DrRocket

I know of one former mathematics professor who is now a lawyer. I have encountered some very stupid lawyers. I taught college algebra to a local anchor man. He was one of the better students in the class. I have encountered reporters who could not count their fingers. Generalizations are usually wrong. A lot of people cannot handle arithmetic. A lot can.

In classical electrodynamics, there is one single electromagnetic field that exists throughout all of space. It can be viewed as a sum (superposition) of the fields associated with each of the charged particles, including their motion, that exist. When electrons move in a copper wire, the position and velocity of those charged electrons changes, and with that change comes a change in the electromagnetic field associated with them. That local change propagates according to Maxwell's equations, and that is what is meant by "the emission of a radio wave".

Friction losses from slippage are relatively easy to calculate and come out of the model directly, if indeed here is slippage given knowledge of the coefficient of friction in a Coulomb friction scenario. Rolling rresistance is a separate ingredient that may or may not be a consideration. One can formulate the problem in a sensible way with the assumption that rolling resistance is zero. This is a reasonable idealization in many circumstances. In others rolling resistance is significant. The loss mechanisms associated with rolling resistance can be very complex, as can the mechanisms for sliding friction -- which are not well understood at the most fundamental level. You can make this problem as difficult as you want by boring down into all possible loss scenarios, but the major point that some energy is embodied in rigid body rotation will be quickly lost in the fog and the important physics will be obscured.

When in doubt it usually helps to fall back to the basic definitions, which in the case of a discrete probability space are: [math] \overline X = \displaystyle \dfrac {1}{N} \sum_{n=1}^N x_n[/math] [math] \sigma_x^2 = \displaystyle \dfrac {1}{N} \sum_{n=1}^N (\overline X - x_n)^2 [/math] From this you should be able to easily show that [math] \overline {aX}= a \overline X[/math], [math]\overline {X + b} = {\overline X} + b[/math], [math] \sigma_{ax}^2 = a^2 \sigma_x^2 [/math] and [math]\sigma_{x+b}^2 = \sigma_x^2[/math]

Now you are getting it. Even in special relativity what one observer views as the time coordinate of spacetime will differ from that of an observer in relative motion, so "coordinate time" varies with the observer -- different observers coordinatize spacetime differently. But the coordinate time of an observer will be the proper time of that observer (or clock) -- you are just talking about a spacetime interval with a constant sparial coordinate in the frame of the observer. The difference between SR and GR is that in SR there is a single global set of coordinates. In GR, with a curved spacetime, coordinates are only local. The key to all of this is that proper time is associated with a timelike curve (aka world line) in spacetime and is just the length of that curve, divided by c, measured in the spacetime metric. A consequence of this is that the 4-velocity of anything, not just light, is c (velocity is distance divided by time, and the distance along a curve is just the time divided by c). Time in relativity is a different animal from time in Newtonian mechanics. Time and space are not separate, but rather are intertwined in spacetime. They are only local concepts in general relativity, and are observer dependent in both SR and GR. There is no absolute time and no absolute space. This thread on spacetime in special relativity might be helpful. http://www.scienceforums.net/topic/54988-spacetime-in-special-relativity/

You argue about a theory, general relativity, of which you have not the slightest understanding. Hence your posts are non sequiturs. Before you can sensibly either support or criticize a theory you must first understand it. Then you can decide whether or not to accept it. But when you attempt to discuss something about which you are clueless, such as "curvature" (which is in fact a rather technical and subtle concept) you wind up just babbling. You cannot hope to eschew mathematics and then intelligently discuss an inherently mathematical subject. There is a reason that general relativity and quantum field theory are not taught in kindergarten.

Not sure what kind of books you are interested in. Two of the best probability books are Loeve's Probability Theory and Feller's An Introduction to Probability Theory and its Applications. Probability and Measure by Billingsley is also very good. For statistics I like van der Waerden's Mathematical Statistics and Cramer's Mathematical Methods of Statistics. However your observation that if the government pays out 50% on a lottery then the government also keeps 50% should not require a book, but should in fact be obvious to the casual observer. The reality that this is not obvious to some people is merely a comment on what is meant by "average intelligence".

There is one other effect to consider. That involves friction, the moment of inertia of the sphere about its center, and the distinction between rolling and moving with slippage. The sphere can slide down the hill, roll without slipping or both slip and slide. The normal force plays a role in determing this, and the final velocity is dependent on the specific scenario since some of the initial potential energy will be realized as translational speed and some as rotational speed.

no This makes no sense. Geodesics in general relativity are determined by the Lorentzian metric. It is a metric on spacetime, not just space, and in units for which c=1, the length os a timelike geodesic is in fact the proper time of that segment of a world line. You measure "distance" withy a clock. SR is GR in flat Minkowski spacetime. There, coordinate time and proper time are the same thing. This is because the spacetime manifold and its tangent space at any point are isometric.

This just plain wrong. Obviously wrong. Patently ridiculous. You need to learn some mathematics. Probability theory is just a small part of the obvious lack.

Clocks measure the proper time of their world line. That is all that any clock measures. Coordinate time is a fiction. It is time as modeled in some choice of a coordinate patch, and would correspond to the time registered by a real clock only in the case of flat spacetime. It is an approximation that is reasonably accurate in approximately flat circumstances; i.e. away from large gravitational fields over moderate distances. Only coordinate time makes sense as a comparison of separated clocks. Special relativity, being a theory that excludes gravity, deals with a situation in which coordinate time and proper time coincide, but in the real world the time of special relativity is coordinate time -- SR is the local approximation to GR.

No, what you are observing is both correct and very simple. But you are talking to lottery players, and that is a group that accepts a very poor bet from an economic perspective, some for the entertainment value of a minor expense, but many from a position of abject ignorance. You will find that almost any logical argument goes over the heads of a great many people. Probability is often not intuitive and goes over the heads of many more.

In a rigorous treatment of the theory of probability, the probability of elementary events is given, not calculated. The starting point for probability theory is a Set, a sigma algebra of subsets and a positive measure of total mass 1 on that sigma algebra. Probabilities based on relative frequency of occurrence are really estimates based on the law of large numbers. Probabilities based on combinatorics are really definitions of a probability measure, and an assumption that some given class of events are of equal probability. This sort of treatment is usually found in very elementary and non-rigorous treatments of probability theory that attempt to give an overview of the subject while avoiding the measure theory on which rigorous probability has been based since the work of Kolmogorov. The heuristic "definition" in terms of relative frequencies is conceptually useful but is not a practical way to determine actual probabilities. It is not strictly speaking correct, hence my qualification of it as heuristic, but rather is roughly a converse to the law of large numbers, modulo some loose language as to the sense in which things are meant to converge (see "convergence in probability" or "convergence in measure".) The only way to do this correctly is to use the general theory of measure and integration. For that see the book of Loeve. Probability is the most misused and incorrectly presented branch of mathematics. A good deal of what one finds in engineering, physics and introductory mathematics texts is not strictly correct.

You are precisely half right. My post is correct. You have no idea what you are talking about. Read my post again. Better yet read Loeve's Probability Theory.

This is not only wrong, but absurd. This formula would have the probability of an event decrease to 0 as the numbeer of outcomes increased without bound. A heurestic treatment, loosely based on the law of large numbers often presents the probability of an event A as [math] P(A) = \lim_{N \to \infty} \dfrac { nummber \ of \ A \ outcomes}{N}[/math] where [math]N[/math] is the number of trials. wrong again. But closer. By definition, [math]P(A|B) =\dfrac {P(AB)}{P(B))}[/math] Which, if [math]A[/math] and [math]B[/math] are independent reduces to [math]P(A|B)=P(A)[/math] Since the probability of the outcomes of heads or btails in separate tosses of a fair coin are independent, conditional probabilities offer no nrw insight (see above). In short this is ridiculous and [math]P(Head) \times P(Head|Head) \times P(Head|Head,Head) = P(Head) \times (\frac{P(Head \cap Head)}{P(Head)}) \times (\frac{P(Head \cap Head \cap Head)}{P(Head) \times P(Head)})= P(Head)^3[/math] There is no such thing as "probability in the complex case". Probability measures are real valued.

This would be more enlightening if either: 1) You were to provide a reference to the literature or 2) It made sense.

There is "nothing remaining to be discoveresd", nor has there ever been. Special relativity is nothing more and nothing less than the localization of general relativity, general relativity on the tangent space to the spacetime manifold. This results in the reduction of general relativity to special relativity in the special case in which spacetime is flat Euclidean space; i.e. in the absence of gravity. There is nothing whatever mysterious about this. It does require understanding of the basic theory of manifolds, so cannot be understood without some facility with the requisite mathematics, but then neither can general relativity be understood without that background.

I'm not so sure. In many classes detailed notes in pdf form are provided by faculty now, but the textbook problem remains. I personally find the electronic format inconvenient for scientific texts which, unlike novels are often not read linearly. I flip back and forth in texts at a rate and frequency that is very difficult with an electronic copy. I like having the books permanently on my shelf, rather than renting them for a limited time. That said, many universities have entered into arrangements with electronic media companies for rentals to students of electronic texts. I also admit to having a large set of electronic files of papers, notes and some public-domain books. I am quite sure this trend will only accelerate. But I am not convinced that it will solve the problem completely. It may spell the end of the most specialized and lowest volume science books.

I think that there are several problems. One is the captive customer base that you note. Another relates to the use of "customized" texts that publishers can produce through the use of computer-based printing. This reduces the number of "standard" books being sold and also severely curtails the used book market. The market size is critical. Technical books simply do not have the volume of novels, and any factor that reduces the market size has major ramifications in cost. There is also the issue of software offerings that come along with some texts that support the instructor and are reflected in the price of the book itself. But all of that aside, the cost of technical books is out of control. Books that sold for $10-$15 years ago now sell for ten times that amount -- basically the same books, by the same authors. This seems to vary substantially with publisher.

This appears to be a homework problem.

It has nothing to do with the number of particles. Classical statistical mechanics applies to a very large number of interacting particles, the more the better. Problems arise when there are too few particles, not too many. We can solve the two-body problem. Statistically we can solve the very many body problem. The general three body problem is intractable. There is no clear demarcation. Three is a problem. Avogadro's number is tractable. This has little to do with quantum mechanics, though there is also the subject of quantum statistical mechanics to consider. The line between what is recognized as "quantum behavior" and macroscopic "classical behavior" is also fuzzy, and not well understood. This comes under the heading of better understanding of "collapse of the wave function" or "quantum decoherence", or maybe something else. The problem is not when classical mechanics stops working, but how, in detail, classical behavior arises from quantum mechanics. A great deal is just not known. At this stage of the game, the role of quantum theory is to try to establish the rules of play at the most fundamental level. It has been fairly successful at doing that, acknowledging that much more research remains to be done before there is a fully mathematically consistent "theory of everything" or even a fully well-defined and rigorous quantum field theory. But even if you accept the success of quantum electrodynamics, the extension of those fundamental rules to everyday phenomena remains beyond our grasp. In theory all of chemistry is just a corollary of QED, but in practice that is not the case. Similarly in theory the forces that bind nucleons, the "residual strong force" are a consequence of quantum chromodynamics, but no one has been able to derive those forces from the underlying field theory. There is a hell of a lot yet to be learned.

Classical statistical mechanics not stochastic. I provides deterministic predictions of the aggregate properties of large numbers of particles. There is no inherent randomness to classical statistical mechanics. Neither is there any attempt to predict the behavior of any single particle. Statiostics is used simply because of our inability to either determine the initial conditions of the myriad of particles or to solve in closed form the many-body problem. It is a tool used to model complexity, not randomness. Quantum mechanics is fundamentally different in that the impossibility of a deterministic prediction is accepted, and only probabilities are predicted. In fact what quantum mechanics describes is the evolution of probability measures. Operator theory is secondary and incidental to this fundamental aspect of the theory. Quantum mechanics is fundamentally stochastic.

There is nothing magic about dimension 2. The general theory applies in n dimensions in a straightforward manner -- the real part of the eigenvalues of the "A" matrix determines stability. Any good book on ordinary differential equations should present this. Roger Brockett's Finite Dimensional Linear Systems is an excellent text on linear synamical systems. The non-linear case is much more difficult. for that see Lyapunov Theory.

"Collapse of gravity" within the current understanding of gravity makes no sense. Black holes are about as "collapsed" as one could even imagine. The closest thing to this is modified Newtonian dynamics (MOND) theories used as alternatives to "dark matter" and Beckenstein's corresponding version of general relativity. This is VERY loosely reminiscent of an idea called the "holographic" principle that has been considered and continues to be considered. However, the holographic principle is much more subtle, though about equally speculative. There are all sorts of very speculative popularizations in which you can read about this stuff. Or just "Google" some of the buzz words.

Which is precisely what I described, and what is given by the expression that I provided.