About Me
First a short biography …. so Who am I?
Like the contents page of a book, I present an overview of various series of posts I have published (plus a few in planning).
I hope to develop a few parallel pages, one being about my favourite mathematicians, click here.
Follow the linked title to the first post of any series which I hope will be the starting point for you to enjoy the whole series as it unfolds ….
1. Simple Random Walks
- Simple Random Walks
Introducing the Drunkard’s Walk
A Simple Random Walk in 1-D where solutions to questions about terminating probabilities, expected length of walk are provided. A lot of interesting diversions are made along the way.
… followed by the Gambler’s Ruin
A different interpretation of essentially the same SRW. A standard approach to solutions is used. The two differing solution methods are shown to be identical.
2. Sums of Integer Powers
- Sums of Integer Powers
An introduction to the investigation of the sums of powers of \( \small {n} \) integers and the formulas that have been discovered to enable quick calculation.
Starting with Sums of Powers – a series of 10 posts
… followed by Bernoulli Numbers – a series of 5 posts
… and finishing with a series of 4 posts on the Riemann Zeta Function
3. On Population Growth
- Population Growth
As an Australian I witnessed the aftereffects of releasing a few tame rabbits on a defenceless environment. A literal population explosion, which was repeated with foxes, camels, water buffalo, horses, donkeys, cane toads … poor country.
- Competing Populations
Darwin saw it, so did Wallace (the joint authors of the theory of evolution). Nature is “red in tooth and claw”. Out in the big, wide world species are not benign, they compete for survival so that they can pass on their genes to future generations.
- Spread of Epidemics
The growth and spread of infectious diseases is an example of a living species competing for survival with us as their target. With successful intervention epidemics may die out – smallpox and polio are good examples. But a few still lurk in dark places and remain a scourge around the world today – malaria, dengue fever,
- Population in Decline
In my lifetime the human population of the world has doubled! Yes from 4 billion when I was young to some 8 billion today and still growing. But that growth is slowing. What will happen in the future as we literally “pollute” ourselves to death is a key question here.
4. Arches, Bridges & Chains
- Arches
Perhaps unsurprisingly arches became the first mass-produced support structure that helped us ford rivers and valleys. The Romans were adept at constructing aqueducts 2000+ years ago, some of which still function to this day!
- Bridges
By stretching an arch into a parabola they can and do support many arch bridges like the Sydney Harbour Bridge. Invert them and suspend support from cables give us suspension bridges like the Golden Gate Bridge in San Francisco.
- Chains
If you take away the road deck of a suspension bridge you get a hanging chain. These have a parabolic look about them but are subtly different – hanging in a shape called a catenary. We see them stretch across the skyline as power lines bringing us the electricity which runs our lives.
- and some strange shaped wheels
An oddity that you may be aware of – catenaries arise as the segment shape of tracks for some odd shaped wheeled vehicles as you will see here
5. Bayesian Musings
- Bayes & Probability
A quiet, well-mannered English non-conformist minister asks if it is possible
“to find out a method by which we might judge concerning the probability that an event has to happen, in given circumstances, upon supposition that we know nothing concerning it but that, under the same circumstances, it has happened a certain number of times, and failed a certain other number of times”
The answer lies in probability, especially the conditional.
- Statistics & Inference
A goal of statisticians is to be able to make confident predictions about future happenings based on data collected in the presence. The classic frequentist approach is complemented by a Bayesian approach, with key words – prior, likelihood, marginal, posterior
- Conjugateness
Until the power of electronic computers arrived on the scene Bayesian inference was held back the intractability of calculating difficult integrals for the marginals – solution conjugateness, where the posterior distribution and the posterior combination of likelihood function and prior distribution have similar structure so avoiding a sometimes impossible integral that ultimately is constant.
- Markov Chains & Monte Carlo Processes
Computing power was the key to overcoming the integration difficulties mentioned. Starting at Los Alamos, where roomfuls of predominantly women enabled the success of the Manhattan project, through to modern electronic computing machines that enable tens and hundreds of thousands of calculations to be performed in realistic time posterior distributions can be accurately approximated to allow Bayesian inference to blossom as it has today. I credit George Slade for the image.
6. Cosmic Tales
- Cosmic Visions (is close to being ready)
The Cosmos has always fascinated humans. Gazing skywards during the dark night skies a movie plays out above us. The Milky Way and myriad of stars wheels across the heavens from horizon to horizon. No wonder we imagined “gods” living up there among them. The first explanations were like dreaming to explain the everyday with the supernatural … until on the islands of Greece the first scientific theories began their
- Copernicus,Brahe & Kepler
Three giants of the Renaissance contributed first the key idea of the Solar System with the Sun at the centre, then the data with meticulous pre-telescope recording of the stars and planets, and then the insight to put the pieces together to give the planets their courses through our skies.
- Keplerian Orbits
Kepler overturned the geocentric order of the day to demonstrate the elliptical nature of planetary orbits – a paradigm shift of mega proportion.
- The Expanding Universe & the Solar System
It began as a Bang but will it end in a Whimper?
7. Problems in Basel
- Basel, Switzerland
Why Basel? Just coincidence? Basel was one of the important settlements that blossomed into a thriving city and cultural melting pots in the beginning years of the Renaissance. In the 18th century it was home to some of the great mathematical minds of the time.
- The Basel Problem and solutions
The BP was proposed by the Italian Mengoli and it was 100 years later before Euler, a native Basilian (sorry if I offend), proposed his left field solution. Subsequently there have been many and varied alternative proofs that show equal genius and insight.
- The Zeta Function and its Zeros
Euler generalised the BP and expressed it as what we know as the Zeta function (for positive integer inputs). Surprisingly he linked it to the primes with his Euler Product formula. But it was the genius of Riemann that extended it to its modern form that
- The Riemann Hypothesis and Prime Distribution
Riemann’s 10 page paper has spawned the search for non-trivial zeros that he claims all have real part of ½ and, with a final twist to the story, also have a lot to say about the prime numbers and their distribution.
8. Transforming Mathematics
- The Problem of Heat Propagation
- Fourier Series
- Fourier & Laplace Transforms
9. Shining Light on Maxwell’s Equations
- Electricity and Magnetism
We have known about electricity (rubbing fur on amber) and magnetism (direction pointing lodestones) for a long time. We haven’t know a lot more until the 17th century. Then as the 19th century dawned a series of experimentalists started to get serious. Coulomb, Volta, Ørsted, Ampère, Ohm and Michael Faraday. Now was the time for a mathematical unification.
- Maxwell’s Equations
image from Wikimedia Commons
Richard Feynman once said “From a long view of the history of mankind, seen from, say, ten thousand years from now, there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics.“
The Scot James Clerk Maxwell was the right man in the right place. assembling all information known into 20 equations in 20 unknowns. It took a devoted follower Oliver Heaviside to reduce and simplify Maxwell’s equations into what we recognise today, along the way introducing to the mathematical world Vector Analysis.
- Light Speed
Experimentalists in the 19th century too were honing in on the speed of that wave/particle stuff called light. Behold, Maxwell predicted it all with remarkable accuracy.
- Electro-Magnetic Radiation
There was a whole range of electromagnetic phenomena to be discovered. From the long to the short of it … visible light is obvious (not as being necessarily electromagnetic, it took Maxwell et al. to show this), it is all around. Herz followed this up by discovering radio waves, as useless (!) as he thought “it’s of no use whatsoever … this is just an experiment that proves Maestro Maxwell was right“. Röntgen discovered some mysterious waves next that could penetrate flesh, must be U-rays (U for unknown), no let’s call them X-rays. There were still microwaves, near light IF and UV radiation and then gamma radiation.
My Favourites
Like all students, and even teachers of Mathematics, I have a list of favourite mathematicians. I don’t want to repeat information that can be easily googled but I do want to present some background and personal points of view that I don’t think is included in the average search.
And first up is, who else, but Pythagoras … just click on the title above.
Donations
My goals in establishing this site are twofold …
The first is to delve into all the questions I have accumulated throughout my pedagogical career but did not have the time to explore or develop in full.
And the second is to share my findings with the broader mathematically interested public.
In doing so I have committed hours of labour, albeit enjoyable. And the site itself costs.
So if you have enjoyed and/or benefitted from my style and contents please consider consider a small donation.
p.s. the image in the form is taken on a recent trip I took in 2022, any guesses?
My View on Copyright
As a writer I am concerned about the proper use of the hard work of others. As an educationist I like to draw on ideas from the broadest possible field that requires me to “borrow” from sources more authoritative than I am or just express ideas more succinctly than I do.
Further, I am a lousy illustrator so I use Google a lot to find images that best illustrate what I want to say … “a picture is worth a thousand words” …to enhance the appeal of each piece of work I produce.
Hence, I wonder how copyright affects me. I have consulted several key sources and none put it better than Encyclopaedia Britannica when they state that “you may display, reproduce, print or download content, including images, … only for your personal, non-commercial use“. This blog site is never intended to be commercial so I use images (not from pay for image sites except Adobe Stock for which I have a subscription) from several sites including EB throughout my posts.