studiot

So far no one has ridiculed you, though you have been asked for more detail. A short answer is that yes in some circumstance there are (serious) applications of probability to the physics of time. For instance given the probability of the radioactive decay of an atom in a bunch of atoms, you can calculate the probability of simultaneous decay of 2,3 or more atoms. You can also discuss the meaning of simultaneous in terms of the time taken and the uncertainty principle. For a longer answer you need to propose a less flippant example.

Why is it so important to you that your expression be an identity? It may well be an expression that correctly models some phenomenon. But an identity relationship does not depend upon the use you put the expression to. Either the expression always equal one, whatever you use it for and it is then an identity. Or the expression doesn't always equal one, in which case it is not an identity. I am not sure what you are trying to achieve with this projective stuff. Can you elaborate?

Actually it doesn't. The equation actually shows that the relativistic mass v speed curve becomes asymptotic to infinity at the speed of light so is not determinable from that equation.

1) Why unfortunately? Is this personal? 2) No 3) Neither pzkpfw's nor Kodak's image formation on a screen show the screen at the focal distance from the lens optical centre. 4) I didn't suggest it did, I was going to post some very excellent pictures from Nikon about that. 5) If Kodak's calculation is correct (suprise suprise) why are you bitching about it? 6) It's basic that you can start a fire by focusing the Sun's rays onto an imflammable substance with a converging lens. That is what happens if you put the substance at the focus of the lens. A project wants to make the image (much) larger, a camera wants to make it (much) smaller 7) The focal distance is determined by the lens not the light approaching it. If Kodak say their lens has a focal distance of 86 mm I would rather believe them than you, without much stronger evidence than personal attack. I particularly liked this +1 and +1 tp pzkpfw for all the hard (and good) work put in on this thread.

Very smart. However, it's not how a lens forms an image, and it explains why this thread is getting nowhere. I think you misunderstand both Dalo and pzkpfw's diagram. Here is a kodak carousel projector calculator which shows the truth. For instance the focal length of a lens to project a 2m image on a screen 5 metres away is 86mm Kodak projector lenses are in the range 75mm to 200mm focal length, and zoom between these values to focus at a fixed screen. Note that projectors usually achieve Dalo's parallel ray requirement with a collimator between the light source and the slide. Note that projector greatly magnify the object in the mage, whereas cameras greatly reduce the object size in the image on film or sensor. Damm I lost all my text again https://www.digitalslides.co.uk/wp-2013/faq-items/what-focal-length-lens-do-i-need-for-my-slide-projector/

With respect, there's room for theory speculation? hypotheses?

Because Dalo want's to display an image on the screen, not burn a hole in it ??

Do not be modest about your diagram,. It is not only better than no diagram, it is perfectly fit for purpose ie adequate. So +1

[math]{\sin ^2}\nu + {\cos ^2}\nu = 1[/math] Is a trigonometric identity since there are no values of nu that do not satisfy the equality. So is [math]\sin (\alpha + \beta ) = \sin \alpha \cos \beta + \cos \alpha \sin \beta [/math] again since there are no (pairs of) values of alpha and beta that fail to satisfy the equality. But there are still values of (triples of) alpha, phi and lambda that fail to satisfy your equation 6. So it is still not an identity. For instance put [math]\alpha = \frac{\pi }{4}\quad \varphi = \frac{\pi }{2}\quad \lambda = \frac{\pi }{2}[/math] Then your expression 6 becomes [math]{\left( {\sqrt {\frac{2}{4}} *\sqrt {\frac{2}{4}} } \right)^2} + {\left( {1*\sqrt {\frac{2}{4}} } \right)^2} = \frac{3}{4} \ne 1[/math]

What better place to start then the two books from the man who did the original experiments? The Mathematical Theory of Relativity Space Time and Gravitation. Both by Eddington Cambridge University Press.

You made an assertion concerning three numeric variables. You gave them Greek letters which makes writing about them more difficult, they could just as easily be called a, b and c or x1, x2 and x3. Whatever, fixing the first two has a greater impact than you think on the third variable, you call alpha. Fixing phi and lambda to the values I gave reduces alpha to numbers which make [math]\cos \alpha [/math] zero or the contents of the second brakcet cannot reduce to the square root of 1. There is only one value of alpha that satisfies this (along with its cyclic values). Dimensions have no bearing upon identities. Perhaps you should look up the meaning of the term in mathematics. Your formula is an equation not an identity. As such it has solutions for some phi, lambda and alpha, but not all.

Grease and oils form a barrier that is more easily removed by hot (soapy) water.

OK so if it is an identity the equation asserts that this is true for any value of phi, lambda and alpha so let us try some [math]{\left( {\cos \frac{\varphi }{2}\sin \frac{\lambda }{2}} \right)^2} + {\left( {\cot \frac{\varphi }{2} + \cos \alpha } \right)^2} = 1[/math] Put [math]\varphi = \frac{\pi }{2}\quad \lambda = 0\quad \alpha = \pi [/math] This computes to [math]{\left( {\sqrt {\frac{2}{4}} *0} \right)^2} + {\left( {1 + \left( { - 1} \right)} \right)^2} = 0 \ne 1[/math] OOps so the equation is not an indentity.

pi is pi. As such it is one of the infinity of what we call real numbers that cannot be represented in decimal format. Since it is so important to us we give it a special name - pi. This is no different from asking what is one half exactly, when using the counting numbers. One half is one of the infinity of what we call rational numbers that cannot be represented in integer format. Since it is so important to us we give it a special name - one half.

What's to reconcile? Draw a line four inches long. Before your pencil reached 4 inches and after it reached 3 inches from the start, you must have drawn a straight line exactly pi inches long.

Perhaps if you cleaned and polished your genius spectacles beofre you read my posts you would refrain from posting such rubbish. Of course it is distinguished and measured, if that is what you mean by capture (how do you capture curvature?) within the manifold. That is the definition of intrinsic curvature. There are indeed more esoteric curvatures that can't be so determined. Again the point is that in Physics we are seeking reasonable manifolds that do not exhibit pathogenic properties in order to model reality. The Lorenz relationship is an additional constraint, as is the invariant s2 = x2 + y2 + z2 + (ict)2 The imaginary i provides the nencessary orthogonal rotation and leads to the negative sign usually seen in the formula. This invariant is commonly known as the Minkowski interval.

The whole point about these mappings, from the point of view of Physics, is that each different model of Rn leads to a different geometry, characterised by it intrinsic curvature. So in R2, the plane and the surface of the sphere serve as suitable models. The flatlanders could determine whether their universe was planar or spherical by measurement made purely within their universe (ie in 2D).

That was the simpler version I was looking for. Thank you Strange, +1 1) Yes 2) Yes, there are many types of mapping with fancy words to describe them. This one can't be one to one. 3) Either could be done, but see not (2). An ismorphism is not only one-to one it also preserves the structure of the set, eg to order of the elements (if that is important)

Would you agree that , taken by itself, 1 is a random number? Something like this came up elsehwere. Although you could compare random intervals, this really has nothing to do with the first point. That is why I underlined your first assertion and labelled it (1) and placed your second assertion, about subspaces, in italics and labelled it (2). My humble apologies if that failed to identify that I was makeing two quite separate points. I appreciate the pace this thread has gathered makes for hasty replies. I am fond of observing that reality has more weird quirks than we can dream up. For instance so far as we know the electron is indivisible, but we cannot prove it. I only know of one truly indivisible object - you cannot have half a hole. Or here is another one. Take a simple light switch and light. Start and switch the switch and repeat switching one and off or off and on an infinite number of times. What is the final state of the light (ignoring longevity) ?

Since I expect that comment was mine here is some backup. Any Rn has the cardinality (loosely same number of points) as R and so Rn has the same cardinality as Rm. Note that this does not say that the mapping is unique. https://math.stackexchange.com/questions/966645/cardinality-of-rn-and-r-is-equal The non uniqueness is brought out in the link. Here is a finite example. Consider two (infinite) sets : {1,3,5...} and {0,2,4...} You can make the obvious pairing 1→0,3→2,5→4... But others are available [math]1 \to 2,3 \to 4,5 \to 6...[/math]

The Kolmogorov definition? What about my other points?

Are you dictating or discussing?

Now I'm even more mystified as I can't make any connection to the points I made.? I agree with Koti that random has different meanings to different people and would add that so does order

Not entirely sure what you mean here. 1) How do these assertions play with the phenomenon of emergence? 2) How does this play with the fact you can put any interval of R in one-to-one correspondence with the whole of R?

Well I liked this bit as it reminds me of a simpler example I owe Prometheus and explanation for, The difference between analysis and synthesis. So thanks +1