studiot

Good that you have this sorted, is the pace better now ? So onto the next bit, here we see how non linearity arises quite naturally in a simple way. Keeping to my idea of worlds, we change from the world of material objects (fruit) to the world of abstract or theoretical objects, shapes. Out selection this time are simple squares and we consider the area of these squares, not as the number of squares increases but as the size as measured by length of side increases. When we plot the relationship between area and length of side the graph is found to be a curved line. This is defined as non linear since it is not a straight line. Comparing our linear apple equation with the formula for area we then find a new way of denoting linear and non linear. This new way involves looking at the powers of the variables involved, and new terminology is introduced ready for future use. So linear equationsa are of the first degree non linear equations are of the second, third, fourth or whatever degree, so long as it is greater than first.

Perhaps I have been trying to go to fast. Remember that we are working our way towards the formula the @KJW gave for linearity ? I like apples because the great man, Newton, liked them and I have already introduced the counting idea using them. Remember I said that one important idea is to draw a boundary between the subject of interest and the rest of the universe ? Well the box in the first picture attached does just that. The boundary is the dashed line. The subject of interest, also called the system, is a collection of apples. Note that my world is just 'The World of Fruit'. This is meant to illustrate that the whole arrangement is just an imaginary way of organising real things and that I can limit the 'world' to something useful. Working now just on my subject of interest - the collection of all apples. This collection allows me to create the imaginary 'average apple' that I can use to represent every apple. Ley us say for convenience that this average apple weighs 0.1 kilogrammes I have tabulated the weight of several different numbers of these apples. And then plotted a schoolboy graph of the weight of apples against number of apples so I can read off the weight of any number of apples. We say two things about this graph. That the weight of apples is proportional to the number of apples and I have given a simple formula, which we can examine in more detail next time. We also say that the the relationship between weight and numbers of apples is Linear. This relationship is the simplest form of linear and the plot or graph is a straight line. A linear relationship is one of proportionality and has the equation or formula of a constant (1/10 in this case) times one of the two related variables. How are we doing ? Yes, but I think you will find that the original statement referred to changes in the observed system, not the observer.

Well as I have no idea who or what they are I really don't have a choice. As to leonardo, what Maths is he responsible for ?, apart from a few extensions to Euclid ?

'the kid' doesn't appear to have come back or I would have refrained from commenting on your post so as not to confuse him. But we can certainly expand the discussion if you like. For your information Newton originally invented the calculus of finite differences, before going on to the differential calculus of a single continuous variable. For my money, he was also the greatest genius that ever lived because not only did he largely invent mathematical physics he had to invent the mathematics as well. Others who came after always then largely had the benefit of sufficient mathematics to work with.

I'm glad you worked that out for yourself. Well done. Tell me, do pictures help you understand ? It may do and it may not. @joigus already told you that. An example of a measurement that makes no changes would be to count the apples on a tree. Whilst I like genady's fishy example, I think it quickly sowed it was too complicated so I will stick with apples.

I agree with joigus, a very interesting viewpoint. That is more like the old Eise we know and love. +1 I have always had a great respect for Smullyan, and I sadly left his excellent book about Godel on a train a few years ago. But I'm sorry his dualist story I find rather contrived, though I agree that there is more than a mechanistic meat machine to human ( and perhaps other) thought.

I think you mean what effects ? The impoetant thing to remember is that we define linearity and evrything else is non linear. I will do the next bit after lunch.

You really should get out and about more. 'differential' is both a noun and an adjective much used in technical subjects and generally some sort of difference, which can be very large. This has little or nothing to do with infinitesimals. Perhaps you should look in the fields of automotive engineering, medicine, hydroelectric engineering, clean room engineering and many more besides.

A perfect fluid is one with no turbulence and no viscosity.

What about It ? Hve you done no work on it wither from a physics or statistical or even medical point of view ? You first post seems to me to be more like a blog, summarising other people's blogs. 5,500 deaths ? Let us put that in context of the 8 billion or so people on the planet, that's almost 1 in 2 million. How does this compare with other causes of death ? https://ourworldindata.org/causes-of-death That's just slightly less than the bottom line of the longest list I could quickly find 'Natural Disasters'.

And I am weak in my French, although my time at the lycee Francois Villion in Paris helped me immensely with my schoolboy French. But I think both of us are intelligent, after all you have mastered at least French and English, n'est ce pas ? I do not think the world (universe) tells us anything as that would imply it was deliberately trying to communicate with us. Certainly it interacts with us an we interact with it. In doing so the interaction influences us and quantum theory is still debating how much we influence it. We learn not to put out hand into the fire. The passage from Dicke I posted has more to say on this. But in your statement you have done something very scientific. A very important theoretical scientific techique is to theoretically separate the 'subject of interest' into two parts and draw a line or boundary between them. Then to examine and study whatever passes across that boundary. You have drawn the boundary between us and the rest of the universe. Well done. You have mentioned non linearity twice in the last post and before that so let us look more deeply into the subject of linearity and non linearity. Not only do I agree with you, but I like the sentiment with which you posted something intended to be helpful. +1 But there is much more to linearity and linear Maths and linear Science than this so let us see how we get there. Essentialy linear means ' arranged in a (sensibly straight) line' Luc do you understand what I mean when I say 'sensibly straight' ? For example in particle physics the particles in a linear accelerator travel in a straight line whereas those in a cyclotron travel in a circle or spiral (ie a curved line). If I take the numbers 1, 2, 3, 4, 5, 6, 7, 8 , 9, 10, 11, 12, they are arranged in a straight line or linear order. But if I look at a clock face those same numbers are arranged around a circle so something is different. I have stuff to do now, but I will expand on this in my next post, where we will find the simplest of (schoolboy maths is non linear) Did I mention that we define 'linear' and declare that everything else is non linear. This is much much simpler than the other way round.

Other feature ? Other than what ? Here is some important ones. Firstly the photoelectric effect Classically there should be no threshold frequency to this effect. Yet observationally it is very sharply defined, as required by QM basic tenet that energy can only be accepted in certain quanta. So if the incoming photon does not have enough energy for the electron to transition to participate in a current, there is no current. In the Bohr atom and subsequent models, QM overcomes the classical problem of why an electron accelerating in the electrostatic field of the nucleus does not radiate its energy and fall into the nucleus. QM explains why the nucleus holds together (shell theory) Qm explains radioactive elements and radioactivity. QM explains observed spectra. QM explains band theory, metallic and semiconductor bonding and results in the non linear Konig Penny equation. These are all huge gains in our knowledge of how our world works. There are many other more specialist details but I note a discussion between @Luc Turpin and @Genady about 'our world' The core of the two major modern theories in Physics, relativity and QM were done and dusted and ironed out during the first half of the 20th century. This allowed many applications and also refinements and technology improvements to be made during the second half. Here is a very clear explanation of 'our world' from just after this half way point from two Princeton Professors, Dicke and Witke This clarity about Our world and physics (models) should also be of interest to @mar_mar In their book there is consider help transitioning from a classical mechanics to a quantum mechanics viewpoint. Also included is a whole chapter on the 'correspondence principle', using an interesting view of going backwards from QM to classical as a limit of very simple cases.

Why small? That is not always so.

Yes you are right. +1 Thanks for all the extra detail. But please remember tha many of these equations are 'linearised' (approximated linearly) in order to be able to solve them. Even special relativity employs a linearised quadratic for this. It's an inequality.

Good question, especially if you actually know what linear means ? Vector spaces are the backbone of linear mathematics. I already mentioned a while back that with the Schrodinger equation (which is linear) we are working in the vector space of square integrable functions of class C∞ However the relativistic version of Schrodinger is non linear ( Dirac equation, Klein Gordon equation etc) https://en.wikipedia.org/wiki/Nonlinear_Dirac_equation This bears out Seth's comment that combining linear operators may result in a non linear equation. +1 It should also be remembered that of the famous Four Laws (of Thermodynamcs) , only the First Law is linear and even has q and w as incomplete differentials. Back to Luc, Heisenberg's Uncertainty Principle is not even an equation - It is an inequality like the Second Law of Thermodynamics. Pauli Matrices are linear.

I have posted core information several times, but you seem to have little to ask or discuss about it. Lasr time spin and entanglement were new. Both are key quantum features. I have also drawn on the classical world, where possible, as it is easier and also easier to not be suprised by parallel features there.

Indeed +1. Luc, there are always those who try to 'push the envelope'. This is good and part of the Ccientific Method. But the successful ones which eventually add something to body of knowledge generally know the core of the subject they are hoping to expand. And I thought that it was the core of Quantum Mechanics you are hoping the get to grips with not the fringes. There is a whole Wikipedia article on the subject of Bell's Tests with quite a list of isolated esoteric experiments that generally have not been independently repeated. It is this repeatability that is important so the as near as every time we go to the cupboard the same thing happens, rather than the odd instance where the mouse has got into the cookie jar. Take entanglement. There are questions, but this happens every day all over the universe in an entirely predictable way. We rely on it, our existence relies on it. This is the entanglement of two electrons in a molecular orbital forming a bond. This exploration of conceivable, but fringe, effects has been going on for a long time in many disciplines. In classical mechanics, it is known that a spinning top has two stable positions - upright and upside down. And that it is theoretically possible for the spinning top to spontaneously flip between these states. Such flipping has been invoked to explain geological process from Noah's flood to plate techtonics on Earth, but we have never found evidence that this is what actually happened. However such flipping has been observed as a frequent and regular occurence in the sub atomic world of bonding orbitals so is not 'pie in the sky'

Becasue it's by Deepak Chopras's mates ? I'm sorry Luc, that blog reminds me of the early days of the science of Geology. Lots of discoveries were made by people whose motivation was to prove the glory of God.

Because we have had 6 pages of listening to failed attempts to reconcile a series of self contradictory statements. In particular "Gravity is still a force." do you mean a newtonian force following all the Newtonian rules ? or Do you mean a special Relativity 4 force following the rules of special relaticity? or Do you mean something following some other as yet undefined rules ? Since these choices are not compatible you need to pick one and stick to it, not pick n mix

If you knew the meaning of the basic terms you are bandying about, you would know exactly what my simple question is asking. You want to introduce imaginary hobgoblin 'forces' when you don't actually know what a force is.

If folks want to have a ding dong about judicial deterrence, prisons, sentences and so on can we have a proper thread for it please, rather than dragging this thread about free will further and further off topic ?

The answer to a person's difficulties in understaning existing names for natural processes is not by introducing fresh imaginary extra processes with fancy names but by putting in the effort to properly understand the ones we already work with. Once that has been done and there is still a phenomenon that cannot be explained is the time to introduce new ones. Now one thing about forces, and I don't see any evidence that you understand what is meant by a force, is that all known forces except one, can be shielded against. How would your proposals work in the case of shielding or not ?

But, just like the number pi we can do calculations involving the wave function. 😀

There has been lots of, probably confusing, argument about quantum interpretation, calculation and 'measurement' . A fundamental question that need to be addressed before any of this can be done concerns the wave function. Consider a photon or electron just poddling along. It has a wave function Ψa , Now let the particle interact with the rest of the universe. What is the wave function now ? Let's call it Ψb Is Ψb the same as Ψa or is it a new wave function ? If the wave function for the part of the universe that it interacts with is Ψc , can Ψb be constructed from some combination of Ψa and Ψc ? In other words when there is an interaction does a new wave function appear which now includes the 'observer' in the quantum system ? Please note this extremely important comment from swansont. You only get one dot. Moreover this one dot 'contains' all the quantum of energy of the one photon or electron. This is not wave behaviour. (The italics were mine, the bold was swansont.) We will see why this is important when we fully examine the mechanism of the double slit from both classical and quantum points of view. But I keep saying, and I hope you can now begin to see, that the double slit experiment is difficult and complicated for an good intructory explanation of QM. Here is an excellent short extract from a Cambridge University text, Optical Physics by Lipson and Lipson whcih introduces QM via the PhotoElectric Effect, which is much simpler and more understandable.

You are the musician, not I. But I have always understood that in English, and internationally, we borrow from the Italian, not the German. And I see that there is a whole range of terms for playing vigorously, available, but none that I can see related to my version of coerce. the middle one meaning emphasise would by marcato https://en.wikipedia.org/wiki/List_of_Italian_musical_terms_used_in_English The OED is not very helpful but seems to go along with my notion that coerce is reserved for some animate entity imposing its will on another as opposed to an inanimate entity the actions of an animate one.