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studiot

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Everything posted by studiot

  1. Thanks for the reply. I really can't evaluate the 'intelligence' of octopussys, tomatoes or similar. That would involve know exactly what intelligence is . However the more we study Nature the more examples of 'intelligent ' activity we discover, even in the plant kingdom. David Attenborough's latest 'planet' series has some striking new examples, some based on time lapse photography. This is ideal for plants, which operate over a generally longer timescale than more mobile 'creatures'. But Attenborough's time lapse video shows many plant behaving in a similar way to creatures over months or even years rather than minutes. Such a tolerance of many years of timescale would give space travelling plants an advantage. On another tack, it is well known that ants are livestock farmers. But (again thank you Attenborough) on Sunday I found out that they are also arable farmers. Apparantly they collect and compost (readers of the compost thread please note) suitable leaves and then feed them to certain fungi which produce substances the ants want.
  2. Let me offer a few choice ones. Courant and Robbins What is Mathematics ? Matt Parker Things to make and Do in the FourthDimension. David Wells The Penguin Dictionary of Curious and Interesting Geometry. Acheson From Calculus to Chaos. Mark Levi The Mathematical Mechanic. J E Spice Chemical Binding and Structure. Pauling and Pauling Chemistry Fred Hoyle Ice Hermann Bondi Relativity and Common Sense Robert Millikan Electrons ( + and -) Protons, Photons, Neutrons, Mesotrons and Cosmic Rays. PS thanks for the thread, +1 Swinnerton Solving Earth's Mysteries Steven Vogel Cats Paws and Catapaults
  3. Not exactly an explosion, but then it is difficult to ignite some hydrocarbon deposits. https://en.wikipedia.org/wiki/Pitch_Lake
  4. I have been asked by a member to comment on this worked example question encountered in an online tigonometry course. The temperature during the week oscillates daily between 48o and 74o. (presumabably Farenheit). If the minimum occurs at 5am, at what time is the temperature at 65oF ? The lecturer chooses a sinusoidal model for the variation, but uses a very overcomplicated method where he arrives at and solves the following equation. [math]f\left( \theta \right) = - 13\cos \left( {\frac{\pi }{2}\left( {\theta - 5} \right)} \right) + 61[/math] Here are my comments. The lecturer has chosen to use a cosine wave as the model, using radian measure for the angles, and has come up with a 'grand unified' formula which is fraught with the dnager of miscalculations as will become clear. It is vey important to solve and understand the Physics as well as the Mathematics of a model. I have shown the Physics in Fig1, covering both the day in question and part of the day before and part of the day after. The lecturer failed to note the characteristics of a sinusoidal wave - that it passes throuhg its speial points (its peaks and zeros) at spacings of 90o or 6 hours or 'quadrants'. This is possibly because he thought to work in radians. It is worth noting that most scientific instruments that measure angle use the degrees, minutes and seconds units. Almost none read directly in radians. Maybe this is why he made the incorrect statement that we cannot locate the peaks, troughs and zeros of the model wave, only the minimum at 5am. I have labelled the important ones A, B and C in green in Fig1. I have also shown where the target 65oF intersects the model curve at P and Q within the target day and at R on the day before. It is also worth noting that working in DMS (degrees etc) has advantages in some disciplines since time, latitude and longitude in geography and azimuth and right ascension in astronomy are measured in these. So 15o corresponds to 1 hour. The lecturer also points out that there are actually many angles that satisfy the inverse trigonometric equation, but goes on to make very heavy weather of choosing the right one. He doesn't stress enough that a calculator or set of tables will only provide one of these angles, not necessarily the one you want. So I have shown in Fig2 and Fig3 the standard way of overcoming this issue. If positive values are taken the calculator or table will always give you the first quadrant anngle (I have called A to distinguish from his theta). A mnemonic based on the english word CAST is shown to help remember which trig functions are positive in the other quadrants as well. Fig3 shows how to establish the angles to be used in each quadrant. So here are 3 separate ways of working this out, without solving that overblown equation. We will see how my comments pan out as we work through them. So starting from point A which is the normal starting point of a cosine wave, where the cosine has a value of +1. This must be two quadrants of 6 hours ie 2x6 =12 hours back from the minimum at 5am (point B). That is 5pm the previous day. Counting from point A Sorry I see I've used A for both the angle and the point, hopefully this will not cause too much confusion. P is in the 4th quadrant, Q is in the 5th quadrant, but one day later than R which is in the first quadrant and therefore at the same time of day as Q. for point R, cos A = 4/13 therefore A = 72o or 72/15 = 4.8 hours later than point A So R is the point at 1700 + 4.8 hours or 21.8 hours, the previous day. So Q is the point at 21.8 hours on the target day in question. P is in the 4th quadrant so is (360-A) = (360- 72) = 288o . 288 degrees is 288/15 = 19.2 hours So P is the point at 1700 + 19.2 hours or 1700 +6+6+6+1.2 hours = 12.2 hours the following day. Counting from point B In relation to point B, P is in the second quadrant and Q is in the third quadrant. Thus both P and Q will have negative cosines. Using a calculator to calculate the inverse cosine of -(4/13) will correctly give the correct angle of 108o. However there is a trap here as this will not give the value for Q directly. Better to ignore the -ve and follow the standard procedure of Fig3. For P Angle = [math]180 - {\cos ^{ - 1}}\left( {\frac{4}{{13}}} \right)[/math] ie (180 - 72) ie 108o or 108/15 = 7.2 hours For Q Angle = [math]180 + {\cos ^{ - 1}}\left( {\frac{4}{{13}}} \right)[/math] ie (180 + 72) = 252o or 252/15 = 16.8 hours So P is the point at 0500 + 7.8 = 12.2 hours and Q is the point at 0500 + 16.8 = 21.8 hours. Counting form point C Since C is a zero it would be the natural point to count from if modelling with a sine wave, rather than a cosine wave. P is then in the first quadrant and Q in the second so both are then positive. So Angle is [math]{\sin ^{ - 1}}\left( {\frac{4}{{13}}} \right) = {18^o}[/math] ie 18/15 = 1.2 hours So P is the point 1100 + 1.2 = 12.2 hours For Q we have Angle = (180 - 18) = 162o ie 162/15 = 10.8 hours So Q is the point 1100 + 10.8 = 21.8 hours. So all three methods come up with the same answer (and the same one the lecturer obtained) But it shows the value of solving the Physics as well as the fancy trigonometry.
  5. A very recent study by workers at Bristol University that is awaiting its peer review suggest that covid looses its infectivity as it dries out. The study concerned particles in the air and found that at 50% RH the particles quickly dried out compared to 90% RH, as they spread, and so the effective range was decreased at low RH. I wonder if this is part of the reason why countries with persistent high RH, such as the UK and Eire experienced greater infection rates than drier ones. https://www.theguardian.com/world/2022/jan/11/covid-loses-90-of-ability-to-infect-within-five-minutes-in-air-study
  6. Maybe not a great example, but I meant it as a ballpark example along the following lines. Classically, (without QM) bonds are just links of electrostatic origin. That is the bond energy is contained in an electrostatic field of some sort between the atoms. Si is 4 valent and B is 3 valent. So one can measure or look up the bond energies. And also the mass difference between a silicon atom and a boron atom. So one can get the energy difference in substituting 1 in X silicons by a boron and reducing the bond energy by four Si-Si bonds and adding back three Si-B bonds. However this substitution will also introduce strain energy into the lattice which will tend to zero as X tends to infinity.
  7. If you want a classical answer you could consider the decrease in mass due to changing a silicon atom for a boron one in a silicon lattice, since this would decrease the charge by 1,thus reducing the field slightly. Since we are then talking about a solid lattice, momentum would not be involved the simple e = mc2 would suffice.
  8. Thank you for your thoughts. Since we have not yet heard from exchemist, I hope he is OK. However there is a clue in the name. I suspect he is more interested in the field of the nucleus than the fields of astrophysics at the scale of the universe. Particularly as he specified 'static' fields. If the universe is actually infinite then the global mass integral must also be infinite so adding mass in any way does not change that result.
  9. A few quick suggestions. 1) Get to know (buy them a pint ?) your nearest CD/DVD repair shop. Some will have laser calibration equipment. 2) Hire a laboratory light source, you can get ones from NPL standard to school grade. Don't know if Griffin and George are still in business. 3) I think Radiospares or CPC offer calibration diodes for suggestion (1) at reasonable cost. 4) You are right, using the Sun presents special difficulties. This is why the earliest observers (Tyndall and then Piazzi Smythe) went up high mountains to observe the solar spectrum. It is worth reading the science of this in Chapter 3 of Sarah Dry's book "Waters of the World". She also references the original material.
  10. I was looking up Earnshaw's original paper. The contents list of the 7th volume of the Proceedings of the Cambridge Philosophical Society is quite stunning. Earnshaw (Paper 5 page 97) is in the company of Green, Stokes, De Morgan, Airy, Challis and O'Brien to list but a few and the subjects covered as well as the presentations could be offered today. And yes any force field has a source, which itself has inertia. So much can indeed be done classically, but the Lagrangian calculus of variations refers to a conservative field using continuous linear mathematics so obviously has limitations in a quantum sense. However I am waiting for exchemist to spell out exactly what he wants to achieve.
  11. Details discussed in the full article here. https://www.bbc.co.uk/news/world-us-canada-59960949
  12. Quite right, here is a lovely chapter from Frank Wilczek "The Lightness of Being." You will need to get the book for the previous chapter containing the calculations.
  13. A wise comment. +1 Leaves, whether whole or chopped up, contain a high proportion of beneficial organic material. As such gardeners add the material as a soil conditioner rather than a source of nutrients per se. I am not sure at what fineness this effect is nullified by 'grinding to dust'. Certainly simply adding carbon particles does not improve the soil.
  14. From the beginning of the text in your link which brings us straight back to my comment about static v dynamic. This needs clearing up before looking at the question of how to increase the stored energy , without motion of something. And of course there are no relativistic effects when there is no motion.
  15. I think first you would have to explain what a static EM field looks like. I have heard of static E fields and static H fields separately. But aren't all EM fields dynamic ?
  16. Du Sautoy is definitely into symmetry. He has written at least three books. The Alhambra stories are in the Finding Moonshine one. https://www.amazon.co.uk/Symmetry-Journey-Into-Patterns-Nature/dp/0060789417 Here are the relevent pages from Ball's Tapestry about mixing drums. My apologies for the poor scans due to the spine of the book.
  17. If I might be permitted to pass comment on this discussion. I rather get the impresson that CharonY is a frustrated instructor who would genuinely like to do better by their students but is hamstrung by the system. Yes I blame the system or rather what it has developed into, rather than either the the instructors or the students. Yes there are good instructors and poor ones, but they have to teach to the system and they are human. I hold that it is fundamentally wrong and counterproductive to grade the learning process. The ultimate goal of learning is to reach competence at the end of the process. Clearly at the beginning the student is not competent and so need to follow the path of learning, practise, making mistakes, finding out what can go wrong and how it can go right. None of this should be graded except as an aid to learning for both the student and instructor. And none of the learning process should count towards the final result. The whole point of having an instructor is to assess where the student gets it right and where they get it wrong and point out the correct way and to help/guide where the competence is understanding rather than skill ie to promote that 'light bulb moment'. Since this may lead to personality clashes, it is important that the instructor has no say in the final assessment of competence. This should be made by some independent means. You are right in saying there is a big difference between turning out competent humans and self-sealing stem bolts.
  18. Great summary +1. @Trần Thành 2022 I can only add that I suggest you look up 'mass defect and chemical reactions'. Here is a good reference. https://chem.libretexts.org/Courses/Grand_Rapids_Community_College/CHM_120_-_Survey_of_General_Chemistry/2%3A_Atomic_Structure/2.07_Mass_Defect_-_The_Source_of_Nuclear_Energy Note this also explains swansont's good point further.
  19. Yes, I think he did say that, or something very like it. Did you read his delightful book on symmetry containing references to his visit to the Alhambra ? The interesting thing about patterns is that they can be algebraicised for incidence using incidence matrices. Another related point is that in studying patterns maths uses much the same techniques as other disciplines. In particular idealisations are extracted from observations of reality, many of these idealisations are also of great use in Physics. Another author much concerned with pattern is Philip Ball - The self Made Tapestry. Di you know the pattern to determine the optimum level to fill a cement mixer ?
  20. And yet, as I noted, it is the first statement to appear in Euclid! You can't have Pythagoras, without algebra. I don't see how much Geometry you can do without numbers. There is no similarity theorem in curvilinear space eg in spherical triangles. You can't have vector geometry without the zero vector as it is involved in four of the vector axioms. Therefore the link between extrinsic and intrinsic geometry (holonomy) fails as you can't have vector fields without vectors. Without a side of zero length, Aubel's theorem in plane geometry is incomplete. What I am saying is that 'zero' is buried very deep in Geometry so that we don't normally think about it, On the other hand I fully support this comment. There is plenty of archeological evidence that maths was in at the beginning of language. (see John Derbyshire "Unknown Quantity"). However perhaps 'counting' is too sophisticated a word and a deeper, but incredibly useful, mathematical technique of tallying (putting in one-to-one correspondence) came first. Perhaps this process even came before real words separated out from grunts which simply meant "Attention!" I don't disagree with this at all. Maths is a tool. And we'd rather use maths to make a hammer than use hammers to do maths. I even think maths is at the basis of language. Even people who say they hate maths, I think, have a simpler, more basic way of mathematically understanding the world. Perhaps less sophisticated, refined, or whatever. May I respectfully remind both you and @MigL that this thread was started by a professor of Mathematics, in the Mathematics section ? Whilst I would wholeheartedly agree that maths is an incredibly useful tool in many disciplines, especially Physics, this is surely a question of Mathematics?
  21. But that is part of the definition of a distance function (the Mathematical term for a metric) Without zero you have no metric. But that is not the only use of zero in Mathematics. You probably know the four colour theorem, and the two colour theorem. Can you draw a map with zero colours ? Or perhaps you would like this poem I have highlighted the use of zero bend (curvature or the reciprocal of radius) to indicate a straight line. You also need zero in projective geometry for the ratio theorem to indicate the 'missing' ratio.
  22. I do understand you and please be clear. That is just not possible. There are many branches of maths. You need a bit of almost each and evry one to study fully any one branch. That is why Maths in particular is taught in what I call a spiral approach. At each turn of the spiral (say each year) you learn some more of some or all the branches, based on all that you learned in each branch the precious year. Mathematicians carry this on through their working lives. Here is a true story. When I was in junior high I was pretty good at Geometry. I thought I knew everthing there was to know about Geometry and could solve any problem. So when I move up to senior high (college for you like the american system) I was suprised to find the first thing on the curiculum was - Geometry. But Coordinate Geometry. I was shocked and blown away by all this new stuff. But it underlies calculus, applied maths, differential equations, physics and so much more.
  23. Hello, Riba and welcome. Maths on your own huh ? That's both ambitious and admirable. So is asking questions so remember to come here and ask if you get stuck. Quadratic equations and basic trig ? So you are in junior high then. You haven't said which country you are in, but in the UK you will be doing some algebra (those pesky equations), some geometry, which goes with the trig, some modern maths such as sets and logic. There will also be lots of practice applications questions (both in maths symbols and in words). This is all good stuff that leads into later material, normally encountered in senior high. This is where you will encounter calculus first and perhaps a taste of differential equations. But you will also encounter coordinate geometry and other topics. There is no one order to 'do' the different topics in. This is because Maths hangs together so you need a bit of one branch to be able to work in another branch. With that under you belt you can use your new knowledge to expand both branches and perhaps a third, fourth etc. One word of warning since your ambition is (astro)physics. Any course whether at school, college or online will be designed for the questions asked to be solvable using the material taught. But no course teaches it all so they leave out the exceptions and difficult pieces, and never ask questions about them. So it is easy for a student to be lulled into a sense of false security, thinking they know enough. But in real Physics it is often the exception that you need. Go well in your studies, I look forward to periodic questions from you.
  24. Euclid Definition#1 "That which hath no part." This is an early version of a metric which you cannot have without a zero.
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