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Saturday, October 31, 2009

Mathematics Video Lectures

Hi all!

This month I have a nice collection of math video lectures.

Mathematics topics include: calculus, vector calculus, tensors, the most important concepts of mathematics, basic mathematics, numerical methods, p=np problem, randomness, fractals and splines, lectures from advanced institute for study and a video on indian super 30 mathematics school.

Mathematics Illuminated

Course description:
Mathematics Illuminated is a 13-part, integrated-media resource created for adult learners and high school teachers. The series covers the broad scope of human knowledge through the study of mathematics and its relevance in the world today. It reaches beyond formulas and computations to explore the math of patterns, symmetry, relationships, multiple dimensions, and more, all the while uncovering the secrets and hidden delights of the ever-evolving world of mathematics.

Course topics:
The Primes. Combinatorics Counts. How Big is Infinity? Topology's Twists and Turns. Other Dimensions. The Beauty of Symmetry. Making Sense of Randomness. Geometries Beyond Euclid. Game Theory. Harmonious Math. Connecting with Networks. In Sync. The Concepts of Chaos.

Calculus (Math 1231, University of New South Wales)

Course topics:
Lectures presented by Dr. Chris Tisdell. Introduction to calculus. How to sketch functions of two variables. Partial derivatives. Normal vector. Tangent plane to a surface. The Chain rule. What is a Differential Equation? Separable differential equations. Linear differential equations. Exact differential equations. Series - The Comparison Test. Series: Limit Comparison Test.

Vector Calculus (Math 2111, University of New South Wales)

Course topics:
Lectures presented by Dr. Chris Tisdell. Several variable calculus: Applications of double integrals. Path integrals - how to integrate over curves. Vector calculus - what is a vector field? What is the divergence? What is the Curl? What is a line integral? Applications of line integrals. Fundamental theorem of line integrals. What is the Green's theorem? More on Green's theorem. Parametrised surfaces. What is a surface integral? More on surface integrals. Surface integrals. Vector fields. How to solve PDEs via separation of variables + Fourier series.

What is the Taylor Series?

Lecture description:
In this lecture the Taylor's Theorem is discussed and a proof is given using only elementary calculus methods based on the mean value theorem for integrals. The approach is simple and straightforward and meant to be accessible.

Tensors and Differential Geometry

Lecture description:
What are Contravariant and Covariant Components of a Vector? The contravariant and covariant componets of a vector is central concept of differential geometry. Lecture gives a brief look at applications of differential geometry and the concept of contravariant and covariant components of a vector. It is shown that in the simple case of an oblique coordinate system in two dimensional Euclidean space the formula for the length requires covariant and contravariant components of a vector. The metric tensor is introduced and its components found using coordinate transformation matrices. Mysterious upper and lower vector indices are explained. Transformations and the Metric Tensor. Vector Fields and Tensors Differential Geometry.

The Language of Mathematics

Course topics:
The Real Number Set. Zero and Infinity. Adding and Subtracting. Dealing with Negative Numbers. Multiplying and Dividing. Prime Numbers. Finding Prime Factors. Reducing Fractions. Multiply and Divide Fractions. Adding and Subtracting Fraction. Introduction to Trigonometry. Right Triangles. Solving Right Triangles. Congruent Similar Triangles. Cartesian Coordinate System. Slope and Midpoint of a Line. Distance of a Line. Introductions to Proofs. Proofs Involving a Line. Proofs Involving Triangles. Parallel Lines. Coordinate Geometry. Exponents and Radicals. Rules for Exponents to Exponents. Reducing Radicals. Multiplying Radicals. Simplifying Radicals. Simplifying Large Radicals. Solving Equations. Linear Equations. Quadratic Equations. Graphing Equations.

Numerical Methods

Introduction to Numerical Methods

Anecdotes on Interpolation

Sources of Error

Binary Representation

Floating Point Representation

Interpolation Background

Direct Method of Interpolation

Spline Interpolation

Primer on ODEs

Finite Difference Method for ODEs

Euler's Method of Solving ODEs

Runge Kutta 2nd order method

Runge Kutta 4th Order Method

Higher Order Ordinary Differential Equations

Integrating Discrete Functions

Trapezoidal Rule

Simpson's 1/3 rule of integration

Measuring Errors

Linear Regression

Primer on Matrix Algebra

Taylor Series

Newton Raphson Method

Secant Method

Bisection Method

Gaussian Elimination

Nonlinear Regression

LU Decomposition Method

Background for Simultaneous Linear Equations

Numerical Differentiation of Continuous Functions

Differentiation of Discrete Functions

The "P vs. NP" Problem.

Lecture description:
The "P vs. NP" problem is a central outstanding problem of computer science and mathematics. In this talk, Professor Wigderson attempts to describe its technical, scientific, and philosophical content, its status, and the implications of its two possible resolutions. Efficient Computation, Internet Security, and the Limits of Human Knowledge.

Search for Randomness

Lecture description:
Although the concept of randomness is ubiquitous, it turns out to be difficult to generate a truly random sequence of events. The need for "pseudorandomness" in various parts of modern science, ranging from numerical simulation to cryptography, has challenged our limited understanding of this issue and our mathematical resources. In this talk, Professor Jean Bourgain explores some of the problems of pseudorandomness and tools to address them.

Lectures from Advanced Institute for Study 2009/2010

Lectures include:
Algorithms for Unique Games. Complexity of Circuit Satisfiability. Detectability Lemma and Quantum Gap Amplification. Span Programs and Quantum Query Algorithms. The Completeness of the Permanent. Twice-Ramanujan Sparsifiers. Affine Dispersers from Subspace Polynomials.

The Marriage of Fractals and Splines

Video description:
Fractals and splines have very different geometric features. Fractals can be continuous everywhere, yet differentiable nowhere. Fractals are often self-similar curves with fractional dimension. And fractals are also attractors, fixed points of iterated function systems. In contrast, splines are piecewise polynomial curves, so well behaved that they are often used for large scale industrial design and manufacture. Splines are essentially polynomials, so splines are one dimensional curves that are differentiable everywhere. Splines have control points -- polynomial coefficients -- that can be used to control the shape of the spline in an intuitive fashion. Moreover, unlike fractals, splines have parametrizations.

Nevertheless, the goal of this talk is to marry fractals and splines: to demonstrate that fractals and splines share many geometric properties and algorithms. We shall show that just like splines, fractals can be parametrized and fractals have control points that allow us to adjust the shape of the fractal in an intuitive manner. Moreover, just like fractals, splines are attractors, fixed points of iterated function systems. We shall show how to apply fractal algorithms to generate splines and spline algorithms to generate fractals. We conclude that fractals and splines are not really that different after all.

Super 30 Indian Mathematics Competition

Video description:
Each year children from around India try to get in the most prestigious Indian Institute of Technology. This video shows a story about a school where the 30 best students are trained to get in the IIT.

Have fun with these!

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