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Dr. Philippe H. Trinh \\ | Dr. Philippe H. Trinh \\ | ||
- | Departmental Lecturer in Mathematical Modelling \\ | + | University of Bath \\ |
- | Mathematical Institute \\ | + | Department of Mathematical Sciences |
- | University of Oxford | + | p.[my-last-name]@bath.ac.uk |
- | Oxford, Oxfordshire, | + | |
- | [my-last-name]@maths.ox.ac.uk | + | |
- | [[https://www.maths.ox.ac.uk/people/profiles/philippe.trinh|Mathematical Institute Profile]] \\ | + | /* |
- | [[http:// | + | [[https://researchportal.bath.ac.uk/en/persons/philippe-trinh|University of Bath profile]] \\ |
+ | [[:collaborators|Collaborators and students]] \\ | ||
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* You might also be interested in learning a bit about [[: | * You might also be interested in learning a bit about [[: | ||
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- | ==== 25 May 2017: A tax on those who can't do maths ==== | + | /* |
+ | ==== Edit in progress | ||
- | It' | + | It' |
- | Here's a typical situation that will be familiar to a lot of our readers. You would like to take out a mortgage of a certain amount, let's say $L = L_0$. Currently, Halifax has a deal where they will charge you a fixed rate of $r_1 = 2.11\%$ interest on the first $t = n_1$ months. For the remainder of the time, up to $t = n_e$, they will charge the variable rate, which for simplicity we assume to be at $r_2 = 3.74\%$. | ||
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- | Now when you fill out the details of the mortgage on their calculators, | ||
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- | This turns out to be a question of recurrence relations. Let $L_n$ be the current loan amount in the nth month. During the first period, $0 < n < n_1$ we can verify that | ||
- | \[ | ||
- | L_n = L_{n-1}(1 + r_1/12) - m_1, | ||
- | \] | ||
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- | where $r_1 = 0.0211$ is the interest rate and it is assumed to be compounded monthly. From this, it follows that | ||
- | \[ | ||
- | L_n = k_1^n L_0 - m_1 \left(\frac{1 - k_1^n}{1- k_1}]\right), | ||
- | \] | ||
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- | where we have set $k_1 = 1 + r_1/12$. In the same vein, we reason that in the second period, where $n_1 < n \leq n_e$, it follows that | ||
- | \[ | ||
- | L_n = k_2^{(n-n_e)} L^* - m_2 \left(\frac{1 - k_2^{n-n_e}}{1- k_2}\right), | ||
- | \] | ||
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- | where we have set $k_2 = 1 + r_2/12$. The key parameter here is the value of $L^*$, which is the loan amount that exists in the changeover month, $t = n_1$. By solving the above equations for $m_1$ and $m_2$, then these fixed monthly payments can be determined as a function of $L^*$ and all the other parameters of the problem. | ||
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- | As a test, I found that Halifax was quoting me monthly figures of $m_1 = \pounds 620.87$ and $m_2 = \pounds 755.81$ for a loan of $\pounds 176, | ||
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- | {{ : | ||
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- | Based on the image, you see two things. First, there is a critical point of intersection where you would pay exactly the same every month, and where $m_1 = m_2$. This point occurs at $£723.432$. | ||
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- | ==== 30 January 2017: On reduced models for gravity waves generated by moving bodies ==== | ||
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- | I'm happy to announce a recent [[https:// | ||
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- | {{ : | ||
- | (Left) Ernie Tuck (1939--2009) (Right) Marshall Tulin (1926--) | ||
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- | Since around 2007--2010, I'd often play with certain reduced models for studying gravity wave generation by two-dimensional bodies. These reduced models you can derive using some more modern techniques in asymptotics, | ||
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- | A few years ago, I spotted a curious question that was written in a transcription of audience questions in a conference where Tuck had presented his research (in fact, such transcriptions are quite rare in this day and age). [[https:// | ||
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- | //" | ||
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- | Tuck had replied that he didn't know the answer, and the matter was apparently left at that. However, Tulin' | ||
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- | Tulin was quite pleased to have been asked for more details (as it had been over two decades since that conference!). He told me that he had, in fact, published a report in 1983 for the 14th Symposium on Naval Hydrodynamics where he laid out a particularly involved reduction of the water wave equations. | ||
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- | He explained that nobody had really picked up on the 1983 paper (1 current citation!), even though there were a series of questions he had asked and a series a results he had presented that had seemed of some importance. He encouraged me to look up the manuscript and close the chapter, if I could. | ||
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- | And so I did. The result is this most recent paper. | ||
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- | {{ : | ||
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- | ==== 8 September 2016: A topological study of gravity free-surface waves generated by bluff bodies using the method of steepest descents ==== | ||
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- | This paper, now published in the Proceedings of the Royal Society A (PRSA) has a few interesting distinctions. It's the first paper I've published in PRSA---but hopefully not the last as it's certainly a strong journal with an illustrious history. It's the first solo paper I've published. And it has the longest title of any other paper I've worked on. | ||
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- | In any case, it's a paper where I explore exponential asymptotic techniques for free-surface flows (now well known) from a slightly different viewpoint. It turns out that the situation of gravity waves permits the governing equations to be re-formulated in a particularly simple way: that of a first-order nonlinear differential equation. In this paper, I show how the differential equation is studied using steepest descents. What results is a visual and beautiful way of understanding wave-structure interactions through a correspondence with the topology of certain Riemann surfaces (seen above). | ||
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- | You can download a copy of the paper {{: | ||
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- | ==== 01 June 2016: New singularities in Stokes waves ==== | ||
< | < | ||
<iframe src=' | <iframe src=' | ||
</ | </ | ||
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- | I'm happy to announce the publication of a paper in collaboration with [[collaborators# | ||
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- | Interestingly, | ||
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- | ==== 18 May 2016: Jet flows from angled nozzles ==== | ||
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- | [[this> | ||
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- | A late congratulations to second-year student [[collaborators# | ||
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- | ==== Jan 2016: Spot patterns on the surface of the sphere ==== | ||
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- | I'm happy to announce the publication of my paper in the journal // | ||
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- | ==== June 2015: Fluids and elasticity in France | ||
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- | I'll be attending the Fluid and Elasticity 2015 conference, from June 22-24 in Biarritz, France, and presenting some joint work with Stephen K. Wilson (Strathclyde University) and Howard A. Stone (Princeton University). | ||
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- | ==== May 2015: Two new papers published ==== | ||
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- | {{ : | ||
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- | I'm happy to announce the publication of two new papers. The [[http:// | ||
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