We consider a spectral line formed with a linear combination of monochromatic scattering and complete frequency redistribution. A previous theoretical and numerical investigation by Hemsch and Ferziger in 1971 and 1972 suggests that a very small percentage of complete frequency redistribution is sufficient to significantly modify a coherent scattering process. This article reexamines this result with numerical solutions of the radiative transfer equation calculated with an iterative method, introduced by Paletou and Auer in 1995 for radiative transfer with partial frequency redistribution. The partial frequency redistribution process is defined by a branching ratio b with values in [0,1], with 0 for complete frequency redistribution and 1 for monochromatic scattering. It is characterized by two dispersion functions, which are derived here by a Fourier transform of an integral equation constructed with the radiative transfer equation. The calculations are performed for a plane-parallel semi-infinite medium, with a uniform internal source of photons and a uniform single scattering albedo 1-& varepsilon;. They show that complete frequency redistribution dominates the scattering process as soon as b becomes smaller than a critical value bcrit similar or equal to 1-& varepsilon;, given by the intersection the two dispersion functions.
Data-driven surrogate models become physically inadmissible when the data are sparse. Physics-informed neural networks impose a governing differential equation as a residual, which presupposes that the equation is known. This work concerns the case in which it is not, and physical insight is qualitative. A physics-constrained surrogate modeling framework is introduced in two stages. In the first, derivative signs prescribed by transport theory for the critical slab under linearly anisotropic scattering are imposed as hinge penalties on the surrogate derivatives. In the second, the signs need not be known in advance: each is estimated point-wise with a Savitzky-Golay filter and may reverse within a variable, extending the constraint to non-monotone responses. Only the sign is constrained, never the magnitude. On the criticality data the unconstrained network satisfies the inequalities unaided and the constraint is inactive, a negative control. On the 14C cross section it reduces the mean violation threefold over 25 random splits without loss of accuracy. A single global sign makes matters worse, as does gradient boosting under hard monotonicity: both meet the constraint by suppressing the derivative. Calibration of the sign estimator proves decisive; a density scaling rule for the smoothing window is given.
The radiative transfer equation is one of the most fundamental relationships for describing macroscopic thermal radiation transfer processes. Based on this, the main contribution of this paper is to provide new trends into an unbounded block 3 & times;3 operator matrix defined with a non-maximal domain, offering new insights into the phenomena of radiative transport equations. More precisely, we formulate sufficient and necessary conditions for the entries of the above mentioned model: of the operator matrix to provide an explicit formula for its associated Frobenius-Schur decomposition. Some new spectral properties of such matrix model are also derived to relate its eigenvalues to their two associated Schur complements. As an illustrative example of our findings, we state a model of 3 & times;3 groups of radiative transfer equations in a channel with specific boundary conditions. Our findings build upon and enhance the recent advancements of Bouzidi et al. (Georgian Mathematical Journal, 32, 1, (2025), 37-50). In particular, this study expands the theoretical scope to the realm of 3 & times;3 operator matrix form with a non-maximum domain.
