Ralph Greco’s Logic and Public Speaking Notes

For the New Year we start off with a document: the eighth grade logic and public speaking notes of Ralph Greco, Palm Beach Day School/Academy’s long-time English teacher. How long? This was the introduction which Walter Butler, then PBDS’s headmaster, gave him and his wife when they started in 1968:

Mr. Greco, a native of Pennsylvania, will teach sixth and eighth grade English. A graduate of the University of Maryland in 1955, Mr. Greco taught English four years in the Pittsburgh area prior to teaching English in the Palm Beach County schools for the past seven years. Mrs. Greco, also a native of Pennsylvania, will teach in one of the Second Grade sections. She received her B.S. degree from Pennsylvania State, majoring in Elementary Education and Home Economics.

He’s still teaching there, something which was underscored by the Washington Post’s Catherine Rampell’s piece just before Christmas last year. And he’s become something of the school’s “historian,” as this page attests.

As for me, let’s start with the public speaking part: it was immensely useful for that, although it was some time before I really got to put it to use. The logic was even more complicated: I was certainly receptive to what he was saying, but in an environment of hidden agendas and a lot of nice-sounding but self-serving rhetoric, it was hard to implement. It wasn’t until undergraduate school at Texas A&M that I took a full-blown course in logic, complete with syllogistic logic, which was missing from this presentation. (I also spent much time with Thomas Aquinas in those days, which is a logic course in and of itself.)

But even that had its moments:

One of the things that the Mechanical Engineering department required its majors to take was Logic, which was offered by the Philosophy Department.  Most of the engineers did pretty well in this course, which was doubtless a source of secret frustration to liberal arts’ professors.

One day I went up to pick up a test from the professor.  The professor looked at the grade, noted that I had nearly aced it, looked at me, and exclaimed, “You’re not as dumb as you look!”

I’m not sure that Mr. Greco would agree with that sentiment, but hey, that’s what happens when you go “beyond the gates.”

Teaching Around the Christmas Tree

Most of you who follow this blog know that this site, in addition to all of my family history, is also the open online host for my Fluid Mechanics Laboratory material, which includes pages for the teaching videos, which I’ve adopted in the wake of COVID. Recently I had to update one of those videos and did it with the following background:

The Christmas tree to my right has a little bit of family history associated with it: it was Chet and Myrtle’s, and probably dates back to the 1950’s. It’s been passed down and now graces these videos. For a better view of it, you can watch the video itself.

In spite of the fact that this year has been a general ordeal, it has been a good one for all of my sites and the YouTube channel. Lord willing, there will be more to come.

Let me take the opportunity to wish all my visitors a very Merry Christmas and a Happy and Prosperous New Year. Or, as my great grand-uncle James N. Warrington put it:

Using LINEST for Least Squares Regression With More Than One Independent Variable

The use of least squares regression and curve fitting is well established in the applied sciences. It is discussed in detail in the monograph Least Squares Analysis and Curve Fitting. Most analyses of this type, however, are done with only one independent variable (the classic linear fit is a good example of this.)

For some problems it is necessary to consider two or more independent variables (a recent example is here.) A way to perform regression on such data is to use the LINEST function, which can be used for linear/planar types of correlations. It can be found in most of the current spreadsheet packages. It is tricky to use; about the only way to illustrate its use is through a video, and one from Dr. Todd Grande is featured here.

Fluid Mechanics Laboratory Video: Flow Meters, Gate Valve and Pipe Losses (Moody Chart)

Our main fluid mechanics laboratory page is here.  Other resources relating to this laboratory is here:

Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra

Theorem 1 (Eigenvector-eigenvalue identity) Let {A} be an {n \times n} Hermitian matrix, with eigenvalues {\lambda_1(A),\dots,\lambda_n(A)}. Let {v_i} be a unit eigenvector corresponding to the eigenvalue {\lambda_i(A)}, and let {v_{i,j}} be the {j^{th}} component of {v_i}. Then

\displaystyle |v_{i,j}|^2 \prod_{k=1; k \neq i}^n (\lambda_i(A) - \lambda_k(A)) = \prod_{k=1}^{n-1} (\lambda_i(A) - \lambda_k(M_j))

where {M_j} is the {n-1 \times n-1} Hermitian matrix formed by deleting the {j^{th}} row and column from {A}.

When we posted the first version of this paper, we were unaware of previous appearances of this identity in the literature; a related identity had been used by Erdos-Schlein-Yau and by myself and Van Vu for applications to random matrix theory, but to our knowledge this specific identity appeared to be new. Even two months after our preprint first appeared on the arXiv in August, we had only learned of one other place in the literature where the identity showed up (by Forrester and Zhang, who also cite an earlier paper of Baryshnikov).

Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv a completely rewritten version of our previous paper, now titled “Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra“. This paper is now a survey of the various literature surrounding the following basic identity in linear algebra, which we propose to call the eigenvector-eigenvalue identity:

Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra