Let Hn be the Heisenberg group of dimension 2n + 1. Let ℒ1,…,ℒn be the partial sub-Laplacians on Hn and T the central element of the Lie algebra of Hn. We prove that the operator m (ℒ1,…,ℒn,−iT) is bounded on Lp (Hn), 1 < p < +∞, if the function m satisfies a Marcinkiewicz-type condition in Rn+1.

Determining meaningful lower bounds on the supremal strict p-negative type of classes of finite metric spaces is a difficult nonlinear problem. In this paper we use an elementary approach to obtain the following result: given a finite metric space (X,d) there is a constant ζ>0, dependent only on n=∣X∣ and the scaled diameter 𝔇=(diamX)/min{d(x,y)∣x⁄=y} of X (which we may assume is >1), such that (X,d) has p-negative type for all p∈[0,ζ] and strict p-negative type for all p∈[0,ζ). In fact, we obtain A consideration of basic examples shows that our value of ζ is optimal provided that 𝔇≤2. In other words, for each 𝔇∈(1,2] and natural number n≥3, there exists an n-point metric space of scaled diameter 𝔇 whose supremal strict p-negative type is exactly ζ. The results of this paper hold more generally for all finite semi-metric spaces since the triangle inequality is not used in any of the proofs. Moreover, ζ is always optimal in the case of finite semi-metric spaces.

This paper gives a very general sufficient condition for the existence of constants B(n), C(n) for which either almost surely or almost surely, where Sn = X1 + X2 + … + Xn and Xi are independent and identically distributed random variables. The theorem is closely connected with results of Klass and Teicher on the one-sided boundedness of Sn, with the relative stability of Sn, and with a generalised law of the iterated logarithm due to Kesten. For non negative Xi the sufficient condition is shown to be necessary, and the results are partially generalised to the case when Xi form a stationary m-dependent sequence. Some connections with a generalised type of regular variation and with domains of partial attraction are also noted.

Let f: T → T be a tree map with n end-points, SAP(f) the set of strongly almost periodic points of f and CR(f) the set of chain recurrent points of f. Write E(f,T) = {x: there exists a sequence {ki} with 2 ≤ ki ≤ n such that and g = f\CR(f). In this paper, we show that the following three statements are equivalent:

(1) f has zero topological entropy.

(2) SAP(f) ⊂ E(f,T).

(3) Map ωg: x → ω(x,g) is continuous at p for every periodic point p of f.

In the paper “The topological degree of A-proper mapping in the Menger PN-space (I)”, the new concept of A-proper topological degree has been given. Now, utilising the new concept, we give the corresponding definitions of convex A-proper, P*-compact and Pγ-compact in Menger PN-space. As an application of these new concepts, we prove the existence of solution for some equations.

The third order delay equation

y‴(t) + a(t)yτ(t) = 0

is studied for its nonoscillatory nature under the general condition in which a(t) has been allowed to oscillate. It is shown by way of a differential inequality that if g(t) is a thrice differentiable and eventually positive function then

g‴(t) + t2|a(t)|g(t) ≤ 0

is sufficient for this equation to have bounded nonoscillatory solutions.

We say that a C*-algebra X has the approximate n-th root property (n ≥ 2) if for every a ∈ X with ∥a∥ ≤ 1 and every ɛ > 0 there exits b ∈ X such that ∥b∥ ≤ 1 and ∥a − bn∥ < ɛ. Some properties of commutative and non-commutative C*-algebras having the approximate n-th root property are investigated. In particular, it is shown that there exists a non-commutative (respectively, commutative) separable unital C*-algebra X such that any other (commutative) separable unital C*-algebra is a quotient of X. Also we illustrate a commutative C*-algebra, each element of which has a square root such that its maximal ideal space has infinitely generated first Čech cohomology.

The purpose of this paper is to give some necessary and sufficient conditions for p-nilpotent groups. We extend some results, including the well-known theorems of Burnside and Frobenius as well as some very recent theorems. We also apply our results to determine the structure of some finite groups in terms of formation theory.

We establish an existence theorem for weak saddle points of a vector valued function by making use of a vector variational inequality and convex functions.

In this paper we discuss the existence of solutions for a class of abstract degenerate neutral functional differential equations. Some applications to partial differential equations are considered.

Let be a commutative algebraic group defined over a number field K. For a prime ℘ in K where has good reduction, let N℘,n be the number of n-torsion points of the reduction of modulo ℘ where n is a positive integer. When is of dimension one and n is relatively prime to a fixed finite set of primes depending on , we determine the average values of N℘,n as the prime ℘ varies. This average value as a function of n always agrees with a divisor function.

For three points , and in the n-dimensional space 𝔽nq over the finite field 𝔽q of q elements we give a natural interpretation of an acute angle triangle defined by these points. We obtain an upper bound on the size of a set 𝒵 such that all triples of distinct points define acute angle triangles. A similar question in the real space ℛn dates back to P. Erdős and has been studied by several authors.

We study a special class of quasi-cyclic codes, obtained from a cyclic code over an extension field of the alphabet field by taking its image on a basis. When the basis is equal to its dual, the dual code admits the same construction. We give some examples of self-dual codes and LCD codes obtained in this way.

Let R be an associative ring. We prove that if for each finite subset F of R there exists a positive integer n = n(F) such that (xy)n − yn xn is in the centre of R for every x, y in F, then the commutator ideal of R is nil. We also prove that if n is a fixed positive integer and R is an n(n + 1)-torsion-free ring with identity such that (xy)n − ynxn = (yx)n xnyn is in the centre of R for all x, y in R, then R is commutative.

Let L denote the subvariety lattice of the variety of distributive double p-algebras, that is, the lattice whose universe consists of all varieties of distributive double p-algebras and whose ordering is the inclusion relation. We prove in this paper that each proper filter in L is uncountable. Moreover, we prove that except for the trivial variety (the zero in L) and the variety of Boolean algebras (the unique atom in L) every other element of L, generated by a finite algebra, has infinitely many covers in L, among which at least one is not generated by any finite algebra. The former result strengthens a result of Urquhart who showed that the lattice L is uncountable. On the other hand, both of our results indicate a high complexity of the lattice L at least in comparison with the subvariety lattice of the variety of distributive p-algebras, since a result of Lee shows that the latter lattice forms a chain of type ω + 1 and every cover in it of the variety generated by a finite algebra is itself generated by a finite algebra.

Nonsmooth calculus using the approximate subdifferential of Mordukhovich and loffe admits a sharper chain rule, hence sharper applications in optimisation, than does the generalised gradient of Clarke. We observe, however, that at a local minimum point of the composition of nonsmooth vector valued and real valued functions, the generalised gradient admits a special, relatively sharp chain rule, that yields sharper results than have been seen before in the context of the generalised gradient.

Let $G$ be a group and $\unicode[STIX]{x1D70E}=\{\unicode[STIX]{x1D70E}_{i}\mid i\in I\}$ some partition of the set of all primes. A subgroup $A$ of $G$ is $\unicode[STIX]{x1D70E}$-subnormal in $G$ if there is a subgroup chain $A=A_{0}\leq A_{1}\leq \cdots \leq A_{m}=G$ such that either $A_{i-1}\unlhd A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is a finite $\unicode[STIX]{x1D70E}_{j}$-group for some $j=j(i)$ for $i=1,\ldots ,m$, and it is modular in $G$ if $\langle X,A\cap Z\rangle =\langle X,A\rangle \cap Z$ when $X\leq Z\leq G$ and $\langle A,Y\cap Z\rangle =\langle A,Y\rangle \cap Z$ when $Y\leq G$ and $A\leq Z\leq G$. The group $G$ is called $\unicode[STIX]{x1D70E}$-soluble if every chief factor $H/K$ of $G$ is a finite $\unicode[STIX]{x1D70E}_{i}$-group for some $i=i(H/K)$. In this paper, we describe finite $\unicode[STIX]{x1D70E}$-soluble groups in which every $\unicode[STIX]{x1D70E}$-subnormal subgroup is modular.

In this paper, the concept of a fitted semigroup of binary relations is introduced. The binary operation for these semigroups is ordinary composition of relations. A particular type of homomorphism is investigated which maps a certain kind of subsemigroup of one fitted semigroup into another. The main result states that these homomorphisms are injective and gives a representation for them. This and several related results are then applied to the semigroup of all binary relations on a topological space which have a certain prescribed type of topological property. Compactness is one of these properties and the semigroup SK[X] of all compact binary relations on a Hausdorff k–space X is given special attention. It is shown that X and Y are homeomorphic if and only if a number of statements are true which relate the semigroups SK[x] and SK[y].For example, in order that X and y be homeomorphic it is both necessary and sufficient that some nonzero ideal of SK[X] be isomorphic to some nonzero ideal of SK[y].