Table of contents

The main result of this paper is an enumeration (given in the Table) of all mosaics  among the 165 mosaics presented in Chavey's catalogue [2] (which includes the most important classifications of mosaics) such that either  or the dual mosaic  is embeddable in  isometrically or with scale . We also study some interesting infinite-dimensional hypermetric spaces.

We study the asymptotic behaviour of the codimension growth sequence  of a finite-dimensional Lie algebra  over a field of characteristic zero. It is known that the growth of the sequence  is bounded by an exponential function of , and hence there exist the upper and lower limits of the th roots of , which are called the upper and lower exponents. By Amitsur's conjecture, the upper and lower exponents should coincide and be integers. This conjecture has been confirmed in the associative case for any PI-algebra. For finite-dimensional Lie algebras, a positive solution has been found for soluble, simple and semisimple algebras and also for algebras whose soluble radical is nilpotent. For infinite-dimensional Lie algebras, the problem has been solved in the negative. In this paper we give a proof of Amitsur's conjecture for arbitrary finite-dimensional Lie algebras.

The paper deals with a generalization of Rivoal's construction, which enables one to construct linear approximating forms in 1 and the values of the zeta function  only at odd points. We prove theorems on the irrationality of the number  for some odd integers  in a given segment of the set of positive integers. Using certain refined arithmetical estimates, we strengthen Rivoal's original results on the linear independence of the .

We introduce the construction of a -(co)module over a -(co) algebra and study its main homotopy properties. We establish a connection between -(co)modules over -(co)algebras and spectral sequences, and thus obtain the structure of an -comodule over the Milnor -coalgebra on the homology of any spectrum directly from the differentials of the Adams spectral sequence of this spectrum.

We study derived categories of coherent sheaves on Abelian varieties. We give a criterion for the equivalence of the derived categories on two Abelian varieties and describe the autoequivalence group for the derived category of coherent sheaves of an Abelian variety.

Expansions with semimultiplicative estimates for the remainders are found for the solutions of the multidimensional renewal equation. The effect of the roots of the characteristic equation on the asymptotic expression for the solution is taken into consideration. These results are used to investigate the asymptotic behaviour of the average number of particles in relation to the age of the branching processes in several cases.

In this paper, which is a continuation of [12], we develop the idea of applying Abelian Lagrangian algebraic geometry (see [3], [4], [10], [11]) to geometric quantization. The Dirac correspondence principle holds for this ALG(a)-quantization. The known models of geometric quantization involving the choice of real or complex polarizations are presented as reductions (or linearizations) of the proposed quantization. This enables us to link the results of known constructions that use polarizations.

We define the concept of I-stable ideals in the ring of commutative polynomials over a field, generalizing the so-called stable ideals, which arise as ideals of higher terms under general linear changes of variables. The interest in ideals of this type is motivated by the fact that certain problems concerning homogeneous ideals (for example, the problem of obtaining upper estimates for the graded Betti numbers) can be reduced to the study of stable ideals. I-stable ideals retain many interesting properties of stable ideals. In particular, the minimal resolutions of I-stable ideals constructed in this paper enable us to obtain an explicit formula for the graded Betti numbers, which turn out to be independent of the characteristic of the ground field. Factor rings by I-stable ideals generated by monomials of degree ≥2 are Golod rings. We also consider other analogues of stable ideals (strongly and weakly I-stable ideals) and give conditions sufficient for the factor ring by an I-stable ideal to be Cohen-Macaulay or Gorenstein.