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

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

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

Fused 1,2,3-dithiazoles: convenient synthesis, structural characterization, and electrochemical properties

A new general protocol for synthesis of fused 1,2,3-dithiazoles by the reaction of cyclic oximes with S2Cl2 and pyridine in acetonitrile has been developed. The target 1,2,3-dithiazoles fused with various carbocycles, such as indene, naphthalenone, cyclohexadienone, cyclopentadiene, and benzoannulene, were selectively obtained in low to high yields. In most cases, the hetero ring-closure was accompanied by chlorination of the carbocyclic moieties. With naphthalenone derivatives, a novel dithiazole rearrangement (15→13) featuring unexpected movement of the dithiazole ring from α- to β-position, with respect to keto group, was discovered. Molecular structure of 4-chloro-5H-naphtho[1,2-d][1,2,3]dithiazol-5-one 13 was confirmed by single-crystal X-ray diffraction. Electrochemical properties of 13 were studied by cyclic voltammetry and a complex behavior was observed, most likely including hydrodechlorination at a low potential

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

Digestive Disease Mortality Dynamics During New Coronavirus Infection Pandemic in Different Subjects of Russian Federation

Aim. An analysis of digestive disease mortality dynamics in different subjects of the Russian Federation in course of the new coronavirus infection pandemic.Key points. In most subjects of the Russian Federation, the first half of 2021 enduring the COVID-19 pandemic has witnessed a higher overall mortality from digestive diseases and from peptic ulcer, liver and pancreatic illnesses compared to the same period in 2020. This situation may have roots in both the adverse impact of coronavirus infection on pre-existing digestive diseases and shortages in providing specialty medical aid to gastroenterological patients during the pandemic. Improved outpatient care and remote counselling, as well as successful educational measures, may reduce gastroenterological disease-associated mortality.Conclusion. Most regions of Russia have registered a growing mortality from digestive diseases at the new coronavirus infection pandemic due to the SARS-CoV-2 adverse impact on illness progression, as well as imposed difficulties in providing specialty medical aid

СИНТЕЗ КОНЪЮГАТА 24-ЭПИКАСТАСТЕРОНА С ФЛУОРЕСЦЕНТНЫМ КРАСИТЕЛЕМ BODIPY

To carry out studies of molecular events involving brassinosteroids in a living cell, the synthesis of the phytohormone 24-epicastasterone labeled with a fluorescent dye BODIPY was performed. A distinctive feature of the conjugate obtained is the preservation in its molecule of native functional groups of brassinosteroids responsible for the physiological effects of this class of compounds, which is intended to ensure its effective binding to the corresponding receptor. The reaction of azide-alkyne [3 + 2]-cycloaddition of steroid azide with an acetylene BODIPY-containing dye fragment was used as a key stage in chemical synthesis.Для проведения исследований молекулярных событий с участием брассиностероидов в живой клетке осуществлен синтез меченного флуоресцентным красителем BODIPY фитогормона 24-эпикастастерона. Отличительной особенностью полученного конъюгата является сохранение в его молекуле нативных функциональных группировок брассиностероидов, ответственных за физиологические эффекты данного класса соединений, что призвано обеспечить его эффективное связывание с соответствующим рецептором. В качестве ключевой стадии химического синтеза использована реакция азид-алкинового [3 +  2]-циклоприсоединения стероидного азида с содержащим ацетиленовый фрагмент BODIPY красителем