In this paper, we investigate finite solvable tidy groups. We prove that a solvable group with order divisible by at least two primes is tidy if all of its Hall subgroups that are divisible by only two primes are tidy.

In this paper, it is shown that there exists a connected topological group which is not homeomorphic to any $\omega $-narrow topological group, and also that there exists a zero-dimensional topological group $G$ with neutral element $e$ such that the subspace $X = G\setminus \{e\}$ is not homeomorphic to any topological group. These two results give negative answers to two open problems in Arhangel’skii and Tkachenko [Topological Groups and Related Structures (Atlantis Press, Amsterdam, 2008)]. We show that if a compact topological group is a $K$-space, then it is metrisable. This result gives an affirmative answer to a question posed by Malykhin and Tironi [‘Weakly Fréchet–Urysohn and Pytkeev spaces’, Topology Appl. 104 (2000), 181–190] in the category of topological groups. We also prove that a regular $K$-space $X$ is a weakly Fréchet–Urysohn space if and only if $X$has countable tightness.

It is shown that if a partially-ordered topological space X admits a finite-point T2-ordered compactification, then it admits a countable T2-ordered compactification if and only if it admits n−point T2-ordered compactifications for all n beyond some integer m.

Given a poset $P$ and a standard closure operator $\unicode[STIX]{x1D6E4}:{\wp}(P)\rightarrow {\wp}(P)$, we give a necessary and sufficient condition for the lattice of $\unicode[STIX]{x1D6E4}$-closed sets of ${\wp}(P)$ to be a frame in terms of the recursive construction of the $\unicode[STIX]{x1D6E4}$-closure of sets. We use this condition to show that, given a set ${\mathcal{U}}$ of distinguished joins from $P$, the lattice of ${\mathcal{U}}$-ideals of $P$ fails to be a frame if and only if it fails to be $\unicode[STIX]{x1D70E}$-distributive, with $\unicode[STIX]{x1D70E}$ depending on the cardinalities of sets in ${\mathcal{U}}$. From this we deduce that if a poset has the property that whenever $a\wedge (b\vee c)$ is defined for $a,b,c\in P$ it is necessarily equal to $(a\wedge b)\vee (a\wedge c)$, then it has an $(\unicode[STIX]{x1D714},3)$-representation.

Let ${ \mathcal{T} }_{X} $ be the full transformation semigroup on a set $X$ and $E$ be a nontrivial equivalence relation on $X$. Denote

$$\begin{eqnarray*}{T}_{\exists } (X)= \{ f\in { \mathcal{T} }_{X} : \forall x, y\in X, (f(x), f(y))\in E\Rightarrow (x, y)\in E\} ,\end{eqnarray*}$$

${T}_{\exists } (X)$

${ \mathcal{T} }_{X} $

${T}_{\exists } (X)$

This paper presents two properties of the Julia set of a random iteration system of rational functions, which are similar to the well-known results in the classical case.

Let Γ.𝒜 be the semi-direct product group of a nuclear Lie group Γ with the additive group 𝒜 of a real nuclear vector space. We give an explicit description of all the continuous representations of Γ.𝒜 the restriction of which to 𝒜 is a cyclic unitary representation, and a necessary and sufficient condition for the unitarity of such cylindrical representations is stated. This general result is successfully used to obtain irreducible unitary representations of the nuclear Lie groups of Riemannian motions, and, in the setting of the theory of multiplicative distributions initiated by I.M. Gelfand, it is proved that for any connected real finite dimensional Lie group G and for any strictly positive integer k there exist non located and non trivially decomposable representations of order k of the nuclear Lie group (M;G) of all the G-valued test-functions on a given paracompact manifold M.

In this note, we characterise completely the ideals of the groupoid C*-algebra arising from the asymptotic equivalence relation on the points of a Smale space and show that the related Ruelle algebra is simple when the Smale space is topologically transitive.

We show that the Murasugi conditions for the Alexander polynomial of a cyclically periodic knot imply a modified form of the Burde-Trotter condition.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a graph and ${{\tau }}$ be an assignment of nonnegative thresholds to the vertices of $G$. A subset of vertices, $D$, is an irreversible dynamic monopoly of $(G, \tau )$ if the vertices of $G$ can be partitioned into subsets $D_0, D_1, \ldots, D_k$ such that $D_0=D$ and, for all $i$ with $0 \leq i \leq k-1$, each vertex $v$ in $D_{i+1}$ has at least $\tau (v)$ neighbours in the union of $D_0, D_1, \ldots, D_i$. Dynamic monopolies model the spread of influence or propagation of opinion in social networks, where the graph $G$ represents the underlying network. The smallest cardinality of any dynamic monopoly of $(G,\tau )$ is denoted by $\mathrm{dyn}_{\tau }(G)$. In this paper we assume that the threshold of each vertex $v$ of the network is a random variable $X_v$ such that $0\leq X_v \leq \deg _G(v)+1$. We obtain sharp bounds on the expectation and the concentration of $\mathrm{dyn}_{\tau }(G)$ around its mean value. We also obtain some lower bounds for the size of dynamic monopolies in terms of the order of graph and expectation of the thresholds.

Let $S$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We say that $I$ is MHC (that is, $I$ satisfies the maximal height condition for the associated primes of $I$) if there exists a prime ideal $\mathfrak{p}\in {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I$ for which $\mathrm{ht} (\mathfrak{p})$ equals the number of indeterminates that appear in the minimal set of monomials generating $I$. Let $I= { \mathop{\bigcap }\nolimits}_{i= 1}^{k} {Q}_{i} $ be the irreducible decomposition of $I$ and let $m(I)= \max \{ \vert Q_{i}\vert - \mathrm{ht} ({Q}_{i} ): 1\leq i\leq k\} $, where $\vert {Q}_{i} \vert $ denotes the total degree of ${Q}_{i} $. Then it can be seen that when $I$ is primary, $\mathrm{reg} (S/ I)= m(I)$. In this paper we improve this result and show that whenever $I$ is MHC, then $\mathrm{reg} (S/ I)= m(I)$ provided $\vert {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I\vert \leq 2$. We also prove that $m({I}^{n} )\leq n\max \{ \vert Q_{i}\vert : 1\leq i\leq ~k\} - \mathrm{ht} (I)$, for all $n\geq 1$. In addition we show that if $I$ is MHC and $w$ is an indeterminate which is not in the monomials generating $I$, then $\mathrm{reg} (S/ \mathop{(I+ {w}^{d} S)}\nolimits ^{n} )\leq \mathrm{reg} (S/ I)+ nd- 1$ for all $n\geq 1$ and $d$ large enough. Finally, we implement an algorithm for the computation of $m(I)$.

In a previous paper, the author sought to classify those solutions of second order nonlinear ordinary differential equations which can be expressed as sums of solutions of related equations. In that paper one sub-class of solutions was overlooked. This paper is to remedy that defect.

Let {Gr,i} be a sequence of r-generator Kleinian groups acting on . In this paper, we prove that if {Gr,i} satisfies the F-condition, then its algebraic limit group Gr is also a Kleinian group. The existence of a homomorphism from Gr to Gr,i is also proved. These are generalisations of all known corresponding results.

The deep Preiss theorem states that a Lipschitz function on a nonempty open subset of an Asplund space is densely Fréchet differentiable. However, the simpler Fabian–Preiss lemma implies that it is Fréchet intermediately differentiable on a dense subset and that for a large class of Lipschitz functions this dense subset is residual. Results are presented for Asplund generated spaces.

Let M be any bounded set in n-dimensional Euclidean space. Then almost all n-dimensional lattices L with determinant 1 have the following property: There exists a diagonal transformation D with determinant 1 (depending on L) such that L does not cover space with DM. Moreover, if M has non-empty interior, the exceptional (null-) set contains at least enumerably many diagonally non-equivalent lattices.

A refinement of the Newton-Kantorovich Theorem, which has many-potential applications in existence - uniqueness theory, is used to strengthen a result of Lancaster and Rokne concerning existence and uniqueness regions for zeros of operator polynomials.

In this note we point out that a recent result of Leung concerning hypersurfaces of a Euclidean space has a simple generalisation to hypersurfaces of complete simply-connected Riemannian manifolds of non-positive constant sectional curvature.

It is shown in this note that if Q is an algebra of uniformly bounded mean-square continuous real-valued random functions indexed in a compact set T, containing all bounded random variables and separating points of T (i.e., given t1 and t2 in T, there is a random function Xt in Q such that , then given any mean square continuous random function, there is a sequence in Q converging in mean square to the given random function uniformly on T.