This work implements a stable algorithm in Field-Programmable Gate Arrays (FPGAs) for which two architectures (unrolling the loop) were developed and analyzed. The algorithm recovers sources located at the boundary separating two homogeneous media that make up a two-dimensional non-homogeneous region from measurements on the boundary of such region. The problem of recovering these sources is an ill-posed inverse problem as small errors in the measurement can produce important changes in the source location. Inverse source problems have many applications in different areas, such as engineering and medicine, making the proposed implementation important. The first architecture (mode one) allows considering different operating speeds, which is an advantage depending on whether we work with fast or slow signals. The second one (mode two) reduces resource consumption by exploiting the characteristics of the source identification algorithm, which is an advantage for multichannel problems such as inverse electrocardiography or electroencephalography. The architectures were tested on four FPGAs of the 7 Series of Xilinx: Spartan-7 xc7s100fgga484, Virtex-7 xc7v585tffg1157, Kintex-7 xc7k70tfbg484, and Artix-7 xc7a35tcpg236. The two FPGA implementations of the algorithm were validated using synthetic examples implemented in MATLAB. The results presented here can be extended to concentric spheres and complex geometries.

The article considers an internal boundary value problem of the distribution of an electrostatic field in a lens formed by two identical semi-infinite circular cylinders coaxially located inside an infinite external cylinder. The problem is reduced to solving a system of singular Wiener-Hopf integral equations, which is further solved by the Wiener-Hopf method using factorized Bessel functions. Solutions to the problem for each region inside the infinite outer cylinder are presented as exponentially converging series in terms of eigenfunctions and eigenvalues.

The present paper introduces a method of basis transformation of vector fields that is specifically applicable to polynomials space and differential equations with certain polynomials solutions such as Hermite, Laguerre and Legendre polynomials. The method based on separated transformation of vector space basis by a set of operators that are equivalent to the formal basis transformation and connected to it by linear combination with projection operators. Applying the Forbenius covariants yields a general method that incorporates the Rodrigues formula as a special case in polynomial space. Using the Lie algebra modules, specifically , on polynomial space results in isomorphic algebras whose Cartan sub-algebras are Hermite, Laguerre and Legendre differential operators. Commutation relations of these algebras and Baker-Campbell-Hausdorff formula gives new formulas for special polynomials

The thymus hosts the development of a specific type of adaptive immune cells called T cells. T cells orchestrate the adaptive immune response through recognition of antigen by the highly variable T-cell receptor (TCR). T-cell development is a tightly coordinated process comprising lineage commitment, somatic recombination of Tcr gene loci and selection for functional, but non-self-reactive TCRs, all interspersed with massive proliferation and cell death. Thus, the thymus produces a pool of T cells throughout life capable of responding to virtually any exogenous attack while preserving the body through self-tolerance. The thymus has been of considerable interest to both immunologists and theoretical biologists due to its multiscale quantitative properties, bridging molecular binding, population dynamics and polyclonal repertoire specificity. Here, we review mathematical modelling strategies that were reported to help understand the flexible dynamics of the highly dividing and dying thymic cell populations. Furthermore, we summarize the current challenges to estimating in vivo cellular dynamics and to reaching a next-generation multiscale picture of T-cell development.

The article proposes various new approximate analytical solutions of the boundary value prob-lem for the non-stationary system of Nernst-Planck-Poisson (NPP) equations in the diffusion layer of an ideally selective ion-exchange membrane at overlimiting current densities. As is known, the diffusion layer in the general case consists of a space charge region and a region of local electroneutrality. The proposed analytical solutions of the boundary value problems for the non-stationary system of Nernst-Planck-Poisson equations are based on the derivation of a new singularly perturbed nonlinear partial differential equation for the potential in the space charge region (SCR). This equation can be reduced to a singularly perturbed inhomogeneous Burgers equation, which, by the Hopf-Cole transformation, is reduced to an inhomogeneous singularly perturbed linear equation of parabolic type. Inside the extended SCR, there is a suffi-ciently accurate analytical approximation to the solution of the original boundary value prob-lem. The electroneutrality region has a curvilinear boundary with the SCR, and with an un-known boundary condition on it. The article proposes a solution to this problem. The new ana-lytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transfer in membrane systems.

Fractional differential systems have received much attention in recent decades likely due to its powerful ability in modeling memory processes, which are mostly observed in real world. Fractional models have been widely investigated and applied in many fields such as physics, biochemistry, electrical engineering, continuum and statistical mechanics. It is shown by many researchers that fractional derivatives can provide more accurate models than integer order derivatives do. Obviously, bang-bang property is of a significant importance in optimal control theory. In this paper a Bang-Bang property and time-optimal control problem for time-Fractional Differential System (FDS) is under consideration. First we formulate our problem and we prove the existence theorem. We state and prove the bang-bang theorem. Finally we give the optimality conditions that characterized the optimal control. Some illustrate applications are given to clarify our results.

This paper presented a two-dimensional, well-balanced hydrodynamic and sediment transport model based on the solutions of variable density shallow water equations (VDSWEs) for sediment-laden flows, and the Exner equation for bed changes. Those equations are solved in a coupled way by the first-order Godunov-type finite volume method. The Harten-Lax-van Leer-Contact (HLLC) Riemann solver is extended to find the local Riemann fluxes in order to maintain the exact balance between the momentum term and the bed slope term. The advantage of a well-balanced scheme over an unbalanced scheme is demonstrated by the synthetic standing contact-discontinuity test case. Then, the model is employed to simulate two laboratory experiments. At last, a field case, the 1996 Lake Ha! Ha! flood event (Canada), is simulated. Results of cross sectional geometries and profiles of longitudinal thalweg agree well with measurements. The accuracy and simplicity of the numerical model, together with the robust implementation, make the model a good candidate for practical engineering applications.

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.

Recent studies have indicated that terahertz time-domain spectroscopy (THz-TDS) can stably and efficiently acquire output spectra using an affordable and compact multimode laser diode (MMLD) with delayed optical feedback as the light source. This research focused on a numerical analysis of the optimal conditions for employing an MMLD with delayed optical feedback (chaotic oscillating laser diode) in THz-TDS, utilizing multimode rate equations. The findings revealed that the intermittent chaotic output generated by the MMLD, characterized by concurrent picosecond pulse oscillations lasting several tens of picoseconds, proves to be highly effective for THz-TDS. By appropriately setting the amounts of injection current and optical feedback, and the delay time of optical feedback, intermittent chaotic oscillation can be attained within a considerably broad parameter range. Moreover, both the MMLD output spectrum and the THz-TDS output spectrum exhibit a consistently stable shape at the microsecond scale, demonstrating the attractor properties inherent in an MMLD with delayed optical feedback.

In this paper, mean-field type games between two players with backward stochastic dynamics are defined and studied. They make up a class of non-zero-sum differential games where the players' state dynamics solve backward stochastic differential equations (BSDEs) that depend on the marginal distributions of player states. Players try to minimize their individual cost functionals, also depending on the marginal state distributions. Under some regularity conditions, we derive necessary and sufficient conditions for existence of Nash equilibria. Player behavior is illustrated by numerical examples, and is compared to a centrally planned solution where the social cost, the sum of player costs, is minimized. The inefficiency of a Nash equilibrium, compared to socially optimal behavior, is quantified by the so-called price of anarchy. Numerical simulations of the price of anarchy indicate how the improvement in social cost achievable by a central planner depends on problem parameters.

Circles, semicircles and its derived circular sectors are ubiquitous both in human creations and in the natural realm. However, mathematically speaking they have always represented a certain level of enigma or even mysticism since ancient times for being such a perfect shape that enclosed the irrational value that we now call π. Such figure was relatively unknown until the seventeenth century; polymaths like Guarini even equated it, after so many centuries, to the Archimedes fraction of 22/7 or sometimes 25/8 which, giving a value 3.125 introduces a considerable degree of error; Eventually, Lambert had to demonstrate, well advanced the eighteenth century that π was actually an irrational number. In recent years, the author has worked with sections of spheres and other curved bodies as related to radiative heat transfer and applied to the finding of form factors pertaining to such special geometries, to the point of defining new postulates. The main theorems so far enunciated refer to the radiative exchange between circles and half disks, but recently the possibility to treat circular sectors has arrived, thanks to the research already conducted in which, by virtue of Cabeza-Lainez first postulate, we are able to know the form factor of any spherical segment over itself, discarding in this way, the former uncertainties that appeared when dealing with the inner side of spherical surfaces. As it is known, to find the exact expression of the configuration factor by integration alone is mostly a tedious task and the results are not readily available as it has been frequently proved. In the said problem of the circular sectors, the author could integrate the first two steps of the basic formulation for radiant exchange. Subsequently, the novelty of the procedure lies in introducing a finite differences approximation for the third and fourth integrals which still remain unsolved, once that we have been able to find the first and second integrals of the form factor. This possibility had not been identified by former research. This sole output provides us with an ample variety of scenarios previously unforeseen. As a consequence, we would be able to analyze with more precision but freely, the spatial transference of radiant heat for figures composed of circular sectors such as spherical sectors like quarters or octaves of spheres. Many architectural, industrial or aircraft modules and objects fall into this category. We already know that spherical shapes cannot be discretized with any accuracy as there are no spherical tiles of the same size, therefore we hope to overcome with our finding a persistent extent of error. In this situation we are able to reduce a considerable amount of hindrances in the progress of thermal radiation science, due to the advances hereby presented. As the new formulations found can be adapted for algorithms, they will be integrated in simulation procedures with significant avail. Our achievement may be considered timely in the effort invested to square the elements of mathematics for radiative transfer. Important sequels are already being derived for radiation in the entrance to tunnels, aircraft design, low-energy building construction and also lighting and HVAC industries to cite just a few.

Several problems of radiative transfer are yet unsolved because of the difficulties of the calculations involved in them, especially if the intervening shapes are geometrically complex. The main goal of our investigation in this domain is to convert the formulas that were previously derived, into a graphical interface based on the projected solid-angle principle. Such procedure is now feasible by virtue of several widely diffused programs for Algorithms Aided Design (AAD). Accuracy and reliability of the process is controlled by means of the analytical software DianaX developed at an earlier stage by the authors. With this new approach the often cumbersome procedure of lighting and thermal exchange calculations can be simplified and made available for the neophyte, with the undeniable advantage of reduced computer time.

Solving Linear equations with large number of variable contains many computations to be performed either iteratively or recursively. Thus it consumes more time when implemented in a sequential manner. There are many ways to solve the linear equations such as Gaussian elimination, Cholesky factorization, LU factorization, QR factorization. But even these methods when implemented on a sequential platform yield slower results as compared to a parallel platform where the time consumption is reduced considerably due to concurrent execution of instructions. The above mentioned linear equation solving methods can be implemented on the parallel platform using the direct approaches such as pipelining or 1D and 2D Partitioning approach. Vedic mathematics is a very ancient approach for solving mathematical problems. These Vedic mathematical approaches are well known for quicker and faster computation of mathematical problems. Vedic Mathematics provides a very different outlook towards the approach of solving linear Equation on parallel platform. It could be considered as a better approach for reducing space consumption and minimizing the number of algebraic operations involved in solving linear equation. In future Vedic Mathematics might serve as a viable solution for solving linear equation on parallel platform.

This paper presents a comprehensive approach to enhancing submersible safety through mathematical modeling and decision-making frameworks. We develop models for submersible location prediction, emergency preparedness, and scenario extrapolation, including a Seawater Density Model, Submersible Mechanical Model, Bayesian Searching Model, and Extended Kalman Filter. Sensitivity analysis confirms the robustness of these models under varying conditions, making them adaptable for different marine environments.In the first section, we focus on accurately locating a submersible after communication loss. The Seawater Density Model employs a hyperbolic tangent function to model depth-dependent density in the Ionian Sea, while the Submersible Mechanical Model simulates underwater dynamics using the 4th order Runge-Kutta method. Uncertainties are addressed using an Extended Kalman Filter, enhancing accuracy, as shown by trajectory and Mean Squared Error (MSE) comparisons.In the second section, we develop a Bayesian Searching Model to efficiently locate a missing submersible. This model iteratively updates location probabilities using a bimodal Gaussian distribution. The search zone is discretized into grids, and simulations demonstrate the model's ability to effectively narrow down the search area and improve detection success.In the third section, we adapt the models to different environments, such as the Caribbean Sea. A warning and obstacle avoidance system is introduced to manage multiple submersibles in close proximity, dynamically adjusting their paths to avoid collisions.Reliability analysis demonstrates the robustness of the models against changes in density and sudden environmental shifts, showing minimal deviation from neutral buoyancy. These results confirm the reliability and adaptability of the models for enhancing submersible safety across various marine settings.

Large values and gradients of stresses and deformations, triggering concentrations of stresses and deformations, arise in the corner areas of a structure. The action of forced deformations, leading to the finite rupture of the contact between the elements of a structure, also triggers the concentration of stresses, while the rupture reaches an irregular point, a line on the area boundary. The theoretical analysis of the stress-strain state (SSS) of areas with angular cutouts in the boundary under the action of discontinuous forced deformations is reduced to the study of singular solutions to the homogeneous problem of the elasticity theory that has power-related features. The calculation of stress concentration coefficients in the domain of a singular solution to the elastic problem makes no sense. It is experimentally proven that the zone, that is close to the vertex of the angular cutout in the area boundary, has substantial deformations, rotations, and it corresponds to rising values of the first and second derivatives of displacements along the radius in cases of sufficiently small radii in the neighbourhood of the irregular point of the boundary. For such areas, it is necessary to consider the plane problem of the elasticity theory, taking into account the geometric nonlinearity under the action of forced deformations. This will allow analyzing the effect of relations between orders of values of deformations, rotations, and forced deformations on the form of the equation of equilibrium. The purpose of this work is to analyze the effect of relations of deformation orders, rotations, forced deformations on the form of the equilibrium equation in the polar coordinate system for a V-shaped area under the action of forced temperature-induced deformations with regard for the geometrical non-linearity and physical linearity.

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 the relationship between stability and the Gaussian curvature, as well as the connections between queueing theory, information geometry, , Riemannian geometry, and the Theory of Relativity.

Robots in space are necessarily extremely light and lack structural stiffness resulting in natural frequencies of resonance so low as to reside inside the attitude controller’s bandwidth. A variety of input trajectories can be used to drive a controller’s attempt to ameliorate the control-structural interactions where feedback is provided by low-quality, noisy sensors. Traditionally, step functions are used as the ideal input trajectory. However, step functions are not ideal in many applications, as they are discontinuous. Alternative input trajectories are explored in this manuscript and applied to an example system that includes a flexible appendage attached to a rigid main body. The main body is controlled by a reaction wheel. The equations of motion of the flexible appendage, rigid body, and reaction wheel are derived. A feedback controller is developed to account for the rigid body modes. Additional filters are added to compensate for the system’s flexible modes. Sinusoidal trajectories are autonomously generated to feed the controller. Whiplash compensation is additionally implemented for comparison. The control method without random errors with the smallest error is the sinusoidal trajectory method, which showed a 97.39% improvement when compared to the baseline response when step trajectories were commanded, while the sinusoidal method was inferior to traditional step trajectories when sensor noise and random errors were present.

L\'evy walks represent important modeling tools for variety of real life processes. Their natural scaling limits are known to be described by the so-called material fractional derivatives. So far these scaling limits were derived for spatially homogeneous walks, where Fourier and Laplace transforms represent natural tools of analysis. Here we derive the corresponding limiting equations in the case of position depending times and velocities of walks, where Fourier transforms cannot be effectively applied. In fact, we derive three different limits (specified by the way the process is stopped at an attempt to cross the boundary), leading to three different multi-dimensional versions of Caputo-Dzherbashian derivatives, which correspond to different boundary conditions for the generators of the related Feller semigroups and processes. Some other extensions and generalisations are analysed.

We point out an inconsistency in Newton's equations of celestial mechanics. A set of differential equations implied by Newton's equations are shown to be free of this inconsistency. We then investigate the integrals of motion associated with this relative difference system.

In the present paper, the effect of the non-linear thermal radiation on the neutral gas mixture in the unsteady state is investigated for the first time. The unsteady BGK technique of the Boltzmann kinetic equations for a neutral non-homogenous gas is solved. The solution of the unsteady case makes the problem more general significance than the stationary one. For this purpose, the moments' method, together with the traveling wave method, is applied. The temperature and concentration are calculated for each gas component and mixture for the first time.Furthermore, the study is held for aboard range of temperatures ratio parameter and a wide range of the molar fraction. The distribution functions are calculated for each gas component and the gas mixture. The significant non-equilibrium irreversible thermodynamic characteristics the entire system is acquired analytically. That technic allows us to investigate the consistency of Boltzmann's H-theorem, Le Chatelier principle, and thermodynamics laws. Moreover, the ratios among the different participation of the internal energy alteration are evaluated via the Gibbs formula of total energy. The final results are utilized to the argon-helium non-homogenous gas at different magnitudes of radiation force strength and molar fraction parameters. 3D-graphics are presented to predict the behavior of the calculated variables, and the obtained results are theoretically discussed.

This work introduces the mathematical framework of the novel “First-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations (1st-CASAM-NODE)” which yields exact expressions for the first-order sensitivities of NODE decoder-responses to the NODE parameters, including encoder initial conditions, while enabling the most efficient computation of these sensitivities. The application of the 1st-CASAM-NODE is illustrated by using the Nordheim-Fuchs reactor dynamics/safety phenomenological model, which is representative of physical systems that would be modeled by NODE while admitting exact analytical solutions for all quantities of interest (hidden states, decoder outputs, sensitivities with respect to all parameters and initial conditions, etc.). This work also lays the foundation for the ongoing work on conceiving the “Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations (2nd-CASAM-NODE)” which aims at yielding exact expressions for the second-order sensitivities of NODE decoder-responses to the NODE parameters and initial conditions while enabling the most efficient computation of these sensitivities.

Advanced methods of treatment are needed to fight the threats of virus-transmitted deseases and pandemics. Often they are based on an improved biophysical understanding of virus replication strategies and processes in their host cells. For instance, an essential component of the replication of the Hepatitis C virus (HCV) proceeds under the influence of non-structural HCV proteins (NSPs) that are anchored to the endoplasmatic reticulum (ER), such as the NS5a protein. The diffusion of NSPs has been studied by in-vitro fluorescence recovery after photobleaching (FRAP) experiments. The diffusive evolution of the concentration field of NSPs on the ER can be described by means of surface partial differential equations (sufPDE). Knodel et al, Viruses 2018, 10, 28 estimated the diffusion coefficient of the NS5a protein by minimizing the discrepancy of an extended set of sufPDE simulations and experimental FRAP time series data. Here, we provide a scaling analysis of the sufPDEs that describe the diffusive evolution of the concentration field of NSPs on the ER. This analysis provides an estimate of the diffusion coefficient that is based only on the ratio of the membrane surface area in the FRAP region and its contour length. The quality of this estimate is explored by comparison to numerical solutions of the sufPDE for a flat geometry and for ten different 3D embedded 2D ER grids that are derived from fluorescence z-stack data of the ER. Finally, we apply the new data analysis to the experimental FRAP time-series data analyzed in our previous paper, and we discuss the opportunities of the new approach.

It is shown that the 5D geodetic equations and 5D Ricci identities give us a way to create a new viewpoint on some problems of Modern Physics, Astrophysics, and Cosmology. Specifically, the application of the 5D geodetic equations in (4+1) and (3+1+1) splintered forms obtained with the help of the monad and dyad methods made it possible to introduce a new, effective generalized concept of the rest mass of the elementary particle. The latter leads one to novel connections between the General Relativity and quantum field theories, as well as all of that, including the (4+1) splitting of the 5D Ricci identities, brings about a better understanding of the magnetic monopole problem and the vital difference in the origins of the Maxwell equations and gives rise to surprising connections between them. The obtained results also provide new insight into the mechanism of the 4D Universe’s expansion and its following acceleration.

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.

This work introduces the mathematical framework of the novel “First-Order Features Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations” (1st-FASAM-NODE). The 1st-FASAM-NODE methodology produces and computes most efficiently the exact expressions of all of the first-order sensitivities of NODE-decoder responses with respect to the parameters underlying the NODE’s decoder, hidden layers, and encoder, after having optimized the NODE-net to represent the physical system under consideration. Building on the 1st-FASAM-NODE, this work subsequently introduces the mathematical framework of the novel “Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations (2nd-FASAM-NODE).” The 2nd-FASAM-NODE methodology yields and computes most efficiently the exact expressions of the second-order sensitivities of NODE decoder-responses with respect to the NODE parameters. Since the physical system modeled by the NODE-net necessarily comprises imprecisely known parameters that stem from measurements and/or computations subject to uncertainties, the availability of the first-and second-order sensitivities of decoder responses to the parameters underlying the NODE-net are essential for performing sensitivity analysis and quantifying the uncertainties induced in the NODE-decoder responses by uncertainties in the underlying uncertain NODE-parameters.

We propose a novel framework for understanding quantum phenomena through the lens of a modified electromagnetic spacetime governed by Einstein field equations. By assuming the existence of an electromagnetic spacetime that adheres to these modified geodesic and Einstein field equations, provided the field equations satisfy the Coulomb force law in weak-field approximations, we explore the derivation of Maxwell’s equations and highlight the necessity of incorporating spin to validate these equations. We derive the wave equation for the normalized transverse metric and establish its equivalence with the de Broglie wave proposition, suggesting a relationship between this metric and the particle's wavefunction. This approach offers a classical explanation for several quantum phenomena, including the double-slit experiment, entanglement, tunneling, non-locality, and wavefunction collapse. Additionally, it posits a method to visualize atomic orbital structures in line with quantum mechanical predictions. These findings suggest that the foundational principles of quantum mechanics can be interpreted as linearized approximations of general relativity applied to electromagnetic spacetime. While this theory does not yet extend to the correlations between general relativity, quantum electrodynamics (QED), and quantum chromodynamics (QCD), it provides a valuable visualization tool for physicists and advances our understanding of the unresolved challenges linking quantum mechanics with general relativity.

We dive into the axion-dilaton model within the supergravity framework, with a specific focus on the intricacies of domain wall construction and stability. Employing holographic vitrification, we unravel the dynamics of domain wall formation in gauge theories featur- ing periodic vacuum structures. Our model, incorporating a QCD-like axion term and a stabilizing dilaton, undergoes scrutiny for conductivity variations under weak disorder. The investigation reveals the model’s resilience, manifesting near-perfect conductivity under mild disorder conditions. However, the rigorous mathematical motivation for the holographic setup demands further elucidation. The scattered nature of our results prompts the neces- sity for a more systematic interpretation of QCD phenomena and conductivity transitions. This study contributes to the mathematical understanding of the axion-dilaton model’s be- havior, highlighting the imperative for a refined holographic framework and a more coherent interpretation of observed phenomena.

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 .

This work presents an illustrative application of the newly developed “Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations (2nd-FASAM-NODE)” methodology to determine most efficiently the exact expressions of the first- and second-order sensitivities of NODE-decoder responses to the neural net’s underlying parameters (weights and initial conditions). The application of the 2nd-FASAM-NODE methodology will be illustrated using the Nordheim-Fuchs phenomenological model for reactor safety, which describes a short-time self-limiting power transient in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted. The representative model responses that will be analyzed in this work include the model’s the time-dependent total energy released, neutron flux, temperature and thermal conductivity. The 2nd-FASAM-NODE methodology yields the exact expressions of the first-order sensitivities of these decoder responses with respect to the underlying uncertain model parameters and initial conditions, requiring just a single large-scale computation per response. Furthermore, 2nd-FASAM-NODE methodology yields the exact expressions of the second-order sensitivities of a model response requiring as few large-scale computations as there are features/functions of model parameters, thereby demonstrating its unsurpassed efficiency for performing sensitivity analysis of NODE-nets.

We introduce the linear subgroup of volume-preserving diffeomorphism as the underlying symmetry to construct an action within the effective field theory framework and the Schwinger-Keldish formalism. The formulated action is posited to effectively incorporate dissipative effects into the Navier-Stokes equations.

Abstract:Shown is the derivation of Lorentz-Einstein k-factor in SRT as an amplitude-term of oscillation-differential equations of second order.This case is shown for classical Lorentz-factor as solution of an equation for undamped oscillation as well as the developed theorem as a second solution for advanced SRT of fourth order with an equation for damped oscillation-states.This advanced term allows a calculation for any velocities by real rest mass

We mathematically derived a comprehensive theoretic framework “Asymmetry Theory”, based solely on the well-established principle of “the constancy of the velocity of light” without any other assumptions, like the ether. This paper proves that Asymmetry Theory encompasses all results of Special Relativity (STR) as a special case when the observer moves perpendicularly to the “source-observer” line. It explains everything STR does but extends beyond it, offering novel and groundbreaking insights. The longstanding controversies surrounding STR arise from its application outside its intended special case, hence, they are all resolved in Asymmetry Theory. Asymmetry Theory is supported by all experiments supporting STR, like Kaufmann, Ives-Stilwell, muon lifetime, nuclear reaction…., and those contradicting STR, like the Lunar range and GPS tests of light speed and the lack of transverse Doppler shift. Despite its mathematical rigorousness and all experimental evidence, we propose two new experiments to decisively test predictions from both theories.The mathematically derived Asymmetry Theory is a more comprehensive theoretical framework covering STR as a special case. It is self-consistent and theoretically proved by Maxwell’s equations. It does not contradict any established physical observations but instead resolves century-old controversies by providing a unified, mathematical and paradox-free explanation of key phenomena. Furthermore, its groundbreaking results offer a unified perspective across various distinct laws of physics, including: A formula for light velocity mathematically explains the Sagnac effect, one-way light speed, stellar aberration, Michelson-Morley, Hafele-Keating and Optic clocks. Electrodynamics formulas encompass particle acceleration, Mass-Energy and matter wave. A unified formula for all cases of classical and transverse Doppler effects covers time-varying Doppler effect, cosmological redshift, and Cherenkov radiation. A generalized form of Maxwell's equations for moving observers directly derives the formulas for light speed, Doppler effect and Sagnac effect.