studiot

Well I make it that there are 12 possible outcomes for n = 4. I also make it that 6 of these result in an equal number of heads and tails. But then I don't rely on dodgy information from the net. Have you considered Strange's tree diagram?

This is why I asked you what your definition of probability is. I don't understand this. Let me work it forwards a bit more fore you Let P(n) be the probability that n trials will contain an equal number (n/2) of heads and tails. p(0) = 1 since there are 0 heads and zero tails P(1) = 0 since there is either one head or one tail P(2) = 0.5 since there are 4 possibilities, two of which have an equal number of heads and tails P(3) = 0 since there are no possibilities with an equal number of heads and tails over to you

So how are you getting on with my sequence?

2) I apologise, I didn't finish the second statement. The comment is linked to the first, however. 1) I asked for a mathematical proof of your assertion. Something anyone (especially a student) is entitled to ask of a maths teacher. self explanator is not good enough. I would expect something along the lines given the probability of n coin tosses P(n) = ?, where The probability of P(n+1) is (by induction) some smaller value You would need to start with n = 0 - equal probability P(0) = 1 p(1) = zero, yes 0. p(2) = ? And show that as n increases the mathematical function you have proposed does indeed tend to 0.5

Can't see how you can do Bragg without a gamma detector. How do you think a gamma detector works, if not by energy? At the risk of another silly argument over the word interact I don't know of any detector that doesn't interact with the gamma energy. If someone else does know of one please tell us, I would pleased to learn about it.

If you are teaching probability, how would you show this assertion to be true? Remember that every second toss represents an odd number of tosses so for every second toss as Ghideon says. As a matter of interest what definition of 'probability' are you teaching your students since you mention the correspondence to reality?

Why is this not direct? What do you mean by direct? Frequency analysis of gamma rays is called gamma ray spectroscopy https://en.wikipedia.org/wiki/Gamma_spectroscopy Note gamma and X ray frequencie ranges sunstantially overlap. The difference is the nature of source.

Since you ask this as a question. No. If you insist on only discussing interference between two lasers as an interaction then read this https://www.nature.com/articles/198255a0

I'm not sure why you are concentrating on African animals. One of the interesting things that has come out of the many wildlife televesion programmes is the behaviour of african animals around a water hole in times of shortage. The hostility between species that normally fight on contact is suspended. Many more good points showing just how complicated and varied life forms are, leading to complicated and varied activity. Howsois, I say again there is no simple answer to your question, nor did any transition happen suddenly. Further we are still a long way from having enough evidence to fit the jigsaw together.

"Mix" "Effect" The OP says interact, not interfere. How is an effect not an interaction? If you want interference, what about beat frequencies between nearby radio stations? The OP asks about light, then laser beams. Perhaps the first need is to explain the diference between light in general and laser beams in particular.

Who says it can't ? How do you think colour projectors work? Mix red light and green light, what colour light do you get and is that not an interaction?

Yeah. +1

Yes but did I not say that sometimes there are opposing factors to balance? And if you are to defend yourself when caught by a more powerful predator, having a weapon is better than no weapon. And weapons can only be used ( I would say wielded here) by standing on two legs. Please try to look at all my points, just as I have tried to look at all yours. Thus you did not answer my question how many lions or wolves are there in the jungle? Another opposing factor is Do these animals hunt by sight or sense of smell? Whatever, when dealing with these animals, humans have learned to cooperate. That is another factor.

Yes. To go with this I like the spectroscopist's version of the HUP to explain line broadening. Using the Δt / ΔE version The transition take a finite time which leads to time uncertainty. So there must be an assiciated corresponding energy uncertainty.

1 Of course they used branches. It is likely that pointed sticks, sharpened and hardened in a fire, wire* were used before stone tools/weapons were made. Edit* 2 Yes that's true. Why is this relevent to the question of why they walk upright? 3 I have already pointed out one advantage. They can see further from the increased height. How many lions or wolves are there in the jungle?

I do not have a view because I think that 1) Your hypothesis is about a time after any spread to the savannah and much of your argument is about the time after hominids had mastered tools. I think they may well have been walking upright before these times. Clearly one cannot hold and use tools very well when one is on all fours. 2) As I have pointed out we do not have enough information as to what hominid we are descended from and cannot therefore say whether that ancestor was upright or not. 3) I think the process of was a regenerative gradual process. Some ancestor(s) found advantage in standing upright sometimes. Evolution suggests that they will prosper. Further evolution improved their uprightness and perhaps altered their physiology a little at a time to adapt. So they stayed upright for longer periods. and so on. It has been suggested that this happened to the dinosaurs, ending in the tyrannosaurus. 4) Of course there is the question of when (and why) did humanoids loose their tails? Was this part of the standing upright process? You might also like to contrast my suggestion about standing up to see better with other creatures. Some developed long and mobile necks. Some developed better eyes.

time is implicit in B-T because this is a construction. So you start with one ball Then you have two and so on. So there is a point when you have only one ball Then another when you more, developed from the original. So the original no longer exists.

You seem determined that your hypothesis is the only possible chain of events. And whilst there are some logical parts to your hypotheses, There are many gaps in our knowledge and there are many considerations and factors, some of which oppose each other. Balancing such opposing factors is most often the way of the world. So a good test of ideas is to answer why you considered and rejected altenative hypotheses. Another useful avenue of enquiry is to look at animals that sometimes stand up, but also work on all fours. Bears, for instance stand up to fight and appear more imposing. They also stand up to reach up trees to get at desired objects. Soem also stand in water an fish with their front paws. Now consider is their visual apparatus set up to see best (look straight ahead) when they are standing up or on all fours? How does this compare with say dogs or pigs? Dogs are interesting because they hold bones with their front paws ti gnaw at them. You mentioned the use of tools. To make tools man needs to sit or squat. Even today workers are often hunched over their work say a lathe. You don't chip flint arrowheads standing up. But you use weapons standing up, the spear or bow and arrow. As regards to food, It may not be a pleasant thought to our modern ways but it is logical that early man ate a lot of worms, grubs, insects and the like. In a way like badgers. Evidence for this can be obtained by studying the evolution of our teeth, which is determined by the food we eat. Talking of swamps, marshes and other very soft ground, There is no evidence that early man did not occupy these. This could be because the timescale is not long enough for genuine fossils to have formed from those lands, as we have for say dinosaurs. But bodies will have decayed completely. Our evidence is based on remains rather than fossils. And the sort of semi arid regions or cooler caves are the best bet for preservation of remains. It is also true that neolithic Man lived and built substantial communities in swampland. The lake villages of Somerset are a fine example. Today tourists go for excursions in replica dugout canoes. But remember the time period of your transition / migration is really before the stone age proper. It's the time period that led to the stone age.

I really like that setting of Noether in a Physics context. +1 As regards to a relation have you considered Gauss flux theorem and its relation to the fundamental theorem of calculus? Here Gauss flux = your 'current'.

When you start working in multidimensional frames (matrix, tensor etc) many quantities which are a plain numerical constant in simpler constructs become themselves matrix or tensor objects. This is true for instance in GR.

In you obstinancy to turn a deaf ear to others who are offering the simplest, yet perfectly sound, scientific/mathematical explanations you have missed this very important point, which I made earlier. An n dimensional sphere is the set of all the accumulation (or limit) points of a closed ball in an (n+1) dimensional manifold. An important property of surfaces is that they divide the manifold into two parts. For a sphere these are the inside and the outside.

Note, strictly a sphere is not a solid object. A sphere does not have a surface, it is a surface. The correct mathematical term of the solid object is a ball, but there is much more to it than that. There is also mathematical justification for this, But it is not what you think. Being a true surface a sphere has two sides. An inside and an outside, with inward and outward normals.

Thank you for your response. Please do not answer in that fashion as the quote function on this site cannot handle it properly. There have been heated arguments here as a result of this with members being accused of deliberately falsifying or misreporting the words of others. So please separate your words from those of others as in the examples in this thread. You say this Which directly contradicts your failed answer to this Do you know what the difference in repulsion between a proton and another proton and the repulsion between a proton and a neutron actually is ? Can you draw one of your diagrams to explain how either or peferably both these forces are generated? And yet when I asked the selfsame question with less detail before you said yet now you say you don't understand or are working on it. I respectfully suggest you find out what has already been discovered over the past couple of centuries of human investigation into electrical phenomena, instead of guessing.

Enlighten us what you mean since I don't think you are correct, in strictly mathematical terms. But then neither are those who talk about the surface of a sphere either.

https://en.wikipedia.org/wiki/Incandescent_light_bulb Note the halogen scavengers against filament evaporation.