De Morgan's Laws in Action

Augustus De Morgan, originally uploaded by Mike Lynch.

Logic: It's a beautiful thing. It's not everyday, however, that I get to implement it so explicitly as I did today.

One of the hallmarks of mathematical logic are some rules set out by Augustus De Morgan called, appropriately, De Morgan's Laws. They go something like this: Take two (or more) statements A and B. These can be things like "The sky is blue" and "We are on Earth." These statements have complements, namely, "The sky is not blue" and "We are not on Earth."

De Morgan gives an equivalence between the relation of these statements' complements and the complement of the relation of these statements. In (better) English, we could say that the complement of A and B is the complement of A or the complement of B. So, in our example, De Morgan tells us that the complement of "The sky is blue AND we are on Earth" is the same thing as saying "The sky is not blue OR we are not on Earth." Confused? Try out the formal statements for a bit of clarification:

where we read "c" as complement,

and

Why would this ever arise in practice, you might ask? In fact, just today I wanted to add a stopping condition to a WHILE loop I had in a program. A WHILE loop runs (loops) for as long as a given statement is true. My loop ran for as long as the variable called FLAG was true. Essentially, I wanted to cheat a little and add a different stopping criterion that would bounce out of the loop if a different variable hadn't been changing for a while.

So in my head, the loop should run while FLAG was true, and it should stop when COUNTER was bigger than 20 and W(COUNTER)==W(COUNTER-20). But, just my luck, you can't really tell a WHILE loop when not to run, you have to tell it when it has to run. But guess, what? These conditions are complements! Enter De Morgan.

My situation was that I want to stop the loop when

is true. Which means that I want to NOT stop (i.e. run) when

is true.

Bonus! We get to use both flavors of De Morgan. First apply the first equation from up top, then apply the second to the B parentheses. After all is said and complemented, I want to run my WHILE loop while FLAG is true AND (COUNTER is less than 20 OR W(COUNTER)~=W(COUNTER-20)). Cool huh?

Stay Logical.

How to Peek Around a Corner - Part 1

around the corner,, originally uploaded by ~Faiz.

Occasionally in my reading, I come across a particularly well-posed question that I feel is not only accessible to "lay" readers but also interesting to them. The problem of how to best look (or "peek") around a corner is just one of these questions.

As a warning, I will pose the question in this post and then solicit ideas and answers from the readers. Then next week I'll post a solution and hopefully an example.



First, the setting: You are Bruce Willis in Die Hard. You are pursuing and being pursued by bloodthirsty terrorists in a strange office building. You find yourself walking in a smoky room with your hand against a wall in order to not get lost. All of a sudden, you notice that your wall comes to an end exactly one meter in front of you. You pause and analyze how to best approach this corner in the wall in order to be able to see down the adjoining wall.





If you knew the angle that the adjoining wall intersected your wall (and this angle was acute), the quickest path would be one that departed the wall you were following at an angle of
90-\phi and traveled in a straight line for \sin\phi meters. This would take you directly to the "extension" of the line of the adjoining wall and would afford you an unobstructed view down that wall. Conversely, if the angle is obtuse, it would be quickest to just go directly along the wall to the corner. This concept is illustrated in the figure below (courtesy R. Dorrigiv and A. Lopez-Ortiz):




The tension between these two approaches (curl out into the room versus proceed directly to the corner) represent the crux of this path-planning problem.

Let me finally pose the question: Starting from one unit "south" of a corner with unknown acute angle (between 0 and 90 degrees, inclusive), what is the best approach path, and what does "best" mean to you?

A solution---and my take on it---coming soon.

2010 ARCS-Honolulu Kresser Award in Engineering

Every year, the Honolulu chapter of the ARCS Foundation (Achievement Rewards for College Scientists) presents scholarships to a select number of graduate students working on research in the various sciences at UH Manoa.

This year, there were 12 awards given to students ranging from the physical and natural sciences to health sciences and engineering. I was one of two students (both from Electrical Engineering) to receive an award from the College of Engineering.

In exchange for the award, the ARCS Foundation asks the students to prepare short semi-technical talks and posters to share their research with the donors and coordinators. I took this as a great opportunity to hone my (slightly rusty) public speaking skills, and prepared the slideshow below to augment my ten minute talk.



(edit: It looks like the video player has cut off the title slide. You're not missing much, but the whole thing is available via the links below)
The slides and poster are both also available as PDF's.

In addition to practicing my speaking skills, the presentation session was enjoyable because it shed some light on the interesting research being done by 11 other bright scientists around the campus. Some particularly interesting work is being done by Jim Baker (EE) in compact high-frequency radio antenna design, Brendan Bowler (Astro) in the search for extra-solar planets, and Tiffany Anderson (Oceanography) in shoreline erosion due to storms.

Wait, Colorado?

_DSC8545.jpg, originally uploaded by kcmckell.

Originally published Monday, February 22, 2010.

I could have sworn that URL thingy up in the top of my browser says “hawaii.edu”, right?

Well, it’s true. I am a bona fide graduate student at the University of Hawaii at Manoa. That does not mean, however, that I spend all my time there. Case in point: This semester, I am living in Boulder, Colorado doing research under the auspices of “Visiting Researcher” and/or “Campus Guest.”

There are several reasons why a graduate student may want to spend some time away from his or her home institution. If the research is “portable” enough, a change of scenery may increase productivity if one goes somewhere with fewer distractions such as friends, family, and other non-academic commitments.

There are potential long term benefits as well. Working in different locations will only broaden the pool of colleagues that you know. While making new friends is always exciting, one never quite knows where future collaborations will come from, so knowing more---and different---people is a plus. Further, in this world of social networking on steroids (now there’s a Google hit just waiting to happen), we can maintain contacts like never before.

Lastly (and this plays into my love of travel), if one is lucky enough to land a job at an institution that grants sabbaticals---almost unheard of in industry---your opportunities for extended travel reduce to once every seven or ten years. Thus, why not get it in when you can?

Stay warm,

Clay