These lectures are from a graduate level course taught by Dr. Lawrence Sirovich at the Courant Institute for Mathematical Sciences in Fall 2008.

**Course Description**

The purpose of this course is to present an extensive, integrated treatment of applied mathematics useful to science students who wish to include mathematical modeling and simulation in their future research. In this course many years of graduate applied mathematics are compressed into a traditional one semester course. To accomplish this, rigor is replaced by convincing arguments, intuitive concepts and the development of a geometrical perspective. Within this looser framework of proof the course is self-contained. Computation, through the use of Matlab, plays a central role in the teaching and learning process. Although many course illustrations come from Biology, in its wider definition, students in other research areas should also be able to profit from the novel treatment presented in this course. Amongst others the topics that will be covered are: Linear Algebra with Applications to Data Analysis and Modeling; Complex Analysis; Fourier Methods; Probabilistic & Stochastic Modeling; Dynamical Systems and Applications to Chemical Kinetics with Applications to Biochemical Systems; Dimension Reduction & Low Dimensional Systems. All topics will be considered within a Matlab framework.

Calculus (Part I)

Differentiation; Integration; Finite Differences; Euler and Trapezoidal Sums; Taylor Expansion; Binomial Expansion; General Function Definition

Calculus (Part II)

Second Law and Differential Equations; Inverse Functions; Histograms; Lesbeque Integral; Simple Monte Carlo Calculations

Higher Dimensions

Surfaces; Volumes; Partial Derivatives; Extrema; Green's Theorems; Modeling of Diffusion Phenomena

Linear Algebra (I)

Matrices; Determinants; Least Squares: Moore-Penrose Inverse; Eigentheory; Functions of Matrices; Linear Systems Analysis

Analysis of Data: Empirical Eigenfunction Methods

Lagrange Multipliers; Singular Value Decomposition; Principal Component Analysis; Optimal Coordinates; Image Analysis and the Rogues Gallery Problem

Modeling Input/Output Data: Partial Least Squares (PLS)

Input/Output Relations; Partial Least Squares; Empirical Modeling; Applications

Complex Analysis

Complex Plane; Roots of Unity; Power Series; Cauchy-Riemann Equations; Cauchy Theorem; Complex Linear Algebra; Hermitian Matrices

Fourier Methods

Fourier Series; Function Space; Aperiodic Functions; Gibbs' Phenomenon; Discrete Fourier Series; Sampling Theory; Aliasing; Nyquist Criterion

Population Dynamics

Markovian Systems; Mass Action Equations; Lotka-Volterra Dynamics

Dynamical Systems

One and Two Dimensional Phase Spaces; Bifurcations; Stability of Equilibria

Probabilistic and Stochastic Modeling (Part I)

Coin Tossing: Methods of Most Probable Distribution; Brownian Motion; Chapman-Kolmogorov Equation;
Einstein and Langevin Formulations

Probabilistic and Stochastic Modeling (Part II)

Random Arrivals, Waiting Times; Poisson Process; Poisson Distributions; Gamma Distribution; Gaussian
Distribution; Central Limit Theorem; Applications: Quantal Phenomena in Physiology; Population Models
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