studiot

Not at all, why do people always assume the worst of others? Being old fashioned I asked because I wanted to know the answer. I think equivalence relations has relevence to the subject of tautologies, but need go no further if you already understand this.

Do you understand the mathematical concept and application of equivalence relations?

Thank you swansont for sorting my spoiler. Robbitybob, It is not necessary to know if all the digits are unique in order to solve this one. You can work on this one from the outside in, like peeling an onion, First A, then E, then B then............................. Thank you commander for a neat problem, I assume you have established a set of simultaneous equations as a method since you mention the general case. +1 will head you off into the right direction (the green) I look forward to more useful and entertaining posts in the future.

Fiveworlds, since both have 5 digits A cannot be greater than 2 ie A = 0 or A = 1 or A = 2 A=0 leads to the trivial solution (00000) x 4 = (00000), there are no other solutions containing zero.

I'm not sure what your prof means but the resting potential of neurons is not 10mV. http://en.wikipedia.org/wiki/Resting_potential#Generation_of_the_resting_potential

No one is accusing you of 'borrowing' anything. You would not be the first to independently discover something already known elsewhere so all credit for that. However I am suprised that since you put that effort into this subject you seem uninterested and defensive when someone else points to what others have also found in the same subject.

Not quite. I'm suggesting that there is a wealth of previous human experience out there and offering some, because it's probably happened before more than once. Don't undervalue theoretical physics, the ability to think logically is valued in all sorts of fields, besides theoretical physics. One of my contemporaries (now a millionaire) was told when he applied to a firm of accounts to start accountancy training "We prefer entrants with a technical background such as yours, we find they make better high level professionals"

Just to clarify a point. You are presenting part of the real zeta function, which is upper schoolboy/entry university stuff. eg Mathematical Analysis : Quadling : Oxford University Press. The series [math]\sum {\frac{1}{{{r^k}}}} [/math] Where is is a positive integer not equal to zero. is convergent for all [math]k > 1[/math] and divergent for all [math]k \le 1[/math]

I did read the next line, as well as the document, and decided that, as written, it was at best ambiguous, but strictly referred to only to the second part of the sentence I quoted and not the first. That is it only referred to "all possible integers that come after 1" as demonstrated in your document. So strictly your statement "all possible powers" includes 1 and so I respectfully suggested you amend it. Note I have not argued for or against your thesis, like Ophiolite, only asked what you want us to disuss about it.

Students (like everyone else) suffer a wide range of personal issues, but one of their biggest problems is the tendency to think they are unique and that whatever it is has never happened to anyone else. This is an observation, not a criticism, and I think it is due to youthful lack of general experience of the world. I originally started the wrong course at the wrong university due to this and the experience certainly affected my career. My brother in law studied Electronic Engineering at Liverpool, where all technical subjects are highly mathematical and theoretical. He came out well and went straight into a large multinational where he thrived. Another friend studied Physics at Liverpool, but left after year 1 because he could not hack the mathematics. He later completed a London External in Physics at Portsmouth, where they are much more hands on than theoretical. He also left and joined a large multinational, where he also thrived. I would suggest that at least part of your question is best dealt with by PM.

1) You should be more careful with your wording since this is not quite true The infinite series [math]\frac{1}{{{2^1}}} + \frac{1}{{{3^1}}} + \frac{1}{{{4^1}}}....................................[/math] is divergent, although it conforms strictly to your above wording., which therefore needs to be changed to exclude exceptions. 2) Since there is speculation in this post I expect it will be moved to the speculations forum where it belongs. 3) Like Ophiolite I am confused as to what aspect of your paper you want to discuss. If you want to connect it to quantum theory then say so, but be aware that your late countryman and genius (Ramanujan) did this many years ago. 4) There is considerable evidence that all creatures except one use the same chemical reactions and colour receptors to detect colours so only those with a neurological abnormality will see red as green or blue.

Only if we return to the topic?

You said Which means they are reflexive or work both ways. Your gave a definition of a bachelor If I can find an unmarried man who is not a bachelor your definition is not a tautology since it is not reflexive. Since I can find an unmarried man (a widower) who is not a bachelor, the relationship is not reflexive and therefore not a tautology. Therefore your definition is not a definition, or it is defective, or your claim that a definition is a tautology is untrue. Suprisingly this refutation of an off-topic example, invokes time (on topic) since time is necessary for a man who was once a bachelor to become married and then widowed.

If you must construct examples please at least make sure that your use of the terms is correct. All (male in modern parlance) bachelors are men, true but Not all unmarried men are bachelors. So where is the tautology?

I find it refreshing for someone to acknowledge their limitations. +1

So your condition for a chain of definitions to become a circle is that it is unbounded. Sounds like the definition a straight line is an arc of a circle of radius infinity.

Well someone recognised it Actually Imatfaal is too modern for my liking. The phraseology I used is the 1908 translation to English of the third definition ever in human (scientifc) history. Edit I see more posts whilst I was composing the above. Eise you offer a chain of definitions required not a circle (or other closed loop). It is often the case that something rests on something else. So what? That still does not make it circular. Euclid is the prime model for later thought system construct. It has all the main ingredients. Axioms, definitions and what he called 'common notions' But the construct is not circular, it is not one chain but several, ie it is branching. It rests on certain unproven propositions, again so what? The whole point is that it is not self reflexive, despite literally millenia of attempts to prove it so.

Eise, perhaps you missed my post#155 in all the chaff? Please explain the circularity of this famous definition to me as I can't see it. "The Extremities of a Line are Points"

Yes, you have reached into the crux of the energy question. The work done on the balloon by the lift forces not only provides potential energy to the balloon, it also provides kinetic energy. Some of this kinetic energy is lost overcoming friction and slows the balloon. Some may also be used powering John Cutherber's generator. This needs tidying up, however, in the light of my post#196 since the lift force derives from the internal gas in the case of non rigid balloons.

In what way does that make the definition circular?

Please explain the circularity of this famous definition to me as I can't see it. "The Extremities of a Line are Points"

If anyone is actually interested in the mechanics of flight, the mechanics of balloon flight is special. For rigid bodies, including hollow ones, the upwards bouyant lift force is applied to the underside of the body by the external air. This applies whether the body is 'heavier-than-air' or 'lighter-than-air'. Of course, for heavier-than-air bodies an additional (non bouyant) source of lift is also required for flight. For inflatable (non rigid balloons) the upwards lift force is applied, not to the underside of the balloon or by the external air, but by the gas inside the balloon to the inside surface of the top of the balloon. This has implications for the centre of bouyancy and the stabilty of the body in flight. I can post a working through the mechanics of this if anyone is interested.

As many times as necessary until you actually offer a mechanism that conforms to the known laws of Physics Merely stating "It gets under the balloon thereby pushing the balloon up which increases the balloons potential energy." does not cut it. How and why? By what law of Physics does this happen? Why would any air have any reason to move at all, before the balloon has started rising? Or are you suggesting the Enron philosophy "buy before you have the money" That was a new philosophy in the last 15 years and look where it got them. In Physics for money read potential energy. Final question, "what lifts the heavier than air container off the ground in both the cases of rigid and flexible containers?" .

This does indeed address situations when Archimedes principle is obeyed, but not the question I asked which was, to paraphrase, What happens if and when it is not?

As a pilot(!) Talos should be aware of the ICAN (International Commission for Air Navigation) standard formulae for atmospheric pressure. From zero to 11km they offer a formula that is temperature compensated (derivations available) P = P0(1 - 0.0226z)5.26 Where height, z in km, is above sea level where P = P0 z is measured in km and other figures are in cgs units. Above 11km temperature is assumed constant at 216.5 K and the formula becomes P = 1.26P0e-0.158z If we can take the simpler 'schoolboy' formula for pressure [math]P = {P_0}{e^{ - \frac{{gz}}{k}}}[/math] Where k is a constant. A spherical balloon, radius r, Weight W, floats at neutral bouyancy at a height given by [math]{e^{\frac{{gz}}{k}}} = \frac{{4\pi {P_0}k}}{{gW}}\left( {r\cosh \frac{{gr}}{k} - \frac{k}{g}\sinh \frac{{gr}}{k}} \right)[/math]