We construct the field generated by n algebraically independent elements, and show that the linear space of derivations over this field is faithfully represented by the linear space of the n − th fold Cartesian product of this field acting through inner product on the gradient of this field. We prove also that functional independence of a set in this field is equivalent to linear independence o...

One way to view Theorem 1.1 is as a statement that the algebraic structure of a finitely generated profinite group somehow also encodes the topological structure. That is, if one wishes to know the open subgroups of a profinite group G, a topological property, one must only consider the subgroups of G of finite index, an algebraic property. As profinite groups are compact topological spaces, an...

Contents 1. Introduction 1 2. Ergodic theory: some background 4 3. Entropy of dynamical systems: some more background 6 4. Conditional Expectation and Martingale theorems 12 5. Countably generated σ-algebras and Conditional measures 14 6. Leaf-wise Measures, the construction 19 7. Leaf-wise Measures and entropy 37 8. The product structure 61 9. Invariant measures and entropy for higher rank sub...

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A| where ε depends only on the Lie rank of L, or AAA = L. This implies that for a family of simple groups L of Lie type of bounded rank the diameter of any Cayley graph is polylogarithmic in |L|. Combining our result on growth with known results of Bourgain, Gamburd and Va...

a subgroup $h$ is said to be $nc$-supplemented in a group $g$ if there exists a subgroup $kleq g$ such that $hklhd g$ and $hcap k$ is contained in $h_{g}$, the core of $h$ in $g$. we characterize the supersolubility of finite groups $g$ with that every maximal subgroup of the sylow subgroups is $nc$-supplemented in $g$.

Let L∈K(z)[∂] be a linear differential operator, where K is an effective algebraically closed subfield of C. It can be shown that the differential Galois group of L is generated (as a closed algebraic group) by a finite number of monodromy matrices, Stokes matrices and matrices in local exponential groups. Moreover, there exist fast algorithms for the approximation of the entries of these matri...

Let L be a commutative Moufang loop (CML) with multiplication group M, and let F(L), F(M) be the Frattini subgroup and Frattini subgroup of L and M respectively. It is proved that F(L) = L if and only if F(M) = M and is described the structure of this CLM. Constructively it is defined the notion of normalizer for subloops in CML. Using this it is proved that if F(L) 6= L then L satisfies the no...

The concordance group of algebraically slice knots is the subgroup of the classical knot concordance group formed by algebraically slice knots. Results of Casson and Gordon and of Jiang showed that this group contains in infinitely generated free (abelian) subgroup. Here it is shown that the concordance group of algebraically slice knots also contain elements of finite order; in fact it contain...