Posted on 01/20/2011 7:35:04 AM PST by decimon
“So, what are the practical, real world applications/uses of being able to determine the number of partitions of large number?”
It gives us ADD kids another way to count numbers and now I don’t have to remember up to 65536 as I double numbers from 2.
Doesn’t everybody know this?
The Laplace transform is used frequently in engineering and physics; the output of a linear time invariant system can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication; the latter being easier to solve because of its algebraic form. For more information, see control theory.
The Laplace transform can also be used to solve differential equations and is used extensively in electrical engineering. The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra. The original differential equation can then be solved by applying the inverse Laplace transform. The English electrical engineer Oliver Heaviside first proposed a similar scheme, although without using the Laplace transform; and the resulting operational calculus is credited as the Heaviside calculus.
wikipedia.com
As a side note, this is why I have always liked the English based system of measurement, because the number 12 (1 foot) has several equal partitions (i.e. 6+6, 4+4+4, 3+3+3+3, 2+2+2+2+2+2), the yard (divisible in inches to an integer by 1,2,3,4,6,9,12,18) and the mile (divisible in feet by every number from 1 to 12 except for 7 and 9).
If you need to, you can multiply feet by inches to give you even more divisible numbers for greater flexibility. That also equals out to fractions the brain understands, like 1/2, 1/4, 1/8, 1/3, 1/6, 1/9 etc. It’s so much easier for me anyway to think in those terms v. the crappy decimal system, which I despise.
I think someone asked a similar question to Einstein when he came up with his Theory of Relativity.
Maybe there is a pattern to the number of partitins of prime numbers, which would help in the search for larger prime numers?
He probably has to beat the babes off with a stick.
I love partitions and rarely post, so please forgive my long explanation of this article.
What is a partition? As already explained by previous posters, given any positive whole number n=1,2,3,... it is possible to write it as the sum of smaller positive whole numbers in different ways. This is called “partitioning” the number. For example, there are 3 ways to partition n=3:
3 = 2+1 = 1+1+1.
Note that the number n is counted as a trivial partition of itself, so 3 is a partition of 3; and the order of the summands is neglected in calculating partitions, so 2+1=1+2 are not considered different partitions.
The number of different partitions of n is called p(n).
The first few values of p(n) and the corresponding partitions it counts are
p(1) = 1, counts 1
p(2) = 2, counts 2 = 1+1
p(3) = 3, counts 3 = 2+1 = 1+1+1
p(4) = 5, counts 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1
p(5) = 7, counts 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
and so forth...
Unfortunately the number of partitions p(n) gets unwieldly large very quickly.
P(1) = 1
...
p(100) = 190,569,292
...
p(1000) = 24,061,467,864,032,622,473,692,149,727,991 (Wikipedia)
...
Around the time of WWI Hardy and Ramanujan found an asymptotic approximation for p(n), but they failed to find an exact formula. In the 1930s Rademacher did find an exact formula for p(n), but it involved a complicated infinite series of irrational numbers that only converged to p(n). Ken Ono claims that he has found a *finite* formula for p(n). If so, in the words of VP Biden, that’s a Big F’ing Deal.
Finding an exact result for p(n) is so hard that mathematicians have been happy merely to find certain general patterns true about p(n). For example, Ramanujan found some division “congruences” in p(n) for special values of n:
For any k=0,1,2,...
p(5*k+4) is always divisible by 5.
p(7*k+5) is always divisible by 7.
p(11*k+6) is always divisible by 11.
For example, in the simplest case, p(4)=5 and p(5+4)=p(9)=30, and both are divisible by 5.
The values 5,7,11 turn out to be very special: It can be shown that there is no similar divisibility congruence statement of the form
p(A*k+B) is always divisible by A.
for any prime number A other than A=5,7, or 11. This is the content of Ramanujan’s “mysterious quote”.
In the past, Ono has proved there are more complicated divisibility congruences for every prime number, and later he showed that there are congruences for every number not divisible by both 2 and 3. So Ono generally knows what he is talking about when it comes to partitions. For example, one of his complicated congruences is
p(4063467631*k + 30064597) is always divisible by 31.
The article says that Ono has found a systematic way of finding/explaining these divisibility congruences.
The one aspect of this article that I consider to be slightly incorrect is saying that the “partition numbers are fractal”. I know that Ono means he has a recursive proscription for the partitions, but in my opinion the term “fractal” (used to describe geometric objects of fractional dimension) is not exactly right here.
So, what are the practical, real world applications/uses of being able to determine the number of partitions of large number?
Although advances in number theory generally won’t help us do things like build a better door hinge, partitions are fundamental to many problems in combinatorics. The “practical” applications that may benefit from advances in our understanding of partitions are things like algorithm theory in computer science, elliptic curve cryptography, solutions to certain types of problems in statistical physics such as the Ising Model, and Feynman diagram expansions in quantum field theory.
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How many ways can those tally marks be grouped or partitioned, keeping the total number at ten?
He probably has to beat the babes off with a stick.
solitinic's cast-offs. ;-)
Wow. I just had a deja vu. I felt like I was reliving a high school moment. By the third paragraph I could barely stay awake.
He does it with fractals. Nature is a series of repeating patterns that appear even in things that appear to be random.
What these guys did was tie yet something else that seemed to spin off into infinity into something that could be understood through a pattern that repeats.
The golden ratio is something you’ll be able to ‘hook into’ mentally. You’ll find this ratio in everything from how a sunflower’s seeds sit in the center of the flower to how a nautilus shell’s growth from it’s center has the same proportions.
It appears this legendary East Indian mathmatician understood this, but hadn’t explained it fully on paper. Tesla was notorious for this failing. What these guys might find is that the moduli consisting of these three primes - 5, 7, and 11 are at the heart of the pattern.
What fascinates me is that we have a term for numbers called ‘the partition’ and that we’ve discovered a significance for it in nature and science. I love that.
If you are not doing math at 8PM on Friday night you do not have what it takes to be a great mathematician. Of course that probably means you are a normal human being.
If you’re not sleeping wioth your calculator, you’re not a mathematician.
I wasn't asking mockingly. I really was curious.
My understanding is that if someone were to discover a quick and easy method to determine the prime factors of very large numbers, our current methods of public-key cryptography would become obsolete. I wanted to know if this discovery had similar implications.
the Pythagorean says, "all is number", I say...
- What about letters?
- Must have been a big number that you smoked.
- How many finger am I holding up?
- What's the number for 9-1-1?
- Other(please specify)
Though tests said I should be highly otherwise, my performance was not great in math.
Would you be willing to relate, in simple terms, perhaps graphically??? the golden ration with the 5, 7, 11 thing you were mentioning?
I'm going to begin an immediate study of flocks of demoncrats...(Or is that "Flocks of flockers"?)
I'm glad I'm not the only one to ask that question. Somehow I doubt it has any real world use.
Brilliant!
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