I think Shiro Hamaguchi typically does the orchestral arrangements for Final Fantasy. He's credited on the 20020220 Final Fantasy Concert CD.
Masashi Hamauzu typically does the piano arrangements. He's one of the 3 composers credited on the FFX soundtrack as well along with Junya Nakano, and you can usually tell which of the pieces were his by the use of piano in the scoring.
"[The sieve of Eratosthenes] is so fast that there is no reason to store a large list of primes on a computer--an efficient implementation can find them faster than a computer can read from a disk.";)
Hm, I guess you could say it's a matter of consensus that 1 is not prime, but it really is the most useful classification. If you must put 1 into a box, you can call it a unit, because this provides the most logical extension to general number fields where the integers can be classified into one of {zero, units, primes, composites}. http://www.utm.edu/research/rimes/notes/faq/one.ht ml goes into more detail.
Hm, ok, as long as you were aware of that. But you don't need to introduce the complex domain to get non-unique factorization! Consider the set of positive integers congruent to 1 mod 4.
S = {1, 5, 9, 13,...}
This is closed under multiplication. 9, 21, and 49 are irreducible in this domain, but 9.49 = 21.21 = 441.
I guess what I'm getting at here is that unique factorization is a truly nifty result that is not at all a tautology. It gives the group of integers some nice structure that we can't count on in the general case. As a matter of fact, Fermat's Last Theorem was almost proved in the 19th century (by both Lamé and Cauchy), but the fatal flaw in their reasoning was that the generalized integers in a field generated from the roots of a polynomial do not necessarily form a UFD.
No, this is only true in a unique factorization domain (UFD). There are lots of number fields which are not UFDs, in which there can be multiple decompositions into irreducibles.
For example, in the ring formed from sqrt(-5) and the integers closed under addition and multiplication, 6 has two factorizations.
6 = 2.3 6 = (1+sqrt(-5)).(1-sqrt(-5))
None of the factors above can be decomposed any further into non-units in the number field.
I wish I could google up some good links to explain this, but the short answer is that silence can be a conversational cue in Japanese dialogue, signalling the other person to refine his suggestion, or withdraw an invitation, or... whatever. Whereas in English such signals would be given more vocally.
EV79 = EV7 microarchitecture on the CMOS9 process. Historically the numbers have matched up on each new generation. EV6 came out on the CMOS6 process; the shrink EV67 was on the CMOS7 process. However, so much time passed between EV6 and EV7 that EV7 is actually on the CMOS8 process, with the shrink EV79 on the CMOS9 process. So the usage of the term EV78, while a bit odd, is not really incorrect.
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I think Shiro Hamaguchi typically does the orchestral arrangements for Final Fantasy. He's credited on the 20020220 Final Fantasy Concert CD. Masashi Hamauzu typically does the piano arrangements. He's one of the 3 composers credited on the FFX soundtrack as well along with Junya Nakano, and you can usually tell which of the pieces were his by the use of piano in the scoring.
From http://www.utm.edu/research/primes/prove/prove2_1. html
;)
"[The sieve of Eratosthenes] is so fast that there is no reason to store a large list of primes on a computer--an efficient implementation can find them faster than a computer can read from a disk."
Nuts, the correct link is http://www.utm.edu/research/primes/notes/faq/one.h tml
Hm, I guess you could say it's a matter of consensus that 1 is not prime, but it really is the most useful classification. If you must put 1 into a box, you can call it a unit, because this provides the most logical extension to general number fields where the integers can be classified into one of {zero, units, primes, composites}. http://www.utm.edu/research/rimes/notes/faq/one.ht ml goes into more detail.
Hm, ok, as long as you were aware of that. But you don't need to introduce the complex domain to get non-unique factorization! Consider the set of positive integers congruent to 1 mod 4.
...}
S = {1, 5, 9, 13,
This is closed under multiplication. 9, 21, and 49 are irreducible in this domain, but 9.49 = 21.21 = 441.
I guess what I'm getting at here is that unique factorization is a truly nifty result that is not at all a tautology. It gives the group of integers some nice structure that we can't count on in the general case. As a matter of fact, Fermat's Last Theorem was almost proved in the 19th century (by both Lamé and Cauchy), but the fatal flaw in their reasoning was that the generalized integers in a field generated from the roots of a polynomial do not necessarily form a UFD.
No, this is only true in a unique factorization domain (UFD). There are lots of number fields which are not UFDs, in which there can be multiple decompositions into irreducibles.
For example, in the ring formed from sqrt(-5) and the integers closed under addition and multiplication, 6 has two factorizations.
6 = 2.3
6 = (1+sqrt(-5)).(1-sqrt(-5))
None of the factors above can be decomposed any further into non-units in the number field.
I wish I could google up some good links to explain this, but the short answer is that silence can be a conversational cue in Japanese dialogue, signalling the other person to refine his suggestion, or withdraw an invitation, or... whatever. Whereas in English such signals would be given more vocally.
EV79 = EV7 microarchitecture on the CMOS9 process. Historically the numbers have matched up on each new generation. EV6 came out on the CMOS6 process; the shrink EV67 was on the CMOS7 process. However, so much time passed between EV6 and EV7 that EV7 is actually on the CMOS8 process, with the shrink EV79 on the CMOS9 process. So the usage of the term EV78, while a bit odd, is not really incorrect.