Saturday, January 30, 2010

A gift of the heavens.

Aloha kakou,
Sorry for the long delay between posting. I have been quite busy with homework recently, and when done, I don't really feel like doing anything (actually, being truly 'done' is unlikely to happen till Spring Break, at least).
 
There has been one interesting tidbit of news recently, however: I have the opportunity to do research with one of my professors, Dr. Marianne Takamiya, on one of her research projects on the effects of extinction on the reported rate of star formation in low- to medium-redshift galaxies. She mentioned that any students interested in helping should come talk to her, and so while I was seeing her about a homework problem, I casually asked what kind of research she was doing. The result was an impromptu half-hour presentation and an on-the-spot offer to get involved. Needless to say, I was blown away. Not every day does an opportunity to do undergraduate research with long-term involvement present itself, and especially not an opportunity that practically forces itself upon you. She's already talking about teaching me UNIX and IRAF (a package of data-reduction tools used by professional astronomers). And from what I gather, it will involve me working with quite a few pictures of big, beautiful spiral galaxies, matching and overlaying data from the visible, near infrared, and radio regions. This really is a remarkable opportunity; all I can say is soli Deo gloria. I will be sure to keep you all informed about how this progresses (and when I know more about it myself).
 
 A hui hou!

Tuesday, January 26, 2010

Wake up and smell the vog.

I realized today that I hadn't written for a while, mostly due to lots of homework. But since doing homework for hours at a time is driving me crazy, I'm going to take a little time and catch up on what I've been doing.

Saturday we had some of the worst vog I've seen yet here in Hilo. Vog, in case you don't know, is volcanic fog. It's basically clouds of sulfur dioxide that react to form sulfuric acid upon contact with water. Thankfully, it doesn't bother me much, but a lot of people get sniffles, coughs, sore throats, and headaches from it.  It was so thick on Saturday that you could see it as a thick haze if you looked down the road. I knew it was bad when I could see the sunlight being scattered by it as it came down through breaks in the trees.

Interestingly enough, however, Sunday was very clear in Hilo. I went up to the Visitor Station on Mauna Kea last night, and as we climbed the Saddle between Mauna Kea and Mauna Loa we encountered increasingly heavy vog. Upon reaching the visitor center at 9,200 feet, we could look out over the Saddle and towards Hilo and see a huge bank of vog hanging in the air. But thankfully it was nice and clear up there last night.

While eating supper at Hale Pohaku, we noticed some Erckel's Francolins hanging around. Being a kind-hearted and generous soul, or maybe just a little crazy, I started tossing food off the balcony where we were eating for them. One little guy in particular (I'm pretty sure it was a male, because francolins are unusual in that the males have two spurs on each leg) hung around for three quarters of an hour, devouring everything I threw at him: hard-boiled egg, steamed broccoli, croûtons, French fries, lettuce, blue cheese (after I tried some for the first time and discovered I could not stand the taste), corn, even a little turkey with gravy. I think he may even have eaten the candied cherry I tossed him from my cheesecake. I wonder how many weeks of doing that it will take for me to have them eating out of my hand? I managed to snap a picture of him, and here it is:


It was a very relaxing and happy time – I felt just like I was back home feeding chickens off the porch.

After supper, we helped get the telescopes all set up for the evening, and I resumed learning how to operate the imaging system belonging to the Visitors center. Have I mentioned that it's $30,000 worth of equipment, all free for volunteers to use? The guy who's teaching me, Nathan, is an absolute master of astrophotography. He's been doing it for five years, and regularly makes images that would knock your socks off with their beauty. Last night we were focused on getting image data, but next Sunday, if I can make it up, he's going to teach me how the process and 'reduce' the data to get a passable picture. So stay tuned – you might get to see some pretty neat stuff in the near future!

Unfortunately, I need to get back to work,  and don't know when I will be able to write next because I just got my first homework assignment from Dr. Crowe in Electromagnetism, and if it's anything like Classical Mechanics with him last semester, it's going to eat a LOT of my time. So, a hui hou! (until next time!)

Thursday, January 21, 2010

On the method of finding roots of unity.

Remember that method of root-finding I worked out last night? Turns out it was an important part of an attempt to prove Fermat's Last Theorem, one of the most famous theorems in the history of mathematics. In other words, probably every mathematician worth their salt already knows about it. (Fermat's Last Theorm says that if you have the equation \(x^n + y^n = z^n\),  there are no integer values for x, y, and z (other than 0) that will solve the equation for \(n\ge3\). It was proposed in 1637, but not actually proven until 1995. Yes, that recently.) I found this out because it got brought up in a talk by our Mathematics department chair Dr. Lee that I attended this afternoon, complete with illustration on the board of the method I was using. I learned a few other useful tidbits of information about it while listening, so I'm going to set down a formal notation of the process here, before I forget it.

Start with your equation, \(z^n=\pm1\) Depending on whether 1 is positive or negative, the formulas for the roots take different forms. Since the roots will share several features in common, it is helpful to denote them by a common symbol. I'm going to use omega, \(\omega\), for variety (and 'cause Dr. Lee was using it).

In both cases the roots are given by \[\omega = e^{(i\pi k/n)}\] however, if \(z^n=+1\), the number k is given by \(k = 0, 2, 4, ... , (2n-2)\), whereas if \(z^n=-1\), k is given by \(k = 1, 3, 5,  ... , (2n-1)\).

I had originally thought that the evenness or oddness of n would effect the formulas, but upon looking over the examples I worked out, I see that it doesn't. The only thing that changes is the range of the number k. Note that one interesting thing happens if n is even and 1 is negative: all the roots will be complex numbers, which makes them very difficult to find without using an explicit method like this.

 Well, once again, it looks like I have managed to independently discover something that was already worked out long before me. Perhaps, someday, I will actually be able to produce something original.

Wednesday, January 20, 2010

A novel method for finding nth roots.

By dint of hard work, I have tonight come across a previously unknown (to me at least) method by which one can find higher order roots of equations of the form x to the nth power equals a constant. Since in several hours of Internet searching for just such a solution I did not come across anything comparable, I'm going to set it down here so I don't forget it later on.

The problem arose while I was working on my Complex Analysis homework. I needed to find the roots to the equation z to the 6th power equals 1. In other words, I needed to find 6 roots of 1, each of which, when raised to the 6th power, would equal 1. Now, I know that I learned procedures to do just that back in algebra, but I could not for the life of me remember them tonight. After some fruitless searching on the Internet, I came across a Wikipedia article on roots of one. It too did not contain what I was looking for, but it did contain some very useful pieces of information, among them that the n roots of one were always evenly spaced on the unit circle in the complex plane, and that they formed a regular n-sided polygon within the unit circle.

Now, if you remember complex numbers from algebra, you know that they can be written in the form \(a+bi\), where a and b are real numbers. However, if you do some playing around with very complicated math like infinite series, you can derive Euler's formula, which says that e to the \(i\times x\) power equals the cosine of x plus i times the sine of x, or, in symbols, \(e^{ix} = \cos(x) + i \sin(x)\). This allows complex numbers to be written in the equivalent polar form \(re^{ix}\), where r is the absolute distance (positive) from the origin to the point in question, and x is the angle from the real axis in the complex plane.

Since the vertices of the regular n-sided polygon formed by the roots of the equation you're looking at are evenly spaced, it's a trivial matter to quickly write down all the roots of the equation in polar form just by reading the values of the angles for the polar form off a well-drawn picture. You can then undertake the almost-as-simple process of converting them back into the more familiar rectangular form of \(a+bi\).

As an example, take my original problem of \(z^6 = 1\). The picture, hopefully, will make things a little clearer. There, I've marked the polygon that the 6 roots make inscribed on the unit circle. In the complex plane, the x-axis is referred as the "real" axis and the y-axis as the "imaginary" axis. With a number of the form \( a+bi\), a is the real part, b the imaginary part. When working the problem, start off by checking if 1 is a solution. In this case it is, so one vertex of the polygon must be at the point (1,0), and the coordinates of the rest of the vertices can be determined from there. With the polar form, only the angle is necessary, and I've marked the angles of all the vertices for you. Once you have the polar form, you can convert it to a rectangular form using the equations
Real part = \(r \cos(x)\)
Imaginary part = \(r \sin(x)\)
where \(r = 1\) since we're using a unit circle.

Anyway, to cut a long process short, since it's getting late, I will simply give you the 6 roots of the equation given first in their polar forms, derived simply by reading off the diagram, then in their more familiar rectangular forms, arrived at using the definitions just given.
\[z=1,e^{i\pi/3},e^{i2\pi/3},-1,e^{i4\pi/3},e^{i5\pi/3}\\
z=1,\frac{1+\sqrt{3}i}{2},\frac{-1+\sqrt{3}i}{2},-1,\frac{-1-\sqrt{3}i}{2},\frac{1-\sqrt{3}i}{2}\]When I have a bit more time, I'll work out a formal notation for the process, so it can be a bit more rigorous and useful to people encountering it for the first time. Until then, a hui hou!

Final musing: what do you think the chances are that this has probably been worked out hundreds of years ago already by much greater mathematicians than I? I'll ask my professor tomorrow...

Tuesday, January 19, 2010

Quiet Monday.

Not much to report today, except that my cold seems to be getting better. It mainly takes the form of a persistent cough now, and I should be feeling well enough to get to classes tomorrow.

I'm very glad I've already taken Calc IV, because my teacher assigns a lot of homework, and it would probably take me at least twice as long if I didn't already have an idea of what I was doing.

I've also been reading the material for my Cosmology class, which involves an explanation of tensors. Fascinating stuff -- I'm looking forward to finally understanding Einstein's field equations -- but extremely complicated and mystifying to me at the moment. We'll see how that goes...

Monday, January 18, 2010

Quiet Sunday.

Just a quick update to say that the cold I had yesterday intensified somewhat today. I stayed home all day, and will do so as much as possible tomorrow. I'm very thankful tomorrow is a holiday...

I am also still working on the homework from last week (I didn't do as much because I was sick). This could prove to be a very interesting semester in that regard.

Sunday, January 17, 2010

Some thoughts on dates.

And I mean the historical kind, not the fruit. This blog is called Daniel's Musings, after all, not Daniel's Boring Everyday Life (although there will be some of that occasionally. I beg your indulgence at those times). But, about those musings. If you're like me, you've heard plenty about the "new decade" in the last two or three weeks. And frankly, I am sick and tired of it.

The reason I am tired of hearing about how 2010 is the start of the new decade, is because it's not. It's the end of the last decade. Just like the year 2000 wasn't the first year of the third millennium; it was the last year of the second.

This may sound a little strange to you, if you haven't thought about it before. The reason for it is because there was no year zero. Our dating system goes directly from 1 B.C. to A.D. 1. (a diversion on the use of A.D.: It typically goes before the date it represents and not after because it is a Latin abbreviation for Anno Domini, roughly translated "year of the Lord", and it makes more sense to have it in front [unless you are discussing centuries. The 'nth' century A.D. is perfectly fine] I'm not certain about C.E. [Common Era], never having used it much, but I believe, just to muddy the waters, it goes behind the date.)

Anyway, short diversion aside, what was the first decade of the A.D.'s? It would be the years 1-10. Think about it. That makes A.D. 10 the last year of the first decade. Likewise for the first century: the year 100 would be the last year of the first century, and the year 101 would be the first year of the second century. Ditto for millennia: the year 1000 was the last year of the first millennium, just as the year 2000 was the last year of the second (maybe that's why Arthur Clarke made it 2001: A Space Odyssey. He wanted it to be a next-millennium thing).

Once you grasp this, the number of centuries makes more sense. Start again with the first century A.D. The first century ended with the year 100. The second century ended with the year 200. But it would consist of the 100's, numerically. The 3rd century, ending in 300, would consist of the 200's. And so on and so forth. The 11th century (the 1000's) ended with the year 1100, the 17th (the 1600's) with the year 1700, and the 20th (the 1900's) with the year 2000. Basically, when you hear the number of a century, subtract one from it to get the numbers of the years in it. When you remember this simple little tidbit of information, deciphering history become much easier, because you can place what happened in a particular century into its proper place in time. Anyway, I hope it works for you. It's just a little trick I use to help keep tabs on history. Personally, I find history fascinating and can't understand how people could find it boring, but whether you do or not, hopefully this trick will help you make sense of it.

And now you know why calling 2010 the first year of the second decade bugs me so much. Now you, too, know the secret of the numbering of time. Please use it responsibly.

In other, boring-everyday news, I seem to have contracted a cold over the last few days. Nothing major, as usual with me and colds, but a petty annoyance that refuses to go away. At least it happened over a long weekend. I hope that you, my readers, are doing better than I am, and until next time, a hui hou!

Thursday, January 14, 2010

New classes, new teachers...(part II)

Aloha kākou!
I am back, after a busy two days. Monday night, after posting, I went up to the Visitor Center on Mauna Kea, a trip I don't regret, but one that did cut into my studying/sleeping time. I was to busy to post last night because of homework. I nearly didn't get the chance to post tonight for the same reason. When I decided to take 21 credits this semester, I did it for two reasons: 1) I need to get credits quickly in order to graduate in a decent amount of time, especially with what I'm going for, and 2) I know that if I have too much free time, I get lazy and slack off. So I aimed to prevent that by simply not having any free time, and it may work even better than I'd planned.... If I don't post for a couple of days, that's probably what happened.

 Monday night at the Vis was very nice. Despite the cold, there were a lot of people, and I got to point out and see many beautiful sights. I especially loved seeing the Great Nebula in Orion through the 14-inch scope we have up there. There's just something about seeing something through a telescope that pictures cannot capture. I've seen more pictures than I can count of the Orion Nebula, and it doesn't show up anywhere near as much detail in the eyepiece, but experiencing the photons from it hitting my eye after thousands of years' travel through interstellar space is indescribable. I wish I could take a picture of what I saw, but it's too faint for my camera to be able to pick it up by holding it to the eyepiece.

There are other objects in the night sky, however, that are brighter, one of them being the planet Mars, which I saw Monday night through the same telescope. I managed to catch a picture of it. It's a little hard to see, but on the left you can just make out one of the polar icecaps of Mars, and there's a suggestion of a darker marking on the lower right. They weren't much easier to see through the eyepiece, but I could make them out, so I know they are real markings.

Tuesday I attended the rest of my classes, and met my new professors.  I think they will work out all right. Since I didn't take a specific 'easy' class this semester, it will be interesting to see which class turns out to be the easiest. For my Introduction to Modern Physics class, I have to do an experiment at the end of class demonstrating some effect or principle of modern physics. Toxic and flammable are both out, which unfortunately also rules out the only one I had on the top of my head, but I have plenty of time to get one worked out. I just have to pick a partner now.

It's getting late, so I need to finish up here. I hope to give a fuller exposition of how my classes are going a little later on, when I've had the chance to observe them for a while. In the mean time, I have a little something for you to do. On the right side of this blog, I've added a little poll. The reason I'm asking what level of math you're comfortable with is so that if I feel like discussing things of a mathematical nature later on, I'll know how much I need to explain for you, my readers. But that will come later...
Aloha aumoe! (good night!)

Tuesday, January 12, 2010

New classes, new teachers... (part I)

Well, my first day of classes has gone well so far. I don't have time to write much because Callie, who coordinates the volunteers at the Mauna Kea visitor center, called apologizing for yesterday and asked if I wanted to come up another time. I found out tonight was open, so I jumped at the chance to get up there before having too much homework on my plate. Hence this post will be short.

 My classes look like they will go well. I guess it's a good sign that I lost track of time in my Electromagnetism class, despite the fact that we were merely covering basic vector calculus formulae which I've already seen before. Dr. Crowe made an interesting point, though, when he said that of the four fundamental forces of the universe, only Electromagnetism can we make any sort of claim to understand fully. Having had a taste of the subject last semester, I'm looking forward to more fully exploring it this time around.

 I have more to say, but I've run out of time to say it today. I think I'll try to keep this blog updated pretty frequently, perhaps even daily. Sort of like a journal. Tomorrow I'll try to give you a little run-down on my trip to Mauna Kea tonight.
A hui hou! 

Monday, January 11, 2010

A new beginning for the new year.

Aloha everyone, and welcome to my new blog!
Where to begin...so much has happened the last few weeks. I finished up my first semester at UH Hilo with A's in all of my classes. I then immediately flew home to spend three weeks with my family. Much of the time, I was working at my old job at Food 4 Less, which kept it from being a true vacation, but on the plus side earned me more than enough money to pay off my books for this semester. Since I am taking seven classes, six of which require books, that's a good thing.

Since someone will probably ask if I don't put it up here now, these are the classes I'm taking this semester: Principles of Astronomy II, Gravitation and Cosmology, Electromagnetism, Introduction to Modern Physics, Calculus IV, Partial Differential Equations, and Complex Variables and Analysis. (Note that Electromagnetism is not the same as the Electricity & Magnetism course I took last semester)

The last few days have been a bit of a jumble as I flew back to Hawai`i and have been getting adjusted and ready for school to start. I can still hardly believe it's tomorrow. Thankfully, I think I have everything ready. (I hope so, because I'm going up to Mauna Kea tonight for stargazing and imaging)

Going from cold to warm quickly is always a shock to me, and this time was no exception. The weather here is so different from Northern California. It's very dry this winter, I've been told, and the sun shines pretty much every day (whereas, in CA, I think I hadn't seen the sun since the 23rd). The air, however, is actually cool, and the Trade Winds are blowing, often at a decent speed. All in all, very nice, but still a bit of a shock to the system.

Although I realized pretty quickly last semester that I was essentially blogging in my emails, writing an 'actual' blog is still slightly intimidating. I've decided to start this up for a few reasons, however:

  1. I don't like spamming your inboxes with too many emails, so I'd wait a long time between writing my email updates. That meant I'd sometimes be trying to recall details about something that happened over a week ago, which usually doesn't go so well. With a blog, you check it as frequently as you like, so I don't mind writing more frequently.
  2. Because I was writing about so many things, my emails would usually be monstrously long. That made me a bit unwilling to write them, and they needed a large chunk of free time. Shorter updates more frequently is the goal of this blog.
  3. Finally, with a blog I'll be able to weave pictures and even videos directly into my narrative, which should prove a bit more interesting.
Well, having said all that, I need to go get ready to head up Mauna Kea now. I hope you are all doing well, and keep watching for new things as I figure how this whole blogging thing works.

Aloha, a hui hou! (Goodbye, until next time!)

Edit: Well, turns out I'm not going up to Mauna Kea after all. Just as I was walking out the door, I got a call saying that the driver who was supposed to be there today couldn't make it, so I had no ride. I don't mind too much; I can certainly put the newly-free time to good use. Maybe next week.