Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications

We provide a theory to establish the existence of nonzero solutions of perturbed Hammerstein integral equations with deviated arguments, being our main ingredient the theory of fixed point index. Our approach is fairly general and covers a variety of cases. We apply our results to a periodic boundary value problem with reflections and to a thermostat problem. In the case of reflections we also discuss the optimality of some constants that occur in our theory. Some examples are presented to illustrate the theory.Comment: 3 figures, 23 page

Six results on Painleve VI

After recalling some of the geometry of the sixth Painleve equation, we will describe how the Okamoto symmetries arise naturally from symmetries of Schlesinger's equations and summarise the classification of the Platonic Painleve six solutions. A key observation is that Painleve VI governs the isomonodromic deformations of certain Fuchsian systems on rank \emph{three} bundles.Comment: 16 pages, written for Angers 2004 International Conference on Asymptotic Theories and Painleve Equations (updated, added Remark 7 giving simple formulae for isomonodromic families of rank three systems in terms of PVI solutions and as an example the full family of "Klein connections" are written down

Continuous Symmetries of Difference Equations

Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalized symmetries acting on solutions of difference equations, but leaving the lattice invariant. The other is to restrict to point symmetries, but to allow them to also transform the lattice.Comment: Review articl

A modeling framework and toolset for simulation and characterization of the cochlea within the auditory system

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Biological Engineering, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 50-53).Purpose: This research develops a modeling approach and an implementation toolset to simulate reticular lamina displacement in response to excitation at the ear canal and to characterize the cochlear system in the frequency domain. Scope The study develops existing physical models covering the outer, middle, and inner ears. The range of models are passive linear, active linear, and active nonlinear. These models are formulated as differential algebraic equations, and solved for impulse and tone excitations to determine responses. The solutions are mapped into tuning characteristics as a function of position within the cochlear partition. Objectives The central objective of simulation is to determine the characteristic frequency (CF)-space map, equivalent rectangular bandwidth (ERB), and sharpness of tuning (QERB) of the cochlea. The focus of this research is on getting accurate characteristics, with high time and space resolution. The study compares the simulation results to empirical measurements and to predictions of a model that utilizes filter theory and coherent reflection theory. Method We develop lumped and distributed physical models based on mechanical, acoustic, and electrical phenomena. The models are structured in the form of differential-algebraic equations (DAE), discretized in the space domain. This is in contrast to existing methods that solve a set of algebraic equations discretized in both space and time. The DAEs are solved using numerical differentiation formulas (NDFs) to compute the displacement of the reticular lamina and intermediate variables such as displacement of stapes in response to impulse and tone excitations at the ear canal. The inputs and outputs of the cochlear partition are utilized in determining its resonances and tuning characteristics. Transfer functions of the cochlear system with impulse excitation are calculated for passive and active linear models to determine resonance and tuning of the cochlear partition. Output characteristics are utilized for linear systems with tone excitation and for nonlinear models with stimuli of various amplitudes. Stability of the system is determined using generalized eigenvalues and the individual subsystems are stabilized based on their poles and zeros. Results The passive system has CF map ranging from 20 kHz at the base to 10 Hz at the apex of the cochlear partition, and has the strongest resonant frequency corresponding to that of the middle ear. The ERB is on the order of the CF, and the QERB is on the order of 1. The group delay decreases with CF which is in contradiction with findings from Stimulus Frequency Otoacoustic Emissions (SFOAE) experiments. The tuning characteristics of the middle ear correspond well to experimental observations. The stability of the system varies greatly with the choice of parameters, and number of space sections used for both the passive and active implementations. Implication Estimates of cochlear partition tuning based on solution of differential algebraic equations have better time and space resolution compared to existing methods that solve discretized set of equations. Domination of the resonance frequency of the reticular lamina by that of the middle ear rather than the resonant frequency of the cochlea at that position for the passive model is in contradiction with Bekesys measurements on human cadavers. Conclusion The methodology used in the thesis demonstrate the benefits of developing models and formulating the problem as differential-algebraic equations and solving it using the NDFs. Such an approach facilitates computation of responses and transfer functions simultaneously, studying stability of the system, and has good accuracy (controlled directly by error tolerance) and resolution.by Samiya Ashraf Alkhairy.M.Eng