Meandering of trajectories of polynomial vector fields in the affine n-space

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in Rn and an affine hyperplane. The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned by consecutive derivatives. This exposition constitutes an extended abstract of a forthcoming paper: only the basic steps are outlined here, with all technical details being either completely omitted or at best indicated

Hopf Term for a Two-Dimensional Electron Gas

In this Comment on the paper by W. Apel and Yu. A. Bychkov, cond-mat/9610040 and Phys. Rev. Lett. 78, 2188 (1997), we draw attention to our prior microscopic derivations of the Hopf term for various systems and to shortcomings of the Apel-Bychkov derivation. We explain how the value of the Hopt term prefactor $\Theta$ is expressed in terms of a topological invariant in the momentum space and the quantized Hall conductivity of the system. (See also related paper cond-mat/9703195)Comment: RevTeX, 1 page, no figure

Дослідження напружено-деформованого стану циліндра з мікроструктурними перетвореннями за умов імпульсного навантаження

Axisymmetric dynamic problem of thermomechanical loading of a steel cylinder is considered. Volume strain caused by the microstructural transformations of the martensitic type under cooling solids as well as into consideration and dependence inelastic characteristics of material for multiphase state are taken. The thermomechanical nonlinear behavior of an isotropic material is described by unified flow model generalized for the case of multiphase material state. The problem is solved numerically by the implicit step-by-step time integration method, by the iterative method and by the finite element method. The investigation of the stress-strain state of an inelastic material with regard for the dependence of parameters of the flow model on the phase composition of a material is carried out by using of numerical simulation. We established that microstructural transformations significantly reduce residual inelastic strain and promote the appearance of compressive stresses. The results obtained in the work can be used in calculations of parameters of surface hardening technologies. Pages of the article in the issue: 176 - 179 Language of the article: UkrainianРозглядається осесиметрична задача про імпульсне термомеханічне навантаження сталевого циліндра. Враховуються об’ємні деформації, які супроводжують мікроструктурні перетворення мартенситного типу при охолодженні тіл, а також залежність непружних характеристик матеріалу від фазового складу. Термомеханічна поведінка ізотропного матеріалу описується моделлю течії Боднера-Партома. Задача розв’язується чисельно методом покрокового неявного інтегрування за часом, ітераційним методом та методом скінченних елементів. За допомогою чисельного моделювання проводиться дослідження напружено-деформованого стану за умов врахування залежності параметрів моделі від мікроструктури матеріалу. Встановлено, що мікроструктурні перетворення суттєво зменшують залишкові деформації та сприяють появі стискальних напружень. Отримані результати можуть бути використані в інженерних розрахунках на міцність елементів конструкцій, а також параметрів технологій поверхневого зміцнення

Oscillation of linear ordinary differential equations: on a theorem by A. Grigoriev

We give a simplified proof and an improvement of a recent theorem by A. Grigoriev, placing an upper bound for the number of roots of linear combinations of solutions to systems of linear equations with polynomial or rational coefficients.Comment: 16 page

Interlayer tunneling spectroscopy of graphite at high magnetic field oriented parallel to the layers

Interlayer tunneling in graphite mesa-type structures is studied at a strong in-plane magnetic field $H$ up to 55 T and low temperature $T=1.4$ K. The tunneling spectrum $dI/dV$ vs. $V$ has a pronounced peak at a finite voltage $V_0$. The peak position $V_0$ increases linearly with $H$. To explain the experiment, we develop a theoretical model of graphite in the crossed electric $E$ and magnetic $H$ fields. When the fields satisfy the resonant condition $E=vH$, where $v$ is the velocity of the two-dimensional Dirac electrons in graphene, the wave functions delocalize and give rise to the peak in the tunneling spectrum observed in the experiment.Comment: 6 pages, 6 figures; corresponds to the published version in Eur. Phys. J. Special Topics, Proceedings of the IMPACT conference 2012, http://lptms.u-psud.fr/impact2012

On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem

We prove that the number of limit cycles generated by a small non-conservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing tangential Hilbert 16th problem. The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection (Gauss-Manin connection) with a quasiunipotent monodromy group.Comment: Final revisio

Deconfined Fermions but Confined Coherence?

The cuprate superconductors and certain organic conductors exhibit transport which is qualitatively anisotropic, yet at the same time other properties of these materials strongly suggest the existence of a Fermi surface and low energy excitations with substantial free electron character. The former of these features is very difficult to account for if the material possesses three dimensional coherence, while the latter is inconsistent with a description based on a two dimensional fixed point. We therefore present a new proposal for these materials in which they are categorized by a fixed point at which transport in one direction is not renormalization group irrelevant, but is intrinsically incoherent, i.e. the incoherence is present in a pure system, at zero temperature. The defining property of such a state is that single electron coherence is confined to lower dimensional subspaces (planes or chains) so that it is impossible to observe interference effects between histories which involve electrons moving between these subspaces.Comment: 31 pages, REVTEX, 3 eps figures, epsf.tex macr