An investigation is made of the stability of block LU-decomposition of matrices A arising from boundary value problems of differential equations, in particular of ordinary differential equations with separated boundary conditions. It is shown that for such matrices the pivotal growth can be bounded by constants of the order of ‖A‖ and, if solution space is dichotomic, often by constants of order one. Furthermore a method to estimate the growth of the pivotal blocks is given. A number of examples support the analysis.

The hyper-archimedean kernel Ar(G) of a lattice-ordered group (hence forth l–group) is the largest hyper-archimedean convex l–subgroup of the l–group G. One defines Arσ (G), for an ordinal σ as if a is a limit ordinal, and as the unique l–ideal with the property that Arσ(G)/Ar.σ–1(G) = Ar(G/Arσ–1(G)), otherwise. The resulting "Loewy"-like sequence of characteristic l–ideals, Ar(G) ⊆ Ar2(G) ⊆ … ⊆ Arσ (G) ⊆ …, is called the hyper-archimedean kernel sequence. The first result of this note says that each Arσ(G) ⊆ Ar(G)”.

Most of the paper concentrates on archimedean l–groups; in particular, the hyper-archimedean kernels are identified for: D(X), where X is a Stone space, a large class of free products of abelian l–groups, and certain l–subrings of a product of real groups.

It is shown that even for archimedean l–groups the hyper-archimedean kernel sequence may proceed past Ar(G).

In 1987 Sullivan determined the elements of the semigroup N(X) generated by all nilpotent partial transformations of an infinite set X; and later in 1997 he studied subsemigroups of N(X) defined by restricting the index of the nilpotents and the cardinality of the set. Here, we describe the ideals and Green's relations on such semigroups, like Reynolds and Sullivan did in 1985 for the semigroup generated by all idempotent total transformations of X. We then use this information to describe the congruences on certain Rees factor semigroups and to construct families of congruence-free semigroups with interesting algebraic properties. We also study analogous questions for X finite and for one-to-one partial transformations.

An N-semigroup is a commutative, cancellative, archimedean semigroup with no idempotent element. This paper obtains a representation of finitely generated N-semigroups as the subdirect product of an abelian group and a subsemigroup of the additive positive integers.

Bear's characterisation of central-by-finite groups as groups possessing a finite covering by abelian subgroups if the starting point for this invertigation. We characterise groups with a finite covering by 2-Engel subgroups as groups for which the subgroup of right 2-Engel elements has finite index; and the groups having a finite covering by normal 2-Engel subgroups are exactly the 3-Engel groups among those having a finite covering by 2-Engel subgroups. The second centre of a group having a finite covering by class two subgroups does not necessarily have finite index. However, a group has a finite covering by subgroups in a variety containing all cyclic groups if the margin of thes variety in the group has finite index.

Let $a, b, c$ be relatively prime positive integers such that ${a}^{2} + {b}^{2} = {c}^{2} $. In 1956, Jeśmanowicz conjectured that for any positive integer $n$, the only solution of $\mathop{(an)}\nolimits ^{x} + \mathop{(bn)}\nolimits ^{y} = \mathop{(cn)}\nolimits ^{z} $ in positive integers is $(x, y, z)= (2, 2, 2)$. In this paper, we consider Jeśmanowicz’ conjecture for Pythagorean triples $(a, b, c)$ if $a= c- 2$ and $c$ is a Fermat prime. For example, we show that Jeśmanowicz’ conjecture is true for $(a, b, c)= (3, 4, 5)$, $(15, 8, 17)$, $(255, 32, 257)$, $(65535, 512, 65537)$.

Let $A_{3}(n)$ denote the number of bipartitions of $n$ with 3-cores. Recently, Lin [‘Some results on bipartitions with 3-core’, J. Number Theory139 (2014), 44–52] established some congruences modulo 4, 5, 7 and 8 for $A_{3}(n)$. In this paper, we prove several infinite families of congruences modulo 3 and 9 for $A_{3}(n)$ by employing two identities due to Ramanujan.

In this paper finite versions of Hardy's discrete, Hardy's integral and the Levin–Cochran–Lee inequalities will be considered and some new generalisations of these inequalities will be derived. Moreover, it will be shown that the constant factors involved in the right-hand sides of the integral results obtained are the best possible.

This paper concerns the problem of description of the set of rings containing a given ring as an essential ideal. The results obtained are applied to some problems of ring theory and radicals.

The aim of this paper is to give a new precise formula for h(n, A), where A is a finite non-abelian simple group, h(n, A) is the maximum number such that Ah(n, A) can ke generated by n elements, and n ≥ 2. P. Hall gave a formula for h(n, A) in terms of the Möbius function of the subgroup lattice of A; the new formula involves a concept called cospread associated with that of spread as explained in Brenner and Wiegold (1975).

A duality theory for finite abelian hypergroups over fairly general fields is presented, which extends the classical duality for finite abelian groups. In this precise sense the set of conjugacy classes and the set of characters of a finite group are dual as hypergroups.

The Euclidean group E(3) is the noncompact, semidirect product group E(3)≅ℝ3⋊SO(3). It is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. The Euclidean algebra 𝔢(3) is the complexification of the Lie algebra of E(3). We embed the Euclidean algebra 𝔢(3) into the simple Lie algebra and show that the irreducible representations V (m,0,0) and V (0,0,m) of are 𝔢(3)-indecomposable, thus creating a new class of indecomposable 𝔢(3) -modules. We then show that V (0,m,0) may decompose.

In this paper we prove a result concerning the Cauchy functional equation, that is the functional equation f(x + y) = f(x) + f(y), in skew fields with characteristic not two.

For a large class of operator inclusions, including those generated by maps of pseudomonotone type, we obtain a general theorem on existence of solutions. We apply this result to some particular examples. This theorem is proved using the method of difference approximations.

This study is a continuation of an earlier paper on the regularity series of a convergence space. The notions of a R-Hausdorff series and the T3-modification of a convergence space are introduced, and their relationship with the regularity series is studied. The concept of a symmetric space is shown to be useful in studying T3-compactifications. Several examples are given; one being a Hausdorff convergence space with an arbitrarily large regularity series.

Let f(x) ε Lipα, 0 < α < 1, in the range (-π, π), and periodic with period 2π, outside this range. Also let

.

We define the norm as

and let the degree of approximation be given by

where Tn (x) is some n–th trigonometric polynomial.

We establish a new sufficient condition under which a monoid is nonfinitely based and apply this condition to Lee monoids $L_{\ell }^{1}$, obtained by adjoining an identity element to the semigroup generated by two idempotents $a$ and $b$ with the relation $0=abab\cdots \,$ (length $\ell$). We show that every monoid $M$ which generates a variety containing $L_{5}^{1}$ and is contained in the variety generated by $L_{\ell }^{1}$ for some $\ell \geq 5$ is nonfinitely based. We establish this result by analysing $\unicode[STIX]{x1D70F}$-terms for $M$, where $\unicode[STIX]{x1D70F}$ is a certain nontrivial congruence on the free semigroup. We also show that if $\unicode[STIX]{x1D70F}$ is the trivial congruence on the free semigroup and $\ell \leq 5$, then the $\unicode[STIX]{x1D70F}$-terms (isoterms) for $L_{\ell }^{1}$ carry no information about the nonfinite basis property of $L_{\ell }^{1}$.

To George Szekeres on his ninetieth birthday

We give new proofs of some sum–to–product identities due to Blecksmith, Brillhart and Gerst, as well as some other such identities found recently by us.

In 1978 Anderson and Gardner investigated semisimple classes and recently Buys and Gerber developed the theory of special radicals in Andrunakievich varieties. In this note we continue the study of radical theory in Andrunakievich varieties. Sharpening some of the results of Anderson and Gardner we prove versions of Sands' Theorem characterizing semisimple classes by regularity, coinductivity and being closed under extensions. In the proof we follow a new method which avoids calculations with defining identities of the variety. We generalize van Leeuwen's Theorem characterizing semisimple classes of hereditary radicals as classes being regular and closed under essential extensions and subdirect sums.

Our main result is that if G(x1, …, xn) = 0 is a system of homogeneous equations such that for every partition of the positive integers into finitely many classes there are distinct y1,…, yn in one class such that G(y1, …, yn) = 0, then, for every partition of the positive integers into finitely many classes there are distinct Z1, …, Zn in one class such that

In particular, we show that if the positive integers are split into r classes, then for every n ≥ 2 there are distinct positive integers x1, x1, …, xn in one class such that

We also show that if [1, n6 − (n2 − n)2] is partitioned into two classes, then some class contains x0, x1, …, xn such that

(Here, x0, x2, …, xn are not necessarily distinct.)