studiot

I think Michel is trying to display the logical connective for implication which is not necessarily symmetric . In boths maths and logic the symbol is some sort of arrow. https://en.wikipedia.org/wiki/Material_conditional

Yes I agree with what you are saying although I would say that the statement x equals 2 does indeed imply that x2 equals 4 and makes perfect sense. What you are describing is the difference (in Mathematics) between an equality and an identity. The statement x2 = 4 is true only for particular values of x and is called an equality. The statement x2 = a, where a is a positive number is true for any value of x and is called an identity.

Obvious to some perhaps. A small point about Topology. It is more than reasonable to have a topological space without a metric, it is fundamental. This is an example of endowing (this mathematically term is perhaps kinder than imposing ?) additional structure on some mathematics to produce richer mathematics as a result. Endowing a space with a metric makes it a metric space. Another example would be endowing a vector space (which is defined as having a single binary operation) with a second binary operation, permitting the construction of tensors, and physical Field theory amongst other things. However the extra structure (position) imposed by a coordinate system is different in that it comes in a package deal with the coordinate system. You can't cut it out.

I do not know that textbook, but various solutions may be found derived in many Physics and Physical Chemistry textbooks. Modern Physics by Eisberg Physical Chemistry by Moore It is important to remember that Schrodinger is a partial differential equation. As such the boundary conditions are needed to provided any useful solution since general solutions of PDEs involve arbitrary functions.

You are getting closer to understanding my thinking. 1) the OP assumes an absolute space as is simply says As is stands this makes no sense since the important velocity should be relative to something else, in the same universe as the ship, as you have correctly noted. If the ship was isolated in its own universe the statement "the ship has velocity v" has no meaning at all. I asked two questions, one one which you addressed in a Barnum sort of way and the other you have ignored twice, although I repeated it in my first reply to you. Here is is again. The quote clearly shows the question mark at the end of the sentence. So I picked out a part of the universe for more detailed examination, and as you again correctly point out that is only part of the known universe. We don't know how 'big' the whole universe is and probably never will. So you are right to encourage the OP to clarify the question, to one that can be answered.

So I will try to answer my own question since you didn't. Google tells us that the diameter of our visible universe is 8.8 x 1026 metres. Suppose this ship flew left to right across our visible universe, what would its relative velocity be to us to compress the diameter of our universe to 1m ? IOW at what speed would it bypass us ? Applying the Lorenz contraction [math]\sqrt {\left( {1 - \frac{{{v^2}}}{{{c^2}}}} \right)} \left( {8.8*{{10}^{26}}} \right) = 1[/math] Square both sides [math]\left( {1 - \frac{{{v^2}}}{{{c^2}}}} \right)\left( {77.4*{{10}^{52}}} \right) = 1[/math] Rearrange [math]\left( {1 - \frac{{{v^2}}}{{{c^2}}}} \right) = \left( {\frac{1}{{77.4}}*{{10}^{ - 52}}} \right) = 1.3*{10^{ - 54}}[/math] Rearrange again [math]{v^2} = {c^2}\left( {1 - 1.3*{{10}^{ - 54}}} \right)[/math] Take square root both sides [math]v = \left( {\sqrt {1 - 1.3*{{10}^{ - 54}}} } \right)c[/math] Wolfram alpha cannot give me this square root the result is so close to c

Yes I think you have. And someone who is stationary relative to a different pair of stars will come ro a different measurement. That is relativity. Where do you think this absolute space is that the ship is travelling so fast that all of it is 'compressed' to 1m ? What exactly is the ship travelling so fast relative to that his gamma factor will be that large? It is like the Muon falling to Earth.

OOps so sorry, I didn't finish the sentence. 😊 It should have been I think in simplified terms what Markus has been saying (I think correctly) is that there is nothing inherent in mathematical structures that provide a space with a 4 dimensional coordinate system with the observed properties of time. I should perhaps add there is nothing to prevent them either. Hopefully that makes more sense now. Thanks for spotting the error +1.

I think the OP scenario is much more simplistic than replies so far. Winterlong, the fly in your ointment argument is that the pilot does not see a contracted universe, that is the view of some observer travelling relative to him. As the pilot masures things, not only does the ship have its normal 10m length, but the universe has its 'normal' length, whatever that is (we don't have a good number for this).

You can often get examples from google by selecting the 'images tab for example Fourier series https://www.google.co.uk/search?q=Fourier+series&source=lnms&tbm=isch&sa=X&ved=2ahUKEwjQ-IDVo4zrAhVWiFwKHSePAjIQ_AUoAnoECBIQBA&biw=1366&bih=646 This should keep you busy.

I think in simplified terms what Markus has been saying (I think correctly) is that there is nothing inherent in mathematical structures that provide a space with a 4 dimensional coordinate system. Therefore you have to look elsewhere for the controlling relationships that provide the properties/characteristics attributable to time. Thus we are back to constitution and compatibility again.

Look here at the 'four rules' https://www.masterorganicchemistry.com/2017/03/03/is-this-molecule-aromatic-some-practice-problems/ Look down the examples to cyclobutene (could you name your compound ? ) Can you find a cyclobutene that is aromatic?

You are welcome to demonstrate this in practice, which is why I invited you to do just that (theatrically large flywheels are unneccessary). What would be refrigerated?

So you will have no difficulty providing the answers by using a kitchen thermometer and winding your stirling engine backwards. When you boil the water in your #dewar, mine is two sugars and cream please.

Again not sure what you seek here. The best book I know for understanding is Ahlfors "Complex Analysis" A wide ranging use of complex analysis is Churchill's "Complex Variables and Applications" Also lots of detailed worked examples in Alan Jeffrey's "Complex Analysis and Applications"

Not sure what you are trying to achieve here, seems like a very tall order to me. Have you looked in standard texts such as Titchmarsh "theory of functions" or Knopp "Infinite series and Sequences" ?

What do you mean by direction ? What happens if you reverse the hot and cold plates in your machine ; Does the machine then run counter clockwise, if it ran clockwise before ? Now what happens if you do nothing but monitor the plate temperatures and drive the flywheel first clockwise and then counter clockwise with say a small electric motor? I don't see the difficulty measuring these temperatures - for a first ball park run you do not need to be very accurate. You will also gain so much more by conducting these simple experiments for yourself, than by posting details of exotic machinery.

I should also have said that there are two types of atoms. Those which are support these processes and are 'radioactive'. And those which are do not support these processes and are called 'stable'. The end of all radioactive processes are stable atoms, but on the way other radioactive atoms are formed and, which in turn, eventually decay to stable atoms.

Another example of change outside of time. +1

(Atomic) Radiation comes in 3 types, called alpha, beta and gamma. The lifetime of these radiations themselves is quite short as they interact with material the encounter (hit). It is this interaction which makes them dangerous. A piece of tissue paper will stop most alpha rays. Beta rays are more penetrating, and are the same type that are used in electron beam welding. Gamma rays are the most penetrating and have been used in X ray machines. The point is that there is a continual production and dispersion of this radiation due to processes inside the atom. In NW Wales, where you are (Trawsfynydd decomissioned 1919 or Wylfa decomissioned in 2015 ) are both located near natural granite rocks. Granite (as in Cornwall) produces natural radiation by these atomic internal processes. These processes can take many thousands of years to complete, and have been going on since the rocks were formed. So all this time natural rocks have been generating radiation. We call this background radiation. However the natural background is, we hope, safe. Mankind has sifted and concentrated the rocks that have a high level of activity, raising the radiation levels to way above the natural abckgoround. At these raised radiation levels the radiation can create / initiate new processes in material that is not of itself radioactive (such as concrete, pipes etc) but can become radioactive by these new processes. These secondary radioactive sources are generally much more short lived than the ones in the original rocks, but the concentration makes them still dangerous untill they have died away. Does this help ?

So don't you owe it to us to share the details of your insight ?

On my way from Bristol to Nottingham I had to change buses in Birmingham. Both buses were red routemasters, the only difference being the first was the no. 7 service, the second a no. 12.

A further example, The measurement of goods for sale varies, apples by weight, milk by volume but now the the measurement of apples has changed to number (a pack of six).

Yes it is interesting observing several foreigners (no offence at all meant) discssing a finer point of my native language. They have useful material to add. But here is my take on vary v change. Let us use the example "What is considered the ideal female form ?" This varies from location to location and ethnic group to ethnic group. It has also changed at one location / ethnic group over time. The models of Rubens were plumper than the models of today. Another example John travels frequently from London to Amsterdam. He varies his route, sometimes going by ferry to France and sometimes flying direct. Yesterday he intended to fly, but changed his route to the ferry at the last moment.

Thank you for your reply studiot, but I couldn't follow the second last paragraph of your explanation, starting with Your book must have assumed.... You see guys, even here I am getting mixed answers, John says the data is insufficient and studiot says the question's fine. I see you are here again so I will post half an answer. Actually John is quite right and I also siad the same thing. Did you understand that part? It would be helpful to say whether you are studying Engineering or Physics, as their view is slightly different. Here are a few pages from the UK standard intoductory text for at least 50 years. As you can see it only goes so far in explanation. I expect it say similar things to your book. Please compare them and let us know.