Let G be a group. A subset N of G is a set of pairwise noncommuting elements if xy⁄=yx for any two distinct elements x and y in N. If ∣N∣≥∣M∣ for any other set of pairwise noncommuting elements M in G, then N is said to be a maximal subset of pairwise noncommuting elements. In this paper we determine the cardinality of a maximal subset of pairwise noncommuting elements in a three-dimensional general linear group. Moreover, we show how to modify a given maximal subset of pairwise noncommuting elements into another maximal subset of pairwise noncommuting elements that contains a given ‘generating element’ from each maximal torus.

Recently Calegari and Emerton made a conjecture about the 3-class groups of certain pure cubic fields and their normal closures. This paper proves their conjecture and provides additional insight into the structure of the 3-class groups of pure cubic fields and their normal closures.

In this paper we determine the metric dimension of $n$-dimensional metric $(X,G)$-manifolds. This category includes all Euclidean, hyperbolic and spherical manifolds as special cases.

I shall consider the following problem: given a stable, conservative, single-exit q-matrix, Q, over an irreducible state-space S and a μ-subinvariant measure, m, for Q, determine all Q-processes for which m is a μ-invariant measure. I shall provide necessary and sufficient conditions for the existence and uniqueness of such a process.

Given a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

This paper is an attempt to provide a partial answer to the following question put forward by Bernhard H. Neumann in 2000: “Let G be a finite group of order g and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. What relations between g, m, n guarantee that G is Abelian?” We find an exponential function f(m,n) such that every such group G is Abelian whenever |G| > f(m,n) and this function can be taken to be polynomial if G is not soluble. We give an upper bound in terms of m and n for the solubility length of G, if G is soluble.

We show that for reflexive spaces X the density of numerical radius attaining operators in L(X) is equivalent to the density of numerical radius attaining operators in L(X*). As a consequence of this fact and of a result of Berg and Sims, we prove that for uniformly smooth spaces X the numerical radius attaining operators are dense in L(X).

If A is a Dedekind domain and f generates a prime ideal of A[X] which is not maximal, then the domain A[X]/(f) is Dedekind if and only if f is not contained in the square of any maximal ideal of A[X]. This criterion is used find the ring of integers of a cyclotomic field, and to determine when a plane curve is normal.

It is shown that a simple trick involving Legendre polynomials readily yields the irrationality of ea, , π2, and of the zeros of Bessel functions of integer order. Generalisation of this idea yields the irrationality of ζ(3).

An abstract orthogonality relation is defined, a closure operation and a corresponding lattice of closed sets are associated with it. Necessary and sufficient conditions are obtained for the orthomodularity of a sub-ortholattice of the lattice of closed sets.

The following Gelfand-Mazur like theorems are proved in this paper:

(1) A complex Banach algebra which is locally finite, and which is also an integral domain, is isomorphic to the complex field .

(2) A complex Banach algebra which is a noetherian domain is isomorphic to .

(3) A complex Banach algebra which is a principal ideal domain is isomorphic to .

An application is given to the algebra of all complex formal power series in several variables.

Every group is the automorphism group of an orthomodular lattice.

We show that if a continuous bump function on a Banach space X locally depends on finitely many elements of a set F in X*, then the norm closed linear span of F equals to X*. Some corollaries for Markuševič bases and Asplund spaces are derived.

This paper examines conditions for the monotone convergence of general Newton-like methods generated by point-to-point mappings. The speed of convergence of such mappings is also examined.

Let a function be regular in the disk |z| < 1. The radius of univalence 0.164 … of the family of f with |an| ≤ n (n ≥ 2) is, actually, the radius of star-likeness. The radius of univalence 1 - [k/(l+K)]½ of the family of f with |an| ≤ k (n ≥ 2), where K > 0 is a constant, is, actually, the radius of starlikeness. The radii of convexity of the two families are estimated from below.

Our main concern is the existence of maximal compact subgroups in a locally compact topological group. If G is a locally compact group such that P(G/G o), the set of periodic points of G/Go, is a compact subgroup of G/Go, than G has maximal compact subgroups K such that G/N is a Lie group where N = ∩K, the intersection of the collection K of all maximal compact subgroups of G. Also every compact subgroup of G is contained in a maximal compact subgroup. We given an example of a discrete group which has maximal finite subgroup and has finite subgroups not contained in maximal finite subgroups. We note that the above result on P(G/Go) is an extension of the well-known corresponding result for almost connected groups.

The max-plus algebra is well known and has useful applications in the investigation of discrete event systems and affine equations. Structural matrix rings have been considered by many authors too. This article introduces more general structural matrix semirings, which include all matrix semirings over the max-plus algebra. We investigate properties of ideals in this construction motivated by applications to the design of centroid-based classification systems, or classifiers, as well as multiple classifiers combining several initial classifiers. The first main theorem of this paper shows that structural matrix semirings possess convenient visible generating sets for ideals. Our second main theorem uses two special sets to determine the weights of all ideals and describe all matrix ideals with the largest possible weight, which are optimal for the design of classification systems.

Given $f(x,y)\in \mathbb Z[x,y]$ with no common components with $x^a-y^b$ and $x^ay^b-1$, we prove that for $p$ sufficiently large, with $C(f)$ exceptions, the solutions $(x,y)\in \overline {\mathbb F}_p\times \overline {\mathbb F}_p$ of $f(x,y)=0$ satisfy $ {\rm ord}(x)+{\rm ord}(y)\gt c (\log p/\log \log p)^{1/2},$ where $c$ is a constant and ${\rm ord}(r)$ is the order of $r$ in the multiplicative group $\overline {\mathbb F}_p^*$. Moreover, for most $p\lt N$, $N$ being a large number, we prove that, with $C(f)$ exceptions, ${\rm ord}(x)+{\rm ord}(y)\gt p^{1/4+\epsilon (p)},$ where $\epsilon (p)$ is an arbitrary function tending to $0$ when $p$ goes to $\infty $.

We apply Straßmann's theorem to p–adic power series satisfying linear differential equations with polynomial coefficients and note that our approach leads to our estimating the number of integer zeros of polynomials on a given interval and thence to an investigation of the number of p–adic small values of a function on such an interval, that is, of the number of solutions of a congruence modulo pr.

We establish existence results concerning solutions to multipoint boundary value problems for weakly coupled systems of second order ordinary differential equations with fully nonlinear boundary conditions.