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Motivating Mathematical Concepts through Problems

Motivating Mathematical Concepts through Problems

by Roland S McIntire

Season 1

Diversification Constraints in Convex Optimization

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.

A New Property of a Minimal Unbiased Estimator

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.

Develops a Systematic Formula for Computing the Probability of Variables Transformed in a Singular Way

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.

Deriving the Heat and Fokker-Plank Equations from Balance Laws

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

Modern Conditional Expectation in a Discrete Setting

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.

Why Lebesgue Integration

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.

What is Fractal Dimension

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.

Where does the Inner Product Come From

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.

What is the Determinant

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.

Why_Matrix_Multiplication

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.

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