A natural number $n$ is called multiperfect or $k$-perfect for integer $k\ge 2$ if $\sigma (n)=kn$, where $\sigma (n)$ is the sum of the positive divisors of $n$. In this paper, we establish a theorem on odd multiperfect numbers analogous to Euler’s theorem on odd perfect numbers. We describe the divisibility of the Euler part of odd multiperfect numbers and characterise the forms of odd perfect numbers $n=\pi ^\alpha M^2$ such that $\pi \equiv \alpha ~({\rm mod}~8)$, where $\pi ^\alpha $ is the Euler factor of $n$. We also present some examples to show the nonexistence of odd perfect numbers of certain forms.

A generically embedded real submanifold of codimension two in a complex manifold has isolated complex points that can be classified as either elliptic or hyperbolic. In this paper we show that a pair consisting of one elliptic and one hyperbolic complex point of the same sign can be cancelled by a $\mathcal {C}^{0}$small isotopy of embeddings.

The notion of BSE algebras was introduced and first studied by Takahasi and Hatori and later studied by Kaniuth and Ülger. This notion depends strongly on the multiplier algebra $M( \mathcal{A} )$ of a commutative Banach algebra $ \mathcal{A} $. In this paper we first present a characterisation of the multiplier algebra of the direct sum of two commutative semisimple Banach algebras. Then as an application we show that $ \mathcal{A} \oplus \mathcal{B} $ is a BSE algebra if and only if $ \mathcal{A} $ and $ \mathcal{B} $ are BSE. We also prove that if the algebra $ \mathcal{A} \hspace{0.167em} {\mathop{\times }\nolimits}_{\theta } \hspace{0.167em} \mathcal{B} $ with $\theta $-Lau product is a BSE algebra and $ \mathcal{B} $ is unital then $ \mathcal{B} $ is a BSE algebra. We present some examples which show that the BSE property of $ \mathcal{A} \hspace{0.167em} {\mathop{\times }\nolimits}_{\theta } \hspace{0.167em} \mathcal{B} $ does not imply the BSE property of $ \mathcal{A} $, even in the case where $ \mathcal{B} $ is unital.

Khovanov homology, an invariant of links in ${ \mathbb{R} }^{3} $, is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda et al. [‘Categorification of the Kauffman bracket skein module of $I$-bundles over surfaces’, Algebr. Geom. Topol. 4 (2004), 1177–1210] generalised this construction by defining a double graded homology theory that categorifies the Kauffman bracket skein module of links in $I$-bundles over surfaces, except for the surface $ \mathbb{R} {\mathrm{P} }^{2} $, where the construction fails due to strange behaviour of links when projected to the nonorientable surface $ \mathbb{R} {\mathrm{P} }^{2} $. This paper categorifies the missing case of the twisted $I$-bundle over $ \mathbb{R} {\mathrm{P} }^{2} $, $ \mathbb{R} {\mathrm{P} }^{2} \widetilde {\times } I\approx \mathbb{R} {\mathrm{P} }^{3} \setminus \{ \ast \} $, by redefining the differential in the Khovanov chain complex in a suitable manner.

We study the top left derived functors of the generalised $I$-adic completion and obtain equivalent properties concerning the vanishing or nonvanishing of the modules ${L}_{i} {\Lambda }_{I} (M, N)$. We also obtain some results for the sets $\text{Coass} ({L}_{i} {\Lambda }_{I} (M; N))$ and ${\text{Cosupp} }_{R} ({ H}_{i}^{I} (M; N))$.

We give a generalisation of the Cartan decomposition for connected compact Lie groups of type B motivated by the work on visible actions of Kobayashi [‘A generalized Cartan decomposition for the double coset space $(U(n_{1})\times U(n_{2})\times U(n_{3})) \backslash U(n)/ (U(p)\times U(q))$’, J. Math. Soc. Japan59 (2007), 669–691] for type A groups. Suppose that $G$ is a connected compact Lie group of type B, $\sigma $ is a Chevalley–Weyl involution and $L$, $H$ are Levi subgroups. First, we prove that $G=LG^{\sigma }H$ holds if and only if either (I) both $H$ and $L$ are maximal and of type A, or (II) $(G,H)$ is symmetric and $L$ is the Levi subgroup of an arbitrary maximal parabolic subgroup up to switching $H$ and $L$. This classification gives a visible action of $L$ on the generalised flag variety $G/H$, as well as that of the $H$-action on $G/L$ and of the $G$-action on $(G\times G)/(L\times H)$. Second, we find an explicit ‘slice’ $B$ with $\dim B=\mathrm {rank}\, G$ in case I, and $\dim B=2$ or $3$ in case II, such that a generalised Cartan decomposition $G=LBH$holds. An application to multiplicity-free theorems of representations is also discussed.

Let $p(z)= z{f}^{\prime } (z)/ f(z)$ for a function $f(z)$ analytic on the unit disc $\mid z\mid \lt 1$ in the complex plane and normalised by $f(0)= 0, {f}^{\prime } (0)= 1$. We provide lower and upper bounds for the best constants ${\delta }_{0} $ and ${\delta }_{1} $ such that the conditions ${e}^{- {\delta }_{0} / 2} \lt \mid p(z)\mid \lt {e}^{{\delta }_{0} / 2} $ and $\mid p(w)/ p(z)\mid \lt {e}^{{\delta }_{1} } $ for $\mid z\mid , \mid w\mid \lt 1$ respectively imply univalence of $f$ on the unit disc.

The Kirchhoff elastic rod is one of the mathematical models of equilibrium configurations of thin elastic rods, and is defined to be a solution of the Euler–Lagrange equations associated to the energy with the effect of bending and twisting. In this paper, we consider Kirchhoff elastic rods in a space form. In particular, we give the existence and uniqueness of global solutions of the initial-value problem for the Euler–Lagrange equations. This implies that an arbitrary Kirchhoff elastic rod of finite length extends to that of infinite length.

We investigate the topological and metric properties of attractors of an iterated function system (IFS) whose functions may not be contractive. We focus, in particular, on invertible IFSs of finitely many maps on a compact metric space. We rely on ideas of Kieninger [Iterated Function Systems on Compact Hausdorff Spaces (Shaker, Aachen, 2002)] and McGehee and Wiandt [‘Conley decomposition for closed relations’, Differ. Equ. Appl. 12 (2006), 1–47] restricted to what is, in many ways, a simpler setting, but focused on a special type of attractor, namely point-fibred invariant sets. This allows us to give short proofs of some of the key ideas.

We use a generalisation of Vinogradov’s mean value theorem of Parsell et al. [‘Near-optimal mean value estimates for multidimensional Weyl sums’, arXiv:1205.6331] and ideas of Schmidt [‘Irregularities of distribution. IX’, Acta Arith. 27 (1975), 385–396] to give nontrivial bounds for the number of solutions to polynomial congruences, when the solutions lie in a very general class of sets, including all convex sets.

In this paper we develop a Kantorovich-like theory for Chebyshev’s method, a well-known iterative method for solving nonlinear equations in Banach spaces. We improve the results obtained previously by considering Chebyshev’s method as an element of a family of iterative processes.

We prove an upper bound on sums of squares of minors of $\{+1, -1\}$-matrices. The bound is sharp for Hadamard matrices, a result due to de Launey and Levin [‘$(1,-1)$-matrices with near-extremal properties’, SIAM J. Discrete Math.23(2009), 1422–1440], but our proof is simpler. We give several corollaries relevant to minors of Hadamard matrices.

The commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The object of this paper is to compute the commutativity degree of a class of finite groups obtained by semidirect product of two finite abelian groups. As a byproduct of our result, we provide an affirmative answer to an open question posed by Lescot.

We consider a code to be a subset of the vertex set of a Hamming graph. We examine elusive pairs, code-group pairs where the code is not determined by knowledge of its set of neighbours. We construct a new infinite family of elusive pairs, where the group in question acts transitively on the set of neighbours of the code. In these examples, the alphabet size always divides the length of the code. We show that there is no elusive pair for the smallest set of parameters that does not satisfy this condition. We also pose several questions regarding elusive pairs.

The $p$-length of a finite $p$-soluble group is an important invariant parameter. The well-known Hall–Higman $p$-length theorem states that the $p$-length of a $p$-soluble group is bounded above by the nilpotent class of its Sylow $p$-subgroups. In this paper, we improve this result by giving a better estimation on the $p$-length of a $p$-soluble group in terms of other invariant parameters of its Sylow $p$-subgroups.

Let $G$ be a general weighted graph (with possible self-loops) on $n$ vertices and $\lambda _1,\lambda _2,\ldots ,\lambda _n$ be its eigenvalues. The Estrada index of $G$ is a graph invariant defined as $EE=\sum _{i=1}^ne^{\lambda _i}$. We present a generic expression for $EE$ based on weights of short closed walks in $G$. We establish lower and upper bounds for $EE$in terms of low-order spectral moments involving the weights of closed walks. A concrete example of calculation is provided.

Two arithmetic functions $f$ and $g$ form a Möbius pair if $f(n)= {\mathop{\sum }\nolimits}_{d\mid n} g(d)$ for all natural numbers $n$. In that case, $g$ can be expressed in terms of $f$ by the familiar Möbius inversion formula of elementary number theory. In a previous paper, the first-named author showed that if the members $f$ and $g$ of a Möbius pair are both finitely supported, then both functions vanish identically. Here we prove two significantly stronger versions of this uncertainty principle. A corollary is that in a nonzero Möbius pair, one cannot have both ${\mathop{\sum }\nolimits}_{f(n)\not = 0} 1/ n\lt \infty $ and ${\mathop{\sum }\nolimits}_{g(n)\not = 0} 1/ n\lt \infty $.

Let $G$ be a locally compact group. In this paper, we show that if $G$ is a nondiscrete locally compact group, $p\in (0, 1)$ and $q\in (0, + \infty ] $, then $\{ (f, g)\in {L}^{p} (G)\times {L}^{q} (G): f\ast g\text{ is finite } \lambda \text{-a.e.} \} $ is a set of first category in ${L}^{p} (G)\times {L}^{q} (G)$. We also show that if $G$ is a nondiscrete locally compact group and $p, q, r\in [1, + \infty ] $ such that $1/ p+ 1/ q\gt 1+ 1/ r$, then $\{ (f, g)\in {L}^{p} (G)\times {L}^{q} (G): f\ast g\in {L}^{r} (G)\} $, is a set of first category in ${L}^{p} (G)\times {L}^{q} (G)$. Consequently, for $p, q\in [1+ \infty )$ and $r\in [1, + \infty ] $ with $1/ p+ 1/ q\gt 1+ 1/ r$, $G$ is discrete if and only if ${L}^{p} (G)\ast {L}^{q} (G)\subseteq {L}^{r} (G)$; this answers a question raised by Saeki [‘The ${L}^{p} $-conjecture and Young’s inequality’, Illinois J. Math. 34 (1990), 615–627].

Let $G$ be a finite $p$-solvable group and let ${G}^{\ast } $ be the set of elements of primary and biprimary orders of $G$. Suppose that the conjugacy class sizes of ${G}^{\ast } $ are $\{ 1, {p}^{a} , n, {p}^{a} n\} $, where the prime $p$ divides the positive integer $n$ and ${p}^{a} $ does not divide $n$. Then $G$ is, up to central factors, a $\{ p, q\} $-group with $p$ and $q$ two distinct primes. In particular, $G$ is solvable.

It is proved that every remainder of a nonlocally compact semitopological group $G$ is a Baire space if and only if $G$ is not Čech-complete, which improves a dichotomy theorem of topological groups by Arhangel’skiǐ [‘The Baire property in remainders of topological groups and other results’, Comment. Math. Univ. Carolin. 50(2) (2009), 273–279], and also gives a positive answer to a question of Lin and Lin [‘About remainders in compactifications of paratopological groups’, ArXiv: 1106.3836v1 [Math. GN] 20 June 2011]. We also show that for a nonlocally compact rectifiable space $G$ every remainder of $G$ is either Baire, or meagre and Lindelöf.