SUSY transformations with complex factorization constants. Application to spectral singularities

Supersymmetric (SUSY) transformation operators corresponding to complex factorization constants are analyzed as operators acting in the Hilbert space of functions square integrable on the positive semiaxis. Obtained results are applied to Hamiltonians possessing spectral singularities which are non-Hermitian SUSY partners of selfadjoint operators. A new regularization procedure for the resolution of the identity operator in terms of continuous biorthonormal set of the non-Hermitian Hamiltonian eigenfunctions is proposed. It is also shown that the continuous spectrum eigenfunction has zero binorm (in the sense of distributions) at the singular point.Comment: Thanks to A. Sokolov a number of inaccuracies are correcte

Universal Drinfeld-Sokolov Reduction and Matrices of Complex Size

We construct affinization of the algebra $gl_{\lambda}$ of ``complex size'' matrices, that contains the algebras $\hat{gl_n}$ for integral values of the parameter. The Drinfeld--Sokolov Hamiltonian reduction of the algebra $\hat{gl_{\lambda}}$ results in the quadratic Gelfand--Dickey structure on the Poisson--Lie group of all pseudodifferential operators of fractional order. This construction is extended to the simultaneous deformation of orthogonal and simplectic algebras that produces self-adjoint operators, and it has a counterpart for the Toda lattices with fractional number of particles.Comment: 29 pages, no figure

Разработка бизнес-плана открытия кафе-кондитерской "SweetCake" в г. Юрга

В работе были проанализированы теоретические основы бизнес-планирования; выявлены преимущества малого бизнеса и рассмотрены новые льготы для его поддержки; разработан бизнес-проект предприятия общественного питания "Кафе-кондитерская "SweetCake" в г. Юрга, а так же были просчитаны мероприятия, направленные на развитие кафе-кондитерской "SweetCake".In work theoretical bases of business planning have been analyzed; the advantages of small business are revealed and new benefits for its support are considered; the business project of the public catering enterprise "Cafe-confectionery" SweetCake "in Yurga was developed, as well as the activities aimed at the development of the cafe-confectionery" SweetCake "

Poisson-Lie group of pseudodifferential symbols

We introduce a Lie bialgebra structure on the central extension of the Lie algebra of differential operators on the line and the circle (with scalar or matrix coefficients). This defines a Poisson--Lie structure on the dual group of pseudodifferential symbols of an arbitrary real (or complex) order. We show that the usual (second) Benney, KdV (or GL_n--Adler--Gelfand--Dickey) and KP Poisson structures are naturally realized as restrictions of this Poisson structure to submanifolds of this ``universal'' Poisson--Lie group. Moreover, the reduced (=SL_n) versions of these manifolds (W_n-algebras in physical terminology) can be viewed as subspaces of the quotient (or Poisson reduction) of this Poisson--Lie group by the dressing action of the group of functions. Finally, we define an infinite set of functions in involution on the Poisson--Lie group that give the standard families of Hamiltonians when restricted to the submanifolds mentioned above. The Poisson structure and Hamiltonians on the whole group interpolate between the Poisson structures and Hamiltonians of Benney, KP and KdV flows. We also discuss the geometrical meaning of W_\infty as a limit of Poisson algebras W_\epsilon as \epsilon goes to 0.Comment: 64 pages, no figure

Formality theorems for Hochschild complexes and their applications

We give a popular introduction to formality theorems for Hochschild complexes and their applications. We review some of the recent results and prove that the truncated Hochschild cochain complex of a polynomial algebra is non-formal.Comment: Submitted to proceedings of Poisson 200

Minimal Liouville gravity correlation numbers from Douglas string equation

We continue the study of (q, p) Minimal Liouville Gravity with the help of Douglas string equation. We generalize the results of [1,2], where Lee-Yang series (2, 2s + 1) was studied, to (3, 3s + p 0) Minimal Liouville Gravity, where p 0 = 1, 2. We demonstrate that there exist such coordinates \u3c4 m,n on the space of the perturbed Minimal Liouville Gravity theories, in which the partition function of the theory is determined by the Douglas string equation. The coordinates \u3c4 m,n are related in a non-linear fashion to the natural coupling constants \u3bb m,n of the perturbations of Minimal Lioville Gravity by the physical operators O m,n . We find this relation from the requirement that the correlation numbers in Minimal Liouville Gravity must satisfy the conformal and fusion selection rules. After fixing this relation we compute three- and four-point correlation numbers when they are not zero. The results are in agreement with the direct calculations in Minimal Liouville Gravity available in the literature [3-5]. \ua9 2014 The Author(s)

Oscilatory modules

Developing the ideas of Bressler and Soibelman and of Karabegov, we introduce a notion of an oscillatory module on a symplectic manifold which is a sheaf of modules over the sheaf of deformation quantization algebras with an additional structure. We compare the category of oscillatory modules on a torus to the Fukaya category as computed by Polishchuk and Zaslow.Comment: To appear in the proceedings of Moshe Flato Memorial Conference, November, 2008, Ben Gurion Universit

Highlights From the Annual Meeting of the American Epilepsy Society 2022

With more than 6000 attendees between in-person and virtual offerings, the American Epilepsy Society Meeting 2022 in Nashville, felt as busy as in prepandemic times. An ever-growing number of physicians, scientists, and allied health professionals gathered to learn a variety of topics about epilepsy. The program was carefully tailored to meet the needs of professionals with different interests and career stages. This article summarizes the different symposia presented at the meeting. Basic science lectures addressed the primary elements of seizure generation and pathophysiology of epilepsy in different disease states. Scientists congregated to learn about anti-seizure medications, mechanisms of action, and new tools to treat epilepsy including surgery and neurostimulation. Some symposia were also dedicated to discuss epilepsy comorbidities and practical issues regarding epilepsy care. An increasing number of patient advocates discussing their stories were intertwined within scientific activities. Many smaller group sessions targeted more specific topics to encourage member participation, including Special Interest Groups, Investigator, and Skills Workshops. Special lectures included the renown Hoyer and Lombroso, an ILAE/IBE joint session, a spotlight on the impact of Dobbs v. Jackson on reproductive health in epilepsy, and a joint session with the NAEC on coding and reimbursement policies. The hot topics symposium was focused on traumatic brain injury and post-traumatic epilepsy. A balanced collaboration with the industry allowed presentations of the latest pharmaceutical and engineering advances in satellite symposia