Friday, October 14, 2022

A new matrix multiplication record

In Chapter 5, we mentioned Strassen’s algorithm for matrix multiplication. Not many improvements have been made since 1969 when Strassen discovered how to multiply a \pair of 4 by 4 matrices with 49 multiplications - a reduction of 15 multiplications from the  basic definition. A further reduction by one multiplication was announced by two Austrian researchers at Johannes Kepler University Linz in October 2022. 

Monday, September 26, 2022

Does learning about quantifiers help students understand limits?

 A recent thread on the MAA member web site discussed how limits should be taught in Calculus I/II. One comment was that students who take a discrete math course, where quantifiers are discussed, might better understand the definition of a limit.  What follows is a possible example that could be added to our section on quantifiers. Background:  I taught a calculus workshop for mostly middle school teachers several years ago and I recall the most spirited discussion being around the idea that $0.999…  = 1$.

Example: What does it mean that 0.999… = 1?  The ellipsis (…) implies that there are an infinite number of 9’s on the left of the equals sign.  After many years of struggling with what this means, mathematicians have come up with a universally accepted interpretation involving quantifiers.  It is that

$$(\forall \epsilon)_{\mathbb{R}^+} ((\exists N)_{\mathbb{P}})(n\geq N \Rightarrow  |1- 0.\underbrace{99..9}_{n\,9’s}| \lt \epsilon))$$

In calculus, the symbol $\epsilon$ is usually reserved for small positive real numbers. Let’s pick a value for $\epsilon$ and peel the universal quantifier off the statement above.  Let’s try  $\epsilon$ equal to $\frac{1}{2^{10}}=\frac{1}{1024}$.  In addition we note that $0.\underbrace{99..9}_{n\,9’s}=1-\frac{1}{10^n}$.  With our choice of $\epsilon$ we get 

$$ (\exists N)_{\mathbb{P}}(n\geq N \Rightarrow  |1- 0.\underbrace{99..9}_{n\,9’s}| \lt \frac{1}{1024}) $$

$$(\exists N)_{\mathbb{P}}(n\geq N \Rightarrow \frac{1}{10^n} \lt \frac{1}{1024}) $$

This last statement is true - one value of  $N$ that would work is $11$. You just have to convince yourself that any positive value of $\epsilon$, no matter how small, will produce a true statement.  If you see that, you’ve convinced yourself that $0.999…  = 1$!

Thursday, August 5, 2021

UMass Lowell Faculty Word Search

 We have another month before the semester gets started, so here is a word search puzzle you can do while you relax in your back yard or the beach. Most of the names are the last names of UMass Lowell Math Faculty and Staff. A few individuals had names too long for the puzzle app to handle, so their first names are used instead. Also, a few of the names are spelled right to left or bottom to top.

Monday, June 14, 2021

Version 3.8 of Applied Discrete Structures is now available!

All forms of Applied Discrete Structures - Version 3.8 are available.  As always, the web and pdf versions are free.  The print copies increased in price due to increases in paper costs passed on by Lulu.  The full version is now $41.00.  I'm not so concerned since very few students buy it.

Cover of full version

I didn't keep track of the changes in the new version, but here is a list some of the changes/additions that I recall:

  • Exercises in Section 7.1 were rearranged and augmented
  • A new exercise in Section 8.2 reviewing quantifiers in the context of sequences.
  • A few operations tables were added in Section 11.4 - Modular Arithmetic, including one multiplicative group.
  • An exercise similar to one of our final exam questions this spring was added to section 13.1 on posets and lattices.
  • A new example was added to the section on algebraic coding (15.5) with a follow-up exercise.
If course there were a few minor typos, but not nearly as many as in previous years!  

Wednesday, April 28, 2021

UML OERscars - 2021

 The UMass Lowell chapter of MASSPIRG is active in promoting the use of OER materials for students.  I fully agree that it's worth a try for any course.   I'm happy to have been recognized by them at this year's inaugural  OERscars awards ceremony for Applied Discrete Structures. The good news for UML students is that there were faculty from every college recognized for developing and/or using OER materials for their courses. Congratulations to all OERscar winners! 

Applied Discrete Structures has be