The purpose of this paper is to study the influence of $c$-supplemented minimal subgroups on the $p$-nilpotency of finite groups. We obtain ‘iff' and ‘localized' versions of theorems of Itô and Buckley on nilpotence, $p$-nilpotence and supersolvability.

Asymptotic expansions for an incomplete Bessel function of large argument are derived when the parametric point (a) is well away from any saddle point, (b) coincides with a saddle point and (c) is in the neighbourhood of a saddle point.

This paper presents new existence results for the singular boundary-value problem

\begin{gather*} -u''+p(t)u=f(t,u),\quad t\in(0,1),\\ u(0)=0=u(1). \end{gather*}

In particular, our nonlinearity $f$ may be singular at $t=0,1$ and $u=0$.

We characterize algebra homomorphisms from the Lebesgue algebra $L^1_\omega(\mathbb{R})$ into a Banach algebra $\mathcal{A}$. As a consequence of this result, every bounded algebra homomorphism $\varPhi:L^1_\omega(\mathbb{R})\to\mathcal{A}$ is approached through a uniformly bounded family of fractional homomorphisms, and the Hille–Yosida theorem for $C_0$-groups is proved.

By using Lebesgue’s dominated convergence theorem and constructing a suitable Lyapunov functional, we study the following almost-periodic Lotka–Volterra model with $M$ predators and $N$ prey of the integro-differential equations

\begin{alignat*}{2} \dot{x}_i(t)\amp=x_i(t)\biggl[b_i(t)-a_{ii}(t)x_i(t)-\sum_{k=1,k\neq i}^{N}a_{ik}(t)\int_{-\infty}^tH_{ik}(t-\sigma)x_k(\sigma)\,\mathrm{d}\sigma\\ \amp\hskip45mm-\sum_{l=1}^{M}c_{il}(t)\int_{-\infty}^tK_{il}(t-\sigma)y_l(\sigma)\,\mathrm{d}\sigma\biggr],\amp\quad i\amp=1,2,\dots,N,\\ \dot{y}_j(t)\amp=y_j(t)\biggl[-r_j(t)-e_{jj}(t)y_j(t) +\sum_{k=1}^{N}d_{jk}(t)\int_{-\infty}^tP_{jk}(t-\sigma)x_k(\sigma)\,\mathrm{d}\sigma \\ \amp\hskip45mm-\sum_{l=1,l\neq j}^{M} e_{jl}(t)\int_{-\infty}^tQ_{jl}(t-\sigma)y_l(\sigma)\,\mathrm{d}\sigma\biggr],\amp\quad j\amp=1,2,\dots,M. \end{alignat*}

Some sufficient conditions are obtained for the existence of a unique almost-periodic solution of this model. Several examples show that the obtained criteria are new, general and easily verifiable.

In this paper, the dual $L_p$ John ellipsoids, which include the classical Löwner ellipsoid and the Legendre ellipsoid, are studied. The dual $L_p$ versions of John's inclusion and Ball's volume-ratio inequality are shown. This insight allows for a unified view of some basic results in convex geometry and reveals further the amazing duality between Brunn–Minkowski theory and dual Brunn–Minkowski theory.