In this work, the scattering problem pertaining to the excitation of a dielectric cylinder surrounded by a cluster of significantly smaller, assumed point-like scatterers, by a field due to a line-source is considered. In particular, by considering a line-source parallel to the z-axis, the corresponding electric fields are perpendicular to the xy-plane, allowing the original 3D vector problem to be reduced to a 2D scalar problem. Furthermore, the multiple-scattering problem is reduced to a single-scattering problem of excitation of a dielectric sphere by an arbitrary distribution of electric dipoles, by means of the Foldy-Lax approximation. Specifically, by assuming that each point-like scatterer elicits fields similar to that of line-sources that disect the xy-plane on a point in the interior of each scatterer, the recovery of the electric fields of the point-like scatterers is enabled through an inverse-problem scheme utilizing measurements of the surface electric fields. To that end, the surface-field measurements at specific angles of observation are manipulated into a matrix form, so that a linear-system for the unknown point-scatterer fields can be devised. Furthermore, a dense cluster approximation of the scattering cross section attributed to the cluster's scatterers is obtained when certain conditions are met, while explicit numerical results demonstrating the accuracy and stability of the method, are presented.

In this paper, we propose a new iterative method inspired by recent developments in inertial techniques. The scheme is constructed by modifying the Mann-type iteration and incorporating inertial terms to improve the speed of convergence. The idea is to use information from both the current and the previous iterates so that the generated sequence approaches the solution more efficiently than the standard Mann process. We establish the strong convergence of the proposed algorithm under mild and natural assumptions on the underlying mappings. The proof relies on suitable auxiliary results and careful analysis of the sequence generated by the method. The assumptions imposed are standard in the literature and do not restrict the applicability of the algorithm to special cases. As a result, the convergence result holds for a broad class of nonlinear problems in Hilbert spaces. To highlight the usefulness of the method, we apply our main theorem to equilibrium problems, variational inclusion problems, and convex minimization problems. These applications show that the proposed iteration provides a unified framework for treating several important models in nonlinear analysis and optimization. In the final section, we include a numerical example to illustrate the convergence behavior of the algorithm and to support the theoretical results. The numerical outcome confirms that the method converges as predicted and demonstrates its practical effectiveness.

This work presents a numerical study of the coupling between the Burgers equation with a source term and the advection–diffusion–reaction equation in one spatial dimension, employing an explicit finite difference scheme. The formulation considers the Burgers equation as the governing model for the evolution of the velocity field, whose solution is subsequently used in the transport equation as the convective velocity at the current time step, thereby characterizing a sequential coupling between nonlinear advection and diffusion phenomena. The adopted discretization is based on the forward Euler method for the transient term, combined with second-order centered approximations for the first- and second-order spatial derivatives. The study focuses on the investigation of the consistency, stability, and convergence properties of the numerical scheme, with a particular emphasis on the influence of the spatial (dx) and temporal (dt) discretization steps on solution accuracy. The error analysis is conducted using standard norm-based metrics, enabling the quantification of convergence rates and the identification of regimes in which temporal and spatial truncation errors dominate the solution. Stability constraints associated with convective and diffusive terms are examined, along with the effects of inter-equation coupling on the propagation of numerical errors. The results demonstrate the sensitivity of the explicit scheme to the discretization parameters and highlight conditions under which the coupled system preserves stable and convergent behavior. This study contributes to a deeper understanding of the performance of explicit schemes in nonlinear coupled systems relevant to transport phenomena, simplified fluid dynamics, and reactive process modeling.

Traditional aerospace navigation often decouples attitude (orientation) from position (translation), using quaternions for the former and Cartesian vectors for the latter. However, in modern proximity operations such as autonomous docking, satellite servicing, and formation flying, the physical coupling between these two motions becomes significant. Our research proposal addresses the critical challenge of precise relative navigation within large-scale distributed spacecraft systems, where traditional decoupled representations of translation and rotation often fail to capture the complex kinematic coupling inherent in six-degree-of-freedom (6-DOF) maneuvers. In others words, our project aims to exploit the algebraic properties of dual quaternions to directly model these coupled dynamics, thereby improving the accuracy, computational efficiency, and robustness of state estimation in large space systems. The first phase of the project involves the rigorous formulation of a relative state error model. By applying the dual velocity transformation, we combine angular velocity and linear velocity into a single dual vector. The second task focuses on the development of a Multiplicative Extended Kalman Filter (MEKF) adapted specifically for the dual quaternion manifold. Unlike standard Euclidean filters, this architecture preserves the unit constraint of the dual quaternion through an error-state approach, facilitating distributed state estimation across large swarms. The final task establishes theoretical rigor through a stochastic stability analysis. By constructing a Lyapunov-like candidate function based on the dual quaternion geodesic distance, we prove that the estimation-control loop remains bounded under Gaussian perturbations. The proposed result is a robust, computationally efficient estimation framework that achieves a notable improvement in convergence rates and significantly lower steady-state error compared to traditional decoupled Extended Kalman Filter methods, providing a scalable solution for autonomous proximity operations in complex satellite constellations.

Our research establishes a rigorous mathematical framework for the six-degree-of-freedom (6-DOF) relative navigation of interconnected spacecraft using unit dual quaternions. Unlike traditional methods that decouple translational and rotational kinematics, often leading to singularities or inaccuracies in highly dynamic proximity operations, dual quaternions provide a unified, compact, and singularity-free representation of rigid body motion. By leveraging the Principle of Transference, we extended classical quaternionic rotation theory to encompass the full Euclidean group SE(3), treating the spacecraft's pose as a single algebraic entity. This approach inherently respects the natural coupling between orientation and position, which is critical for the precision required in docking and formation flying. Our project, focusing on the analytical derivation of relative motion equations without the use of stochastic filters, ensuring a deterministic foundation for state estimation, is structured around three primary tasks. Firstly, we developed the derivation of the dual quaternionic kinematic and dynamic equations of motion for a leader–follower spacecraft configuration. Secondly, we provided a proposal formulation of a closed-form analytical solution for relative pose estimation based on line-to-line and point-to-point correspondences. Finally, we developed a coordinate-invariant interpolation scheme for smooth trajectory generation between discrete states. Results demonstrate that the dual quaternion framework reduces the computational overhead associated with homogeneous transformation matrices by eliminating the need for frequent orthogonalization, replacing it with a simpler normalization process. Furthermore, the implementation in MATLAB for numerical validation and Python for high-fidelity visualization confirms that the proposed mathematical foundations provide superior accuracy in representing screw-based displacements, effectively mitigating gimbal lock and ensuring robust relative navigation for small satellite clusters.

We investigate finite semigroups equipped with a preorder relation that is compatible with the semigroup operation. We introduce the concept of a compatible preorder and define the associated structure of a preordered semigroup, which captures the interaction between the semigroup operation and order-theoretic properties. To classify such structures, we develop an enumeration methodology based on automorphism group actions. For a fixed finite semigroup S, the automorphism group Aut(S) acts naturally on the set Pre(S) of compatible preorders via pushforward, and the orbits of this action correspond exactly to isomorphism classes of preordered semigroups. To address anti-isomorphism, we incorporate the converse relation, which allows us to identify preorders equivalent up to inversion, effectively reducing the computational complexity. By considering the quotient of Pre(S) modulo, the converse relation, and extending the action via pushforward to the equivalence classes, we obtain a systematic approach to enumerate preordered semigroups up to both isomorphism and anti-isomorphism. Using known classifications of semigroups of a small order, together with exhaustive generation of compatible preorders, our method provides a complete enumeration of preordered semigroups up to an order of five. Finally, this framework opens a computational window toward the study of more complex algebraic systems, notably EL-hyperstructures arising in hypercompositional algebra, where preordered semigroups serve as a fundamental tool for their construction.

Introduction

This work establishes a novel connection between the relativistic Hamilton–Jacobi equation (HJE) and the relativistic massive Schrödinger equation (RSE) in free space, achieved without relying on the semiclassical limit (ℏ → 0). The approach operates mode-by-mode on spectral fibers associating with any Minkowski momentum bra S_p = <p, . > the de Broglie wave η_p. The families S and η generate distinct subspaces in tempered distribution spaces: the former spans a real four-dimensional vector space isomorphic to Minkowski momentum space, while the latter comprises the full space of complex tempered distributions on Minkowski space.

Methods

The key families of distributions — Minkowski bras (S) and de Broglie basis (η) — span (via suitable subfamilies) the solution spaces of the HJE’s and RSE’s, respectively. Employing Schwartz linear algebra for complex tempered fields, we apply a von Neumann-style linear-continuous operator extension to lift the HJE (formulated in complex variables) from certainty momentum states |p⟩ to complex amplitude-probability states ψ. This extension mirrors procedures used in game theory and follows standard von Neumann techniques. The construction is further extended to the Maxwell–Schrödinger formalism - in complex tempered distribution 3-field - through de Broglie-Maxwell isomorphisms (Fe), which map wave distributions to corresponding electromagnetic-like fields while preserving translation representations, dispersion relations, and polarization structures.

Results

The principal finding demonstrates that the relativistic massive Schrödinger equations are von Neumann-like linear extensions of the relativistic HJE in complex form. These equations are uniquely determined spectrally by the Einstein energy-momentum relation within the tempered distribution framework.

Conclusions

This framework provides a vast, concrete (although partial) unification of classical relativistic mechanics, relativistic quantum mechanics for massive particles, and Maxwellian field theory - all within the setting of tempered distributions - offering new insights into the foundational relationships among these domains.

This talk concerns the development and interrelations between geometry and physics since the development of Euclidean geometry. It briefly describes the ideas behind Euclid's work and moves quickly through to the groundbreaking work of Copernicus, Kepler, Galileo and Newton during which the foundations of classical (Newtonian) physics were laid down. These advances used the geometry of Euclid, as applied to 3-dimensional space, to act as the arena where the activities of physics took place and to measure and describe, for example, the motion of particles. In such a philosophy, the geometry was absolute and fixed (and Euclidean) and uninfluenced by the physics. Such ideas were developed in a different way by Lagrange, Hamilton, Routh and others who employed the use of "generalised" coordinates to shake off the idea of preferred inertial observers. However, Euclidean geometry still played the role of absolute background geometry. With the advent of Einstein's general relativity theory in the 20th century the sources of the physics (the matter and fields, etc) were now allowed to influence the geometry and, conveniently, the resulting (differential) geometry needed was then being developed by the Italian school. For Einstein the geometry was no longer background and Euclidean but rather a Riemannian-type geometry determined by the physics and, in this sense, dynamical and so there arose reciprocal actions of the geometry and the physics upon each other. This was the approach of Einstein and it led to his field equations which, given the physical situation, gave (somewhat complicated) field equations for determining the geometry and, from this, the physics. This is now the best available theory of gravitation and, when applied on a global scale, led naturally to the study of cosmology.

The Random Domino Automaton (RDA)—a slowly driven system in the form of a one-dimensional stochastic cellular automaton—was introduced as a stylized simple model of earthquake statistics to provide a basis for the interrelation of the Gutenberg–Richter law and Omori law with the waiting time distribution for earthquakes. The Gutenberg–Richter distribution provides a universal relationship between the frequency of earthquakes and their size. If the earthquake magnitude is measured by their energy (or seismic moment), it has the form of an inverse power-law distribution.

In the RDA model, energy-related clusters can grow, merge, and disintegrate (trigger avalanches) depending on specific system parameters, which in stationary conditions is described in terms of the coupled recurrence relation for cluster-size statistics. This formulation proves appropriate for studying the role of these mechanisms in the formation of discrete inverse power-law distributions, or discrete Zeta distributions.

By asymptotic analysis of the relationship between the avalanche probability and the resulting stationary cluster-size distribution, we show that the convolution term in the governing equation, which encodes cluster merging, plays a decisive role in generating inverse power-law relations for a wide regime of parameters.

We conclude by pointing out an interesting connection of RDA-type systems with well-known Catalan-like integer sequences (Catalan, Motzkin, and Schröder numbers) and also mention the generalization of RDA to the geometry of the Bethe lattice.

A significant and increasing number of problems related to radiative heat transfer, demand accurate if not exact solutions to fourfold integral equations. In mathematical terms, this is a complex and cumbersome issue even acknowledged by famous polymaths like J. H. Lambert, since the 18^th century. The present author demonstrated and published along his research, the whole procedure for rectangular radiative emitters some 25 years ago. However, in the case of circular and curved sources, the usual analytical approach remains exceedingly difficult and not even suitable for numerical calculus. Recently, the author has elaborated on integral equations applied to fragments of spheres which usually appear in radiative heat transfer and has been fortunate to define new postulates which may solve, with required accuracy, the question of energy transfer between them. In this way not only spherical sectors but also circles, semicircles and circular sectors become affordable. As it is widely accepted, to find the exact expression for radiative exchange by virtue of integration alone is considered no mean feat. The novelty of the process developed lies in that firstly we solve the two initial sets of integrals demanded by analytical methods and then employ the exact results obtained as a kind of iterative algorithm in order to be able to provide the final solution for the remaining double integral equations. This method, has not been described by former researchers and it provides us with a considerable repertoire of scenarios for simulation of energy transmission in the void or in gaseous media. Incidentally, within the operation, new unexpected geometric forms appear and they are subject to exhaustive mathematical assessment revealing interesting findings and a plethora of applications for diverse fields such as architecture, tunnelling or space vessel construction.