Probability Theory - I

The course objectives were to develop probabilistic reasoning and problem solving approaches, to provide a rigorous mathematical basis for probability theory, and to examine several important results in the theory of probability.

Topics included axiomatic probability, independence, random variables and their distributions, expectation, integration, variance and moments, probability inequalities, and modes of convergence of random variables. The course included introductory measure theory as needed. Students were expected to have previous study of both analysis and probability.

This course was the first half of a yearlong sequence. The second semester's course, 550.621 Probability Theory II, would cover classical limit theorems, characteristic functions, and conditional expectation.

Texts: P. Billingsley, Probability and Measure, 3rd ed., Wiley 1995
K. L. Chung, A Course in Probability Theory, 2nd ed., Academic Press,

Instructor: Prof John Wierman, Department of Mathematical Science, JHU  

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Foundations of Optimization

This course was the first in a two-semester sequence on the theory, algorithms, and applications of optimization. The first semester focused mainly on linear programming and the geometry of linear systems. Topics included the simplex method, revised simplex method, linear programming duality, theorems of the alternative, sensitivity analysis, and interior point methods for linear programming. In parallel with our theoretical development it was considered how to formulate mathematical programs for a variety of applications.

Text: D. Bertsimas & J. N. Tsitsiklis, Introduction to Linear Optimization, Athena Scientific

Instructor: Prof Jong-Shi Pang, Department of Mathematical Science, JHU

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Matrix Analysis and Linear Algebra

The objective of the course was to study interesting topics in matrix analysis and linear algebra that are not ordinarily included in a first course in linear algebra but are useful in the study of statistics, stochastic processes, optimization, mathematical economics, computer science, and numerical analysis. The course included a review of linear algebra, and the following topics were discussed: decomposition and factorization, positive definite matrices, norms and convergence, eigenvalue characterization and location, variational methods, positive and nonnegative matrices, generalized inverses.

Text: R.A. Horn & C.R. Johnson, Matrix Analysis, Cambridge University Press,
1985

Instructor: Prof James Allen Fill, Department of Mathematical Science, JHU

One-dimensional wave motion, Linear elasticity, Elastodynamics (Helmholtz decomposition of vectors, Integral representation of state, Boundary value problems, Boundary element method), Elastic waves in unbounded media, Plane harmonic waves (Basic concepts, SH/P/SV waves, Rayleigh, Stoneley Waves), Harmonic Waves in waveguides (Energy transport, group velocity, Love waves, Approximate theories for rods and plates), Forced motion in half-space (Integral transforms, Lamb's problem, Transient waves), Transient waves in layers and rods, Acoustic waves, Electromagnetic waves

Text: (1) J. A. Achenbach, Wave propagation in elastic solids, North-Holland, 1973 (2) K. F. Graff, Wave motion in elastic solids, Dover, 1991

Instructor: Prof Roger Ghanem, Department of Civil Engineering, JHU

Spatial statistics, Bayesian networks, Principal components analysis, Principal oscillation pattern analysis, Bayesian conjugate distributions, Dempster-Shafer non-probabilistic theory of evidence 

Text: No texts (based on lecture notes)

Instructor: Prof Takeru Igusa, Department of Civil Engineering, JHU