tag:blogger.com,1999:blog-78972354238122776832024-03-27T18:42:48.805+00:00Ramblings of a Short Fat Failed PhysicistChris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.comBlogger193125tag:blogger.com,1999:blog-7897235423812277683.post-74280142332006616112022-08-29T20:32:00.000+01:002022-08-29T20:32:54.807+01:00Notes on Blackbody radiation <p> Hi folks sorry I have been silent for a while anyway I have beeen busy for about a year writing up some notes on Blackbody radiation. This is a fairly straightforward run through of the basic ideas of Statistical Mechanics and how Planck managed to finally obtain the correct formula by making the guess that electromagneitic energy is quantised. For Planck this was just a ruse and he thought little about it. However it took Einstein to take the idea seriously and he applied it to calculate the specific heat of solids and almost got the correct low temperature behaviour. It was left to Debye to improve on Einstein's model by performing a numerical integration over all frquencies. So Debye is a better physicist than Einstein ๐ ๐ Well at least in that respect. </p><p>I must admit I have had a love hate relationship with Statistical Physics over the years we had a pretty bad lecturer a certain Dr Jones at Exeter university who was (is) a really clever guy but his lectures bore no real resemblance to conventional statistical physics texts and so I and a friend who I was studying with at the time were left scratching our heads and wondering what the subject was all about. With these set of notes I have finally laid the ghost of that experience to reat and now feel I have enough background in statistical physics to understand its applications in Astro-physics especially the physics of the early universe and the calculation of the Helium abundance in the universe by Peebles and other people. </p><p>Of course everyone knows that the Cosmic Background radiaton is a black body and I produce a graph comparing Planck's formula with the FIRAS data obtained from the COBE satellite. I also show how to do the Numerical Integration to get Debye's prediction of the specific heat of silver at low temperatures. </p><p>There are some pretty interesting integrals which invove the Product of the Riemann Zeta function and the Gamma Function. and I show how these are derived unlike most books which just state the formula. As this formula is ubiquitous in Astro-Physical applications then I have enclosed it in a red box at the end of a fairly long appendix at the end. </p><p>Anyway I hope you find the notes useful I now have all the building blocks in place to understand the Peebles calculation and over the next year or two I hope to finally finish this culminating in a code which traces the abundance of the Light elements during the first three minutes after the biig bang </p><p>Here is a link to the file </p><p><a href="https://drive.google.com/file/d/11tpWAP4vh1ywl9ITv8opKCCsiJBws2oP/view?usp=sharing">https://drive.google.com/file/d/11tpWAP4vh1ywl9ITv8opKCCsiJBws2oP/view?usp=sharing</a></p><p><br /></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-77617844835275559402021-10-20T19:31:00.002+01:002021-10-20T19:31:59.984+01:00Cambridge NST Maths year 1 2019 Paper 2 solutions<p> Here is the second in a sequence of my solutions to the Cambridge NST maths papers, This one is for the second of the papers set for the first year students in 2019 the last year before COVID. You can access it here </p><p><a href="https://drive.google.com/file/d/1SSePjGdcGjWatVFNHh5kVzYORGgaBUrD/view?usp=sharing">https://drive.google.com/file/d/1SSePjGdcGjWatVFNHh5kVzYORGgaBUrD/view?usp=sharing</a></p><p>I have previously published my solutions to the first of the papers and for convenience I post the link again here for those who missed it first time around</p><p><a href="https://drive.google.com/file/d/1SSePjGdcGjWatVFNHh5kVzYORGgaBUrD/view?usp=sharing">https://drive.google.com/file/d/1SSePjGdcGjWatVFNHh5kVzYORGgaBUrD/view?usp=sharing</a></p><p>So there you have it a complete set of answers to the first year papers for NST students at Cambridge for 2019. Most of them will have sat there finals last summer and I hope they did well. </p><p>I apologise in advance for any typos spelling errors etc. A review of the paper follows </p><p>Again just like the first paper this was quite challenging and unfortunately I was unable to answer a question on Parseval's theorem properly. On the whole though I think I got the questions out, but I doubt if I could do well under exam conditions. </p><p>Anyway just like the first paper there were 10 short questions which were relatively straightforward once you decoded what the examiner was getting at. The questions included solving a first order differential equation by the integrating factor method. Deriving a recurrence relation for an Integral (something they love testing people on). Calculating the stationary values of a function f(x,y). A slightly confusing question on pronability I really need more practice at questions involviing conditional probability. A volume integral and a surface integral. All this should take no longer than 30 mins but I suspect I would take a lot longer. Again there is no time to think and in the rush you would probably end up making silly mistakes,</p><p>The core of the paper is 10 questions of which answers to five must be submitted. The last two questions are reserved for those students deemed clever enough to do some advanced topics although I didn't think they were particularly difficult. </p><p>Anyway here are the questons </p><p><u>Question 11</u> was a geometric one involving the equation of a plane and finding the volume of the parallipped enclosed by 3 vectors. This was relatively straightforward once I had reminded myself of the vector equation of a plane. But there were quite a few parts. Although the question asked for a few diagrams I had the luxury of using MATLAB to draw the relevant pictures/ Not very exciting I must admit</p><p><u>Question 12</u> . Involved fiinding the stationary points of a function in f(x,y) and drawing a contour plot showing the function and the gradient. This was tedious but relatively straightforward and again I was able to use MATLAB to draw some pretty pictures. </p><p><u>Question 13 </u>Involved calculating the line integral of a vector function F for various paths and also finding a function satisfying curl F = 0. (a conservative function). I hadn't done a question like this for ages so I had to remind myself of how you go about calculating such things but it was relatively straightforward although finding the conservative function involved a little guesswork so not very satisfactory.</p><p><u>Question 14 </u> Involved some questions on probability density functions. and evaluating their products and change of variables. It was ok but not a very exciting topic. </p><p><u>Question 15 </u> This was a set of questions on solving second order differential equations with constant coefficients my favourite topic at this level. The first question was a homogeneous equation with boundary conditions and relatively straightforward to solve. The second part was an inhomogenous equation and whilst finding the complementary function was relatively straightforward. In order to find the particular integral you had use a function of the form x^n f(x) and increase n until you found one that worked. This took a couple of goes and so would have been quite time consuming under exam conditions how nice of them ๐. The last part involved solving two differential equations simultaneously some what surprisingly they don't seem to teach how to solve such systems using matrices and their eigenvectors unlike the open university courses MST210 or MST224 so this is one occasion wihere the open university is better than Cambridge. Anyway compared to the tedium of the last few questions this was a delight to do. </p><p><u>Question 16 </u> This question was all about calculating various surface integrals and the flux of a field through a surface. It got a bit fiddly but again was relatively straight forward. Another boring topic though I much prefer solving differential equations </p><p><u>Question 17</u> This was a boring question on matrices again pretty straighrforward but you have to know the definitions and again there were so many parts to the question. Give me calculus questions any day</p><p><u>Question 18 </u> This was a question on Fourier series you had to find the Fourier series for cosh(x) then differentiate to get the Fourier series for sinh(x). For this topic you really need to be on top of integration by parts. The last part of the question then asked you to use Parsevals theorem to show that the integral of (cosh(x)-sinh(x))^2 over the interval was sinh(2) . I tried this a couple of times but I couldn't get the expansions to cancel out to leave sinh(2) so unfortunately I was unable to complete the paper properly. However if you integrate the function directly it comes out relatively straightforwardly. </p><p>So the two questions for the so called advanced students were as follows </p><p><u>Question 19</u> was on Lagrangian multipliers and you had to find the optimum volume of a cylinder the optimum volume of a cone inscrbed in a sphere and then prove that the Arithmetic mean is >= to the geometric mean. I confess to nor really understanding this topic although I can go through the motions and I find it difficult to tell whether I am finding a minimum or a maximum. For the cone inscribed inside a sphere I found it easier to just finding the maximum volume directly. I'll let you solve this question using Lagrangian Multipliers fot your self. </p><p><u>Question 20 </u> A relatively straightforward question on solving partial differential equations using separation of variables. Two first order ones and a question on the diffusion equaton. This is really a warm up for what comes next year so you aren't asked to solve the differential equations in spherical or cylindrical coordinate systems or use Lagrange Polynomials or Bessel functions or any of the other exotic functions out there. So a bit boring really </p><p>Overall conclusion is that this was an exercise worth doing to remind myself and extend my mathematical knowledge a bit. I think on the whole I preferred the first paper as it seemed to cover slightly more interesting topics. Apart from the two quesitions on differential equations this paper could be described as worthy but dull. </p><p>The second year papers for this year beckon next and I hope that I find them a bit more interesting than this one. Hopefully I can finish them by June next year</p><p>I would urge you to have a go for yourself and I hope you find these solutions useful </p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-48085069747785827222021-10-04T19:58:00.000+01:002021-10-04T19:58:09.182+01:00Thermodynamics <p> The laws of thermodynamics are fundamental for understanding the structure of matter and the transfer of heat. Remarkably they stand by themselves and have no need of any microscopic underpinning This point is often missed in treatments of thermodynamics which quickly move onto statistical physics and don't encourage physicists to develop their powers of thermodynamic reasoning. </p><p>The best account of thermodynamics I know of is given in Longairs book </p><p><a href="https://www.amazon.co.uk/Theoretical-Concepts-Physics-Second-Alternative/dp/052152878X">https://www.amazon.co.uk/Theoretical-Concepts-Physics-Second-Alternative/dp/052152878X</a></p><p>Which then goes onto discuss how the attempts to model black body radiation broke down using classical physics thus paving the way for Planck and Einstein to introduce quantum mechanical ideas. It really is a fascinating story and shows that there is more to quantum mechanics than the development of Schrodinger's equation </p><p>In general terms a good overall book on Thermodynamics is the classic by Zemansky </p><p><a href="https://www.amazon.co.uk/Heat-and-Thermodynamics-Fifth-edition/dp/B00X4VPZ0C/ref=sr_1_12?crid=32NLWLWHKDY3O&dchild=1&keywords=heat+and+thermodynamics+zemansky&qid=1633373359&s=books&sprefix=Zemansky+%2Cstripbooks%2C173&sr=1-12">https://www.amazon.co.uk/Heat-and-Thermodynamics-Fifth-edition/dp/B00X4VPZ0C/ref=sr_1_12?crid=32NLWLWHKDY3O&dchild=1&keywords=heat+and+thermodynamics+zemansky&qid=1633373359&s=books&sprefix=Zemansky+%2Cstripbooks%2C173&sr=1-12</a></p><p>Anyway thermodynamics has many applications Chandresekhar used it to work out the General equations of stellar structure without any need to know the internal structure of a star </p><p>Here is my tribute to Thermodynamics and I would encourage people to study it in it's own right </p><p class="MsoNormal"><span style="font-size: 16.0pt;"> </span><span style="font-size: medium;">Thermodynamics
<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;"> </span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Three
laws oh so neat,<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Describing
the nature of heat. <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">The
first says you cannot win,<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">You
wont get back more than you put in.<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;"> </span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">But
if itโs heat, thereโs a permanent loss, <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">That
is only regained at greater cost. <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Finally
there will come a great big chill,<o:p></o:p></span></p><p>
</p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Where
all that there is, will stand still.</span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;"><br /></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;"></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-size: medium;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5kDQN42wSA6UmD9f6wkrUYgz8UHvKg3t5GsPLXUQoRu7Cz2ZC71bEkpGvsTZRQarvkC2rXj3EYY5j_V6yLY1Bz8nRHvjwVm3Ggf1ssySvfgWhxiM5Qhl_IxcEnVsU4t8ADxei0sfPkwlk/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="279" data-original-width="454" height="197" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5kDQN42wSA6UmD9f6wkrUYgz8UHvKg3t5GsPLXUQoRu7Cz2ZC71bEkpGvsTZRQarvkC2rXj3EYY5j_V6yLY1Bz8nRHvjwVm3Ggf1ssySvfgWhxiM5Qhl_IxcEnVsU4t8ADxei0sfPkwlk/" width="320" /></a></span></div><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;"><span style="font-size: medium;"><br /></span></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;"><span style="font-size: medium;"><br /></span></span></p><span style="font-size: medium;"><br /><br /></span><p></p><p><br /></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-88604711053693080852021-09-22T22:08:00.001+01:002021-10-03T13:58:23.889+01:00Lasers <p> I was challenged by a work colleague to see if I could write a poem about lasers. The result is given below. Just like supeconductivity lasers are another successful application of quantum mechanics which again no agonising about it's meaning will ever produce a laser Ironically given that Einstein rejected the later formulation of quantum mechanics it was Einstein who worked out the basic theory of spontaneous emission on which the laser is based in 1918 This is a purely statistical argument, which again might surprise people as Einstein is allegedly supposed to have claimed that God does not play dice. Well in the early days of quantum mechanics it was Einstein who used statistical arguments to work out the consequences of the photo-electric effect and applied statistical reasoning to work out the Heat capacity of solids. So the idea that Einstein didn't like statistical reasoning is just incorrect. Indeed in his final years when he surveyed his debates with Bohr, he actually made the statement</p><p><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"><i>The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems.</i></span></p><p><br /><br />So the only way to make sense of quantum mechanics according to Einstein is to endorse a statistical interpretation.<br /><br />Anyway lasers are extremely useful devices but of course in the wrong hands can be used as terrifying weapons. Also idiots shine laser pens in pilots eyes, these people should be forced to face the consequences of their actions. In the right hands of course lasers are a benefit to mankind and a testimony to the ingenuity of scientists all over the world. Here is my tribute to them. <br /><br /><br /><br /><br /><span style="font-size: 16pt;"> Lasers</span></p><p class="MsoNormal"><o:p> </o:p></p><p class="MsoNormal" style="margin-left: 108pt;"><span style="font-size: 12pt;"> Purest light that shines so bright, </span><span face=""Calibri",sans-serif" style="font-size: 12pt;"><o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 108pt;"><span style="font-size: 12pt;"> All because a photon takes flight.<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 108pt;"><span style="font-size: 12pt;">Channelled
by some clever means,<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 108pt;"><span style="font-size: 12pt;"> Into a set of very intense beams.<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 108pt;"><span style="font-size: 12pt;"> </span></p><p class="MsoNormal" style="margin-left: 108pt;"><span style="font-size: 12pt;">Once
the stuff of science fiction,<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 108pt;"><span style="font-size: 12pt;"> Itโs now part of our jurisdiction. <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 108pt;"><span style="font-size: 12pt;">The
wise will put you to good use, <o:p></o:p></span></p><p>
</p><p class="MsoNormal" style="margin-left: 108pt;"><span style="font-size: 12pt;">But
we must guard against abuse</span></p><p class="MsoNormal" style="margin-left: 108pt;"><span style="font-size: 12pt;"><span style="color: #1f497d;">.<o:p></o:p></span></span></p><p class="MsoNormal" style="margin-left: 108pt;"><span style="font-size: 12pt;"><span style="color: #1f497d;"></span></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIn2k3ONvkrJQ2TX15ScbhnyGlzZTkxUpYLvW3XzrUVvPfv6S7LHjb8RFD3UX_jhjMOyeAXcZ6QbTpfPaxsDTRKHxasoREqiZ8IKLzqHgJ8uS5TIQcv-QXtey6a8nHf34z-DvjaFaMnh_S/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="254" data-original-width="454" height="179" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIn2k3ONvkrJQ2TX15ScbhnyGlzZTkxUpYLvW3XzrUVvPfv6S7LHjb8RFD3UX_jhjMOyeAXcZ6QbTpfPaxsDTRKHxasoREqiZ8IKLzqHgJ8uS5TIQcv-QXtey6a8nHf34z-DvjaFaMnh_S/" width="320" /></a></div><br /><p></p><p><span style="font-family: inherit;">James Bond concerned that
Goldfingerโs Laser might destroy his manhood. Iโll leave you to decide whether
or not that would have been a good thing</span><span face=""Arial",sans-serif" style="font-size: 12pt; mso-ansi-language: EN-GB; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;"> ๐</span></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-91269078200617332402021-09-20T22:06:00.003+01:002021-09-20T22:06:41.942+01:00Super Conductivity<p> One of the most amazing applications of the formalism of quantum mechanics was the explanation by Baarden, Cooper and Schrieffer (BCS) of the phenomenon of Superconductivity. At low temperatures roughly about 4K it was noticed by Onnes that some metals appeared to have a dramatic reduction in their resistance. Whilst classical models were developed describing this phenomenon it wasn't until the 1950's that a microscopic version of the theory was developed. At first sight given the Pauli exclusion principle the phenomenon would seem impossible as it implies many electrons are occupying the same state. Which electrons being Fermions was impossible. Cooper realised at low temperatures there was the possiblity of an interaction between the lattice of the solid and the electrons causing them to effectively pair off. At low temperatures this interaction would be stable as the lattice vibrations would be relatively low. If electrons pair off their total spin becomes zero and they now behave like bosons for which it is possible for many bosons to occupy the same state. Thus the resistance of the metal is lowered BCS quickly realised that the properties of superconductors could be explained and they were awarded the Nobel prize for this work in 1972. </p><p>Compared to the endless debates about the meaning or not of quantum mechanics which are going nowhere. This gives us a real insight into how nature works and ia a triumph of Mankinds ability to understand nature, something that will never come from discussing the meaning of the wave-function. As an interesting foot-note it was discovered in 1986 that some cuprates exhibited Superconductivity at much Higher temperatures than the 'normal ones' As yet there is no convincing explanation for this phenomenon so if you want a Nobel prize get cracking ๐ </p><p>Here is my poem </p><p align="center" class="MsoNormal" style="text-align: center;"><span style="font-size: 16.0pt;">Super Conductivity <o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="text-align: center;"><span style="font-size: 12.0pt;"> </span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">When
it becomes very cold,<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Nature
becomes extremely bold.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">All
resistance suddenly dies, <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">An
electric current really flies. <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;"> </span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Electrons
interact with the grid, <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Combined
in pairs they are hid. <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">This
really ingenious tactic,<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Was
explained by a quantum mechanic<sup>1)</sup></span></p><p class="MsoListParagraph" style="margin-left: 29.9pt; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -18.0pt;"><!--[if !supportLists]-->1)<span style="font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal;">
</span><!--[endif]-->Someone who uses
their knowledge of quantum mechanics to explain a feature of nature. In this
case it was Cooper (what a clever fellow <span style="font-family: Wingdings;">๐ ๐ </span> <span style="font-size: 12.0pt;">).</span></p><p class="MsoListParagraph" style="margin-left: 29.9pt; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -18.0pt;"><span style="font-size: 12.0pt;"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmqhzHmRQGlD19iI8XkNGDPRGkM4_zpn5YjXpDIpHwLp2mxxb3-infonxbl4CfKa7fKwrzeEGGG-AeovAo6BuDAXvV2z0QTidn1WRoCUAAFWAVwUuoof_VcJssY3uuopIR11fOLZ4CqmM3/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="377" data-original-width="602" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmqhzHmRQGlD19iI8XkNGDPRGkM4_zpn5YjXpDIpHwLp2mxxb3-infonxbl4CfKa7fKwrzeEGGG-AeovAo6BuDAXvV2z0QTidn1WRoCUAAFWAVwUuoof_VcJssY3uuopIR11fOLZ4CqmM3/" width="320" /></a></div><br /> <o:p></o:p><p></p>
<p><span style="font-size: 12pt;"> </span> </p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-31446899522169451192021-09-19T17:44:00.002+01:002021-09-19T17:44:39.836+01:00Quantum Debates part II <p> Ok I realise in the last post some people will say well what about the collapse of the wavefunction and the violation of the Bell inqualities don't they show that quantum mechanics is weird. I have covered this in many posts before but to save you looking here is a recap </p><p>Let's take the so called collapse of the wavefunction first. This is usually illustrated by the so called cat paradox. If a quantum system can occupy a number of possible states then prior to measurement the sysrem is said to be in a superposition of states and on the usual story this means that when a measurement is made the so called wave function of the system collapses into one of the possible states with a given probability. Those who see the so called wave function as something more than the square root of a probability density function then make the leap to say that the measurement has caused the system to change from it's superposition to a single state. So there is something special about measurement and indeed quantum mechanics via this so called collapse vindicates idealism. So that we create reality by acts of measurement something which is alleged to be part of the Copenhagen interpretation which in fact it isn't. This should really be called the 'California Interpretation' and if it were true would indeed be weird and mysterious and all those who like to link Buddhism or Transcendental meditation with quantum mechanics would be vindicated. </p><p>But this is totally unnecessary, lets take the classical situation for any statistical event if I know the underlying probability density function which summarises all the possiblities and I pick a given sample then I can calculate the probability that that person has a given height or a particular number will turn up if I throw a dice or spin a roulette wheel. Prior to the outcome I did not know what the outcome would be after the outcome I do. Thus the probability density function has 'collapsed' to give the particular outcome but all that is saying is that for classical systems the probability density function summarises all the possible outcomes and after the event has occured the outcome was realised with the given probability. I would argue the same is true of quantum mechanics before a measurement is made I do not know what the result would be but only the total outcomes with a given probability which if I know the appropriate solution to Schrodinger's wave equation I can calculate via the Born rule. Thus the superposition is not a real superposition but just a summary of possible outcomes of a measurement with the appropriate outcomes. The cat is definitely alive or dead before I open the box all I have done is updated my knowledge of the situation before hand. Opening the box hasn't caused the cat to be alive or dead but whether the radioactive poison was released or not. Something you could estimate if you know the half life of the radioactive material. All of this is consistent with quantum mechanics and how it is applied to calculate the probabilities of certain things happening and there is no need to invoke the collapse of the wavefunction to explain this process. </p><p>In contrast to other realist interpretations such as those invoking hidden variables or the Many Worlds interpretaton nothing is added to the formalism. There is no need to invoke many worlds in a desperate attempt to maintain realism. My solution is robustly realist because the entities to which quantum mechanics is applied electrons, atoms and all the various exotic particles and there interactions are seen as real. Once one accepts that the so called wavefunction isn't anything physical but related to the probability density function via the Born rule, then there is no need to agonise whether or not the wave function is a physical object defined in (3N+1)*S space-time-spin dimensions for an N body system it makes no sense as a physical object but as a probability density function it makes a lot of sense. So by sticking to the statistical interpretation of Born then we can retain a fairly robust realism about the entities to which quantum mechanics is applied to and there is no need to worry about any form of idealism or mysticism. </p><p>I'll talk about the Bell inequalities in a later post. But I defy anyone who disagrees with my perspective to show that my position is inconsistent with quantum mechanics. One can never observe a physical superposition of states because by definition any observation would collapse the wavefunction into one of the possible eigenstates. Those who claim otherwise are simply collating the reuslts of various measurements and claiming that this represents a real superposition. </p><p> </p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-37134715407036597782021-09-15T18:41:00.001+01:002021-09-15T20:34:43.401+01:00Quantum Debates <p> Ok this poem is a rant against the endless debates about the alleged meaning of quantum mechanics. A debate I regard as totally pointless as the main issues were all settled by the following two rules of interpretation</p><p>1) The Borm rule which says that the modulus squared of the solution to Schrodinger's equation, when suitably normalised gives rise to a probability density function from which wc can calculate the expectation value of any variable. Conversely this means that the solution to Schrodinger's equation is effectively the square root of a probability density function not a real wave.</p><p>2) The link between the eigenvalues of the solution to Schrodinger's equation and the energy levels of the system under consideration. </p><p>These rules are readily extendible to relativistic equations </p><p>That really is all you need. plus the ability to use Schrodinger's equation or it's relativistic generalisation to understand the results of experiments and from which all the great successes of quantum mechanics have been applied to and which help us understand how nature works. </p><p> The rest whether or not the solution to Schrodinger's equation is a real wave in 3N+1 dimensional space time or merely a mathematical object which enables the probability that certain events will happen to be calculated or the energy levels of a system to be obtained. Whether there are multiple universess and so forth have nothing to do with physics at all. To coin a phrase quantum mechanics isn't mystical it is just statistical. </p><p>Once the implications of this are accepted then much of the so called debate just dissolves. If there are no hidden variables then we can only have a statistical solution. But this does not mean that particles or their interactions are not real. just that their so called wave functions are not real, but only as real as say the Gaussian Distribution describing the distribution of heights of a collection of human beings in a sufficiently large enough sample.</p><p> Instead of worrying about the meaining of the wave function try and apply the well trodden path of discovering the correct Hamiltonian to describe particles and how they interact with each other once this has been obtained just get on and calculate the consequences. If your Hamiltonian agrees with experimental results, then brilliant you have understood a part of nature. No amount of agonising about whether or not the solution to Schrodinger's equation is a real wave,. or signals travel faster than the speed of life is going to add one iota to the new knowledge. So let's celebrate all those physicists and chemists who daily apply quantum mechanics to understanding the world around us. Rather than the parasitic journalists and philosophers and I am afraid some prominent scientists such as Jim Al Khalili who jump on the band wagon of claiming that quantum mechanics is difficult to understand or mysterious. </p><p>If you want to understand how quantum mechanics works, learn how to solve Schrodinger's equation or any of the relativistic equations don't waste your time reading what is essentially gibberish. Anyway here is my rant </p><p class="MsoNormal" style="margin-left: 155.9pt;"><span style="font-size: 16pt;">Quantum
Debates <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: 12pt;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12pt;"> <span> </span><span> </span> </span><span style="font-size: medium;">Many thoughts about you
abound,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: medium;"> Some of them seeming
quite profound.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: medium;"> But itโs all an illusion,
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: medium;"> Just adding to our
confusion. <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: medium;"> </span></p>
<p class="MsoNormal"><span style="font-size: medium;"> Itโs not mystical,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: medium;"> Itโs just statistical. <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: medium;"> Time to end this silly
debate, <o:p></o:p></span></p>
<p><span style="font-size: medium;"> And get on and calculate. </span></p><p><span style="font-size: medium;"><br /></span></p><p></p><div class="separator" style="clear: both; font-size: large; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-Zy-NAq55c6maUHa0um2MbXruqq5f39MJrBnhuh6bUNaOcVPwKeTXOTg7ioamPziINXrtL36ct9zxY6KmL-kzcBG1W_Y-TAPpzX8-UEFN1JcdpsIP7hESka3nSNmLuRvxKiWjQxIK601L/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="233" data-original-width="454" height="164" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-Zy-NAq55c6maUHa0um2MbXruqq5f39MJrBnhuh6bUNaOcVPwKeTXOTg7ioamPziINXrtL36ct9zxY6KmL-kzcBG1W_Y-TAPpzX8-UEFN1JcdpsIP7hESka3nSNmLuRvxKiWjQxIK601L/" width="320" /></a></div><div class="separator" style="clear: both; font-size: large; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><span face="Arial, sans-serif">Example of a silly debate
about the meaning of quantum mechanics, that is going nowhere IMHO and there are plenty more around unfortunately. </span></div><br /><br /><p></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-35027481178194666172021-09-13T20:40:00.004+01:002021-09-14T23:16:00.276+01:00Spherical Harmonics <p>Todays poem is about Spherical Harmonics, just as any function in rectangular coordinates can be expressed as a Fourier Series in Trigonometric functions and in Cylindrical coordinates we can expand any function in terms of Bessel Functions. So in spherical coordinates it is possible to express any function as a series in Spherical Harmonics. Naturally then this expansion has been applied to spherical waves, the calculation of the fields outside a sphere which leads to a series of Polynomials called Legendre polynomials. One of the most exciting application is to the calculation of the angular dependence of the wave function of the hydrogen atom. It is often thought that because the "waves" are quantised that this is unique to quantum systems however this is mistaken. Any spherical wave be it classical or quantum is quantised in the sense that depending on the size of the sphere only a certain set pf standing waves are available. </p><p>Of course the biggest misconception about the so called wavefunction of the hydrogen atom is the fact that many people think they are really waves. Well as I have argued many times in quantum mechanics the so called waves are really probability density functions. Despite the misleading pictures of orbitals of the hydrogen atom the electron is not spread out over all space as they would seem to imply. Like any probability density function they just represent the probability that an electron of a certain energy will be seen at a certain place with a given probability. The surfaces represent the boundary at which the electron is likely to be found upto 95% of the time. But an electron is essentially a small point like particle with a definite mass, intrinsic spin and charge. Quantum mechanics does not change this picture. Also it should be remembered that the probability density functions are three dimensional functions so the electron is not in an orbit at all. But can be anywhere within that three dimensional region. Calling these three dimensional probability density functions orbitals is just so misleading. Matters are not helped by energy level diagrams showing the electron jumping from one energy level to another. But it must be remembered that these are e<i>nergy</i> level diagrams not positional diagrams. An electron does not jump from one energy level to another. It just changes energy and with each energy level the form of the probability density function changes. </p><p>Anyway having said that Spherical Harmonics just like Bessel Functions are truly amazing functions. There are a whole load of other 'Special Functions' associated with the differential equations of mathematical physics, but anyone who masters the mathematical properties of Spherical Harmonics and Bessel functions will be well placed to understand a lot of physics. </p><p>Finally it is of interest to note that the Spectrum of the Cosmic Microwave Background radiation has been expressed in terms of Spherical Harmonics. Also via the Spherical Harmonic addition theorem there is a close link between Spherical Harmonics and Bessel Functions another example of the interconnectedness of mathematical ideas. Here is my tribute to them </p><p><br /></p><p class="MsoNormal"> <span style="font-size: 16pt;">Spherical Harmonics <o:p></o:p></span></p>
<p class="MsoNormal"><b><span style="font-size: 12pt;"> </span></b></p>
<p class="MsoNormal" style="margin-left: 36pt;"><b><span style="font-size: 12pt;"> </span></b><span style="font-size: 12pt;"> </span><span style="font-size: medium;"> Many
uses we have found, <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 36pt;"><span style="font-size: medium;"> For your
functions so profound.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 36pt;"><span style="font-size: medium;"> The fields
inside a sphere,<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 36pt;"><span style="font-size: medium;"> Have become
very clear. <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: medium;"> </span></p>
<p class="MsoNormal" style="margin-left: 36pt;"><span style="font-size: medium;"> You describe
the spectrum,<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 36pt;"><span style="font-size: medium;"> Of an atoms
angular momentum.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 36pt;"><span style="font-size: medium;"> And the
background radiation,</span><span style="font-size: 12pt;"><o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 36pt;"><span style="font-size: medium;"><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> That comes from natures creation. </span><br /></span></p><p class="MsoNormal" style="margin-left: 36pt;"><span style="font-size: medium;"><span><br /></span></span></p><p class="MsoNormal" style="margin-left: 36pt;"></p><div class="separator" style="clear: both; font-size: large; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4A44DwGIItPnGRKU5i8LXsCUSstKLhRC2TErnEFOR2Ogy6kNoAgBdfteKvqQV-74jPahqC6CbSWWRjj-UoVPlEcw4CTjSZwhe-K_23wfyL-dPMtyl-HfQJIJNWYl1vN3qgF2KTdD3QMZ1/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="316" data-original-width="316" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4A44DwGIItPnGRKU5i8LXsCUSstKLhRC2TErnEFOR2Ogy6kNoAgBdfteKvqQV-74jPahqC6CbSWWRjj-UoVPlEcw4CTjSZwhe-K_23wfyL-dPMtyl-HfQJIJNWYl1vN3qgF2KTdD3QMZ1/" width="240" /></a></div><br /><span style="font-size: large;"> </span><span style="font-size: large;"> </span>Probability density funtions of an electron outside a hydrogen atom <p></p><p class="MsoNormal" style="margin-left: 36pt;"><span><br /></span></p><p class="MsoNormal" style="margin-left: 36pt;"><span style="font-size: medium;"><span><br /></span></span></p><p align="center" class="MsoNormal" style="margin-left: 36pt; text-align: center;"><br /></p>
<p><span face="Arial, sans-serif" style="font-size: 12pt;"> </span><span style="font-size: medium;"><span face="Arial, sans-serif"> <span> </span><span> </span><span> </span><span> </span></span> </span></p><p> </p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-54117442759185123092021-09-12T19:37:00.000+01:002021-09-12T19:37:10.053+01:00Ode to Bessel Functions <p>Ok here is what I believe is the first ever poem to be published on Bessel Functions. Those who studied partial differential equations will know that if a function is expressed in cylindrical coordinates the separation of variables technique will usually end up with one of the equations resulting in Bessel's equations, named after Bessel who first came across it when investigating planetary ,motion. If the equation is solved by assuming a series solution by the Froebenius method a set of orthogonal functions is obtained called Bessel Functions. These are essentially similar to cosines and sines in rectangular coordinates and any function in cylindrical coordinates can be expressed in terms of them. </p><p>Since their discovery (or invention) they have been used for all sorts of purposes to model the acoustic waves in a cylindrical tube, the electromagnetic waves in a cylinder, heat conduction in a cylinder and many other things so it is important to become familiar with them, Unfortunately the Open University Applied maths and physics courses only mention them in passing and their properties are hardly developed. A good little book although out of print but easy to find on Amazon which discusses Bessel functions and their properties is by Sneddon </p><p><a href="https://www.amazon.co.uk/Special-functions-mathematical-physics-chemistry/dp/B00316KJ0Q">https://www.amazon.co.uk/Special-functions-mathematical-physics-chemistry/dp/B00316KJ0Q</a></p><p> Perhaps the most amazing application of Bessel Functions was their use by Crick to deduce the diffraction pattern of the Double Helix. He realised that if DNA was a helix then it's diffraction pattern could be expressed in terms of Bessel Functions. Resulting in a 'Saltire' like pattern as shown in the photograph below. What is even more amazing was the fact that if there were two helices intertwined with each other then the 4 peak would be minimised and again this can be seen in the photograph below. So there we have it almost definitive proof that DNA the molecule of life is a double helix and ultimately dating back to Bessel's Functions</p><p>I have to add a sorry note to this marvelous tale, the woman who did all the hatd work in obtaining the diffraction photographs was a French woman called Rosalind Franklin, Being a women she was treated really appallingly and Watson in his book makes some quite sexist remarks about her. Alsu unfortunately she died at a fairly young age too late to get any credit when the Nobel prize was awarded. So just as in the case of Jocelyn Bell, sexist males got all the credit for a woman's work. There was a very good Horizon programme about this made in the mid 1980's I don't know if it is still available if you can get a copy it is definitely worth trying to track down. </p><p>Anyway here is the tribute to Mr Bessel and his amazing set of functions </p><p></p><p class="MsoNormal"><span style="mso-tab-count: 5;"> </span><b style="mso-bidi-font-weight: normal;"><span style="font-size: 14.0pt;">Bessel <o:p></o:p></span></b></p>
<p class="MsoNormal"> <span style="font-size: medium;"> </span><span style="font-size: medium;">What
an amazing function,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: medium;"><span style="mso-tab-count: 4;"> </span>We
can use it without compunction.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: medium;"><span style="mso-tab-count: 4;"> </span>Any
field thatโs in a tube,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: medium;"><span style="mso-tab-count: 4;"> </span>You
have put it in the groove. <o:p></o:p></span></p>
<p class="MsoNormal"><o:p><span style="font-size: medium;"> </span></o:p></p>
<p class="MsoNormal"><span style="font-size: medium;"><span style="mso-tab-count: 4;"> </span>Aided
by your Function,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: medium;"><span style="mso-tab-count: 4;"> </span>Franklin,
Crick and Watson,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: medium;"><span style="mso-tab-count: 4;"> </span>After
a long, long trial,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: medium;"><span style="mso-tab-count: 4;"> </span>Deduced
lifeโs double spiral. <o:p></o:p></span></p>
<p class="MsoNormal"><o:p><span style="font-size: medium;"> </span></o:p></p><div class="separator" style="clear: both; text-align: center;"><span style="font-size: medium;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhgzaUQfuGnGCiG4TP3C5r9e_6h5XGxjBwYNGLZ8QEp6RgCqbZUve7-lyHK1ruA7511B_HT8uo1pGcF4QfpI6my7alkb3AmA6X0kG1W8bUnbAK_EZFxXWEylKiXNKYvuy3Wlbhyphenhyphen50wR8ow/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="288" data-original-width="283" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhgzaUQfuGnGCiG4TP3C5r9e_6h5XGxjBwYNGLZ8QEp6RgCqbZUve7-lyHK1ruA7511B_HT8uo1pGcF4QfpI6my7alkb3AmA6X0kG1W8bUnbAK_EZFxXWEylKiXNKYvuy3Wlbhyphenhyphen50wR8ow/" width="236" /></a></span></div><span style="font-size: medium;"><br /></span><p></p>Rosalind Franklin's photograph of DNA, note the missing peak this indicates that there are two helices interwined with each other. <p></p><p><br /></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com1tag:blogger.com,1999:blog-7897235423812277683.post-37645191414041726502021-09-10T20:15:00.002+01:002021-09-10T20:15:58.266+01:00Mr Fourier and his series <p> One of the marvels of mathematics is the Fourier series which shows how to either decompse a complex function into a sum of sines and cosines or vice versa build up a complex function by summing over various sines and cosine functions. This idea has many applications in partial differential equations and any function expressed in rectangular co-ordinates can be expressed in terms of them. The separate sine and cosine terms can be seen as vectors in a function space. In that they are orthogonal to each other thus there is a an amazing analogy between geometric vectors and vectors in function space. This can be extended to other coordinate systems as we shall see in the next two poems. One of the most amazing application of Fourier series is the ability to synthesize complex sounds and make new music. An application that I bet Fourier never thought of. So next time you are struggling to solve a problem in Fourier series just remember how useful they can be. Here is the tribute. </p><p class="MsoNormal" style="margin-left: 180.0pt;"><span style="font-size: 16.0pt;">Mr
Fourier <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt;"> </span></p>
<p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">You
discovered something quite profound,<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">How
to create a complex sound.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">From
itโs component bits, <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">To
make a wave that fits. <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;"> </span></p>
<p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">All
these waves combine,<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">Into
a sound thatโs fine. <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">The
possibilities are endless, <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">To
create sounds quite stupendous</span><span style="font-size: 12.0pt;">.<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: 12.0pt;"><br /></span></p><p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: 12.0pt;"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaFfTVPM8uGbcJJa8dm8xe-N95BObdLHysOI-AzESsg0FpvJFkN7aY3Hq16HB6eFbspTRBWTVGJHV_aXqCO7UlFkx3Mtr-qsAj8z46i0aj0wZpXbVvTFwb5kMgSmXQnlTHbH_vUtPjoRyp/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="340" data-original-width="454" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaFfTVPM8uGbcJJa8dm8xe-N95BObdLHysOI-AzESsg0FpvJFkN7aY3Hq16HB6eFbspTRBWTVGJHV_aXqCO7UlFkx3Mtr-qsAj8z46i0aj0wZpXbVvTFwb5kMgSmXQnlTHbH_vUtPjoRyp/" width="320" /></a></div><p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: 12.0pt;"><br /></span></p><p></p><p>A Moog synthesiser must be much more fun playing around with one of these than the digital synthesisers which you can generate using a computer A concrecte application of Fourier series. </p><p><br /></p><p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: 12.0pt;"><br /></span></p>
<p><span style="font-size: 12pt;"> </span> </p><p><br /></p><p><br /></p><p><br /></p><p><br /></p><p><br /></p><p> </p><p><br /></p><p><br /></p><p><br /></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-90068270197047528342021-09-07T22:09:00.000+01:002021-09-07T22:09:28.706+01:00Ode To Calculus <p> The next few posts will be a tribute to mathematics or more strictly the branches of maths that are useful to physicists. For physicists as a whole mathematics is secondary to nature. It is a moot question whether maths is discovered or created. I tend to the view that it is essentially created which no doubt will surprise people after all I am a scientitst and I use mathematics every day and it works so it must be true. Well to some extent, but mathematics can only ever provide a good approximation to an underlying reality. Anyone who has solved a differential equation will know that of all the possible solutions only one will represent the reality that one is trying to model. Certainly it is amazing how given a few basic facts about the world mathematics can extend the relationships and maybe even uncover relatiohnships that we didn't think were there. Maxwell's equations for example. </p><p>But let's not get carried away there are certain constraints on the world. Signals cannot travel faster than the speed of light, there are only 4 space time dimensions, there is only one universe. Any theory such as superstrings or the many worlds interpretation of quantum mechanics which contradict those basic facts is just fantasy and should be called out as such.Also no amount of mathematical reasoning will explain why the Gravitational coupling constant is what it is or the mass of the electron or any fundamental particle. These are givens and any mathematics which claims otherwise is just over reaching itself. There is no need to invoke as some people do God or the anthtopic principle to explain why the world is the way it is, ii just is. Of course once we know some basic facts we can use mathematics to deduce inter-relationships between those facts and that is amazing. One of the reasons it works so well is that nature is approximately linear and calculus exploirs this linearity</p><p>Without the invention (discovery) if you prefer of calculus there would be no explanation of the world as it is. As it was when Newton and Leibniz invented calculus the laws of motion were explained and the fact that planets moved in ellipses was also deduced thus being one of the greatest intellectual achievements of mankind. The Open university used to include a couple of units in the old course MST209 which has been superseded by MST210 from which this has been dropped. I think that is a pity if anyone reading this blog would like a copy of the missing units please contact me on </p><p><a href="mailto:chrisf19572002@yahoo.co.uk">chrisf19572002@yahoo.co.uk</a> </p><p>as it is one of the greatest intellectual achievements of mankind. If you want to understand how physics works then you really must learn some calculus I think it a shame that the currrent Open university course in physics S217 minimizes the use of calculus in it's units. It is often claimed by Arts students that you are ignorant if you haven't read a certain book or watched say a certain Shakespeare play, Well conversely I would argue that if you don't know calculus to at least A level maths and why physicists use it to explain how nature works you are equally just as ignorant. Anyway here is my tribute to one of the greatest inventions of mathematics </p><p></p><p align="center" class="MsoNormal" style="text-align: center;"><b style="mso-bidi-font-weight: normal;"><span style="font-size: 14.0pt;">Calculus <o:p></o:p></span></b></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">You explain how nature works, <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Itโs a task we should not shirk.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Integration and Differentiation<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Gives me such exhiliration<sup>1)</sup>
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><o:p><span style="font-size: medium;"> </span></o:p></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">All the laws of motion,<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Are explained without commotion.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">When weโve mastered you,<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">There is so much we can do. </span><o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoListParagraph" style="margin-left: 29.9pt; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -18.0pt;"><!--[if !supportLists]--><span style="mso-bidi-font-family: Arial; mso-fareast-font-family: Arial;"><span style="mso-list: Ignore;">1)<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]-->I realise not everyone feels the same way <span style="font-family: Wingdings; mso-ascii-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">J</span></span> <o:p></o:p></p><p class="MsoListParagraph" style="margin-left: 29.9pt; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -18.0pt;"><span style="font-family: Wingdings; mso-ascii-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;"><br /></span></span></p><p class="MsoListParagraph" style="margin-left: 29.9pt; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -18.0pt;"><span style="font-family: Wingdings; mso-ascii-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;"></span></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPde6tCEonTpQS7h_4iBUMwThwrVIWgV65USLx3XJnfHy396JIwbA2G0Ho5U8BQNTdyNLKY1SXRA5JAlZainJ8ALFySJKnzveMao4zm2sMV8sjCvhkqyWu3nAT4uJQ1HWDcXTEv4GPYeRW/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="377" data-original-width="440" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPde6tCEonTpQS7h_4iBUMwThwrVIWgV65USLx3XJnfHy396JIwbA2G0Ho5U8BQNTdyNLKY1SXRA5JAlZainJ8ALFySJKnzveMao4zm2sMV8sjCvhkqyWu3nAT4uJQ1HWDcXTEv4GPYeRW/" width="280" /></a></div><br /><br /><p></p><br /><p></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-23403414967850657602021-09-05T13:12:00.000+01:002021-09-05T13:12:38.592+01:00Jocelyn Bell <p> This poem pays a tribute to Jocelyn Bell who with the aid of her phased array antenna the she built and maintained (Radio telescope) discovered Pulsars At first she noticed a small amount of persistent noise in her plots which she jokingly referred to as LGM (little Green men). The pulses were regularly and quite rapid. For a while there was speculation that they may indeed be signals from outer space and of course the media latched on to it. However it was soon realised that they could in fact be neutron stars and this was later confirmed. A neutron star is a much more compact object than a white dwarf and represents another stable state of stellat collapse. Oppenheimer and his student Volkoff had worked out that one of the implications of General relativity was indeed this new stable state. One of the signals of a neutron star is the fact that it rotates rapidly (due to conservation of angular momentum) hence the short bursts that Jocelyn Bell measured. In one of the great scandals despite having done all the work it was her supervisor that got the credit and the subsequent Nobel prize. This is typical of the sexism that prevailed in those days and probably still lingers on today. Jocelyn Bell continued to develop her career as an Astrophysicist and was for a time Professor of Astronomy and Astrophysics for the Open University. As I am an antenna engineer it would be interesting if I can find out the details of Jocelyn Bell's antenna and plot some typical radiation patterns watch this space</p><p>Anyway her is my tribute to her </p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 16.0pt;">Jocelyn
Bell <o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12.0pt;"> </span></p><p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">What
was that strange pulse from the stars,<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">Could
it be little men from Mars.<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">No
it was a star so dense, <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">Made
of neutrons with mass immense. <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;"> </span></p><p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">Spinning
round so fast, <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">Emitting
pulses in a blast. <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">This
marvel you found,<o:p></o:p></span></p><p>
</p><p class="MsoNormal" style="margin-left: 108.0pt;"><span style="font-size: medium;">With
your antenna on the ground. <o:p></o:p></span></p><p><span style="font-size: medium;"> </span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-size: medium;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9BKIcsvhZtNGEGEKQycz8fXkDnDufjEL-kTHlRZUI_pAPTP26ZMSNJRGYJmUaehCdKCzrEzZYDclz5MdhLDJX3FDyBlkSguvDTkvhxMJd1hPaRmJJTcZu_bw-J5lWOlca8nvpuNLQIFNn/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="385" data-original-width="454" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9BKIcsvhZtNGEGEKQycz8fXkDnDufjEL-kTHlRZUI_pAPTP26ZMSNJRGYJmUaehCdKCzrEzZYDclz5MdhLDJX3FDyBlkSguvDTkvhxMJd1hPaRmJJTcZu_bw-J5lWOlca8nvpuNLQIFNn/" width="283" /></a></span></div><span style="font-size: medium;"><br /></span><p></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com1tag:blogger.com,1999:blog-7897235423812277683.post-62241041786310148082021-09-04T13:45:00.001+01:002021-09-04T14:07:41.849+01:00Chandrasekhar<p> This poem is a tribute to Chandrasekhar, probably one of the greatest astro-physicists ever. He was a prolific Astro-physicist probably an Astrophysical equivalent of Ramanujan the Indian mathematician. Like Ramanujan Chandrasekhar came to Cambridge to Study amongst other people Eddington who had demonstrated that Einstein's theory of relativity was correct by measuring the bending of light by the sun during an eclipse in 1919. Also he amongst others worked out the basic equations governing stellar structure which are still used today and was able to reproduce the Herzsprung Russell diagram. However Eddington did not have any idea what happened to a star when it died out. Chandresekhar using the relatively new ideas of statistical physics as applied to fermions and the Pauli exclusion principle worked out that there was a tension between the radiation pressure of a collection of fermions and their graviational attraction. he worked out that if the mass of the star was less than about 1.4 solar masses the resulting configuration would be stable and it's radious would much smaller than the radius when it was active. The matter would be ejected in a glorious explosion giving rise to a supernovae. The final state would be a white dwarf. All this was fine, but the consequence of this was that if the mass was greater than 1.4 solar masses the star would continue to collapse indefinitely and become a black hole (although that name hadn't been used back then). Chandrasekhar presented this idea at a meeting of the Royal Astronomical society. Somewhat unfairly even though Eddington knew about Chandrasekhar's results he waited until Chandrasekher had given his presentation to raise his objections. Of course Eddington's objectons, being who he was, prevailed and as a result the idea of black holes was suppressed for about 30 years. Chandrasekhar after he got his PhD moved to Chicago and continued to do work in Astrophysics and contributed to the ideas of stability in Fluids leading to some early investigations of phenomenon that would be later recognised as prefiguring chaos theory. In the end of course Chandrasekhar was vindicated. He wrote a book on Stellar structure which is still relevant today </p><p>An introduction to the Study of Stellar Structure </p><p><a href="https://www.amazon.co.uk/Introduction-Study-Stellar-Structure-Astronomy/dp/0486604136/ref=sr_1_1?dchild=1&keywords=An+introduction+to+Stellar+structure+Chandrasekhar&qid=1630760779&s=books&sr=1-1">https://www.amazon.co.uk/Introduction-Study-Stellar-Structure-Astronomy/dp/0486604136/ref=sr_1_1?dchild=1&keywords=An+introduction+to+Stellar+structure+Chandrasekhar&qid=1630760779&s=books&sr=1-1</a></p><p><br /></p><p>And hid definitive text on black holes including a full derivation of the relativistic equations governing the Kerr metric (one day I'll get round to understanding this :) )</p><p><a href="https://www.amazon.co.uk/Mathematical-Theory-Classic-Physical-Sciences/dp/0198503709?asin=0198503709&revisionId=&format=4&depth=1">https://www.amazon.co.uk/Mathematical-Theory-Classic-Physical-Sciences/dp/0198503709?asin=0198503709&revisionId=&format=4&depth=1</a></p><p>Although Chandrasekhar does include the health warning that at times the line of reasoning will be quite obscure as there are many leaps in some of the derivations. Perhaps with the aid of a symbolic manipulator such as Maple or Mathematics it might be possible to understand these derivations. </p><p>There is also a summary of his work on instabilities in fluids </p><p><a href="https://www.amazon.co.uk/gp/product/B00C59C7ZA/ref=dbs_a_def_rwt_hsch_vapi_tkin_p1_i0">https://www.amazon.co.uk/gp/product/B00C59C7ZA/ref=dbs_a_def_rwt_hsch_vapi_tkin_p1_i0</a></p><p>Finally a good general overview of the history of the development of Black holes and stellar stability is given by Kip Thornes book </p><p><a href="https://www.amazon.co.uk/Black-Holes-Time-Warps-Commonwealth/dp/0393312763">https://www.amazon.co.uk/Black-Holes-Time-Warps-Commonwealth/dp/0393312763</a></p><p>Anyway here is my tribute to this fine Astrophysicist </p><p class="MsoNormal"><span style="font-size: 16pt;"><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span>Chandrasekhar<o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 16pt;"> </span></p><p class="MsoNormal"><span style="font-size: 12pt;"> </span><span style="font-size: medium;"> When a star collapses,<o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: medium;"> There are no relapses. <o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: medium;"> If itโs mass is really
great, <o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: medium;"> There is no final resting
state.<o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: medium;"> </span></p><p class="MsoNormal"><span style="font-size: medium;"> This idea made Eddington
frown,<o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: medium;"> After his great big put
down.<o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: medium;"> In science there was a
great lack,<o:p></o:p></span></p><p>
</p><p class="MsoNormal"><span style="font-size: medium;"> Until people found holes
that are black. <o:p></o:p></span></p><p><span style="font-size: medium;"></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-size: medium;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiYk0DdsB-SZbisdEvIgRfwZFoHnc03tIuhuqQq3Z0p-wtModhWEpirBJUN2q-31JG6DjB92mqNUHybXmTFwiEmJF9kh3epsWr2OPbvJWfLjuwgHRCwSTxJIkgeFLEfgJr0eIWYZV396y0Y/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="380" data-original-width="302" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiYk0DdsB-SZbisdEvIgRfwZFoHnc03tIuhuqQq3Z0p-wtModhWEpirBJUN2q-31JG6DjB92mqNUHybXmTFwiEmJF9kh3epsWr2OPbvJWfLjuwgHRCwSTxJIkgeFLEfgJr0eIWYZV396y0Y/" width="191" /></a></span></div><span style="font-size: medium;"><br /><br /></span><p></p><p><span style="font-size: medium;"> </span></p><p><br /></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-55891845120993548502021-09-01T20:31:00.000+01:002021-09-01T20:31:12.006+01:00Alan Turing <p> This is a short poem about Alan Turing, who as many people will know played a part in helping decode the Enigma machine and thus helping to save many lives in the war of the Atlantic. He became interested in the whole idea of computing and whether machines could think just at the right time. It is still an open question whether or not a machine can think but my chess computer regularly thrashes me at chess and I get computers to perform long tedious calculations not just involving numbers but stuff like integrals as well. I believe you can get computers to evaluate Feynman Integrals so for many well defined problems computers definitely can think even if it is not in the way that we do. Has this solved the problem of consciousness well not exactly but it is at least a step in the right direction. </p><p>Unfortunately whilst Turing was good at decoding secrets he wasn't able to hide from the world that he was Gay and indulged in sexual activity which at that time was considered illegal/ Had he just kept in within a Cambridge college or at public school he would have probably got away with it. But it was the fact that he indulged in sex with working class men that seemed to alarm the authorities. So despite being a war hero Alan Turing was convicted and forced to take oestrogen which had the effect of making him impotent. As he couldn't live with this he committed suicide a tragic loss both to the country and the mathematical community. The narrow mindedness of the authorities really deserves condemnation although I suspect that there are still a substantial number of people Evangelical Christians in particular who would want to make Homo-Sexuality illegal again. Well let's hope they don't succeed. </p><p></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="font-size: 16.0pt;">Alan Turing <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt;"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Build
a machine that can think, <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Otherwise
the ships will sink.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Many
lives were saved by you, <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">And
your talented Bletchley crew.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><o:p><span style="font-size: medium;"> </span></o:p></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Yes
you cracked Enigma,<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">But
because of societies stigma,<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">The
secret you could not hide, <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;">Led
ultimately to your suicide.</span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;"><br /></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: medium;"></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-size: medium;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjp96PWqGXNomVq3gryieFe2Z8fcnXb1EC0S6V72ScCHPeo0sfP7F0dfK5gcdfsxOo_fSMK8QS7rp28Fawp2t-q0x_RQGn3A-FI9Y_Z1puW2tLHNHaGSG7Xk0SowrUPDMUC-X9iUQcy4C52/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="340" data-original-width="340" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjp96PWqGXNomVq3gryieFe2Z8fcnXb1EC0S6V72ScCHPeo0sfP7F0dfK5gcdfsxOo_fSMK8QS7rp28Fawp2t-q0x_RQGn3A-FI9Y_Z1puW2tLHNHaGSG7Xk0SowrUPDMUC-X9iUQcy4C52/" width="240" /></a></span></div><span style="font-size: medium;"><br /><br /></span><p></p><br /><p></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-26106122738124225542021-08-31T13:24:00.002+01:002021-08-31T23:09:24.178+01:00James Clerk Maxwell <p> Ok as the first tribute (of many) this poem is about the greatest physicist of the 19th century namely James Clerk Maxwell. Whilst his main contribution was in unifying electromagnetism (the first unification theory) he also developed the Kinetic theory of gases and the theory of colour. When he was 17 he deduced that the rings around Saturn could not be solid but had to consist of small bits of rock. For this he was awarded the Adams prize essay.</p><p> Anyway his main achievement is of course summarising Faraday's work on electromagnetism into 4 equations and then he realised that applying some mathematical transformations using quaternions that electromagnetic waves had to exist. He missed a trick though in that it was thought like all waves there had to be a medium for which the waves had to be propagating through. It wasn't until Einstein came along when it was realised that the so called Ether was not necessary. Indeed electromagnetic waves are self generating in that an oscillating electric field generates an oscillating magnetic field which in turn generates an oscillating electric field so the wave self propagates without any underlying media. Also whilst Maxwell used quaternions it wasn't until Heaviside reformulated them in terms of vector calculus that the structure of Maxwell's equations became apparent. </p><p>Whilst Maxwell's equations are seen as part of classical physics, they are relativistically invariant and are still used today in many fields such as antenna engineering which is my specialist field of study. Furthermore despite the alleged importance of quantum mechanics in understanding the two slit experiment there is no quantum mechanical explanation of the two slit experiment. Indeed it is difficult to see how there could be as Schrodinger's equation is a scalar equation whereas Maxwell's equations involve vector quantities such as Electric and Magnetic fields. There is no concept of the near field from Schrodinger's equation I will expand on these points later. In the mean time here is a verbal summary of the four Maxwell equations </p><p>1) The electic field of a source is proportional to the total charge enclosed by a surface surrounding that source </p><p>2) There is no magnetic charge </p><p>3) If I wiggle a magnetic in front of a conductor there will be an electric current associated with it and the faster the magnetic is wiggled then the greater the electric current will be </p><p>4) An electric current passing through a wire generates a magnetic field </p><p>Put 3 and 4 together and you end up with a wave equation the velocity of which is the same as the speed of light so light is in fact an electromagnetic wave. Truly magical </p><p>Anyway here is the poem hope you enjoy it </p><p class="MsoNormal"><b><span style="font-size: 16pt;"> James
Clerk Maxwell <o:p></o:p></span></b></p><p class="MsoNormal"><b> </b></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">Four basic facts all combine,<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">To show how fields intertwine. <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">Static or Dynamic,<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">It really is magic. <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144pt;"><o:p><span style="font-size: medium;"> </span></o:p></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">A wiggly field generates another,
<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">Which in turn regenerates the
other. <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">With this great insight,<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">We now understand light.</span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;"><br /></span></p><p class="MsoNormal" style="margin-left: 144pt;"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDKWeCR63Phc7O9l7hGYXK11xsfELY5WJ7AqsYTYlgMjP0X_8jnm10PLJNBOlNKjGz7mhP9f0B6f6gYYa76teUploxJmsonOjJCWAhk_pnLDYnV5Nw8F97Bdzq_fY4kTq1iWp_I06OXS1u/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="354" data-original-width="283" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDKWeCR63Phc7O9l7hGYXK11xsfELY5WJ7AqsYTYlgMjP0X_8jnm10PLJNBOlNKjGz7mhP9f0B6f6gYYa76teUploxJmsonOjJCWAhk_pnLDYnV5Nw8F97Bdzq_fY4kTq1iWp_I06OXS1u/" width="192" /></a></div><br /><br /><p></p><p>
</p><p class="MsoNormal"><o:p> </o:p></p><p> </p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-21756431120089512092021-08-30T23:08:00.004+01:002021-09-01T00:24:34.788+01:00 Tribute to science <p>Ok as promised this is the first of many poems which pays a tribute to science and various scientists, The first poem is a general tribute to how science has played an important part in changing life for the better and challenging vested interests. I find it quite depressing that even today there are people who dispute the basic facts of science because it challenges pre-conceived ideas about say creation as in fundamentalist evangelical circles creating a suspicion of science. </p><p>Far closer to home are those people who deny climate change and who are given a platform from the media in the interests of so called balance. Well I am sorry if 95% of scientists have strong evidence for the fact that mankind is accelerating climate change then there is no way a person such as say Nigel Lawson or Donald Trump should be allowed equal media coverage because it suits their interests in order to do so. Their views are minority ones and have nothing to do with science. </p><p>Similarly for those idiots who refuse to take vaccinations the scare about the possible link between MMR and autism has meant that measles has come back when to all intents and purposes it was eradicated. Those journalists such as Melanie Philips who supported Andrew Wakefiled should hang their heads in shame. Similarly the only way we have any hope of defeating the Corunna virus is to develop vaccines and if need be take boosters as and when necessary. If you want to end the lockdown restrictions then make sure you take the vaccine. </p><p>Finally I strongly resist the current trend of the transgender lobby to defy science and for men to claim that because they feel like a woman they should be allowed to invade women's private spaces such as toilets or womens showers. Or take part in women's sports or women only short list/ It is biologically impossible for a man to change into a woman no matter how many hormones they take and anyone who claims otherwise is just denying science. I hope this nonsense will stop soon but the way things are going it seems highly unlikely. I have a number of female colleagues who are genuinely worried about this and I hope they win their battles against this unscientific nonsense. </p><p> Anyway here is my Tribute to science in general I hope you enjoy it. </p><p></p><p class="MsoNormal"><span style="mso-tab-count: 4;"> </span><span style="font-size: 14pt;">Ode To Science <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">In the midst of Manโs inanity,<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">Science brings a voice of sanity.
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">Getting rid of superstition,<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">You improve our condition. <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144pt;"><o:p><span style="font-size: medium;"> </span></o:p></p>
<p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">Vested interests they despair,<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">When the truth is bought to bear.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">Only with you will we find, <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: medium;">What it is that drives Mankind</span>. <o:p></o:p></p><br /><p></p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJR2U3nmF36scq5Aj4z9JKyQWWSk5sufEe0iVXy6kn_TyEAM-wVYvlpN9GdEEk4ka93eDcIkZjVXPBPwWi-W6aJoEq5iTp6Yfd7ZAmNou29M0FeTKsg2h4R7Nf3rZXigwwB1eFEOzgXs0J/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="302" data-original-width="454" height="213" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJR2U3nmF36scq5Aj4z9JKyQWWSk5sufEe0iVXy6kn_TyEAM-wVYvlpN9GdEEk4ka93eDcIkZjVXPBPwWi-W6aJoEq5iTp6Yfd7ZAmNou29M0FeTKsg2h4R7Nf3rZXigwwB1eFEOzgXs0J/" width="320" /></a></div><br /><br /><p></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-53251214599247506882021-08-25T17:37:00.002+01:002021-08-25T17:37:23.782+01:00Tribute to David Attenborough part 2 <p> This is the second batch of poems paying tribute to David Attenborough's life series. This time we cover </p><p>1) Reptiles </p><p>2) Birds </p><p>3) Mammals this one focuses on our nearest primate the chimpanzee</p><p>I dare say others will follow in the fullness of time But 'Uncle David' for your achievement in helping bring nature close to us. I salute you </p><p>The first poem is on reptiles which are closely related to dinosaurs what happened to the dinosaurs is an interesting scientific problem and as yet there is still no answer. It is amazing to think that they ruled the earth for 150 million years whilst human civilisation has only been around for about 100,000 years and recorded history about 20000 at the most. Anyway here is the poem </p><p class="MsoNormal"><span style="font-size: 16.0pt;"> Reptiles <o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12.0pt;"> </span></p><p class="MsoNormal"><span style="font-size: 12.0pt;"> Relics of an age long
gone,<o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12.0pt;"> Nature still allows you
to throng.<o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12.0pt;"> From the lizard to the
crocodile,<o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12.0pt;"> Youโll still be here for
a while.<o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12.0pt;"> </span></p><p class="MsoNormal"><span style="font-size: 12.0pt;"> When Dinosaurs ruled the
earth,<o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12.0pt;"> You were there, at their
birth.<o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12.0pt;"> Now they are gone, I
wonder why,<o:p></o:p></span></p><p>
</p><p class="MsoNormal"><span style="font-size: 12.0pt;"> What it was, caused them
to die.<o:p></o:p></span></p><p class="MsoNormal"><span style="font-family: inherit;">And here is a picture of a magnificent crocodile best keep your distance </span></p><p class="MsoNormal"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlsAb87wJivJShPKtOK8WsqGCNYn7F4-ytKRSriWOOKPT7Prc-jCRC-R1gqAgi2aVFjbf0xRES6kkVFJoUpWHDBpaSp6dKB2YepTVIora1Twcuz6sQinGcuYvAJF7I8L7_7BB0EHQwiowK/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="302" data-original-width="454" height="213" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlsAb87wJivJShPKtOK8WsqGCNYn7F4-ytKRSriWOOKPT7Prc-jCRC-R1gqAgi2aVFjbf0xRES6kkVFJoUpWHDBpaSp6dKB2YepTVIora1Twcuz6sQinGcuYvAJF7I8L7_7BB0EHQwiowK/" width="320" /></a></div><br /><br /><p></p><p class="MsoNormal">The second one is on birds very inspiring creatures with their ability to fly, they have always inspired us long may they do so.</p><p class="MsoNormal" style="margin-left: 180.0pt;"><span style="font-size: 16.0pt;">Birds
<o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12.0pt;"> </span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">From
the eagle that soars,<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">To
the talking Macaws. <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">In
variety there are plenty,<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">Without
you life would be empty. <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;"> </span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">When
you take flight, <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">Itโs
a truly wonderful sight.<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">Helping
us to aspire, <o:p></o:p></span></p><p class="MsoNormal">
</p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">To
all that we desire.</span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-family: inherit;"><br /></span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-family: inherit;">Here is a picture of a an Eagle wonderful </span></p><p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-family: inherit;"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPAnlYy_rn-Ia7P3fvw5IOxdGk-Q3irgSmmIAUThf1HWyfEi184bFfKgdluqp-AKMBUiguQttPc4-Pm2PyRgwLzhq5_Wlaj_luP4LAMBTweBpF5By_tqqdnQHDofrELDZaw6buOswtSa53/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="255" data-original-width="454" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPAnlYy_rn-Ia7P3fvw5IOxdGk-Q3irgSmmIAUThf1HWyfEi184bFfKgdluqp-AKMBUiguQttPc4-Pm2PyRgwLzhq5_Wlaj_luP4LAMBTweBpF5By_tqqdnQHDofrELDZaw6buOswtSa53/" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><p></p><p>Finally here is a poem about the species closest to us namely the chimpanzee. I hope the days when chimps were used to entertain us via chimpanzee's tea parties are long gone. Never to return </p><p></p><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><p class="MsoNormal"><span style="font-size: 16.0pt;"><span style="mso-tab-count: 5;"> </span>Chimpanzees<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt;"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">We
used to laugh at the Chimpanzee,<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">Making
them drink our tea.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">But
now we know this primate, <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">Is
our species closest mate. <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">Closer
to them than we thought,<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">Let
us treat them as we ought.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">Open
the zoos and let them free, <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 144.0pt;"><span style="font-size: 12.0pt;">Swinging
from tree to tree.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 0cm;"><span style="font-size: 12.0pt;"><o:p> </o:p></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6hXC5-Ish5Wx3SKmvvN0RSHBZ_SWn61QtdKP6aFoJyozmUNx4poMKz2kedOdGUzqpgu-3l-fkKOzWQeH0jqEpbn7v2iA-JHJSRIksuutB-9B447xPLiJkwgJRY76abxrO1IHvRxvJxZgv/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="420" data-original-width="602" height="223" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6hXC5-Ish5Wx3SKmvvN0RSHBZ_SWn61QtdKP6aFoJyozmUNx4poMKz2kedOdGUzqpgu-3l-fkKOzWQeH0jqEpbn7v2iA-JHJSRIksuutB-9B447xPLiJkwgJRY76abxrO1IHvRxvJxZgv/" width="320" /></a></div><br /><p></p><p class="MsoNormal" style="margin-left: 0cm;"><span style="font-size: 12.0pt;"><o:p><span style="font-family: inherit;">I will start publishing a whole load of poems about science and mathematics soon </span></o:p></span></p><p class="MsoNormal" style="margin-left: 0cm;"><span style="font-size: 12.0pt;"><o:p><span style="font-family: inherit;"><br /></span></o:p></span></p><br /><p></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-84916244075863650972021-08-21T17:58:00.001+01:002021-08-21T18:14:32.216+01:00Tribute to David Attenborough Part 1 <p> Hi these two poems have been inspired by David Attenborough's 'Life series' a truly amazing set of DVD's There are five specialist titles in the series </p><p>Life of Plants</p><p>Life in the Undergrowth</p><p>Life in Cold Blood </p><p>Life of Birds </p><p>Life of the Mammals </p><p>These two poems are about flowers and insects. Which complement each other nicely. It is normal to stress how nature is essentially about competing species fighting for scarce resources. However plants and insects cooperate with each other in a way that is beneficial to both. One doesn't have to be a creationist or particularly religious to admire the ingenuity in which both insects and plants cooperate with each other. How does a particular orchid know for example how to disguise it's flowers as a female wasp so that it will attract male wasps. Truly amazing and without 'Uncle David' this amazing fact about nature would be hidden from us. So a truly amazing series and if you haven't seen it then I would urge you to get hold of it. Start with life on earth the best introduction to evolution there is even if the photography is a bit dated.</p><p class="MsoNormal" style="margin-left: 36pt;"> <span style="font-size: 16pt;">Flowers <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: 12pt;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12pt;"> Your blooms are such a
sight,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: 12pt;"> Giving us all great
delight.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: 12pt;"> But all that, you have
planned, <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: 12pt;"> To get insects, on you to
land. <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: 12pt;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12pt;"> The little bug might
think, <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: 12pt;"> They have had a nice free
drink.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: 12pt;"> But for you, itโs also
great, <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: 12pt;"> Now you have a chance to
mate. <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: 12pt;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12pt;"> <span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span></span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_fu_YoJ1UGowdARZPCfGX9jrfe1zgYxShGEGoDW3ejptsA_vevvmr8TFmj5pRRuwtXJlMCgMmizkQ0D6IUjriID5F6a6tb8fGulRNgZzKJ6okUzZwsulVO9FSRfBOHHvd56gz4llRnG4q/" style="font-size: 12pt; margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="" data-original-height="301" data-original-width="302" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_fu_YoJ1UGowdARZPCfGX9jrfe1zgYxShGEGoDW3ejptsA_vevvmr8TFmj5pRRuwtXJlMCgMmizkQ0D6IUjriID5F6a6tb8fGulRNgZzKJ6okUzZwsulVO9FSRfBOHHvd56gz4llRnG4q/" width="241" /></a><span style="font-size: 12pt;"><span><br /></span></span></p>
<p><span style="font-size: 12pt;"> </span> </p><p>And here is the one on insects generally speaking we avoid insects and indeed David Attenborough shows some pretty gruesome ones including a giant millipede which preys on bats. However without them the soil would not be aereated and of course the flowers would not be pollinated.</p><p class="MsoNormal"><span style="font-size: 16pt;"> <span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span>Insects</span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: 12pt;">Nasty
horrible little things,<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: 12pt;">That
quite often stings. <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: 12pt;">But
if you did not toil,<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: 12pt;">The
earth would have no soil. <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: 12pt;"> </span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: 12pt;">The
flowers would not grow,<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: 12pt;">They
would not put on a show. <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: 12pt;">On
you so much depends, <o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: 12pt;">To
help nature meet itโs ends.</span></p><p class="MsoNormal" style="margin-left: 144pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguH4ASXbxep4Fy2bB0TMF27q90iKLDo6RjDdHaKFWyeZSbR76ZGHgaV16Es1fgJdFbSwbAgYupXvToEkEpfdgUyy5j4wQEKCFOdWkRWftVpF0wyHZ3I7uptlC9iRP6xPuwlhTcVdUgnz_P/s360/Ants.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="151" data-original-width="360" height="134" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguH4ASXbxep4Fy2bB0TMF27q90iKLDo6RjDdHaKFWyeZSbR76ZGHgaV16Es1fgJdFbSwbAgYupXvToEkEpfdgUyy5j4wQEKCFOdWkRWftVpF0wyHZ3I7uptlC9iRP6xPuwlhTcVdUgnz_P/s320/Ants.jpg" width="320" /></a></p><p class="MsoNormal" style="margin-left: 144pt;"><br /><span style="font-size: 12pt;"><br /></span></p><p class="MsoNormal" style="margin-left: 144pt;"><span style="font-size: 12pt;"><br /></span></p><p class="MsoNormal" style="margin-left: 144pt; text-align: justify;"><br /></p><p class="MsoNormal" style="margin-left: 144pt;"> </p><p><br /></p><p><br /></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com6tag:blogger.com,1999:blog-7897235423812277683.post-69413620739075521192021-08-19T18:19:00.001+01:002021-08-19T18:33:50.669+01:00Three more poems <p> For some inexplicable reason I have been inspired to write some poems on all sorts of topics which I will post over the next few weeks Today's batch are three more poems about animals </p><p>1) Urban Foxes </p><p>2) Little Mousey</p><p>3) Boris the Spider </p><p>Here is the first one/ Unlike seagulls which are a menace it is nice to see foxes in the urban areas so here is my tribute to them </p><p><br /></p><p align="center" class="MsoNormal" style="text-align: center;"><b><span face=""Arial",sans-serif" style="font-size: 14pt; line-height: 107%;">Urban
Foxes</span></b></p><p class="MsoNormal" style="text-align: center;"><span face="Arial, sans-serif" style="font-family: arial; font-size: medium;">Escaping to the Towns,</span></p><p class="MsoNormal" style="text-align: center;"><span face=""Arial",sans-serif" style="font-family: arial; font-size: medium;"> Running from the nasty hounds</span></p><p class="MsoNormal" style="text-align: justify;"><span style="font-family: arial;"><span style="font-size: medium;"> And the men in red,</span></span></p><p class="MsoNormal" style="text-align: center;"><span face=""Arial",sans-serif" style="font-family: arial; font-size: medium;">Who just want you dead.</span></p><p class="MsoNormal" style="text-align: center;"><span style="font-family: arial;"><span style="font-size: medium;"><br /></span></span></p><p class="MsoNormal" style="text-align: justify;"><span style="font-family: arial;"><span style="font-size: medium;"> Coming out, just at night,</span></span></p><p class="MsoNormal" style="text-align: center;"><span face=""Arial",sans-serif" style="font-family: arial; font-size: medium;">So that you donโt get a fright.<o:p></o:p></span></p><p class="MsoNormal" style="text-align: center;"><span face=""Arial",sans-serif" style="font-family: arial; font-size: medium;">Let us hope you have found, <o:p></o:p></span></p><p>
</p><p class="MsoNormal" style="text-align: center;"><span face=""Arial",sans-serif" style="font-family: arial; font-size: medium;">A place thatโs safe and sound</span></p><p class="MsoNormal" style="text-align: center;"><br /></p><p class="MsoNormal" style="text-align: justify;">And a cute picture </p><p class="MsoNormal" style="text-align: justify;"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIQJmzXc4TQTX_bNDIAFeaSqUZ8wOZ1AE4d0gJWyL6G1O9TYejIQqbJOkjgVPjLks2dPpxC118jWfG2zzn9vfJxgaOwXLO2CDvR2OfEt7F6fksN1Fx9-4lWi1CiuomTssYvbkOdbjAnXxN/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="302" data-original-width="454" height="213" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIQJmzXc4TQTX_bNDIAFeaSqUZ8wOZ1AE4d0gJWyL6G1O9TYejIQqbJOkjgVPjLks2dPpxC118jWfG2zzn9vfJxgaOwXLO2CDvR2OfEt7F6fksN1Fx9-4lWi1CiuomTssYvbkOdbjAnXxN/" width="320" /></a></div><br /><br /><p></p><p style="text-align: justify;">The second one is about mice, unfortunately round about the start of Autumn or Spring the little critters seem to want to come out of their hiding holes and I usually see one or two of them during this time of year. Unfortunately I am not allowed to have a cat in my house so I just have to do my best to catch them. I usually lay open traps and then let them free but I dare say they come back again. Anyway here we are (with apologies to Rabbie Burns :) </p><p class="MsoNormal"><b><span style="font-size: 16pt;"> Little Mousey <o:p></o:p></span></b></p><p class="MsoNormal"><span style="font-size: 12pt;"> Help I saw a mouse,</span></p><p class="MsoNormal"><span style="font-size: 12pt;"> Running
about the house.</span></p><p class="MsoNormal" style="text-align: left;"><span style="font-size: 12pt; text-align: center;"> Now you see me, Now you don't </span></p><p class="MsoNormal" style="text-align: left;"><span style="font-size: 12pt; text-align: center; text-indent: 24.1pt;"> Try to catch me, but you wonโt</span></p><p class="MsoNormal" style="text-align: left;"><span style="font-size: 12pt; text-align: center;"><br /></span></p><p class="MsoNormal" style="text-align: left;"><span style="font-size: 12pt; text-align: center;"> Under the floors, through the doors</span></p><p class="MsoNormal"><span style="font-size: 12pt;"> Up the
wall and in the hall.<o:p></o:p></span></p><p class="MsoNormal" style="margin-left: 119.9pt;"><span style="font-size: 12pt;"> Think I need a cat.<o:p></o:p></span></p><p style="text-align: justify;">
</p><p class="MsoNormal"><span style="font-size: 12pt;"> But one
thatโs not too fat. </span><span style="font-family: Wingdings; font-size: 12pt; mso-ascii-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Wingdings;">J</span></p><p class="MsoNormal"><br /></p><p class="MsoNormal">and here is a not so cute picture </p><p class="MsoNormal"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGlhefpTIKiZp69cRoryRVEbTiM26oMd7-Rpd1iOvQfqalGvP5Puc5ZVIyiCagqM8-tfS1XTi33omI9j_JP0yUIgPQaAyR4hN5NSdgsg6fycVQtYS0f_09J1pjLyb5YyPowGN9eo_Lrwp9/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="269" data-original-width="368" height="234" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGlhefpTIKiZp69cRoryRVEbTiM26oMd7-Rpd1iOvQfqalGvP5Puc5ZVIyiCagqM8-tfS1XTi33omI9j_JP0yUIgPQaAyR4hN5NSdgsg6fycVQtYS0f_09J1pjLyb5YyPowGN9eo_Lrwp9/" width="320" /></a></div><br /><br /><p></p><p class="MsoNormal">Finally here is a satirical one about spiders and our useless Prime Minister Boris Johnson who basically helped deceive the people into voting for Brexit. Just so he could be come leader of the Conservative party and prime minister. The full effects have been masked by COVID but we are already seeing shortages in the shops and it can only get worse as time moves on. I have written a whole series of poems about Fake nationalism both English and Scottish and will publish them in due course. I believe quite strongly that both the SNP and the Tories are destroying this country by appealing to a false sense of patriotism to disguise the unfairness of society and to deflect peoples attention from what is truly wrong with our society. Dr Johnson said that patriotism is the last refuge of a scoundrel and even though he was a High Tory for once I agree with him. </p><p class="MsoNormal"> <span style="font-size: 16pt;"> Boris the
Spider<o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12pt;"> What a tangled web you weave, <o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12pt;"> Hoping for the Flies to deceive.<o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12pt;"> Hanging there by a thread, <o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12pt;"> Wishing they were all dead. <o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12pt;"> </span></p><p class="MsoNormal"><span style="font-size: 12pt;"> Your nickname is Boris, <o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12pt;"> Do you have a friend called Doris? <o:p></o:p></span></p><p class="MsoNormal"><span style="font-size: 12pt;"> But something far more sinister,<o:p></o:p></span></p><p class="MsoNormal">
</p><p class="MsoNormal"><span style="font-size: 12pt;"> Youโre the same as our prime
minister.</span></p><p class="MsoNormal"><span style="font-size: 12pt;"><br /></span></p><p class="MsoNormal"><span style="font-family: inherit;">And here is one of the best pictures of 2019 </span></p><p class="MsoNormal"><span style="font-size: 12pt;"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEil_nBnE6KBBqHsedtXlKIxlYaj_HmKl7O21zwQpZ1XpVWo5fE8e1xA9iUIEYy7iVKMJQ6pYhiTZdYhQgEB6AjUGt2H1PXuH0ocPJZuMadKFx7YCbaxePSLJJGwIbxV2Tz1HYRch1l-qz5l/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="401" data-original-width="426" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEil_nBnE6KBBqHsedtXlKIxlYaj_HmKl7O21zwQpZ1XpVWo5fE8e1xA9iUIEYy7iVKMJQ6pYhiTZdYhQgEB6AjUGt2H1PXuH0ocPJZuMadKFx7YCbaxePSLJJGwIbxV2Tz1HYRch1l-qz5l/" width="255" /></a></div><br /><br /><p></p><p class="MsoNormal"><span style="font-family: inherit; font-size: 12pt;"> </span><span style="font-family: inherit;">Of course the fact that Boris Johnson illegally prorogued parliament, didn't stop people voting for him or the opposition parties falling into his trap of calling for an early election. It is arguable had they resisted we would have had another European Referendum and could possibly still be part of the EEC. As it is we will just have to make it on our own, and unlike the two world wars there will be no one to help us out. </span></p><p class="MsoNormal"><span style="font-family: inherit;">Anyway hope you enjoy these poems. The next sequence will be ones which celebrate the David Attenborough 'Life Series'. Then after that I will publish ones on maths and science including and I believe this is unique one on Bessel Functions ๐๐. I am about 2/3 the way through finishing the second of the Cambridge 1st year maths papers and hope to get it out early in September. </span></p><p class="MsoNormal"><span style="font-family: inherit; font-size: 12pt;">Regards Chris </span></p><p class="MsoNormal"><span style="font-size: 12pt;"><br /></span></p><p class="MsoNormal"><br /></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-47038063386208345282021-07-19T19:38:00.000+01:002021-07-19T19:38:17.651+01:00Anti-Ode to a Seagull<p> In contrast to the cute squirrels I wrote an Ode to last week Edinburgh is plagued by Sea Gulls who make a llot of noise and scavenge amongst the bins. Mind you if the council got it's act together and made bin collections more regularly the problem wouldn't be as great as it is. Anyway here is an Anti-Ode to a seagull</p><p><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span></p><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span> <span> </span></span><span style="font-family: Arial, sans-serif; font-size: 16pt;">Anti-Ode to a Seagull</span><p></p>
<p class="MsoNormal"><span style="font-family: "Arial",sans-serif; font-size: 16.0pt; line-height: 107%;"> <span> </span></span><span style="font-family: "Arial",sans-serif; font-size: 12.0pt; line-height: 107%;">Stomping
on the roofs <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: "Arial",sans-serif; font-size: 12.0pt; line-height: 107%;"> With
your great big hooves <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: "Arial",sans-serif; font-size: 12.0pt; line-height: 107%;"> What
a noise you make <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: "Arial",sans-serif; font-size: 12.0pt; line-height: 107%;"> It
is more than I can take <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: "Arial",sans-serif; font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><span style="font-family: "Arial",sans-serif; font-size: 12.0pt; line-height: 107%;"> Rummaging
around the bags <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: "Arial",sans-serif; font-size: 12.0pt; line-height: 107%;"> Looking
out for scrags <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: "Arial",sans-serif; font-size: 12.0pt; line-height: 107%;"> Why
not let us be <o:p></o:p></span></p>
<p><span style="font-family: Arial, sans-serif; font-size: 12pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 12pt;">And
return back to the sea.</span> </p><p>And here is not such a pretty picture </p><p><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcQXp_RSnukIPlphqPHjd06tXAD_00BAJ3K67fLEiyQzdzxGSvHd7L_oqSNZfAFJiP1uRRtcBZlCIvbQt3sqXS6gGi3yi8v6phX44woLBxje_7xpkRmRlz_hOdkf8ACnQHODJLzQf86sc0/s265/seagull.jpe" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="190" data-original-width="265" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcQXp_RSnukIPlphqPHjd06tXAD_00BAJ3K67fLEiyQzdzxGSvHd7L_oqSNZfAFJiP1uRRtcBZlCIvbQt3sqXS6gGi3yi8v6phX44woLBxje_7xpkRmRlz_hOdkf8ACnQHODJLzQf86sc0/s0/seagull.jpe" /></a></div><p></p><p><span>Hope this gave you a laugh anyway. Wonder if I have invented a new genre of poetry </span></p><p><span><br /></span></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-49037897737657088132021-07-08T22:44:00.001+01:002021-07-08T22:44:26.955+01:00Ode to a Squirrel <p>During lockdown I have been working from home. However from my living room window there are some trees close by I often see squirrel's running about and I was inspired to write this little ode which I hope brightens your day </p><p><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> Ode to a squirrel </span><br /></p><p><span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> Little squirrel running around </span><br /></span></p><p><span><span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> In the trees above the ground</span><br /></span></span></p><p><span><span><span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> What a pleasant sight to see </span><br /></span></span></span></p><p><span><span><span><span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> You bring such delight to me.</span><br /></span></span></span></span></p><p><span><span><span><span><span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> O what fun it must be </span><br /></span></span></span></span></span></p><p><span><span><span><span><span><span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> Running around so gay and free</span><br /></span></span></span></span></span></span></p><p><span><span><span><span><span><span><span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> Sometimes with a friend or two</span><br /></span></span></span></span></span></span></span></p><p><span><span><span><span><span><span><span><span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> Now the trees belong to you </span><br /></span></span></span></span></span></span></span></span></p><p><span><span><span><span><span><span><span><span><span>And here is a pretty picture </span></span></span></span></span></span></span></span></span></p><p><span><span><span><span><span><span><span><span></span></span></span></span></span></span></span></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg961Wv2mTQ1jZaSIrjADMFtXugGKGDut3QcsYNLR8xtMvorC5BkkHlxN6r11sICK3J-TAxLN1Wlyzerdf2MhmZZOe9w4dTMbVuTIigmjyC4On__blji3F6fdDLDcJU1lI241wsxjKU7vbz/s640/Squirrel.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="480" data-original-width="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg961Wv2mTQ1jZaSIrjADMFtXugGKGDut3QcsYNLR8xtMvorC5BkkHlxN6r11sICK3J-TAxLN1Wlyzerdf2MhmZZOe9w4dTMbVuTIigmjyC4On__blji3F6fdDLDcJU1lI241wsxjKU7vbz/s320/Squirrel.jpg" width="320" /></a></div><br /><span>Take care soon this nightmare will be over </span><p></p><p><span><span><span><span><span><span><span><span><span><br /></span></span></span></span></span></span></span></span></span></p><p><span><br /></span></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-72716837106301383272021-06-24T20:29:00.008+01:002021-06-25T22:31:29.159+01:00Solutions to Cambridge 1st Year NST Maths paper 1A 2019<p>Hi every one I have finally finished my first set of solutions to a Cambridge mathematics paper the paper in question was the first paper of 2019 which was given to the science students. The solutions are given here and I hope you enjoy reading through them</p><p><a href="https://drive.google.com/file/d/1vLNuqpxyomuiHsR8nYJ1P-AGPG-XTSTh/view?usp=sharing">https://drive.google.com/file/d/1vLNuqpxyomuiHsR8nYJ1P-AGPG-XTSTh/view?usp=sharing</a></p><p>Some general comments </p><p>On the whole I thought this paper was quite challenging certainly because the questions were a lot more tricky and involved than say a typical OU paper such as MS224 (although this years paper by all accounts seems to have been really difficult as the OU are still experimenting with the format for remote exams. My sympathies to those who found it difficult and when it eventually comes out I will have a look at it myself and make comments on it) </p><p>So what was in the paper the first set of questions was a set of 10 which were essentially A level topics and on the whole they were relatively straightforward. Although you do have to have integration techniques and trig identities at the tip of your fingers. There is really no time for much thought and one question involved a tricky integration by parts. Something which this paper seems to love testing people on I think there must have been about 6 or 7 times that I had to use this technique. As a result my grasp of this technique improved a lot and I can almost just write down the answer now whereas before I would struggle. Candidates have to answer all of these questions and a bit like multiple guess questions in the MS224 papers no marks are given for showing the working</p><p>The next section is a set of 10 questions from which the candidates can choose up to 5 the last two questions being reserved for those who took a more advanced option. The elite of the elite as it were it must be a bit disappointing to be told right at the outset of your undergraduate career that you are not part of this special group. I can see many potential traitors to our country arising from this slight ๐ <span> </span><span> Anyway the system is what it is and for us working through these papers we don't have to worry about such distinctions. Unfortunately there was one part of this section which I just could not see what the question was asking and hence I just have to accept that I will remain a pleb for the rest of my life ๐ I digress </span></p><p><u>Question 11</u> had four parts on complex variables. I really enjoyed the first part which involved using De Moivre's theorem to reduce a complex trigonometric identity to a product of powers of cos a and sin b. In the course of this I found a quick way of writing the expansion of cos(nx) or sin(nx) in terms of powers of cos(x) or sin(x) using Pascal's triangle. I have described this in detail in an earlier post and now feel I can tackle this sort of problem with ease. The other parts were straightforward enough but one involved a set of trigonometric identities for the Tan Function something I am not sure I would remember in an exam/ The final part was a bit difficult to decode what the question was asking but it involved the mapping between two complex variables and I got it out eventually. </p><p><u>Question 12</u> was a long tedious question on volume integrals with tricky boundary conditions, It went ok but it is not the sort of question I enjoy and it took me two or three attempts to arrive at the correct answer.</p><p><u>Question 13</u> was a question on first order differential equations and exact differentials. I generally prefer second order differential equations but that will have to wait till the next paper Two parts involved using the integrating factor method and one involved a not so obvious substitution in order to get the final integral out. The last part involved a lot of partial differentiation to show that a function obeyed Laplace's equation in spherical polar coordinates. So just like question 12 above this would have been tedious and I doubt whether I would have got it out in time </p><p><u>Question 14</u> was divided into two parts the first part asked you to calculate the variation in the relativistic formula for energy in terms of the variations for mass and energy and then given the velocities of two particles work out the relative mass of these particles wrt to each other. This was OK if a bit novel to me and it was difficult to tell whether I had got the signs correct. The second part of question 14 involved a really tedious exercise in partial differentiation of functions of functions. Whilst in principle this type of question is relatively straightforward in practice ir can get extremely tedious so best avoided in an exam and I found myself losing the will to live</p><p><u>Question 15</u> was a question on Taylor series and expansions. Again relatively straightforward and tedious you were allowed to use known Taylor expansions but could anyone off the top of their head remember if they ever knew the Taylor expansion for sech(x). I had to derive this myself. </p><p><u>Question 16</u> was an unusual one about probability involving the various probabilities of drawing a blue ball from a bag and then deciding to return it or not with a certain probability and then drawing another ball. Why anyone would think this is something useful to do is beyond me. It certainly wouldn't help you win the lottery๐ I hadn't really done much of this type of question before so it was useful to learn the various formulae for conditional probability and how to draw a probability tree using MATLAB. So whilst the actual activity described in the question is quite pointless, it was quite instructive to work out the various probabilities involved again this would have been really difficult to complete in the time given</p><p><u>Question 17</u> was a whole bunch of questions on integration some of which seemed impossible at first sight but you just have to persist and I strengthened my integration skills a lot by doing this question</p><p><u>Question 18</u> was a long boring question on matrices the first part solving a matrix equation for 2x2 matrices. The second part at first sight was quite intimidating as it involved a long introduction to set the scene for testing a hypothesis about the size of worm lengths. It turned out that in it's long winded way the question was taking the examinee through the method of least squares fit. So it turned out to be relatively interesting but it really is not the sort of question that any normal human being can just rattle off under exam conditions</p><p>Now we come to the questions for the elite. </p><p><u>Question 19</u> This was split in two parts the first question being a question on the convergence or not of various series. This was quite straightforward if you are up to speed with regards to the techniques but this is really the sort of question that belongs in a pure paper not a mathematical methods one. I suppose the idea is to get their students to think a bit about real analysis. But they are better off doing a full blown course on it rather than spending just a couple of lectures or so. I would recommend the OU course M208 as giving a far better understanding of real analysis than this sort of rapid overview.</p><p>The second part was my nemesis for this paper it involved relating the Newton Rhapson method to a Taylor expansion and estimating the errors with each iteration. I was unable to understand what was going on. However given that the function was a quadratic equation for which the roots are known exactly I was able to provide an estimate of the error at each stage but the resulting expressions looked nothing like the expressions in the paper. If anyone knows the answer to this question I would be grateful. I can't see the point of applying the Newton Rhapson method to a quadratic equation anyway </p><p><u>Question 20</u> This was the final question Yippee and was on Integration the first part was showing how the area of a parabola could be approximated by a series of rectangles under the parabola and showing how in the limit the exact expression could be obtained. I got this question out but again it involved a lot of tricky algebra the thing to do here as Corporal Jones from Dad's Army would say is 'DON'T PANIC' and just see it through even if at first sight your answer appears to be going nowhere.</p><p>The last two parts of the question involved something which I hadn't really come across before namely differentiating functions under an integral sign involving variable limits. Apparently Leibniz derived a formula for this many years ago (what a clever fella that man was ๐ ) and once you know this formula the question was quite straightforward. The final part involved yet another tricky substitution making use of the obvious fact ๐ that sin(x) = tan(x)/sec(x) and the derivatives of both sec(x) and tan(x). </p><p>So what would it take to do really well at this exam essentially a real fluency in techniques of integration, trigonometric identities, hyperbolic functions, their derivatives and identities, Taylor series, differentiation using the product and quotient rules, Being able to partially differentiate functions of functions using the chain rule very quickly and a good deal of persistence. Not for the faint hearted but an exercise well worth doing. I certainly got a lot from this exercise even though I am not particularly fluent at the techniques. I guess like good musicians you have to just practice practice until the techniques just become routine. I look forward to doing the second paper in this series which will cover such topics as vector calculus surface and line integrals, and second order differential equations (yippee my favourite technique at this level) I am not very confident with things like the divergence theorem and surface and line integrals I hope by the time I have finished the second paper I will become more so and no doubt I will learn a whole load of new techniques as well I encourage any one reading this blog to have a go themselves. The last paper took me about 6 months to complete and write up I hope the second one wont take so long </p><p> </p><p><br /></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-57126287412200801642021-05-25T19:44:00.003+01:002021-05-27T23:51:43.987+01:00Now for Something Different (Shakespeare again ) <p> Hi about 10 years ago I signalled my intention to watch all the Shakespeare Plays However this project stalled. Recently however a friend of mine asked me to list my favourite Shakespeare plays and give a recommended order of what ones to watch. Whilst I am no expert I have loved Shakespeare since my student days over 40 years ago when I was enthralled by the BBC Shakespeare Series especially Derek Jacobi playing Hamlet. It was the first time I had really understood a Shakespeare play. Anyway as a bit of a distraction I will attempt to resume my project but narrow the focus down to concentrate on the more well known ones following the list below then maybe fill the rest later. Anyway as a start here is the list</p><p><b>Comedies</b> </p><p><u>A MidSummer Nights Dream</u></p><p><u>Twelfth Night</u>. We did this at school and I played Sir Toby Belch, the drama teacher obviously thought it would bring me out of myself as I was one of the school swots. </p><p>There are of course plenty more such as As you like it or Loves Labours lost, but these are representative and whilst the Comedies are entertaining, they aren't really what Shakespeare is about. </p><p><u>Dark Comedies (Problem plays)</u></p><p><u>The Merchant of Venice</u>. This is a truly problematic play one wonders why it was classed as a comedy it raises all sorts of issues to do with the Anti-Semitism of the time. In mitigation Shylock has the best speech where enraged by the taunts of two Viennese citizens he utters the immortal lines 'If you prick us do we not bleed...'; Which can be seen as Shakespeare endorsing the common humanity of all people. On the other hand Shylock is definitely a caricature of the Jews as they were perceived in late Elizabethan England. I will explore the issues raised in a later post. </p><p><u>Measure for Measure </u></p><p>Another problematic play, in the light of the Harvey Weinstein scandal, it shows up the abuse of those in postions of power have those in power with regard to those in more vulnerable positions especially women. Probably my favourite of the problem plays. </p><p><u>The so called Histories </u></p><p>Shakespeare obvously knew what side his bread was buttered on and wrote these plays essentially to justify the Tudor line. Thus Richard III is made to be a villain rightly deposed by the Duke of Richmond, who later became Henry VII and who wasn't exactly free from Tyranny himself. However the plays whilst having dubious claims to historical accuracy make good drama. I have just invested in the Hollow Crown and look forward to watching it. Despite their obviously sympathetic view of the Tudors it is interesting that Richard II was staged just before the Essex rebellion against Queen Elizabeth in her later years.</p><p>Shakespeare wrote his histories in two installments the first set written in his early career as a playwright involved the three parts of \Henry VI followed by his greatest history Richard III, then a few years later he wrote the series starting with Richard II then the two parts of Henry IV and culminating in Henry V. These plays are far more mature than the parts of Henry VI and also introduce a bunch of rogues Bardolph, Pistol and Shakespeare's greatest comedy character John Falstaff, These characters are seen as drinking companions of the young Prince Hal and often make Falstaff the butt of their jokes. Having said that for all Falstaff's buffonery he probably has the most pragmatic attitude to war and the dubiousness of military glory enshrined in his speech which includes the lines </p><p>"Who hath honor then, Why he who died on Wednesday, doth he feel it No, doth he hear it No 'Tis insensible then. Yea to the Dead ... I'll have none of it" </p><p>Needless to say despite those sentiments, it doesn't prevent Falstaff from falsely claiming that he killed Hotspur to get a bit of glory for himself.</p><p>Of course when Prince Hal becomes King Henry V he has to disown himself from Falstaff and his companions which he does most brutally. Henry V relates how he won the battle of Agincourt and defeated the French, It borders on bombastic Nationalism but is mitigated by scenes before the battle of Agincourt where soldiers debate the fact that whilst they know nothing of the rights or wrongs as to why they are fighting, they will bear no responsibility but the responsibility lies with the Monarch who has led them into battle. </p><p> As a break between the end of Henry IV part II and Henry Vth, the Merry Wives of Windsor provides comic relief where Falstaff short of money (as always) attempts to persuade two rich women Mistress Page and Mistress Ford to marry him even though they are already married, Needless to say his attempts fail badly. In itself this play isn't that significant but as light relief to the history plays it is worth watching now and again </p><p>Jumping ahead chronologically to Richard III then, Shakespeare portrays him as a villain. but attributes this to his jealousy compared to his other brothers that he is not attractive to women due to his physical deformities. In one of the most brilliant opening speeches, Richard III decries the fact that whilst he is not made for 'sportive tricks' his brother Edward IV, currently King, capers nimbly in his mistresses chamber. Thus long before Freud, Shakespeare has identified a psychological motive for a persons evilness. Of course historically it's all tosh. The recent finding of the body of Richard III in Leicester has established that he had no real deformity, but of course the Tudors needed someone to justify their somewhat dubious ascension, to the throne. Despite the lack of historical accuracy, no one can deny the powerfulness of the play as drama, just take it with a pinch of salt. </p><p>So in my humble opinion I would suggest that those wanting to familiarise themselves with the Shakespeare historical plays should start with Richard II then work systematically through the plays culminating in Henry V. Then jump to Richard III. The parts of Henry VI can come later to fill in background to the character of Richard III, It is interesting to note that the Hollow crown has reduced the three parts of Henry VI to two so I suspect that there will be many cuts I will let you know </p><p><br /></p><p><u>The Roman Plays</u> </p><p><u>Julius Ceasar</u> and <u>Cleopatra</u>, Whilst these plays are interesting they aren't exactly mainstream but still worth watching or reading. I will give more details as to what I think in future posts </p><p><br /></p><p><u>The Tragedies </u></p><p>Now we come to the core of Shakespeare, In comedy despite a temporary disruption everything returns to so called normal reinforcing the status quo however unjust. Thus Midsummer nights dream poses no real challenge to the patriarchal view that daughters should marry men whom the father chooses. The Merchant of Venice ends with a conversion of Shylock to Christianity and no challenge to the Anti-Semitism of Venice at the time and so forth. </p><p>Tragedy blows this consensus apart, there is no normality to return to as practically everyone is killed at the end. Thus if the Arts are to be more than entertainment, tragedy has the potential to make people question the status quo. Hence in my opinion tragedy, speaks more powerfully to us, than comedy ever will. Shakespeare's tragedies provide a powerful critique of society as it was without the sycophancy of the history plays or the facile resolutions in many of the comedies. In recommended order I would single the following </p><p><u>Romeo and Juliet </u>- Shakespeare's first big hit, but fairly youthful it has inspired many composers such as Berlioz and the theme formed the basis for Leonard Bernstein's West side story </p><p><u>Macbeth</u> Written in 1606 just after the gunpowder plot and the ascension of James VI of Scotland at first sight it might seem surprising that Shakespeare managed to escape censure for this until one realises that James claimed to be descended from Banquo and not Macbeth. So another sycophantic play. </p><p><u>Othello</u> We have seen Shakespeare as Anti-Semitic now he is racist. Having said that Shakespeare shows how jealousy arises from a persons basic insecurity and the tragic consequences that follow In Iago we have along with Richard III Shakespeare's greatest villain. There is a good production from the RSC with Iain McKellan as Iago </p><p><u>Hamlet.</u> It is a cliche but this is probably Shakespeare's greatest play the psychological depth of the characters prefigures Freud by almost 300 years. My favourite version is that by Kenneth Branagh with a close second being the BBC production that got me into Shakespeare in the first place with Derek Jacobi. The advantage of the Brannagh Version as opposed to the one starring Laurence Olivier is that it doesn't cut the political scenes involving Fortinbras. Plus some absolutely gorgeous music.</p><p><u>King Lear</u> Probably the bleakest of all Shakespeare's tragedies. The BBC production has Michael Horden playing King Lear, There is also a Russian Version which emphasises the bleakness by the director Kozintsev. Just as Hamlet is something that anyone who aspires to be a leading actor must pass through King Lear is for actors at the end of their career. </p><p><u>Pastoral Plays</u> </p><p>This is a genre that Shakespeare came too late in life, and represent the autumn of his working life the two main ones are a <u>Winter's tale</u> and the Tempest. In the <u>Tempest</u> Prospero can be seen as Shakespeare leaving the stage. I will write more about them in later posts </p><p>So there we have it a rough guide to the greatest sexist, racist, anti-Semitic, royalist sycophantic, bombastic nationalistic playwright that ever lived ๐๐</p><p><br /></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-34958515917507815192021-02-14T20:49:00.005+00:002021-02-14T21:53:34.397+00:00Cambridge MAFFS <p>How does one make sure one's mathematical skills are up to date and relevant? One way would be to aim to do past papers from relevant mathematics courses once a year as they come out. Such a set of papers and resources for study of these papers is provided by the Cambridge University mathematics department. This post collects together a few resources for self study of the Cambridge Mathematical Methods courses for Scientists (NST). I am continually amazed at the generosity of the Cambridge Maths site in offering to the public these resources</p><p><br /></p><p>The past papers are collated here. </p><p><a href="https://www.maths.cam.ac.uk/undergradnst/pastpapers">https://www.maths.cam.ac.uk/undergradnst/pastpapers</a></p><p> The syllabus for both years is given here </p><p><a href="https://www.maths.cam.ac.uk/undergradnst/files/misc/NSTschedules.pdf">https://www.maths.cam.ac.uk/undergradnst/files/misc/NSTschedules.pdf</a></p><p>There are two courses one for first year students and one for second year science students who want to study mathematics in more depth. </p><p><br /></p><p>The recommended text book is Riley Benson and Hobson </p><p><a href="https://www.amazon.co.uk/Mathematical-Methods-Physics-Engineering-Comprehensive/dp/0521679710">https://www.amazon.co.uk/Mathematical-Methods-Physics-Engineering-Comprehensive/dp/0521679710</a></p><p>Which has a solution manual which covers the odd problems in great detail so ideal for self study </p><p><a href="https://www.amazon.co.uk/Student-Solution-Mathematical-Methods-Engineering/dp/0521679737/ref=pd_lpo_14_t_0/262-4198048-8655769?_encoding=UTF8&pd_rd_i=0521679737&pd_rd_r=c25e3e37-5d85-425d-ae5f-e2b662f55a36&pd_rd_w=5uBAB&pd_rd_wg=upWCR&pf_rd_p=da0677f5-a47b-4543-8b54-10be576b8f26&pf_rd_r=2N0058F97ARMT25NE1M6&psc=1&refRID=2N0058F97ARMT25NE1M6">https://www.amazon.co.uk/Student-Solution-Mathematical-Methods-Engineering/dp/0521679737/ref=pd_lpo_14_t_0/262-4198048-8655769?_encoding=UTF8&pd_rd_i=0521679737&pd_rd_r=c25e3e37-5d85-425d-ae5f-e2b662f55a36&pd_rd_w=5uBAB&pd_rd_wg=upWCR&pf_rd_p=da0677f5-a47b-4543-8b54-10be576b8f26&pf_rd_r=2N0058F97ARMT25NE1M6&psc=1&refRID=2N0058F97ARMT25NE1M6</a></p><p>Investing in these two books would cover most mathematical methods that a physicist is likely to need in their careers. However it would probably take a good few years to do all the examples in Riley Benson and Hobson so it is more for reference than anything. </p><p>Buried in the vaults of the Cambridge Mathematics department are the following resources </p><p>Answers to first year past papers can be found from this website </p><p><a href="https://www.robinson.cam.ac.uk/iar1/teaching/index.html#nst1a_maths">https://www.robinson.cam.ac.uk/iar1/teaching/index.html#nst1a_maths</a></p><p>Unfortunately I have not been able to find any lecture notes or the example sheets. However if you have Riley Bence and Hobson you should be able to find the techniques that you don't know already covered there. However having the answers to the exam questions is obviously helpful </p><p>Year II is much better served the lectures occur in three parts </p><p>The first part is covered by a set of lectures from Dr Simon Cowley who from his web site seems quite an engaging person</p><p><a href="http://www.damtp.cam.ac.uk//user/sjc1/index.html">http://www.damtp.cam.ac.uk//user/sjc1/index.html</a> </p><p>Lecture notes for the various courses he has taught over the years can be downloaded from here </p><p><a href="http://www.damtp.cam.ac.uk//user/sjc1/teaching/">http://www.damtp.cam.ac.uk//user/sjc1/teaching/</a></p><p>The ones of main interest are the notes for the first part of the Cambridge NST Part IB </p><p><a href="http://www.damtp.cam.ac.uk//user/sjc1/teaching/">http://www.damtp.cam.ac.uk//user/sjc1/teaching/</a></p><p>This covers things such as vector calculus, matrices partial differential equations Fourier Transforms and solution of differential equations, by series including a survey of Legendre Polynomials, Green's Function and a bit on real analysis (for physicists not mathematicians ๐ ). This is more or less equivalent to MS224 and the mathematical methods part of MST326 but also includes Green's Functions and goes into more detail about the solution of differential equations by Series solution especially the Frobenius Method. </p><p>Notes for the second lecture course in Part IB are given by Dr Hunt </p><p><a href="https://www.damtp.cam.ac.uk/user/reh10/lectures/">https://www.damtp.cam.ac.uk/user/reh10/lectures/</a></p><p>Scroll down to the Mathematical Methods part </p><p>He also gives the examples sheets that were part of the course and hints for their solution </p><p>This second part covers variational principles, solutions to Poisson's equation and a summary of the basic techqniques of Complex Analysis again taught as Physicists would use them rather than a Pure Mathematician. There is a bit of overlap with the OU course MS327 but neither MST326 or MS327 cover inhomogeneous partial differential equations in any depth. </p><p>Finally the third part covers group theory and representation theory for physicists including an analysis of multi-mode oscillations. MS327 covers multi-mode oscillations but do not relate it to group theory. </p><p><a href="https://www.damtp.cam.ac.uk/user/examples/N23L.pdf">https://www.damtp.cam.ac.uk/user/examples/N23L.pdf</a></p><p>The examples sheets for the whole year are also given scroll down to the bottom </p><p>Ok I aim to have generated a complete set of solutions to the 2019 papers by the end of the year. This is a nice way IMHO to extend ones mathematical ability and keep it topped up. Overall the topics covered here in terms of mathematical knowledge overlap with MS224, MS327 and MST326 but also extend it to cover topics such as</p><p>1) Solution by Series of Differential equations including the Frobenius Method</p><p>2) Green's Functions and their application to Partial Differential equations </p><p>3) Complex Analysis (although to be fair if you do M337 you will do far more than is given here) </p><p>4) More detail on Special Functions including Legendre Polynomials </p><p>5) Group Theory </p><p>Also I suspect the questions will be harder than the OU exam questions </p><p>Anyway I hope this post will encourage you to extend and revise your mathematics skills. </p><p><br /></p><p><br /></p><p><br /></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com2tag:blogger.com,1999:blog-7897235423812277683.post-47148199484002692012020-12-31T18:55:00.002+00:002021-01-02T21:46:05.915+00:00De Moivre's Theorem and Trig Identities <p> Hi Guys and Gals Hope you have a great New Years eve party if you are having one. Hopefully 2021 will see the end of Covid and life can get back to normal. IMHO the restrictions have been unnecessarily punitive and the shut down of the pubs, theatres and concert halls has been totally unnecessary anyway I'm not here to gripe about the restrictions laid down by Mrs Oliver Cromwell (aka Nicola Sturgeon) up in Scotland during the lockdown. I have been amusing myself to see if I can do some Cambridge Maths past papers set for their Natural science students and available here </p><p><a href="https://www.maths.cam.ac.uk/undergradnst/pastpapers/2019">https://www.maths.cam.ac.uk/undergradnst/pastpapers/2019</a></p><p>In doing the 2019 1st paper I came across this question which at first sight looks nigh impossible the question was </p><p>Use De Moivre's theorem to write out the expression </p><p>$$ 8 cos 6\theta + 15sin 4\theta sin 2\theta $$ in terms of $ cos \theta$ and $ sin \theta $ </p><p>However I discovered for myself a simple method to write out the expansion of $cos (n\theta)$ or $sin (n\theta) $ Almost instantly from Pascal's triangle </p><p>First let us recall De Moivre's theorem it states that </p><p>$$exp(in\theta) = [exp(i\theta)]^n $$ using </p><p>$$ exp(in\theta) = cos(n\theta) + i sin(n\theta) $$ </p><p>it is seen that </p><p>$$cos(n\theta) = Re [cos(\theta) + i sin(\theta)]^n $$ and </p><p>$$sin(n\theta) = Im [cos(\theta) + i sin(\theta)]^n $$ </p><p>Ok that looks really messy how do I remember the coefficients in a binomial expansion and how do I pick out the Real and Imaginary parts of each coefficient and all in about 5-10 mins which is all I have time to answer the question</p><p>Well a short cut is as follows Recall that the binomial coefficients in a binomial expansion are given by the rows of Pascal's triangle . Namely </p><p><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> n = 0</span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> 1</span></p><p><span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> n = 1</span></span> 1 1</p><p><span><span> n = 2</span></span> 1 2 1</p><p><span><span> n = 3 1 3 3 1</span></span></p><p><span><span> n = 4 1 4 6 4 1</span></span></p><p><span><span> n = 5 1 5 10 10 5 1</span></span></p><p><span><span> n = 6 1 6 15 20 15 6 1</span></span></p><p>where each number in a given row is given by adding the two numbers above it and the coefficient corresponds to the following expansion </p><p>$$ (a + b)^n = C_1 a^n + C_2 a^{n-1} b + C_3 a^{n-2} b^2 + ...........C_{n-1}.a b^{n-1} + C_n b^n$$</p><p>The coefficients are just the corresponding element in the nth row of Pascal's triangle. So for example </p><p>$$(a + b)^5 = a^5 + 5 a^4 b + 10 a^3 b^2 + 10 a^2 b^3 + 5 a^1 b^4 + b^5 $$</p><p>Now to apply this to De Moivre's theorem we would have </p><p>$$[(cos(\theta) + i sin(\theta)]^n = C_1 cos^n (\theta) + C_2 cos ^{n-1}(\theta) (i sin(\theta)) + $$</p><p> $$ C_3 cos^{n-2}(\theta) (i sin(\theta))^2 + ..... $$</p><p>So the odd coefficients in the expansion will correspond to the real part and the even coefficients will correspond to the imaginary part of the expansion, Furthermore as $i^2 = -1$ and $i^4 = +1$ the coefficients will alternate in sign </p><p>So this gives us a nice quick method of writing out $cos(n\theta)$ and $sin(n\theta)$ in terms of $cos(\theta)$ and $sin(\theta)$</p><p>For a given n work out the corresponding row in Pascal's triangle then the expansions of $cos(n \theta)$ and $sin(n \theta)$ are as follows </p><p>$$ cos(n\theta) = C_1 cos^n(\theta) - C_3 cos^{n-2}(\theta) sin^2(\theta) + $$</p><p>$$ C_5 cos^{n-4}(\theta) sin^4(\theta)+ ...$$</p><p>$$ sin(n\theta) = C_2 cos^{n-1}(\theta) sin(\theta) - C_4 cos^{n-3}sin^3(\theta)+ .... $$ </p><p>Hence given a horrendous expression involving $cos(n\theta)$ and $sin(n\theta)$ the expansions of these functions in individual powers of $cos (\theta)$ or $sin(\theta)$ can be written down by sight </p><p>So for our problem we have </p><p>$$sin(2\theta) = 2 sin(\theta) cos(\theta) $$ as the second row of pascals triangle is 1 2 1 and the even coefficient is 2 </p><p>$$sin(4\theta) = 4 cos^3(\theta) sin(\theta) - 4 cos(\theta) sin^3(\theta) $$ as the fourth row of Pascal's triangle is 1 4 6 4 1 and the even terms are 4 and 4 </p><p>Finally </p><p>$$cos(6\theta) = cos^6(\theta) - 15 cos^4(\theta) sin^2(\theta) + 15 cos^2(\theta) sin^4(\theta) - sin^6(\theta) $$ </p><p>as the 6th row of Pascal's triangle is 1 6 15 20 15 6 1 and the odd numbered coefficients are 1 15 15 and 1 </p><p>So </p><p>$$8 cos(6\theta) =8[ cos^6(\theta) - 15 cos^4(\theta) sin^2(\theta) + 15 cos^2(\theta) sin^4(\theta) -$$ $$sin^6(\theta) ]$$ </p><p>and </p><p>$$15sin4\theta sin 2\theta=15 [4 cos^3(\theta)sin(\theta)-4cos(\theta)sin^3(\theta) ] $$</p><p>$$ \times 2 sin(\theta)cos(\theta)$$</p><p>This simplifies to </p><p>$$ 15 \times 8 [cos^4(\theta)sin^2(\theta) - cos^2(\theta) sin^4(\theta)] $$ </p><p>and it is seen that the mixed terms in $cos(\theta)$ and $sin(\theta)$ cancel leaving </p><p>$$ 8 cos 6\theta + 15sin 4\theta sin 2\theta = 8(cos^6(\theta) -sin^6(\theta)) $$</p><p>So next time you are faced with a seemingly horrendous expression involving $sin(n\theta)$ or $cos(n\theta)$ just remember Pascal's triangle and De Moivre's theorem.</p><p>Hope every one reading this has a great new year I will write another post over the weekend summarising my plans for 2021 and hopefully achieving them :) </p><p><br /></p><p><br /></p><p><br /></p><p><br /></p><p><br /></p><p><br /></p><p><br /></p>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0