Let ${\mathcal{A}}$ be a unital torsion-free algebra over a unital commutative ring ${\mathcal{R}}$. To characterise Lie $n$-higher derivations on ${\mathcal{A}}$, we give an identity which enables us to transfer problems related to Lie $n$-higher derivations into the same problems concerning Lie $n$-derivations. We prove that: (1) if every Lie $n$-derivation on ${\mathcal{A}}$ is standard, then so is every Lie $n$-higher derivation on ${\mathcal{A}}$; (2) if every linear mapping Lie $n$-derivable at several points is a Lie $n$-derivation, then so is every sequence $\{d_{m}\}$ of linear mappings Lie $n$-higher derivable at these points; (3) if every linear mapping Lie $n$-derivable at several points is a sum of a derivation and a linear mapping vanishing on all $(n-1)$th commutators of these points, then every sequence $\{d_{m}\}$ of linear mappings Lie $n$-higher derivable at these points is a sum of a higher derivation and a sequence of linear mappings vanishing on all $(n-1)$th commutators of these points. We also give several applications of these results.

In characteristic two, some criteria are obtained for a symmetric square-central element of a totally decomposable algebra with orthogonal involution, to be contained in an invariant quaternion subalgebra.

Let $M$ be an $n$-dimensional closed oriented smooth manifold with $n\equiv 4\;\text{mod}\;8$, and $\unicode[STIX]{x1D702}$ be a complex vector bundle over $M$. We determine the final obstruction for $\unicode[STIX]{x1D702}$ to admit a stable real form in terms of the characteristic classes of $M$ and $\unicode[STIX]{x1D702}$. As an application, we obtain the criteria to determine which complex vector bundles over a simply connected four-dimensional manifold admit a stable real form.

Let $\unicode[STIX]{x1D70E}=\{\unicode[STIX]{x1D70E}_{i}\mid i\in I\}$ be a partition of the set of all primes $\mathbb{P}$. Let $\unicode[STIX]{x1D70E}_{0}\in \unicode[STIX]{x1D6F1}\subseteq \unicode[STIX]{x1D70E}$ and let $\mathfrak{I}$ be a class of finite $\unicode[STIX]{x1D70E}_{0}$-groups which is closed under extensions, epimorphic images and subgroups. We say that a finite group $G$ is $\unicode[STIX]{x1D6F1}_{\mathfrak{I}}$-primary provided $G$ is either an $\mathfrak{I}$-group or a $\unicode[STIX]{x1D70E}_{i}$-group for some $\unicode[STIX]{x1D70E}_{i}\in \unicode[STIX]{x1D6F1}\setminus \{\unicode[STIX]{x1D70E}_{0}\}$ and we say that a subgroup $A$ of an arbitrary group $G^{\ast }$ is $\unicode[STIX]{x1D6F1}_{\mathfrak{I}}$-subnormal in $G^{\ast }$ if there is a subgroup chain $A=A_{0}\leq A_{1}\leq \cdots \leq A_{t}=G^{\ast }$ such that either $A_{i-1}\unlhd A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $\unicode[STIX]{x1D6F1}_{\mathfrak{I}}$-primary for all $i=1,\ldots ,t$. We prove that the set ${\mathcal{L}}_{\unicode[STIX]{x1D6F1}_{\mathfrak{I}}}(G)$ of all $\unicode[STIX]{x1D6F1}_{\mathfrak{I}}$-subnormal subgroups of $G$ forms a sublattice of the lattice of all subgroups of $G$ and we describe the conditions under which the lattice ${\mathcal{L}}_{\unicode[STIX]{x1D6F1}_{\mathfrak{I}}}(G)$ is modular.

We consider stable and almost stable points of autonomous and nonautonomous discrete dynamical systems defined on the closed unit interval. Our considerations are associated with chaos theory by adding an additional assumption that an entropy of a function at a given point is infinite.

Let $G$ be a finite solvable group and let $p$ be a prime. In this note, we prove that $p$ does not divide $\unicode[STIX]{x1D711}(1)$ for every irreducible monomial $p$-Brauer character $\unicode[STIX]{x1D711}$ of $G$ if and only if $G$ has a normal Sylow $p$-subgroup.

In this paper, we give an explicit construction of a quasi-idempotent in the $q$-rook monoid algebra $R_{n}(q)$ and show that it generates the whole annihilator of the tensor space $U^{\otimes n}$ in $R_{n}(q)$.

Let $w$ be a group-word. For a group $G$, let $G_{w}$ denote the set of all $w$-values in $G$ and let $w(G)$ denote the verbal subgroup of $G$ corresponding to $w$. The group $G$ is an $FC(w)$-group if the set of conjugates $x^{G_{w}}$ is finite for all $x\in G$. It is known that if $w$ is a concise word, then $G$ is an $FC(w)$-group if and only if $w(G)$ is $FC$-embedded in $G$, that is, the conjugacy class $x^{w(G)}$ is finite for all $x\in G$. There are examples showing that this is no longer true if $w$ is not concise. In the present paper, for an arbitrary word $w$, we show that if $G$ is an $FC(w)$-group, then the commutator subgroup $w(G)^{\prime }$ is $FC$-embedded in $G$. We also establish the analogous result for $BFC(w)$-groups, that is, groups in which the sets $x^{G_{w}}$ are boundedly finite.

The family ${\mathcal{F}}_{\unicode[STIX]{x1D706}}$ of orientation-preserving harmonic functions $f=h+\overline{g}$ in the unit disc $\mathbb{D}$ (normalised in the standard way) satisfying

$$\begin{eqnarray}h^{\prime }(z)+g^{\prime }(z)=\frac{1}{(1+\unicode[STIX]{x1D706}z)(1+\overline{\unicode[STIX]{x1D706}}z)},\quad z\in \mathbb{D},\end{eqnarray}$$

$\unicode[STIX]{x1D706}\in \unicode[STIX]{x2202}\mathbb{D}$

${\mathcal{F}}_{\unicode[STIX]{x1D706}}$

The Laub–Ilani inequality [‘A subtle inequality’, Amer. Math. Monthly97 (1990), 65–67] states that $x^{x}+y^{y}\geqslant x^{y}+y^{x}$ for nonnegative real numbers $x,y$. We introduce and prove new trigonometric and algebraic-trigonometric inequalities of Laub–Ilani type and propose some conjectural algebraic-trigonometric inequalities of similar forms.

We prove that a finite coprime linear group $G$ in characteristic $p\geq \frac{1}{2}(|G|-1)$ has a regular orbit. This bound on $p$ is best possible. We also give an application to blocks with abelian defect groups.

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

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

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

$a(x)$

$0<q<1$

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

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

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

$f$

Let $X$ and $Y$ be two normed spaces over fields $\mathbb{F}$ and $\mathbb{K}$, respectively. We prove new generalised hyperstability results for the general linear equation of the form $g(ax+by)=Ag(x)+Bg(y)$, where $g:X\rightarrow Y$ is a mapping and $a,b\in \mathbb{F}$, $A,B\in \mathbb{K}\backslash \{0\}$, using a modification of the method of Brzdęk [‘Stability of additivity and fixed point methods’, Fixed Point Theory Appl.2013 (2013), Art. ID 285, 9 pages]. The hyperstability results of Piszczek [‘Hyperstability of the general linear functional equation’, Bull. Korean Math. Soc.52 (2015), 1827–1838] can be derived from our main result.

We prove a necessary and sufficient condition for embeddability of an operator system into ${\mathcal{O}}_{2}$. Using Kirchberg’s theorems on a tensor product of ${\mathcal{O}}_{2}$ and ${\mathcal{O}}_{\infty }$, we establish results on their operator system counterparts ${\mathcal{S}}_{2}$ and ${\mathcal{S}}_{\infty }$. Applications of the results, including some examples describing $C^{\ast }$-envelopes of operator systems, are also discussed.

Assume that $(G,+)$ is a commutative semigroup, $\unicode[STIX]{x1D70F}$ is an endomorphism of $G$ and an involution, $D$ is a nonempty subset of $G$ and $(H,+)$ is an abelian group uniquely divisible by two. We prove that if $D$ is ‘sufficiently large’, then each function $g:D\rightarrow H$ satisfying $g(x+y)+g(x+\unicode[STIX]{x1D70F}(y))=2g(x)$ for $x,y\in D$ with $x+y,x+\unicode[STIX]{x1D70F}(y)\in D$ can be extended to a unique solution $f:G\rightarrow H$ of the generalised Jensen functional equation $f(x+y)+f(x+\unicode[STIX]{x1D70F}(y))=2f(x)$.

In this paper, we prove some new reverse dynamic inequalities of Renaud- and Bennett-type on time scales. The results are established using the time scales Fubini theorem, the reverse Hölder inequality and a time scales chain rule.

In this paper, we first give a description of the holomorphic automorphism group of a convex domain which is a simple case of the so-called generalised minimal ball. As an application, we show that any proper holomorphic self-mapping on this type of domain is biholomorphic.

We carry out an in-depth study of some domination and smoothing properties of linear operators and of their role within the theory of eventually positive operator semigroups. On the one hand, we prove that, on many important function spaces, they imply compactness properties. On the other hand, we show that these conditions can be omitted in a number of Perron–Frobenius type spectral theorems. We furthermore prove a Kreĭn–Rutman type theorem on the existence of positive eigenvectors and eigenfunctionals under certain eventual positivity conditions.

Let $\ell >0$ be arbitrary. We introduce the extremal quantities

$$\begin{eqnarray}G(\ell ):=\sup _{f}\int _{-\ell }^{\ell }f\,dx\,\bigg/\int _{-1}^{1}f\,dx,\quad C(\ell ):=\sup _{f}\sup _{a\in \mathbb{R}}\int _{a-\ell }^{a+\ell }f\,dx\,\bigg/\int _{-1}^{1}f\,dx,\end{eqnarray}$$

$\ell =2$

$\overline{\lim }_{\unicode[STIX]{x1D700}\rightarrow 0+}G(k+\unicode[STIX]{x1D700})$

$\overline{\lim }_{\unicode[STIX]{x1D700}\rightarrow 0+}C(k+\unicode[STIX]{x1D700})$

$(k\in \mathbb{N})$

$\ell$

A Littlewood–Paley operator associated with the reflection part of the Dunkl operator is introduced and proved to be of type $(p,p)$ for $1<p<\infty$, based on boundedness of a generalised vector-valued singular integral. This fills a gap for $2<p<\infty$ concerning the boundedness of a $g$-function in the Dunkl setting. The paper also supplies new proofs for $1<p<\infty$ on the $(p,p)$ boundedness of various $g$-functions associated with the Dunkl operator.