Vol 86, No 4 (2022)

The paper is devoted to the study of metric properties of semiregular polytopesin Euclidean spaces $\mathbb{R}^n$ for $n\geqslant 4$ (Gosset polytopes). Theresults obtained here enable us to complete the classification of regular andsemiregular polytopes in Euclidean spaces whose sets of vertices form normalhomogeneous or Clifford–Wolf homogeneous metric spaces.

For the $n$-dimensional multi-parameter quantum torus algebra $\Lambda_{\mathfrak q}$ over a field $k$ defined by a multiplicativelyantisymmetric matrix $\mathfrak q = (q_{ij})$ we show that, in the case whenthe torsion-free rank of the subgroup of $k^\times$ generated by the $q_{ij}$is large enough, there is a characteristic set of values (possibly with gaps)from $0$ to $n$ that can occur as the Gelfand–Kirillov dimensions of simplemodules. The special case when $\mathrm{K}.\dim(\Lambda_{\mathfrak q}) = n - 1$and $\Lambda_{\mathfrak q}$ is simple, studied in A. Gupta, $\mathrm{GK}$-dimensions of simple modules over $K[X^{\pm 1},\sigma]$, Comm. Algebra, 41(7) (2013), 2593–2597, is considered withoutassuming the simplicity, and it is shown that a dichotomy still holds for theGK dimension of simple modules.

A frame in $\mathbb{R}^d$ is a set of $n\geqslant d$ vectors whose linear spancoincides with $\mathbb{R}^d$. A frame is said to be equal-norm if the normsof all its vectors are equal. Tight frames enable one to represent vectorsin $\mathbb{R}^d$ in the form closest to the representation in an orthonormalbasis. Every equal-norm tightframe is a useful tool for constructing efficient computational algorithms. The construction of such frames in $\mathbb{C}^d$ uses the matrix of the discrete Fourier transform, and the first constructions of equal-norm tight frames in $\mathbb{R}^d$ appeared only at the beginning of the 21st century. The present paper shows that Maltsev's note of 1947 was decades ahead of its time and turned out to be missed by the experts in frame theory, and Maltsev should be credited for the world's first design of an equal-norm tight frame in $\mathbb{R}^d$. Our main purpose is to show the historical significance of Maltsev's discovery.We consider his paper from the point of view of the modern theory of frames in finite-dimensional spaces.Using the Naimark projectors and other operator methods, we study important frame-theoretic properties of the Maltsevconstruction, such as the equality of moduli of pairwise scalar products (equiangularity) and the presence of full spark, that is, the linear independence ofany subset of $d$ vectors in the frame.

We prove that, after lifting to some finite ramified covering of a smooth projective curve $C$, the Grothendieck standard conjecture of Lefschetz type holds for the Künnemann compactification of the Neron minimal model of a 4-dimensional principally polarized Abelian variety over the field of rational functions on the curve $C$ provided that the endomorphism ring of the generic geometric fibre of the Neron model coincides with the ring of integers, all bad reductions are semi-stable and have toric rank 1 and, for any places $\delta,\delta'\in C$ of bad reductions, the Hodge conjecture on algebraic cycles holds for the product $A_\delta\times A_{\delta'}$ of the Abelian varieties $A_\delta,A_{\delta'}$ which are the quotients of the connected components of neutral elements in special fibres of the Neron minimal model modulo toric parts.