Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems

We study the problem of placing effective upper bounds for the number of zeros of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of given dimension n having m singular points. As a function of n,m, this bound turns out to be double exponential in the precise sense explained in the paper. As a corollary, we obtain a solution of the so called restricted infinitesimal Hilbert 16th problem, an explicit upper bound for the number of isolated zeros of Abelian integrals which is polynomially growing as the Hamiltonian tends to the degeneracy locus. This improves the exponential bounds recently established by A. Glutsyuk and Yu. Ilyashenko.Comment: Will appear in Annales de l'institut Fourier vol. 60 (2010

Bounded decomposition in the Brieskorn lattice and Pfaffian Picard--Fuchs systems for Abelian integrals

We suggest an algorithm for derivation of the Picard--Fuchs system of Pfaffian equations for Abelian integrals corresponding to semiquasihomogeneous Hamiltonians. It is based on an effective decomposition of polynomial forms in the Brieskorn lattice. The construction allows for an explicit upper bound on the norms of the polynomial coefficients, an important ingredient in studying zeros of these integrals.Comment: 17 pages in LaTeX2

Details of large-panel buildings seismic analysis

The normative requirements of different European countries, USA, CIS, Canada, etc. codes on ensuring of buildings and structures safety at earthquakes are analyzed. The methodology based on non-elastic response spectrum of buildings and allows taking into account non-linear behaviour of structure are proposed in elaboration of Eurocode 8 requirements. The report provides the calculation examples of non-linear displacements of framed and frameless concrete buildings with application of that methodology

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

Simulation of fuel injection through a nozzle having different position of the spray holes

In the article, a method of hydraulic calculation of working process of a diesel fuel feed system having a nozzle with different positions of its spray holes was investigated. A research of diesel engine injector nozzle design which had two groups of holes was carried out. Entering edges of the first group with the coefficient of flow μhl were located in the sack volume and entering edges of the second group (coefficient of flow μhu) – on the locking taper surface of the nozzle body. The coefficients of flow μhl and μhu differ considerably and depend on the valve needle position. This enables to distribute rationally the injection quantity by injection holes taking into account operating conditions of the diesel engine and hence – by the combustion chamber zones