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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)

- Ilya Kapovich, Richard Weidmann, Alexei G. Myasnikov
- IJAC
- 2005

We introduce a combinatorial version of Stallings-Bestvina-Feighn-Dunwoody folding sequences. We then show how they are useful in analyzing the solvability of the uniform subgroup membership problem for fundamental groups of graphs of groups. Applications include coherent right-angled Artin groups and coherent solvable groups.

- Alexei G. Myasnikov, Andrey Nikolaev, Alexander Ushakov
- Math. Comput.
- 2015

We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems. We show that these problems, as well as the bounded submonoid membership problem, are P-time decidable in hyperbolic groups and give various examples of finitely presented groups where the subset sum problem is… (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)

- Alexei D. Miasnikov, Alexei G. Myasnikov
- ArXiv
- 2000

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)

We use context free languages to analyze solution sets to one variable equations over free groups.

- Alexei G. Myasnikov, Alexander Ushakov
- J. Mathematical Cryptology
- 2008

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)

- Robert L. Mach, Bernhard Seiboth, +4 authors Christian Peter Kubicek
- Molecular microbiology
- 1995

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)

- Joel David Hamkins, Alexei G. Myasnikov
- Notre Dame Journal of Formal Logic
- 2006

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)