Let W be an infinite word over finite alphabet A. We get combinatorial criteria of existence of interval exchange transformations that generate the word W.

A non-degenerate toric variety X is called S-homogeneous if the subgroup of the automorphism group Aut(X) generated by root subgroups acts on X transitively. We prove that maximal S-homogeneous toric varieties are in bijection with pairs (P,A), where P is an abelian group and A is a finite collection of elements in P such that A generates the group P and for every a∈A the element a is contained in the semigroup generated by A∖{a}. We show that any non-degenerate homogeneous toric variety is a big open toric subset of a maximal S-homogeneous toric variety. In particular, every homogeneous toric variety is quasiprojective. We conjecture that any non-degenerate homogeneous toric variety is S-homogeneous.

We develop a Galois theory for systems of linear difference equations with periodic parameters, for which we also introduce linear difference algebraic groups. We apply this to constructively test if solutions of linear *q*-difference equations, with *q* ∈ ℂ* and *q* not a root of unity, satisfy any polynomial ζ-difference equations with ζ *t* = 1, *t* ≥ 1.

All classes of integrable cocycles in H2(L,L) are obtained for Lie algebra of type G2 over an algebraically closed field of characteristic 2. It is proved that there exist only two orbits of classes of integrable cocycles with respect to automorphism group. The global deformation is shown to exist for any nontrivial class of integrable cocycles. These deformations are isomorphic to one of the two algebras of Cartan type, one of which being S(3:1,ω) while the other H(4:1,ω).

We investigate the least studied class of differential rings—the class of differential rings of nonzero characteristic. We present the notion of differentially closed quasifield and develop geometrical theory of differential equations in nonzero characteristic. The notions of quasivariety and its morphisms are scrutinized. Presented machinery is a basis for reduction modulo *p* for differential equations.

Let 𝕂 be an algebraically closed field of characteristic zero and *W* _{n be the Lie algebra of all 𝕂-derivations of the polynomial ring R in n variables over 𝕂. It is proved that every Lie algebra of dimension n over 𝕂 can be isomorphically embedded in W n in such a way that any basis of its image (over 𝕂) is a basis of the free module W n over R}

There are only 10 Euclidean forms, that is flat closed three-dimensional manifolds: six are orientable and four are non-orientable. The aim of this paper is to describe all types of *n*-fold coverings over non-orientable Euclidean manifolds *ℬ*1 and *ℬ*2 and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental groups of *ℬ*1 and *ℬ*2 up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index *n*. The manifolds *ℬ*1 and *ℬ*2 are uniquely determined among the other non-orientable forms by their homology groups <img src="/na101/home/literatum/publisher/tandf/journals/content/lagb20/2017/lagb20.v045.i04/00927872.2016.1222396/20161128/images/lagb_a_1222396_ilm0001.gif" alt="" />*H* 1 (B 1 )=Z 2 ×Z 2 H1(ℬ1)=ℤ2×ℤ2 and <img src="/na101/home/literatum/publisher/tandf/journals/content/lagb20/2017/lagb20.v045.i04/00927872.2016.1222396/20161128/images/lagb_a_1222396_ilm0002.gif" alt="" />*H* 1 (B 2 )=Z 2 H1(ℬ2)=ℤ2 .

There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable and four are non-orientable. The aim of this paper is to describe all types of $n$-fold coverings over orientable Euclidean manifolds $\mathcal{G}_{2}$ and $\mathcal{G}_{4}$, and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental groups $\pi_1(\mathcal{G}_{2})$ and $\pi_1(\mathcal{G}_{4})$ up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index $n$. The manifolds $\mathcal{G}_{2}$ and $\mathcal{G}_{4}$ are uniquely determined among the others orientable forms by their homology groups $H_1(\mathcal{G}_{2})=\ZZ_2\times \ZZ_2 \times \ZZ$ and $H_1(\mathcal{G}_{4})=\ZZ_2 \times \ZZ$.

Irreducible representations of finite dimensional Jordan superalgebras of Poisson brackets over algebraically closed field of zero characteristic are examined. If the number of Grassmann generators n∠4 then such a bimodule is isomorphic to regular bimodule or regular bimodule with opposite eveness of homogeneous elements.

We formulate two conjectures about etale cohomology and fundamental groups motivated by categoricity conjectures in model theory. One conjecture says that there is a unique Z-form of the etale cohomology of complex algebraic varieties, up to Aut(C)-action on the source category; put differently, each comparison isomorphism between Betti and etale cohomology comes from a choice of a topology on C. Another conjecture says that each functor to groupoids from the category of complex algebraic varieties which is similar to the topological fundamental groupoid functor, in fact factors through it, up to a field automorphism of the complex numbers acting on the category of complex algebraic varieties. We also try to present some evidence towards these conjectures, and show that some special cases seem related to Grothendieck standard conjectures and conjectures about motivic Galois group.

We develop the technique useful for studying the problem of factoring nonnegative matrices. We illustrate our method, based on the tools from linear algebra over a semiring, by applying it to studying the problem of existence of a rank-three matrix with full nonnegative rank equal to *n*.