This paper examines the suborbital graphs generated by the product actions of the wreath products $S_m \wr S_n$, $A_m \wr S_n$, $C_m \wr S_n$, and $D_{2m} \wr S_n$ on the Cartesian power $X^n$. Using permutation group theory, orbit and stabilizer analysis, combinatorial techniques, and computational verification, the study investigates the classification of suborbits and subdegrees and the structural properties of the resulting graphs. The analysis focuses on the pairing properties of orbitals and graph characteristics, including connectivity, vertex degree, regularity, transitivity, girth, and symmetry. The findings reveal how the orbit structure of each base group, together with coordinate permutations, determines the structure of the associated suborbital graphs. In particular, under suitable conditions, the symmetric and alternating group actions yield adjacency structures governed by Hamming distance, whereas the cyclic and dihedral group actions produce graph structures reflecting their respective permutation actions. The study provides a group-theoretic framework for examining suborbital graphs arising from wreath product actions and contributes to the understanding of the relationship between permutation groups, Cartesian powers, and algebraic graph structures.
This paper proposes a unified operator-theoretic framework for analyzing the Finite Difference Method (FDM) and the Variational Iteration Method (VIM) for solving nonlinear second-order boundary value problems (BVPs). The VIM iteration is interpreted as a Picard-type fixed-point iteration in a Banach space, and a computable a priori error bound is established under a Lipschitz condition on the nonlinearity. Concurrently, FDM is treated as a finite-rank projection method, and its convergence is demonstrated via consistency and stability. The main contribution of this work is a novel hybrid a posteriori error estimator that uses the continuous VIM iterate to bound the global discretization error of the FDM solution without requiring additional PDE solves or mesh-doubling, requiring only the VIM iteration already computed as part of the error estimation process. Unlike traditional residual-based estimators, the proposed estimator relies solely on the discrepancy between the smooth semi-analytical approximation and the discrete numerical solution. Theoretical convergence rates and the efficiency of the estimator are validated through numerical experiments on a nonlinear BVP with a known exact solution and a non-solvable linear BVP. The results indicate that the hybrid estimator provides a reliable and computationally efficient indicator for adaptive mesh refinement.
Edem, Inemesit DavidAfrican Journal of Mathematics, Statistics and Computer Science · 2026
Image denoising is a classical inverse problem in which the recovery of a clean image from its noisy observation is complicated by ill-posedness. This paper develops a functional-analytic framework in which a broad class of denoising methods is expressed as the iteration of firmly nonexpansive or quasi-nonexpansive operators on a Hilbert space. Total variation regularization, wavelet shrinkage, projection onto convex sets (POCS), projection onto epigraph sets (PESC), and regularization by denoising via fixed-point projection (RED-PRO) are each shown to fit this structure, and their convergence is obtained from a single Krasnosel'skii–Mann argument. The proximal nature of wavelet shrinkage is distinguished from metric projection onto the \(\ell^1\)-ball, and the demicontractive hypothesis underlying RED-PRO is stated explicitly. Numerical experiments on standard test images show that RED-PRO with a BM3D denoiser attains the highest PSNR at every noise level tested, that Chambolle's TV projection follows within \(1.2\)–\(1.7\) dB, and that POCS and wavelet shrinkage trade accuracy for speed. The framework clarifies the operator-theoretic content of each method and provides a common language for their analysis and comparison.
Inemesit D. Edem, Donald Goodhead, Utomobong M. OkonKtrend - International Journal of Computational Mathematics and Scientific Computing · 2026
