Analogues of Artin-Wedderburn and Goldie structure theorems are obtained for a class of halfrings which includes the semisubtractive ones. In the semisubtractive case, precise results are obtained which show that non-ring examples of these structures are relatively limited.

We show that a finite loop, whose inner mapping group is the direct product of a dihedral $2$-group and a nonabelian group of order $pq$ ($p$ and $q$ are distinct odd prime numbers), is solvable.

We investigate a dichotomy property for Hardy–Littlewood maximal operators, noncentred $M$ and centred $M^{c}$, that was noticed by Bennett et al. [‘Weak-$L^{\infty }$ and BMO’, Ann. of Math. (2) 113 (1981), 601–611]. We illustrate the full spectrum of possible cases related to the occurrence or not of this property for $M$ and $M^{c}$ in the context of nondoubling metric measure spaces $(X,\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707})$. In addition, if $X=\mathbb{R}^{d}$, $d\geq 1$, and $\unicode[STIX]{x1D70C}$ is the metric induced by an arbitrary norm on $\mathbb{R}^{d}$, then we give the exact characterisation (in terms of $\unicode[STIX]{x1D707}$) of situations in which $M^{c}$ possesses the dichotomy property provided that $\unicode[STIX]{x1D707}$ satisfies some very mild assumptions.

Let p be a prime and let (mk)k<ω be a strictly increasing sequence of positive integers such that m0 = 1 and mk divides mk+1. A field F is said to be of type (p, (mk)k<ω) if it is the union of an increasing sequence (Fk)k<ω of fields such that Fk has pmk elements. A set X is called “finite” if it has n elements for some nonnegative integer n, and “Dedekind-finite” if every injection f: X → X is a bijection. If the Axiom of Choice is rejected, then it is relatively consistent to assume the existence of medial (that is, infinite, Dedekind-finite) sets. In this paper it is shown that given any type (p, (mk)k<ω) as above, it is relatively consistent with the usual axioms of set theory (minus Choice) to assume the existence of a medial field of type (p, (mk)k<ω). Conversely, it is shown that any medial field must be of type (p, (mk)k<ω) for some (p, (mk)k<ω) as above. The paper concludes with a few observations on Dedekind-finite rings. In the first part of the paper, a general knowledge of Fraenkel-Mostowski set theory and of the Jech-Sochor Embedding Theorems is assumed.

For a real quadratic field , let tk be the exact power of 2 dividing the class number hk of k and ηk the fundamental unit of k. The aim of this paper is to study tk and the value of Nk/ℚ(ηk). Various methods have been successfully applied to obtain results related to this topic. The idea of our work is to select a special circular unit ℰk of k and investigate C(k)=〈±ℰk 〉. We examine the indices [E(k):C(k)] and [C(k):CS (k)] , where E(k) is the group of units of k, and CS (k) is that of circular units of k defined by Sinnott. Then by using the Sinnott’s index formula [E(k):CS (k)]=hk, we obtain as much information about tk and Nk/ℚ (ηk) as possible.

We demonstrate a construction that will densely embed a Moore space into a Moore space with the Baire property when this is possible. We also show how this technique generates a new ‘if and only if’ condition for determining when Moore spaces can be densely embedded in Moore spaces with the Baire property, and briefly discuss how this condition can can be used to generate new proofs that certain Moore spaces cannot be densely embedded in Moore spaces with the Baire property.

Let $P(n)$ denote the largest prime factor of an integer $n\geq 2$. In this paper, we study the distribution of the sequence $\{f(P(n)):n\geq 1\}$ over the set of congruence classes modulo an integer $b\geq 2$, where $f$ is a strongly $q$-additive integer-valued function (that is, $f(aq^{j}+b)=f(a)+f(b),$ with $(a,b,j)\in \mathbb{N}^{3}$, $0\leq b<q^{j}$). We also show that the sequence $\{{\it\alpha}P(n):n\geq 1,f(P(n))\equiv a\;(\text{mod}~b)\}$ is uniformly distributed modulo 1 if and only if ${\it\alpha}\in \mathbb{R}\!\setminus \!\mathbb{Q}$.

We show that the restriction of the Dehornoy ordering to an appropriate free subgroup of the three-strand braid group defines a left-ordering of the free group on k generators, k>1, that has no convex subgroups.

In 1987, Sullivan characterised the elements of the semigroup NI(X) generated by the nilpotents in I(X), the symmetric inverse semigroup on an infinite set X; and, in the same year, Gomes and Howie did the same for finite X. In 1999, Marques-Smith and Sullivan determined all the ideals of NI(X) for arbitrary X. In this paper, we use that work to describe all the congruences on NI(X).

Let ${\mathcal{A}}$ be a locally noetherian Grothendieck category. We construct closure operators on the lattice of subcategories of ${\mathcal{A}}$ and the lattice of subsets of $\text{ASpec}\,{\mathcal{A}}$ in terms of associated atoms. This establishes a one-to-one correspondence between hereditary torsion theories of ${\mathcal{A}}$ and closed subsets of $\text{ASpec}\,{\mathcal{A}}$ . If ${\mathcal{A}}$ is locally stable, then the hereditary torsion theories can be studied locally. In this case, we show that the topological space $\text{ASpec}\,{\mathcal{A}}$ is Alexandroff.

This paper contains a precise description of the duality between the formal evolutions of flow charts and of circuits. In addition, it contains a new description of the free category-with-products on a multigraph as a familially representable construction.

Conditions are given, under which a quasi-proximally continuous function is quasi-uniformly continuous, or a continuous function is quasi-proximally continuous. Thus, basic results on uniform and proximal continuity are extended and some new results are obtained. Three results in the literature are shown to be false.

Generalizing the concept of right bounded rings, a module MR is called bounded if annR(M/N)≤eRR for all N≤eMR. The module MR is called fully bounded if (M/P) is bounded as a module over R/annR(M/P) for any ℒ2-prime submodule P◃MR. Boundedness and right boundedness are Morita invariant properties. Rings with all modules (fully) bounded are characterized, and it is proved that a ring R is right Artinian if and only if RR has Krull dimension, all R-modules are fully bounded and ideals of R are finitely generated as right ideals. For certain fully bounded ℒ2-Noetherian modules MR, it is shown that the Krull dimension of MR is at most equal to the classical Krull dimension of R when both dimensions exist.

This paper presents a theoretical investigation into the forced oscillations produced in an elongated lake by wind stresses varying in time. Analysis of the appropriate hydrodynamical equations of motion, in the absence of friction, and the equation of continuity give an estimate of the response function of the longitudinal component of the wind stress onto water level. Two mathematical models are used, one giving an analytical solution and the other requiring numerical methods for solution. The first model assumes that the lake is a homogeneous rectangular body of water and the second uses the mean depth h(x) and area of cross section A(x), considered as functions of distance x directed along the longitudinal axis of the lake.

In this note, we prove that if a torsion group G has an Abelian subgroup B such that G/B is Abelian and R is a G-adapted ring with the property that R (G/B) has only trivial units then G has the normaliser property in RG.

For a hereditary torsion theory τ, a module A is called τ-completedly decomposable if it is a direct sum of modules that are the τ-injective hull of each of their non-zero submodules. We give a positive answer in several cases to the following generalised Matlis' problem: Is every direct summand of a τ-completely decomposable module still τ-completely decomposable? Secondly, for a commutative Noetherian ring R that is not a domain, we determine those torsion theories with the property that every τ-injective module is an essential extension of a (τ-injective) τ-completely decomposable module.

The first four paragraphs of [1, p. 258] are a mildly erroneous over simplification of the situation. A more accurate description follows.

The analysis of two-generator 3-groups of second maximal class goes along the following lines. We first define a class of group whose structure is particularly amenable to theoretical analysis.

In this paper we investigate the relationships among the notions of minimality, characteristic 0, equicontinuity, Lipschitz stability and isometry in dynamical systems. Examples are provided to show that the results obtained are sharp.

Let $G$ be a commutative group and $\mathbb{C}$ the field of complex numbers, $\mathbb{R}^{+}$ the set of positive real numbers and $f,g,h,k:G\times \mathbb{R}^{+}\rightarrow \mathbb{C}$. In this paper, we first consider the Levi-Civitá functional inequality

$$\begin{eqnarray}\displaystyle |f(x+y,t+s)-g(x,t)h(y,s)-k(y,s)|\leq {\rm\Phi}(t,s),\quad x,y\in G,t,s>0, & & \displaystyle \nonumber\end{eqnarray}$$

${\rm\Phi}:\mathbb{R}^{+}\times \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$

${\rm\Phi}(t_{2},s_{2})\leq {\rm\Phi}(t_{1},s_{1})$

$0<t_{1}\leq t_{2}$

$0<s_{1}\leq s_{2}$

$$\begin{eqnarray}\displaystyle u\circ S-v\otimes w-k\circ {\rm\Pi}\in {\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})\quad [\text{respectively}\;{\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})] & & \displaystyle \nonumber\end{eqnarray}$$

$u,v,w,k$

$S(x,y)=x+y,{\rm\Pi}(x,y)=y,x,y\in \mathbb{R}^{n}$

$\circ$

$\otimes$

${\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$

${\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$