Causality and Ordinary Differential Equations

The Steps of Perturbation Theory

Science Text Linearization (Linearization I)

Linearization at an Equilibrium. Peturbation Theory Approach (Linearization II)

Fundamental Theorems of Linear Systems

Zoom Zoom Zoom (Linearization III)

Distinct eigenvalues implies eigenbasis

Phase plane for linear 2 by 2 systems

Ellipse axes, eccentricity, and direction of rotation

Spectral Theory with Repeated Eigenvectors

Asymptotic Stability and Linearization

Asymptotic Stability by Linearization : alternate approach

Undamped Pendulum Energy Landscape

Exercises in 1-d Mechanical Systems

Linearization Unstable and Nonlinear Equilibrium Stable

Derivatives of the Poincare Map

Conjugation of Logistic and Tent Maps

**University of Pisa, ****March-April 2014 **

K.O. Friedrichs, Symmetric Hyperbolic Linear Differential Equations, CPAM 1954

K.O. Friedrichs, Symmetric Positive Linear Differential Equations, CPAM 1958

K.O. Friedrichs and P.D. Lax, Systems of Conservation Equations with a Convex Extension, PNAS 1971.

**Universite de
Paris VII, **

**28 January - 4 February, 2014 **

Proof of Continuity of Pseudodifferential Operators

Sections II.7 and II.8 of Taylor PDE vol. II

Taylor reflection article with diagonalization argument

Mike Taylor's Commutator Lemma for diagonalization

Nicolas Laillet's course notes, 28 January

Fall 2013 Homework Assignments

Fall 2011 Homework Assignments

Image of Circles by 2x2 Matrices

Image of Spheres by Linear Transformations

Outlines of the derivation of everything from Cauchy's Theorem

Laurent Expansion Yields Partial Fractions

Laurent Expansion Yields Fourier Series

Partial Fractions and the Inverse Laplace Transform

More Fourier Analysis from Complex Analysis

The Dirichlet Problem in a Half Space and Corners

The Neumann Problem, Insulator Boundary Conditions

The Dirichlet Problem in the Disk

**
Grand Sasso Science Institute, L'Aquila, Italia,
May 2018
**

Meet Maxwell's Equations Outline

Infinitesimal Mean Value Property

Courant-Friedrichs-Lewy, On the partial difference equations of mathematical physics

Fourier Analysis from Complex Analysis

Introduction of geometric optics from Hyperbolic PDE book

Spring energy from Noether's Theorem

A second example of the approximation of geometric optics

Course information

Daily syllabus

m- files and computer related

Wronskian Theorem statement

Integrating factor review

Matlab tutorials

The Steps of Perturbation Theory

K. Miller's Linear Algebra Lecture Notes

Chapter 9. The first classes.

Chapter 10. April 12.

Chapter 6. Sections 6.4 and 6.6 are cited in Chapter 9.

Chapter 11. April 19,26.

Fall 1994 Homework Assignments

Help with the proof of Theorem 6.2-3.