DrRocket

It is a definition, which is consistent with exponentiation by integers and attendant relations.

You definitely need to read the book.

Of course we can. I just traveled about 30 seconds in the future direction while typing this. Yeah, I took a look at it. It simply showed me that, in addition to not understanding general relativity, you don't understand your own references. For instance, while you assume, using a wild trail of illogic, that the universe must be infinite, it is listed as an open question by your vaunted ISASS. The only point in even addressing your posts is to mitigate the damage that might occur to people who are actually trying to learn a bit about relativity.

One's personal approach to religion, or atheism, does not require the agreement of anyone else. You are trying to apply a scientific perspective to religion. When properly limited there is no conflict between science and religion. Neither is there an overlap. There are excellent scientists and mathematicians who are devout. There are excellent scientists and mathematicians who are atheists. There are atheists who are so dogmatic that they have turned atheism into a religion. Religion has no business making statements of a scientific nature. It is abundantly clear that science has developed impressive theories that very accurately explain many natural phenomena and that new and better theories continue to emerge. Science is utterly reliant on the predictability of nature, and therefore the existence of a god that regularly interferes with natural processes is an anathema. But such a vision of God is more akin to superstition than to a reasoned approach to religion. The existence of God is not a scientific question, once one has dispensed with pure superstition. Science can neither confirm nor refute that existence. "Existence" here is taken in a metaphysical sense and not in the sense of a realist. One of the unexplained aspects of nature, likely never to be explained, is that it appears to be orderly and therefore understandable. Natural phenomena appear to be described by mathematically-formulated laws of great aesthetic beauty. Nobody has a clue why. One possible source of the order in the universe is God. This is not a scientific statement. It is neither provable nor refutable. It has no bearing whatever on science or the progress of scientific research. It is important to separate science from religion, for the benefit of both. There are some recent pseudo-scientific proposals, the string-theory "landscape' for instance that appear to be inspired by atheism cum religion. The proposals are impossible to test and are more akin to religion than to science. Basically the logic is that we have a plethora of competing string theories (forget about the fact that nobody can actually define them) and since there is no means to select any one of them over the others (no, none are clearly identified with what is actually observed) and since an arbitrary choice based on what we do see (if we isolate the choice) might be attributed by some to God, it must therefore follow that ALL such universes actually exist. This is particularly bad science. It isn't very good religion either. Rather it is better to separate religion from science and let each attend to its own "magisteria". http://www.stephenjaygould.org/library/gould_noma.html

It sounds to me as though he means a flat 3-torus. That is one posibility, though not technically globally isotropic.

You need to read a real book on special and general relativity. Rindler's Essential Relativity: Special Genertal and Cosmological would probably be suitable. It is not particularly demanding in terms of mathematics. There are too many misconceptions above to address each individually. Einstein's theory of relativity is in large part a theory regarding the very nature of space and time. The effects of "time dilation"are real, and are not the result of any mechanistic change in anything, atoms included. Time really is observer-dependent, and it is a local, not global, concept. The same statement applies to space. This is not intuitive. It is backed by a mountain of empirical evidence.

You seem to be under the gross misconception that it is the job of a university professor to teach you. It is not. It is the job of the professor to facilitate your learning. It is your job to actually learn. To learn at a high level, it is sometimes useful to have the guidance of an expert. A university is not a high school. Students are there to learn, not to be taught.

Corresponding to any eigenvalue there is an eigenspace, a subspace of eigenvectors, so your "sum" is not well-defined. It is not even well defined if your space comes equipped with a norm and you demand each summand be of unit length. At the very least, if v is an eigenvector, so is -v.

Go back to the proof using the ratio test. You will find convergence when x^2 is less than the limit indicated in the OP which, since the limit is infinite means when |x| is arbitrary. You can also look at the associated series in terms of complex numbers (note that the series is sin (x)). [math] sin(x) = Im ( \sum_{n=0}^ \infty \dfrac {(ix)^n}{n!})[/math] This all boils down to the fact that the power series for the exponential function has an infinitevradius of convergence. [math]exp(z) = \sum_{n=0}^ \infty \dfrac {z^n}{n!}[/math]

[math] a_n= \frac{(-1)^n}{(2n+1)!} [/math] and [math]c=0[/math]

If I recall correctly (no guarantee) there are some old lecture notes from The University of Chicago that present classical theermodynamics from a geometric point of view.

This is a flat contradiction of general relativity unless spacetime were flat, in which case special relativity would apply. In the presence of curvature, hence in any universe that includes matter (most people would agree that our universe qualifies in this regard), time, like space, is local. There is no clear meaning to a comparison of "time here" with "time there". What one can meaningfully compare is the time recorded on two clocks between spacetime intersections of their world lines -- this is the essence of the "twin paradox". That time is, modulo division by c, just the length of the world line of the clock in the Lorentzian metric of spacetime, known as "proper time". You continue to assert expertise in the ontology of spacetime while simultaneously making outrageous and false statements regarding the content of general relativity. How can you make such positive assertions regarding the reality of constructs of general relativity when you have no idea what the theory actually says ?

He seems to have asked this question, and received answers, in several places. http://www.thescienc...5a61c9207f40b1f Slow learner ?

[math] (x_k,x_{k+1})[/math] is an interval on [math]\mathbb R[/math] The normaliization is a map [math](x_k,x_{k+1}) \rightarrow (0,1) [/math] given by [math]x \rightarrow t[/math] where [math] t= \frac{x-x_k}{x_{k+1}-x_k}[/math] "t's are points on the real line, "p's" are given values at the point "m's" are tangents, according to the notation adopted in the article. I don't quite see why this is not clear from the article.

state variables or "extensive and intensive properties" -- see for instance Chemical Thermodynamics by Kirkwood and Oppenheim (any book on classical thermodynamics will do). Isotherms, adiabats, etc. are curves or surfaces in the state space.

Right. That is exactly the way they are treated in classical rhermodynamics.

Those curved trajectories are a bit misleading. An object in freefall, such as an orbit, has a trajectory that is a geodesicc in spacetime. But, for instance, an elliptical orbit is obviously not a geodesic in space. Neither is the familiar parabolic trajectory of an artillery shell. A geodesic in spacetime is quite different from a geodesic in space and the curvature in question is spacetime curvature, not just curvature of "space". In fact there is no clear separation of space and time, except locally. Spacetime is not space and time, but an intertwined amalgamation of both. In this regard special relativity can be misleading. The "referent" is the spacetime trajectory, the world line. The world line is also the source of time. The length of the world line (divided by c), a timelike path, is the tme experienced by the body associated with that world line. Time is not a universal, but rather is associated with each world line and two bodies with different world lines may experience very different times -- this is the essence of the "twin paradox". Owl is indeed saying something very different. He and Einstein are on completely different pages.

If [math] a_n= \frac{(\lambda^nn!)^2}{(2n+1)!} \nonumber\ [/math] Then [math]\frac{a_{n+1}}{a_n}=\frac{(\lambda^{n+1}(n+1)!)^2}{(2n+3)!}\cdot \frac{(2n+1)!}{(\lambda^nn!)^2}[/math] [math]= (\frac{\lambda^{n+1}}{\lambda^n}\frac{(n+1)!}{n!})^2\frac{(2n+1)!}{(2n+3)!} [/math] [math]= (\frac{\lambda}{1}\frac{(n+1)}{1})^2\frac{(1)}{(2n+3)(2n+2)} [/math] [math]= \frac{\lambda^2 (n^2 +2n +1)}{4n^2 +10n +6)} [/math] [math] = \frac {\lambda ^2}{4} \frac {n^2 +2n +1} {n^2 + \frac{5n}{2} + \frac {3}{4}}[/math] [math] \rightarrow \frac {\lambda ^2}{4}[/math] as [math] n \rightarrow \infty[/math]

http://www.isass.org/ Apparently Google has better things to do with its resources also.

This appears to be pure BS from the Peoples Republic of Boulder. 1. I note that there have been zero contributions in any of the listed categories. Maybe you can't fool the people this time. 2. There is no description anywhere in evidence as to yhe nature of the so-called hidden technology for free energy. 3. The web site for the film is a single page giving not much more than the title. 4.. It has the look and feel of a typical absurd conspiracy theory. 5. Anyone who seriously believes that there is some hidden technology that could produce free, oir even just cheap and abundant, energy that is being suppressed by the entire population of academic and industrial engineers and physicists is in serious need of professional help. 6. Tesla was brilliant. He was also a bit of a nut and made outlandish and unsupportable claims. http://en.wikipedia.org/wiki/Nikola_Tesla#cite_note-54 http://books.google.com/books?id=kNhCQPbYIsgC&printsec=frontcover#v=onepage&q&f=false http://www.teslasociety.com/biography.htm

Therein lies the problem. Your notion is very obviously based on the everyday experience, Euclidean geometry. What Einstein discovered is that the model of your everyday experience is wrong. And that is what should be relevant to ontology. You can imagine all sorts of things that are wrong. This is just one of them. You seem to be insisting on an ontological study of erroneous ideas. I have zero interest in the opinion of someone who is clueless regarding the mathematical meaning of curvature on the subject of what curvature "really means". I know what curvature really means. Your question is not the least bit reasonable. It is in fact ridiculous. ridiculous The Calabi-Yau manifolds occur as one term in a Cartesian product, and they are WITHOUT BOUNDARY. They are also compact and are therefore completely irrelevant to the discussion of whether space is compact (aka finite) or open (aka infinite). A fourth or fifth spatial dimension quite obviously refers to a fourth or fifth spatial dimension -- what else could possibly be the referent ? Empirical facts -- the existence of the known list of elementary particles. The extra dimensions of string theories are required for the mathematical consistency of those theories. Whether they are valid theories that describe nature remains unknown. It is amusing to watch you attempt to address the ontology of spacetime when you have no idea what spacetime is and harbor so many gross misconceptions regarding general relativity. It is rather like a critique of French poetry by someone who speaks and reads only Swahili.

AAgreed but for one thing -- the term "expert'. To me an expert in a given area is someone who: 1) knows everything that is known in the area 2) knows what the important problems are and why they are important 3) understands the known avenues of research on the open problems and who is pursuing them and 3) has in-depth knowledge of related areas and understands their relevance, and 4) can quickly evaluate a proposed avenue of research and weed out those that won't work.. In some areas there are no experts.

In my graduate classes in mathematics many classes were conducted by the students with guidance from the professor, but little lecturing in the traditional sense (typical class size 4-12). I am not a big fan of lectures, but I am a huge fan of having classes run by experts in the subject. Student participation is one thing. Replacing experts with video games or students is quite another. I do not agree that even a good (stellar perhaps, but only real exceptions)) undergraduate student has the depth and expertise to be teaching a class that he has just recently taken. But having students present their perspective is a great way to get discussion going and to find areas that need reinforcement.

It surprises not at all that such an approach produces improved results for average students enrolled in a large freshman-level introductory class. Those large classes are so bad that just about any novel approach is likely to realize an improvement. That alone would prompt the student to actually think about the material a bit and that is the key to learning. Changing back at some future point might also again produce improvement just from the novelty effect. What does not seem to be addressed is the better student or classes with more serious content.

http://en.wikipedia....tary_Grand_Tour http://en.wikipedia.org/wiki/Gravitational_slingshot