studiot

Sorry I haven’t done any significant posting for a while, but the crap new forum software doesn’t play nice with my style of posting and I’ve been too busy to cope with that recently. I’m an applied mathematician which means I concentrate on getting results and leave the burdensome task of making sure the theory is ship shape and watertight to the pure mathematicians and trust them to do that. So here’s the deal. I’m offering a unique mix of geometry, topology and mathematical analysis, blended to serve a particular purpose. After al these are branches of the same subject, not separate subjects on their own. You have presented a good opening post and explained your need and goals clearly. A topological view is too general; a geometric one to restrictive and analysis is required because you want to do calculus on surfaces. Intrinsic curvature is firmly placed in the classical differential geometry of Euler. So I am going to start with a few simple ideas, some classical, some modern, and develop them to a situation where you have a useful and understandable object (surface) you can do calculus on. Euclid said By which we mean today that 0) A point is of zero dimensions or is a 0 dimensional space. 1) A line of one dimension or is a 1 dimensional space. 2) A surface is of two dimensions or is a 2 dimensional space. 3) A solid is of three dimensions or is a 3 dimensional space. I will return to this (0) (1) (2) (3) theme a few times from different points of view., but first a theorem. If we take R to mean all the real numbers and p to mean any one particular positive integer we call the result Rp space, which has dimension p. R0 has exactly one point For any p > 0, Rp has equal numbers of points Thus Rm has the same number of points as Rn Or we can establish a one-to-one mapping between any Rm and any Rn This is the underlying reason that allows us to write down (and sometimes solve) functions and equations in Mathematical Physics and do ‘calculus' on them to find and compare properties like curvature. So 0) R0, a zero dimensional space is a point 1) R1, a one dimensional space is a line Notes A line in this sense has no ends and may be straight or curved, looped or open and extended. Parts of a line have ends and are called a line segment or just a segment. 2) R2, a two dimensional space has no boundaries, just as a one dimensional line has no ends 3) R3, a three dimensional space, is the normal everyday space we live and do Physics in and again has no boundaries. Back again to my 0,1,2,3 from a different point of view. 0) A point is indivisible. 1) A point divides a line into two parts. Note the line is considered to extend indefinitely in both directions. 2) A line divides a surface into two parts 3) A surface divides 3 dimensional space into two parts. Summary an (n-1) dimensional space divides an n dimensional space into two parts. We call an (n-1) dimensional space an (n-1) dimensional hypersurface. But can an (n-2) space divide an n dimensional space? Well a single point can’t divide a plane – you can always go around it A line can exist in 3 dimensional space, but you can always go around it. So no, spaces of lower order than (n-1) can’t divide n space and the reason why we single out (n-1) space for special treatment. This last bit introduced some new ideas, That we can embed a lower dimensional space in one of higher dimension, Or equivalently we can select a part of a space to create a complete space of lower dimension. This selection is then called a subspace. So The point in FIG1 is a zero dimensional space embedded in a one dimensional space The line in FIG2 is a one dimensional subspace embedded in a two dimensional space The surface in FIG3 is a two dimensional subspace embedded in a three dimensional space But There is also something different between FIGs 2 and 3 and FIG 1. FIG2 contains a looped line as well as an extended one and FIG3 a closed surface, as well as an extended one. This brings us to the topological definition of a surface. “A surface is a two dimensional boundary between a solid object in a 3 dimensional space” As this is a work in progress We shall see in the next instalment that this is too general for our purposes, and find out that what we need to retain from topology is connectness and compactness to enable continuity and therefore calculus We will also revisit the 0,1,2,3 again to introduce parameters.

A word about your technique. Please don't react the wrong way to this, it is meant to help you progress whatever your goal actually is. The most important thing to learn here would be to stop using terms and notation that already have very well and tightly defined meanings for something entirely different. I'm sure the moderators do not expect this when they said Such presentation makes your posts impenetrable to even the most intelligent members and leads to comments like this So start with your basic object and give it a name. It is not a number so how about duplet or couplet since it appears to be formed of two simpler objects joined together in some way. Then you can explain how the joining process works and what you can do with your object. If you must use the term 'binary expression' (which establishes a connection between two objects) you (and others) will then be in a position to distinguish between your object (see why it needs a name of its own?) and objects already named and defined by you or anybody else. I really hope this helps you progress.

I have been unsure as to your position on the topic, since your comments have all been rather cryptic. But thinking about it, you are right that at many discontinuities some physical, measurable quantity has multiple values (the imperfection of our equipment aside). The hydraulic jump is one such which has a discontinuity with duplicate values in the specific energy line. Other examples would be the electrical output v time of a square wave generator or a staircase generator.

So can you show this for the mathematics of the hydraulic jump since that was one of my OP examples?

Who said anything about Physics? Did you read the thread title or the OP? Here is an example from mathematics that is as clear cut as I can make it The integer line is discontinuous The real number line is continuous But I don't see any multiple values in either.

Yeah, I like it.

Stating there Surely that is a contradiction, (hence Bell's paradox) not a discontinuity ?

Let me try to cool this discussion off a bit. Cladking, can you tell us the purpose of the thermal imaging and the timescale over which it was conducted? I ask because one of the projects I was once associated with occurred when the builders of two 400+ metre bridge girders made them over 120mm too short. The solution involved measuring the thermal profile of that bridge, which was a very tricky operation.

Thank you for replying, but I don't see a discontinuity. Can you pinpoint it ? The endpoint of a load extension graph is not a discontinuity since the graph does not go beyond that point.

Have you lost interest here?

Yes, http://scienceworld.wolfram.com/physics/SpeedofLight.html

Hello geordie, Yes there is more than one definition of a surface in Mathematics, and yes they exist in high dimensions and are called hypersurfaces. To best understand this start with 2 dimensions. A surface is a two dimensional plot or graph in three dimensions, of the function z = f(x,y) or f(x,y,z) = 0 Another definition is that a surface is a connected set of locally flat points. Such a set can be a subset or subspace of any number of higher dimensions, but is still a two dimensional manifold. I will draw some (hopefully helpful) diagrams when I have time.

Then your understanding is seriously flawed. How can a wave have no dimension? And if light gives up its energy, then it no longer exists therefore the wave no longer exists. It's that overmarinaded tuna agian.

There are good reasons to thinks we don't The lack of observable shadows is one such. Poincare had another topological one.

The idea is sound. But I also said that it is not pure maths. So I have been seeking a place in applied maths place for it, where many similar ideas already operate. However none of these run counter to the underlying pure maths - they all conform to it as the master plan. They also all have extra restrictions peculiar to their own application. That is also probably why uncool has spent so much time trying to work it out with you. You should thank him for that. The problem is that you want your idea to be more basic than the underlying maths rather than a restricted application like all the others. I'm sorry but this it can never be.

What do you expect stamping your foot and showing off your temper to achieve? I neither said that I did or that I did not like your explanations I said several times that I did not understand it. That is why I repeatedly asked for further explanation. I am still waiting for this.

I would like to make it quite plain before this thread is closed as unproductive that I have only argued with you once in my seven posts in this thread. You quickly agreed that I was right and that you cannot place all the real numbers in a table of any sort. I congratulated you on this. Apart from that all my post have been questions as I have tried to understand what your proposal actually is. Each time you have failed to answer and finally stated that you cannot answer. Each of my questions have been straightforward technical questions. As the author of a hypothesis, how do you expect it to be accepted if you cannot answer questions about it?

Strange and DrP I fear you will be waiting a very long time for your £20k. I am still waiting for a reasonable and proper response to my reasonable and proper comment and question on the first page of this charade. I am also considering reporting it as not being fit for the Physics section.

Conventional balances are not sensitive enough to measure the mass change effects you want to discuss. They can be observed by mass spectrometer. However as others have pointed out its more complicated than you are making out. One thing to note is that photon radiation is a dynamic phenomenon. So energy lost with escaping photons is at least partially replaced or even outwitghed by energy gained from radiatiion due to the surrounds. How about a simple prediction along the lines of "If a block of X milligrams of material A is cooled by Y degrees K then there will be a corresponding gain/loss of Z picograms of mass"

Oh dear thank you Prometheus +1 for spotting that. My excuse is that this was earlier than I usually post and to adapt the words of Bob Hope I don't think anything in the morning I don't think anyhthing until noon and then it's time for my nap. My apologies to all concerned obviously my equation should have the terms on its left hand side reversed. 1 - (0.5)n NO, it means there is no value, constant or otherwise. Zilch. So no we cannot correctly assert that h will happen, only that it becomes of increasingly greater probability. One thing that should be noted is that the axiom that all the probabilities add to 1 means that the possibilities (event) are disjoint. In other words each throw is entirely independent of the results of any other throw. This is not the case with geological phenomena, in particular the second comment by prometheus states otherwise.

To analyse Yellowstone or San Andreas or Mt St Helens you need a Bayesian statistical analysis, not the type I posted above.

Let us say you want to toss a head- call it H To analyse the possibility of there being some x number of throws by which an H must have been thrown proceed as follows. If you haven't tossed an H in any throw before x then all the tosses are Ts. The probability of tossing H is 0.5 and the probability of tossing T is also 0.5. in one throw. The probability of tossing T in both of two throws is (0.5) x (0.5) = 0.25 The probabiltiy of tossing T in all of three throws is (0.5) x (0.5) x (0.5) = 0.125 can you see the trend? (we need Two things) 1) The probability of tossing T in all of n throws is (0.5) x(0.5) x (0.5) x (0.5)........... n times = (0.5)n 2) The probability tossing at least one H is (the probability of tossing all Ts minus 1) = (0.5)n - 1 To be certain of an H we need this to equal one. (0.5)n - 1 = 1 So (0.5)n must equal zero So we are looking for an n for which (0.5)n = 0 To try to find this we need to investigate what mathematicians call a limit, which is a number (zero in this case) which the result of the expression gets closer and closer to the large n gets. [math]\mathop {\lim }\limits_{n \to x} {\left( {0.5} \right)^n}[/math] Now I hope you can see that this can never be zero since we have [math]\mathop {\lim }\limits_{n \to \infty } {\left( {0.5} \right)^n} = 0[/math] That is no matter how large an n we take the result is always greater than zero all the way to infinity. So there is no x for which the original proposition is true.

Well I think that that the equations found at the end of your post, labelled 85 and 86 tell something important about the phase relationship between the magnetic component and the electric component of an electromagnetic wave. Do you think either of those components could exist on their own?

This thread was inspired by a comment in a recent thread here on optics so I though I'd share my Wiki research on he subject. https://en.wikipedia.org/wiki/List_of_humorous_units_of_measurement post#51

E=hf So observer A sees different energy of light then observer B yet at the source the light is evenly distributed.....You can describe those thing 'classically' but in order to explain it you need to study the particle behavior. Quantum mechanical collapse states that when light does not travel trough vacuum, the photons interact with particles of the transmission medium they are in. I didn't understand this let alone see how it connects to my point. Are you denying that we use the doppler shift to observe and measure many things about distant stars and galaxies? How can anything deflect from its path without an interaction? What do you mean with this: "that every particle is scattered a different variable amount in a different variable direction" The momentum decides how the photon is scattered, how is that random? Yes but we have a saying in English "It takes two to tango " A (momentum related) interaction is the photon striking a particle of the medium. This is random for two reasons. 1) All the medium particles are moving about in random directions, so the resultant momentum vectors will be randomly directed (scattered) 2) All the medium particles have a physical size (the unit of this used to be called rather picturesquely a barn). This means that there will be a random cross section of impact from full head on to just glancing or grazing. I have already mentioned this. Edit, If the phton doesn't strike the medium particles in this fashion but meets them in a regular way because they are all lined up in a crystsal then we get diffraction, not scattering.