Life goes on and I eventually get a new job, though I neglected and later lost another precious friendship along the way.
- How I enjoy music
I reworked a page about the moving around I do when enjoying a band (that some folk consider to be dancing).
- Serotonin failure
I expanded my account of my experiences with a neurochemical's regulatory mechanism failing. This prompted me to gather it and a few other pages from elsewhere into an autobiographic branch of this web-site.
- Generalising conjugation
A look at how the image of {naturals} in any ringlet equips us to recognise how much of the ringlet is
real
and a sketch of how to then study the rest of the ringlet in terms of that, to generalise the notion of conjugation that arises for complex and quaternion.- Prove numerals work
I had specified how they should work, so it was time to show the specification did what we expect of it.
- Platonic solids
Showed that there can be no more than the ones classical antiquity knew about.
- Studied Perrin's iterator
Similar enough to Fibonacci's that a similar trick had to work for it…
- Euclid's proof of Pythagoras
I'd read Anathem, by Neal Stephenson, so had fun making sense of a proof exhibited there.
- Power and polynomials
Repeated multiplication gives us power; scaling and adding the resulting functions gives us polynomials that let ringlets beget ringlets and rings beget rings.
- Pareto's principle
Used to define a statistic that illuminates a distribution; related this to the thermodynamics of solids.
- Thermodynamic solid
Extended my study of the thermodynamics of a simple solid to include distribution of quanta among modes and the heat capacity.
- Ordering a ringlet
The consequences of arithmetic playing nicely with
less than
andgreater than
as orderings.- Reworked my CV
Minor style fixes, restructuring and rephrasing to bring out the main story from the details.
- Homomorphism
A general treatment of mappings that respect mathematical structures.
- Dimensions of angle
Should angles be treated as dimensionless ?
- Reworked arithmetic
Continued my reworking of arithmetic and notation; reorganised the pages between directories. Contrasted binary operator, bulk action and a translation-like representation as three faces of a common topic. Focussed on the flat and the categoric. Subsequently purged lots of material elsewhere made redundant by the new treatment.
- The colour of black holes
It, along with temperature, depends on size. Looked at the possibility that quantum effects could stop very small black holes from finishing their evaporation.
I watched a tragedy play out and had more bad news heaped on with it, losing a nice job and ending the year severely depressed. None the less, I pressed on with my new treatment of arithmetic.
- Counting
Separated out an important use of the natural numbers from the discussion of what it means to be finite.
- Lists
Mappings from a natural number, the concatenation they support, prefixes and suffixes. The notation for them also got a new home.
- Addition
A look at the things that happen when you can combine values, don't care about their order and can cancel matching parts from the sides of an equation. This ends up simplifying the treatment of ringlets, among other things. It also opens the door to modules over ringlets (and thus vector spaces over a field).
I got a new job, turned 50, got head-hunted into another new job and ran into some appalingly bad news. Yet somehow I managed to begin a major re-working of how I treat arithmetic, though I did slow down.
- Pentagonal construction
The details of constructing a pentagon and pentagram.
- Ratios
Inducing ratios of whole numbers by reversing repetition.
- Ring(let)s and ideals
Abstractions that allow a generalised description of arithmetic, notably proving results about prime factorisation.
- Arithmetic from repetition
I used repetition to specify arithmetic on the natural numbers.
- Reworked multi-angle formulae
I found a better way to express sin and cos of multiples of an angle in terms of those of the base angle.
The end of a job left
me with lots of free time on my hands, so I changed quite a lot. Then I got
distracted by Illyriad.
- The Catenary
The shape of a chain hanging freely between two fixed end-points.
- The median and its kin
Revised my treatment of splitting a distribution at values which sub-divide probability evenly.
- A chart for all seasons
Using the chart notion from smooth manifolds to take a different look at how we describe the year's cycle in terms of seasons.
- Open-sourced my study package
It's now available on github for all to play with, under the FSF's GPL.
- Eigenvalues and eigenvectors
A way of analysing linear maps to get a clearer understanding of how they behave.
- Quadratic and Hermitian forms
The generalisations of Pythagorean sums of squares.
- Vector statistics
How to extract information from statistical data with more than one variable being discussed and possibly messy correlations among them.
- Space without rocketry
I expanded and elaborated my older discussion of ways to get out of a gravity-well with less violence than the orthodox solution.
- XHTML conversions
Lots of pages are now in XHTML, mostly so that I can use custom character entities, instead of plain HTML.
- Coin toss distributions
Illustrated Bayesian reasoning with tossing a coin whose bias is unknown.
- Stating the obvious
A few things that are obvious yet get treated as noteworthy.
- An effect of culture
A look at how attempts to do social science run into the impossibility of truly isolated systems, when some components are social animals.
- Binary operators
A major over-haul of how I describe binary operators – and thus arithmetic.
- Complex numbers
Clarified explanation of the algebraic completion of the real continuum. Needed for the Hilbert space formalism of quantum mechanics.
- Grouping statistics
How to divide up the range of a sample of data to get a clear idea of how the data are distributed.
- Special Relativity
Split it out from analysis of the Lorenz transformation, made both more coherent. Also reworked constant acceleration.
- Sub-repositories in git
An ill-informed exploration of how to do something like sub-modules in the version control system I use.
- Refactored notation
Carved up the specification of my notation into several files. Reverted a foolish conflation of end-relations into restrictions.
- Constructibility
An attempt at characterisng a core of entities I expect any foundation to allow my abstraction over foundations to work with.
- Natural Numbers
A reworked account of the numbers we use for counting, along with the positional numeral notation we use to represent them.
The beginings of a re-working of the foundations of arithmetic; and a lot of digressions and maintenance.
- Chebyshev's bound
The probability of a random variate being more than u standard deviations from its mean is never more than 1/u/u.
- Finiteness
Turned the pigeonhole principle into a partial ordering on collections and used this to characterise finiteness.
- DnD caricature of me as a project
Made an on-going project of caricaturing me using DnD and gave some thought to what spells this imaginary creature would learn.
- Conic sections
Studied the shapes one gets when a plane intersects a cone.
- Police riot
Wrote up my first-hand view of a demonstration (in 1994) whose handling by the UK police looked suspiciously like they wanted a riot.
- Position observable
Expanded on an earlier discourse, now applying some of the tools I've built up on other pages.
- The tensor product
Began another attempt at describing this multiplication on vector quantities.
- Optimising on a manifold
A look at interpreting Lagrange/Hamilton optimisation techniques on a smooth manifold, where position isn't a vector quantity.
- Misner, Thorne & Wheeler
Started an area for notes on a big fat book I've made too little progress on reading,
Gravitation
(one of the definitive text books on the topic).- Remedial mathematics
Noticed a project I should attempt – explaining how some topics in mathematics are commonly taught in confusing ways that make them more confusing than they need to be – but have actually made little progress with it since.
- Leibniz Operators
Continuation of earlier work on operators, on a smooth manifold's tensor bundles, obeying the product rule; this entailed reworking of the trace operator, explaining the Lie Bracket and proving some things about permutations.
