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Monday, June 26, 2006

More Mathematics and Theoretical Computer Science Video Lectures

Here are the additional video lectures in mathematics (and theoretical computer science) which were in my bookmarks. I still have many links left to check so visit my blog in a few days for updates.

Lectures are ordered by their complexity.

Algebra Review
  • Video Lectures: Math 160 (University of Idaho)
Course covers: factoring, interval notation, definition of function, functions, piece-wise defined functions, function composition, quadratic functions, solving quadratic functions. Slope of the line, equation of the line, parallel and perpendicular lines. Law of exponents, properties of logarithms. Applications to exponential function, exponential growth and decay. Solving systems of equations by substitution and elimination.

Intermediate Algebra
The primary purpose of Intermediate Algebra is to improve your skills and competency in algebra so that you will be successful in calculus, the other math courses required for your major, and in the courses that use mathematics. Another goal is to help you develop your mathematical learning skills so that you will be more confident in future mathematical courses.

Course covers: the real numbers, linear equations, linear inequalities and absolute value, linear equations and inequalities in two variables, systems of linear equations, exponents, polynomials and polynomial functions, factoring, rational expressions, roots and radicals, quadratic equations and inequalities.

Elementary Statistics
Elementary Statistics is an introduction to data analysis course that makes use of graphical and numerical techniques to study patterns and departures from patterns. The student studies randomness with emphasis on understanding variation, collects information in the face of uncertainty, checks distributional assumptions, tests hypotheses, uses probability as a tool for anticipating what the distribution of data may look like under a set of assumptions, and uses appropriate statistical models to draw conclusions from data.

The course introduces the student to applications in engineering, business, economics, medicine, education, the sciences, and other related fields. The use of technology (computers or graphing calculators) will be required in certain applications.

Course covers: Sampling and data. Statistical graphs, quartiles and percentiles, mean, median, mode, variance and standard deviation. Basic probability, independent and dependent events, addition and multiplication rules. Discrete random variables, discrete probability distribution functions, expected value, binomial probability distribution function. Continuous random variables, continuous probability distribution functions, uniform probability distribution, exponential probability distribution. The normal probability distribution function, standard normal probability density function. Central limit theorem for averages and sums. Confidence intervals. Hypothesis testing. The Chi-Square distribution function. Linear regression and correlation.

Applied Probability (5 lectures)
Focuses on modeling, quantification, and analysis of uncertainty by teaching random variables, simple random processes and their probability distributions, Markov processes, limit theorems, elements of statistical inference, and decision making under uncertainty. This course extends the discrete probability learned in the discrete math class. It focuses on actual applications, and places little emphasis on proofs. A problem set based on identifying tumors using MRI (Magnetic Resonance Imaging) is done using Matlab.

Finite Mathematics with Applications
  • Video Lectures: Math 1313 (University of Houston)
Course covers slopes, equations and graphing of lines, linear depreciation, cost, revenue and profit functions, intersection of lines, break-even analysis, the method of least squares, graphing linear inequalities, graphing systems of linear inequalities, linear programming problems, graphical solution of linear problems, simple interest, future value, present value, and effect rate, annuities, amortization and sinking funds. Set notation and terminology, set operations, Venn diagrams, number of elements in a set, the multiplication rule, permutations and combinations. Experiments, events and sample spaces, definition of probability, rules of probability, use of counting technique, conditional probability, independent events, Bayes' theorem, distributions of random variables, expected value, odds, variance and standard deviation, Chebyshev's inequality, the binomial distribution, the normal distribution, applications of the normal distribution.

Trigonometry for Calculus
The goal of this course is to prepare you for the trigonometry that you will encounter in calculus. During this one credit course in trigonometry, you will learn how to evaluate trigonometric functions, sketch the graphs of the sine, cosine and tangent functions, study the inverse trigonometric functions and much more.

Course covers the Cartesian coordinate system, functions, angle and radian measure, special right triangles, the unit circle, the trigonometric ratios, graphs of trig. ratios, periodic functions, fundamental trigonometric identities and inverse trigonometric functions.

Introduction to Mathematical Computation

Throughout the course we will illustrate application of software in typical undergraduate mathematical subjects such as calculus, probability, linear algebra, and number theory. Further, we will move to structural programming. We conclude the course by illustrating elements of contemporary platform independent language, java. No programming experience required

Course covers: Basic commands in Mathematica, Mathematica in Calculus, Mathematica in Probability, Mathematica and Linear Algebra, Mathematica and Number Theory, Mathematica and structural programming, Introduction to Java.

Note: Links to lectures 6 - 17 are missing. You can access them by changing the last number of the link to the first 5 lectures. Example: to access lecture 12 use address, etc.

Pre-Calculus and Introduction to Analytic Geometry

The primary purpose of Pre-Calculus and Analytic Geometry is to improve your skills and competency in algebra so that you will be successful in calculus, the other math courses required for your major, and in the courses that use mathematics. Another goal is to help you develop your mathematical learning skills so that you will be more confident in future mathematical courses.

Course covers: equations and identities, graphs, functions and their graphs, polynomial and rational functions, exponential and logarithmic functions, analytic geometry.

First Year Calculus (Calculus I)

The central object of the study in calculus is the concept of a function. Functions are used to describe the real world around us. Calculus introduces two fundamental concepts which enable us to describe and investigate functions. These are: the derivative and the integral. The derivative describes the behavior of a function at a particular time. The integral carries information about the history of a function.

Course covers: limits, limit laws, continuity, limits involving infinity, rates of change, derivatives, differentiation rules, product and quotient rules, rates of change in science, derivatives of trigonometric functions, the chain rule, implicit differentiation, logarithmic differentiation, maxima and minima, mean value theorem, L'Hospital's rule, optimization problems, areas and distances, definite integral, fundamental theorem of calculus.

Business Calculus
  • Video Lectures: Math 1314 (University of Houston)

Course covers limits, one-sided limits and continuity, the derivative, basic rules of differentiation, the product and quotient rules, the chain rule, higher order derivatives, basic applications of derivative, marginal functions in economics, applications of the first derivative, applications of the second derivative, curve sketching, absolute extrema, optimization, applications with exponential functions, antiderivatives, integration by substitution, area under the curve - Riemann Sums, the fundamental theorem of calculus, evaluation of definite integral, area between two curves, functions of several variables, partial derivatives, relative extrema.

Mathematical Writing (by Donald E. Knuth!)
Issues of technical writing and the effective presentation of mathematics and computer science. Preparation of theses, papers, books, and "literate" computer programs.

"I also gave a class called Mathematical Writing, just for one quarter," says Knuth. "The lectures are still of special interest because they feature quite a few important guest lecturers." This collection contains thirty-one tapes.

Mathematics and Computer Science Problem Seminar (by Donald E. Knuth!)
During the course students with Professor Knuth solve 5 problems which have not been solved.
According to D. E. Knuth course is given only once in two years because it takes him two years to think of good enough problems. The goal of the course it to understand problem solving in general and not just to solve those 5 problems and to get into as many of the different areas of computer science research as possible.

"This was an experimental project where we'd have three or four cameras in a basement studio and we would film classes of about an hour," says Knuth. "We got a bunch of our brightest students and gave them extremely difficult problems. You could literally see the Aha taking place. People can watch the problem-solving process as it occurred." Over 25 hours of these sessions are available for viewing.

Dynamical Systems and Chaos
The course will provide quick introduction to Dynamical Systems, Ergodic Theory and Chaos. We will start with examples of dynamical systems, with basic notions such as orbits, periodic points, phase portraits, attraction and repulsion, calculus of fixed points, invariant measures, Bernoulli shifts and ergodic theorems of various types.
Then we will study bifurcations on the example of dynamics of quadratic maps. The quadratic family will be used to demonstrate the transition to chaos and the main features of chaotic behaviour. We will touch Sarkovsii's Theorem and Newton's Method.

Elements of Symbolic Dynamics and subshifts of finite type will be considered. Then we will move to fractals and discuss fractal dimension and related topics. After that we will introduce Holomorphic Dynamics and the main objects such as Julia sets and the Mandelbrot set. Time permitting, we will consider some rational maps in dimension two and higher. Henon map will be considered, as well as some maps arising in the theory of fractal groups, and the Smale horse shoe map. We will consider also spectra and spectral measures related to such groups and to fractal sets like Sierpinski gasket or Cantor set.

Computer Musings Lecture Series (by Donald E. Knuth)

“These lectures I'’ve given have been inspired and shaped by the questions and responses of the audiences to whom I spoke, and I want to keep them alive,prof. D.E.Knuth explains. We'’ve got these tapes and the world is going digital; Stanford Centre for Professional Development has the talent and expertise to convert them. I feel that archiving is important. I'’ve learned from archived lectures and classes myself, so I think others can learn from these.

A sampling of musings includes:
  • Dancing Links
  • Fast Input/Output with Many Disks, Using a Magic Trick
  • MMIX: A RISC Computer for the New Millennium
  • The Joy of Asymptotics
  • Bubblesort at random (one-dimensional particle physics)
  • Trees, Forests, and Polyominoes
  • Finding all spanning trees

"Other" Donald E. Knuth Lectures

Also available are two five-session short courses about TeX (1981); twelve lectures about the implementation of TeX (1982); video recordings of eight history sessions about Computer Science at Stanford, taped in 1987 and featuring many alumni of our department; and some reminiscences by Professors Feigenbaum, Floyd, Golub, Herriot, Knuth, McCarthy, Miller, and Wiederhold about the founding of Stanford's Computer Science Department, The Living Legends (1997).

Questions from audience and students are important to the learning process, according to Knuth. Sometimes the expression of a more mature idea isn't the most interesting or effective way to learn you may learn more from how a professor reacts to an idea or a question. He pauses, and then adds, People might learn a lot from watching me fumble around to answer a question.

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