Mathematical Moments, Memories and Prizes - Page 2

mfgoode · 2006-05-18, 03:51

Jean Baptiste Fourier - Born 21st of March 1768 in Auxerre, France
Died 16th of May 1830 in Paris, France

Every periodic function can be expressed as an infinite sum of sine and
cosine functions. This probably is Fourier's most famous result and his
name remains attached to these infinite sums. Fourier analysis, the
process
of splitting a function into its constituent sine and cosine parts, is
invaluable when it comes to analysing anything that travels in waves.
Music, speech and image analysis, as well as the compression of sounds
and
images, and analysis of weather and seismic data are just a few
examples of
the applications of Fourier analysis. It has particularly come into its
own
since the rise of computers, because our digital and finitely-minded
computers can only understand complicated wave forms - like those
coming
from music - if they are approximated by their constituent sine and
cosine
waves.
For more from Plus magazine.

http://us.f521.mail.yahoo.com/ym/Sho...ead=&box=Inbox

Mally

drew · 2006-05-19, 06:21

Quote:

Originally Posted by mfgoode

Jean Baptiste Fourier - Born 21st of March 1768 in Auxerre, France
Died 16th of May 1830 in Paris, France

Every periodic function can be expressed as an infinite sum of sine and
cosine functions. This probably is Fourier's most famous result and his
name remains attached to these infinite sums. Fourier analysis, the
process
of splitting a function into its constituent sine and cosine parts, is
invaluable when it comes to analysing anything that travels in waves.
Music, speech and image analysis, as well as the compression of sounds
and
images, and analysis of weather and seismic data are just a few
examples of
the applications of Fourier analysis. It has particularly come into its
own
since the rise of computers, because our digital and finitely-minded
computers can only understand complicated wave forms - like those
coming
from music - if they are approximated by their constituent sine and
cosine
waves.
For more from Plus magazine.

http://us.f521.mail.yahoo.com/ym/Sho...ead=&box=Inbox

Mally

...and don't forget its utility in convolving the binary representations of very large integers (mod 2^(P-1)).

patrik · 2006-05-21, 21:31

There is a story about this in the local newspaper today, Upsala Nya Tidning.

mfgoode · 2006-05-22, 07:22

[QUOTE=Uncwilly]

Proxy-firewall

I just clicked on that website and found no problem. Its as clear as day.
Maybe it was your off day. Why dont you give it a try again.
Mally

davar55 · 2006-05-22, 10:07

Congrats Lennart Carleson! You deserve the Abel Prize (~1 $million).

I remember the elegance of the proof that every square integrable
(hence every continuous) function equals the sum of its Fourier Series
except at a "small" set of points (finite,nowhere dense,or measure zero?).

Very important result!!!!! Thank you, sir!

-- davar55 (David Yablon).

jasong · 2006-05-24, 23:35

Please excuse my ignorance, but if this stuff was originally made to deal with waves, how does it connect to determining primality?

Please keep in mind, if you try to explain, that while I'm fairly intelligent, my education was cut short near the beginning of my 11th grade school year(by illness)

philmoore · 2006-05-25, 00:18

Quote:

Originally Posted by jasong

Please excuse my ignorance, but if this stuff was originally made to deal with waves, how does it connect to determining primality?

FFT stands for Fast Fourier Transform. Google on "FFT multiplication" for some good descriptions of how FFTs are used for multiplying large numbers.

dsouza123 · 2006-05-25, 00:46

Normally to do a Lucas-Lehmer test the following is done

S (1) = 4
S (2) = (4 * 4 – 2) mod 127 = 14
S (3) = (14 * 14 – 2) mod 127 = 67
S (4) = (67 * 67 – 2) mod 127 = 42
S (5) = (42 * 42 – 2) mod 127 = 111
S (6) = (111 * 111 – 2) mod 127 = 0

With 2^P - 1 being 2^7 - 1 = 127 which is why the steps are mod 127.
Mod 127 means integer division by 127
with a remainder in the range 0 to 126.

Using DWT a form of FFT (Fast Fourier Transforms) instead of multiplying
111*111 then modding (dividing) by 127 for the remainder,
it gets both done at once using considerably less calculations and time.
I ignored the - 2.

For large numbers in the range that are being tested by Prime95
for example 2^37654321 - 1 is a number that is 4706791 bytes long.
Squaring a number of that size then modding it by 2^37654321 - 1
by the math operations of multiplication and integer division
for the remainder, requires alot more calculations then doing a DWT.

2006-05-18, 03:51	#12
mfgoode Bronze Medalist Jan 2004 Mumbai,India 2²×3³×19 Posts	Mathematical moments Jean Baptiste Fourier - Born 21st of March 1768 in Auxerre, France Died 16th of May 1830 in Paris, France Every periodic function can be expressed as an infinite sum of sine and cosine functions. This probably is Fourier's most famous result and his name remains attached to these infinite sums. Fourier analysis, the process of splitting a function into its constituent sine and cosine parts, is invaluable when it comes to analysing anything that travels in waves. Music, speech and image analysis, as well as the compression of sounds and images, and analysis of weather and seismic data are just a few examples of the applications of Fourier analysis. It has particularly come into its own since the rise of computers, because our digital and finitely-minded computers can only understand complicated wave forms - like those coming from music - if they are approximated by their constituent sine and cosine waves. For more from Plus magazine. http://us.f521.mail.yahoo.com/ym/Sho...ead=&box=Inbox Mally

2006-05-21, 21:31	#14
patrik "Patrik Johansson" Aug 2002 Uppsala, Sweden 5²×17 Posts	There is a story about this in the local newspaper today, Upsala Nya Tidning. Last fiddled with by patrik on 2006-05-21 at 21:32

2006-05-22, 07:22	#15
mfgoode Bronze Medalist Jan 2004 Mumbai,India 2²·3³·19 Posts	firewall. [QUOTE=Uncwilly] Proxy-firewall I just clicked on that website and found no problem. Its as clear as day. Maybe it was your off day. Why dont you give it a try again. Mally

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2006-05-22, 10:07	#16
davar55 May 2004 New York City 1000010001011₂ Posts	Congrats Lennart Carleson! You deserve the Abel Prize (~1 $million). I remember the elegance of the proof that every square integrable (hence every continuous) function equals the sum of its Fourier Series except at a "small" set of points (finite,nowhere dense,or measure zero?). Very important result!!!!! Thank you, sir! -- davar55 (David Yablon).

2006-05-24, 23:35	#17
jasong "Jason Goatcher" Mar 2005 6663₈ Posts	Please excuse my ignorance, but if this stuff was originally made to deal with waves, how does it connect to determining primality? Please keep in mind, if you try to explain, that while I'm fairly intelligent, my education was cut short near the beginning of my 11th grade school year(by illness)

2006-05-25, 00:46	#19
dsouza123 Sep 2002 296₁₆ Posts	Normally to do a Lucas-Lehmer test the following is done S (1) = 4 S (2) = (4 * 4 – 2) mod 127 = 14 S (3) = (14 * 14 – 2) mod 127 = 67 S (4) = (67 * 67 – 2) mod 127 = 42 S (5) = (42 * 42 – 2) mod 127 = 111 S (6) = (111 * 111 – 2) mod 127 = 0 With 2^P - 1 being 2^7 - 1 = 127 which is why the steps are mod 127. Mod 127 means integer division by 127 with a remainder in the range 0 to 126. Using DWT a form of FFT (Fast Fourier Transforms) instead of multiplying 111*111 then modding (dividing) by 127 for the remainder, it gets both done at once using considerably less calculations and time. I ignored the - 2. For large numbers in the range that are being tested by Prime95 for example 2^37654321 - 1 is a number that is 4706791 bytes long. Squaring a number of that size then modding it by 2^37654321 - 1 by the math operations of multiplication and integer division for the remainder, requires alot more calculations then doing a DWT.