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Well, I am not sure exactly what you are asking, but in modern geometry one is often interested in how geometric objects on a manifold M (tensor fields, densities, connections etc.) transform, usually under the diffeomorphism group of M or a subgroup of it. So, depending on exactly what you want to prove, being very clear on what type of objects you have to deal with (identified by how they transform) is important.

Use digital. You can do all kinds of image processing, like stacking and changing colours etc. Do you mean astronomy rather than astrology? Binoculars are no good for astrophotography, but are great for viewing the moon, open clusters and some of the brighter deep sky objects like M31 and M13. The main advantage over a telescope is the ease of use and the large field of view.

I never fully undersood it, but basically it is because higher spin particles will not couple to gravity in a consistent way. David Bailin gave a short talk on this at Sussex as part of the theory groups informal seminars. There is a link to the ps notes here

You are talking about T-duality. It is due to the "extended" nature of strings and compactification. A good introcuction can be found in BUSSTEPP Lectures on String Theory by Richard Szabo.

Yes of course, I would say that this is one of the most useful things you can do with complex numbers. If you can't remember multiangle formule for the trigonometric functions etc, you can work them out explicitly from the properties of exponentials.

I suggest that you get a pair of binoculars, say 10x50. These should be light enough to use without a tripod. Any bigger you should mount on a tripod or something similar. You won't see the detail that you would with a telescope, but binoculars are far easier to use and have a wider field of view. In that respect they are far better than a telescope for looking at star clustes. Also, thay are a fraction of the cost of a telescope. They offer you the chance to try astronomy without spending loads of money. If you take to it then you can buy a telescope etc... On the astrophotography side, many of my friends at the local astronomical society have been getting great pictures using digital camers and webcams. These pictures are of great quality, far beyond that of standard photographic techniques employed previously. Infact I would say they are better than the professional pictures of 10-15 years ago. What I strongly recommend is that you get involved with your local astronomical society. I personally own a set of 12x60 binoculars, that I mount on a camera tripod via an "L"-bracket. Total cost was less than £100. Unfortunatly, since I got the tripod, we have had nothing but rain just about every night! I don't have any immediate ambitions to get involved in astrophotography.

I simply plugged it into mathematica and got [math]{\frac{-\arctan ({\frac{3} {{\sqrt{-9 + 4\,{x^2}}}}})}{ 3}}[/math] Maybe you can spot this by differentianting arctan.

When I first come across complex numbers I had no idea just how important they are. On the mathematics side there is a lot of machinery developed for complex analysis, which comes in useful in may branches of mathematics and physics. On the physics side complex numbers are fundamental in quantum mechanics. Generically, you dont see just Plancks constant it is always multiplied by i. Complex numbers are also very useful when describing waves and oscillations. They are used to describe phases. One important thing to realise with complex numbers is that they are algebraically closed. That is any polynomial with complex coefficients has a complex root. This is not true of the real numbers, i.e. you can have a polynomial with real coefficents whose roots are not real numbers. In fact they will be complex numbers. For a simple exmple consider [math]z^{2} +1 =0 [/math]. You see that all the coefficents are real numbers. Question, what are it's roots? You qucikly realise that [math] z = \pm i[/math] are the roots, which are imaginary numbers. If you do something similar with a polynomial with complex coefficient, you will see that the roots are also complex. That is you will never "leave" the field of complex numbers by considering roots of polynomials.

Ok, so it might not be laziness, but with public libraries, reliable web-books and other online resources why don't people read up on the basics? I feel that a lot of the questions posted on this forum could be answered by a quick trip to the library.

What I don't understand is why people insist on working out their own theories before they understand current scientific thinking? Is it just because they cannot be bothered to spend the time and effort that physics and mathematics requires? I fully encourage anybody to think about science, but they should be armed with a basic knowledge. This comes through study and asking the right questions, not by making wild speculations. Any comments?

Classical light rays are waves in the electromagnetic field. Photons (light particles) are the result of quantising the electromagnetic field. Quantum field theory produces from the classical field localised excitations which we interpret as "light particles". These particle are massless. Maxwell's equation tell us that accelerating charges produce electromagnetic radiation. So, I think the answers you are looking for can be found in any book on classical electromagnetic theory and then supplemented by a book on quantum field theory.

Thinking of definite integrals separately from derivaties, the integral is the "area under the curve". In essence this was Riemann's definition. At first glance it is not obvious that intergration and derivatives have anything to do with each other. It is a deep fact from the so called "fundamental theorem of calculus" that they are "inverses" of each other. To answer you question you have to look at this theorem. See for example http://mathworld.wolfram.com/Integral.html http://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html

I though that Mythbusters was about making statements of the plausibility of certian claims, rather than a "full scientific investigation". I think it lives up to that.

Upon quantisation, these modes correspond to states. I'll list some of the lower states for you. For the open bosonic string we have [math]N=0[/math] tachyonic "unstable" vacuum. [math]N=1[/math] massless spin one particle, (Maxwell or Yang-Mills Field). [math]N \geq 2 [/math] infinite tower of massive particles. The closed bosonic string [math]N=0[/math] tachyonic "unstable" vacuum. [math]N=2 [/math] massless spin 2 graviton, spin 2 antisymmetric tensor field "Neveu- Schwartz B-field" and the spin 0 dilaton. For more details I suggest Polchinski "String Theory" two volumes Sazbo "String Theory and D-brane Dynamics" (short but clear book) Zwiebach "A First Course in String Theory" An older classical reference is Green, Witten and Schwartz "Superstring Theory" It is before the development of branes but tells you just about everything else.

Like I said, it is the only book that I can think of. The first few chapters will give you the introduction you want. Before you look at theories with constraints are you happy with Hamiltonian mechanics, Poisson Brackets and symplectic geometry? You should be comfortable with these notions before looking at constraints. If you have any direct questions I will try to help.

you had a look at the book? It is the only one that springs to my mind. Theories with constraints are hard, maybe you just have to bite the bullet! I'll have a think and see if I cna find a more gental book.

The modern way to treat theories with constraints, like gauge theories is via BRST methods. I suggest "Quantization of Gauge Systems " by Henneaux &. Teitelboim. They deal with gauge symmetries in mechanics and field theories using both the canonical BRST and the antifield formulism. Both approches require supergeometry with is just about explained enough in the book. I must warn you that the subject is quite heavy.

I just looked a SPIRES and there is 904 papers written by John Ellis. So I can beleive you! I have seen him twice now, but yesterday was the first time I have spoken to him. I am sure our paths will cross again, even though I do not do phenomenology.

That is funny, but is it ture? You know this area of science better than I do.

You can see that is must be [math]E=m c^{2}[/math] just on dimensional grounds. Just consider what the units on both sides of the equation.

To para-phrase John Ellis "String theory needs supersymmetry more than supersymmetry needs string theory".

Why do you think this is the case?

You can move [math]\delta[/math] through a derivative provided the variation is not with respect to the argument of the derivative. If it is then you do pick up an extra term as you have to consider how the derivative changes. The same is true of the measure for an integral.

The function was something like [math]\frac{x}{x^{2} + y^{2}} [/math]

Severian is correct. If supersymmetry is found then it would be consistent with string theory, but not evidence of string theory. If supersymmetry is not found then one can argue that the energy scale of supersymmetry is higher then expected. This would not be inconsistent with string theory or supersymmetry in general. This means that you cannot "disprove" supersymmetry in nature! So I would say if supersymmetry is found then I would take it as an indication that string theory is on the right lines. Even if supersymmetry is not seen in nature, it's mathematical beauty and power means that "super-tools" will be with us forever, both in physics and mathematics. (Also the BV-BRST formulism should be considers as a supersymmetry, there is a lot of intersting geometry behind this). John Ellis (CERN) is giving a talk called " Novel signatures for supersymmetry" in Manchester. I will be attending and may ask what he thinks about this.