This paper presents a criterion of KUR for sequence Orlicź spaces with Luxemburg's norm. The result also indicates that for any integer k ≥ 1, there exists a k + 1-uniformly rotund Banach space not being k-uniformly rotund.

We prove that the composition of a minimal usco map, defined on a Baire space, with a lower semicontinuous function is single valued and usco at each point of a dense G$ subset of its domain. This extends earlier results of Kenderov and Fitzpatrick. As a first consequence, we prove that a Banach space, with the property that there exists a strictly convex, weak* lower semicontinuous function on its dual, is a weak Asplund space. As a second consequence, we present a short proof of the fact that a Banach space with separable dual is an Asplund space.

Some structural results about quotients and tensor products of hereditary respectively quasi-hereditary algebras are presented. They are related to properties of stratifications of derived module categories. The concept of derived-simplicity for an algebra is introduced and studied.

One result concerning the series representation for the continuous Jacobi transform in Walter and Zayed [1] is improved, the same thought also can be applied to the related results in [1].

We investigate extensions of pro-p-groups 1 —> N —> G —> Γ —> 1 where N is pro-p-free and Nab, is a free Zp[Γ]-module. In case Γ is finite we show that such an extension splits modulo the second derived group N".

Let d = 1,2, or p, prime p ≡ 3 (mod 4). Let Od be the ring of integers of an imaginary quadratic field A complete classification of conjugacy classes of maximal non-elementary Fuchsian subgroups of PSL(2, Od) in PGL(2, Od) is given.

It is shown that the equation xyz = x+y+z has unit solutions in only four not totally real cubic fields: two fields which are real and two fields which are imaginary. These fields are then listed.

Let G be a finite solvable group. Fix a prime integer p and let t be the number of distinct degrees of irreducible Brauer characters of G with respect to the prime p. We obtain the bound 3t — 2 for the derived length of a Hall p'-subgroup of G. Furthermore, if |G| is odd, then the derived length of a Hall p'-subgroup of G is bounded by /.

We show that the partial sums Snf of the Vilenkin-Fourier series of f ∊ L1 are of exponential type off any set where the Hardy-Littlewood maximal function of f is bounded. It then follows that Snkf(x) = o(log log nk) a.e. for any lacunary sequence {nk}. Our results are Vilenkin-Fourier series analogues of those of R. A. Hunt [1].