Alexei G. Myasnikov

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In this paper we prove Implicit Function Theorems (IFT) for algebraic varieties defined by regular quadratic equations and, more generally, regular NTQ systems over free groups. In the model theoretic language these results state the existence of very simple Skolem functions for particular ∀∃formulas over free groups. We construct these functions(More)
This is the second article in the series of papers by the authors on the theory of exponential groups. In the first one [15] we discussed foundations of this theory. Definitions necessary for independent understanding of the present article are given in the introduction and the first section. The theory of exponential groups begins with results of A.Mal’cev(More)
We are now witnessing a rapid growth of a new part of group theory which has become known as “statistical group theory”. A typical result in this area would say something like “a random element (or a tuple of elements) of a group G has a property P with probability p”. The validity of a statement like that does, of course, heavily depend on how one defines(More)
The Andrews-Curtis conjecture states that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of the elementary Nielsen transformations and conjugations. In this paper we describe all balanced presentations of the trivial group on two generators and with the total length of relators ≤ 12. We show that all these(More)
In this paper we discuss generic properties of ”random subgroups” of a given group G. It turns out that in many groups G (even in most exotic of them) the random subgroups have a simple algebraic structure and they ”sit” inside G in a very particular way. This gives a strong mathematical foundation for cryptanalysis of several group-based cryptosystems and(More)
We have investigated the effect of disruption of the bgl1-(beta-glucosidase l-encoding) gene of Trichoderma reesei on the formation of other beta-glucosidase activities and on the induction of cellulases. To this end the bgl1 locus was disrupted by insertion of the Aspergillus nidulans amdS (acetamidase-encoding) gene. The bgl1-disrupted strain did not(More)
The halting problem for Turing machines is decidable on a set of asymptotic probability one. The proof is sensitive to the particular computational model. The halting problem for Turing machines is perhaps the canonical undecidable set. Nevertheless, we prove that there is an algorithm deciding almost all instances of it. The halting problem is therefore(More)