Wednesday, January 19, 2005

Oh yeah, baby ... Guess who's playin' at my school tomorrow?!
The Lincoln Center Jazz Orchestra, headed by Wynton Marsalis!

I originally went just to get tickets to celebrate my dad's birthday, expecting to spend a good $66 ... but they had so few seats in trio at good spots (or even cheap spots) that it was either pay a shitload and sit in the far back, or pay $30 more than a shitload and get into the orchestra pit! So ... I bought the three Orchestra Pit seats! ORCHESTRA PIT, baby! RIGHT THERE! In front!

And my dad doesn't even know! (No, he doesn't read the blog, and besides, his compy got fried (literally ... it smelled like rubber tires in that room... :-D ) and he's still repairing it)

Monday, January 17, 2005

Happy MLK Jr Day !
End the idiocy and stupidity!
Celebrate the dream!

Note to self: don't try any more poems.

Tuesday, January 11, 2005

Run amok, ye who enter my domain! It is of nothing spoken, for all is explained! No sound! No speech! All is expressed for All to see.

When one side says yes, another says no,
And the sad truth says the answer's twenty-fou',

You see in the mind, not truth but "facts", more oft than not -
When humans debate, such is the plot;
boil down translations and the crowd still roars. What still remains when you set all aside? The same bottom line, sung on both sides:
"My facts
              Are better
                              Than yours."

Science is a process - a method of understanding, describing, enhancing, and always further developing our knowledge and applications of the Laws of the Universe. We will never fully have a true Law of the Universe.(erm, well there may be divine intervention on that topic one day, but until then...) All we have are approximations. Even when we do have a fully unified system for all of Science, it will be a theory. Why? BECAUSE A THEORY ISN'T A HYPOTHESIS. IT'S A WORKING DESCRIPTION OF SOME ASPECT OF REALITY! IT'S NOT the same as the word used in polite conversation, and that's something that some people don't get.
Case and Point:
Einstein's Theory of Relativity encompasses (and can calculate to a higher degree of accuracy) Newton's Laws of Physics. How interesting. (They aren't really laws) And Einstein found the key that shows the beauty of how such an incredibly accurate system as Newton's can be derived from one single, powerful equation:
E = mc2

So what's my point? Don't look for your facts; look for the facts, and look for all opinions. Ask "them." Ask about "them." Talk with "them." Your "side" doesn't mean jack if you don't know who or what the "other side" is.

Remember: Nothing is certain except death and prime numbers. So it is written - abre los ojos.

Monday, January 10, 2005

A friend posted these links on his site. I thought I'd share, since they're for a good cause. Donate.
Interesting article about why you should donate to the Salvation Army rather than the Red Cross ... I can't make an informed opinion with my little bit of information, so I won't say what's right or wrong, but I'm still going to donate. Won't you? If you're wondering about the validity of the link, just check the domain name.

Also, I usually flat-out hate stuff like this, but: from another friend: "If there is at least one person in your life whom you consider a close friend, and whom you would not have met without the internet, post this sentence in your journal."
Yep. I know. Sappy, infective, and almost clique-type reaction, coupled with a pleasing dopamine-response in the brain, but... oh well.
Near and dear... I have friends in this category as well.

However, there are also several people near and dear that I have neglected in real life ... by simply not mailing a damn letter to them. Here's what can happen: You feel guilty, and more guilty as years go by. They give up contacting you at some point, and move on with their life/lives. They forget about you. You, however, don't forget about them, and it tears at you. Just because you didn't send a damn letter. Go out and get in touch with that friend in Washington, maybe in Brazil, in Maryland, in Anchorage, in Ohio, in Portland, in Singapore. Do it before you regret not doing it sooner or at all.

Now, back to my next set of ramblings.

I just bought some new books on mathematics that will make interesting reads. I'll let you all know if they're any good ... after all, if *I* can read this stuff and learn from it (e.g. the person who takes an average of slightly over 5 hours to complete a 2-hour final exam) then chances are in your favor that you can, too.

For those of you who care, here are some of my somewhat more-favored NP-Complete problems (well - there are many other incredibly interesting problems and much more than those on the page, but these are just a subset of the most well known problems that are primarily housed in mathematics and the storage/interpretation of numbers) taken from an Annotated List of Selected NP-Complete Problems as maintained by Paul E. Dunne, a professor in the Computer Science Department at the University of Liverpool:
Number: 9

Name: Comparative Divisibility [AN4] 3

Input: A (strictly increasing) sequence A=< a1,a2,...,an> and a (strictly increasing) sequence B=< b1,b2,...,bm> of positive integers.

Question: Is there an integer, c, such that Divides(c,A)> Divides(c,B), where Divides(x,Y) (Y being a sequence of positive integers) is the number of elements, y in Y, for which x is an exact divisor of y?

Comments: You may think that this has an obvious fast algorithm, and, indeed the algorithm in question is obvious: what it is not is efficient. Consider: how many bits are needed to store the input data (assuming, without loss of generality, that an>=bm)? How many steps, however, is this `obvious method' taking in the worst-case? It is important to realise that representing integer values in unary is not considered to be a `reasonable' approach (the number 250-1 requires 250 digits in unary but only 50 digits in binary).

Number: 11

Name: Quadratic Diophantine Equations [AN8] 3

Input: Positive integers a, b, and c.

Question: Are there two positive integers x and y such that (a*x*x)+(b*y)=c?

Comments: The comments regarding Problem 9 (Comparative Divisibility) are also pertinent with respect to this problem. Again, there is an `obvious' algorithm that, on the surface, appears to be efficient and is seen not to be so only once one compares the input size (space needed to represent the input data) to the actual computation time in the worst-case.

Number: 53

Name: Quadratic Congruences [AN1] 3

Input: Positive integers a, b, and c.

Question: Is there a positive integer x whose value is less than c and is such that x2==a(mod b), i.e. the remainder when x2 is divided by b is equal to a?

Comments: The comments made with respect Problem 9 and Problem 11 are also relevant with respect to this problem.

Number: 57

Name: Simultaneous incongruences [AN2] 3

Input: A set of n ordered pairs of positive integers {(a1,b1),...,(an,bn)} where ai<=bi for each 1<=i<=n.

Question: Is there a positive integer x such that: for each i, ai does not equal the remainder when dividing x by bi?

Comments: As with most number-theoretic problems, the comments regarding Problems 9, 11, and 53 apply.

School starts in One day !

Friday, January 07, 2005

Happy New Year's Day. 2004 was the first year in many that felt somewhat like a full year - an eventful one.

'tis good tidings and a busy schedule that bids me to write my updates so few and far in between - didn't a buddhist once mention that the spaces between the notes make the music ?

More Random Thoughts For You All:
Infinite sets are fun, and even weirder are infinite sets derived from non-repeating irrational numbers such as pi ... in fact, for those of you who know of Edmund Landau's notation for the Prime Counting Function (pi(x)), here's an interesting rationalization (read: rationalization, NOT a proof) for his use of that particular symbol:

Take an irrational non-repeating number such as pi, and treat that number as an infinite sequence - one that can be written as a partitioned set of strings, where all strings in the set hold a trait in common. In particular, assume each string to represent a prime number, created by appending digits from pi (as found in the sequence) into the current result until the result is a prime not yet counted, whereupon it is added to the count, and we move on to the next string of digits. Since this set can be represented as the set of all prime numbers, we may order this set, starting with the smallest prime - thus giving us a mapping of the set of prime numbers to the set of positive integers.
We may call this function (when it is found at last) "The xth Holy Grail of Mathematics" where x is a variable (after all, there are far more than just one of these in mathematics) .
But less dramatically, since we have shown that the sequence of digits from pi can be associated with the set of all prime numbers, we can all (or maybe just all of one person) sleep more easily knowing that there is some justification in the use of the symbol pi as the Prime Counting Function (pi(x)). (which is simply a function that gives the number of primes less than a given integer.)

Even weirder: Take a second number in that category of irrational non-repeating numbers, such as phi ( for the function phi(x) ), and partition its equivalent sequence in the same manner as shown above ... we now have a 1-to-1 mapping from one set to the other, showing them to be of equal size ... bizarre, eh ? Just what meaning could this hold for phi and pi ? Well, as you will see below, my conjecture is that it doesn't prove anything, because we can map these irrational numbers in any function that we want ... allowing the range of phi(x) to be the set of all even numbers, and the range of pi(x) to be the set of all odd numbers ... then of course, the set of all positive integers must be equal to the union of these sets phi and pi, thus showing them to be subsets where the positive integer set is equal to the partition { range(phi) , range(pi) } !
This is just idle wondering.

Something a little more concrete (and perhaps more productive), but not by much:
Now here's something more concrete (and redundant) I've been wondering about for awhile:
Using an array of logarithms, where the lookup is based off of the index of the array, how does a computer's built-in multiplication compare to the equivalent via logarithm addition and lookup ? And what about treating these numbers as logarithm polynomials ? And what about programming structures created for arbitrarily large numbers? (grouped with modular arithmetic and bit-shifting, too?) How does it affect prime number verification ? Especially coupled with the knowledge that all prime numbers above ten have a least significant digit of 1, 3, 7, or 9 ?
If you haven't yet guessed by my overly specific questions, I'm going to find out. I know it's been done many many times before by other programmers, but I haven't been able to find in-depth results online anywhere - hence, I'm going to experiment and post the results. Just keep in mind, it's another project added to the stack (the stack that should be a queue)
and, if you-know-who is willing, then it may just get done. After all, I had so many projects I wanted to finish before the start of Spring semester, which now looms over the hills less than a week away.

Happy New Year, everyone ! And remember :
A good teacher makes boring stuff exciting, and a bad teacher makes exciting stuff boring
(I didn't create this particular aphorism; I heard it from someone else and had to share.)