Can i know the definition of conformal invariance when applied to Complex Brownian motion?  Thank you.
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„input harvmac „overfullrule=0pt {„it Exam} (1) Prove via path \n„def„R{{„bf R}} „def„S{{„bf S}} (a) Work in four-dimensional (flat) Euclidean space $„R^4$ and show that conformal invariance implies...\nhttp://www.math.ias.edu/QFT/fall/exam1.tex \n\n \nhttp://www.math.ias.edu/QFT/spring/stringPS8.tex\n...def„det{{„rm det}} „def„Det{{„rm Det}} „def„tr ... B$ of the Wess-Zumino-Witten model, satisfy the equations for conformal invariance and for\nhttp://www.math.ias.edu/QFT/spring/stringPS8.tex \n\n \nComments: Latex(2e), 34 pages, 4 figures (eps) On some UNIX machines\nSome private shortcuts: „def„be{„begin{equation}} „def„ee{„end{equation}} „def„bea{„begin{eqnarray}} „def„eea{„end{eqnarray}} „def...\nhttp://www.ma.utexas.edu/mp_arc/e/96-436.latex.mpmix \n\n \n=4 19| >0 63| >1 65| >2 434| >4 525| 0 1773| 1 630| 1/2 2530| 10 \n\nhttp://www-library.desy.de/lists/spihep.k \n\n \nStardrive.org - - Article/Content Template\nStardrive.org is dedicated to fostering increased public awareness and education in the theoretical and technological aspects of interstellar travel.\nhttp://www.stardrive.org/Sarmail1-3-01.shtml \n\n \n„documentstyle[12pt]{article} %„renewcommand{„baselinestretch}{2.0}\n...oddsidemargin 0 mm „def„ee{„end{equation}} „def ... correlation functions have a power law behaviour, indicates that conformal invariance may be\nhttp://theory.ipm.ac.ir/papers/src/ipm63.tex \n\n \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%...\n...def „pa {„partial} „def „ti {„widetilde} „def „se {{„prime„prime}} „def ... „subsection{Pre-big bang inflation and conformal frames}\nhttp://www.ba.infn.it/~gasperin/tex/ba437.tex \n\n \nhttp://www.ba.infn.it/~gasperin/tex/cer96267.tex\n...def „pa {„partial} „def „ra {„rightarrow} „def ... discussed in „cite{6a}, to the conditions of $„sg$-model conformal invariance represented by\nhttp://www.ba.infn.it/~gasperin/tex/cer96267.tex \n\n \n%%%%%%%%% LATEX FILE FOR FORMATTING GUIDELINES \n...def„xp{X^{+}} „def„xm{X^{-}} „def„Yp{Y^+} „def„Ym ... must be obeyed (in a conformal gauge this is the same as maintaining conformal\nhttp://theory.tifr.res.in/group/colloq/das.tex \n\n \n„documentstyle[twoside,psfig]{article}\n...def„ds{„displaystyle} „parindent=0pt „def ... out that this model is advantageous for the study of the conformal invariance conjecture for\nhttp://www.math.utah.edu/~ejpecp/EcpVol1/paper8.tex
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Expressions such as yours is called a power tower. They are differentiable by writing them as exponentials but they are not integrable in closed form. You can find the series form at the source link however.
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I won't do your homework for you, but you should look here:\nhttp://hyperphysics.phy-astr.gsu.edu/hbase/carcr.html#cc1\n\nGood luck. :)
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A step by step method to understand cramer's rule:\n\nhttp://www.purplemath.com/modules/cramers.htm\n(basically its the first result found by google; you can do more googling)
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That is the central limit theorem.  Each distribution can be uniform, but the distributions themselves are normally distributed in their range, their value, their frequency.\nRead the attached link for more details.
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Hi,\nCheck this one too:\n\nhttp://www.cs.brown.edu/research/ai/dynamics/tutorial/home.html
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Ott, Edward (2002). Chaos in Dynamical Systems. Cambridge University Press New, York. ISBN 0521010845. \nGutzwiller, Martin (1990). Chaos in Classical and Quantum Mechanics. Springer-Verlag New York, LLC. ISBN 0387971734. \nMoon, Francis (1990). Chaotic and Fractal Dynamics. Springer-Verlag New York, LLC. ISBN 0471545716. \nTufillaro, Abbott, Reilly (1992). An experimental approach to nonlinear dynamics and chaos. Addison-Wesley New York. ISBN 0201554410. \nGollub, J. P.; Baker, G. L. (1996). Chaotic dynamics. Cambridge University Press. ISBN 0521476852. \nBaker, G. L. (1996). Chaos, Scattering and Statistical Mechanics. Cambridge University Press. ISBN 0521395119. \nAlligood, K. T. (1997). Chaos: an introduction to dynamical systems. Springer-Verlag New York, LLC. ISBN 0387946772. \nKiel, L. Douglas; Elliott, Euel W. (1997). Chaos Theory in the Social Sciences. Perseus Publishing. ISBN 0472084720. \nStrogatz, Steven (2000). Nonlinear Dynamics and Chaos. Perseus Publishing. ISBN 0738204536. \nSprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 0198508409. \nHoover, William Graham (1999,2001). Time Reversibility, Computer Simulation, and Chaos. World Scientific. ISBN 981-02-4073-2. \n Beauty of Fractals, by H.-O. Peitgen and P.H. Richter \nChance and Chaos, by David Ruelle \nComputers, Pattern, Chaos, and Beauty, by Clifford A. Pickover \nFractals, by Hans Lauwerier \nFractals Everywhere, by Michael Barnsley \nOrder Out of Chaos, by Ilya Prigogine and Isabelle Stengers \nChaos and Life, by Richard J Bird \nDoes God Play Dice?, by Ian Stewart \nThe Science of Fractal Images, by Heinz-Otto Peitgen and Dietmar Saupe, Eds. \nExplaining Chaos, by Peter Smith \nChaos: Making a New Science, New York: Penguin, by James Gleick \nComplexity, by M. Mitchell Waldrop \nChaos, Fractals and Self-organisation, by Arvind Kumar \nChaotic Evolution and Strange Attractors, by David Ruelle \nSync: The emerging science of spontaneous order, by Steven Strogatz \nThe Essence of Chaos, by Edward Lorenz \nDeep Simplicity, by John Gribbin \nThe Road To Chaos, by Yoshisuke Ueda \nThe Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory, Edited by Ralph H. Abraham and Yoshisuke Ueda
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Most introductory books on differential geometry discuss this. Looks at 'From Calculus to Cohomology' and 'Introduction to Smooth Manifolds' as references.
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Let u=ln t, e^u=t, du=dt/t, dt=t du\nThen:\n∫ln² t dt = ∫t u² du = ∫u² e^u du\nIntegrating by parts:\nu² e^u - ∫2u e^u du\nu² e^u - 2u e^u + ∫2e^u du\nu² e^u - 2u e^u + 2e^u + C\nt ln² t - 2t ln t + 2t +C\nAnd we are done.
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A partial answer.\nThe case of W_Y(t) is easy: this is the sum of Y(t) variables that are exponentially distributed and independent of Y. So the sum's expectation is the exponential expectation (1/c) times the expectation of Y(t), which is ct (Y(t) has the Poisson distr with para ct). So\n\nE(W_Y(t)) = (1/c)ct = t.\n\nThat is, the average wait in the process X until its number of events is equal to the number of events in Y at time t is the average inter-event time (1/c) times the number of events in Y at time t (which is ct), and this is simply t.\n\nThe case of W_X(t) is not so easy because the variables and their number in the sum are no longer independent.\nEW_X(t) equals the sum over n of the probability that that X(t)=n (which is exp(-ct)(ct)^n/n!, the Poisson probability) times the expectation of W_n conditional on W_n being less than t and W_n+1 being greater than t (this is a way of saying that X(t)=n). So EW_X(t) is certainly less than t. Also, W_n has the gamma distribution, and in principle the conditional expectation could be calculated from that.
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Well, this was one aspect of my research thesis in Physics of about 10 years ago. The big picture is quite wide, so let's focus on a particular property of classical thermodynamics that is challenged by the chaotic behavior of certain systems: the equipartition of energy.\nClassical thermodynamics tells us that, providing enough time, a system goes to equipartition, i.e. its energy is apportioned in equal parts along all the degrees of freedom. It is well known that the equipartition law has been disproved by quantum mechanics, but until some years ago it was believed a necessary consequence of classical mechanics.\nThe news is that the recent studies (including my little contribution) have shown that there are certain systems for which, under given conditions, neither classical mechanics prescribe the equipartition of energy.\nA class of such systems is constituted by the nonlinear Hamiltonian lattices, that is, N-dimensional spring/mass chains with nonlinear springs. To make it simple, consider that some solid gasses have their atoms stuffed in crystals that can be modeled as such lattices.\nThese systems are classified as chaotic (or also stochastic) for mathematical reasons too complex to be explained here.\nSuppose you freeze Helium at the point that it solidifies. Then you have such a crystal. Its degrees of freedom are the harmonic modes of oscillation of the crystal (just like the harmonic sounds in a guitar string). From the equipartition law one should expect that, after waiting some time, all harmonics go to the same energy. But this is not the case: most energy remains instead concentrated in the lower harmonics. The effect is stronger as the temperature decreases.\nObviously these results are mainly obtained thanks to computer simulations, because the involved equations have no analytic solution.\nSurprisingly, some of these results match with the prediction of quantum mechanics. This is clearly a debated issue but it suggests anyway that the "old-fashioned" classical mechanics has still something to say in modern physics!\n\nLater addition (in relation to the "Additional Details" above): yes, of course I'm speaking of deterministic chaos here, in the mathematical sense (thought it was obvious).
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For a simple matrix 'trace ' is the sum of all main diagonal elements.\n\nFor a tensor, it is a scalar given by the contracted mixed tensor equal to T and it satisfies the conditon\n\nTr[M(x)^(-1)  d/dx^(lambda) M(x)] = d/dx^(lambda) ln det[M(x)]\n\nThe derivative is partial, in this case.\n\nI suggest you write this equation by hand, since its not easy for me to write legibly without equation editor. Sorry for that.
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That's an elliptic integral.  Check out the link.
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I assume delta(x, y, z) = 1 is the mass density ... isn't rho usually used for that?  Anyway, you need to integrate mass times square of perpendicular distance from axis of rotation.\n\nLooks like it will be messy.\n\nMaybe outer x from 0 to 1, middle y from 0 to 1 - x^2, inner z from 0 to (12 - 4x - 3y)/2, of delta(x, y, z) (x^2 + y^2) dz dy dx.
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theace0804:  (i) the second law of thermodynamics states that the entropy of a closed system will tend to a maximum; (ii) what has non-linear dynamics got to do with anything here?\n\ncurious_chemical: the 2nd law implies that the disorder of a closed system tends to a maximum, not the other way round.  You need to start understanding about closed thermodynamical systems or none of it's going to make any sense.  Ice may be more ordered than water, but during crystallisation heat was radiated into the environment and this becomes more disordered.\n\nNot relevant to the discussion here, but a fantastic mind-mangling thought experiment nonetheless, is Maxwell's Demon.  Feynman gives a nice account of it in his Lectures on Physics (IIRC).\n\nEdit:\nWe do live in a very real closed system, called the universe.  But on a more practical / local level, your thermodynamic arguments regarding order need to be referenced against an approximation of a closed system.  Your arguments regarding ice cubes forming are just bung.  You're making the same mistake as thousands of perpetual motion machine designers, and as a result your arguments are invalid.\n\nAnother Edit: So let me get this straight.  At the big bang, an infinitely large space-time was instantly created for the rest of perpetuity, and this space-time is fixed and absolute?  That's abit 19th Century don't you think?  Also,  if the edge of the universe is expanding outwards at the speed of light, then there exists a causation boundary as the interior cannot causally affect the exterior.  So ... the universe is a closed thermodynamic system.\n\nYour grasp of cosmology is pitiful but your grasp of thermodynamics is even worse.  What you say is in direct conflict with the 1st and 2nd laws, two fairly simple laws.\n\nI don' care if it isn't your "idea to make thermodynamics an issue in evolution".  You made a horribly incorrect remark about thermodynamics.  If your physics is so way off, why should your biology be any better?\n\nYou want to argue biology?  Okay, let's start with "Evolution has always been a case-by-case event".  Wrong.  Evolution is about trends not events.  Our "Our ape ancestors" is wrong; apes and humans have a common ancestor, but we most certainly don't have an "ape ancestor".  How about "Evolution [...] guarantees [...] adaptation."?  Adaptation is never guaranteed, as witnessed by the continual extinction of species.  One could probably argue that becoming more adaptable is an "arrow of progress" for the species concnered.  I know you are trying to argue against popular misconceptions of darwinian thought here, but you're doing it really badly.\n\nAlthough it is assumed by most that the same mechanisms are responsible for both macro- and micro-evolution (gene mixing and mutation), the way they are studied scientifically differ.  As a result, it is specious at best and unscientific at worst to claim that "micro is macro" in such a loud voice.  You also fail to show how the same mechanism does result in two very different phenomenom, that genes between species usually diverge and genes within species usually combine, making it very easy for creationists to pick holes in your arguments.\n\nIn short, you haven't got a clue what you are talking about and are thereby doing scientists a large disservice.
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log into www.wikipedia.org and search for lie group...there is a detailed description and all the formulas as well and its free.
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ja :), that's a good one. E=mc^2 talks about the equivalence between matter and energy; c is just the constant that joins them both ..you won't attain a c ¨velocity¨out of it ; you'll just have to use \n E=mc^2/(sqrt(1-(v/c)^2) ( Is it Lorenz transf?) and I think Xav was pretty clear with his explanation! :)