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Supersymmetry
Published in Harish Parthasarathy, Supersymmetry and Superstring Theory with Engineering Applications, 2023
[f] If an integral of the form ∫μ∧f(k,g)dk that represents a propagator correction or some scattering amplitude diverges as μ ↓ 0 (infrared divergence) of as ⋀ → ∞ (ultraviolet divergence) or both where ɡ is a coupling constant, or some parameter in the field theory Lagrangian like mass, charge or some parameter in the propagator, then we redefine a renormalized coupling constant ɡR= ɡR(μ, ⋀) so that this integral assumes a definite finite value say c and then we express all the ampltiudes etc in terms of this renormalized parameter ɡR:c=∫μΛf(k,gℛ)dk
Turbulent flow over urban-like fractals: prognostic roughness model for unresolved generations
Published in Journal of Turbulence, 2018
It is important to remember that for multiscale topographies, the small scales have a pronounced influence on the aggregate momentum fluxes, thus motivating the need for high-fidelity prognostic models. For natural fractals, such as fluvial landscapes, this can be appreciated via consideration of the spectral density of the gradient of surface height. The variance of the gradient (surface slope) is derived from the integration of its spectral density, , where . It follows that, for , the entire range of spatial scales are aero-/hydro-dynamically relevant; moreover, for , (ultraviolet divergence). Interestingly, the preceding scaling arguments and deductions on the importance of the small scales are supported by experimental data from Mejia-Alvarez and Christensen [17], who performed an ensemble of experimental measurements of inertial-dominated turbulent boundary layer flows over a complex, multiscale geometry. These authors decomposed the original topography with proper-orthogonal decomposition, and then reconstructed reduced-order cases from the first five modes or the first sixteen modes. This approach enabled a clear indication of how a wide spectrum of topographic modes affected the turbulence statistics. For all cases, the outer layer (that is, above the roughness sublayer) exhibited only weak sensitivity to the exclusion of higher-order modes. Within the roughness sublayer, however, turbulence statistics were clearly modulated by the range of topographic scales present. Note also that, as per Orey's formula, and correspond with D=1 and D=2, respectively, corresponding with the physical upper and lower bounds for affine geometries (D>2 represents a ‘space-filling’ fractal, which is physically ill-defined).