Monthly Archives: October 2011

Why is Separation of Variables Valid?

One of the most confusing aspects of the calculus sequence concerns the Leibniz notation for the derivative: .  It sure looks like a fraction.  But we tell our students it’s not really a fraction.  But then sometimes we treat and … Continue reading

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Binomial Coefficient Identities via Complex Numbers

Binomial coefficient identities are normally proved by combinatorial arguments, by manipulation of other known binomial coefficient identities, by generating functions, or even sometimes by finding the right integral representation.  This post is going to look at a couple of binomial … Continue reading

Posted in binomial coefficients, combinatorics, complex numbers | Leave a comment

Tail Bounds of the Normal Distribution

The question of bounding the tails of the normal distribution has popped up a couple of times on math.SE lately.  This is an easy-to-prove but useful result, and so it’s worth talking about. The standard normal probability density function is … Continue reading

Posted in probability, statistics | 6 Comments

A satisfying weekend

This past weekend was a satisfying one for me, mathematically: I had two papers accepted. “On Solutions to a General Combinatorial Recurrence” was accepted by The Journal of Integer Sequences after being rejected by the first two journals I submitted … Continue reading

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Random Assignment Problems

The most fundamental problem in combinatorial optimization is probably the assignment problem.  Here we have, say, a set of n resources and a set of n tasks, and we want to assign each resource to exactly one task and exactly … Continue reading

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The Euler-Maclaurin Summation Formula

One of my favorite formulas in all of mathematics is the Euler-Maclaurin summation formula.  This post will talk a little about the formula and give several references for applications of it. Here’s one way to express the infinite series version … Continue reading

Posted in asymptotics, sequences and series | 3 Comments

Perfect numbers and Pythagorean triples

For my first entry, let’s talk about a problem I posed in Mathematics Magazine several years ago.  (Plus it gives me a chance to mention Fermat, which I really should.) Two of the more famous topics in elementary number theory … Continue reading

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