Grassmannians Gr(N-1,N+1), closed differential N-1 forms and N-dimensional integrable systems

Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by the requirement of closedness of the differential N-1 forms $\Omega_{N-1}$ of rank N-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of these flows, given by the systems of the N-1 quasi-linear differential equations, describe coisotropic deformations of (N-1)-dimensional linear subspaces. For the class of solutions which are Laurent polynomials in one variable these systems coincide with N-dimensional integrable systems such as Liouville equation (N=2), dispersionless Kadomtsev-Petviashvili equation (N=3), dispersionless Toda equation (N=3), Plebanski second heavenly equation (N=4) and others. Gauge invariant part of the forms $\Omega_{N-1}$ provides us with the compact form of the corresponding hierarchies. Dual quasi-linear systems associated with the projectively dual Grassmannians Gr(2,N+1) are defined via the requirement of the closedness of the dual forms $\Omega_{N-1}^{\star}$. It is shown that at N=3 the self-dual quasi-linear system, which is associated with the harmonic (closed and co-closed) form $\Omega_{2}$, coincides with the Maxwell equations for orthogonal electric and magnetic fields.Comment: 26 pages, references adde

On the identification of Ti-Ta-Nb-oxides in “wiikites” from Karelia

Purpose. With respect to the nature of Ti-Ta-Nb-oxides, which are included in the mineral associations that have historically gone by the now outdated name of “wiikites”, there is no unanimity of opinion. The main difficulty in identifying them is associated with the complexity of their chemical composition, their metamict structure and the substantial changes in their initial mineral form due to the effects of natural weathering. The aim of this work is the development of methodology to identify Ti-Ta-Nb-oxides corresponding to the mineral formulas AB2O6 and A2B2O7. Methods. The methodology developed in the work includes two experimental approaches: 1) electron probe microanalysis of phases revealed through use of the SEM-BSE method. Based on the resulting data, a hypothesis about the nature of the Ti-Ta-Nb-oxides in a number of wiikite samples is proposed. 2) XRD analysis of those phases that are formed in the course of thermal annealing of samples from 200 to 1000°C. Based on the resulting data here, the previous hypotheses are either accepted or rejected for each sample. Findings. Using this methodology, three “wiikite” samples were analyzed (the experimental codes were Wk-2, Wk-3 and Wk-7). The Ti-Ta-Nb-oxides in two of these wiikites (Wk-2 and Wk-3) were preliminarily determined to be hydroxyyttropyrochlore and hydroxycalciobetafite, respectively. In the third sample (Wk-7), the Ti-Ta-Nb-oxide was classified as polycrase. The results of XRD phase analysis of the annealed samples were compatible with the primary hypotheses that hydroxycalciobetafite and polycrase were the initial minerals in “wiikites” 3 and 7, respectively. Originality. The paper considers the issue of the chemical conditions necessary for the recovery of the original Ti-Ta-Nb-oxide during recrystallization of the metamict structures. This is a problem that has not been substantively addressed in the literature. Also for the first time experimentally proved the existence of the polycrase in “wiikites”. Practical implications. Completed work expands our knowledge of the mineral composition of the granitic pegmatites of the Baltic shield and the possibility of their use both for research purposes and for needs of national economy.Мета. Розробка методології ідентифікації Ti-Ta-Nb-оксидів складного хімічного складу з метаміктною структурою, що відповідають мінеральним формулами АВ2О6 та А2В2О7. Методика. Для розробки методології ідентифікації у роботі використано два експериментальних підходи: 1) рентген-спектральний мікроаналіз фаз, виявлених методом скануючої електронної мікроскопії зразків у назад відображених електронах. На основі отриманих результатів висунута гіпотеза щодо природи Ti-Ta-Nb-оксидів. 2) рентген-дифракційний аналіз фаз, що утворюються в процесі термічного відпалу зразків від 200 до 1000°С. На основі отриманих результатів для кожного зразка первинна гіпотеза приймалася або відкидалася. Результати. З використанням даної методики були вивчені три зразки “віікітів” (шифри зразків: Wk-2, Wk-3 і Wk-7). Наявність Ti-Ta-Nb-оксидів попередньо визначено e двох зразках (Wk-2 і Wk-3) відповідно, як гідроксііттропірохлор та гідроксікальціобетафіт. У третьому зразку (Wk-7) Ti-Ta-Nb діагностований як полікраз. Результати рентген-дифракційного аналізу відпалених зразків підтвердили гіпотези про те, що початковими мінералами у Wk-3 і Wk-7 були гідроксікальціобетафіт і полікраз відповідно. Наукова новизна. Визначено хімічні умови, необхідні для утворення початкового Тi-Ta-Nb під час рекристалізації метаміктної структури, що раніше не розглядалось у відповідній тематичній науковій літературі. Вперше експериментально виявлено присутність у “віікітах” полікраза. Практична значимість. Отримані результати розширюють й доповнюють знання щодо мінерального складу гранітних пегматитів Балтійського щита та надають можливість їх використання в дослідницьких цілях і потребах народного господарства.Цель. Разработка методологии идентификации Ti-Ta-Nb-оксидов сложного химического состава с метамиктной структурой, соответствующих минеральным формулам АВ2О6 и А2В2О7. Методика. Для разработки методологии идентификации в работе использовано два экспериментальных подхода: 1) рентген-спектральный микроанализ фаз, выявленных методом сканирующей электронной микроскопии образцов в обратно отраженных электронах. На основе полученных результатов выдвинута гипотеза о природе Ti-Ta-Nb-оксидов. 2) рентген-дифракционный анализ фаз, образующихся в процессе термического отжига образцов от 200 до 1000°С. На основе полученных результатов для каждого образца первоначальная гипотеза принималась или отвергалась. Результаты. С использованием данной методики были изучены три образца “виикитов” (шифры образцов: Wk-2, Wk-3 и Wk-7). Наличие Ti-Ta-Nb-оксидов предварительно определено в двух образцах (Wk-2 и Wk-3) соответственно, как гидроксииттропирохлор и гидроксикальциобетафит. В третьем образце (Wk-7) Ti-Ta-Nb диагностирован как поликраз. Результаты рентген-дифракционного анализа отожженных образцов подтвердили гипотезы о том, что первоначальными минералами в Wk-3 и Wk-7 были гидроксикальциобетафит и поликраз соответственно. Научная новизна. Определены химические условия, необходимые для образования первоначального Тi-Ta-Nb во время рекристаллизации метамиктной структуры, что ранее не рассматривалось в соответствующих тематических научных литературе. Впервые экспериментально выявлено присутствие в “виикитах” поликраза. Практическая значимость. Полученные результаты расширяют и дополняют знания о минеральном составе гранитных пегматитов Балтийского щита и дают возможности их использования в исследовательских целях и нуждах народного хозяйства.The study was conducted using equipment of the Center of X-ray Diffraction Studies and the Center for Microscopy and Microanalysis of the Research Park at Saint Petersburg State University, as well as the analytical laboratory of OJSC MMC Norilsk Nickel in Saint Petersburg

Current sheets at three-dimensional magnetic nulls:effect of compressibility

The nature of current sheet formation in the vicinity of three-dimensional (3D) magnetic null points is investigated. The particular focus is upon the effect of the compressibility of the plasma on the qualitative and quantitative properties of the current sheet. An initially potential 3D null is subjected to shearing perturbations, as in a previous paper [Pontin et al., Phys. Plasmas, in press (2007)]. It is found that as the incompressible limit is approached, the collapse of the null point is suppressed, and an approximately planar current sheet aligned to the fan plane is present instead. This is the case regardless of whether the spine or fan of the null is sheared. Both the peak current and peak reconnection rate are reduced. The results have a bearing on previous analytical solutions for steady-state reconnection in incompressible plasmas, implying that fan current sheet solutions are dynamically accessible, while spine current sheet solutions are not.Comment: to appear in Physics of Plasmas. This version contains updated figures and references, additional discussion, and typos are fixed. This is the second in a series of papers - the first of which (by the same authors) is located at astro-ph/0701462. A version with higher quality figures can be found at http://www.maths.dundee.ac.uk/~dpontin

New Perturbation Theory for Nonstationary Anharmonic Oscillator

The new perturbation theory for the problem of nonstationary anharmonic oscillator with polynomial nonstationary perturbation is proposed. As a zero order approximation the exact wave function of harmonic oscillator with variable frequency in external field is used. Based on some intrinsic properties of unperturbed wave function the variational-iterational method is proposed, that make it possible to correct both the amplitude and the phase of wave function. As an application the first order correction are proposed both for wave function and S-matrix elements for asymmetric perturbation potential of type $V(x,\tau)=\alpha (\tau)x^3+\beta (\tau)x^4.$ The transition amplitude ''ground state - ground state'' $W_{00}(\lambda ;\rho)$ is analyzed in detail depending on perturbation parameter $\lambda$ (including strong coupling region $$ $\sim 1$) and one-dimensional refraction coefficient $\rho$.Comment: LaTeX, 13 page

Chiral Skyrmionic matter in non-centrosymmetric magnets

Axisymmetric magnetic strings with a fixed sense of rotation and nanometer sizes (chiral magnetic vortices or Skyrmions) have been predicted to exist in a large group of non-centrosymmetric crystals more than two decades ago. Recently these extraordinary magnetic states have been directly observed in thin layers of cubic helimagnet (Fe,Co)Si. In this report we apply our earlier theoretical findings to review main properties of chiral Skyrmions, to elucidate their physical nature, and to analyse these recent experimental results on magnetic-field-driven evolution of Skyrmions and helicoids in chiral helimagnets.Comment: 13 pages, 7 figures, invited talk - JEMS-2010 ( 23-28 August, Krakow, Poland

A New View on Worst-Case to Average-Case Reductions for NP Problems

We study the result by Bogdanov and Trevisan (FOCS, 2003), who show that under reasonable assumptions, there is no non-adaptive worst-case to average-case reduction that bases the average-case hardness of an NP-problem on the worst-case complexity of an NP-complete problem. We replace the hiding and the heavy samples protocol in [BT03] by employing the histogram verification protocol of Haitner, Mahmoody and Xiao (CCC, 2010), which proves to be very useful in this context. Once the histogram is verified, our hiding protocol is directly public-coin, whereas the intuition behind the original protocol inherently relies on private coins

Complex itinerant ferromagnetism in noncentrosymmetric Cr11Ge19

The noncentrosymmetric ferromagnet Cr11Ge19 has been investigated by electrical transport, AC and DC magnetization, heat capacity, x-ray diffraction, resonant ultrasound spectroscopy, and first principles electronic structure calculations. Complex itinerant ferromagnetism in this material is indicated by nonlinearity in conventional Arrott plots, unusual behavior of AC susceptibility, and a weak heat capacity anomaly near the Curie temperature (88 K). The inclusion of spin wave excitations was found to be important in modeling the low temperature heat capacity. The temperature dependence of the elastic moduli and lattice constants, including negative thermal expansion along the c axis at low temperatures, indicate strong magneto-elastic coupling in this system. Calculations show strong evidence for itinerant ferromagnetism and suggest a noncollinear ground state may be expected