Monday, April 26, 2010

Sights of spring in Hawai`i.

Wow...I can't believe it's been a whole week since I last wrote anything here. Rest assured, I am still here and have no intentions to stop blogging, I've just been really, really, busy. And when I had a brief respite, I really didn't feel like sitting down and writing. However! I'm writing now, and that's what counts.

Where to start...the main reason I was busy this week is because I had  Modern Physics homework due on Tuesday, an Electromagnetism take-home test due on Wednesday, my third and final Calculus 4 exam on Thursday, and Complex Analysis homework due as usual on Friday (which, thankfully, was moved to Monday, but I didn't know that until Friday morning).

Tuesday after turning in the Modern Physics homework I made a final push on the EM test. After the previous homework in that class, I was pleasantly surprised: it only took me 9 hours to do the last 8 problems on the test, and I got to go to bed just after midnight.
My feelings of mild happiness were quickly crushed the next morning when I handed it in, because not only did we get another homework assignment, but we got the last one back, where I found to my extreme dismay that I had made only 70%. That may possibly be the lowest grade I have ever got in college, and I was...displeased. I can only hope that the most recent test felt easy because it was and I was doing it right, not because I was doing it wrong...

Thankfully the calculus test went fairly well, small compensation for the worry I poured into it ahead of time, and as I mentioned the Complex Analysis homework was postponed till Monday. Equally thankfully, this week looks to be rather quiet, about the only big project I have coming up is finishing my paper and presentation for my project in Partial Differential Equations (which overlaps nicely with the test we have in Modern Physics on Tuesday).

Finally, in a lighter vein than recounting all my woes, I offer a sight of Hawaiian spring. Most of the plumeria trees have been blooming more than ever the last few weeks, including the one in our backyard. Here's a shot from the kitchen window, with the backyard and plumeria tree (yes, I know, our backyard is a bit of a mess...).

Monday, April 19, 2010

Building a working cloud chamber, part 3.

Once again, things are never dull around here. Yesterday Joe, our new housemate, went to the Wailuku river and found a Jackson's Chameleon in the back of a riverside cave where it had no reason to be. He brought it back, and I got some really cool pictures. Those things are amazing
Samuel L. the Jackson's Chameleon
Where to begin with this incredible little animal? The eyes that swivel independently? The tongue mechanism so excellently designed for catching insects? The remarkable color-changing ability? The prehensile tail and powerful gripping claws? It's just absolutely astonishing to see one up close (for the record, Jackson's Chameleons, like all chameleons, are native to Africa, not Hawai`i). And Sam is a rather large specimen, probably a good 7-8 inches from horn tip to tail. He reminds me a lot of a dinosaur from the Triceratops genus.

We also ran our cloud chamber experiment again yesterday, and finally, some results! With an old slide projector to provide a powerful light source and a modified bicycle pump to provide a vacuum, we managed to produce a thick cloud almost as soon as we had the setup complete. Now, the experiment was not 100% successful because of one fact: we were unable to catch a particle trail on camera. The reason for this is because our radiation source seems to be a dud. We did see several streaks that we believe to be cosmic ray tracks (high-speed particles from outer space that shower the earth), but nary a track did we see coming from our americium. This happening left us fairly puzzled, as americium is supposed to put out copious amounts of alpha particles (helium nuclei). Scratching the surface of the americium to remove any possible coating had no effect, so we were left scratching our heads and had to conclude that our radiation source was either a fraud, or else that we really don't know what to look for. I'd like to get ahold of a Geiger-counter and independently verify its radioactivity, but in the meantime we'll see if we can borrow a radioactive sample from the school (one of the professors has a lump of uranium he offered to lend us).
Still, seeing cosmic ray tracks (if that is indeed what they were) is pretty cool. These are high-speed (~99% of the speed of light) protons and electrons that travel through space, and constantly shower down on the earth. We saw one every couple of minutes, on average.

I should log off and get some sleep soon, but I thought I'd mention the interesting coincidences surrounding our house at the moment. I've already related my computer troubles, but I neglected to mention that Jonathan's laptop suddenly refused to turn on about a day after mine started having trouble, and the fact that John's laptop's power cord actually caught fire two days ago, leaving him without a computer as well. Thankfully, I can still use my new computer well enough if it manages to avoid locking up while starting up, and I have my old computer as well, so between me and Josh we've been able to provide internet access for them so far. So, coincidence? Or some electro-magnetic anomaly plaguing our house? As the saying goes, once is chance, twice is coincidence, but three times...three times makes you wonder...

Saturday, April 17, 2010

Excursions into the infinite complex plane.

So much has happened since I last wrote ... I'm still working on making sense out of it all.

Homework has been keeping me busy, as usual. I once again have several things due over the weekend, so I will be busy working on that. Between electromagnetism, partial differential equations, modern physics, and calculus IV, I have plenty to keep me occupied.
As if that weren't enough, I got my computer back from the campus tech guy, who was unable to find anything wrong with it other than the fact that the computer simply refuses to acknowledge the wireless module, and it still locks up, with no pattern that I can seem to figure out, yet in an oddly non-random way. I'm unsure which way to go at this point ... I hardly have time to deal with warranties and such, but I could really use my computer working fully as the end of the semester approaches.

Before I forget, I'd like to say a big 'thank you' to my aunt and uncle who were out here this week for everything -- the dinners were great, and the conversation and catching up were awesome. It was really nice to take a break and focus on something other than school for a while. So, mahalo nui loa! (thank you very much)

 In other news, I managed to solve another problem for my Complex Analysis professor today, which was the question of the convergence or lack thereof of the geometric series in the complex plane on the unit circle. If you don't know what I'm talking about, don't worry, you're not missing much. Infinite series and sequences are some of my least favorite subjects from Calculus II, and from the people I've talked to, many others feel exactly the same way. They inspire...strong feelings, to put it mildly. This is because they are very complicated, and most of the time you work with them is spent trying to figure out if a particular series converges or not. (Convergence means that if you add up an infinite number of terms from the series, you will get a finite number. Divergence simply means that you get infinity. This can be a surprisingly difficult thing to do) The geometric series is one of the few series for which you can actually find what it converges to; its terms look like \(x^n\), for n from zero to infinity. (Its first few terms are \(1 + x + x^2 + x^3 + ...\)) It is well known that for any x less than 1, the series  converges, while for any x greater than 1 it diverges. This is because its radius of convergence is equal to 1, and it turns out that every infinite series has such a radius, inside which it converges and outside which it diverges. On the real number line, the word 'radius' doesn't seem to make much sense, because there are only two points for the boundary, but in the complex plane it makes perfect sense because you are dealing with a plane in which numbers can approach from any direction, instead of merely from the left or right.
Anyway, in the real case, it can be shown that the geometric series diverges at \(x = 1\) and \(x =-1\), for different reasons. But that's only two points. The question in the complex case is, what happens for all the other infinite number of points on the circle of radius 1 around the origin? (written symbolically as \(|z|=1\)) However, there is a rather simple (once you see it) and elegant proof that the geometric series diverges for all \(|z|=1\), and I was able to be the inspiration for my professor that enabled him to discover it today (though as usual, the proof he came up with was much more elegant and simple than the clumsy one I had cobbled together). 

Tuesday, April 13, 2010

Computer woes.

Remember what I said about being able to use my computer, even without the Internet? Well, scrap that. My computer has taken to randomly freezing up on startup, in a way that suggests it's a hardware issue like a loose connection that I can't fix. I have some suggestions about on-campus computer repair, but for now it's back my old computer (what is it about the end of the semester and computers? That's the only time I hear people talk about their computers breaking down).

Monday, April 12, 2010

The falls of Hawai`i.

Today my aunt and uncle from Delaware flew into Hilo, so we went out for lunch and had a lovely catching-up chat. After that we went out to Rainbow Falls which is just outside of Hilo, and which had a lot more water going over it than I remember from the last time. I didn't have my camera with me or I would have taken pictures, but the volume of water going over the falls was probably four or five times greater than it was when I saw it in December. Water was bursting from several channels instead of just one like last time, and plunging in great, overflowing dollops into the pool below, churning up the water and generating massive amounts of mist. No rainbows though, because the sun was in the wrong part of the sky. It was really quite spectacular (no doubt a result of the heavy rains we've been having off and on the last few weeks).

Sunday, April 11, 2010

Problems in paradise.

Hello everyone

Once again I have been too busy to write for a few days, mostly working on my electromagnetism homework that was due on Friday, for which I stayed up till 2 in the morning to finish. And then, fittingly enough as we approach the end of the semester, my new computer's wireless module decided to conk out on me yesterday. It appears to be a bona fide hardware problem, so it looks like I'm stuck till I can find someone to fix it. This morning when I turned on my old computer I thought for a while that it had suddenly stopped working as well, and was starting to wonder if I had inadvertently grown an anti-Internet aura, but thankfully a simple restart fixed the problem, else you would not be reading this post right now.

So…

Expect fewer posts from me for a time, due both to my busyness and this. Thankfully, my new computer continues to function fine in other respects, so I can at least use it for non-Internet related school things, like working on my project for PDE's. And the take-home test for that class, and the new take-home test for electromagnetism, and…yeah. I suppose, if I can live through the end of the semester I can make it through anything, but that's a pretty big 'if' with the seemingly ever-accelerating pace of things.

My consolation for doing all this homework is that I can typeset it with \(\LaTeX\). It makes doing the work almost fun, with the expectation of seeing beautifully typeset math come out of what you do. When I have time and opportunity, I'll attach some pictures of the gorgeous results of \(\LaTeX\) typesetting from my homework, so you can see what it's like.

Finally, I'll be heading up to Mauna Kea tonight with some other UAC members since it's the second Saturday of the month. It's close to new moon, so if the weather will cooperate, I hope to get some great astrophotos that maybe I'll have time to process and show you all some day.

Wednesday, April 7, 2010

Back to the daily grind.

I haven't had time to write for a while because I've been rather busy this week with two homeworks due on Friday and a Partial Differential Equations take-home test due next Wednesday. It's not helping my efficiency that I'm trying to learn some of the new programs I installed over spring break at the same time, like \(\LaTeX\) for typesetting homework. Typesetting things is immensely fun, almost addicting, but the learning curve is...not gentle. Still, the rewards are great, and it's an investment that will continue to pay off throughout my life. It's very satisfying to watch a long, complicated plain text document get turned into an elegant PDF file with everything arranged in a professional looking manner. I'll try to insert a sample from my homework of \(\LaTeX\)'s beautiful formatting when I get a chance.

Monday, April 5, 2010

Happy Easter!

ούκ ’έστιν ‛ώδε, ’ήγέρθη γάρ καθώς ’είπεν! δευτε ’ίδετε τόν τόπον ‛όπου ’έκειτο  - Matthew 28:6
'He is not here, for He was raised, as He said! Come, see the place where He lay.'

For those of you who know Koine Greek, apologies for the pathetic accents. I thought it'd be a simple matter of opening up Character Map and picking the right symbols, but there really aren't a whole lot to choose from. There aren't even rough and smooth breathing marks! I had to use apostrophes for those. And forget about circumflex accents. Anyway, it's hard to be annoyed on such a glorious day. Happy Easter everyone!

(I also took the liberty of adding an exclamation point because the Greeks didn't have those, but I think it's certainly implied by the text.)

Sunday, April 4, 2010

Forrays into movie criticism.

Although I neglected to mention it in my last post, I spent part of yesterday evening watching the latest Star Trek movie (mostly because Josh and Jonathan put it on while I was doing homework). I am not a movie critic, so I will not critique the movie (although it had way too much lens flare for my liking), but I am a physicist, so I will critique the physics. And let me tell you, if the level of astrophysical knowledge displayed in the movie was in any way indicative of the general populace's knowledge of the subject, I will suddenly understand where some of the odd questions we get at the Vis now and then come from.
Let's start with the big one: General Relativity. Where is it? I'm not going to critique such things as faster-than-light travel because that's part of the willing suspension of disbelief, and I'm willing accept it. But the black holes were completely unrealistic! In reality, you would never see something fall into a black hole like that. This is because, in GR, curved space time causes time dilation, and infinitely curved space (a.k.a. a singularity) causes infinite time dilation. If you watched something fall into a black hole, you would never see it hit the event horizon and pass beyond -- it would simply become dimmer and redder, forever. In addition, gravitational tidal forces would stretch it into spaghetti long before it reached the event horizon, especially with a small black hole (counter-intuitively, large black hole have small tidal forces and vice-versa).
And all that part about the singularity consuming a planet in a matter of seconds. Come on! If the planet had any amount of rotational motion when it happened (virtually certain), the Conservation of Angular Momentum would spin it up into a rapidly rotating accretion disk which would take a very long time to lose its angular momentum through frictional heating before it could pass beyond the event horizon (admittedly, still not optimal for the inhabitants).
Plus, you have to worry about Hawking radiation. In essence, the intense gravitational energy around a black hole constantly switches between being energy and matter, creating virtual particle/antiparticle pairs in the vicinity of the black hole. If one of the pair happens to fall into the black hole, the other can escape, taking with it a tiny amount of the black hole's mass. Over time (tens of billions of years for a typical few-solar-mass black hole) the black hole would evaporate if it received no additional matter. But the picture is very different for smaller black holes. In fact, if you were to create a black hole with the mass of a massive atom, it would evaporate in such a short time you'd never even know it was there. The Large Hadron Collider may have already created such tiny black holes, and we wouldn't be able to tell (so now you know how to refute those people who think it's going to create a black hole that will engulf the world).
And then there's the whole matter of the supernova engulfing the Romulan planet. Where to begin with that? For a supernova to actually engulf and destroy a planet like that, it would have to be the star that the planet was orbiting in the first place (barring some weird globular cluster configuration, which didn't seem to be the case). And yet they have no idea that the star is going to go supernova? Stars undergo enormous changes in temperature and physical size before the end of their life, changes that make it quite obvious that they're going to blow. We know of several stars in the Milky Way that are candidates for popping off at any time (Betelguese and Eta Carina are two I can think of off the top of my head). Granted that we can't predict exactly when, it certainly doesn't happen overnight. You'd think the Romulans would have evacuated their planet at the first sign of their sun going supernova. There are certainly very real dangers from supernovae, even at great distances, such as gamma ray bursts, but this obviously wasn't such an event.
I should probably stop speaking my mind and picking nits like this, and point you to the Bad Movie Physics entry for the movie, which brought up a bunch of points I hadn't considered. I hope, that if you enjoyed the movie, this doesn't decrease your enjoyment of it. Though the physics may be bad, I find it a subject more for laughter than anything else (and I did break into laughter a few times at odd points).
A hui hou! (until next time)

Saturday, April 3, 2010

Building a working cloud chamber, part 2.

Today was a rather busy day for me. I and my partners in Modern Physics met again today to try and get our cloud chamber project working. It was a rather mixed session. Initially, we had high hopes, as we had corrected several of the problems that plagued our first generation attempts. However, despite doing everything right that we could think of, we were still unable to see even the faintest wisp of cloud. After repeated attempts, we were about to give up in frustration and think about finding another project altogether, when I suggested for lack of any better idea that we invert the setup to have the heat source on the bottom and the dry ice on top, to create conditions more in line with the earth's hydrological cycle. Lo and behold, almost immediately after flipping everything over and applying a slight vacuum we observed some extremely faint clouds. However, this success was tempered by the fact that the clouds were too faint to actually observe any particle tracks, even after repeated attempts fiddling with various parameters.
We finally stopped again after a while, with several more ideas for improvements and the fact that we had managed to observe something. I'm attaching a humorous picture of our setup below; let me stress that the vapor in the picture is not the cloud we saw, it's from some dry ice I put in the container to create a nice visual effect. We never saw anywhere near that much cloud from the alcohol.

It's the intrepid radiation-carrying Lego man of science!

Yes, that's our radiation source that he's carrying on his head. It fits perfectly into Lego-sized openings, so we used a Lego man to hold it up and keep us from losing it. What you actually see is just the aluminum casing, the americium is only exposed in a tiny portion on the very top.
My classmate whose apartment we were running the experiment at lives right on the beach, and I actually saw a whale breaching while I was there. It was out in the ocean a couple hundred meters offshore, just expelling a huge cloud of water vapor (a lot more cloud than we ever saw in our experiment!).

Doing my latest round of complex analysis homework today, I had some really mixed feelings. Now that I've reached this point in the semester, I know enough theorems and propositions to actually accomplish some impressive things. I'm struck by how often a really complicated looking integral can be evaluated very simply, and turn out to be quite easy in the end. It's taken two months to get to this point, two months of building the scaffolding theorem by theorem, problem by problem, with little (up to this point) to show for it, so it's nice to finally be able to really do stuff.

Friday, April 2, 2010

The yards of Hilo.

Went for a walk to go pay my rent today and was struck with the fact of how distinct all the yards in Hilo are. I mean, I'm sure yards are distinct in most places in the world, but in Hilo there seems to be an extra layer of distinctness. I think part of this distinctness is because of the huge range of exotic tropical plants and trees you see in every yard. But what really makes different yards stand out for me, and which you don't see most places, are the distinct lava outcrops (or lack of them) in each yard.

Hilo is really not a flat town, not even down by the sea shore. The streets are deceptively flat down here, but the yards on either side of them are often full of bumps and protuberances, with rocky outcroppings of black basalt poking up through the grass and gleaming dully in the sun. That, I feel, gives each yard a totally different and distinct flavor, because each one has different outcroppings and formations. Or a yard may have no outcroppings, merely bumps and swells like the bulging muscles of some immense beast where the lava beneath has been conquered by the grass above, broken into a layer of growth-supporting soil. I wouldn't go so far as to say that most yards in Hilo have lava outcroppings because I can't claim to be familiar with that much of the city, but certainly a good portion do. Our yard, for instance, has an outcropping in the front running parallel with the road that stands about a foot and a half about the level of the street. It's not much more than a miniature cliff face, however; the grass quickly takes over a little ways in from the road. The basalt fades on one side into the driveway, and on the other into the neighbor's yard where it suddenly loses its layer of top-soil and becomes bare rock for about half of his property before diving below the grass again, only to reappear later on down the street in another form, like a playful pod of dolphins frozen half-in and half-out of the water. I have a suspicion that having lived here I will forever be slightly bored with cities whose yards do not display such variety and originality!

In the category of "learning something new every day", while writing this post I discovered that 'proturbance' is not a recognized English word, at least not according to the dictionaries I consulted. Apparently the correct term is 'protuberance'. I was a little shocked to discover that a word that has been in my vocabulary for I don't know how long would turn out to be a chimera of my imagination like that. I was a little sad, as well, because 'proturbance' sounds much nicer to my ear than 'protuberance'. Though I have not let a word's prior non-existence prevent me from inventing and using it before, I bow to tradition in this case to prevent someone becoming hopelessly confused by trying to look up 'proturbance' in the dictionary.

Thursday, April 1, 2010

Tools and toys.

First off, apologies for the lack of posts recently, I've been somewhat more busy than I expected getting back into the swing of things here as school starts up again. I'm sure glad we get Good Friday off, as I could us the extra day off to get homework done.

I was musing on my way home today about the difference between applied and theoretical mathematicians, since often both kinds use very similar levels of math.  What I realized, is that applied mathematicians (which includes physicists, in my opinion) view math as a tool; theoretical mathematicians view it as a toy. Both physicists and mathematicians play around with math, but mathematicians play with it like you play with any other toy, while physicists play with it like you play with a tool. When I say 'like a tool', I mean in the same way that an accomplished carpenter might 'play around' with his tools when designing a fancy piece of furniture to give it an interesting or beautiful design, or a chef might 'play around' in decorating a cake he baked. The focus is not on the tools; it's on what can be produced with those tools. The tools may be valuable and precious objects in their own right, but they're not visible in the final product, usually.

I hope I'm not sounding unfair to theoretical mathematicians, because even though I am not one, I sometimes envy the state of mind that regards pushing back the frontiers of ignorance on the cutting edge of mathematical research as something fun. After all, a carpenter without tools would be able to produce practically nothing, and the tools of mathematics are precious and valuable in their own right. In fact, the picture as I have drawn it is a bit simplistic. Historically, theoretical and applied mathematicians have worked together, though perhaps without always being aware of it. Scientific research (and particularly physics) has often revealed a profound lack in the mathematics of the time period, prompting investigation and extension into new areas (perhaps even the discovery of a whole new field, as with the calculus). This investigation may be done initially by the scientists who are affected by it, but it usually gets absorbed by theoretical mathematicians who may have no interest whatsoever in the physical problem. Eventually they probe it even further than the scientists needed it taken, and everything seems happy again until someone discovers that taking this new math and applying it somewhere in science makes new discoveries possible. People look into this, and eventually, after a while, discover that the existing math is not enough, and the cycle begins again.
I have unfortunately forgotten whatever point I intended to make when I started writing this, so I will leave you with this marginally-useful bit of advice: if you want to know whether someone is an applied or theoretical mathematician, look at how they use math. Do they use it as a tool, or as a toy?