studiot

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

Yes indeed. Dalo, What do you mean by "field of view"? Do you mean depth of field as Strange talks about? Then you should google "Hyperfocal distance and perhaps add the word charts or calculator This determines how much of the scene is in focus. Alternatively do you mean angle of view? This is set by the focal length and determines how much of the scene the lens 'sees'. BTW, irrelevant here, the sensitivity of the film or electronic sensor does not affect the amount of light falling. Only the lens system through the shutter and aperture can do this. The sensitivity affects the response of the sensor to a given quantity of light.

I smell troll cooking up in the microwave. Not only rude enough to fail to thank Strange for putting in all that effort to search for and find that delectable method, but also to flatly refuse to read it properly twice in a row.

No, it is not. Complete rubbish. I just looked at Strange's link (+1 for finding this) and I note that is aimed at 11 year olds. They are asked to measure (with a ruler) the half wavelength at about 6cm. You can't get much more direct than that. The (wave)length is the only part of the experiment that is an actual measurement and not taken on trust.

I already gave you one. Lecher Lines.

Yes in the linear case there need be no acceleration. In the rotating case you cant do without it.

Yes that is another name for them, though some reserve it for when they model particle like behaviour after the originators (Zebrusky and Kruskal) of that term about 100 years after Russell.

So you have at least two objects.

This is a good time to examine the formula Velocity = frequency x wavelength. Not all waves have a velocity, some are stationary or standing waves. Not all waves have a frequency. Frequency has no meaning for something that only happens once. Yes solitary waves only happen once and therefore have no frequency. Solitary waves is the original name given by the person (Russell) who identified them and they are not single cycles chopped out of a solution to the usual linear wave equation. Solitary waves are non linear and the mathematics constructs a single event. Often the Korteweg-de Vries equation. This last bit is deeper than I really meant for Dalo but Strange also deserves an answer. All waves occupy some space so we can define or identify the region occupied with a space measurement i.e. a wavelength.

You have two objects in your diagram. Either side of the free body diagram, because both objects are contained within the same body.

You always require at least two objects to generate a potential.

Yes misunderstanding the sine curve was prophetic, +1 I try to avoid talking about a sine wave unless I am absolutely certain of my audience because it is loose terminology. I would add a very basic description of simple harmonic motion (SHM) to my three prerequisites for waves since it helps explain the difference between oscillation and wave motion. Dalo please note that both oscillation always has a frequency, but never has a wavelength. Wave motion has a frequency, except for solitary waves, but always has a wavelength.