A holiday puzzle

I am afraid of gifts, both receiving and giving.  Luckily, I have been largely spared having to confront this challenge.  I am often (rightly) criticized that in rare occasions when I give, the gifts are often what I like, not what the receivers would.  People say gifting is an art — no wonder I’m bad at it.   It is therefore a pleasant surprise that I received a holiday gift a few days ago, and it is a fun puzzle.

Consider an infinite grid where each branch (solid blue line segment) has a resistance of 1 ohm, as shown in the figure below.

What is the equivalent resistance between any pair of adjacent nodes?   In other words, take an arbitrary pair of adjacent nodes, labeled + and − in the figure, and apply an 1-volt voltage source to the pair (the dotted line connecting the voltage source to the grid is idealized and has zero resistance). Denote the current through the voltage source by I_0.  What is the value of the equivalent resistance R := 1/I_0?

Chances are such an interesting problem must have been solved.  But instead of researching on prior work and its history, why not have some fun with it.  We don’t have to worry about (nor claim any) credits or novelty with a holiday puzzle!

…. But I would appreciate any pointer to its history or solution methods if you do know.  Even a random guess of the answer will be welcome.

In the next post, I’d describe two methods: one is a simple symmetry argument for a special case, and the other a numerical solution for the general case.   Meanwhile, have fun and happy holidays!

Do you attend more or fewer talks in larger conferences?

In the last few months, the conferences I’ve attended have been on the extremes in terms of sizes.  A few small targeted workshops (like NEGT, which I blogged about), and a few massive conferences (CDC & INFORMS).  The contrast made me realize that I get very different things out of these two types of venues.

In particular, for me, there is a clear relationship between the number of talks I go to and the size of the conference.

There is a clear peak in the number of talks I attend around the size of a smallish single or double track conference… When there are 30-40 talks and 100-200 people, it seems that I end up spending most of my time actually in the talks, whereas in these massive mega-conferences like CDC and INFORMS, where there are 50+ parallel tracks, I end up spending almost no time in talks, and instead end up spending all of my time in the halls chatting/working with people.  CDC this year was an extreme case for me — I ended up scheduling meetings with collaborators and prospective students/postdocs nearly all day every day, so I only managed to make it to ~1 talk per day!  (Hence the lack of a “Report from CDC” blog post.)

If the 20th century was “the engineered century,” what will the 21st century be?

In 2000, Neil Armstrong gave a compelling speech to the National Press Club where he laid out the idea that the 20th century could, in some sense, be thought of as “the engineered century.”  (PhD Comics has a great illustration of excerpts from the speech.)  In particular, engineering came into its own during the 20th century, and engineers really changed the whole fabric of society.

In his speech, he lays out the results of an attempt by the National Academy of engineering to identify the 20 most significant engineering contributions to quality of life in the 20th century, and the list is quite impressive, including things like lasers and fiber optics, radio and television, transistors and integrated circuits, airplanes and automobiles, and (of course) electrification.   When one reads the list, it’s truly an amazing illustration of how magical the 20th century was, and the changes in life driven by engineering.

So then, the question, of course, is now that we are 15 years into the next century, can we start to imagine what the 21st century will be known as?  Personally, I’d say it’s looking like it will be “the century of information technology.”

Now, it’s quite early, of course, to make such a guess, but IT is certainly the early leader… Information (and computation) is increasingly pervasive, and it is now almost impossible to find a scientific discipline that is not being altered by this theme as tools from algorithms, machine learning, big data, etc., are invading areas like physics, economics, biology, …  The trend is unavoidable as computer science enrollments are doubling, tripling, and even more at top schools across the country.  For example, the computer science undergraduate option at Caltech is now larger than any other two options combined!  Nearly every student on campus takes our freshman-level CS courses, and nearly half the undergraduate body takes at least one (often more) of the sophomore level courses in algorithms and complexity.

More broadly, it is interesting to see how IT is now pervasive in most lists of the top societal accomplishments in recent years.  A great example is Slate’s recent series on the seven wonders of the modern world.   Nearly all of the mentioned “wonders” are made possible by recent IT advancements!

Business case for DER and utility

Climate and energy are critical, massive, and complex issues.  Whatever we talk about, it will be just a small piece of the overall puzzle and, by definition, unbalanced.  This post collects some tidbits that point to an underlying trend, focusing on the most commonly asked question “is there a business case for smart grid?” This trend suggests an indispensable role for distribution utility of the future.

Accelerating pace of DER (distributed energy resources)

I’m pleasantly surprised by the NYT report today (Dec 1, 2014) that one of the world’s largest investor-owned electric utilities, E.On of Germany, has decided to split itself into two, one focusing on the less (!) risky business of renewables and distribution, and the other on the more risky conventional generation business of coal, nuclear and natural gas.   “We are seeing the emergence of two distinct energy worlds,” E.On’s CEO said.  In case you think this is an irrational impulsive move, a financial analyst estimated that of E.On’s 9.3 billion euro in pretax profits in 2013, more than half came from the greener, more predictable businesses. The utility industry has entered a period of disequilibrium in recent years, contemplating how best to leverage emerging technologies and evolve their business models (we will return to this point below).  Initial response to E.On’s decision: its share price rose about 5% today.  E.On said it will present a plan in 2016 to spin off most of the unit that currently holds the conventional generation.