Category Archives: General Mathematics

Ever wondered how to calculate a given number?

How you ever wondered what is special about your favourite number? As we all know, the answer to the ultimate question of life, the universe and everything is \(42 = 2100/50 \approx sin(\pi 3/26)-sin(\pi 11/39)\) and can also be approximated very well by many other expressions.

By using the The Inverse Symbolic Calculator (ISC) you can take numerology to another level by finding closed expressions that well approximate the number (any truncated decimal) you think has some special meaning. It used a mixture of lookup tables and integer relation algorithms. The tables were first compiled by S. Plouffe.

Whatever you do please have some fun with it.

Link
The Inverse Symbolic Calculator
Simon Plouffe homepage

Nuclear theory research in the UK to be exanded

Flag A new nuclear theory group is going to be set up at the University of York. The Science and Technology Facilities Council (STFC) will make a special funding award to set up the group and will provide funding to appoint a nuclear physics theory chair and PhD studentships. Furthermore, the university of York will fund a nuclear physics theory lectureship.

The need to expand the UK’s capability in theoretical nuclear physics was part of the Institute of Physics review in October 2012. For sure, although the UK has some good researchers in this field, the numbers of people in theoretical nuclear physics is small. One number that has been suggestion is that there are about seven permanent researchers in the UK working on theoretical nuclear physics.

The establishment of a new group must be welcome news for the UK nuclear physics community.

Link
Gap in nuclear physics research identified by IOP is to be filled.

Assume spherical sheep in vacuum…

sheep Scientists have now used GPS to uncover the rules that describe how sheepdogs are able to herd sheep.

Strömbom et al [1] have shown that there are surprisingly few rules here; in fact they suggest just two rules.

  1. The sheepdog learns how to make the sheep come together in a flock.
  2. Whenever the sheep are in a tightly knit group, the dog pushes them forwards.

The sheepdogs make the most of what is know as “selfish herd theory”, that is the tendency of a given sheep to want to be near the centre of the flock when under threat.

There is a Welsh connection here. One of the authors, Dr. A. King is based at Swansea University, which is where I studied for my undergraduate degree.

Now, anyone know any good jokes about Welsh people and sheep? Can’t say that I have herd many…

References
[1] Strömbom et al, Solving the shepherding problem: heuristics for herding autonomous, interacting agents, J. R. Soc. Interface 11(100) (2014).

Happiness is a long equation

As you can imagine as a mathematician, the bigger and harder the equations the happier I am. Not really, we look for pattens and elegance rather than just difficult equations, though of course difficult equations can be elegant and contain a lot of interesting structure.

Anyway, scientists now have an equation for happiness and here it is

happy
Taken from [1].

Now we just need to apply some calculus to find the maxima (local or global I’m not fussy) and find out just how happy a mathematician can be!

Reference
[1] Robb B. Rutledge, Nikolina Skandali, Peter Dayan, and Raymond J. Dolan, A computational and neural model of momentary subjective well-being, PNAS 2014 : 1407535111v1-201407535.

Link
Equation ‘can predict momentary happiness’ BBC News

More experiments with random walks

I have again been playing with some random walks, using the same method as here. This time I used 1000000 iterations and added some colour.

Below are random walks, on the plane (not a lattice) for which step size gets (on average) smaller and smaller with each step. I pick the step size using the Maxwell-Boltzman distribution (with a =1) and a suitable scaling which depends on the iteration parameter. I the add a opacity depending on how many times the points are visited: bright white means a lot, while grey means not many and black never.

walk

walk

walk

walk

Once again, these images are rather for artistic purposes than scientific purposes.

This is like so random…

Below are random walks on the plane (not a lattice) for which step size gets (on average) smaller and smaller with each step. I pick the step size using the Maxwell-Boltzman distribution (with a =1) and a suitable scaling which depends on the iteration parameter. I the add a opacity depending on how many times the points are visited: bright white means a lot, while grey means not many and black never.

I may play with these further, but they make some interesting pattens. We have approximate self-similarity and so these patterns have fractal-like properties. Anyway, enjoy….

random10

random1

random2

random5

These images were created for artistic rather than scientific reasons. That said, random walks are have been applied to many fields including ecology, economics, psychology, computer science, physics, chemistry, and biology.

Probably the most famous application of a random walk is to Brownian motion, which describes the trajectory of a tiny particle diffusing in a fluid. I have no idea if there is anything scientific in these images, but I would not be surprised if for small step sizes we have approximately Brownian motion. However, I would need to think a lot more about this before making concrete statements.

Turing's Birthday

Turing Everybody reading this post should be aware that today is Alan Turning’s birthday. He was born on the 23th June 1912.

I say everyone reading this should be aware of this fact as Turing is considered the farther of theoretical computer science.

Memorial at Manchester
Turing was involved in the development of the Manchester computers, which were the first series of stored-program electronic computers. They were developed during the 30-year period between 1947 and 1977 by a small team at the University of Manchester.

Near the Sackville street buildings of the University of Manchester there is a memorial. The plaque reads as follows;

turing

I won’t say much about the way he was treated, different times and so different ways of thinking. You can find out more via the link below.

Link
Alan Turing: the enigma