This paper consists of two parts. The first is to study the existence of a point a at the intersection of the Julia set and the escaping set such that a goes to infinity under iterates along Julia directions or Borel directions. Additionally, we find such points that approximate all Borel directions to escape if the meromorphic functions have positive lower order. We confirm the existence of such slowly escaping points under a weaker growth condition. The second is to study the connection between the Fatou set and argument distribution. In view of the filling disks, we show nonexistence of multiply connected Fatou components if an entire function satisfies a weaker growth condition. We prove that the absence of singular directions implies the nonexistence of large annuli in the Fatou set.

We consider the influence on a group G of the condition that every infinite set of cyclic subgroups of G contains a pair that permute and prove (Theorem 1) that finitely generated soluble groups with this condition are centre-by-finite, and (Theorem 2) that torsion free groups satisfying the condition are abelian.

These notes provide a concise introduction to the representation theory of reductive algebraic groups in positive characteristic, with an emphasis on Lusztig's character formula and geometric representation theory. They are based on the first author's notes from a lecture series delivered by the second author at the Simons Centre for Geometry and Physics in August 2019. We intend them to complement more detailed treatments.

A group is called an s-group if it is locally finite and all its Sylow subgroups are abelian. Kovács [4] has shown that, for any positive integer e, the class se of all s-groups of exponent dividing e is a (locally finite) variety. The proof of this relies on the fact that, for any e, there are only finitely many (isomorphism classes of) non-abelian finite simple groups in se; and this is a consequence of deep results of Walter and others (see [6]). In [2], Christensen raised the finite basis question for the laws of the varieties se. It is easy to establish the finite basis property for an se which contains no non-abelian finite simple group; and Christensen gave a finite basis for the laws of the variety s30, whose only non-abelian finite simple group is PSL(2,5). Here we prove Theorem For any positive integer e, the varietysehas a finite basis for its laws.

Assume that $A$ is a finite-dimensional algebra over some field, and assume that $A$ is weakly symmetric and indecomposable, with radical cube zero and radical square nonzero. We show that such an algebra of wild representation type does not have a nonprojective module $M$ whose ext-algebra is finite dimensional. This gives a complete classification of weakly symmetric indecomposable algebras which have a nonprojective module whose ext-algebra is finite dimensional. This shows in particular that existence of ext-finite nonprojective modules is not equivalent with the failure of the finite generation condition (Fg), which ensures that modules have support varieties.

In [1] it was shown if S is a stable semigroup and H an ℋ-class of S then there is a congruence (H) on S in which H is a single class. After considering some consequences of this result for abstract semigroups, we consider some analogous questions for compact semigroups.

The completions of certain nilpotent groups with respect to some ascending sequences of integral domains are constructed. These completions are generalizations of Lazard completions for the groups under consideration and they are Lie algebras over the first integral domain in the sequence. The construction is possible in particular for finite p-groups of exponent p and class < p.

A close analogue for some hypergroup measure algebras of the structure semigroup theorem of J. L. Taylor for convolution measure algebras is constructed: a structure semihypergroup representation is made for the hypergroup measure and its spectrum. This is done for those hypergroup measure algebras that satisfy a condition known as the structure-strong condition. This condition is that the norm-closure of the linear span of the spectrum of the hypergroup measure algebra is a commutative B*-algebra. Then examples of hypergroups whose measure algebras satisfy this condition are given. They include the space of B-orbits of G, where B is a finite solvable group of automorphisms on a locally compact abelian group G. (The hypergroup measure algebra may be identified with the algebra of B-invariant measures on G.) Other examples are the algebra of central measures on a compact, connected, semisimple Lie group, and the algebra of rotation invariant measures on the plane.

Let k and r be fixed integers such that 1 < r < k. It is well-known that a positive integer is called r-free if it is not divisible by the r-th power of any integer > 1. We call a positive integer n, a (k, r)-integer, if n is of the form n = a kb, where a is a positive integer and b is a r-free integer. In the limiting case, when k becomes infinite, a (k, r)-integer becomes a r-free integer and so one might consider the (k, i) integers as generalized r-free integers.

Let G be a unitary group of rank one over a non-archimedean local field K (whose residue field has a characteristic ≠ 2). We consider the action of G on the projective plane. A G(K) equivariant map from the set of points in the projective plane that are semistable for every maximal K split torus in G to the set of convex subsets of the building of G(K) is constructed. This map gives rise to an equivariant map from the set of points that are stable for every maximal K split torus to the building. Using these maps one describes a G(K) invariant pure affinoid covering of the set of stable points. The reduction of the affinoid covering is given.

This paper is concerned with the three-parameter generalized gamma distribution (g.g.d.) which is widely employed as a model in life testing. The structural probability distributions of the parameters and a number of structural prediction densities for specific future measurements have been derived based on type-Il progressively censored sample.