arXiv subjects
Publications and source records attributed to V. Sokolov.
A collection of miscellaneous continuous, semi-discrete, and discrete integrable systems can be associated with each integrable evolution equation of the KdV type. We give them for the Schwarz--KdV equation and generalize to the vector case. The existence of these vector generalizations is a non-trivial experimental fact for which no mathematical explanation is yet known.
Using the nonabilinization procedure, we find an integrable matrix version of the Euler top on $\mathfrak{so}_3$
We review nonabelian Poisson structures on affine and projective spaces over $\mathbb{C}$. We also construct a class of examples of nonabelian Poisson structures on $\mathbb{C} P^{n-1}$ for $n>2$. These nonabelian Poisson structures depend on a modular parameter $τ\in\mathbb{C}$ and an additional descrete parameter $k\in\mathbb{Z}$, where $1\leq k<n$ and $k,n$ are coprime. The abelianization of these Poisson structures can be lifted to the quadratic elliptic Poisson algebras $q_{n,k}(τ)$.
When compiling databases, for example to meet the needs of healthcare establishments, there is quite a common problem with the introduction and further processing of names and last names of doctors and patients that are highly specialized both in terms of pronunciation and writing. This is because names and last names of people cannot be unique, their notation is not subject to any rules of phonetics, while their length in different languages may not match. With the advent of the Internet, this situation has become generally critical and can lead to that multiple copies of e-mails are sent to one address. It is possible to solve the specified problem by using phonetic algorithms for comparing words Daitch-Mokotoff, Soundex, NYSIIS, Polyphone, and Metaphone, as well as the Levenshtein and Jaro algorithms, Q-gram-based algorithms, which make it possible to find distances between words. The most widespread among them are the Soundex and Metaphone algorithms, which are designed to index the words based on their sound, taking into consideration the rules of pronunciation. By applying the Metaphone algorithm, an attempt has been made to optimize the phonetic search processes for tasks of fuzzy coincidence, for example, at data deduplication in various databases and registries, in order to reduce the number of errors of incorrect input of last names. An analysis of the most common last names reveals that some of them are of the Ukrainian or Russian origin. At the same time, the rules following which the names are pronounced and written, for example in Ukrainian, differ radically from basic algorithms for English and differ quite significantly for the Russian language. That is why a phonetic algorithm should take into consideration first of all the peculiarities in the formation of Ukrainian last names, which is of special relevance now.
We find all homogeneous quadratic systems of ODEs with two dependent variables that have polynomial first integrals and satisfy the Kowalevski-Lyapunov test. Such systems have infinitely many polynomial infinitesimal symmetries. We describe all possible non-commutative generalizations of these systems and their symmetries. As a result, new integrable quadratic homogeneous systems of ODEs with two non-commutative variables are constructed. Their integrable non-commutative inhomogeneous generalizations are found. In particular, a non-commutative generalization of a Hamiltonian flow on the elliptic curve is presented.
In this paper we provide a comprehensive description of the internal dynamics of G0.253+0.016 (a.k.a. 'the Brick'); one of the most massive and dense molecular clouds in the Galaxy to lack signatures of widespread star formation. As a potential host to a future generation of high-mass stars, understanding largely quiescent molecular clouds like G0.253+0.016 is of critical importance. In this paper, we reanalyse Atacama Large Millimeter Array cycle 0 HNCO $J=4(0,4)-3(0,3)$ data at 3 mm, using two new pieces of software which we make available to the community. First, scousepy, a Python implementation of the spectral line fitting algorithm scouse. Secondly, acorns (Agglomerative Clustering for ORganising Nested Structures), a hierarchical n-dimensional clustering algorithm designed for use with discrete spectroscopic data. Together, these tools provide an unbiased measurement of the line of sight velocity dispersion in this cloud, $σ_{v_{los}, {\rm 1D}}=4.4\pm2.1$ kms$^{-1}$, which is somewhat larger than predicted by velocity dispersion-size relations for the Central Molecular Zone (CMZ). The dispersion of centroid velocities in the plane of the sky are comparable, yielding $σ_{v_{los}, {\rm 1D}}/σ_{v_{pos}, {\rm 1D}}\sim1.2\pm0.3$. This isotropy may indicate that the line-of-sight extent of the cloud is approximately equivalent to that in the plane of the sky. Combining our kinematic decomposition with radiative transfer modelling we conclude that G0.253+0.016 is not a single, coherent, and centrally-condensed molecular cloud; 'the Brick' is not a \emph{brick}. Instead, G0.253+0.016 is a dynamically complex and hierarchically-structured molecular cloud whose morphology is consistent with the influence of the orbital dynamics and shear in the CMZ.
For a positive integer $n>1$ denote by $ω(n)$ the maximal possible number $k$ of different functions $f_1,\dots,f_k:\mathbb{Z}/n\mathbb{Z}\mapsto \mathbb{Z}/n\mathbb{Z}$ such that each function $f_i-f_j,i<j$, is bijective. Recently A. Sidorenko conjectured that $ω(n)$ equals to the minimal prime divisor of $n$. We disprove it for $n=15,21,27$ by several counterexamples found by computer.
An integrable non-abelian generalization of a Hamiltonian flow on an elliptic curve is presented. A Lax pair for this non-abelian system is found.
We find all non-equivalent constant solutions for classical associative Yang-Baxter equation for $gl(3)$. New examples found in the classification yield the corresponding quadratic trace Poisson brackets, double Poisson structures on free associative algebra with three generators and anti-Frobenius associative algebras.
We discuss double Poisson structures in sense of M. Van den Bergh on free associative algebras focusing on the case of quadratic Poisson brackets. We establish their relations with an associative version of Young-Baxter equations, we study a bi-hamiltonian property of the linear-quadratic pencil of the double Poisson structure and propose a classification of the quadratic double Poisson brackets in the case of the algebra with two free generators. Many new examples of quadratic double Poisson brackets are proposed.
Using the method of hydrodynamic reductions, we find all integrable infinite (1+1)-dimensional hydrodynamic-type chains of shift one. A class of integrable infinite (2+1)-dimensional hydrodynamic-type chains is constructed.
We present multiwavelength (X-ray/optical/near-infrared/millimetre) observations of GRB 051022 between 2.5 hours and ~1.15 yr after the event. It is the most intense gamma-ray burst (~ 10^-4 erg cm^-2) detected by HETE-2, with the exception of the nearby GRB 030329. Optical and near infrared observations did not detect the afterglow despite a strong afterglow at X-ray wavelengths. Millimetre observations at Plateau de Bure (PdB) detected a source and a flare, confirming the association of this event with a moderately bright (R = 21.5) galaxy. Spectroscopic observations of this galaxy show strong [O II], Hbeta and [O III] emission lines at a redshift of 0.809. The spectral energy distribution of the galaxy implies Av (rest frame) = 1.0 and a starburst occuring ~ 25 Myr ago, during which the star-forming-rate reached >= 25 Msun/yr. In conjunction with the spatial extent (~ 1'') it suggests a very luminous (Mv = - 21.8) blue compact galaxy, for which we also find with Z Zsun. The X-ray spectrum shows evidence of considerable absorption by neutral gas with NH, X-ray = 3.47(+0.48/-0.47) x 10^22 cm^-2 (rest frame). Absorption by dust in the host galaxy at z = 0.809 certainly cannot account for the non-detection of the optical afterglow, unless the dust-to-gas ratio is quite different than that seen in our Galaxy (i.e. large dust grains). It is likely that the afterglow of the dark GRB 051022 was extinguished along the line of sight by an obscured, dense star forming region in a molecular cloud within the parent host galaxy. This galaxy is different from most GRB hosts being brighter than L* by a factor of 3. We have also derived a SFR ~ 50 Msun/yr and predict that this host galaxy will be detected at sub-mm wavelengths.
We present multiwavelength (optical/near infrared/millimetre) observations of a short duration gamma-ray burst detected by Swift (GRB 050509b) collected between 0 seconds and ~18.8 days after the event. No optical, near infrared or millimetre emission has been detected in spite of the well localised X-ray afterglow, confirming the elusiveness of the short duration events. We also discuss the possibility of the burst being located in a cluster of galaxies at z= 0.225 or beyond. In the former case, the spectral energy distribution of the neighbouring, potential host galaxy, favours a system harbouring an evolved dominant stellar population (age ~360 Myr), unlike most long duration GRB host galaxies observed so far, i.e. thus giving support to a compact binary merger origin. Any underlying supernova that could be associated with this particular event should have been at least 3 magnitudes fainter than the type Ib/c SN 1998bw and 2.3 magnitudes fainter than a typical type Ia SN.
GRB 021004 is one of the best sampled gamma-ray bursts (GRB) to date, although the nature of its light curve is still being debated. Here we present a large amount (107) of new optical, near-infrared (NIR) and millimetre observations, ranging from 2 hours to more than a year after the burst. Fitting the multiband data to a model based on multiple energy injections suggests that at least 7 refreshed shocks took place during the evolution of the afterglow, implying a total energy release (collimated within an angle of 1.8 deg) of ~ 8x10^51 erg. Analysis of the late photometry reveals that the GRB 021004 host is a low extinction (Av ~ 0.1) starburst galaxy with M_B ~ -22.0.
Using the symmetry approach, we find a class of integrable nonlinear PDEs with dispersion law $ω(k)=k^{\frac32}$. All these equations turn out to be linearizable by means of a differential parametrization.
A three-parameter family of integrable quadratic Hamiltonians on $e(3)$ and $so(4)$ is presented. When one of the parameters vanishes, the Hamiltonian coincides with the Kowalevski Hamiltonian with the girostatic term. If the other two parameters are equal to zero then the Hamiltonian reduces to a new integrable case of the Kirchhoff equations found recently by the author.