Vol 84, No 5 (2020)

We study the covariant bundles (Mostow bundles) for simply connected homogeneous manifolds, establishtheir relation to homogeneous bundles and consider classes of homogeneous manifolds for which theMostow bundle is trivial (resp. non-trivial). We also give a classification of non-compact simply connectedhomogeneous manifolds of dimension not exceeding seven.

We consider subdivision schemes, which are used for the approximation of functions and generation of curves on the dyadic half-line.In the classical case of functions on the real line, the theory of subdivision schemes is widely known becauseof its applications in constructive approximation theory and signal processing as well as for generatingfractal curves and surfaces. We define and study subdivision schemes on the dyadic half-line (the positive half-lineendowed with the standard Lebesgue measure and the digit-wise binary addition operation), where the role ofexponentials is played by Walsh functions.We obtain necessary and sufficient conditions for the convergence of subdivision schemes in terms of the spectralproperties of matrices and in terms of the smoothness of solutions of the corresponding refinement equation. We also investigate the problem of convergence of subdivision schemes with non-negative coefficients. We obtain an explicit criterion for theconvergence of algorithms with four coefficients. As an auxiliary result, we define fractal curves on the dyadic half-lineand prove a formula for their smoothness. The paper contains various illustrative examples and numerical results.

We obtain a class of explicit formulae each of which gives an expression for the remainder term in the asymptotic equation for the Chebyshev function in terms of the spectrum of the Laplace operator on the fundamental domain of the modular group.

We study homotopes of alternative algebras over an algebraically closed field of characteristic different from $3$. We prove an analogue of Albert's theorem on isotopes of associative algebras: in the class of finite-dimensional unital alternative algebras every isotopy is an isomorphism. We also prove that every $(a,b)$-homotope of a unital alternative algebra preserves the identities of the original algebra. We also obtain results on the structure of isotopes of various simple algebras, in particular, Cayley–Dixon algebras.