Maths at Bondi Beach
Mathematics without the mystery!
Mathematics without the mystery!
9 July 2026
Some time ago I produced a paper which was intended to go through how the Laplacian figures in mathematics and physics with detailed derivations. I left out two things: one was the full tensor approach which gives you the Laplacian in a few lines after you have built the whole tensor edifice. I gave hints of that approach but never went down that track simply because I did not want to make an already large paper even larger with the abstractions of tensors when I suspect most readers will prefer the more concrete approaches. The second thing I omitted to do was to explain how rotation matrices can be used to derive the gradient and hence the Laplacian in cylindrical and spherical coordinates. Now if you could simply use 3 dimensional rotation matrices to painlessly knock out the Laplacian in spherical coordinates, for instance, the observable universe would be full of such proofs. But it isn't. Keep that in mind.
Fábio M. S. Lima and Pedro G. F. Jordão did a paper for the Mathematics Magazine explaining how to get the gradient in cylindrical and spherical coordinates using rotation matrices and I have used it to go through the logic of the process and the detailed calculations. Readers can assess whether that approach gives them a better feel for the Laplacian which really is a creature of averaging, rotational invariance and spherical symmetry. I remain to be convinced that the rotation matrix approach the authors develop gives any greater insights than the other methods but some people may prefer it. The self-contained rotation matrix paper is here and I have added a link to it at the end of the bigger paper dealing with other techniques here
26 May 2026
I recently published an article on how ChatGPT as it is employed in Mathematica's Notebook Assistant could not without much effort produce Mathematica code for a discrete Fast Fourier Transform: https://gotohaggstrom.com/Mathematicas%20Notebook%20Assistant%20(ChatGPT).pdf
I actually thought before the experiment that ChatGPT could produce the code simply because the concept is so ubiquitous and would have been in the original training.
I was wrong and a more recent test on something so preposterously banal makes me even more worried about the reliability of these models.
The article can be downloaded here
12 April 2026
Using Large Language Models (LLMs) such as ChatGPT and Claude to do serious mathematics gives rise to all sorts of problems. On the one hand these models can provide some astonishing insights even though they actually do not perform logical deductions like a theorem prover. Yet they can also generate some preposterous results. You have to be extremely vigilant with these models and if you let your guard down you can be caught out.
Being a long time Mathematica user I was interested in seeing how Notebook Assistant (NA) worked within Mathematica. It is essentially ChatGPT and is able to produce Mathematica code within the Mathematica environment. I tested it out on what I thought was a problem it could nail simply because the issue I put to it was so ubiquitous it would have been in the original training set and is referred to in many contexts. What I asked Notebook Assistant to do is to generate Mathematica code for a discrete Fast Fourier Transform to multiply two large numbers. Well it demonstrated that it was not up to it and it took a lot of work to get it to understand some Mathematica code that I knew worked. I then tested Claude and Deepseek on the same problem.
All the responses are set out and it is interesting to note that the free version of Deepseek actually produced an astonishing 82 page stream of consciousness.
While you can get some useful work out of these models you actually have to be relatively sophisticated to be aware of their limitations. It is a bit like giving a loaded AK 47 to a young child.
The full story can be found here
05 January 2026
Carlson’s inequality is as follows: if the a_k are positive real numbers not all zero then
He proved it by rather intricate methods in 1935 and then shortly after Hardy came along with a truly gobsmacking proof of a few lines that can only be described as magical. I have gone through how Carlson proved the inequality (and there is a continuous version as well) and then compare it with Hardy’s proof. Having done the hard work of getting into Carlson’s mindset I have to say that Hardy’s proof is just immediately convincing while Carlson’s approach is so intricate it is easy to get lost.
To understand Carlson’s approach you will need to know something about the Cauchy-Schwarz and Holder inequalities as well as harmonic/subharmonic theory, Fourier theory and Mittag-Leffler expansions. You have been warned! The paper is here
04 October 2025
About 15 years ago I published 4 articles on induction along with problem and solution articles. These are still available here. It appears no-one reads the 4 theoretical articles so I have put them together in one pdf file. There is an enormous amount of intellectual capital in those articles and I am surprised that no-one reads them. Actually I was surprised until I realised that I had pitched it too high for secondary teachers and their students and I moved the website content to more undergraduate level material. The combined pdf can be accessed here
24 September 2025
I have produced a lengthy paper titled “The basics of Tauberian theory”. This paper is really only of interest to hard core analysis students probably at the Cambridge Tripos level.
This paper deals with the history of Tauberian methods in analysis in the context of Abelian and Cesaro methods of summation. Hardy’s book on “Divergent Series” is used as the foundation of this history. Tauber’s original proof of his first theorem is given based on a translation of his paper. This is followed by an undergraduate analysis course style of proof. Hardy’s “epsilon free” proof of Tauber’s first theorem is explained in detail and its relationship to Laplace transform theory is fleshed out to lay the foundation for the use of Laplace transforms and Tauberian techniques to obtain an asymptotic limit for the Renyi Parking Problem. This process unites analysis and Laplace transform theory with probability theory. The original papers of Tauber, Renyi and Karamata are included in the Appendix. I give a heuristic explanation for Karamata's 1931 simplification of the earlier Hardy-Littlewood Tauberian theorem. I sent the Rigour Police on a long boozy holiday for the purposes of that explanation.
The paper can be accessed here
It is a lengthy and intricate paper so there will be typos and perhaps infelicities of expression so I will not be offended if you point them out.
02 July 2025
This website is devoted to things mathematical however I felt I had to say something about a rather absurd Australian government policy on the taxation of superannuation which involves some basic mathematical reasoning. In essence the government wants to raise additional revenue from people with large superannuation asset balances. “Large” is implicitly defined as AUD 3 million but there are about 80,000 people currently with assets of at least AUD 3 million. There is a handful of people with balances in excess of AUD 20 million.
Australia’s superannuation industry is large by international standards essentially because of its compulsory nature. Currently the industry has an asset base of about AUD 4 trillion. This asset base represents millions of people with assets of much less than AUD 3 million.
The government wants to apply an extra tax effectively on assets greater than AUD 3 million even though those assets have not been realised. In terms of tax theory it is offensive to tax someone on an unrealised gain since they don’t have the cash to pay the tax and critics have obviously locked on to that issue. What is utterly bizarre is that no Australian government has ever actually legislated a global maximum for what you can have in superannuation. In my asset management days in the early 2000s I was asked about a proposed transfer of AUD 1 billion dollars into a self-managed superannuation fund. That transaction did not eventuate but it was technically lawful and possible at the time.
This policy is bizarre for another reason that goes to the analytical competence of the economists in Treasury. There is a simple way to impose extra tax on those with balances defined to be offensive from a policy point of view and it just involves a non-linear tax rate that applies to taxable income which itself reflects realised gains so that there can be no criticism that unrealised gains will be taxed. This method has other policy benefits that I would have thought the government would want to pursue. I am puzzled as to why the “bleeding obvious” was not done and in the paper I offer 3 hypotheses. Because I don’t have access to the data sets that Treasury has I cannot model how much revenue the proposed policy will generate in comparison to my proposal so I am unable to determine whether this policy is just a cynical and inefficient revenue raising exercise or Treasury never thought of my simpler method. Interestingly self-interested industry commentators have not suggested my alternative which might mean that even though some have actuarial training they are stuck in some sort of intellectual black hole.
What is even more fascinating is that when I asked ChatGPT o3 mini to solve the abstract problem it came up with the formula I worked out. If I were a Treasury economist that would worry me.
In any event the detailed paper is here
23 May 2025
I have updated my “Fooling juries with statistics” article to add in how ChatGPT o3 explained the coin tossing problem. The problem was this: you toss an unbiased coin many times and count the number of tosses until you get the pattern ”HTH”. You then calculate the average. You repeat the experiment but this time you look for the pattern ”HTT”. The question is: ”Is the average number of trials required to get ”HTH” greater than, less than or equal to the average number of trials required to get ”HTT”?
ChatGPT o3 first started to write some Python code to work out the expected value but realised that it got the wrong answer and self-corrected. This happened dynamically and I didn’t get a screen shot. It then used Markov chain methods to get the correct result and gratuitously offered a very clear high level explanation of why the expected number of throws for “HTH” is 10 while that for “HTT” is 8. My paper was based on generating function methods rather than Markov chains but the ChatGPT o3 solution is correct and is appropriate for the problem. ChatGPT o3’s performance was impressive.
Download the updated article here
13 April 2025
Laplace transforms can be used to evaluate integrals in ways which might offer some simplification. They can also be used to perform abstract approximations of integrals and Renyi’s parking problem is an example of that type of use. The following article deals with an integral from the American Mathematical Monthly which is solved using a Laplace Transform and for comparison by another method. Download the article here.
07 January 2025
I am republishing an old article from about 2008 which deals with the question: “How long will my savings last? “ For reasons explained in the article, the humble annuity formula can be used in a practical way to get bounds on this question without probability theory. Download the article here
01 January 2025
Let’s get 2025 off to a rip roaring start with a question that beach goers really want to know - what is the area of Bondi Beach, Sydney, Australia. Because I live about 800 m from the beach and I am an official “beach bum” , I thought I would apply technical rigour to the problem. Now, I have to say at the outset that if you are the sort of person who says “ Duh, just use Google Earth” please leave now. There were cabin boys in the age of Captain Cook who had memorised log tables and knew enough spherical trigonometry that it could advance them up the nautical management tree. Ironically, a lot of these skills have atrophied because of GPS based algorithms that underpin Google Earth and Google Maps. and are inbuilt in phones giving people the “knowledge” but none of the understanding. I have quite intentionally set up this problem this way because at the 1 km level the earth is locally flat (so the dopey Flat Earthers are right locally, but hopelessly wrong globally) and you can effectively use plane geometry with great accuracy. I cover Flat Earth approaches, basic cabin boy spherical geometry, some planimeter algorithms from some JPL/NASA mathematicians and a really funky use of Stokes’ Theorem by academic mathematicians Yuliy Barishnikov and Robert Ghrist who published a paper in the American Mathematical Monthly titled “Stokes's Theorem, Data and Polar Ice Caps “. What was funky was that they used the abstraction of Stokes’ Theorem to derive an approximation formula for a closed polygonal path that was very similar to one of the JPL/NASA planimeter algorithms, However, what I found when I did numerical examples in Mathematica is that the Stokes’ Theorem method required coding rigour even in my simple benchmark case - in fact you wouldn't use it for such cases because you can do a double integral. When you pay attention to detail the method is really cool as it should be for analytical reasons. One sneaky method which is more pedagogically useful than useful generally, is to characterise the area as a “polar” cap and slice off bits. Because there are really simple double integrals for polar caps all we have to do is a rotation to the North Pole and apply a simple formula that arises for the area of a polar cap. I give a summary of the results of the various methods and derivations of the algorithms. Download the paper here
30 August 2024
The geodesic equation is a rite of passage for general relativity. Its derivation can be approached in various ways but it is possible to do a basic “bare hands” derivation which has been done for instance by Alan Guth in his MIT “Early Universe” course which can be found here: ocw.mit.edu/courses/8-286-the-early-universe-fall-2013/resources/lecture-14-the-geodesic-equation/
Guth leaves several important gaps in the derivation (because they would have been covered in tutorials or homework) but a complete development of the derivation is set out in the this paper. Because of all the indices there may be some typos despite several proof readings.
11 June 2024
Order statistics concern random samples of random variables which are arranged in order of increasing magnitude. It is useful to obtain the distributions of the maximum and minimum of samples as well as other more complicated relationships. The basic principles are set out in this paper
12 April 2024
L’Hopital’s rule is the workhorse of many Youtube maths problem solving videos and it is a workhorse that has been flogged to death. There are substantial classes of limit problems that can be solved without this mechanistic rule which dulls the senses for more subtle analytical estimates and the time has come to ask the question: “Is L’Hopital’s rule the tool of the slob mathematician?” To get the cynical history of the rule, the original proof and much more read this paper.
10 November 2023
Dirichlet’s convergence test is very useful for proving convergence of oscillatory series. It can be proved by first proving the summation by parts formula. To see the details click here
30 September 2023
I have published a 61 page paper on generating functions which goes into quite a bit (actually a lot) of analytical detail as to why they work and how they are applied to problems. There is also a set of solved problems at Masters of Biostatistics level that some may find of interest. To view the paper click here.
15 July 2023
The following articles are in preparation:
(1) A paper which deals with generating functions in a mathematical statistics context with detailed analytical proofs of some fundamental properties.
(2) A paper which estimates the area of Bondi Beach using various approaches, the funkiest of which uses Stokes’ Theorem followed by a 2007 NASA estimation method.
(3) A truly gargantuan paper which seeks to demonstrate why the Gaussian function is so ubiquitous. The paper covers, among other things, Gauss’ original work on least squares, Laplace’s original proof of the Central Limit Theorem, Paul Levy’s work on function spaces which Norbert Wiener used in his development of Brownian motion, Einstein’s work on Brownian motion, entropy maximisation, the Heisenberg uncertainty principle, Maxwell’s velocity distribution law and why it is reasonable to hypothesise that the cosmic background radiation is Gaussian in nature.
O6 March 2023
The Arithmetic Mean – Geometric Mean inequality is an endless source of problems. A particularly sly use of it is contained in the attached paper
15 February 2023
I have done a lengthy paper on the basics of Bessel functions which explains the historical genesis of them in the context of astronomy as well as giving many very detailed proofs covering equivalence of various forms of the functions, uniform convergence, analyticity and so on. It is an understatement to say that I have only touched the surface in what is a 66 page paper. Anyone who has ever read Watson’s tome on the subject will understand just how sprawling the subject is. To access the paper click here.
03 October 2022
Bernoulli’s inequality is a useful little tool for various aspects of analysis and the following short article shows how to prove it and use it in a uniform convergence proof. Download the article here
27 August 2022
Exponentials and logarithms are fundamental to calculus and analysis. The following article explores some fundamental properties: Exponentials and logarithms
14 July 2022
Mean square convergence of functions plays important roles in mathematical physics and many other areas. To learn more about mean square convergence and its relationship to pointwise convergence even where discontinuities are involved read this short paper.
10 June 2022
Bessel functions are usually introduced in undergraduate Fourier analysis or engineering courses in the context of hanging chains and vibrating circular membranes, for example. However, the integral form of Bessel’s function actually arose from Bessel’s analysis of the eccentric anomaly in elliptic planetary motion and yet a modified form of Bessel’s function, known as a spherical Bessel function figures in the solution of a certain radial equation derived from the Schrodinger wave equation. A detailed paper on Bessel functions is coming soon. To learn more see this article: Spherical Bessel functions in quantum mechanics.pdf
17 March 2022
I have updated the paper on the Laplacian to contain a proof of a property that I had glossed over to my shame in previous iterations. A further expansion of the material covering the use of rotation matrices to obtain gradients and the Laplacian is coming. The updated paper can be accessed here
06 February 2022
The Gaussian is ubiquitous throughout mathematics and science. The fact that it maximises entropy is one of the reasons for this ubiquity. It also has a role in the Heisenberg Uncertainty Principle. To understand this in more detail read the following article.
17 July 2021
I recently noticed an integration substitution trick in a Youtube video https://www.youtube.com/watch?v=BfZObnTIsYk&t=110s that apparently came from an Indian high school exam. What was interesting was reverse engineering the trick (which was actually unnecessary anyway) and then showing how powerful simple substitutions are for the integral form of Bessel functions. View the article here.
12 April 2021
The Koide formula in particle physics is a fascinating experimental proposition that may suggest deeper things. In this article I explore some of the maths. Download here.
19 March 2021
Serious students of analysis will need to make various estimates of trigonometrical quantities and the following paper may assist in either refreshing or expanding knowledge. Download the paper here.
01 September 2020
In 1905 Einstein produced a remarkable paper on Brownian motion. In this paper he derived from meagre physical and probabilistic assumptions a partial differential equation for the heat equation which had a well known solution at the time. I have gone through his succinct derivation in detail in this article.
26 July 2020
The scientists at Hitachi did a double slit experiment some years ago and it is worth viewing the video they made. This is the quintessential quantum experiment. Follow this link
23 July 2020
Updated paper on the Laplacian by adding some material on natural frames for cylindrical coordinates and a section on how differential forms can be used to work out elementary areas and volumes purely algebraically. The updated paper is here.
23 May 2020
Trigonometrical integration is absolutely fundamental in higher mathematics and physics yet it is often treated in a superficial way. Salomon Bochner, who was an expert in Fourier theory, did a series of lectures in the 1950s in which he developed a really basic, yet rigorous approach, to trigonometric integrals which are, of course, at the core of Fourier theory. This is an “old school” approach which is not in favour today. It is reminiscent of how Frigyes Riesz did functional analysis almost like writing an airplane novel. My functional analysis professor, the late Alan McIntosh (of Kato’s square root fame), was taught using Riesz’s works, and that is the sort of flavour I bring to this article. There is no blizzard of epsilons, rather it is all about making some basic observations about trigonometric behaviour which any serious student will appreciate. The view the article click here.
26 April 2020
As a complement to my earlier paper on the intuition behind the Fourier and Laplace transforms I have done a detailed paper explaining how Fourier integrals (transforms) arise in the context of solving the heat equation for both discrete and continuous eigenvalues. The paper emphasises the basic point that the Fourier integral or transform owes its existence to solving differential equations. To read the paper click here
09 April 2020
Engineering and maths students frequently seek an explanation for the “intuition” behind the Fourier and Laplace transforms. The following paper explains the roots of the Fourier and Laplace transforms and provides some insights into why they exist. Download the paper here
18 February 2020
It is a standard homework problem in undergraduate physics to show how the Stefan-Boltzmann law can be derived from Planck’s radiation law. Underpinning the derivation one has to evaluate a certain type of integral and in this article I go through all the analytical steps involved in the evaluation. You need to know some analysis, Fourier theory and the properties of the gamma function. I also provide some historical information how Planck derived his law. Download the article here.
06 February 2020:
That the Gaussian can be extracted from an integral of cosines was proved by French functional analyst and probability theorist, Paul Levy, back in 1922 in his book “Lessons on Functional Analysis” ( this is the translated title but as far as I can tell there is no translation of the original French work). To see how this was done and appreciate that Norbert Wiener used Levy’s theory to develop his approach to Brownian motion, read this article.
18 January 2020:
The Gram-Schmidt orthogonalization process is an important tool in linear algebra and much more. It is based on a recursive process which can be visualised in 2 and 3 dimensions and inductively extended to n dimensions. To learn more about why it works read this article.
20 November 2019:
The theory of matrix exponentiation is covered in linear ordinary differential equations courses usually. If you want some practice at how the theory works, there are several problems in the Cambridge Tripos Part 1A exam from 30 May 2019 which may be of interest. Full solutions can be found here.
08 November 2019:
In the May 2019 Part 1A Cambridge Mathematical Tripos examination a couple of problems caught my eye for being like a “cheeky” white wine - inviting you to quaff them quickly in a mathematical sense. See what you think. As usual there are plenty of problems that require proficiency in technique to get through them in the space of 3 hours.
To view the problems and solutions click here.
27 August 2019:
The inequalities of Holder and Minkowski are fundamental to analysis and J E Littlewood devoted a whole book ( “Lectures on the Theory of Functions” ) to squeezing the mathematical pips out of them. To see how Littlewood proved these inequalities download this short article here.
11 July 2019:
Singular Value Decomposition (SVD) is an important part of machine learning algorithms and this article goes through the mechanics of SVD in a tutorial format. Download the article here.
A Powerpoint presentation can be accessed here
21 April 2019:
How did Maxwell derive his famous velocity distribution law, one of the foundational elements of statistical mechanics? As you will see, he used geometry and functional equation concepts to derive the law in only a few lines. To see how the gaps are filled in read the detailed paper here.
11 February 2019:
The Math Stack Exchange seems to be a refuge for some arrogantly offensive types so students who want some illumination through that forum do so at their own risk. Don’t be surprised if someone humiliates you. And mathematicians wonder why many people don’t like mathematics! For my views on this forum, read this paper
29 January 2019:
Chebyshev’s sum inequality is an important inequality and can be proved in various ways. Emile Picard proved it in the 1880s via concepts of centre of gravity. It can be proved easily once one has proved a more fundamental inequality. To see how read here.
Some time ago I produced a paper which was intended to go through how the Laplacian figures in mathematics and physics with detailed derivations. I left out two things: one was the full tensor approach which gives you the Laplacian in a few lines after you have built the whole tensor edifice. I gave hints of that approach but never went down that track simply because I did not want to make an already large paper even larger with the abstractions of tensors when I suspect most readers will prefer the more concrete approaches. The second thing I omitted to do was to explain how rotation matrices can be used to derive the gradient and hence the Laplacian in cylindrical and spherical coordinates. Now if you could simply use 3 dimensional rotation matrices to painlessly knock out the Laplacian in spherical coordinates, for instance, the observable universe would be full of such proofs. But it isn't. Keep that in mind.
Fábio M. S. Lima and Pedro G. F. Jordão did a paper for the Mathematics Magazine explaining how to get the gradient in cylindrical and spherical coordinates using rotation matrices and I have used it to go through the logic of the process and the detailed calculations. Readers can assess whether that approach gives them a better feel for the Laplacian which really is a creature of averaging, rotational invariance and spherical symmetry. I remain to be convinced that the rotation matrix approach the authors develop gives any greater insights than the other methods but some people may prefer it. The self-contained rotation matrix paper is here and I have added a link to it at the end of the bigger paper dealing with other techniques here
26 May 2026
I recently published an article on how ChatGPT as it is employed in Mathematica's Notebook Assistant could not without much effort produce Mathematica code for a discrete Fast Fourier Transform: https://gotohaggstrom.com/Mathematicas%20Notebook%20Assistant%20(ChatGPT).pdf
I actually thought before the experiment that ChatGPT could produce the code simply because the concept is so ubiquitous and would have been in the original training.
I was wrong and a more recent test on something so preposterously banal makes me even more worried about the reliability of these models.
The article can be downloaded here
12 April 2026
Using Large Language Models (LLMs) such as ChatGPT and Claude to do serious mathematics gives rise to all sorts of problems. On the one hand these models can provide some astonishing insights even though they actually do not perform logical deductions like a theorem prover. Yet they can also generate some preposterous results. You have to be extremely vigilant with these models and if you let your guard down you can be caught out.
Being a long time Mathematica user I was interested in seeing how Notebook Assistant (NA) worked within Mathematica. It is essentially ChatGPT and is able to produce Mathematica code within the Mathematica environment. I tested it out on what I thought was a problem it could nail simply because the issue I put to it was so ubiquitous it would have been in the original training set and is referred to in many contexts. What I asked Notebook Assistant to do is to generate Mathematica code for a discrete Fast Fourier Transform to multiply two large numbers. Well it demonstrated that it was not up to it and it took a lot of work to get it to understand some Mathematica code that I knew worked. I then tested Claude and Deepseek on the same problem.
All the responses are set out and it is interesting to note that the free version of Deepseek actually produced an astonishing 82 page stream of consciousness.
While you can get some useful work out of these models you actually have to be relatively sophisticated to be aware of their limitations. It is a bit like giving a loaded AK 47 to a young child.
The full story can be found here
05 January 2026
Carlson’s inequality is as follows: if the a_k are positive real numbers not all zero then
He proved it by rather intricate methods in 1935 and then shortly after Hardy came along with a truly gobsmacking proof of a few lines that can only be described as magical. I have gone through how Carlson proved the inequality (and there is a continuous version as well) and then compare it with Hardy’s proof. Having done the hard work of getting into Carlson’s mindset I have to say that Hardy’s proof is just immediately convincing while Carlson’s approach is so intricate it is easy to get lost.
To understand Carlson’s approach you will need to know something about the Cauchy-Schwarz and Holder inequalities as well as harmonic/subharmonic theory, Fourier theory and Mittag-Leffler expansions. You have been warned! The paper is here
04 October 2025
About 15 years ago I published 4 articles on induction along with problem and solution articles. These are still available here. It appears no-one reads the 4 theoretical articles so I have put them together in one pdf file. There is an enormous amount of intellectual capital in those articles and I am surprised that no-one reads them. Actually I was surprised until I realised that I had pitched it too high for secondary teachers and their students and I moved the website content to more undergraduate level material. The combined pdf can be accessed here
24 September 2025
I have produced a lengthy paper titled “The basics of Tauberian theory”. This paper is really only of interest to hard core analysis students probably at the Cambridge Tripos level.
This paper deals with the history of Tauberian methods in analysis in the context of Abelian and Cesaro methods of summation. Hardy’s book on “Divergent Series” is used as the foundation of this history. Tauber’s original proof of his first theorem is given based on a translation of his paper. This is followed by an undergraduate analysis course style of proof. Hardy’s “epsilon free” proof of Tauber’s first theorem is explained in detail and its relationship to Laplace transform theory is fleshed out to lay the foundation for the use of Laplace transforms and Tauberian techniques to obtain an asymptotic limit for the Renyi Parking Problem. This process unites analysis and Laplace transform theory with probability theory. The original papers of Tauber, Renyi and Karamata are included in the Appendix. I give a heuristic explanation for Karamata's 1931 simplification of the earlier Hardy-Littlewood Tauberian theorem. I sent the Rigour Police on a long boozy holiday for the purposes of that explanation.
The paper can be accessed here
It is a lengthy and intricate paper so there will be typos and perhaps infelicities of expression so I will not be offended if you point them out.
02 July 2025
This website is devoted to things mathematical however I felt I had to say something about a rather absurd Australian government policy on the taxation of superannuation which involves some basic mathematical reasoning. In essence the government wants to raise additional revenue from people with large superannuation asset balances. “Large” is implicitly defined as AUD 3 million but there are about 80,000 people currently with assets of at least AUD 3 million. There is a handful of people with balances in excess of AUD 20 million.
Australia’s superannuation industry is large by international standards essentially because of its compulsory nature. Currently the industry has an asset base of about AUD 4 trillion. This asset base represents millions of people with assets of much less than AUD 3 million.
The government wants to apply an extra tax effectively on assets greater than AUD 3 million even though those assets have not been realised. In terms of tax theory it is offensive to tax someone on an unrealised gain since they don’t have the cash to pay the tax and critics have obviously locked on to that issue. What is utterly bizarre is that no Australian government has ever actually legislated a global maximum for what you can have in superannuation. In my asset management days in the early 2000s I was asked about a proposed transfer of AUD 1 billion dollars into a self-managed superannuation fund. That transaction did not eventuate but it was technically lawful and possible at the time.
This policy is bizarre for another reason that goes to the analytical competence of the economists in Treasury. There is a simple way to impose extra tax on those with balances defined to be offensive from a policy point of view and it just involves a non-linear tax rate that applies to taxable income which itself reflects realised gains so that there can be no criticism that unrealised gains will be taxed. This method has other policy benefits that I would have thought the government would want to pursue. I am puzzled as to why the “bleeding obvious” was not done and in the paper I offer 3 hypotheses. Because I don’t have access to the data sets that Treasury has I cannot model how much revenue the proposed policy will generate in comparison to my proposal so I am unable to determine whether this policy is just a cynical and inefficient revenue raising exercise or Treasury never thought of my simpler method. Interestingly self-interested industry commentators have not suggested my alternative which might mean that even though some have actuarial training they are stuck in some sort of intellectual black hole.
What is even more fascinating is that when I asked ChatGPT o3 mini to solve the abstract problem it came up with the formula I worked out. If I were a Treasury economist that would worry me.
In any event the detailed paper is here
23 May 2025
I have updated my “Fooling juries with statistics” article to add in how ChatGPT o3 explained the coin tossing problem. The problem was this: you toss an unbiased coin many times and count the number of tosses until you get the pattern ”HTH”. You then calculate the average. You repeat the experiment but this time you look for the pattern ”HTT”. The question is: ”Is the average number of trials required to get ”HTH” greater than, less than or equal to the average number of trials required to get ”HTT”?
ChatGPT o3 first started to write some Python code to work out the expected value but realised that it got the wrong answer and self-corrected. This happened dynamically and I didn’t get a screen shot. It then used Markov chain methods to get the correct result and gratuitously offered a very clear high level explanation of why the expected number of throws for “HTH” is 10 while that for “HTT” is 8. My paper was based on generating function methods rather than Markov chains but the ChatGPT o3 solution is correct and is appropriate for the problem. ChatGPT o3’s performance was impressive.
Download the updated article here
13 April 2025
Laplace transforms can be used to evaluate integrals in ways which might offer some simplification. They can also be used to perform abstract approximations of integrals and Renyi’s parking problem is an example of that type of use. The following article deals with an integral from the American Mathematical Monthly which is solved using a Laplace Transform and for comparison by another method. Download the article here.
07 January 2025
I am republishing an old article from about 2008 which deals with the question: “How long will my savings last? “ For reasons explained in the article, the humble annuity formula can be used in a practical way to get bounds on this question without probability theory. Download the article here
01 January 2025
Let’s get 2025 off to a rip roaring start with a question that beach goers really want to know - what is the area of Bondi Beach, Sydney, Australia. Because I live about 800 m from the beach and I am an official “beach bum” , I thought I would apply technical rigour to the problem. Now, I have to say at the outset that if you are the sort of person who says “ Duh, just use Google Earth” please leave now. There were cabin boys in the age of Captain Cook who had memorised log tables and knew enough spherical trigonometry that it could advance them up the nautical management tree. Ironically, a lot of these skills have atrophied because of GPS based algorithms that underpin Google Earth and Google Maps. and are inbuilt in phones giving people the “knowledge” but none of the understanding. I have quite intentionally set up this problem this way because at the 1 km level the earth is locally flat (so the dopey Flat Earthers are right locally, but hopelessly wrong globally) and you can effectively use plane geometry with great accuracy. I cover Flat Earth approaches, basic cabin boy spherical geometry, some planimeter algorithms from some JPL/NASA mathematicians and a really funky use of Stokes’ Theorem by academic mathematicians Yuliy Barishnikov and Robert Ghrist who published a paper in the American Mathematical Monthly titled “Stokes's Theorem, Data and Polar Ice Caps “. What was funky was that they used the abstraction of Stokes’ Theorem to derive an approximation formula for a closed polygonal path that was very similar to one of the JPL/NASA planimeter algorithms, However, what I found when I did numerical examples in Mathematica is that the Stokes’ Theorem method required coding rigour even in my simple benchmark case - in fact you wouldn't use it for such cases because you can do a double integral. When you pay attention to detail the method is really cool as it should be for analytical reasons. One sneaky method which is more pedagogically useful than useful generally, is to characterise the area as a “polar” cap and slice off bits. Because there are really simple double integrals for polar caps all we have to do is a rotation to the North Pole and apply a simple formula that arises for the area of a polar cap. I give a summary of the results of the various methods and derivations of the algorithms. Download the paper here
30 August 2024
The geodesic equation is a rite of passage for general relativity. Its derivation can be approached in various ways but it is possible to do a basic “bare hands” derivation which has been done for instance by Alan Guth in his MIT “Early Universe” course which can be found here: ocw.mit.edu/courses/8-286-the-early-universe-fall-2013/resources/lecture-14-the-geodesic-equation/
Guth leaves several important gaps in the derivation (because they would have been covered in tutorials or homework) but a complete development of the derivation is set out in the this paper. Because of all the indices there may be some typos despite several proof readings.
11 June 2024
Order statistics concern random samples of random variables which are arranged in order of increasing magnitude. It is useful to obtain the distributions of the maximum and minimum of samples as well as other more complicated relationships. The basic principles are set out in this paper
12 April 2024
L’Hopital’s rule is the workhorse of many Youtube maths problem solving videos and it is a workhorse that has been flogged to death. There are substantial classes of limit problems that can be solved without this mechanistic rule which dulls the senses for more subtle analytical estimates and the time has come to ask the question: “Is L’Hopital’s rule the tool of the slob mathematician?” To get the cynical history of the rule, the original proof and much more read this paper.
10 November 2023
Dirichlet’s convergence test is very useful for proving convergence of oscillatory series. It can be proved by first proving the summation by parts formula. To see the details click here
30 September 2023
I have published a 61 page paper on generating functions which goes into quite a bit (actually a lot) of analytical detail as to why they work and how they are applied to problems. There is also a set of solved problems at Masters of Biostatistics level that some may find of interest. To view the paper click here.
15 July 2023
The following articles are in preparation:
(1) A paper which deals with generating functions in a mathematical statistics context with detailed analytical proofs of some fundamental properties.
(2) A paper which estimates the area of Bondi Beach using various approaches, the funkiest of which uses Stokes’ Theorem followed by a 2007 NASA estimation method.
(3) A truly gargantuan paper which seeks to demonstrate why the Gaussian function is so ubiquitous. The paper covers, among other things, Gauss’ original work on least squares, Laplace’s original proof of the Central Limit Theorem, Paul Levy’s work on function spaces which Norbert Wiener used in his development of Brownian motion, Einstein’s work on Brownian motion, entropy maximisation, the Heisenberg uncertainty principle, Maxwell’s velocity distribution law and why it is reasonable to hypothesise that the cosmic background radiation is Gaussian in nature.
O6 March 2023
The Arithmetic Mean – Geometric Mean inequality is an endless source of problems. A particularly sly use of it is contained in the attached paper
15 February 2023
I have done a lengthy paper on the basics of Bessel functions which explains the historical genesis of them in the context of astronomy as well as giving many very detailed proofs covering equivalence of various forms of the functions, uniform convergence, analyticity and so on. It is an understatement to say that I have only touched the surface in what is a 66 page paper. Anyone who has ever read Watson’s tome on the subject will understand just how sprawling the subject is. To access the paper click here.
03 October 2022
Bernoulli’s inequality is a useful little tool for various aspects of analysis and the following short article shows how to prove it and use it in a uniform convergence proof. Download the article here
27 August 2022
Exponentials and logarithms are fundamental to calculus and analysis. The following article explores some fundamental properties: Exponentials and logarithms
14 July 2022
Mean square convergence of functions plays important roles in mathematical physics and many other areas. To learn more about mean square convergence and its relationship to pointwise convergence even where discontinuities are involved read this short paper.
10 June 2022
Bessel functions are usually introduced in undergraduate Fourier analysis or engineering courses in the context of hanging chains and vibrating circular membranes, for example. However, the integral form of Bessel’s function actually arose from Bessel’s analysis of the eccentric anomaly in elliptic planetary motion and yet a modified form of Bessel’s function, known as a spherical Bessel function figures in the solution of a certain radial equation derived from the Schrodinger wave equation. A detailed paper on Bessel functions is coming soon. To learn more see this article: Spherical Bessel functions in quantum mechanics.pdf
17 March 2022
I have updated the paper on the Laplacian to contain a proof of a property that I had glossed over to my shame in previous iterations. A further expansion of the material covering the use of rotation matrices to obtain gradients and the Laplacian is coming. The updated paper can be accessed here
06 February 2022
The Gaussian is ubiquitous throughout mathematics and science. The fact that it maximises entropy is one of the reasons for this ubiquity. It also has a role in the Heisenberg Uncertainty Principle. To understand this in more detail read the following article.
17 July 2021
I recently noticed an integration substitution trick in a Youtube video https://www.youtube.com/watch?v=BfZObnTIsYk&t=110s that apparently came from an Indian high school exam. What was interesting was reverse engineering the trick (which was actually unnecessary anyway) and then showing how powerful simple substitutions are for the integral form of Bessel functions. View the article here.
12 April 2021
The Koide formula in particle physics is a fascinating experimental proposition that may suggest deeper things. In this article I explore some of the maths. Download here.
19 March 2021
Serious students of analysis will need to make various estimates of trigonometrical quantities and the following paper may assist in either refreshing or expanding knowledge. Download the paper here.
01 September 2020
In 1905 Einstein produced a remarkable paper on Brownian motion. In this paper he derived from meagre physical and probabilistic assumptions a partial differential equation for the heat equation which had a well known solution at the time. I have gone through his succinct derivation in detail in this article.
26 July 2020
The scientists at Hitachi did a double slit experiment some years ago and it is worth viewing the video they made. This is the quintessential quantum experiment. Follow this link
23 July 2020
Updated paper on the Laplacian by adding some material on natural frames for cylindrical coordinates and a section on how differential forms can be used to work out elementary areas and volumes purely algebraically. The updated paper is here.
23 May 2020
Trigonometrical integration is absolutely fundamental in higher mathematics and physics yet it is often treated in a superficial way. Salomon Bochner, who was an expert in Fourier theory, did a series of lectures in the 1950s in which he developed a really basic, yet rigorous approach, to trigonometric integrals which are, of course, at the core of Fourier theory. This is an “old school” approach which is not in favour today. It is reminiscent of how Frigyes Riesz did functional analysis almost like writing an airplane novel. My functional analysis professor, the late Alan McIntosh (of Kato’s square root fame), was taught using Riesz’s works, and that is the sort of flavour I bring to this article. There is no blizzard of epsilons, rather it is all about making some basic observations about trigonometric behaviour which any serious student will appreciate. The view the article click here.
26 April 2020
As a complement to my earlier paper on the intuition behind the Fourier and Laplace transforms I have done a detailed paper explaining how Fourier integrals (transforms) arise in the context of solving the heat equation for both discrete and continuous eigenvalues. The paper emphasises the basic point that the Fourier integral or transform owes its existence to solving differential equations. To read the paper click here
09 April 2020
Engineering and maths students frequently seek an explanation for the “intuition” behind the Fourier and Laplace transforms. The following paper explains the roots of the Fourier and Laplace transforms and provides some insights into why they exist. Download the paper here
18 February 2020
It is a standard homework problem in undergraduate physics to show how the Stefan-Boltzmann law can be derived from Planck’s radiation law. Underpinning the derivation one has to evaluate a certain type of integral and in this article I go through all the analytical steps involved in the evaluation. You need to know some analysis, Fourier theory and the properties of the gamma function. I also provide some historical information how Planck derived his law. Download the article here.
06 February 2020:
That the Gaussian can be extracted from an integral of cosines was proved by French functional analyst and probability theorist, Paul Levy, back in 1922 in his book “Lessons on Functional Analysis” ( this is the translated title but as far as I can tell there is no translation of the original French work). To see how this was done and appreciate that Norbert Wiener used Levy’s theory to develop his approach to Brownian motion, read this article.
18 January 2020:
The Gram-Schmidt orthogonalization process is an important tool in linear algebra and much more. It is based on a recursive process which can be visualised in 2 and 3 dimensions and inductively extended to n dimensions. To learn more about why it works read this article.
20 November 2019:
The theory of matrix exponentiation is covered in linear ordinary differential equations courses usually. If you want some practice at how the theory works, there are several problems in the Cambridge Tripos Part 1A exam from 30 May 2019 which may be of interest. Full solutions can be found here.
08 November 2019:
In the May 2019 Part 1A Cambridge Mathematical Tripos examination a couple of problems caught my eye for being like a “cheeky” white wine - inviting you to quaff them quickly in a mathematical sense. See what you think. As usual there are plenty of problems that require proficiency in technique to get through them in the space of 3 hours.
To view the problems and solutions click here.
27 August 2019:
The inequalities of Holder and Minkowski are fundamental to analysis and J E Littlewood devoted a whole book ( “Lectures on the Theory of Functions” ) to squeezing the mathematical pips out of them. To see how Littlewood proved these inequalities download this short article here.
11 July 2019:
Singular Value Decomposition (SVD) is an important part of machine learning algorithms and this article goes through the mechanics of SVD in a tutorial format. Download the article here.
A Powerpoint presentation can be accessed here
21 April 2019:
How did Maxwell derive his famous velocity distribution law, one of the foundational elements of statistical mechanics? As you will see, he used geometry and functional equation concepts to derive the law in only a few lines. To see how the gaps are filled in read the detailed paper here.
11 February 2019:
The Math Stack Exchange seems to be a refuge for some arrogantly offensive types so students who want some illumination through that forum do so at their own risk. Don’t be surprised if someone humiliates you. And mathematicians wonder why many people don’t like mathematics! For my views on this forum, read this paper
29 January 2019:
Chebyshev’s sum inequality is an important inequality and can be proved in various ways. Emile Picard proved it in the 1880s via concepts of centre of gravity. It can be proved easily once one has proved a more fundamental inequality. To see how read here.