The article describes the tensor products of approximation spaces associated with regular elliptic operators on tensor products of Lebesgue spaces $L_2(\partial\Omega)$, where $\partial \Omega$ considers as smooth manifold that describes in the usual way by local system of local coordinates. We use the quasi-normed approximation spaces and subspaces of exponential type functions associated with such operators.} A connection between the tensor products of approximation spaces and interpolation spaces obtained by the real method of interpolation is showed. We prove the direct and inverse approximation theorems for Bernstein–Jackson type inequalities as well as we give the explicit dependence of constants on parameters of approximation spaces. Such constants are expressed via some normalization factor. Application to spectral approximations on tensor products of interpolation spaces associated with regular elliptic operators on compact manifolds is shown. In the article also consider the spectral approximations (Theorem 2), since the subspaces of entire functions of exponential type of regular elliptic operators on compact manifolds coincide with their spectral subspaces (Lemma 3).

This work presents a mathematical methodology for estimating the ensemble-averaged effective multiplication factor and the neutron scalar flux profile in statistically homogeneous, multiplying random media. The methodology is based on the generalized linear Boltzmann equation (GLBE), a neutral particle transport model that accounts for nonexponential neutron flux attenuation, which may arise in media where spatially correlated scattering centers occur. This nonexponential attenuation is incorporated through a carefully constructed free-path distribution function and a total macroscopic cross section, both of which depend on the neutron’s direction of motion and the distance traveled by the neutron since its last interaction (free-path variable). To solve the GLBE, we employ the spectral approach to handle the free-path independent variable, the discrete ordinates (SN) formulation to handle the angular variable, and the response matrix spectral nodal method to solve the resulting spectral SN approximation numerically. The conventional power method is used to estimate the effective multiplication factor. The numerical results to two test problems, associated with a one-dimensional, multigroup, random periodic medium, illustrate the methodology’s effectiveness in predicting the ensemble-averaged effective multiplication factor and the neutron scalar flux profile. These findings highlight the applicability of the proposed approach, particularly in cases where classical transport models may fail to provide accurate results due to their implicit assumption of exponential neutron flux attenuation.

Spectral densities approximations of incidence-based locally tree-like hypergraph matrices via the cavity method

Physics-informed spectral approximation of Koopman operators

Spectral approximation of Gaussian random graph Laplacians and applications to pattern recognition

This work presents a new numerical approach that uses spectral approximation techniques with Delannoy polynomials to solve linear and nonlinear fractional initial differential equations (FDEs). The suggested approach uses Delannoy polynomials and their shifted variations to effectively create approximation solutions, while the fractional derivatives are stated in the Caputo sense. By converting FDEs into a system of algebraic equations using the tau approach for linear situations and collocation for nonlinear ones, the technique makes computer implementation dependable. It is a great option for fractional calculus applications due to its high accuracy and efficiency. Numerical examples are used to validate and demonstrate the approach’s effectiveness.

Camera traps are powerful tools that facilitate ecological monitoring and behavioral observations of non-human primates. Although supposedly non-intrusive, some models generate sound and illumination that elicit behavioral responses from different species. Reactions of primates to camera traps are poorly documented, including those of tarsiers, despite their distinctive auditory and visual specializations. Here, we described the reactions of wild Philippine tarsiers (Carlito syrichta) to camera traps based on existing video records on Leyte Island (N = 12) and characterized the light and sound emissions of the camera traps used for recording. We observed avoidance, attraction, and inspection behaviors from the tarsiers after their looking impulses. Using a spectrometer and ultrasonic acoustic analysis, we found that the camera traps emitted light at predominantly infrared wavelengths with peaks at ca. 850nm (low glow) and ca. 930nm (no glow). Some low-glow models produced a faint red glow during recording and a detectable clicking sound upon trigger, whereas the no-glow camera traps emitted infrared light with significant ultraviolet components. Based on spectral sensitivity approximations and audible threshold assessments, we found that the emissions of the camera traps are detectable, not only to tarsiers but also other primate species. Our findings suggest that camera traps influence the behavior of tarsiers. Hence, we advise caution when using camera traps since behavioral reactions may induce bias depending on the framing of studies. We also recommend proper planning when accounting for these behaviors, selecting camera trap models, and designing camera trapping studies.

Spectral Approximations Optimized by Flower Pollination Algorithm for Solving Differential Equations

This paper proposes a hierarchical Fourier extension framework for the accurate reconstruction of piecewise smooth functions with mixed-order singularities. To address key challenges in spectral approximation–namely boundary-induced artifacts, instability in edge detection, and loss of accuracy near discontinuities–the method integrates three main components: (1) boundary-focused Fourier extensions that isolate endpoint effects while preserving internal structures; (2) a multi-stage edge detection strategy combining spectral mollifiers and coordinate transformations to identify discontinuities in function values and their derivatives; (3) adaptive domain partitioning followed by localized Fourier extensions to retain spectral accuracy on smooth segments. Numerical results demonstrate near machine-precision accuracy (∼10−14–10−15) with significantly improved stability and performance over traditional global methods.

This paper presents a nonlinear spectral framework for analyzing monotone and nonexpansive operators in Banach and Hilbert spaces. We construct a nonlinear spectral resolution for maximal monotone operators using Yosida approximations and Fitzpatrick functions, leading to a family of nonlinear projections and an associated spectral measure. For nonexpansive mappings, we establish an iterative spectral approximation based on Krasnoselskii iterations, with proven convergence and recovery of nonlinear eigenvectors. We further extend this framework to ReLU-based neural networks, analyzing spectral bounds, depth-dependent scaling, and gradient alignment. These results bridge nonlinear operator theory and neural architectures, offering new tools for theoretical analysis and applications in optimization, physics, and machine learning.

Spectral Approximation of Convolution Operators of Fredholm Type

Hyperspectral image reconstruction through a subspace-based spectral group sparsity tensor approximation model

On this note, we investigate the quantitative aspects of norm-attainability in operators, focusing on distances to the set of norm-attainable operators, rates of convergence, and approximation properties. Key results include the structural characterization of norm-attainable operators, convexity of the distance function, and convergence rates for sequences of approximations. We also establish optimality conditions and error bounds for norm approximations, providing new insights into their geometric and analytical behavior. Applications in approximation theory, including spectral and compact operator approximations, are explored, emphasizing practical relevance and computational efficiency.

In this paper, an hp spectral element approximation for distributed optimal control problem governed by an elliptic equation is investigated, whose objective functional does not include the control variable. And the constraint set on control variable is stated with $L^2$ -norm. Optimality condition of the continuous and discretized systems are deduced. In order to solve the equivalent systems with high accuracy, $hp$ spectral element method is employed to discretize the constrained optimal control systems. Based on the property of some interpolation operators, a posteriori error estimates are also established by using some properties of some interpolation operators carefully. Finally, a projection gradient algorithm and a numerical example are provided, which confirm our analytical results. Such estimators guarantee the construction of reliable adaptive methods for optimal control problems.

We study McKean–Vlasov PDEs obtained as the mean field limit of interacting particle systems driven by noise, modelling phenomena such as opinion dynamics. We are interested in systems that exhibit phase transitions, i.e. non-uniqueness of stationary states for the corresponding PDE, in the mean field limit. We develop an efficient numerical scheme for identifying all steady states (both stable and unstable) of the mean field McKean–Vlasov PDE, based on a spectral Galerkin approximation combined with a deflated Newton’s method to handle the multiplicity of solutions. Having found all possible equilibria, we formulate an optimal control strategy for steering the dynamics towards a chosen unstable steady state. The control is computed using iterated open-loop solvers in a receding horizon fashion. We demonstrate the effectiveness of the proposed steady state computation and stabilization methodology on several examples, including the noisy Hegselmann–Krause model for opinion dynamics and the Haken–Kelso–Bunz model from biophysics. The numerical experiments validate the ability of the approach to capture the rich self-organization landscape of these systems and to stabilize unstable configurations of interest. The proposed computational framework opens up new possibilities for understanding and controlling the collective behaviour of noise-driven interacting particle systems, with potential applications in various fields such as social dynamics, biological synchronization and collective behaviour in physical and social systems.

The integration of functional, integral and delay components of pantograph Volterra–Fredholm integro-differential equations provides a powerful framework for modeling complex systems with interdependent dynamics, particularly where nonlinearity influences and proportional feedback are essential. Numerical approximations to multi-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations present significant challenges due to the integration of nonlinearity, proportional delays and mixed integral terms, necessitating adaptive methods to achieve highly accurate approximations. In this work, we extend the Legendre spectral approximation to the one- and two-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations. In this method, the Legendre differentiation matrix and the pantograph operational matrix are used to manage the proportional delay terms inherent in these equations. To demonstrate the superiority of the proposed scheme, we present comparisons with other established spectral methods, highlighting the advantages of the Legendre spectral approach.

In the paper, two algorithms that allow identification of a parametric models of random time-series from binary-valued observations of their realizations, as well as from quantized measurements of their values, are proposed. The proposed algorithms are based on the idea of time-series decomposition either on a direct power spectral density or autocorrelation function approximation. They use the concepts of randomized search algorithms to recover the corresponding parametric models from calculated estimates of power spectral density or autocorrelation function. The considerations presented in the paper are illustrated with simulated identification examples in which linear and nonlinear block-oriented dynamic models of timeseries are identified from the binary-valued observations and quantized measurements.

Hyperspectral image destriping with spectral tensor sparse approximation

In this paper, we propose and investigate a novel mixed spectral method for solving the fourth-order problem with Neumann boundary conditions. To theoretically investigate the well-posedness of the solutions, we incorporate an auxiliary variable containing parameters and transform the original problem into an equivalent second-order coupled system. Subsequently, we establish a weak form and a discrete variational form. Under specific conditions of the parameter, the well-posedness of the weak form and the discrete variational form is proven. By utilizing the approximation properties of the projection operator in Sobolev space, we further provide an error estimate between them. Specifically, based on the mixed scheme, we extend the algorithm to complex domains and validate its effectiveness and spectral accuracy through extensive numerical examples.

A finite element/polynomial spectral mixed approximation for the Stokes problem