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Showing content with the highest reputation on 09/01/23 in all areas

  1. OK so before I give any detail a couple of important things you need to know. There are two editions of VCA, which was first published in 1997. VDG&F followed in 2021 and was written in the same style and from the same viewpoint. A second edition of VCA (which I have) was published in 2023 and has some important typographic deficiencies remedied. The maths text is the same, but the many diagrams now all have explanatory captions, the index has been much expanded and a conventional referencing system added, and the book is in a larger physical size. The two books contain much common material and frequently reference each other. Between them they develop the author's theme that it is good for understanding to approach fundamental principles in maths from multiple viewpoints. He expresses how reassuring it is to come to the same result from different routes. Both books have Feynman's american ability to pull out the essential statements in a clear and obvious form and to highlight and separate them from the block of the text. I would certainly recommend both as a pair; you really need the 2023 version of VCA though. I already have three books entitled "The Geometry of Complex Numbers", but apart from the usual few diagrams you might find in any work on complex analysis they do not approach it from the geometric viewpoint at all. VCA certainly achieves this. VCA covers a lot of ground with good mathematical insights. But what it is not is a tabulation or treatise on the applications of complex analysis, which is where many readers are coming from. So if you want CA in the solution of differential equations, it only mentions 2, Schrodinger and Dirac, complex Bessel functions are not treated at all, you will have to go to alfhors for that. Complex integration is dealt with at a fundamental level, in relation to measure theory. It is not a textbook of complex integration techniques. (Conformal) mapping again much wanted by engineers is treated at a mathematically fundamental level rather that a catalogue of techniques. As promised the book is not a catalogue of algebraic results and formulae, with the excuse 'we can explain this result geometrically'. The results are there but arise naturally by considering the geometry ( my preferred way ) and also the topology / continuity. Also arising from this geometric approach Tristran delves deeply into noneuclidian geometry. Hope this helps, sorry it has taken so long to reply, but I needed to do justice to the books.
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  2. Essentially, you need to heat and compress some fuel to get it to fuse. You can only focus EM radiation into a spot roughly the size of the wavelength of the radiation. So for the same output power, you can get much higher power densities (power/ area) using lasers than using masers. And that means you can get a much higher temperature and pressure. In principle, you might imagine extrapolating this and using even shorter wavelengths. But to make a laser you need a population inversion. And the shorter the life span of the upper excited state, the harder it is to fill it up before energy "leaks" out of it. And, all other things being equal, the lifespan of the upper state is inversely proportional to the cube of the energy. That's fundamentally, why it's relatively easy to make lasers for visible and very hard for X-rays. So, for a given input power you can get a lot more output power from a longer wavelength laser. (We use this all the time; our "green laser pointers" are actually IR lasers, frequency doubled to get a visible beam. the increased ease of getting an IR laser to work overcomes teh inefficiency that results from frequency doubling) And, of course, the best known property of x-rays is that they go straight through stuff. If all your input energy is in the form of x-rays, it goes right through your target without affecting it. It doesn't even warm it up. Bother! There's also the fact that we have been using light for a long time and are skilled at manipulating it. So, for us, visible light (or near visible IR/ UV), is the "sweet spot" for getting a lot of energy into a target quickly.
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  3. @Duda Jarek It should better work then a MASER fusion reactor
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  4. Being able to build gamma laser ( https://en.wikipedia.org/wiki/Gamma-ray_laser ), fusion could be trivial - e.g. 782 keV photons to reverse neutron decay: producing free neutrons from hydrogen. However, it is technically extremely difficult, for free electron lasers maybe 30keV might be reachable, here is 14.4keV for nuclear transition: https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.4.L032007
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  5. Some are. Some are not. Correct
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  6. This is moot, since fusion does not deal with significant amounts of bulk material. It also depends on the material and wavelength. Some materials are quite transparent at certain laser wavelengths. Fiber optics depends on this But you don’t want the EM radiation to penetrate, you want it to exert pressure. “NIF does this by amplifying and focusing 192 laser beams onto a tiny hollow metal cylinder at the centre of which is a peppercorn-sized capsule containing the hydrogen isotopes deuterium and tritium. X-rays generated from the walls of the cylinder blast off the outer surface of the capsule, forcing the rest of it inwards thanks to momentum conservation and causing the deuterium and tritium nuclei within it to fuse – in the process releasing alpha particles (helium nuclei), neutrons and lots of energy.”
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  7. Sheep are not so unlike antelope that you need special evolution for it. Fruits, including cucurbits, nuts and roots were always part of the hominid diet. Grains, too, though the strains have been modified over time to suit human needs: they have been accelerated in their evolution by us. Cooking food, including flat-breads, has been in our repertoire for quite a long time; it was the norm well before agriculture was. The problem isn't an evolutionary one; it's commercial one. The more processing an ingredient undergoes, the more of its nutrients are lost - and usually the more refined sugar and chemical preservatives are added.
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  8. I think it's bad science. One can make a lot of other guesses on why they avoided the gap. For example, they knew that it is easier to stay on a flat surface.
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  9. You appear to be experiencing a comprehension problem. https://www.scientificamerican.com/article/is-our-tendency-to-experi/#:~:text=The specific form that fear,has a clear genetic component.
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  10. Yes it's very strange. Yes it is allowed. In fact it's the defining property of infinite sets. We can't do that with a finite set! One way to define an infinite set is to say that it's a set that can be placed into bijection with a proper subset of itself. Only infinite sets have that property. And yes it is strange! This was noticed by Galileo in 1638. He observed that we can correspond the natural numbers with the perfect squares: 0 <-> 0, 1 <->1, 2 <-> 4, 3 <-> 9, etc. So the whole numbers must at the same time be more numerous and equally numerous with the squares. https://en.wikipedia.org/wiki/Galileo's_paradox If he had only stuck to math he would not have gotten into trouble with the Pope. There's a lesson in there somewhere.
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  11. More that the role and universality of language assumes a much greater depth and simplicity at the same time, with proper recognition the Objectives and subjective. No, I can't. As stated I'm very far from being a mathematician. It should not be essential to the purpose regardless. It may or may not complicate my attempts at explanation. Its basic language of subjective to objective informs the language we commonly use. Perhaps the disconnect may be occurring through how we define 'value' in this context? I would say through relativity. Any objective considered in relationship is a value application. All relationships are value expressions, Mathematics and spoken/written language are only two ways of expressing value. Training animals is communication, expression of values, and their recognition. The interplays of environment on objects and organisms are value expressions. All subjective. The effects of biological selection and evolution are value expressions, as is our Human condition.
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  12. No, the acceleration is not 0. The air in the wind moves with the constant velocity before it gets to the wall. At this time, its acceleration is 0. But when it hits the wall, the air cannot continue moving as before because it cannot go through the wall. It starts slowing down before it hits the wall, because of the air in front of it, and eventually it stops moving toward the wall. So, its velocity changes during this time. Change in velocity per time is acceleration.
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