Geometry of word equations in simple algebraic groups over special fields

This paper contains a survey of recent developments in investigation of word equations in simple matrix groups and polynomial equations in simple (associative and Lie) matrix algebras along with some new results on the image of word maps on algebraic groups defined over special fields: complex, real, p-adic (or close to such), or finite.Comment: 44 page

Kelvin-Helmholtz instability of AB interface in superfluid 3He

The Kelvin-Helmholtz instability is well-known in classical hydrodynamics, where it explains the sudden emergence of interfacial surface waves as a function of the velocity of flow parallel to the interface. It can be carried over to the inviscid two-fluid dynamics of superfluids, to study different types of interfaces and phase boundaries in quantum fluids. We report measurements on the stability of the phase boundary separating the two bulk phases of superfluid 3He in rotating flow, while the boundary is localized with the gradient of the magnetic field to a position perpendicular to the rotation axis. The results demonstrate that the classic stability condition, when modified for the superfluid environment, is obeyed down to 0.4 Tc, if a large fraction of the magnetic polarization of the B-phase is attributed to a parabolic reduction of the interfacial surface tension with increasing magnetic field.Comment: 14 pages, 14 figure

From Thompson to Baer-Suzuki: a sharp characterization of the solvable radical

We prove that an element $g$ of prime order $>3$ belongs to the solvable radical $R(G)$ of a finite (or, more generally, a linear) group if and only if for every $x\in G$ the subgroup generated by $g, xgx^{-1}$ is solvable. This theorem implies that a finite (or a linear) group $G$ is solvable if and only if in each conjugacy class of $G$ every two elements generate a solvable subgroup.Comment: 28 page

Equations in simple Lie algebras

Given an element $P(X_1,...,X_d)$ of the finitely generated free Lie algebra, for any Lie algebra $g$ we can consider the induced polynomial map $P: g^d\to g$. Assuming that $K$ is an arbitrary field of characteristic $\ne 2$, we prove that if $P$ is not an identity in $sl(2,K)$, then this map is dominant for any Chevalley algebra $g$. This result can be viewed as a weak infinitesimal counterpart of Borel's theorem on the dominancy of the word map on connected semisimple algebraic groups. We prove that for the Engel monomials $[[[X,Y],Y],...,Y]$ and, more generally, for their linear combinations, this map is, moreover, surjective onto the set of noncentral elements of $g$ provided that the ground field $K$ is big enough, and show that for monomials of large degree the image of this map contains no nonzero central elements. We also discuss consequences of these results for polynomial maps of associative matrix algebras.Comment: 22 page

Proof-graphs for Minimal Implicational Logic

It is well-known that the size of propositional classical proofs can be huge. Proof theoretical studies discovered exponential gaps between normal or cut free proofs and their respective non-normal proofs. The aim of this work is to study how to reduce the weight of propositional deductions. We present the formalism of proof-graphs for purely implicational logic, which are graphs of a specific shape that are intended to capture the logical structure of a deduction. The advantage of this formalism is that formulas can be shared in the reduced proof. In the present paper we give a precise definition of proof-graphs for the minimal implicational logic, together with a normalization procedure for these proof-graphs. In contrast to standard tree-like formalisms, our normalization does not increase the number of nodes, when applied to the corresponding minimal proof-graph representations.Comment: In Proceedings DCM 2013, arXiv:1403.768

Analysis of Canon CAPT protocol for Linux printer support improvement

The following paper discusses major deficiencies found in Canon’s own proprietary Advanced Printing Technology(CAPT) driver for Linux distributions. It points out the existence of experimental, but more clean and completely opensource driver based on several previous reverse engineering attempts and poses a problem of its incompatibility with aparticular printer model (LBP3000 in this case) in question. Then it proceeds to describe the effort of analysis throughobservation of captured conversation between Canon's own proprietary driver and the printer to point out thedifferences between the inner workings of original and open source drivers. Finally, it describes the implementation of printer's support in an open source driver and concludes with the successful result of producing a driver that is able to work under modern Linux distributions and share a CAPT printer on a heterogeneous local area network