Mathematical Methods for Physicists

By G. Arfken. 1967

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Citations in the corpus

Differential Privacy Preservation for Deep Auto-Encoders: an Application of Human Behavior Prediction (2016, methodology): “First, the cross-entropy error functions of the data reconstruction and softmax layer are approximated to polynomial forms by using Taylor Expansion (Arfken 1985).”
Learning SO(3) Equivariant Representations with Spherical CNNs (2017, methodology): “The Fourier transform and its inverse are defined on the sphere as follows [33]:”
HiPPO: Recurrent Memory with Optimal Polynomial Projections (2020, background): “[2] George B Arfken and Hans J Weber.”
The Recent Evolution of Climate on Venus (2001, methodology): “…flux equations within each spectral interval o this probability space was achieved using an 8-point Gaus integration (Abramowitz and Stegun 1965, Arfken 1985). calculate radiative equilibrium, net infrared fluxes that balan the observed solar net flux profile were determined, using iterative…”
Resolution of complex tissue microarchitecture using the diffusion orientation transform (DOT) (2006, methodology): “2 l=2 4pD u ð Þt ð Þ 3=2 R 2 0 D u ð Þt l=2 : ð41Þ Note that using the addition theorem for spherical harmonics (Arfken and Weber, 2001), X l m ¼ Àl Y lm r ð ÞY lm u ð Þ4 ¼ 2l þ 1 4p P l ur ð Þ; ð42Þ where P l (x) is the l-th order Legendre polynomial, it is possible to express Eq.”
Chemical bonding in crystals: new directions (2005, background): “According to the divergence theorem [92], the sign of the Laplacian of a scalar function indicates whether a net flux of the gradient of the scalar is entering (sign ) or leaving (sign þ) an infinitesimal volume centered on a given point.”
Evolution of circular, nonequatorial orbits of Kerr black holes due to gravitational-wave emission. II. Inspiral trajectories and gravitational waveforms (2000, methodology): “This equation is in self-adjoint form, and so can be solved with Green’s functions [23].”
divand-1.0: n -dimensional variational data analysis for ocean observations (2013, methodology): “The Fourier transform of a radial function inIRn is given in terms of the Hankel transform by (Arfken, 1985) k n−2 2 F̂ (k) = (2π) n 2 ∞∫ 0 Jn−2 2 (kr)r n−2 2 F(r)r dr, (A2) where Jν(r) is the Bessel function of first kind of orderν.”
Acoustic Metamaterials: Negative Refraction, Imaging, Lensing and Cloaking (2013, methodology): “One way to achieve such a Cosserat-type elastic material is possibly to play with effective properties of composite structures that may behave in the low frequency regime as neutral coated inclusions as shown by Davide Bigoni, Alexander Movchan and their colleagues in [2].”
A consistent refinement of first-order shear deformation theory for laminated composite and sandwich plates using improved zigzag kinematics (2010, methodology): “Presently, the Rayleigh–Ritz method was employed, where the kinematic variables were approximated with suitable Gram–Schmidt polynomials [Arfken 1985] that satisfy the kinematic boundary conditions, (26), exactly (refer to [Tessler et al. 2009] for the particular expressions of the Gram–Schmidt…”
On self-propulsion of micro-machines at low Reynolds number: Purcell's three-link swimmer (2003, methodology): “…Lagrangian markers α1 and α2 in terms of orthogonal polynomials with time-dependent mode amplitudes, to then write the mechanical power Φ as a quadratic form in time derivatives of mode amplitudes, and finally to use the calculus of variations (e.g. Arfken 1985) to conclude that Φ must be constant.”
Characterization of anisotropy in high angular resolution diffusion‐weighted MRI (2002, background): “1/2 Ylm, . [ 18 ] Therefore, the process of rotating a diffusion tensor can be reformulated by expressing the diffusion tensor in an irreducible form in which its individual components transform separately under rotations affected by spherical harmonic components.”
Bound states of spatially dependent mass Dirac equation with the Eckart potential including Coulomb tensor interaction (2014, methodology): “Then we obtain the normalization constant as [71,72]”