We show that, under certain conditions, a soluble variety of groups which does not contain the variety of all metabelian groups is a finite exponent by nilpotent by finite exponent variety.

Let f ∈ ℤ[X] and let q be a prime power pl(l ≥ 2). Hua stated and proved that

for some unspecified constant C > 0 depending on the derivative f′ of f; M denoting the maximum multiplicity of the roots of the congruence p−tf′(x) ≡ 0 (mod p), where t is an integer chosen so that the polynomial p−tf′(x) is primitive. An explicit value for C was given by Chalk for p ≥ 3. Subsequently, Ping Ding (in two successive articles) obtained better estimates for p ≥ 2.

This article provides a better result, based upon a more precise form of Hua's main lemma, previously overlooked.

The rings of this paper are assumed to have relations of orthogonality defined on them. Such relations are uniquely determined by complete boolean algebras of ideals. Using the Stone space of these boolean algebras, and following J. Dauns and K.H. Hofmann, a sheaf-theoretic representation is obtained for rings with orthogonality relations, and the rings of global sections of these sheaves are characterized. Baer rings, f–rings and commutative semi-prime rings have natural orthogonality relations and among these the Baer rings are isomorphic to their associated rings of global sections. A special type of ideal is singled out in commutative semi-prime rings and following G. Spirason and E. Strzelecki, in an unpublished note, a characterization of a class of such rings is obtained.

In this note, we shall prove a new variational inequality in non-compact sets and as an application, we prove a generalisation of the Schauder-Tychonoff fixed point theorem.

The object of this paper is to study common fixed points of mappings of a complete metric space into itself. The results obtained are generalizations of Ray Theorems.

In the investigation of factorised groups one often encounters groups G = AB = AK = BK which have a triple factorisation as a product of two subgroups A and B and a normal subgroup K of G. It is of particular interest to know whether G satisfies some nilpotency requirement whenever the three subgroups A, B and K satisfy this same nilpotency requirement. A positive answer to this problem for the classes of nilpotent, hypercentral and locally nilpotent groups is given under the hypothesis that K is a minimax group or G has finite abelian section rank. The results become false if K has only finite Prüfer rank. Some applications of the main theorems are pointed out.

In this paper, compositions of a natural number are studied. The number of restricted compositions is given in a closed form, and some applications are presented.

In 1956, Jeśmanowicz conjectured that, for any positive integers $m$ and $n$ with $m>n$, $\gcd (m,n)=1$ and $2\nmid m+n$, the Diophantine equation $(m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$ has only the positive integer solution $(x,y,z)=(2,2,2)$. In this paper, we prove the conjecture if $4\nmid mn$ and $y\geq 2$.

This note shows that the variety generated by Murskii's 3-element algebra contains an infinite descending chain of subvarieties, thus falsifying a conjecture made previously by the author and M.R. Vaughan-Lee.

We prove a general identity between power series and use this identity to give proofs of a number of identities proposed by M.D. Hirschhorn. We also use the identity to give proofs of a well-known result of Jacobi, the quintuple-product identity and Winquist's identity.

It is shown that, in a variety which does not contain all metabelian groups and is contained in a product of (finitely many) varieties each of which is soluble or locally finite, every group is an extension of a group of finite exponent by a nilpotent group by a group of finite exponent.

Given an integer d≥2, a d-normal number, or simply a normal number, is an irrational number whosed-ary expansion is such that any preassigned sequence, of length k≥1, taken within this expansion occurs at the expected limiting frequency, namely 1/dk. Answering questions raised by Igor Shparlinski, we show that 0.P(2)P(3)P(4)…P(n)… and 0.P(2+1)P(3+1)P(5+1)…P(p+1)…, where P(n) stands for the largest prime factor of n, are both normal numbers.

It is shown that the Petrov-Plebański classification of the trace-free Ricci tensors of some spherically symmetric metrics is invariant, contrary to an assertion by Takeno and Kitamura concerning these metrics.

Let T be a mapping from a set X into itself and let H(X) be a functional Hilbert space on the set X. Then the composition operator CT on H(X) induced by T is a bounded linear transformation from H(X) into itself defined by CTf = f ∘ T. In this paper composition operators are characterized in the case when H(X) = H2(π+) in terms of the behaviour of the inducing functions in the vicinity of the point at infinity. An estimate for the lower bound of ∥CT∥ is given. Also the invertibility of CT is characterized in terms of the invertibility of T.

In this paper we study the symmetric powers of quantum GLn. These are comodules and we give a complete description of the subcomodule structure.

Let ϕ and ψ be analytic self-maps of the open unit disk. Each of them induces a composition operator, Cϕ and Cψ respectively, acting between weighted Bergman spaces of infinite order. We show that the difference Cϕ−Cψ is compact if and only if both operators are compact or both operators are not compact and the pseudohyperbolic distance of the functions ϕ and ψ tends to zero if ∣ϕ(z)∣→1 or ∣ψ(z)∣→1.

Let A be an algebra of formal power series in one indeterminate over the complex field, D a derivation on A. It is shown that if A has a Fréchet space topology under which it is a topological algebra, then D is necessarily continuous provided the coordinate projections satisfy a certain equicontinuity condition. This condition is always satisfied if A is a Banach algebra and the projections are continuous. A second result is given, with weaker hypothesis on the projections and correspondingly weaker conclusion.

We give a sufficient condition for normal structure more general than the well known ɛ0(X) < 1. Moreover we obtain sufficient conditions for the fixed point property for some B-convex Banach spaces.

We consider the logistic equation − Δu = a (x) u − b (x) up on all of RN with possibly unbounded coefficients near infinity. We show that under suitable growth conditions of the coefficients, the behaviour of the positive solutions of the logistic equation can be largely determined. We also show that certain linear eigenvalue problems on all of RN have principal eigenfunctions that become unbounded near infinity at an exponential rate. Using these results, we finally show that the logistic equation has a unique positive solution under suitable growth restrictions for its coefficients.

We present some of the group theoretic properties of reversing symmetry groups, and classify their structure in simple cases that occur frequently in several well-known groups of dynamical systems.