R. Courant’s research while affiliated with New York State and other places

... After subtracting the two expressions and deriving expression (6), the equations of motion for the system are obtained. This is the first approximation, for the solution of the system of the three bodies. ...

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A brief solution to three-body problem: Newtonian and Hamiltonian versions Una breve solución al problema de los tres cuerpos: Versiones Newtoniana y Hamiltoniana

... In order to make this work self-contained, in this appendix we will briefly summarize general methods to solve nonlinear first-order PDE heavily used in the main text. They can be found in the PDE literature [28][29][30], see also [21] and references therein for some historical remarks. We consider a general nonlinear first-order PDE F(t, r, u, p, q) = 0, (A.1) ...

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Non-Noetherian conformal Cheshire effect

... The solution of the wave equation is explicit for the free case # B 2 t u 0´∆ u 0 " 0, on R`ˆR d , pu 0 , B t u 0 q |t"0 " pf 0 , g 0 q, on R d , (1.6) see for instance [Eva98,CH89]. In particular, if the dimension d is odd, the wave propagates at speed 1 (this is the strong Huygens Principle). ...

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Local decay and asymptotic profile for the damped wave equation in the asymptotically Euclidean setting

... From (3.4) and (3.5), one infers that the sine kernel (3.2) defines a positive trace-class operator. The spectrum {γ n | n ∈ N 0 } of an operator belonging to this class is compact, countable, real positive and satisfies ∞ n=0 γ n < ∞ [103,104]. The spectrum of the sine kernel (3.2) is non-degenerate, it satisfies the following bounds (see [105] and eqs. ...

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Entanglement entropies of an interval in the free Schrödinger field theory at finite density

... From the mathematical perspective, the derivation of these models yields systems of partial di erential equations (PDEs) with boundary inputs and outputs. Since the PDEs are hyperbolic 1 , the method of characteristics [32,117] is applied to obtain corresponding time-delay relations. Although this chapter focuses on the transport of evaporating liquid lms, the presented models may also be applied to other transport processes, e.g., in tra c ow [34,28,148], which is motivated via analogies. ...

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Modeling and Control of the Falling Film Evaporator Process

... (28). As above, integration of these modified Green's functions over the shell uses the axisymmetric surface element S D−2 ϖ α dz for integration between −1 ≤ z ≤ 1 along the direction of the shift x, and the surface weight ϖ α (z) = √ 1 − z 2 2α−1 , cf. [91]. It matches the manifold dimensions when 2α = D − 2. We choose a constant variance σ 2 to eliminate independent scalars in the sound energy density and sound intensity potential ...

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Diffuse Sound Field Synthesis

... We assume that the field has some stationary points {(t n , z n )}, e.g. (±1/ √ 2ω t , 0) for the field (36) or (0, ∼ ±0.7/ω z ) for (40). Let F n = E 3 (t n , z n )/E 0 be the values of the field at the peaks. ...

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Schwinger pair production in spacetime fields: Moir\'e patterns, Aharonov-Bohm phases and Sturm-Liouville eigenvalues

... In principle, G(z, a) can be represented in terms of elliptic functions, but the expressions are somewhat complicated and do not add much of theoretical understanding. See however [28,32] for formulas and series developments useful for numerical purposes. The regular part H(z, a) of the Green function is equally complicated, but some of its derivatives are simple and can be obtained by other means. ...

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Two-dimensional potential theory with a view toward vortex motion: Energy, capacity and Green functions

... It is obviously the same basis for all i with corresponding eigenvalues {d i λ k } ∞ k=0 depending on i (cf. Courant and Hilbert [5]). Also, let {e k } n k=1 be the standard orthonormal basis of R n . ...

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Fast Diffusion Limit for Reaction-Diffusion Systems with Stochastic Neumann Boundary Conditions

... As classical in literature (see e.g. [13]), we call nodal domains of v λ α the maximally connected subsets of R 2 for which v λ α does not change sign. From Proposition 1.3 and Lemma 2.1, we obtain the following statement. ...

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Examples of non-scattering inhomogeneities