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It is indeed compactly-supported Poincare duality via currents, for which the standard reference is de Rham's Differentiable Manifolds. For example, the Dirac delta function is the dual of a point! Anyway, to get to a quick understanding, see the beginning of Section 7.3 of Nicolaescu's notes http://www3.nd.edu/~lnicolae/Lectures.pdf


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Hints: Note that the derivative of the sign function $$ {\rm sgn}^{\prime}(z)~=~2\delta(z) \tag{A}$$ is twice the Dirac delta distribution. This fact seems to be at the heart of OP's question. Repeated differentiations of the Mestel disk potential $$\Phi~:=~ v_0^2 \ln(r+|z|), \qquad r~:=~\sqrt{R^2+z^2}, \tag{B}$$ leads to $$\frac{\partial \Phi}{\partial ...


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I gather your unfortunate edit, which further confused things, to the point of intractability, came from verbatim reproduction of eqn (3.23) of the Duff et al paper on Stochastic Local Operations and Classical Communication. One of about a dozen problems with your malformed question is that you used the wrong coset space to be described by the Cartan pair ...


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It is all over the place, but it's involved. It is normally calculated from the path integral propagator. The most concise source of the radial Green's function you are after is eqn (15) of Grosche 1998, in terms of modified Bessel functions, integral rep, $$ G_l^C(r'',r';E) = \int_0^\infty\dfrac{e^{i e^2s''/\hbar}ds''}{\sqrt{v'v''}} ...


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I do not buy into the view that spacelike geodesics are not physical entities or, at best, can only be understood in tachyon terms. Mathematically spacelike geodesic paths are well understood in 3+1 spacetime. They can be computed in principle given a metric. Given 2 events in a local (but not infinitesimal) part of manifold they can be joined by a unique ...


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A Boltzmann distribution is system dependent--it depends on the energy eigenstates. Moreover, if the system lacks symmetry, then Vx, Vy, and Vz may have very different distributions. However, if you're talking about an ideal gas--then it's the standard Maxwell-Boltzmann distribution. To get just Vx, you can use the equal-parition property, and rewrite a 1-d ...


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In 3 dimensions you can use the cross product to get an appropriate rotation axis. If $\hat{x}$ and $\hat{n}$ are non-parallel 3-dimensional unit vectors then $\vec{s}=\hat{n}\times \hat{x}$ is non-zero. Since $\vec{s}$ is orthogonal to both $\hat{n}$ and $\hat{x}$ there is some rotation about $\vec{s}$ that takes $\hat{n}$ to $\hat{x}$. You can get the ...


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Let $\theta>0$ denote the angle between the $\hat{\mathbf x}$ and $\hat{\mathbf n}$. Notice that $$ \hat{\mathbf u} = \frac{\hat{\mathbf x}\times\hat{\mathbf n}}{\sin\theta} $$ Is a unit vector perpendicular to both $\hat{\mathbf x}$ and $\hat{\mathbf n}$. The desired rotation is a right-handed rotation around $\hat{\mathbf u}$ by the angle $\theta$. ...


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According to sec. 4 in Calculus of Variations, Gelfand, Dover 1991, a theorem due to Bernstein concerns existence and uniqueness of solutions to the equation $y'' = F(x,y,y')$: If the functions $F$, $\partial_y F$ and $\partial_{y'} F$ are continuous at every finite point $(x,y)$ for any finite $y'$, and if a constant $k>0$ and functions $\alpha\equiv ...


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Consider a smooth 2D sphere and a point of positive mass constrained to move on that without friction. The only force is the reactive force normal to the sphere. The trajectories of the motions of the point are geodesics of the spherical surface. Therefore if you fix the north and south poles as boundary conditions you find infinitely many solutions ...


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This is indeed possible only in some situations, e.g. when the continuous spectrum is absent (it may also consist of a single point, see Valter Moretti's comment below). A sufficient condition for that to be true is that either the Hamiltonian is compact or it has compact resolvent. Sadly, very few interesting Hamiltonians satisfy that property (an example ...


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2) Prof Carl E Mungan of the US Naval Academy derives a wave speed of sqrt(gx) by assuming the length of a pulse travelling along the rope is much smaller than the length of the rope. He states that this relation has been verified in experiments (reference 2). http://www.usna.edu/Users/physics/mungan/_files/documents/Scholarship/HangingPulse.pdf


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A standard reference packed to the brim with other references to everything you ever wanted to know about anomalies is "Anomalies in Quantum Field Theory" by Bertlmann. This particular topic is what comprises part of chapter 11 there. I'll highlight the main points, but this is a technical topic for which you'll have to go to the references and follow all of ...


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Because hydrostatic pressure depends on depth, the problem requires integration. The pressure as a function of $y$ according to Pascal's Law is: $$p(y)=p_0+\rho g y\sin \theta$$ Where $p_0$ is the atmospheric pressure and $\theta=45\:\mathrm{degrees}$. $y\sin \theta$ is the depth. On an infinitesimal piece of door of length $dy$, at position $y$ and ...



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