studiot

I have to say that I am not a fan of \prof Al-Khalili's presentations. I find them shallow and glossed over in the difficult bits. Sometimes they are at best just misleading as with the TV 'science'' programme referred to above. In particular the probability explanation provided for the slits just does not work mathematically, it is too much of a simplification that is IMHO misleading. Probabilities are defined as positive real numbers. You have to work with complex numbers to achieve wave cancellation, despite the pretty diagram animation AK shows. Unless, of course, you are prepared to introduce negative probabilities and work with them.

With all that huff and puff do you understand what superposition asserts? It is simply a mathematical shorthand for saying that when two (or more) causes, drivers or activities are present, their effects add up and in particular certain properties combine in a linear fashion according to the normal rules of arithmetic. This has been used widely from engineering stress analysis to electrical circuit theory to chemical bonding to classical and quantum wave mechanics and many more. It is one of the most field tested proceedures in applied mathematics.

Something to remember when discussing mass and other parameters. It is called renormalisation. http://www.volkerschatz.com/science/renorm.html In regards to mass it involves replacing the mass one would measure for a free isolated particle by what is called the 'effective mass' in formulae such as QM or Newton's second law. The effective mass is based on the free mass but modified by the environment. The environment inclues the self interaction by the particle particularly in quantum field theory. The QFT version is difficult. The original proposal by Green in 1830 in regards to replacing mo in Newton's second law by me = (mo + 0.5M) in hydrodynamics. M is the mass of the water displaced by the particle and Newton's second law works properly using me.

I think Strange was referring to the structure which may well not be preserved by the mapping. I think this whole thread has arisen because I failed to make this distinction when I first talked about the one-to-one correspondence. Cantor provides a simple example within one of his proofs. You can put the even (or odd) positive integers into one-to-one correspondence with all the positive integers. The structure of continually increasing magnitude is preserved, but the odd/even characteristic is not.

I note that the treatment for many of these unusual conditions includes "Reassurance".

I think we should separate this into two questions. 1) Should you tell them? Yes, they will find out sooner or later anyway and could sue you if you hide it. Early diagnosis may help treatement if needed. 2) How do you tell them? This is where the rest of their circumstances should be taken into account in the telling.

Since you have posted this question in classical Physics, where bodies are allowed to be 'at rest' I think it should be pointed out that a body at rest has no momentum, but may have mass. Its mass may be modelled as' light' that is insignificant in application.

Thank you Strange, I have learned a new word today. +1

Where exactly did I suggest it was?

Amazing what some folks have on their kitchen tables. +1 I want one.

No it was a joke and quite funny I thought.

I have to observe that TIS is correct here. I meant to add exactly this to my post but it somehow got lost. The verb "to be happy" is a reflexive verb, so the subject and object are the same. So both are clear if someone says "I am happy". On the other hand, the verb to lie is transitive as it requires an object (the subject of the lie). So the question "about what ?" is valid, unless the subject has already been stated, as I said above. The paradox is a somewhat artificial construct in that people don't normally lie about everything.

There is also no paradox if you think harder about the physics of the situation since grains of millet are indifferent to sounds as they have no ears. pun intended

I think this is a tad unfair as TIS has the germ of the resolution, but has just not stated it clearly. However in view of the reception I got in the last thread about these ancient paradoxes, I am disinclined to help further.

+1 for understanding the point. and "I am lying" doesn't refer to anything. The conventional way to phrase this is to first classify everyone as either liars or truthers in everything they say. Then to offer the statement I am a liar.

I'm glad you are thinking about the maths because you asked about surfaces and one of the difference between a sphere and a donut (or teacup or torus) is that you can shrink any circle (or any other curve that forms a single closed loop) to a point on the surface of the sphere but you can't on a torus. This article should be accessible. http://www.maths.ed.ac.uk/~aar/surgery/zeeman.pdfhttp://www.maths.ed.ac.uk/~aar/surgery/zeeman.pdf

That you have had too much Christmas spirit already? Seriously I was unable to determine your beef from that ramblng post. As regards paradoxes, I have not encountered all of them. The ones I have encountered are usually formed by combining two (or more) statements inappropriately. Resolution is achiveved by untangling them and taking one statemen at a time.

Have you tried some algebra? I make it [math]4\int_1^2 {\sqrt {1 + \frac{{\left( {{x^2} - 4} \right)\left( {{x^2} + 4} \right)}}{x}} dx} [/math]

Perhaps you should read this book (it takes between one and two hours) What is Random? Edward Beltrami

Perhaps you should read Conway's new thread to find the answer to that particular question.

Good to hear that. So what happens if you shrink your black circle to a point so that each zero radius circle describes a single position on the sphere?

+1 Yes most likely sulphuric (car battery acid). However hydrochloric is put into "limescale removers" for toilets, by bleach manufacturers. It comes in black rather than blue bottles in the UK, but is easy to mistake. The acid and tha alkali remove different componeents of a blockage. As already described, caustic alkali (yes it is caustic so use rubber gloves) attacks the grease componenent which sticks things together. But cellulose fibres in paper and cloth and other fibres such as hair and roots can be mechanically trapped in the drain. That is where you should use the acid to dissolve them out, as the alkali will not touch that situation. Always remove as much as possible mechanically first, since that will reduce the need for chemical action and is cheaper. If you do use the acid, a bucket of washing soda (sodium carbonate) will flush it away safely afterwards. The build up your referred to creates a highly nutrient soup for plants so they try to break into the drain with their roots. This can be quite a nuisance with old drains. Finally you should keep monitoring the situation and clearing before it gets to bad in future.

Are you not aware of this formula? [math]L = \int_a^b {\sqrt {\left( {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right)} } dx[/math]

Well if this was a serious question and you want serious discussion how about also responding to my serious Physics comment? After all you have posted in quantum physics.

Yeah I agree what's the point ? Edit And what about halucinations, mirages , tromp l'oeil and other deceptions? Love the picture by the way. +1