Let p be a prime and the variety of elementary abelian by elementary abelian p-groups. A result of Brisley and Macdonald is generalized as follows. If H is a finite group in and G is a soluble group of p–power exponent such that no section of G is isomorphic to H, then G is nilpotent and its class is bounded by a function of three variables: H, the exponent of G, and the soluble length of G. It is a corollary that if the variety generated by a soluble group G of finite exponent contains , then each finite group in is isomorphic to some section of G.

Let R be the ring of linear transformations of a right vector space over a division ring D. Three results are proved: (1) if |D|>4, then for any a∈R there exists a unit u of R such that a+u,a−u and a−u−1 are units of R; (2) if |D|>3 , then for any a∈R there exists a unit u of R such that both a+u and a−u−1 are units of R; (3) if |D|>2 , then for any a∈R there exists a unit u of R such that both a−u and a−u−1 are units of R. The second result extends the main result in H. Chen, [‘Decompositions of countable linear transformations’, Glasg. Math. J. (2010), doi:10.1017/S0017089510000121] and the third gives an affirmative answer to the question raised in the same paper.

If (X′, τ′, ≤′) is an ordered compactification of the partially ordered topological space (X, τ, ≤) such that ≤′ is the smallest order that renders (X′, τ′, ≤′) a T2-ordered compactification of X, then X′ is called a Nachbin (or order-strict) compactification of (X, τ, ≤). If (X′, τ′, ≤*) is a finite-point ordered compactification of (X, τ, ≤), the Nachbin order ≤′ for (X′, τ′) is described in terms of (X, τ, ≤) and X′. When given the usual order relation between compactifications (ordered compactifications), posets of finite-point Nachbin compactifications are shown to have the same order structure as the poset of underlying topological compactifications. Though posets of arbitrary finite-point ordered compactifications are shown to be less well behaved, conditions for their good behavior are studied. These results are used to examine the lattice structure of the set of all ordered compactifications of the ordered topological space (X, τ, ≤).

We prove the existence of infinitely many solutions $u\in W_{0}^{1,2}(\unicode[STIX]{x1D6FA})$ for the Kirchhoff equation

$$\begin{eqnarray}\displaystyle -\biggl(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}\int _{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}u|^{2}\,dx\biggr)\unicode[STIX]{x1D6E5}u=a(x)|u|^{q-1}u+\unicode[STIX]{x1D707}f(x,u)\quad \text{in }\unicode[STIX]{x1D6FA}, & & \displaystyle \nonumber\end{eqnarray}$$

$\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{N}$

$a(x)$

$0<q<1$

$\unicode[STIX]{x1D6FC}>0$

$\unicode[STIX]{x1D6FD}\geq 0$

$\unicode[STIX]{x1D707}>0$

$f$

The purpose of the paper is to show that weak order convergence of a net of Dedekind complete Riesz space-valued σ-measures is uniform over uniformly bounded, uniformly equicontinuous classes of functions. The paper ends with generalizing Ulam's theorem for tightness of positive, finite Borel measures to Riesz space-valued σ-measures.

An estimate is obtained on the Hausdorff and fractal dimensions of global attractors of semilinear partial differential equations with delay: ẋ(t) = Ax(t) + f(xt). The method employed is to associate such an equation with a nonlinear semigroup on a product space and then appeal to the upper estimate due to Constantin, Foias and Teman on topological dimensions of global attractors for general nonlinear dynamical systems.

Recently, Lin and Liu [‘Congruences for the truncated Appell series $F_3$ and $F_4$ ’, Integral Transforms Spec. Funct. 31(1) (2020), 10–17] confirmed a supercongruence on the truncated Appell series $F_3$ . Motivated by their work, we give a generalisation of this supercongruence by establishing a q-supercongruence modulo the fourth power of a cyclotomic polynomial.

The factorial conjecture was proposed by van den Essen et al. [‘On the image conjecture’, J. Algebra 340(1) (2011), 211–224] to study the image conjecture, which arose from the Jacobian conjecture. We show that the factorial conjecture holds for all homogeneous polynomials in two variables. We also give a variation of the result and use it to show that the image of any linear locally nilpotent derivation of $\mathbb{C}[x,y,z]$ is a Mathieu–Zhao subspace.

Grafakos and Sansing [‘Gabor frames and directional time–frequency analysis’, Appl. Comput. Harmon. Anal.25 (2008), 47–67] have shown how to obtain directionally sensitive time–frequency decompositions in $L^{2}(\mathbb{R}^{n})$ based on Gabor systems in $L^{2}(\mathbb{R})$. The key tool is the ‘ridge idea’, which lifts a function of one variable to a function of several variables. We generalise their result in two steps: first by showing that similar results hold starting with general frames for $L^{2}(\mathbb{R}),$ in the settings of both discrete frames and continuous frames, and second by extending the representations to Sobolev spaces. The first step allows us to apply the theory to several other classes of frames, for example wavelet frames and shift-invariant systems, and the second one significantly extends the class of examples and applications. We consider applications to the Meyer wavelet and complex B-splines. In the special case of wavelet systems we show how to discretise the representations using ${\it\epsilon}$-nets.

A study of centered composition operators on l2 is made in this paper. Also the spectrum of surjective composition operators is computed. A necessary and sufficient condition is obtained for the closed unit disc to be the spectrum of a surjective composition operator.

We investigate several inclusion relationships and other interesting properties of certain subclasses of p-valent meromorphic functions, which are defined by using a certain linear operator, involving the generalized multiplier transformations.

This paper finds restrictions on the parameters of supplementary difference sets in any group G with a subgroup of index 2, which therefore includes all cyclic groups of even orders. As a corollary to the main theorem, we have that if S1, …, Sr are r − {2v, k1, …, kr; 2λ} supplementary difference sets in such a group, then not all of v, k1, …, kr, λ are odd; also is the sum of r squares.

Motivated by the notion of Ulam stability, we investigate some inequalities connected with the functional equation

$$\begin{eqnarray}f(xy)+f(x\unicode[STIX]{x1D70E}(y))=2f(x)+h(y),\quad x,y\in G,\end{eqnarray}$$

$f$

$h$

$(G,\cdot )$

$(E,+)$

$\unicode[STIX]{x1D70E}:G\rightarrow G$

$G$

$\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D70E}(x))=x$

$x\in G$

$\unicode[STIX]{x1D70E}$

$\unicode[STIX]{x1D70E}$

We show that Matui’s HK conjecture holds for groupoids of unstable equivalence relations and their corresponding $C^{\ast }$-algebras on one-dimensional solenoids.

It is shown in this paper that if  is a class of simple groups such that π() = char , the -saturated formation ℌ generated by a finite group cannot be expressed as the Gaschütz product  ∘  of two non--saturated formations if ℌ ≠ . It answers some open questions on products of formations. The relation between ω-saturated and -saturated formations is also discussed.

It is shown that any sequential subgroup of a free topological group is either sequential of order ω1 or discrete. Hence any metrizable subgroup of a free topological group is discrete.

The object of this present paper is to study a nonlinear perturbed stochastic integral equation of the form

where ω ∈ Ω, the supporting set of the complete probability measure space (Ω A, μ). We are concerned with the existence and uniqueness of a random solution to the above equation. A random solution, x(t; ω), of the above equation is defined to be a vector random variable which satisfies the equation μ almost everywhere.

Let the group H have presentation where m ≥ 3, pi ≥ 2 and (pi, pj) = 1 if i ≠ j. We show that H is a one-relator group precisely if H can be obtained from a suitable group 〈a, b; ap = bp〉 by repeated applications of a (two-stage) procedure consisting of applying central Nielsen transformations followed by adjoining a root of a generator. We conjecture that any one-relator group G with non-trivial centre and G/G′ not free abelian of rank two can be obtained in the same way from a suitable group 〈a, b; ap = bp〉.

In this note, we establish some new generalisations of an H-KKM type theorem which unify and generalise the corresponding results of Horvath, Bardaro-Ceppitelli, Tarafdar, Shioji, Park and others. As applications of our H-KKM type principle, we obtain some new generalisations of the Ky Fan type geometric properties of. H-spaces, minimax inequalities and coincidence theorems in Horvath's abstract setting.

In this note we obtain a characterization of the class of convergence spaces for which Richardson's compactification is the largest Hausdorff compactification and a characterization of the class of convergence spaces which possess largest Hausdorff compactifications.