Online you could fetch the GlassBook-Edition. If you signed up for the email waiting list, you could get it in postscript or pdf - so I did.
Was I disappointed, when I discovered, that I the email I received, only contained a link to a GlassBook Edition?
If they only want to give the GlassBoo Edition away, it is okay, but they should not fool the customers into believing they can versions that they can actually read.
First of all, the empty product equals 1 by _convention_. But of course you do not make convention out of the blue. Here is the reason that the empty product equals one:
The following is true for positive n and m:
(a*...*a) * (a*...*a) = a*...*a (+)
The parenthesises contains n, m and m+n a-s respectively.
[Damn,/. eats all my white space. (I can not write n, m and m+n the proper places.)]
or written using powers:
a^n a^m = a^(n+m) (++)
Now in math we to extend the symbol a^n to the case n=0. Thus we need to give a^0 a value (compare this to (+), the value a^0 is the value of the empty product).
We would like equation (++) to stay true, so we have to require:
a^0 a^m = a^(0+m) = a^m
Thus a^0 _have to_ be 1.
If we formulate it this way:
The empty product equals the neutral element of multiplication.
Then it is clear that the empty sum must equal 0, as 0 is the neutral element of addition.
The Miraculous Baily-Borwein-Plouffe Pi Algorithm
on
Happy Pi Day!
·
· Score: 2
Happy pi day.
This is the perfect occasion to spread the message of
The Miraculous Baily-Borwein-Plouffe Pi Algorithm
It is an algorithm to compute the n'th digit of Pi in any base, in particular it is possible to compute the n'th decimal digit without having to compute the n-1 first digits. This is a truly amazing result. We know that pi is irrational (Euler) and that pi is trancedental (Lindemann, 1982) and thus is highly irregular. That the n'th digit of pi is computable is therefore very surprising. There are only a countable number of computer algorithms and thus there are only countable any numbers that have the property that their n'th number is computable.
one can find an article that explains the algorithm together with an implementaion in c (two pages long). The remarkable thing is that the algorithm uses only normal integers and doubles. That is, one need not implement arbitrary precision arithmetic.
The algorithm is new, 1996. In another thread the corresponding program is shown for base 16, but I much prefer the base 10 version :-)
References:
The original article concerning base 10 is
"On the computation of the n'th decimal digit of various transcendental numbers." by Simon Plouffe, November 30, 1996.
Here they also answer why it is fun to compute many digits of pi. In the beginning the mathematicians wanted to know many digits of pi to find out whether pi was irrational or not. Euler showed that pi was irrational (the proof is not that hard). Later Lindemann in 1882 showed that pi was trancedental, that is pi is root in no polynomial with integer coefficents. Today it is customary to compute many digits of pi on new super computers. In 1982 sun (?) actually found some obscure hardware bug due to a pi program.
-- Jens Axel Søgaard -- http://www.jasoegaard.dk
A Mathematician is a machine for turning coffee into theorems. - Paul Erdös
Isn't this the old trick from the Microsoft public relation department?
They deliberately put out the rumour, that they are going to port Office to Linux - the rumour has the effect that no other companies start to port their word processors, since they know they can not beat Word.
The net result is that Linux-opponents still can argue: "Linux does not have any real word processors".
Off course I could be overly pessimistic. If it on the other hand is true, that they are porting Word - I will say "Thanks, that's just what Linux needed.".
-- A mathematician is a device for turning coffee into theorems. -- P. Erdos
It is reasonable that a user beams CDs from a very limited number of machines, since he must beam his CDs from home.
My guess is therefore, that an account gets blocked if CDs are beamed from several machines. In this way I can not go visit all my friends and beam their CD-collections to my account.
Another thing is accessing the same account from several machines at once. Although some say they don't block it now, it doesn't mean that they won't block it in the future.
Claimed...:-) I think he is right. A 5 min search using google lead me to
http://www.jjj.de/fxt/fftnote.txt ,
which contains "Notes on the FFT" written by C. S. Burrus.
The note gives a thorough presentation of the FFT algorithm starting with Gauss/Cooley-Tukey and up to now, where efforts are made to discover a parallel algorithm. The first paragraph of text follows below, wherin [1] is the reference:
[1] M. T. Heideman, D. H. Johnson, and C. S. Burrus, "Gauss and the history of the FFT," IEEE Acoustics, Speech, and Signal Processing Magazine, vol. 1, pp. 14-21, October 1984. also in IEEE Press FFT Reprints, by P. Duhamel, 1995.
This is a note describing results on efficient algorithms to calculate the discrete Fourier transform (DFT). The purpose is to report work done at Rice University, but other contributions used by the DSP research group at Rice are also cited. Perhaps the most interesting is the discovery that the Cooley-Tukey FFT was described by Gauss in 1805 [1]. That gives some indication of the age of research on the topic, and the fact that a recently compiled bibliography [2] on efficient algorithms contains over 3400 entries indicates its volume. An expanded version of this bibliography is published as a book [2] with the references in a data base on a disk. Four IEEE Press reprint books contain papers on the FFT [3,4,5,6].
Slashdot Mirror
Did you notice the ending of the press release ?
:-)
First came an "About VMware" then followed an "About Microsoft".
Who doesn't know Microsoft ?
I don't get it, why on earth is the above rated as funny? The link
http://www.big.net.au/~silvio
he gives, contains a lot of information concerning virusses on Unices.
Katz is right. It is about attitude.
Online you could fetch the GlassBook-Edition. If you signed up for the email waiting list, you could get it in postscript or pdf - so I did.
Was I disappointed, when I discovered, that I the email I received, only contained a link to a GlassBook Edition?
If they only want to give the GlassBoo Edition away, it is okay, but they should not fool the customers into believing they can versions that they can actually read.
I for one felt cheated.
First of all, the empty product equals 1 by _convention_. But of course you do not make convention out of the blue. Here is the reason that the empty product equals one:
/. eats all my white space. (I can not write n, m and m+n the proper places.)]
The following is true for positive n and m:
(a*...*a) * (a*...*a) = a*...*a (+)
The parenthesises contains n, m and m+n a-s respectively.
[Damn,
or written using powers:
a^n a^m = a^(n+m) (++)
Now in math we to extend the symbol a^n to the case n=0. Thus we need to give a^0 a value (compare this to (+), the value a^0 is the value of the empty product).
We would like equation (++) to stay true, so we have to require:
a^0 a^m = a^(0+m) = a^m
Thus a^0 _have to_ be 1.
If we formulate it this way:
The empty product equals the neutral element of multiplication.
Then it is clear that the empty sum must equal 0, as 0 is the neutral element of addition.
Happy pi day.
n ary
h tml
r Pi.pdf
This is the perfect occasion to spread the message of
The Miraculous Baily-Borwein-Plouffe Pi Algorithm
It is an algorithm to compute the n'th digit of Pi in any base, in
particular it is possible to compute the n'th decimal digit without
having to compute the n-1 first digits. This is a truly amazing
result. We know that pi is irrational (Euler) and that pi is
trancedental (Lindemann, 1982) and thus is highly irregular. That the
n'th digit of pi is computable is therefore very surprising. There are
only a countable number of computer algorithms and thus there are only
countable any numbers that have the property that their n'th number is
computable.
On "Fabrice Bellard's Pi Page":
http://www-stud.enst.fr/~bellard/pi/index.html#bi
one can find an article that explains the algorithm together with an
implementaion in c (two pages long). The remarkable thing is that the
algorithm uses only normal integers and doubles. That is, one need not
implement arbitrary precision arithmetic.
The algorithm is new, 1996. In another thread the corresponding
program is shown for base 16, but I much prefer the base 10 version
:-)
References:
The original article concerning base 10 is
"On the computation of the n'th decimal digit of various
transcendental numbers." by Simon Plouffe, November 30, 1996.
and can be found at
http://www.lacim.uqam.ca/plouffe/Simon/articlepi.
History:
A very readable account of the history of computations of pi is the
"The quest for pi by Bailey, Plouffe and the Borweins." this can be
found at
http://www.lacim.uqam.ca/plouffe/Simon/TheQuestfo
Here they also answer why it is fun to compute many digits of pi. In
the beginning the mathematicians wanted to know many digits of pi to
find out whether pi was irrational or not. Euler showed that pi was
irrational (the proof is not that hard). Later Lindemann in 1882
showed that pi was trancedental, that is pi is root in no polynomial
with integer coefficents. Today it is customary to compute many digits
of pi on new super computers. In 1982 sun (?) actually found some
obscure hardware bug due to a pi program.
--
Jens Axel Søgaard -- http://www.jasoegaard.dk
A Mathematician is a machine for turning coffee into theorems.
- Paul Erdös
Isn't this the old trick from the Microsoft public relation department?
They deliberately put out the rumour, that they are going to port Office to Linux - the rumour has the effect that no other companies start to port their word processors, since they know they can not beat Word.
The net result is that Linux-opponents still can argue: "Linux does not have any real word processors".
Off course I could be overly pessimistic. If it on the other hand is true, that they are porting Word - I will say "Thanks, that's just what Linux needed.".
--
A mathematician is a device for turning coffee into theorems. -- P. Erdos
It is reasonable that a user beams CDs from a very limited number of machines, since he must beam his CDs from home.
My guess is therefore, that an account gets blocked if CDs are beamed from several machines. In this way I can not go visit all my friends and beam their CD-collections to my account.
Another thing is accessing the same account from several machines at once. Although some say they don't block it now, it doesn't mean that they won't block it in the future.
Subject says it all.
The price is $99 - a little expensive I think.
Translation of the post post script:
Jon: If you are reading, then have good luck, but do not come here and day that you did it all alone, that is too dumb.
Claimed... :-) I think he is right. A 5 min search using google lead
/Cooley-Tukey and up to now, where efforts are made to
me to
http://www.jjj.de/fxt/fftnote.txt ,
which contains "Notes on the FFT" written by C. S. Burrus.
The note gives a thorough presentation of the FFT algorithm starting
with Gauss
discover a parallel algorithm. The first paragraph of text follows
below, wherin [1] is the reference:
[1] M. T. Heideman, D. H. Johnson, and C. S. Burrus,
"Gauss and the history of the FFT," IEEE Acoustics, Speech,
and Signal Processing Magazine, vol. 1, pp. 14-21, October 1984.
also in IEEE Press FFT Reprints, by P. Duhamel, 1995.
This is a note describing results on efficient algorithms to calculate
the discrete Fourier transform (DFT). The purpose is to report work
done at Rice University, but other contributions used by the DSP
research group at Rice are also cited. Perhaps the most interesting
is the discovery that the Cooley-Tukey FFT was described by Gauss in
1805 [1]. That gives some indication of the age of research on the
topic, and the fact that a recently compiled bibliography [2] on
efficient algorithms contains over 3400 entries indicates its volume.
An expanded version of this bibliography is published as a book [2]
with the references in a data base on a disk. Four IEEE Press reprint
books contain papers on the FFT [3,4,5,6].