) is the Mellin transform of k(x). Further, the Hankel transform and Hilbert transform can be considered as special cases of Watson transforms.

Let $\sigma $, $\theta $ be commuting involutions of the connected semisimple algebraic group $G$ where $\sigma$, $\theta $ and $G$ are defined over an algebraically closed field $\underset{\scriptscriptstyle-}{k},$ char $\underline{k}$=0. Let $H:={{G}^{\sigma }}$ and $K:={{G}^{\theta }}$ be the fixed point groups. We have an action $\left( H\,\times \,K \right)\,\times \,G\,\to \,G$, where $\left( \left( h,\,k \right),\,g \right)\,\mapsto \,hg{{k}^{-1}},\,h\,\in \,H$ , $k\,\in \,K,g\,\in \,G$. Let $G\,//\,\left( H\,\times \,K \right)$ denote the categorical quotient Spec $\mathcal{O}{{(G)}^{H\times K}}$ . We determine when this quotient is smooth. Our results are a generalization of those of Steinberg [Ste75], Pittie [Pit72] and Richardson [Ric82] in the symmetric case where $\sigma \,=\,\theta $ and $H\,=K$.

We describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.

A group S is said to be metacyclic if it contains a normal cyclic subgroup N such that S/N is cyclic. In this note the following theorem is proved.

THEOREM. Let G be a group, S a metacyclic Sylow 2-subgroup of G. If S has a cyclic normal subgroup N such that S/N is cyclic of order greater than 2, then G is soluble.

Remark. We show that such a group G contains a 2-nilpotent normal subgroup of index a divisor of 6. The solubility of these groups requires the solubility of groups of odd order unavoidably.

Notation. All groups considered will be finite. Let G be a group, S a subset of G, A and B subgroups of G, N a normal subgroup of G.

〈S〉: the subgroup of G generated by S.

NG(S): the normalizer of S in G.

CG(S): the centralizer of S in G.

Let $\mathbf{R}/R$ be a quadratic extension of finite, commutative, local and principal rings of odd characteristic. Denote by ${{\mathbf{U}}_{n}}\left( \mathbf{R} \right)$ the unitary group of rank $n$ associated to $\mathbf{R}/R$. The Weil representation of ${{\mathbf{U}}_{n}}\left( \mathbf{R} \right)$ is defined and its character is explicitly computed.

Let

be an indefinite quadratic form in the integer variables x1, . . . , xn with real coefficients of determinant D = ||ars||(n) ≠ 0. The homogeneous minimum MH(Qn) and the inhomogeneous minimum MI(Qn) of Qn(x) are defined as follows :

The following problem has been suggested by Paul Turán. Let $\Omega $ be a symmetric convex body in the Euclidean space ${{\mathbb{R}}^{d}}$ or in the torus ${{\mathbb{T}}^{d}}$. Then, what is the largest possible value of the integral of positive definite functions that are supported in $\Omega $ and normalized with the value 1 at the origin? From this, Arestov, Berdysheva and Berens arrived at the analogous pointwise extremal problem for intervals in $\mathbb{R}$. That is, under the same conditions and normalizations, the supremum of possible function values at $z$ is to be found for any given point $z\,\in \,\Omega $. However, it turns out that the problem for the real line has already been solved by Boas and Kac, who gave several proofs and also mentioned possible extensions to ${{\mathbb{R}}^{d}}$ and to non-convex domains as well.

Here we present another approach to the problem, giving the solution in ${{\mathbb{R}}^{d}}$ and for several cases in ${{\mathbb{T}}^{d}}$. Actually, we elaborate on the fact that the problem is essentially one-dimensional and investigate non-convex open domains as well. We show that the extremal problems are equivalent to some more familiar ones concerning trigonometric polynomials, and thus find the extremal values for a few cases. An analysis of the relationship between the problem for ${{\mathbb{R}}^{d}}$ and that for ${{\mathbb{T}}^{d}}$ is given, showing that the former case is just the limiting case of the latter. Thus the hierarchy of difficulty is established, so that extremal problems for trigonometric polynomials gain renewed recognition.

Giordano, Putnam and Skau showed that the transformation group ${{C}^{*}}$-algebra arising from a Cantor minimal system is an $AT$-algebra, and classified it by its $K$-theory. For approximately inner automorphisms that preserve $C\left( X \right)$, we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto’s result.

We denote by ZG the integral group ring of the finite group G. S.D. Berman [1] showed that every unit of finite order μ in G is trivial (i.e., μ = ±g for some g in G) if and only if either G is abelian or G is a Hamiltonian 2-group. In this note, we give a new and shorter proof for the “only if” part.

A well-known result of Chebyshev is that if pn ∊ Pn, (Pn is the set of polynomials of degree at most n) and

(1)

then an(pn), the leading coefficient of pn, satisfies

(2)

with equality holding only for pn = ±Tn, where Tn is the Chebyshev polynomial of degree n. (See [6, p. 57].) This is an example of an extremal problem in which the norm of a given linear operator on Pn is sought. Another example is A. A. Markov's result that (1) implies that

(3)

There are also results for the linear functionals pn(k)(x0), x0 real, k = 1, … n – 1 ([8]).

Suppose φ(x) ≧ 0 on [–1, 1] and (1) is generalized to

as suggested by Rahman [4] (polynomials with curved majorants), what can then be said about the analogue of (3) or similar extremal problems?

It is a well known fact that for rapidly increasing nkthe sequence , behaves like a sequence of independent random variables; in particular has a limiting Gaussian distribution as N → ∞. Under a certain critical speed this result breaks down and becomes strongly dependent. The purpose of this paper is to investigate the asymptotic behavior of normed sums in the strongly dependent domain; specifically, we construct a large class of nongaussian limit distributions of such sums.

In order to deal with certain problems in the theory of interpolation spaces, it is convenient to consider operators of the following form:

Let k be a non-negative measurable function on the half-line R+, and let ƒ be a measurable function on R+ with

1

Then the operator T is defined by

2

with the domain of T, D(T), consisting of all ƒ which satisfy (1).

Results of a representation-theoretic nature have played a major role in topos theory since the beginnings of the subject. For example, Deligne's theorem on coherent toposes, which says that every coherent topos has a continuous embedding into a topos of the form SetI for a discrete set I, is a typical result in the representation theory of toposes. (A continuous functor between toposes is the left adjoint of a geometric morphism. For Grothendieck toposes, it is exactly the same as a continuous functor between them, considered as sites with their canonical topologies. By a continuous functor between sites on left exact categories, we mean a left exact functor taking covers to covers.)

A representation-like result for toposes typically asserts that a topos that satisfies some abstract conditions is related to a topos of some concrete kind; the relation between them is usually an embedding of the first topos in the second (concrete) one, for which the embedding satisfies some additional properties (fullness, etc.).