Motivating Mathematical Concepts through Problems

Explains the formula for the determinant of a matrix.

Explains Fractal Dimensions

Explains the reason why Matrix Multiplication is the way it is.

We discuss the derivation of the Heat Equation and the Fokker-Plank Equation via a balance law.

Motivates the definition of Matrix/Vector and Matrix/Matrix multiplication by examining a systematic approach to a generalization of the simple equation a * x = b.

Shows a way to add constraints to an optimization problem which wishes to bound the sum of the 'k' largest components of a given collection whiles minimizing a given objective.

Examines the problem AX = b + epsilon looking for an unbiased estimator. when using "trace" norms.

Shows that the estimator is always the same.

The infinite collection of these results imply that the usual "best" estimator has the property that its Maximum Eigenvalue is the smallest amongst all unbiased linear estimators.

It is known how to compute the probability distribution of a variable that is transformed in a non-singular way.

When this is not the case, people find ad hoc ways of computing the probability distribution for singular transformations.

The paper shows a more systematic process that leads to formulas form complex singular transformations.

Shows how balance laws over arbitrary regions leads to a "local" version which is a Partial Differential Equation.

Looks at measure theoretic conditional expectation in a discrete setting using the familiar example of single 6 sided dice.

Shows how measure theory helps with the notion of conditional expectation when the conditional event has probability 0.