
A Narrow Margin
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I am a math professor at the University of Puget Sound, a liberal arts college in Tacoma, Washington. I use this blog to talk some about my own work, posts that I find interesting on Mathematics Stack Exchange (a math question-and-answer site I used to regularly hang out on), and whatever else I happen to be thinking about mathematically. My background is in operations research, and my..
A Narrow Margin
2y ago
In this post I’m going to discuss two “proofs” by induction. That is, they’ll both be attempts at proving a statement, but they’ll both have flaws in their arguments. My hope is to alert people to potential errors when attempting to prove a statement by induction.
Claim 1: For all n, .
“Proof”: Let denote that claim that . Suppose that is true for some k; that is, suppose . Now, is the claim that . If is true, then , which proves that the truth of implies the truth of . By the principle of mathematical induction, then, is true for all n; that is, for all n.
Claim 2: For all , all people ..read more
A Narrow Margin
2y ago
A few months ago Mathematics Magazine published a paper of mine, “A Combinatorial View of Sums of Powers.” In it I give a combinatorial interpretation for the power sum , together with combinatorial proofs of two formulas for this power sum. (An earlier version of some of the results in this paper actually appeared in a blog post from several years ago.)
Recently, however, I received an email from Michael Maltenfort at Northwestern University, pointing out a couple of errors in the paper. I’m recording those errors here.
The most serious mistake Maltenfort pointed out to me is the paper’s clai ..read more
A Narrow Margin
2y ago
The Riemann zeta function can be expressed as , for complex numbers s whose real part is greater than 1. By analytic continuation, can be extended to all complex numbers except where .
The power sum is given by . (Sometimes M is given as the upper bound, but for this post it’s more convenient to use .)
Recently I learned of a really nice relationship between the zeta function and the power sum for . It’s due to Mináč [1], and I reproduce his proof here.
Theorem: For positive integers a, .
Now, of course M in is a positive integer, so what do we mean by integrating from 0 to 1? Well, it ..read more
A Narrow Margin
2y ago
One of the most important theoretical results in linear programming is that every LP has a corresponding dual program. Where, exactly, this dual comes from can often seem mysterious. Several years ago I answered a question on a couple of Stack Exchange sites giving an intuitive explanation for where the dual comes from. Those posts seem to have been appreciated, so I thought I would reproduce my answer here.
Suppose we have a primal problem as follows.
Now, suppose we want to use the primal’s constraints as a way to find an upper bound on the optimal value of the primal. If we multiply the fi ..read more