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PODCAST

Motivating Mathematical Concepts through Problems

Roland S McIntire

This podcast takes the approach of motivating mathematical concepts by way of a compelling problem. This is NOT the same as providing a definition and then showing that it solve a useful problem. The later has the feeling of someone doing mathematics to someone as opposed to giving one the sense that mathematics is a human endeavor; full of practical problems that need to be solved. Instead people often feel like that are being handed a collection of stone tablets handed down from the math gods.This

science mathematics

All Episodes

12:22

Why_Matrix_Multiplication

Motivating Mathematical Concepts through Problems ·

2024/12/08

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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11:14

Diversification Constraints in Convex Optimization

Motivating Mathematical Concepts through Problems ·

2024/11/12

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.

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11:32

A New Property of a Minimal Unbiased Estimator

Motivating Mathematical Concepts through Problems ·

2024/11/12

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.

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16:37

Develops a Systematic Formula for Computing the...

Motivating Mathematical Concepts through Problems ·

2024/11/12

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.

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11:29

Deriving the Heat and Fokker-Plank Equations from...

Motivating Mathematical Concepts through Problems ·

2024/11/12

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

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17:42

Modern Conditional Expectation in a Discrete Setting

Motivating Mathematical Concepts through Problems ·

2024/11/12

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.

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15:36

Why Lebesgue Integration

Motivating Mathematical Concepts through Problems ·

2024/11/12

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

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11:56

What is Fractal Dimension

Motivating Mathematical Concepts through Problems ·

2024/11/12

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.

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11:18

Where does the Inner Product Come From

Motivating Mathematical Concepts through Problems ·

2024/11/12

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.

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17:41

What is the Determinant

Motivating Mathematical Concepts through Problems ·

2024/11/12

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.

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13:43

What is Matrix Multiplication

Motivating Mathematical Concepts through Problems ·

2024/11/12

Uses a motivationing example to show:How a matrix comes up naturally.How to define matrix/vector multiplication.How to define matrix/matrix multiplication.How to define an identity matrix.

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18:35

What is a Derivative

Motivating Mathematical Concepts through Problems ·

2024/11/12

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.

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8:55

Simple Geometric Proof of a Classic result as well as...

Motivating Mathematical Concepts through Problems ·

2024/11/12

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.

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10:25

Computing EMA and Deriving Formulas for Unbiased...

Motivating Mathematical Concepts through Problems ·

2024/11/12

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

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17:58

Discrete and Continuous Calculus: The Essentials

Motivating Mathematical Concepts through Problems ·

2024/11/12

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

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15 results

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