Narrow Results

In this note we discuss some formal properties of universal linearization operator, relate this to brackets of non-linear differential operators and discuss application to the calculus of auxiliary integrals, used in compatibility reductions of PDEs.

The goal of this paper is to present recent progress on several aspects of an inverse medium problem. A recursive linearization method for solving the nonlinear inverse problem is introduced. Convergence of the method is studied. We address issues on regularity and stability for the scattering map which maps the scatterer to the scattered field. Preliminary computational results are also presented.

A linearized system of hydrodynamics for ideal compressible fluids is considered. It is shown that in the case when the sound speed in the medium is equal to zero the characteristics change their multiplicity. For this case the asymptotic solution of the Cauchy problem with high frequency initial data is constructed. We prove that in this case the linearized system is unstable with respect to the high frequency perturbations.

Lie’s method of converting a nonlinear second order scalar ordinary differential equation (ODE) to linear form with point transformations has already been extended to higher order ODEs and systems of ODEs. For 2^nd order linearizable ODEs a unique equivalence class (with 8 infinitesimal symmetry generators) exists, whereas for the third order there are three classes with 4, 5 and 7 generators. For 2-d systems of second order ODEs there are five classes with 5, 6, 7, 8 and 15 dimensional Lie point symmetry algebras. A complex procedure (explained in §2) has been adopted to linearize a class of 2-d systems of second order ODEs, which is shown to possess 6, 7 and 15 dimensional algebras. Here we use the complex procedure for the symmetry group classification of those 2-d systems of 3^rd order ODEs that correspond to scalar linearizable complex third order ODEs. Five equivalence classes of such systems with Lie algebras of dimension 8, 9, 10, 11 and 13, are proved to exist.

The following sections are included:

• Lyapunov Exponents
• Lyapunov Spectrum for the Springy Pendulum
  • A Symmetry-Breaking Surprise
  • Three Methods for Computing λ₁ (t)
• Computing the Lyapunov Spectrum {λ(t)}
  • Gram-Schmidt Orthonormalization Algorithm
  • Pairing of the Local Hamiltonian Exponents
• Equilibrium and Nonequilibrium Lyapunov Spectra
• Lyapunov Spectra Under Steady Shear
• Equilibrium Baker Map
• Nonequilibrium Baker Map
• Summary of Lecture 9
• References

In this paper it is shown under mild assumptions that the local solvability of an infinite dimensional formal Cauchy problem is equivalent to a set of zero curvature relations. The role this type of Cauchy problems plays in integrable systems is illustrated at the hand of lower triangular Toda hierarchies.

This contribution is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli-Kohn-Nirenberg (CKN) and weighted logarithmic Hardy (WLH) inequalities. These results have been obtained in a series of papers [1–5] in collaboration with M. del Pino, S. Filippas, M. Loss, G. Tarantello and A. Tertikas and are presented from a new viewpoint.

In order to solve the problems in the measurement of the temperature in the city heating system, such as low measurement accuracy, narrow measurement range, low reliability and so on, this paper presents a measurement system based on PT1000. In this paper, the MATLAB is used to Linearized processing Data collected, which makes the temperature calculation simpler. The IFC algorithm is applied to filter out the large temperature error value, and the data processing is realized. Experiments show that the accuracy of the temperature measurement system can reach 0.2°C which is suitable for many kinds of high precision temperature measurement, and is stable and reliable, and has good application value.