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This research introduces a fortified cryptographic paradigm for safeguarding RGB image data, founded upon a substitution–permutation network (SPN) architecture defined over the residue class rings of Eisenstein integers modulo a selected prime element. The framework employs a dual-tier configuration of two independently constructed substitution boxes, each synthesized within the algebraic domain of where represents a primitive cubic root of unity. The generation process for these substitution layers leverages affine transformations and their respective inverses, ensuring the simultaneous attainment of strong nonlinearity, bijective mapping, and algebraic robustness. Within the encryption pipeline, the initial substitution box functions exclusively as the confusion layer, whereas the second is designed to integrate both permutation and diffusion properties. To further amplify cryptographic intricacy, an auxiliary substitution box is formulated through the bitwise XOR amalgamation of the two primary layers, thereby intensifying inter-channel diffusion across the RGB spectrum. By harnessing the unique arithmetic and lattice geometry of Eisenstein integer residue classes, including their modular and inherently non-Euclidean characteristics, the scheme achieves superior confusion–diffusion balance. Compared with conventional image encryption techniques that rely solely on chaotic maps, Gaussian integers, or quaternion-based transformations, the proposed system offers enhanced algebraic structure exploitation, triple-layer substitution for richer nonlinear complexity, and explicit multi-channel diffusion. This leads to higher entropy, lower adjacent-pixel correlation, and greater resilience to both differential and linear cryptanalysis. Empirical assessments confirm the effectiveness of the proposed methodology, yielding near-optimal entropy, negligible adjacent-pixel correlation, and robust defense capabilities against an extensive range of established cryptanalytic strategies, thereby positioning the scheme as a compelling solution for secure imagetransmission in modern communication networks.

The new space movement has seen a rise since the last decade, thanks to CubeSats, which are affordable with fast development times and ease of deployment. One of the major subsystems in a CubeSat is the Electrical Power System (EPS), which generates, stores, and distributes electrical power to various subsystems of CubeSats. The first step in ensuring a productive CubeSat mission is to choose an efficient EPS design. This article’s primary goal is to provide a comprehensive review of all traditional and evolving CubeSat EPS systems. EPS designs have been divided into four types, and the operational elements of these architectures as well as a qualitative comparison have been provided. In addition, in this comprehensive review, CubeSat’s EPS is presented with some prospective research areas for future research work and invention.

This article discusses methods for numerical modeling of geofiltration processes in groundwater, as well as the development and implementation of the proposed methodology for solving hydrogeological problems. The article analyzes the flows of grout waters entering underground aquifers in the form of filtration, and offers optimal solutions for the construction of drainage systems in mountainous areas. The main attention is paid to the development of mathematical models that describe the processes of interaction between groundwater and surface water, taking into account various natural and anthropogenic factors. The article describes methods for predicting changes in the level and state of groundwater, as well as calculating water balance elements. The authors propose mathematical models for describing the movement of groundwater in single-layer media, taking into account the processes of saturation and infiltration. Attention is paid to the use of information technology, mathematical modeling systems and geoinformation technologies to improve the accuracy of hydrogeological research results. The work demonstrates the relevance and prospects of the application of the proposed methods for solving problems of rational use and management of water resources, as well as for the development of new approaches in the field of melioration and water supply. As examples, studies of geofiltration processes in the mountainous regions of the Republic of Uzbekistan are given, where recommendations for the optimal use of groundwater in drinking water supply and irrigation have been developed.

This work introduces the Legendre cardinal functions for the first time. Based on Jacobi and Lobatto grids, two approaches are employed to determine these basis functions. These functions are then utilized within the pseudospectral method to solve the fractional Klein–Gordon equation (FKGE). Two numerical schemes based on the pseudospectral method are considered. The first scheme reformulates the given equation into a corresponding integral equation and solves it. The second scheme directly addresses the problem by utilizing the matrix representation of the Caputo fractional derivative operator. We provide a convergence analysis and present numerical experiments to demonstrate the convergence of the schemes. The convergence analysis shows that convergence depends on the smoothness of the unknown function. Notable features of the proposed approaches include a reduction in computations due to the cardinality property of the basis functions, matrices representing fractional derivative and integral operators, and the ease of implementation.

This paper investigates and enhances unmanned aerial vehicle (UAV) relay-assisted free-space optics (FSO) optical wireless communication (OWC) systems under the effects of pointing errors (PEs) and atmospheric turbulences (ATs). The incorporation of UAVs as buffer-aided moving relays in the conventional FSO (CFSO) relay-assisted systems is proposed for enhancing the performance of PEs through AT. Using M-PSK (phase shift keying) and M-QAM (quadrature amplitude modulation), the impact of PEs on transmission quality is evaluated in this work. We evaluate and optimize the symbol error rate, outage probability (OP), and signal-to-noise ratio (SNR) for the UAV-to-ground station-based FSO communications systems. The spatial diversity-based relay-assisted CFSO systems can enhance the performance of the UAV-UAV FSO links. In this paper, a new FSO (NFSO) channel model for the hovering UAV-FSO OWC fluctuations under the PEs, AT effects, jitter, deviation, receiving an error, and wind resistance effects are established. To improve the performance of hovering UAV-based FSO relay OWC systems. We reduce the influence of UAV-FSO OWC fluctuations under PEs and AT effects. By receiving incoherent signals at various locations, the spatial diversity-based relay-assisted NFSO systems can significantly increase the system's redundancy and enhance connection stability. Numerical results show that to achieve a bit-error-rate (BER) of ≤ 10 - 5 , the required SNR is ≥ 23 dB when the wind variance of the UAVs σ α 2 increases from 0 to 7 mrad with FSO link distance L = 2000 m. The required SNR is ≥ 25 dB when the wind variance σ α 2 is 1 mrad at an OP of 10 - 6 . To obtain an average BER of 10 - 6 , the SNR should be 16.23 dB, 17.64 dB, and 21.45 dB when σ α 2 is 0 mrad, 1 mrad, and 2 mrad, respectively. Using 8-PSK modulation without PEs requires 23.5 dB at BER of 10 - 8 while 16-QAM without PEs requires 26.5 dB to maintain the same BER of 10 - 8 . Compared with 16-QAM without PEs, the SNR gain of 8-PSK without PEs is 3 dB. The results show the relay-assisted UAV-FSO system with five stationary relays can achieve BER 10 - 8 at 25 dB SNR in the ideal case and 10 - 5 at 27 dB SNR with AT and PE at FSO length 1000 m. The results show the relay UAV-FSO system outperforms the CFSO at the BER and SNR performance. The effects of UAV’FSO s fluctuation increase when the UAV-FSO link length, L fso increases. The results of the weak turbulence achieve better SER compared with MT and ST. The obtained results show that decreasing L fso can compensate for the effects of UAV-FSO link fluctuation on the proposed system. Finally, we investigated the CFSO relay-assisted UAV-FSO system with aided NFSO-UAVs spatial diversity-based relay-based on NFSO OWC and revealed the benefits of the resulting hybrid architecture.

We address the inverse spectral problem for a PT-symmetric Dirac operator with discontinuity conditions imposed along the entire real axis—a configuration that has not been explicitly solved in prior literature. Our approach constructs fundamental solutions via convergent recursive series expansions and establishes their linear independence through a constant Wronskian. We derive explicit formulas for transmission and reflection coefficients, assemble them into a PT-symmetric scattering matrix, and demonstrate how both spectral and scattering data uniquely determine the underlying complex-valued, discontinuous potentials. Unlike classical treatments, which assume smoothness or limited discontinuities, our framework handles full-axis discontinuities within a non-Hermitian setting, proving uniqueness and providing a constructive recovery algorithm. This method not only generalizes existing inverse scattering theory to PT-symmetric discontinuous operators but also offers direct applicability to optical waveguides, metamaterials, and quantum field models where gain–loss mechanisms and zero-width resonances are critical.

The potential integration of unmanned aerial vehicles (UAVs) with free space optical (FSO) communication systems stands as a promising innovation in the realm of wireless network infrastructures. This study presents a comprehensive investigation into the application of orthogonal frequency division multiplexing (OFDM) in conjunction with UAV-based FSO technology, with a specific focus on establishing robust wireless communication links to ground sites within the evolving landscape of 5G networks. The research introduces a pioneering 4-level quadrature amplitude modulation (4-QAM)-OFDM-FSO framework tailored for UAV-to-ground communication, revolutionizing the prospects for seamless and high-throughput data transmission within dynamic network environments. Through comprehensive simulations and theoretical analyses, we unveil the system's efficacy in mitigating atmospheric turbulence, achieving heightened signal integrity, and ensuring performance adaptability over varying link distances, thus significantly addressing present limitations in traditional wireless communication models. Anchored within the context of modern wireless network infrastructures, this work serves as a crucial stepping stone for the practical application of OFDM-UAV-FSO communication systems, representing a paradigm shift in fostering resilient and agile wireless connectivity in the era of 5G networks. The inception of cutting-edge wireless networks expected to outperform the capabilities of 5G necessitates an infrastructure that can handle vast amounts of data. This infrastructure must be not only cost-effective and simple to deploy but also readily scalable to accommodate the diverse demands of front-haul and backhaul applications. Motivated by the growing interest in harnessing UAVs to extend the reach and enhance the operational efficacy of conventional cellular networks, this work introduces a novel application of UAV-ground station connections. The concept employs FSO to facilitate network traffic within both the segments. To optimize throughput, resilience, and spectral efficiency, the application of OFDM is proposed. The research considers the transmission of a 20 Gbps 4-QAM data signal across various channel conditions. It thoroughly assesses the performance implications of factors such as transmission distance and beam divergence. The study explores the correlation between pointing error, scintillation, beam divergence angle, and average spectral efficiency. A slight increase in pointing error results in a rapid rise in the scintillation index, while a larger beam divergence angle can help minimize the impact of scintillation. Adapting the beam's divergence angle based on the pointing error between the optical transceivers can reduce the effects of scintillation and improve the average spectral efficiency and channel capacity. Additionally, the relationship between pointing error, scintillation, and the determination of the optical beam divergence angle in terms of beam divergence and average spectral efficiency and channel capacity is examined, and theoretical evaluations further confirm the method's effectiveness in reducing scintillation in the presence of pointing errors. Furthermore, the simultaneous use of OFDM adaptive beam divergence control and modulation could significantly enhance the data rate. This approach aims to reduce the impact of scintillation in UAV FSO links, which often experience significant losses due to unpredictable fluctuations in the atmosphere's refractive index. The results of the simulations indicate that the integrated 4-QAM-OFDM-FSO framework can realize high data transmission rates, efficiently serving front-haul and backhaul needs, thereby signifying a significant evolutionary leap for the next generation of wireless technology. The numerical findings demonstrate the significant impact of the coherent FSO OFDM optical wireless communication (OWC) setup in UAV wireless communications to ground links, particularly in mitigating the effects of strong turbulence and pointing errors (PEs). Through the integration of spatial coherence diversity and adaptive modulation OFDM in the coherent OWC, there has been a noticeable enhancement in the average spectral efficiency (ASE). Notably, our results indicate an ASE of 53 bits/s/Hz and 37 bits/s/Hz achieved at an average transmitted optical power of 10 dBm for an aperture diameter of 10 cm, without and with PEs for the coherent OWC-FSO OFDM UAV technique, respectively. The proposed method was validated through simulations, demonstrating both improved average spectral efficiency and effective reduction of the scintillation effect. This approach holds promise for mitigating scintillation effects in UAV-FSO links.

This study develops a rigorous analytic framework for solving the Cauchy problem of polyharmonic equations in R[sup.n], highlighting the crucial role of symmetry in the structure, stability, and solvability of solutions. Polyharmonic equations, as higher-order extensions of Laplace and biharmonic equations, frequently arise in elasticity, potential theory, and mathematical physics, yet their Cauchy problems are inherently ill-posed. Using hyperspherical harmonics and homogeneous harmonic polynomials, whose orthogonality reflects the underlying rotational and reflectional symmetries, the study constructs explicit, uniformly convergent series solutions. Through analytic continuation of integral representations, necessary and sufficient solvability criteria are established, ensuring convergence of all derivatives on compact domains. Furthermore, newly derived Green-type identities provide a systematic method to reconstruct boundary information and enforce stability constraints. This approach not only generalizes classical Laplace and biharmonic results to higher-order polyharmonic equations but also demonstrates how symmetry governs boundary data admissibility, convergence, and analytic structure, offering both theoretical insights and practical tools for elasticity, inverse problems, and mathematical physics.

The article presents a methodology for generating a simulation Simulink model of a rocket. The use of the MATLAB/Simulink environment for simulating the flight of a rocket and calculating its aerodynamic characteristics is described in detail. The principles of forming blocks for calculating the parameters of a standard atmosphere, aerodynamic characteristics, power plant thrust, flight angles, altitude and flight range are described. The results of numerical experiments carried out using the MATLAB/Simulink environment are presented.

This study develops a rigorous analytic framework for solving the Cauchy problem of polyharmonic equations in Rn, highlighting the crucial role of symmetry in the structure, stability, and solvability of solutions. Polyharmonic equations, as higher-order extensions of Laplace and biharmonic equations, frequently arise in elasticity, potential theory, and mathematical physics, yet their Cauchy problems are inherently ill-posed. Using hyperspherical harmonics and homogeneous harmonic polynomials, whose orthogonality reflects the underlying rotational and reflectional symmetries, the study constructs explicit, uniformly convergent series solutions. Through analytic continuation of integral representations, necessary and sufficient solvability criteria are established, ensuring convergence of all derivatives on compact domains. Furthermore, newly derived Green-type identities provide a systematic method to reconstruct boundary information and enforce stability constraints. This approach not only generalizes classical Laplace and biharmonic results to higher-order polyharmonic equations but also demonstrates how symmetry governs boundary data admissibility, convergence, and analytic structure, offering both theoretical insights and practical tools for elasticity, inverse problems, and mathematical physics.