DrRocket

Is that something like "somewhat pregnant" ? Good idea, so long as one's finger is reasonably dirty to begin with (likely for most folks and almost a certainty for high school students).

You apparently did a bit of research. But you are not questionposter and you did not question the use of wiki articles in providing an explanation. The mass of neutrinos is thought to be exceedingly small. It is only because of this mass, which as you note is required for neutrinos to oscillate between types, that they do not travel at c. Prior to the discover of the neutrino oscillations particle physics texts treated the neutrino as massless, traveling at c. I am not aware of any experiments that have actually measured a neutrino speed below c, and in fact there is one that purports to have evidence of superluminal travel by neutrinos (though I think most people expect that there is a mistake somewhere). The difference between particles and anti-particles lies in a change in sign of certain quantum numbers -- charge and chirality. Assuming that neutrinos are indeed massive, chirality is not the same as helicity. That is the "true physical difference" between a particle and an anti-particle.

Read my earlier post.

Because some people ask endless inane questions without first doing the simple research that ought be expected of them and wiki provides a ready reference that one can use without having to write a textbook. I like to use wiki articles after I have satisfied myself that the article is correct (not all of them are correct, by a long shot).

No, the theorem is very provable -- using the knowledge of the properties of E given in your problem statement. But you did not prove the theorem as it was stated. You proved a very special case involving some additional assumptions on E. You can reduce the problem, as you did, to the case a=0 by noting that the theorem is invariant under translation. Then the critical observation is that the function f is unbounded on the intersection of E with any neighborhood of 0 and from that you can then show that f cannot be uniformly continuous since one can find two point in any such neighborhood such that the difference in the values of f on those points is arbitrarily large.

I own the book. Beiser is not atteptiing to derive an equation for relativistic momentum. He is trying to derive the equation for relativistic mass, and he is using conservation of momentum to do it. Momentum is mass x velocity. I have not idea what you find "inappropriate", but Beiser's analysis is correct. You should be aware that Beiser's book, originally written in the early 1960's teaches the notion of relativistic mass, while many physicists use the term "mass" to mean rest mass. Mass in relativity is a term that is used in different contexts with different meanings, and you need to be sure that you understand what an particular author means when using that term. Rest mass is just what it sounds like -- the mass of a particle in the rest frame of the particle. Relativistic mass of a particle is gamma x rest m ass. Invariant mass is sometimes used to describe a system of particles and is rest mass in a frame of reference in which the net momentum of the system is zero. So invariant mass is what is measured in the laboratory when a macroscopic body is weighed and it includes thermal energy of the constituent molecules.

Your proof almost works. Uniform continuity doesn't have much to do with sequences. What you did was not really to use sequences but rather found a way to find two points arbitrarily close to 0 such that the value of your function at those points was arbitrarily far apart -- and that would show that the function is not uniformly continuous on an interval near 0 -- except for the problem noted below.. What causes the proof to not be completely valid is that the set E is somewhat arbitary and need not be an open interval with 0 as a boundary point. All that you are given is that a (which you can assume to be 0 with very little work) is an accumulation point but not an element in E. Unfortunately that means that your points x_n and y_n may not lie in E and hence may not lie in the domain of f. You have the right idea, but you need a bit more finesse.

I asssume that you are working over the set of real numbers, and that n is a natural number. The sequence that you propose has no limit. In fact the absolute value grows without bound, and the sign oscillates depending on whether n is even or odd.

If any of your theory makes sense to you, then I suggest that you seek professional help.

You did not invent a "new number system". "Number systems" refer to things like the integers, rational numbers, complex numbers, etc. What you are talking about is a representation of a number system -- and even there the various representations in terms of bases are quite well known. There are all sorts of representations of number systems, and lots of different symbols have been used. Roman numerals are one representation. But the decimal representation used today, which was invented by the Arabs has distinct advantages for computation, as does the modification to binary representations that is used by computers at the most fundamental level and the hexadecimal system (base 16) used at higher computational levels. However, the choice of representation or base, has absolutely nothing to do with the underlying mathematics, and most mathematical systems are blind to any particular representation. There are a few exceptions involving rather obscure theorems in number theory which deal with the representations themselves. In short, unless you have some computational scheme in mind there is no clear value to any new representation.

Do what interests you. Biomedical engineering is a solid curriculum, and a good undergraduate major for those interested in medical school. There are in fact dual PhD/MD programs with the PhD in biomedical engineering. Engineers need not be employed only in the specific field in which they received their degree, and biomedical engineering, though somewhat specialized at the academic level, is no exception. Within biomedical engineering there are specialties that involve primarily materials, electronics, robotics, mechanical systems, etc. and those skills are readily transferable. Pharmacy is also a viable career path. There is always local demand for pharmacists. But pharmacy is very different from engineering. It involves different academic disciplines and very different working conditions, particularly with respect to dealing with the public. The thing to do is pursue an area that interests you and in which you have aptitude. The situation on a global scale is irrelevant. Employment is based on the specific conditions is a given industry and in fact in a given enterprise.

That is an absurd statement, that simply demonstrates your lack of acquaintence with science and scientists. There are world-class scientists who are devout. There are world-class scientists who are atheists. There are idiots who are atheists. There are idiots who are devout.

The power, and acceptance, of a physical theory lie in the ability of the theory to make valid predictions. We don't have direct evidence of the existence of atoms (no has no one ever will see an atom). But quantum mechanics and the atomic hypothesis have produced a huge number of predictions of phenomena that have been borne out by experiment -- the entire field of chemistry for instance. String theory either will or will not make predictions of new physical phenomena that are validated by experiment or careful observation of nature. That is no different from any other new theory. Until it does, it is not really a physical theory, but only an avenue of research. Philosophers ask questions that are not only experimentally intractable, but in fact questions that are unanswerable in principle. Their hallmark is asking questions and providing no answers,but rather an exploration of the ramifications of the question itself. That is the crux of the difference between philosophy and science. If one placed all of the philosophers who have ever lived end to end, they would not reach........................................................... a conclusion. You have just equated philosophy with pseudoscience. I do not agree with that characterization. It does a disservice to both philosophy and science.

You have not defined "length of connection". t is not obvious what a natural definition would be for your problem. In general the problem of finding minimal lenght Euler path is called the "traveling salesman problem". It is computationally difficult -- as I recall it is NP-complete. Before you spend a lot of time on your problem you should take the time to learn some graph theory. Graph Theory by Bondy and Murty is one of the classic textbooks. A somewhat unusual,but very nice, treatment oriented towards electrical engineers is Graph Theory with engineering applications by David Johnson and Johnny Johnson.

Since you are talking pure fantasy no speed is too fast. The only way to kill your riders is with a pen.

Your facts are wrong. c is most certainly used correctly -- by those who understand it. c is NOT a number. c is a speed. It has the dimension of length/time. That dimension is important and distinguishes c from dimensionless constants (pure numbers) like the fine structure constant. I agree that this is very confusing to you. I suggest reading a good introductory physics text. Fundamentals of Physics by Walker, Halliday and Resnick should do the trick.

The illustration in your link has nothing to do with physics, but rather relates to neuroscience. Your subsequent questions relate to nothing. If you want a light look here.

You need to read a book on orbital mechanics. Orbital Mechanics by Prussing, Conway and Prussing is a good one.

0.5 c is precisely c/2 just as 0.5 x 10 mph = 5 mph The numerical value depends on the particular system of units that one chooses. It has nothing to do with time.

It is quuite obvious that you have never performed, or indeed been seriously involved in, any significant scientific research. This is just plain rubbish and undeserving of any serious rebuttal. To top it off it is difficult to read because you seem to be unacqainted with the device known as a "spell checker".

There are (exotic) white hole solutions to the vfield equations of general relativity. However, there is not the slightest bit of evide4nce that white holes actually exist. Conservation of mass/energy does not in any way imply that black holes must "go somewhere" and in fact that phrase is meaningless. Imagination is good. But in science imagination must be tempered with knowledge of what is already known and the limitations that imposes on potentially valid theories, else what you have is not imagination but just fantasy. Physicists tend to be rather loose when discussing "conservation of information", but a rigorous treatrment seems to be equivalent to stating that the quantum mechanical state function evolves according to a one-parameter family of unitary transformations. A violation of that principle would be a very big deal. The black hole information problem is that under a scenario proposed by Hawking unitarity would appear to be violated. However a way around that has been proposed, and Hawking has conceded a bet that he make with Preskill. Butr, not everyone accepts Hawking's conclusion, including Kip Thorne who is also a party to the bet. Moreover, the "resolution" involves string theory and the AdS/CFTcorrespondence, which itself is an unproved conjecture of Maldecena that dates from about 1997. There is a very one-sided, biased and misleading account of this problem in Leonard Susskind's book The Black Hole War. You may find it enlightening, but be forewarned that you should read it with a lot more than just a grain of salt.

You have not clearly defined your problem, but it appears to be a variation on the Euler circuit in graph theory (Google "Euler circuit" for more information). The minor wrinkle is that all of your vertices (here lines), except 2, are of degree two while the two exceptions are of degree one. This is a well-known variation. The desired circuit always exists under these conditions. Since you have provided no cost function for optimization, there is no meaning to "best". You might want to read a book on graph theory.

You have not clearly defined your problem, but it appears to be a variation on the Euler circuit in graph theory (Google "Euler circuit" for more information). The minor wrinkle is that all of your vertices (here lines), except 2, are of degree two while the two exceptions are of degree one. This is a well-known variation. The desired circuit always exists under these conditions. Since you have provided no cost function for optimization, there is no meaning to "best". You might want to read a book on graph theory.

Right. The important thing is that one can define exponents, using the exponential function, in such a way that it agrees with the elementary definition for integer powers. Thus the more sophisticated idea is merely an extension of what one learns in grade school, but it is a very important extension with far-reaching ramifications. The fact that the exponential function, as a function of a complex variable, is [math] 2 \pi i[/math] periodic only adds to the depth of the concepts that are involved.

Check your calendar. It would be very difficult to have a birthday in 2012, at least for a few more days.