Motivating Mathematical Concepts through Problems

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.

Tries to motivate Lebesgue Integration by trying to find an improved limit theorem for Riemann Integration.In the process, find that this seems a lot like trying to find limit results when only dealing with rational numbers.End up with a example where we can "see" what the integral is, but Riemann does not have an answer.From the example see that by describing the unusual function by its range rather than its domain, we can compute the integralProblem is that, in general we need to find the length of complicated setsOuter measure is introduced to do just that.Unfortunately, this doesn't always work.Define a subset of all sets called measurable sets.These are the building blocks that can be used to create "simple" functionsAnd these can be used to define any function that is a point-wise limit of these simple functions.

Defines fractal dimensionWhy it is a useful concept.Computes the fractal length of a simple non-fractal setComputes the "length" of a self-similar fractal in its "natural" fractional dimension.

Inner Product can be thought of coming from the notion of Projection.If follows from doing a Mathematical "re-factoring", something that one usually only hears in the context of software engineering.

Rather than giving a formula for the determinant, one tries to derive it from first principles.The principles come about by looking at the volume of a parallelepiped.Uses the properties that such a formula would have to derive it.

Shows that the notion of what a derivative is -- NOT what one learns in first year calculus -- that it is the "best" linear approximation to a function.This idea allows us to extend the derivative beyond what one sees in first year calculus; meaning, extending the idea of derivative to higher dimensional functions.In fact, it can be used in an "infinite" dimensional setting to derive equations of motion in physics.

A new geometric proof of the result that for primitive Pythagorean triples:c is oddone of the legs is divisible by 4the other leg is odd.Conjectures that partition the set of primitive Pythagorean hypotenuses into two sets.Conjectures that try to examine the complexity of these two partitioning sets.

Derives recursive computational formulas for EMA and its standard deviation.Derives formulas for computing higher unbiased moments.

The general theme is to show that the differencing and summing oof sequences of numbers are inverse operations in some sense. This is easy to do and helps us understand the continuous analogs of these; namely, differentiation and integration.Goes further and examines difference equations and relates them to the continuous notion of differential equations. Once again, it uses the easier discrete problem to help understand the continuous one.Places supporting concepts to the appendices. These include:Functions, Formulas, Graphs, and Notation..Straight Line Functions.Infinite Sequences and Limits.Mathematical Induction.Taylor's Theorem.