A subgroup H of a group G is said to be pronormal in G if each of its conjugates H^g$H^g$ in G is already conjugate to it in the subgroup \langle H,H^g\rangle $\langle H,H^g\rangle$. The aim of this paper is to classify those (locally) finite simple groups which have only nilpotent or pronormal subgroups.

The signature four elliptic theory of Ramanujan is provided with a counterpart to the Jacobian modular sine; this counterpart yields natural direct proofs of several hypergeometric identities recorded by Ramanujan, bypassing the signature four transfer principle of Berndt et al. [‘Ramanujan’s theories of elliptic functions to alternative bases’, Trans. Amer. Math. Soc. 347 (1995), 4163–4244].

We consider the growth of the convex viscosity solution of the Monge–Ampère equation \det D^2u=1$\det D^2u=1$ outside a bounded domain of the upper half space. We show that if u is a convex quadratic polynomial on the boundary \{x_n=0\}$\{x_n=0\}$ and there exists some \varepsilon>0$\varepsilon>0$ such that u=O(|x|^{3-\varepsilon })$u=O(|x|^{3-\varepsilon })$ at infinity, then u=O(|x|^2)$u=O(|x|^2)$ at infinity. As an application, we improve the asymptotic result at infinity for viscosity solutions of Monge–Ampère equations in half spaces of Jia, Li and Li [‘Asymptotic behavior at infinity of solutions of Monge–Ampère equations in half spaces’, J. Differential Equations 269(1) (2020), 326–348].

A subgroup A of a group G is said to be hereditarily G-permutable with a subgroup B of G, if AB^x = B^xA$AB^x = B^xA$ for some element x \in \langle A, B \rangle $x \in \langle A, B \rangle$. A subgroup A of a group G is said to be hereditarily G-permutable in G if A is hereditarily G-permutable with every subgroup of G. In this paper, we investigate the structure of a finite group G with all its Schmidt subgroups hereditarily G-permutable.

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.

For a group G and m\ge 1$m\ge 1$, let G^m$G^m$ denote the subgroup generated by the elements g^m$g^m$, where g runs through G. The subgroups not of the form G^m$G^m$ are the nonpower subgroups of G. We classify the groups with at most nine nonpower subgroups.

An example of a nonfinitely based involution monoid of order five has recently been discovered. We confirm that this example is, up to isomorphism, the unique smallest among all involution monoids.

Niven’s theorem asserts that \{\cos (r\pi ) \mid r\in \mathbb {Q}\}\cap \mathbb {Q}=\{0,\pm 1,\pm 1/2\}.$\{\cos (r\pi ) \mid r\in \mathbb {Q}\}\cap \mathbb {Q}=\{0,\pm 1,\pm 1/2\}.$ In this paper, we use elementary techniques and results from arithmetic dynamics to obtain an algorithm for classifying all values in the set \{\cos (r\pi ) \mid r\in \mathbb {Q}\}\cap K$\{\cos (r\pi ) \mid r\in \mathbb {Q}\}\cap K$, where K is an arbitrary number field.

Suppose that X and Y are two real normed spaces. A map f:X\rightarrow Y$f:X\rightarrow Y$ is called a min-phase-isometry if it satisfies

\begin{align*} \min\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\min\{\|x+y\|,\|x-y\|\} \quad (x,y\in X). \end{align*} $$\begin{align*} \min\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\min\{\|x+y\|,\|x-y\|\} \quad (x,y\in X). \end{align*}$$

We present properties of min-phase-isometries in the case that Y is strictly convex and show that if a min-phase-isometry f (not necessarily surjective) fixes the origin, then it is phase-equivalent to a linear isometry, that is, f(x)=\varepsilon (x)g(x)$f(x)=\varepsilon (x)g(x)$ for x\in X$x\in X$, where g:X\rightarrow Y$g:X\rightarrow Y$ is a linear isometry and \varepsilon $\varepsilon$ is a map from X to \{-1,1\}$\{-1,1\}$.

This paper relies on nested postulates of separate, linear and arc-continuity of functions to define analogous properties for sets that are weaker than the requirement that the set be open or closed. This allows three novel characterisations of open or closed sets under convexity or separate convexity postulates: the first pertains to separately convex sets, the second to convex sets and the third to arbitrary subsets of a finite-dimensional Euclidean space. By relying on these constructions, we also obtain new results on the relationship between separate and joint continuity of separately quasiconcave, or separately quasiconvex functions. We present examples to show that the sufficient conditions we offer cannot be dispensed with.

Let \mathcal {K}_u$\mathcal {K}_u$ denote the class of all analytic functions f in the unit disk \mathbb {D}:=\{z\in \mathbb {C}:|z|<1\}$\mathbb {D}:=\{z\in \mathbb {C}:|z|<1\}$, normalised by f(0)=f'(0)-1=0$f(0)=f'(0)-1=0$ and satisfying |zf'(z)/g(z)-1|<1$|zf'(z)/g(z)-1|<1$ in \mathbb {D}$\mathbb {D}$ for some starlike function g. Allu, Sokól and Thomas [‘On a close-to-convex analogue of certain starlike functions’, Bull. Aust. Math. Soc. 108 (2020), 268–281] obtained a partial solution for the Fekete–Szegö problem and initial coefficient estimates for functions in \mathcal {K}_u$\mathcal {K}_u$, and posed a conjecture in this regard. We prove this conjecture regarding the sharp estimates of coefficients and solve the Fekete–Szegö problem completely for functions in the class \mathcal {K}_u$\mathcal {K}_u$.

We consider the two-dimensional minimisation problem for \inf \{ E_a(\varphi ):\varphi \in H^1(\mathbb {R}^2)\ \text {and}\ \|\varphi \|_2^2=1\}$\inf \{ E_a(\varphi ):\varphi \in H^1(\mathbb {R}^2)\ \text {and}\ \|\varphi \|_2^2=1\}$, where the energy functional E_a(\varphi )$E_a(\varphi )$ is a cubic-quintic Schrödinger functional defined by E_a(\varphi ):=\tfrac 12\int _{\mathbb {R}^2}|\nabla \varphi |^2\,dx-\tfrac 14a\int _{\mathbb {R}^2}|\varphi |^4\,dx+\tfrac 16a^2\int _{\mathbb {R}^2}|\varphi |^6\,dx$E_a(\varphi ):=\tfrac 12\int _{\mathbb {R}^2}|\nabla \varphi |^2\,dx-\tfrac 14a\int _{\mathbb {R}^2}|\varphi |^4\,dx+\tfrac 16a^2\int _{\mathbb {R}^2}|\varphi |^6\,dx$. We study the existence and asymptotic behaviour of the ground state. The ground state \varphi _{a}$\varphi _{a}$ exists if and only if the L^2$L^2$ mass a satisfies a>a_*={\lVert Q\rVert }^2_2$a>a_*={\lVert Q\rVert }^2_2$, where Q is the unique positive radial solution of -\Delta u+ u-u^3=0$-\Delta u+ u-u^3=0$ in \mathbb {R}^2$\mathbb {R}^2$. We show the optimal vanishing rate \int _{\mathbb {R}^2}|\nabla \varphi _{a}|^2\,dx\sim (a-a_*)$\int _{\mathbb {R}^2}|\nabla \varphi _{a}|^2\,dx\sim (a-a_*)$ as a\searrow a_*$a\searrow a_*$ and obtain the limit profile.

We introduce a technique that is helpful in evaluating the reflexivity index of several classes of topological spaces and lattices. The main results are related to products: we give a sufficient condition for the product of a topological space and a nest of balls to have low reflexivity index and determine the reflexivity index of all compact connected 2-manifolds.

By defining and applying the restricted topology, we have investigated certain connections between the boundary spectrum, the exponential spectrum, the topological boundary of the spectrum and the connected hull of the spectrum (see Mouton and Harte [‘Linking the boundary and exponential spectra via the restricted topology’, J. Math. Anal. Appl. 454 (2017), 730–745]). We now solve a remaining problem regarding the restricted connected hull.

We give an example of a pair of real Banach spaces such that they are neither linearly isomorphic nor isomorphic with respect to the structure of Birkhoff–James orthogonality, but have mutually homeomorphic geometric structure spaces.

We study curve-shortening flow for twisted curves in \mathbb {R}^3$\mathbb {R}^3$ (that is, curves with nowhere vanishing curvature \kappa $\kappa$ and torsion \tau $\tau$) and define a notion of torsion-curvature entropy. Using this functional, we show that either the curve develops an inflection point or the eventual singularity is highly irregular (and likely impossible). In particular, it must be a Type-II singularity which admits sequences along which {\tau }/{\kappa ^2} \to \infty ${\tau }/{\kappa ^2} \to \infty$. This contrasts strongly with Altschuler’s planarity theorem, which shows that {\tau }/{\kappa } \to 0${\tau }/{\kappa } \to 0$ along any essential blow-up sequence.

We derive conditions for recurrence and transience for time-inhomogeneous birth-and-death processes considered as random walks with positively biased drifts. We establish a general result, from which the earlier known particular results by Menshikov and Volkov [‘Urn-related random walk with drift \rho x^\alpha /t^\beta $\rho x^\alpha /t^\beta$’, Electron. J. Probab. 13 (2008), 944–960] follow.