(via macmankev, scrumptiouspie)
Here’s a quote…
Background
Many models used in theoretical ecology, or mathematical epidemiology are stochastic, and may also be spatially-explicit. Techniques from quantum field theory have been used before in reaction-diffusion systems, principally to investigate their critical behavior. Here we argue that they make many calculations easier and are a possible starting point for new approximations.
Methodology
We review the many-body field formalism for Markov processes and illustrate how to apply it to a ‘Brownian bug’ population model, and to an epidemic model. We show how the master equation and the moment hierarchy can both be written in particularly compact forms. The introduction of functional methods allows the systematic computation of the effective action, which gives the dynamics of mean quantities. We obtain the 1-loop approximation to the effective action for general (space-) translation invariant systems, and thus approximations to the non-equilibrium dynamics of the mean fields.
Conclusions
The master equations for spatial stochastic systems normally take a neater form in the many-body field formalism. One can write down the dynamics for generating functional of physically-relevant moments, equivalent to the whole moment hierarchy. The 1-loop dynamics of the mean fields are the same as those of a particular moment-closure.
(via cromagnon)
New Scientist has an article titled “Ditching binary will make quantum computers more powerful” which talks about using Base-5 numbers for Quantum Computers, rather than the Base-2 numbers used in our Turning Machine Computers of today.
Why do our computers of today use base-2 numbers? Remembering back to my days in University, I remember asking one of my professors why base-2 was used instead of base-3, base-5, base-10, or something else. From what I remember, it was due to problems with creating a stable circuit with more than 2 states. Apparently it was extremely difficult thing to do. (At least that’s what I remember the prof telling me. I never tried verifying it.)
From a mathematics point-of-view, the more (unique) prime numbers they can get into the base, the better. Because then there’s less numbers that become repeating decimals when you divide. So it would be better to have things like a 6-state, or a 30-state, or a 210-state, etc. Of course, you’d need Quantum Computing technology to advance to the level were that many states could be supported. At this point in time AFAIK base-5 is the highest level that anyone has figured out how to do with quantum computing technology.
One who studies math beyond what is found in high school often eventually learns, or figures out on their own, that there are many different kinds of infinity.
When you look at the cardinality of the set of all counting numbers[1] you find out that it is one kind of infinity. This kind of infinity is in fact the smallest infinity. (Well, the smallest that I really know of anyways.[2]) It is often called Aleph-Zero, Aleph-Nought, or Aleph-Null. And is often written as ℵ0
We also find that the cardinality of many many other sets are also ℵ0. Such as the set of all Integers, the set of all Rational numbers, and the set of all Prime numbers. (That’s not meant to be an exhastive list.)
But there are of kinds of infinity. (Actually, there are an infinite number of infinities.)
Another kind of infinity is the cardinality of the set of all Real numbers. We often denote this with a fancy letter “C”. Something like: C
When I was in University I would often purchase books for higher level classes (before I took them). I once bought a book on basic abstract algebra and while reading I learned something very interesting. If found that…
2ℵ0 = C
I found such an equation both fascinating and beautiful at the same time.[3] That these two different kinds of infinity could be related in this way.
At this point I should point out that (in addition to ℵ0), people often talk about the other infinities: ℵ1, ℵ2, ℵ3,…, ℵa,…. Each of these being a different kinds of infinity from each other, such that ℵ0 < ℵ1 < ℵ2 < ℵ3 < … < ℵa < … (That’s not meant to be a definition of what these ℵx are. It’s just to show some properties of them. Keep reading and it will make more sense why they are named this way.)
The interesting thing is that…
C = ℵ1
Which means…
2ℵ0 = ℵ1
In fact, the greater pattern is…
2ℵx = ℵ(x+1)
Why does this comes from?! Well, like with all things in math, it can be derived in many many different ways. But I came across this when dealing with Power Sets. The Power Set of a set is the set of all subsets of that set. If we denote our set as S, then the power set is often denoted as: P(S)
For example, P({1,2,3}) = { {}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}}
The thing to note is that if we denote the cardinality of S as b, then the cardinality of P(S) is 2b . [4]
[1] The “counting numbers” are traditionally called the Natural Numbers and are sometimes defined as the set of positive integers — {1,2,3,…} — and sometimes defined as the set of non-negative integers — {0,1,2,3,…}. The former being the more traditional definition. All “good” computer scientists and software engineers preferring the later, of course :-)
[2] I’ve wondered if new types of infinities could be derived from computational complexity theory.
[3] Although not as beautiful as the Euler’s identity.
[4] And thus the cardinality of ℵx+1 equals the cardinality of P(ℵx)
