In this note I follow the same notation adopted in [2]. Let Qk(x, n) denote the number of k-free integers ≦x which are prime to n. In [2], I proved the following: For o≦s>1/k uniformly, where σ*−s(n) is the sum of the reciprocals of the sth powers of the square-free divisor of n.

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.

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.

In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke L-functions of imaginary quadratic number fields of class number 1. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27\ldots \%$ of the L-functions under consideration do not vanish at 1/2.

We examine what can be said about a polynomial p and an entire function f given that p o f is an even, or an odd, function.

We define potential operators on hyperplanes and give sharp mixed norm inequalities for them. One of the operators coincides with the Radon transform for which mixed norm estimates are known but in reverse order. Those inequalities will be crucial in our proofs.

The biharmonic Green's function of a simply supported plate is generalized to Riemannian manifolds and shown to exist if and only if the harmonic measure of the ideal boundary is square integrable.

There are some well-known laws that the commutator satisfies in groups, and that go by some or all of the names Jacobi, Witt, Hall; and there are also some lesser-known laws. This is an attempt at an axiomatic study of the interdependence and independence of these laws.

In this paper, we define the function space H1P(G) on a LCA group G with the algebraically ordered dual, and construct a multiplier on H1P(G) similar to the one given by Gaudry (1968).

Professor Sir Thomas Cherry, F.A.A., F.R.S., died at his home in Melbourne on 21st November 1966 at the age of 68. He was widely known and highly respected as Australia's most distinguished mathematician and a leader in university affairs. He was associated with the University of Melbourne for most of his life, and latterly with La Trobe University as first chairman of its Academic Planning Board. He was a foundation member and a president of both the Australian Academy of Science and the Australian Mathematical Society. His greatest contributions to knowledge were probably made in the mathematics of air flow in trans-sonic flight, simultaneously with Lighthill in Britain; but he also made major contributions to global differential equation theory and general dynamics, and solved some difficult special problems in various branches of applied mathematics. He was a most distinguished teacher, amongst whose students are numbered two Fellows of the Royal Society and several professors in Australian and overseas universities. He was a man of wide interests and great ability, of keen insight and broad vision. He knew much more than he ever wrote, and his influence will live on in the minds of innumerable people with whom he worked.

Given a semigroup S, we define {N(S), +, ·} to be the ‘free’ distributively generated near-ring. Since all words in N(S) can be expressed as the sum and difference of elements of S, we are able to define a length function on the words of N(S). The following theorems then follow: Theorem 1. N(S) contains a multiplicatice identity e if and only if e ∈ S. Theorem 2. If S is the free semigroup in the cariety of all semi—groups then N(S) is left cancellatice.

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.

A subgroup H of an abelian p–group G is pure in G if the inclusion map of H into G is an isometry with respect to the (pseudo-) metrics on H and G associated with their p–adic topologies. In this paper, those subgroups (called here imbedded subgroups) of abelian groups for which the inclusion is a homeomorphism with respect to the p–adic topologies are studied, the aim being to compare the concepts of imbeddedness and purity. Perhaps the main results indicate that imbedded subgroups are considerably more abundant than pure subgroups. Groups for which this is not the case are characterized.

In this paper, we study the boundary blow-up problem related to the infinity Laplacian

$$ \begin{align*}\begin{cases} \Delta_{\infty}^h u=u^q &\mathrm{in}\; \Omega, \\ u=\infty &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$

where $\Delta _{\infty }^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator associated with the infinity Laplacian arising from the stochastic game named Tug-of-War. When $q>h>1$ , we establish the existence of the boundary blow-up viscosity solution. Moreover, when the domain satisfies some regular condition, we establish the asymptotic estimate of the blow-up solution near the boundary. As an application of the asymptotic estimate and the comparison principle, we obtain the uniqueness result of the large solution. We also give the nonexistence of the large solution for the case $q \leq h.$

A locally convex space E is said to be ordered suprabarrelled if given any increasing sequence of subspaces of E covering E there is one of them which is suprabarrelled. In this paper we show that the space m0(X, Σ), where X is any set and Σ is a σ-algebra on X, is ordered suprabarrelled, given an affirmative answer to a previously raised question. We also include two applications of this result to the theory of vector measures.

An element in a free group is a proper power if and only if it is a proper power in every nilpotent factor group. Moreover there is an algorithm to decide if an element in a finitely generated torsion-free nilpotent group is a proper power.

The aim of the present paper is to introduce a definition of the Perron-Stieltjes integral employing the notion of approximate derivative with respect to a nondecreasing function ω and to study some of the properties of the integral. Various authors have studied the Perron integral and Perron-Stieltjes integral in different ways, most of which can be found in the references appended in the list of the bibliography. Among them Ridder [10] uses the concept of approximate co-derivative but he assumes that the monotone function a associated with co is continuous. Finally we consider a more general type of integral, the co-approximately continuous Denjoy-Stieltjes integral, defined descriptively by the method of Saks [11].

If U and V are toplogies on an abstract set x, then the triple (X, U, V) is a bitopologica space. Using the theorem of Priestley on the representation of distributive lattices, results of Dilworth concerning the normal completion of the lattice of bounded, continuous, realvalued functions on a topological space are extended to include the lattice of bounded, semi-continuous, real-valued functions on certain bitopological spaces. The distributivity of certain lattices is investigated, and the theorem of Funayama on distributive normal completions is generalized.

It has been shown that the assumption of a specified cosmological substratum leads to the Lorentz-equivalencc of observers whose uniform motion in the substratum lies along a common straight line. The result is generalised for any pair of unaccelerated observers thus confirming that only the relative velocity of such observers is relevant in relating their observations of an event.