Quiver varieties and a noncommutative P²

To any finite group Γ ⊂ SL₂(ℂ) and each element t in the center of the group algebra Of Γ we associate a category, Coh(ℙ²_(Γ, τ),ℙ¹). It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed preprojective algebra introduced by Crawley-Boevey and Holland. The category Coh(ℙ²_(Γ, τ),ℙ¹) should be thought of as the category of coherent sheaves on a ‘noncommutative projective space’, ℙ²_(Γ, τ), equipped with a framing at ℙ¹, the line at infinity. Our first result establishes an isomorphism between the moduli space of torsion free objects of Coh(ℙ²_(Γ, τ),ℙ¹) and the Nakajima quiver variety arising from G via the McKay correspondence. We apply the above isomorphism to deduce a generalization of the Crawley-Boevey and Holland conjecture, saying that the moduli space of ‘rank 1’ projective modules over the deformed preprojective algebra is isomorphic to a particular quiver variety. This reduces, for Γ = {1}, to the recently obtained parametrisation of the isomorphism classes of right ideals in the first Weyl algebra, A₁, by points of the Calogero– Moser space, due to Cannings and Holland and Berest and Wilson. Our approach is algebraic and is based on a monadic description of torsion free sheaves on ℙ²_(Γ, τ). It is totally different from the one used by Berest and Wilson, involving τ-functions

Multiplicative slices, relativistic Toda and shifted quantum affine algebras

We introduce the shifted quantum affine algebras. They map homomorphically into the quantized $K$-theoretic Coulomb branches of $3d\ {\mathcal N}=4$ SUSY quiver gauge theories. In type $A$, they are endowed with a coproduct, and they act on the equivariant $K$-theory of parabolic Laumon spaces. In type $A_1$, they are closely related to the open relativistic quantum Toda lattice of type $A$.Comment: 125 pages. v2: references updated; in section 11 the third local Lax matrix is introduced. v3: references updated. v4=v5: 131 pages, minor corrections, table of contents added, Conjecture 10.25 is now replaced by Theorem 10.25 (whose proof is based on the shuffle approach and is presented in a new Appendix). v6: Final version as published, references updated, footnote 4 adde

MATHEMATICAL MODEL OF DIAGNOSTICATION OF FUEL APPARATUS OF AUTOMOTIVE DIESELS

Machinery and tractor park is an important link in agricultural production. From its efficient work depends to a large extent on reducing the cost of production, timely harvesting, transportation to its consumers, harvesting of forages and other production and household processes. Increase of productivity and economy of machine-tractor aggregates is possible due to increase of the resource and terms of trouble-free operation of machines. This is achieved through the development and implementation of effective methods and means of controlling the technical condition of machines. The article presents the mathematical model of diagnostics of fuel equipment of automotive diesel engines. A diagnostic matrix and a block diagram of its synthesis were constructed.Key words: technical condition, maintenance, diagnostics of machines, matrix, fault, block diagram.доктор технічних наук, професор Барановський В. М., кандидат технічних наук, доцент Спірін А. В. Математична модель діагностування паливної апаратури автотракторних дизелів / Тернопільський національний технічний університет імені Івана Пулюя, Україна, Тернопіль; Вінницький національний аграрний університет, Україна, ВінницяМашинно-тракторний парк є важливою ланкою сільськогосподарського виробництва. Від його ефективної роботи значною мірою залежить зниження собівартості продукції, своєчасне збирання врожаю, перевезення його споживачам, заготівля кормів та інші виробничі та господарсько-побутові процеси.Підвищення продуктивності та економічності машинно-тракторних агрегатів можливе за рахунок збільшення ресурсу і термінів безвідмовної роботи машин. Це досягається шляхом розробки та впровадження ефективних методів і засобів контролю технічного стану машин. У статті наведено математичну модель діагностування паливної апаратури автотракторних дизелів. Побудовано матрицю діагностування та блок-схему її синтезу.Ключові слова: технічний стан, технічне обслуговування, діагностика машин, матриця, несправність, блок-схема

FORMATION OF STATE BORDER OF THE BSSR ON THE POLOTSK-SEBEZH SECTION (APRIL – MAY 1924)

ГІСТОРЫЯ БЕЛАРУСКАГА ПАДЗВІННЯ І СУМЕЖНЫХ ТЭРЫТОРЫЙУ артыкуле на падставе архіўных матэрыялаў разглядаецца працэс афармлення дзяржаўнай мяжы БССР на полацка-себежскім участку.=The article based of archival materials describes the process of formation of state border of the BSSR on the Polotsk-Sebezh section

FORMATION OF STATE BORDER OF THE BSSR ON THE POLOTSK-SEBEZH SECTION (APRIL – MAY 1924)

ГІСТОРЫЯ БЕЛАРУСКАГА ПАДЗВІННЯ І СУМЕЖНЫХ ТЭРЫТОРЫЙУ артыкуле на падставе архіўных матэрыялаў разглядаецца працэс афармлення дзяржаўнай мяжы БССР на полацка-себежскім участку.=The article based of archival materials describes the process of formation of state border of the BSSR on the Polotsk-Sebezh section

Reconstruction of one-dimensional chaotic maps from sequences of probability density functions

In many practical situations, it is impossible to measure the individual trajectories generated by an unknown chaotic system, but we can observe the evolution of probability density functions generated by such a system. The paper proposes for the first time a matrix-based approach to solve the generalized inverse Frobenius–Perron problem, that is, to reconstruct an unknown one-dimensional chaotic transformation, based on a temporal sequence of probability density functions generated by the transformation. Numerical examples are used to demonstrate the applicability of the proposed approach and evaluate its robustness with respect to constantly applied stochastic perturbations

Derived coisotropic structures II: stacks and quantization

We extend results about $n$-shifted coisotropic structures from part I of this work to the setting of derived Artin stacks. We show that an intersection of coisotropic morphisms carries a Poisson structure of shift one less. We also compare non-degenerate shifted coisotropic structures and shifted Lagrangian structures and show that there is a natural equivalence between the two spaces in agreement with the classical result. Finally, we define quantizations of $n$-shifted coisotropic structures and show that they exist for $n>1$.Comment: 45 pages. Contains the second half of arXiv:1608.01482v1 with new material adde