Related papers: The power conjugacy problem in Higman-Thompson gro…
It has recently been shown that quantum computers can efficiently solve the Heisenberg hidden subgroup problem, a problem whose classical query complexity is exponential. This quantum algorithm was discovered within the framework of using…
For finitely generated nilpotent groups, we employ Mal'cev coordinates to solve several classical algorithmic problems efficiently. Computation of normal forms, the membership problem, the conjugacy problem, and computation of presentations…
A p-group is called powerful if every commutator is a product of pth powers when p is odd and a product of fourth powers when p=2. In the group algebra of a group G of p-power order over a finite field of characteristic p, the group of…
Recently the third named author defined a 2-parametric family of groups $G_n^k$ \cite{gnk}. Those groups may be regarded as a certain generalisation of braid groups. Study of the connection between the groups $G_n^k$ and dynamical systems…
Group theory is a particularly fertile field for the design of practical algorithms. Algorithms have been developed across the various branches of the subject and they find wide application. Because of its relative maturity, computational…
In the counting Graph Homomorphism problem (#GraphHom) the question is: Given graphs G,H, find the number of homomorphisms from G to H. This problem is generally #P-complete, moreover, Cygan et al. proved that unless the ETH is false there…
The cycling operation endows the super summit set $S_x$ of any element $x$ of a Garside group $G$ with the structure of a directed graph $\Gamma_x$. We establish that the subset $U_x$ of $S_x$ consisting of the circuits of $\Gamma_x$ can be…
We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using,…
We discuss time complexity of The Conjugacy Problem in HNN-extensions of groups, in particular, in Miller's groups. We show that for "almost all", in some explicit sense, elements, the Conjugacy Problem is decidable in cubic time. It is…
The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, yields insight into the hierarchy of complexity classes. In particular, the theory produces algebraic tests for a…
We give an overview of the Hidden Subgroup Problem (HSP) as of July 2010, including new results discovered since the survey of arXiv:quant-ph/0411037v1. We recall how the problem provides a framework for efficient quantum algorithms and…
A complete classification is given of finite groups whose elements are partitioned into three orbits by the automorphism groups, solving the long-standing classification problem initiated by G. Higman in 1963. As a consequence, a…
Traditional power theories and one of their most important concepts --apparent power-- are still a source of debate and, as shown in the literature, they present several flaws that misinterpret the power-transfer phenomena under distorted…
This paper investigates the recent advances in Geometric Algebra-based power theory (GAPoT) and how this tool provides new insights to solve the flaws of one of the most widespread theory in the time domain, the Instantaneous Reactive Power…
We present an algorithm for solving the conjugacy search problem in the four strand braid group. The computational complexity is cubic with respect to the braid length.
This is a survey of some problems in geometric group theory which I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some…
In this paper a class of classical Hamiltonian systems is geometrically formulated. This class is such that a Hamiltonian can be written as the sum of a kinetic energy function and a potential energy function. In addition, these energy…
The matching of multiple objects (e.g. shapes or images) is a fundamental problem in vision and graphics. In order to robustly handle ambiguities, noise and repetitive patterns in challenging real-world settings, it is essential to take…
In his papers [2], [3] Brin introduced the higher dimensional Thompson groups nV which are generalizations to the Thompson's group V of self-homeomorphisms of the Cantor set and found a finite set of generators and relations in the case n =…
This is a survey of the recent work in algorithmic and asymptotic properties of groups. I discuss Dehn functions of groups, complexity of the word problem, Higman embeddings, and constructions of finitely presented groups with extreme…