Let S be an inverse semigroup with semilattice of idempotents E, and let ρ be a congruence on S. Then ρ is said to be idempotent-determined [2], or I.D. for short, if (a, b) ∈ р and a∈E imply that b ∈ E. If, further, ρ is a group congruence, then clearly ρ is the minimum group congruence on S, and in this case S is said to be proper [8]. Let T = S/ρ.

The research presented in this paper started by extending a theorem of Swetits [18]about barrelledness of subspaces of metrizable AK-spaces to general AK-spaces of scalar sequences. The extension reads as follows.

(1) A subspace λ0 of a barrelled AK-space λ such that λ0 ⊃ φ is barrelled if and only if its dualis weak* sequentially complete. If in addition λ0 is monotone, then it is barrelled if and only ifequals the Köthe dualof λ0.

As an easy consequence of this extension, we obtained the following result of Elstrodt and Roelcke [8, Corollary 3.4].

(2) If λ is a barrelled monotone AK-space, then also its subspace ℒ(λ), consisting of all sequences in λ with zero-density support, is barrelled.

1. A discussion of the Erdélyi-Kober operator

and its adjoint operator Kη,α together with an account of some of their applications can be found in the book by Sneddon [5].

In this paper we give new information about the conjugacy vector of the group , the Sylow p-subgroup of GL(n, q) consisting of the upper unitriangular matrices. The first two components of this vector are given in [4]. Here, we obtain the third component, that is, the number of conjugacy classes whose centralizer has qn+l elements. Besides, we give the whole set of numbers which compose this vector:

The study of pure-injectivity is accessed from an alternative point of view. A module M is called pure-subinjective relative to a module N if for every pure extension K of N, every homomorphism N → M can be extended to a homomorphism K → M. The pure-subinjectivity domain of the module M is defined to be the class of modules N such that M is N-pure-subinjective. Basic properties of the notion of pure-subinjectivity are investigated. We obtain characterizations for various types of rings and modules, including absolutely pure (or, FP-injective) modules, von Neumann regular rings and (pure-) semisimple rings in terms of pure-subinjectivity domains. We also consider cotorsion modules, endomorphism rings of certain modules, and, for a module N, (pure) quotients of N-pure-subinjective modules.

In this paper, we investigate linear Weingarten hypersurfaces with two distinct principal curvatures in a real space form Mn+1(c), we obtain two rigidity results and give some characterization of the Riemannian product Sk(a) × Sn−k(), 1 ≤ k ≤ n − 1 in Mn+1(c)(c = 1), the Riemannian product Rk × Sn−k(a), 1 ≤ k ≤ n −1 in Mn+1(c)(c = 0) and the Riemannian product Hk(tanh2 ρ−1) × Sn−k(coth2 ρ−1), 1 ≤ k ≤ n −1 in Mn+1(c)(c = −1).

In a previous paper [7], I have made a study of the ”nilpotent” analogue of Hurwitz theorem [4] by considering a particular family of signatures called ”nilpotent admissible” [5]. We saw however, that if μN(g) represents the order of the largest nilpotent group of automorphisms of a surface of genus g < 2, then μN(g) < 16(g − 1) and this upper bound occurs when the covering group is a triangle group having the signature (0; 2,4,8) which is in its own 2-local form

The restriction to the nilpotent groups enabled me to obtain much more precise information than was available in the general case. Moreover, all nilpotent groups attaining this maximum order turned out to be ”2-groups”. Since every finite nilpotent group is the direct product of its Sylow subgroups and the groups of automorphisms are factor groups of the Fuchsian groups, it is natural for us to study the Fuchsian groups havin p-local signatures to obtain more precise information about the finite p-groups, and hence about the finite nilpotent groups.

This suggests a new problem of determining for each prime p, the “p-group” analogue of Hurwitz theorem. It turns out, as often happens in questions of this nature, that p = 2 and p = 3 are indeed quite exceptional and harder to deal with while computing their lower central series than other primes. Actually, p = 3 is the most difficult, but all the other primes p ≥ 5 can be dealt with at once.

We give an easy diagrammatical description of the parabolic Kazhdan–Lusztig polynomials for the Weyl group Wn of type Dn with parabolic subgroup of type An and consequently an explicit counting formula for the dimension of morphism spaces between indecomposable projective objects in the corresponding category . As a by-product we categorify irreducible Wn-modules corresponding to the pairs of one-line partitions. Finally, we indicate the motivation for introducing the combinatorics by connections to the Springer theory, the category of perverse sheaves on isotropic Grassmannians, and to the Brauer algebras, which will be treated in two subsequent papers of the second author.

In this paper we obtain a Hermite-Hadamard type inequality for a class of subharmonic functions. Our proofs rely essentially on the properties of elliptic partial differential equations of second order. Our study extends some recent results from [1], [2] and [6].

This paper is mainly dedicated to describing the congruences on certain monoids of transformations on a finite chain $X_n$ with $n$ elements. Namely, we consider the monoids $\od_n$ and $\mpod_n$ of all full, respectively partial, transformations on $X_n$ that preserve or reverse the order, as well as the submonoid $\po_n$ of $\mpod_n$ of all its order-preserving elements. The inverse monoid $\podi_n$ of all injective elements of $\mpod_n$ is also considered.

We show that in $\po_n$ any congruence is a Rees congruence, but this may not happen in the monoids $\od_n$, $\podi_n$ and $\mpod_n$. However in all these cases the congruences form a chain.

We give two types of generalisation of the well-known Fuglede–Putnam theorem. The paper is ‘spiced up’ with some examples and applications.

Let Ω: be a compact Hausdorff space, let E be a Banach space, and let C(Ω, E) stand for the Banach space of continuous E-valued functions on Ω under supnorm. It is well known [3, p. 182] that if F is a Banach space then any bounded linear operator T:C(Ω, E)→ F has a finitely additive vector measure G defined on the σ-field of Borel subsets of Ω with values in the space ℒ(E, F**) of bounded linear operators from E to the second dual F** of F. The measure G is said to represent T. The purpose of this note is to study the interplay between certain properties of the operator T and properties of the representing measure G. Precisely, one of our goals is to study when one can characterize nuclear operators in terms of their representing measures. This is of course motivated by a well-known theorem of L. Schwartz [5] (see also [3, p. 173]) concerning nuclear operators on spaces C(Ω) of continuous scalar-valued functions. The study of nuclear operators on spaces C(Ω, E) of continuous vector-valued functions was initiated in [1], where the author extended Schwartz's result in case E* has the Radon-Nikodym property. In this paper, we will show that the condition on E* to have the Radon-Nikodym property is necessary to have a Schwartz's type theorem. This leads to a new characterization of dual spaces E* with the Radon-Nikodym property. In [2], it was shown that if T:C(Ω, E)→ F is nuclear than its representing measure G takes its values in the space (E, F) of nuclear operators from E to F. One of the results of this paper is that if T:C(Ω, E)→ F is nuclear then its representing measure G is countably additive and of bounded variation as a vector measure taking its values in (E, F) equipped with the nuclear norm. Finally, we show by easy examples that the above mentioned conditions on the representing measure G do not characterize nuclear operators on C(Ω, E) spaces, and we also look at cases where nuclear operators are indeed characterized by the above two conditions. For all undefined notions and terminologies, we refer the reader to [3].

We study a non-commutative version of the Kadomtsev-Petviashvili equations and construct a family of solutions generalizing naturally the soliton to the non-commutative setting. From this we derive explicit solution formulas as well for the scalar as for the matrix-Kadomtsev-Petviashvili equation which still depend on operator parameters.

Extending a result of Boyle and Goodearl in [1] on V-rings it was shown in Yousif [11] that a generalized V-module (GV-module) has Krull dimension if and only if it is noetherian. Our note is based on the observation that every GV-module has a maximal submodule (Lemma 1). Applying a theorem of Shock [6] we immediately obtain that a GV-module has acc on essential submodules if and only if for every essential submodule K ⊂ M the factor module M/K has finitely generated socle. Yousif's result is obtained as a corollary.

Let R be an associative ring with unity and R-Mod the category of unital left R-modules. Soc M denotes the socle of an R-module M. If K ⊂ M is an essential submodule we write K⊴M.

An R-module M is called co-semisimple or a V-module, if every simple module is M-injective ([2], [7], [9], [10]). According to Hirano [3] M is a generalized V-module or GV-module, if every singular simple R-module is M-injective. This extends the notion of a left GV-ring in Ramamurthi-Rangaswamy [5].

It is easy to see that submodules, factor modules and direct sums of co-semisimple modules (GV-modules) are again co-semisimple (GV-modules) (e.g. [10, § 23]).

Let Gbe a finite group, and P:G → GL(n, ) be such that for all x, y ∈ G

(i) P(x)P(y) = α(x, y)P(xy), and

(ii)P(l) = In,

where α(x, y) ∈ *; then P is a projective representation of G with cocycle α and degree n. For other basic definitions concerning projective representations see [4].

Let Σ be a σ-algebra of subsets of some set Ω and let μ:Σ→[0,∞] be a σ-additive measure. If Σ(μ) denotes the set of all elements of Σ with finite μ-measure (where sets equal μ-a.e. are identified in the usual way), then a metric d can be defined in Σ(μ) by the formula

here E ΔF = (E\F) ∪ (F\E) denotes the symmetric difference of E and F. The measure μ is called separable whenever the metric space (Σ(μ), d) is separable. It is a classical result that μ is separable if and only if the Banach space L1(μ), is separable [8, p.137]. To exhibit non-separable measures is not a problem; see [8, p. 70], for example. If Σ happens to be the σ-algebra of μ-measurable sets constructed (via outer-measure μ*) by extending μ defined originally on merely a semi-ring of sets Γ ⊆ Σ, then it is also classical that the countability of Γ guarantees the separability of μ and hence, also of L1(μ), [8, p. 69].

Coxeter [1] has studied groups defined by the relations

and gives lists of finite groups known to be completely defined by such sets of relations. In a later paper [2] he shows that G3, n, p is finite if n, p are both even and satisfy

and expresses the conjecture that the restriction to even values may be removed. The only case satisfying this inequality and not already known to be finite is G3, 7, 16. In this note we show that G3, 7, 16 is indeed finite, being of order 21504 = 210.3.7, by showing that its subgroup

of index 2 is finite and of order 10752. Thus we add one entry to each of the lists of finite groups in Coxeter [1].

We prove that the conditions R·R=0 and R·S=0 are equivalent for hypersurfaces of a 5-dimensional semi-Riemannian space form N5(c). This solves a problem by P.J. Ryan in the case of hypersurfaces of dimension 4 in semi-Riemannian space forms.

Semilocal and related classes of group rings have been investigated by many authors (cf. [10]). In particular, the following results have been obtained.

Theorem A[4,10]. Let K be a field and G a group.

(i) If ch K = 0, then K[G] is semilocal if and only if G is finite.

(ii) If ch K = p>0 and G is locally finite, then K[G] is semilocal if and only if G contains a p-subgroup of finite index.

In the case of semigroup rings some stronger conditions have been studied. Munn examined the semisimple artinian situation [6]. Zelmanov showed that if K[G]is artinian then G must be finite [11].

In a recent paper M. Cho [5] asked whether Taylor's joint spectrum σ(a1, …, an; X) of a commuting n-tuple (a1,…, an) of continuous linear operators in a Banach space X is contained in the closure V(a1, …, an; X)- of the joint spatial numerical range of (a1, …, an). Among other things we prove that even the convex hull of the classical joint spectrum Sp(a1, …, an; 〈a1, …, an〉), considered in the Banach algebra 〈a1, …, an〉, generated by a1, …, an, is contained in V(a1, …, an; X)-.