Cohomology of the Lie algebra ℌ2: Experimental results and conjectures

The cohomology with trivial coefficients of the Lie algebra ? of Hamiltonian vector fields in the plane and of its maximal nilpotent subalgebra L ₁? is considered. The cohomology H ₂(L ₁?) is calculated, and some far-reaching conjectures concerning the cohomology of the Lie algebras mentioned above and based on an extensive experimental material are formulated.     

On the strong approximation by the (C, α)-means of Fourier series. II

стАтьь ьВльЕтсь пРОД ОлжЕНИЕМ пРЕДыДУЩЕИ ОДНОИМЕННОИ РАБОты АВтОРА, гДЕ ИжУ ЧАлсь пОРьДОк ВЕлИЧИН пРИ УслОВИьх, ЧтО α>-1/2, Рα >- 1 И ЧтО МАтРИцАt _nk УДОВлЕтВОРьЕт НЕкОт ОРОМУ УслОВИУ РЕгУльРНОстИ. жДЕсь ДОкАжыВАЕтсь, Ч тО ЕслИf∈H ^Ω, тО ВыпОлНь Етсь ОцЕНкА $$\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left| {\sigma _k^\alpha \left( x \right) - f\left( x \right)} \right|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} = O\left( {\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left( {\frac{1}{k}\mathop \smallint \limits_{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}}^{2\pi } \frac{{\omega \left( t \right)}}{{t^2 }}dt} \right)^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} + \left( {\frac{{\lambda _n }}{n}} \right)^\alpha \omega \left( {\frac{1}{n}} \right)} \right)$$ (λ₁=1, λ_n+1-λ_n≦1), А тАкжЕ ЧтО Ёт А ОцЕНкА ОкОНЧАтЕльН А В сВОИх тЕРМИНАх; пОДОБ НыИ РЕжУль-тАт спРАВЕДлИВ тАкжЕ И Дль сОпРьжЕННОИ ФУНкцИИ . ДОкАжыВАЕтсь, ЧтО Усл ОВИьα>?1/2 Иpα>?1, кОтОРыЕ Б ылИ НАлОжЕНы В УпОМьНУтО И ВышЕ ЧАстИ I, сУЩЕстВЕН Ны.     

On the strong approximation by the (C, α)-means of Fourier series. I

Статья продолжает ис следования Г. Суноути, Л. Лейндлера и В. Ленског о; она посвящена изучению средних вида . Здесь α> — 1/2 и рα> — 1. Для а>0 полностью иссл едован случай неотри цательной, регулярной матрицы {t _nk } и классаH ^ω (ω — модуль не прерывности). При 0>а> ?1/2 получены результаты для весьма широкого клас са матриц. Доказано, чт о в каждом случае порядки указа нных величин для функций изH ^ω совпа дают с . Кроме того, устанавли вается, что не существ ует естественных обобще ний этих утверждений на классыW ^r H ^ω (r≧1). Из результатов стать и вытекают решения не которых ранее поставленных з адач.     

On model-theoretical properties in the sense of Peretyat’kin,o-minimality,and mutually interpretable theories

We prove that o-minimality is not a model-theoretical property in the sense of Peretyat’kin. We also prove that existence of a prime models need not be preserved under a passage between mutually interpretable theories.     

Some results of the implementation of the “podmodeli” program for the gas dynamics equations

The results of investigations using the PODMODELI (SUBMODELS) program, which is aimed at the exhaustive use of the symmetry of the gas dynamics equations to construct classes of exact solutions (submodels) of these equations, are summarized. The starting point is the fact that any Lie group of transformations of the basis space (of all independent and dependent variables) which is admitted by the gas dynamics equations can generate certain submodels. In order to describe the infinite set of submodels which arises from this in a compact manner, a number of techniques for ordering them is proposed: with respect to the equation of state of a gas, using a similarity criterion, according to their types (rank, defect), with respect to the property of evolutionarity and according to the criterion of regularity. Examples of new submodels are presented. The most significant work carried out using the PODMODELI program up to the present time is indicated in a list of references.     

On Basic Features of the “Mathematics in Nine Sections”

The "Mathematics in Nine Sections" is a famous mathematical treatise trans mitted down from ancient China. The first author of the treatise is not know to us, nor can the date of its composition be exactly determined. Liu Hui, who lived in the third century of the Christian era, points out in the preface of his book, "All books were burned in the reign of the despotic emperor Shi Huang     

Addendum to J. Cilleruelo,D.S. Ramana,and O. Ramaré’s paper “Quotient and product sets of thin subsets of the positive integers”

The purpose of this note is to prove two results on the quotient sets A/A of finite sets A ? [1, n] of positive integers. They complement the results from the paper by J. Cilleruelo, D.S. Ramana, and O. Ramaré.     

Supplement to the paper by A. B. Kiselev “Analytical solutions of problems of the adiabatic compression of thick-walled spherical and cylindrical shells made of incompressible viscoplastic material”. JAMM Vol. 76, No. 4, pp. 489–492, 2012

The problem, which differs from that formulated previously,1, 2 by having an external dynamic load, is considered. Exact solutions are obtained using the same assumptions and the same method.     

Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps

We consider the 2d and 3d many body Schrödinger equations in the presence of anisotropic switchable quadratic traps. We extend and improve the collapsing estimates in Klainerman and Machedon (2008) [25] and Kirkpatrick, Schlein and Staffilani (2011) [23]. Together with an anisotropic version of the generalized lens transform in Carles (2011) [3], we derive rigorously the cubic NLS with anisotropic switchable quadratic traps in 2d through a modified Elgart–Erdös–Schlein–Yau procedure. For the 3d case, we establish the uniqueness of the corresponding Gross–Pitaevskii hierarchy without the assumption of factorized initial data.