The current study characterizes the transient M/M/1 queue manifold info-geometrically, through devising Fisher Information matrix(FIM) and its inverse(IFIM). Additionally, the impact of stability on the existence of IFIM and explore the geodesic equations of motion has been revealed. More potentially, the paper discusses some info-geometric applications to Machine Learning are highlighted. Closing remarks with some challenging open problems combined with the next phase of research are given.

A particular challenge in the conceptual design of direct-from orbit delivery systems is the seemingly well-known kinematics retain fallacies whose efficacies are reduced over great distances. Rotation about the local wing of an aerospace vehicle is almost never the pitch angle, yet modern application of kinematics often assumes such (with accompanying angular error). The same assertion is usually true about the nature of roll and yaw angles. Expressing motion in coordinates of rotating reference frames necessitates transformation between reference frames, and one such transformation is embodied in the Direction Cosine Matrices (DCM) formed by a sequence of three successive frame rotations. One of two ubiquitous sequences of three rotations used to construct the DCM involves first a rotation around the inertial z-axis, then intermediate y-axis, then finally about the body’s x-axis; a sequence commonly called the “aerospace sequence” or the a “3-2-1 rotation sequence”. The second ubiquitous sequence is the so-called “orbital sequence” or “3-1-3 rotation sequence” with rotations about the inertial z axis first, then about the intermediate x-axis, then finally about the body z-axis. This manuscript evaluates which sequence is the most advantageous for an object starting in space and then travels through the atmosphere to a target on the Earth’s surface. Six degrees of freedom of vehicle motion were simulated starting in orbit with a given thrust and commanded maneuver. The simulation performs all twelve possible rotation sequences (transforming inertial coordinates to body coordinates) with comparison by computational burden and error representing rotations about body axes (roll, pitch, and yaw respectively). Simulation precision is validated using the quaternion normalization condition indicating near machine precision (0.9\times{10}^{-15}) and reveals the so-called 132 rotation is the most accurate with an average error of 0.14° and a computational time of 0.013 seconds: resulting in a 97.95% increase in accuracy over the so-called 321 rotation and a 99.84% increase over the so-called 313 rotation.

From a differential geometric perspective, information geometry (IG) aims to characterise the structure of statistical geodesic models. The research done for this paper offers a novel method for modelling the information geometry of a queuing system. From the perspective of IG, the manifold of the transient M/M/infinity queue is described in this context. The Fisher Information matrix (FIM) as well as the inverse of (FIM), (IFIM) of transient M/M/infinity QM are devised. In addition to that, new results that uncovered the significant impact of stability of QM on the existence of (IFIM) are obtained. Moreover, the Geodesic equations of motion of the coordinates of the underlying transient are obtained. Also, it is revealed that stable QM is developable (i.e., has a zero Gaussian curvature) and has a non-zero Ricci Curvature Tensor (RCT). Novel stability dynamics of transient M/M/iInfinity queue manifold is revealed by discovering the mutual dual impact between the behaviour of (RCT) and the stability(instability) phases of M/M/Infinity QM. Furthermore, a new discovery that presents the significant impact of stability of queue manifold and the continuity of the unique representation between M/G/1 queue manifold and Ricci Curvature Tensor (RCT). The information matrix exponential (IME) is devised. Also, it is shown that stability of the devised (IME) enforces the instability of M/M/ Infinity QM. More interestingly, novel relativistic info-geometric queueing theoretic links are revealed .

We delve into nonrelativistic quantum electrodynamics within a background magnetic field, ensuring gauge invariance via a vector potential. Our exploration extends to the Lagrangian, which incorporates electron self-interactions and electromagnetic field interactions. Through the path integral formalism, we elucidate the effective action, with a specific focus on the photon propagator and screening effects. Examining density and current components reveals Laurent expansions in the hydrodynamic limit. The equations of motion in real space are derived, leading to discussions on phenomena such as Poiseuille-like flow and the Navier-Stokes equation. To ground our theoretical framework at practical contexts, we consider applications such as the fluid dynamics at binary star systems. Depending on the velocity of the expelled fluid, our analysis allows for both nonrelativistic and relativistic treatments, providing a versatile tool for understanding the intricacies of fluid behavior at astrophysical scenarios. Additionally, we recognize that at the presence of a magnetic field, magnetohydrodynamics becomes crucial. While our current focus is at nonrelativistic quantum electrodynamics, the insights gained here contribute to a broader understanding, offering a comprehensive foundation for quantum electrodynamics analysis at diverse physical systems.