We classify flag complexes on at most 12 vertices with torsion in the first homology group. The result is moderately computer-aided.

As a consequence we confirm a folklore conjecture that the smallest poset whose order complex is homotopy equivalent to the real projective plane (and also the smallest poset with torsion in the first homology group) has exactly 13 elements.

In this paper we show that every module of a table algebra can be considered as a faithful module of some quotient table algebra. Also we prove that every faithful module of a table algebra determines a closed subset that is a cyclic group. As a main result we give some information about multiplicities of characters in table algebras.

Given an integer $n\,\ge \,3$, a metrizable compact topological $n$-manifold $X$ with boundary, and a finite positive Borel measure $\mu$ on $X$, we prove that for the typical homeomorphism $f:\,X\,\to \,X$, it is true that for $\mu$-almost every point $x$ in $X$ the restriction of $f$ (respectively of ${{f}^{-1}}$) to the omega limit set $\omega \left( f,\,x \right)$ (respectively to the alpha limit set $\alpha \left( f,\,x \right)$) is topologically conjugate to the universal odometer.

Consider the wreath product $H\,\wr \,G$, where $H\,\ne \,1$ is finite and $G$ is finitely generated. We show that the Assouad–Nagata dimension ${{\dim}_{AN}}\left( H\,\wr \,G \right)$ of $H\,\wr \,G$ depends on the growth of $G$ as follows: if the growth of $G$ is not bounded by a linear function, then ${{\dim}_{AN}}\left( H\,\wr \,G \right)\,=\,\infty$; otherwise ${{\dim}_{AN}}\left( H\,\wr \,G \right)\,=\,{{\dim}_{AN}}\left( G \right)\,\le \,1$.

The unitary extension principle $\left( \text{UEP} \right)$ by A. Ron and Z. Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the $\text{UEP}$-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one $\text{UEP}$-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators and prove that this condition is also sufficient to ensure that an extension with just two generators is possible.

Let $p$, $q$ be prime numbers with ${{p}^{2}}\,<\,q,\,n\,\in \,\mathbb{N}$, and $H$ a semisimple Hopf algebra of dimension $p{{q}^{n}}$ over an algebraically closed field of characteristic 0. This paper proves that $H$ must possess one of the following two structures: (1) $H$ is semisolvable; (2) $H$ is a Radford biproduct $R\#kG$, where $kG$ is the group algebra of group $G$ of order $p$ and $R$ is a semisimple Yetter–Drinfeld Hopf algebra in $_{kG}^{kG}y\mathcal{D}$ of dimension ${{q}^{n}}$.

Let ${{H}^{2}}\left( \Omega \right)$ be the Hardy space on a strictly pseudoconvex domain $\Omega \,\subset \,{{\mathbb{C}}^{n}}$, and let $A\,\subset \,{{L}^{\infty }}\left( \partial \Omega \right)$ denote the subalgebra of all ${{L}^{\infty }}$-functions $f$ with compact Hankel operator ${{H}_{f}}$. Given any closed subalgebra $B\,\subset \,A$ containing $C\left( \partial \Omega \right)$, we describe the first Hochschild cohomology group of the corresponding Toeplitz algebra $\mathcal{T}\left( B \right)\,\subset \,B\left( {{H}^{2}}\left( \Omega \right) \right)$. In particular, we show that every derivation on $\mathcal{T}\left( A \right)$ is inner. These results are new even for $n\,=\,1$, where it follows that every derivation on $\mathcal{T}\left( {{H}^{\infty }}\,+\,C \right)$ is inner, while there are non-inner derivations on $\mathcal{T}\left( {{H}^{\infty }}\,+\,C\left( \partial {{\mathbb{B}}_{n}} \right) \right)$ over the unit ball ${{\mathbb{B}}_{n}}$ in dimension $n\,>\,1$.

Let $G$ be a finite group and $H$, $K$ two subgroups of $G$. A group $G$ is said to be a mutually $m$-permutable product of $H$ and $K$ if $G\,=\,HK$ and every maximal subgroup of $H$ permutes with $K$ and every maximal subgroup of $K$ permutes with $H$. In this paper, we investigate the structure of a finite group that is a mutually $m$-permutable product of two subgroups under the assumption that its maximal subgroups are totally smooth.

For Dewitt super groups $G$ modeled via an underlying finitely generated Grassmann algebra it is well known that when there exists a body group $BG$ compatible with the group operation on $G$, then, generically, the kernel $K$ of the body homomorphism is nilpotent. This is not true when the underlying Grassmann algebra is infinitely generated. We show that it is quasi-nilpotent in the sense that as a Banach Lie group its Lie algebra $\kappa$ has the property that for each $a\,\in \,\kappa ,\,\text{a}{{\text{d}}_{a}}$ has a zero spectrum. We also show that the exponential mapping from $\kappa$ to $K$ is surjective and that $K$ is a quotient manifold of the Banach space $\kappa$ via a lattice in $\kappa$.

In a paper from 1976, Berg, Christensen, and Ressel prove that the closure of the cone of sums of squares $\sum{\mathbb{R}{{\left[ \underline{X} \right]}^{2}}}$ in the polynomial ring $\mathbb{R}\left[ \underline{X} \right]\,:=\,\mathbb{R}\left[ {{X}_{1}},\,.\,.\,.\,,\,{{X}_{n}} \right]$ in the topology induced by the $~{{\ell }_{1}}$-norm is equal to $\text{Pos}\left( {{\left[ -1,\,1 \right]}^{n}} \right)$, the cone consisting of all polynomials that are non-negative on the hypercube ${{\left[ -1,\,1 \right]}^{n}}$. The result is deduced as a corollary of a general result, established in the same paper, which is valid for any commutative semigroup. In later work, Berg and Maserick and Berg, Christensen, and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted $~{{\ell }_{1}}$-seminorm topology associated with an absolute value. In this paper we give a new proof of these results, which is based on Jacobi’s representation theorem from 2001. At the same time, we use Jacobi’s representation theorem to extend these results from sums of squares to sums of $2d$-powers, proving, in particular, that for any integer $d\,\ge \,1$, the closure of the cone of sums of $2d$-powers $\sum \mathbb{R}{{\left[ \underline{X} \right]}^{2d}}$ in the $\mathbb{R}\left[ \underline{X} \right]$ topology induced by the $~{{\ell }_{1}}$-norm is equal to $\text{Pos}\left( {{\left[ -1,\,1 \right]}^{n}} \right)$.

Answering a question of H. Petersson, we provide a class of examples of a pair of octonion algebras over a ring having isometric norms.

We show that the manifold resulting from $r$-surgery on the knot ${{5}_{2}}$, which is the two-bridge knot corresponding to the rational number $3/7$, has a left-orderable fundamental group if the slope $r$ satisfies $0\,\le \,r\,\le \,4$.

Let $R$ be an arbitrary ring and let ${{\left( - \right)}^{+}}\,=\,\text{Ho}{{\text{m}}_{\mathbb{Z}}}\left( -,\,{\mathbb{Q}}/{\mathbb{Z}}\; \right)$, where $\mathbb{Z}$ is the ring of integers and $\mathbb{Q}$ is the ring of rational numbers. Let $\mathcal{C}$ be a subcategory of left $R$-modules and $\mathcal{D}$ a subcategory of right $R$-modules such that ${{X}^{+}}\,\in \,\mathcal{D}$ for any $X\,\in \,\mathcal{C}$ and all modules in $\mathcal{C}$ are pure injective. Then a homomorphism $f:\,A\to \,C$ of left $R$-modules with $C\,\in \,\mathcal{C}$ is a $\mathcal{C}$-(pre)envelope of $A$ provided ${{f}^{+}}:\,{{C}^{+}}\,\to \,{{A}^{+}}$ is a $\mathcal{D}$-(pre)cover of ${{A}^{+}}$. Some applications of this result are given.

Nontrivial pairs of zero-divisors in group rings are introduced and discussed. A problem on the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of odd exponent $n\,\gg \,1$ is solved in the affirmative. Nontrivial pairs of zero-divisors are also found in group rings of free products of groups with torsion.

We prove the following result announced by the second and third authors: Any homogeneous, metric $ANR$-continuum is a $V_{G}^{n}$-continuum provided ${{\dim}_{G}}X\,=\,n\,\ge \,1$ and ${{\overset{\vee }{\mathop{H}}\,}^{n}}\left( X;\,G \right)\,\ne \,0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum is a ${{V}^{n}}$-continuum in the sense of Alexandroff. We also prove that any finite-dimensional cyclic in dimension $n$ homogeneous metric continuum $X$, satisfying ${{\overset{\vee }{\mathop{H}}\,}^{n}}\left( X;\,G \right)\,\ne \,0$ for some group $G$ and $n\,\ge \,1$, cannot be separated by a compactum $K$ with ${{\overset{\vee }{\mathop{H}}\,}^{n-1}}\left( K;\,G \right)\,=\,0$ and ${{\dim}_{G}}K\,\le \,n\,-\,1$. This provides a partial answer to a question of Kallipoliti–Papasoglu as to whether a two-dimensional homogeneous Peano continuum can be separated by arcs.

The method for computing the $p$-localization of the group $\left[ X,\,\text{U}\left( n \right) \right]$, by Hamanaka in 2004, is revised. As an application, an explicit description of the self-homotopy group of $\text{Sp}\left( 3 \right)$ localized at $p\,\ge \,5$ is given and the homotopy nilpotency of $\text{Sp}\left( 3 \right)$ localized at $p\,\ge \,5$ is determined.

Let ${{\Gamma }_{1}}$ and ${{\Gamma }_{2}}$ be Bieberbach groups contained in the full isometry group $G$ of ${{\mathbb{R}}^{n}}$. We prove that if the compact flat manifolds ${{\Gamma }_{1}}\backslash {{\mathbb{R}}^{n}}$ and ${{\Gamma }_{2}}\backslash {{\mathbb{R}}^{n}}$ are strongly isospectral, then the Bieberbach groups ${{\Gamma }_{1}}$ and ${{\Gamma }_{2}}$ are representation equivalent; that is, the right regular representations ${{L}^{2}}\left( {{\Gamma }_{1}}\backslash G \right)$ and ${{L}^{2}}\left( {{\Gamma }_{2}}\backslash G \right)$ are unitarily equivalent.

Let $X$ and $Y$ be metric spaces and $E$, $F$ be Banach spaces. Suppose that both $X$ and $Y$ are realcompact, or both $E$, $F$ are realcompact. The zero set of a vector-valued function $f$ is denoted by $z\left( f \right)$. A linear bijection $T$ between local or generalized Lipschitz vector-valued function spaces is said to preserve zero-set containments or nonvanishing functions if

$$z\left( f \right)\,\subseteq \,z\left( g \right)\,\,\,\,\Leftrightarrow \,\,\,\,z\left( Tf \right)\,\subseteq \,z\left( Tg \right),\,\,\,\,\,\text{or}\,\,\,\,z\left( f \right)\,=\,\varnothing \,\,\,\Leftrightarrow \,\,\,z\left( Tf \right)\,=\,\varnothing ,$$

respectively. Every zero-set containment preserver, and every nonvanishing function preserver when $\dim\,E\,=\,\dim\,F\,<\,+\infty$, is a weighted composition operator $\left( Tf \right)\left( y \right)\,=\,{{J}_{y}}\left( f\left( \tau \left( y \right) \right) \right)$. We show that the map $\tau \,:\,Y\,\to \,X$ is a locally (little) Lipschitz homeomorphism.

In 1970, Kotzig and Rosa defined the concept of edge-magic labelings as follows. Let $G$ be a simple $\left( p,\,q \right)$-graph (that is, a graph of order $p$ and size $q$ without loops or multiple edges). A bijective function $f:\,V\left( G \right)\cup E\left( G \right)\,\to \,\left\{ 1,\,2,\,.\,.\,.\,,\,p\,+\,q \right\}$ is an edge-magic labeling of $G$ if $f\left( u \right)\,+\,f\left( uv \right)\,+f\left( v \right)\,=\,k$, for all $uv\,\in \,E\left( G \right)$. A graph that admits an edge-magic labeling is called an edge-magic graph, and $k$ is called the magic sum of the labeling. An old conjecture of Godbold and Slater states that all possible theoretical magic sums are attained for each cycle of order $n\,\ge \,7$. Motivated by this conjecture, we prove that for all ${{n}_{0}}\,\in \,\mathbb{N}$, there exists $n\,\in \,\mathbb{N}$ such that the cycle ${{C}_{n}}$ admits at least ${{n}_{0}}$ edge-magic labelings with at least ${{n}_{0}}$ mutually distinct magic sums. We do this by providing a lower bound for the number of magic sums of the cycle ${{C}_{n}}$, depending on the sum of the exponents of the odd primes appearing in the prime factorization of $n$.

We study analytic properties function $m\left( z,\,E \right)$, which is defined on the upper half-plane as an integral from the shifted $L$-function of an elliptic curve. We show that $m\left( z,\,E \right)$ analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for $m\left( z,\,E \right)$ in the strip $\left| \Im z \right|\,<\,2\pi$.