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## Asymmetric six-vertex model and the classical Ruijsenaars–Schneider system of particles

Theoretical and Mathematical Physics. 2017. Vol. 192. No. 2. P. 1141-1153.

*We discuss the correspondence between models solved by the Bethe ansatz and classical integrable systems of the Calogero type. We illustrate the correspondence by the simplest example of the inhomogeneous asymmetric six-vertex model parameterized by trigonometric*(*hyperbolic*) *functions*.

Publication based on the results of:

Hutsalyuk A., Liashyk A, Pakuliak S. Z. et al., Nuclear Physics B 2017 Vol. 923 P. 277-311

We study scalar products of Bethe vectors in the models solvable by the nested algebraic Bethe ansatz and described by superalgebra. Using coproduct properties of the Bethe vectors we obtain a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of Bethe parameters. We also obtain recursions for ...

Added: October 26, 2017

Hutsalyuk A., Liashyk A., Pakuliak S. Z. et al., Russian Mathematical Surveys 2017 Vol. 72 No. 1 P. 33-99

Bethe vectors are found for quantum integrable models associated with the supersymmetric Yangians in terms of the current generators of the Yangian double . The method of projections onto intersections of different types of Borel subalgebras of this infinite-dimensional algebra is used to construct the Bethe vectors. Calculation of these projections makes it possible to express the ...

Added: October 26, 2017

Hutsalyuk A., Liashyk A., Pakuliak S. Z. et al., Nuclear Physics B 2018 Vol. 926 P. 256-278

We study quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl(m|n)-invariant R-matrix. We compute the norm of the Hamiltonian eigenstates. Using the notion of a generalized model we show that the square of the norm obeys a number of properties that uniquely fix it. We also show that a Jacobian of the system ...

Added: September 13, 2018

Hutsalyuk A., Liashyk A., Pakuliak S. Z. et al., SciPost Physic (Нидерланды) 2018 Vol. 4 No. 006 P. 1-30

We obtain recursion formulas for the Bethe vectors of models with periodic boundary conditions solvable by the nested algebraic Bethe ansatz and based on the quantum affine algebra U_q(gl_n). We also present a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of the Bethe parameters, ...

Added: September 13, 2018

V. A. Poberezhny, Journal of Mathematical Sciences 2013 Vol. 195 No. 4 P. 533-540

We consider systems of linear differential equations discussing some classical and modern results in the Riemann problem, isomonodromic deformations, and other related topics. Against this background, we illustrate the relations between such phenomena as the integrability, the isomonodromy, and the Painlevé property. The recent advances in the theory of isomonodromic deformations presented show perfect agreement ...

Added: February 14, 2014

Liashyk A., Slavnov N. A., Journal of High Energy Physics 2018 Vol. 06 No. 018 P. 1-31

We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl_3-invariant R-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided ...

Added: September 13, 2018

Glutsyuk A., Shustin E., Mathematische Annalen 2018 Vol. 372 P. 1481-1501

We show that every polynomially integrable planar outer convex billiard is elliptic. We also
prove an extension of this statement to non-convex billiards. ...

Added: June 29, 2018

Glutsyuk A., / Cornell University. Series "Working papers by Cornell University". 2021. No. 2104.01362.

Reflection in strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve C whose tangent lines are reflected by the billiard to lines tangent to C. The famous Birkhoff conjecture states that the only strictly convex billiards with a foliation by closed ...

Added: November 4, 2021

Gontsov R. R., V.A. Poberezhnyi, Helminck G. F., Russian Mathematical Surveys 2011 Vol. 66 No. 1 P. 63-105

This article concerns deformations of meromorphic linear differential systems. Problems relating to their existence and classification are reviewed, and the global and local behaviour of solutions to deformation equations in a neighbourhood of their singular set is analysed. Certain classical results established for isomonodromic deformations of Fuchsian systems are generalized to the case of integrable ...

Added: September 27, 2013

Poberezhny V. A., / ИТЭФ. Series "Препринты ИТЭФ". 2013. No. 18/13.

We review the modern theory of isomonodromic deformations, considering linear systems of
differential equations. On that background we illustrate the natural relations between
such phenomena as integrability, isomonodromy and Painlev\'{e} property. The recent
advances in the theory of isomonodromic deformations we present show perfect agreement to
that approach. ...

Added: March 31, 2014

Lyashik A., Pakuliak S. Z., Ragoucy E. et al., Journal of Statistical Mechanics: Theory and Experiment 2019 Vol. 2019 No. 4 P. 1-23

We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl(N)-invariant R-matrix. We study two types of Bethe vectors. The first type corresponds to the original monodromy matrix. The second type is associated to a monodromy matrix closely related to the inverse of the monodromy matrix. We show that these two types of ...

Added: June 6, 2019

V.A.Vassiliev, Arnold Mathematical Journal 2020 Vol. 6 No. 2 P. 291-309

V. Arnold’s problem 1987–14 from his Problems book asks whether there exist bodies with smooth boundaries in R^N (other than the ellipsoids in odd-dimensional spaces) for which the volume of the segment cut by any hyperplane from the body depends algebraically on the hyperplane. We present a series of very realistic candidates for the role ...

Added: August 17, 2020

Bialy M., Fierobe C., Glutsyuk A. et al., Arnold Mathematical Journal 2022 Vol. 8 P. 411-422

This is a collection of problems composed by some participants of the workshop “Differential Geometry, Billiards, and Geometric Optics” that took place at CIRM on October 4–8, 2021. ...

Added: February 1, 2024

Zabrodin A., (Mathematical Sciences 2013 No. 596 P. 7-12

We review the role of the Hirota equation and the tau-function in the theory of classical and quantum integrable systems. ...

Added: February 16, 2013

Povolotsky A. M., Journal of Statistical Mechanics: Theory and Experiment 2023 Article 033103

This work continues the study started in Povolotsky (2021 J. Phys. A: Math. Theor. 54 22LT01), where the exact densities of loops in the O(1) dense loop model on an infinite strip of the square lattice with periodic boundary conditions were obtained. These densities are also equal to the densities of critical percolation clusters on ...

Added: March 25, 2024

Feigin B. L., Jimbo M., Mukhin E., Journal of Physics A: Mathematical and Theoretical 2017 Vol. 50 No. 46 Article 464001

We identify the Taylor coefficients of the transfer matrices corresponding to quantum toroidal algebras with the elliptic local and non-local integrals of motion introduced by Kojima, Shiraishi, Watanabe, and one of the authors.
That allows us to prove the Litvinov conjectures on the Intermediate Long Wave model.
We also discuss the (gl(m),gl(n)) duality of XXZ models in ...

Added: November 5, 2020

Takebe T., Kuroki G., Journal of Physics A: Mathematical and Theoretical 2001 Vol. 34 No. 11 P. 2403-2413

We construct a Gaudin type lattice model as the Wess-Zumino-Witten model on elliptic curves at the critical level. Bethe eigenvectors are obtained by the bosonisation technique. ...

Added: August 14, 2014

Glutsyuk A., Journal of Fixed Point Theory and Applications 2022 Vol. 24 No. 2 Article 35

For a given closed convex planar curve γ with smooth boundary and a given p>0, the string construction yields a family of curves Γp for which γ is a caustic. The action of the reflection Tp on the tangent lines to γ≃S1 induces its action on the tangency points: a circle diffeomorphism Tp:γ→γ. We say ...

Added: November 4, 2021

Glutsyuk A., / Cornell University. Series "Working papers by Cornell University". 2019.

For a given closed convex planar curve γ with smooth boundary and a given p>0, the string construction yields a family of curves Γp for which γ is a caustic. The action of the reflection Tp on the tangent lines to γ≃S1 induces its action on the tangency points: a circle diffeomorphism p:γ→γ. We say ...

Added: November 12, 2019

Derbyshev A. E., Povolotsky A. M., Priezzhev V. B., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2015 Vol. 91 P. 022125

The generalized totally asymmetric exclusion process (TASEP) [J. Stat. Mech. (2012) P05014] is an integrable generalization of the TASEP equipped with an interaction, which enhances the clustering of particles. The process interpolates between two extremal cases: the TASEP with parallel update and the process with all particles irreversibly merging into a single cluster moving as ...

Added: February 19, 2015

Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020

Borzykh D., ЛЕНАНД, 2021

Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...

Added: February 20, 2021

В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72-80

Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...

Added: May 3, 2017

Красноярск : ИВМ СО РАН, 2013

Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...

Added: November 18, 2013