Motivating Mathematical Concepts through Problems

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 integral

Problem is that, in general we need to find the length of complicated sets

Outer 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" functions

And these can be used to define any function that is a point-wise limit of these simple functions.

Defines fractal dimension

Why it is a useful concept.

Computes the fractal length of a simple non-fractal set

Computes 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 odd

one of the legs is divisible by 4

the 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.