My interests
The latest version of the Turing Machine program is dated 2023-10-04
The latest version of Turing Machine analysis is dated 2023-11-30
The latest version of the pdf or tex version of the draft paper relating analytic functions to their behaviours at their singular points is dated 2023-08-30
txt is part of the previous version of the paper on analytic functions probably not needed there and
pdf is the same in pdf format and is dated 2023-02-16
My interests are divided into two categories, my own personal ones (hobbies etc.) listed here and my work-related
interests. These overlap these somewhat and are connected with analysis and presentation of genomics data from
high throughput sequencing of RNA and DNA from plant genetics and breeding experiments. The recent papers in this area in which I have been involved
can be found from the Researchgate link at the bottom.
- These are some of the most inspirational books I have come across. Of particlar note is The Wayfarer that has recently come out of copyright.
- Some time ago I heard about the Kolbrin Bible discovered in England and
I decided to buy one from Amazon.
After searching more on the internet about it I stumbled upon a vast
collection of historical literature related
to the Bible.
Of course there may be a lot here that isn't really worth reading here but it seems that there is a vast amount of
information here from ancient sources that is not well known. What really surprised me was that Britain has many
connections with the Old Testament prophets and with the family of Jesus.
- Here is a famous dialogue promoting faith in God. If you have not seen this before it is a thought-provoking read.
- Somebody who knew that I believe in God tried to catch me out after I said that God created all things. He asked me who created God. To his great surprise I said without too much thought that God did! If God created all things, God created God, and therefore this could be symbolised by something similar to the ouroboros, the snake that appears to be eating its tail, but with the important difference that instead of eating it is creating, which might be represented
as something coming out of its mouth. Could it be that because of this the ouroboros originally represented the ultimate mystery of the divine?
Similar thoughts might have motivated this idea from computer science: a set of instructions which when carried out creates the same program i.e. a self-replicating program.
- Turing Machine
analysis i.e. how to describe the behaviour of small Turing Machines (the simplest lowest
level description of computations) in a higher level "language" giving an overall understanding
of the computation. It gets very complicated for most large examples but there seems to be no
fundamental limit to complexity of a TM that could be analysed. There is no general analysis algorithm
here but some general principles are becoming more clear as larger examples are studied.
Very complex examples are expected to often behave like interpreters of other arbitrary
TM's encoded in the initial state of the "tape". The
first paper describes the basic concepts. I now have a second paper
taking this much further to examples that are too complex for explicit formulae
for the important computation rules that define the machine behaviour.
See my latest study still under development pdf or tex of an even larger 5 state,
5 symbol random Turing Machine (TM) whose analysis is helping me with more efficient methods for extracting
from a TM the information describing its higher level behaviour. I recently separated out from this a lot of results and newer results
that may or may not be needed in the final version. A tentative version (beta) of a new algorithm that generates
the complete triplets of the IRR (i.e. including the origins) has just been obtained. Some tidying up of the code is needed e.g.
, deleting commented out statements etc. , updating fully all the comments, removing redundant code, and final checking of the results.
I plan to re-visit a TM called a 'counter' because unlike most examples I studied, the number of TM steps involved in its short-cut computation rules
increases exponentially with the number of symbols in the rule.
- The book A=B amazing for the simplicity of its ideas and the far-reaching consequences of its results. For example it shows how many mathematical identities can be derived mechanically, including the simplification of definite and indefinite sums of hypergeometric terms. I found a simpler way derive some of its results. [pdf, LaTeX]
- Some of my publications in the area of mathematical physics. I was particularly interested in extending calculus to spaces of functions, extending the calculus of variations. I hoped to thereby to extend the set of one dimensional models in classical statistical mechanics for which there are exact finite dimensional equations for the
thermodynamic properties and structure factors, but I convinced myself that there are no more solutions than those found by what looked like a clever trick (R.J.Baxter 1964-5).
However the beautiful mathematical ideas here look to me likely to find many other uses.
- Functionals of higher order
- A recent paper of mine on Algebraic functions. Again this seems to be a new approach to an old field of mathematics that appears to open up many possibilities for mathematical discoveries.
- A work-related paper on statistical methods I was recently working on.
Continuing this work resulted in a paper describing how to optimally select
a subset of instances (that each have a data set of the same type) most likely to be "interesting data" for further analysis if
each instance in the entire set of data follows one of two models say "interesting data" and uninteresting or "null" data
and it is not known which is which, the only evidence being in the data itself (that is subject to uncertainty e.g. with random error).
This method is likely to be most useful when the number of instances in the entire data set is very large and each instance is not
very informative for example if a single instance has noisy or only few data values.
The separate instances are assumed to be independent, but the method could be combined with other methods that take dependence into account.
- Related to this is a method for estimating probability densities from data such that the integrated mean square error (IMSE) is minimised. This is not yet complete and probably has errors but it is a radically new approach that seems to promise a theoretically optimal solution to this old problem. Some potential uses are discussed.
- A few ideas about some other areas of mathematics that I want to explore further.
A programming language Yacas which could encode a lot of our knowledge of mathematics, for example how to solve certian types of differential equations, evaluate sums and integrals etc. This has been implemented as is free to download. The possibilities seem virtually limitless, it just needs mathematicians to learn the language and code it all!
The more I read the more I like it. Being open source, I think it will develop to the point where it has no equal. Already there is an impressive collection of algorithms implemented, and documention.
Unfortunately I was unable to upload an algorithm I developed because some of the tests failed because of other code that conflicted with mine.
I haven't used it for quite a few years now.
- Some fun with interactive dynamic graphics creating a Ball game. To appreciate this you will need the free Flash player.
If you want to develop it further, here is the source file. There are no resctrictions on its use and no warranty.
- Another little application for Flash designed for visualising earthquake data.