Maths at Bondi Beach
Mathematics without the mystery!
Mathematics without the mystery!
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 or Youtube
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.
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 or Youtube
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.