Partial Differential Equations (PDEs) constitute a fundamental framework for modeling various physical phenomena across diverse fields such as physics, engineering, and finance. Solving PDEs accurately and efficiently remains a significant challenge, particularly when dealing with complex systems or high-dimensional data. In this paper, we present a novel approach based on deep learning techniques to investigate the effective factors in solving PDEs. Our methodology leverages the expressive power of deep neural networks to learn and represent the underlying relationships between input parameters and the solutions of PDEs. By incorporating domain knowledge into the network architecture and training process, our approach aims to provide insights into the key factors influencing the solution of PDEs. We demonstrate the effectiveness of our method through numerical experiments on various PDEs, showcasing its capability to identify important features and improve predictive accuracy. Our findings suggest that deep learning techniques offer a promising avenue for understanding and analyzing the complex dynamics inherent in PDEs, paving the way for enhanced computational methods in scientific and engineering applications.

The Schwarzschild metric emerges independent of Einstein's field equations, offering a straightforward derivation solely reliant on Newtonian mechanics and Minkowskian proper acceleration. This approach provides a clear path to understanding the gravitational field around a spherically symmetric mass without the need for the complexities of Einstein's full theory of General Relativity. Transitioning to our exploration of the cosmic Friedmann equations, we adopt a novel perspective rooted in a Lagrangian formulation grounded in Newtonian mechanics and the first law of thermodynamics. Our investigation operates under the assumption that the universe is populated by either a perfect fluid or a scalar field. By elucidating the intricate interplay between the Lagrangian formulation and the cosmic Friedmann equations, we uncover the fundamental principles governing the universe's dynamics within the framework of these elemental constituents. In our concluding endeavor, we embark on the task of harmonizing the classical equations—namely, the conservation, Euler, and Poisson equations—with the principles of General Relativity. This undertaking seeks to extend these foundational equations to encompass the gravitational effects delineated by General Relativity, thus providing a comprehensive framework for understanding the behavior of matter and spacetime in the cosmic context.

We primarily investigate the existence of solutions for fractional neutral integro-differential equations subjected to Neumann-type boundary conditions, which is crucial for understanding natural phenomena. Taking into account factors such as neutral type, fractional-order integrals, and fractional-order derivatives, we employ probability density functions, Laplace transforms, and resolvent operators to formulate a well-defined concept of a mild solution for the specified equation. Following this, by integrating fixed point theorems, we establish the existence of mild solutions under more relaxed conditions.

This paper brings to fruition recently found \cite{migdal2023exact} exact reduction of decaying turbulence in the Navier-Stokes equation in $3 + 1$ dimensions to a Number Theory problem of finding the statistical limit of the Euler ensemble. We reformulate the Euler ensemble as a Markov chain and show the equivalence of this formulation to the \textbf{quantum} statistical theory of $N$ fermions on a ring, with an external field related to the random fractions of $\pi$. We find the solution of this system in the turbulent limit $N\to \infty, \nu \to 0$ in terms of a complex trajectory (instanton) providing a saddle point to the path integral over the density of these fermions. This results in an analytic formula for the observable correlation function of vorticity in wavevector space. This is a full solution of decaying turbulence from the first principle without approximations or fitted parameters. The energy spectrum decays as $k^{-\frac{7}{2}}$. We compute resulting integrals in \Mathematica{} and present effective index $n(t) = -t\partial_t \log E $ for the energy decay as a function of time Fig.\ref{fig::NPlot}. The asymptotic value of the effective index in energy decay $n(\infty) = \frac{5}{4}$, but the universal function $n(t)$ is neither constant nor linear.

This short paper briefly explores the interplay between quantum mechanics and gravity, drawing from the principle of gravitational redshift and bringing it into quantum mechanical frameworks. The exploration looks through quantum mechanical relations and equations such as Planck-Einstein relation, De-Broglie’s relation and the relativistic wave equations, and propagators, accounting for the presence of gravitational fields through gravitational redshift of its momentum.

A cosmological model described by the perturbation metric is proposed, in which space is entrained by matter expanding as a result of the Big Bang. In the absence of an input of external energy and momentum, the solution of the Einstein equations assumes a pressure equal to zero and an accelerated expansion. The cosmological perturbation model explains the difference in measurements of the Hubble constant using gravitational lensing and a distance ladder.

This article deals with special curves similar to those known in mathematics as Lamé curves. Their equations have a certain similarity with the equations of Lamé curves. But these curves are more like ellipses in appearance. Like ellipses, they have major and minor axes. These are such curves that their major and minor semi-axis coincide with an ellipse but are different than an ellipse. The main feature of these curves is that for any triangle whose base coincides with the axis of a given curve and the vertex is on a given curve, the condition aⁿ+bⁿ=cⁿ is obtained. The number “n” is called the degree or rank of this curve. To distinguish this group of curves, we called them Elbe curves. The equations of these lines are also provided in the paper.

This article describes an approach for optimising sensor-controlled systems through minimal intervention, utilising Fuzzy Linear Systems of Equations (FLSE). Staring with a generalised model of the system behaviour, incorporating an array of control units, environmental sensors, and an expert knowledge base. The described problems of detecting the level of intervention needed to change the system state to another is handled with the help of developed methods for solving the inverse problem for FLSE. By achieving minimal intervention, we ensure that system adjustments effective, economically optimal and non-intrusive. A MATLAB-based implementation is presented.

RNA transcripts play a crucial role as witnesses of gene expression health. Identifying disruptive short sequences in RNA transcription and regulation is essential for potentially treating diseases. Let’s delve into the mathematical intricacies of these sequences. We’ve previously devised a mathematical approach for defining a “healthy” sequence (10.14218/GE.2023.00079). This sequence is characterized by having at most four distinct nucleotides (denoted as nt≤4). It serves as the generator of a group denoted as fp. The desired properties of this sequence are as follows: fp should be close to a free group of rank nt−1, it must be aperiodic, fp should not have isolated singularities within its SL2(C) character variety (specifically within the corresponding Groebner basis). Now, let’s explore the concept of singularities. There are cubic surfaces associated with the character variety of a four-punctured sphere denoted as S24. When we encounter these singularities, we find ourselves dealing with some algebraic solutions of a dynamical second-order differential (and transcendental) equation known as the Painlevé VI equation (10.3390/dynamics4010001). In certain cases, S24 degenerates, in the sense that two punctures collapse, resulting in a “wild” dynamics governed by the Painlevé equations of an index lower than VI. In our paper, we provide examples of these fascinating mathematical structures within the context of miRNAs. Specifically, we find a clear relationship between decorated character varieties of Painlevé equations and the character variety calculated from the seed of oncomirs. These findings should find applications in cancer research.

Based on recent progress in cosmological thermodynamics, we present a new thermody- namic formulation of the Friedmann equations in the new Haug-Tatum cosmology model, as well as for other similar models put forward in the literature from different solutions to Einstein’s field equation. Since the CMB temperature has been measured much more accu- rately than the Hubble constant, we demonstrate that our thermodynamic formulation of the Friedmann equation dramatically narrows the acceptable range for the critical density of the universe. Furthermore, we show how our thermodynamic formulation leads to a more precise cosmological constant estimate.

Euler-Poisson equations of a charged symmetrical body in external constant and homogeneous electric and magnetic fields are deduced starting from the variational problem, where the body is considered as a system of charged point particles subject to holonomic constraints. The final equations are written for the center-of mass-coordinate, rotation matrix and angular velocity. General solution to the equations of motion is obtained for the case of a charged ball. For the case of a symmetrical charged body (solenoid), the task of obtaining the general solution is reduced to the problem of a one-dimensional cubic pseudo-oscillator. Besides, we present a one-parametric family of solutions to the problem in elementary functions.

This paper presents a thorough investigation into the shear strength capacity of reinforced concrete deep beams, with a focus on improving predictive accuracy beyond existing code provisions. Analyzing 198 deep beams from 15 investigations, the study considers parameters such as concrete compressive strength (fc’), shear span to effective depth ratio (av/d), and reinforcement ratios (ρs, ρv, ρh ). Introducing a novel predictive model (Equation 7), the study rigorously evaluates it using nonlinear regression analysis and statistical metrics (MAE, RMSE, R2). The proposed model demonstrates a significant reduction in the coefficient of variation (COV) to 27.08%, surpassing existing codes' limitations. Comparative analyses highlight the model's robustness, revealing improved convergence of data points and minimal sensitivity to variations in key parameters. The findings suggest that the proposed model offers enhanced predictive accuracy across diverse scenarios, making it a valuable tool for structural engineers. This research contributes to advancing the understanding of shear strength in reinforced concrete deep beams, offering a reliable and versatile predictive model with implications for refining design methodologies and enhancing the safety and efficiency of structural systems.

The susceptible-infected-recovered-vaccinated-deceased (SIRVD) epidemic compartment model extends the SIR model to include the effects of vaccination campaigns and time-dependent fatality rates on epidemic outbreaks. It encompasses the SIR, SIRV, SIRD, and SI models as special cases, with individual time-dependent rates governing transitions between different fractions. We investigate a special class of exact solutions and accurate analytical approximations for the SIRVD and SIRD compartment models. While the SIRVD and SIRD equations pose complex integro-differential equations for the rate of new infections and the fractions as a function of time, a simpler approach considers determining equations for the sum of ratios for given variations. This approach enables us to derive fully exact analytical solutions for the SIRVD and SIRD models. For nonlinear models with a high-dimensional parameter space, such as the SIRVD and SIRD models, analytical solutions, exact or accurately approximative, are of high importance and interest: not only as suitable benchmarks for numerical codes, but especially as they allow us to understand the critical behavior of epidemic outbursts as well as the decisive role of certain parameters. In the second part of our study, we apply a recently developed analytical approximation for the SIR and SIRV models to the more general SIRVD model. This approximation offers accurate analytical expressions for epidemic quantities, such as the rate of new infections and the fraction of infected persons, particularly when the cumulative fraction of infections is small. The distinction between recovered and deceased individuals in the SIRVD model affects the calculation of the death rate, which is proportional to the infected fraction in the SIRVD/SIRD cases but often proportional to the rate of new infections in many SIR models using a posteriori approach. We demonstrate that the temporal dependence of the infected fraction and the rate of new infections differs when considering the effects of vaccinations and when the real-time dependence of fatality and recovery rates diverge. These differences are highlighted for stationary ratios and gradually decreasing fatality rates. \Revised{The case of stationary ratios allows one to construct a new powerful diagnostics method to extract analytically all SIRVD model parameters from measured COVID-19 data of a completed pandemic wave.

In this study, we investigate Quantum Entanglement Asymmetry (QEA) as a potential explanation for the matter-antimatter imbalance observed in the universe. By integrating quantum mechanics with cosmological phenomena, the study proposes a model in which quantum entanglement among particle-antiparticle pairs in the early universe leads to an asymmetry favoring matter. Through theoretical development and numerical simulations, we explore how this model predicts unique patterns in Cosmic Microwave Background (CMB) anisotropies and the distribution of matter, setting a foundation for empirical verification. The study aims to enhance the understanding of cosmological asymmetries and highlight the significance of quantum entanglement in the universe's evolution.

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.

o estudo visa explicar os efeitos diferentes entre as equações de multiplicação, soma, divisão e subtração. Nesse sentido, as equações de multiplicação e soma desenvolvem um novo elemento ou fato, enquanto isso, as equações de divisão expressar informações dentro do elemento ou fato, a subtração é a seleção de elemento ou fato Alem disso, para uma equação de multiplicação e soma resulta no novo elemento ou fato é necessario dependência de conceitos físicos específicos, sem alteração extrema, pois resultaria na alteração do elemento ou fato. vale ressaltar, a importância de valores fixos na equação para descreve a natureza específica do elemento ou fato. O estudo possibilita o entendimento das equações em aberta(sem solução) na matematica e fisica, além de compreender equações existentes. O estudo faz referência à teoria da necessidade obrigatória e teoria da diferenças entre elementos.

Our starting point is the covariant coordinate transformation equation of a relativistic positioning system in Minkowski space-time, which maps the receiver’s emission coordinates (proper times broadcast by the emitters) to its coordinates in some inertial reference frame. Bancroft’s analytical (closed-form) solution to the basic pseudorange navigation equations with four emitters is recovered and the subjacent elements are geometrically interpreted. The case of four static beacons is analized as a clarifying situation.

Use of MS Excel as a platform for teaching simple iterative solution of partial differential equations runs into issues of reliability of the computation because of the large number of calculations required. This article documents the problem and an empirical solution. The cause of this unreliability was found by detailed examination of the numerical results from three simultaneous iterations. Computers of several types from different manufacturers were found to fail occasionally in updating variables in the numerical iteration. A simple work-around is suggested and was tested on several platforms.

The main purpose of this article is to consider the smoothness control of a weak solution after some moment if there is known solution regularity until this moment. The necessary tools can be varied. It is possible to control kinetic energy dissipation to fix moment or changing of velocity square or summability of acceleration square to fix at point time.

This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. To illustrate our main findings, we present two numerical algorithms and examples in this paper.

This study presents a novel approach for quantifying the dynamics of monthly hydraulic geometry relationships in the Jingjiang reach of the Yangtze River by integrating leaf area index (LAI) data as an indicator of riparian vegetation. A comprehensive system of ordinary differential equations (ODEs) couples traditional power-law hydraulic geometry equations with LAI, capturing the influence of erosion control on channel morphology. The periodic delayed response theory and automatic differentiation techniques are employed to enhance the modeling framework's capabilities. Calibration and verification demonstrate the models' effectiveness in capturing overall trends in discharge, width, and depth dynamics, outperforming traditional constant-parameter approaches. However, model predictive abilities are limited under extreme climate conditions due to training on normal climate data. Key assumptions, including the periodic delayed response theory's neglect of peak discharges, simplified representation of artificial cut-offs, and reliance on remotely sensed data with limited local validation, introduce uncertainties. The proposed modeling framework shows potential for broader application to other river systems with appropriate adjustments. Future research could incorporate additional environmental factors, integrate models across scales, couple with advanced simulation techniques, and expand stability assessments beyond the riverbed index. By addressing related challenges, this integrated approach lays a foundation for enhancing understanding of river morphodynamics and channel stability.

This study focuses on the optimal deployment problem, and determines the optimal size of a military force to send to the battle field. The decision is optimized, based on an objective function, that considers the cost of deployment, the cost of the time it takes to win the battle, and the costs of killed and wounded soldiers with equipment. The cost of deployment is modeled as an explicit function of the number of deployed troops and the value of a victory with access to a free territory, is modeled as a function of the length of the time it takes to win the battle. The cost of lost troops and equipment, is a function of the size of the reduction of these lives and resources. An objective function, based on these values and costs, is optimized, under different parameter assumptions. The battle dynamics is modeled via the Lanchester differential equation system based on the principles of directed fire. First, the deterministic problem is solved analytically, via derivations and comparative statics analysis. General mathematical results are reported, including the directions of changes of the optimal deployment decisions, under the influence of alternative types of parameter changes. Then, the first order optimum condition from the analytical model, in combination with numerically specified parameter values, is used to determine optimal values of the levels of deployment in different situations. A concrete numerical case, based on documented facts from the Battle of Iwo Jima, during WW II, is analyzed, and the optimal US deployment decisions are determined under different assumptions. The known attrition coefficients of both armies, from USA and Japan, and the initial size of the Japanese force, are parameters. The analysis is also based on some parameters without empirical documentation, that are necessary to include to make optimization possible. These parameter values are motivated in the text. The optimal solutions are found via Newton- Raphson iteration. Finally, a stochastic version of the optimal deployment problem is defined. The attrition parameters are considered as stochastic, before the deployment decisions have been made. The attrition parameters of the two armies have the same expected values as in the deterministic analysis, are independent of each other, have correlation zero, and have relative standard deviations of 20%. All possible deployment decisions, with 5000 units intervals, from 0 to 150000 troops, are investigated, and the optimal decisions are selected. The analytical, and the two numerical, methods, all show that the optimal deployment level is a decreasing function of the marginal deployment cost, an increasing function of the marginal cost of the time to win the battle, an increasing function of the marginal cost of killed and wounded soldiers and lost equipment, an increasing function of the initial size of the opposing army, an increasing function of the efficiency of the soldiers in the opposing army and a decreasing function of the efficiency of the soldiers in the deployed army. With stochastic attrition parameters, the stochastic model also shows that the probability to win the battle is an increasing function of the size of the deployed army. When the optimal deployment level is selected, the probability of a victory is usually less than 100%, since it would be too expensive to guarantee a victory with 100% probability. Some of many results of relevance to the Battle of Iwo Jima, are the following: In the deterministic Case 0 analysis, the optimal US deployment level is 66200, the time to win the battle is 30 days and 14000 US soldiers are killed or wounded. If the marginal cost of the time it takes to win a victory doubles, the optimal deployment increases to 75400, the time to win a victory is reduced to 26 days, and less than 12000 soldiers are killed or wounded. In the stochastic Case 0 analysis, the optimal US deployment level is 65000, the expected time to win the battle is 46 days and almost 25000 US soldiers are expected to be killed or wounded. If the cost per killed or wounded soldier increases from 0 to 5 M $US, the optimal deployment level increases to 75000. Then, the victory is expected to appear after 35 days and 19900 US soldiers are expected to be killed or wounded.

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.

The hard open problem of finding the lower bound for the Fokker Planck Kolmogorov’s time-dependent controller parameter probability density function is addressed for the first time in literature in this work. This revolutionary exposition will put control theory and other related inter-disciplinary fields to a higher level towards contemporary control theory. Notably, based on the influential role of control theory in both engineering and industry, this paper will be of great value to all engineering and industry professionals who seek to know more about advanced trends within control theory settings. On the other remit of the spectrum, Fokker Planck Kolmogorov(FPK) equations are of high importance to physicists as well as mathematicians, based on their multiple applicability to multi-interdisciplinary fields of human knowledge, and beyond. So, this by default adds more taste and credibility to this study. Concluding thoughts are included at the end of the work along with a difficult outstanding problem and the direction of future work.

This work is the first in literature to tackle the difficult open problem of determining the upper bound and threshold theorem for the TDCDP (time-dependent controller parameter) of the (Fokker Planck Kolmogorov) probability density function. This revolutionary exposition will put control theory and other related inter-disciplinary fields to a higher level towards contemporary control theory. Notably, based on the influential role of control theory in both engineering and industry, this paper will be of great value to all engineering and industry professionals who seek to know more about advanced trends within control theory settings. On the other remit of the spectrum, Fokker Planck Kolmogorov(FPK) equations are of high importance to physicists as well as mathematicians, based on their multiple applicability to information theory, graph theory, data science, finance, economics, and beyond. So, this by default adds more taste and credibility to this study. This leads by nature to introducing a different flavor to this ground-breaking research by highlighting the impact of Fokker Planck Kolmogorov(FPK) to revolutionize crypocurrency,which have received its name because it uses encryption to verify transactions, a new debatable digital payment system that doesn't rely on banks to verify transactions. It’s a peer-to-peer system that can enable anyone anywhere to send and receive payments. The paper ends with closing remarks combined with some challenging open problems and the next phase of research.

Polymer electrolyte was used as a medium for testing the performance of microband electrodes under conditions of linear diffusion. Cyclic voltammetry and chronoamperometry experiments were performed in such viscous medium, where diffusion rates are much slower than in fluid solutions. Confirmation of the existence of such conditions from the log i vs. log v or log i vs. log t relationships in the current equation always yielded values lower than the expected value of 0.5. This could indicate impure linear diffusion profile, i. e. some contribution from radial diffusion (edge effect). However, performing these tests in monomeric solvents of similar viscosities, such as glycerol or propylene glycol, gave the desired value of 0.5. These results lead to the conclusion that current equations which are based on Fick’s laws may not be applicable for polymer electrolytes, where various obstruction to diffusion make it a more complicated process than in monomeric solvents.

This study aims to reveal the revolutionary Gamma Distribution Manifold (GDM) info-geometric analysis combined with potential Gamma distribution (GD) applications to advance Satellite Earth Observations (SEOs). An exposition of the GDM Fisher Information Matrix (FIM) was undertaken, together with the analytic investigation of the existence of FIM’s inverse. The derivation of GDM’s Geodesic Equations (GEs) is undertaken. The solution of the obtained Geodesic Equations is optimally complicated, and it provides more extensions to numerous open problems. Notably, the current paper provides a first-time unification between three multi-interdisciplinary fields, namely, information geometry theory, probability theory, and Satellite earth observations. Looking at the other side of the coin, the provided revolutionary approach in this current work reports the significant impact of info-geometric analysis on the probabilistic dynamics as well as the influential role of information geometry on satellite and space industry. This poses numerous open issues that initiates a new form of contemporary unification between these inter-disciplinaries to discover another visionary thinking out of traditional statistical frames. In other words, refraining ourselves from using classical statistical frames of thinking and delving into a new info-geometric vision of statistical distributions as well as examining real life applications of information geometry to advance satellite earth observations. The curtain is drawn on this paper by providing several challenging open problems in combination with a concluding remark and the next phase of research.

The realization of a BRST cohomology of the 4D topological gauge-affine gravity is established in terms of a superconnection formalism. Identification of fields in the quantized theory is occurred directly as usual in terms of superconnection and its supercurvature components with the double covering of the general affine group GA¯(4,R). Then, by means of an appropriate decomposition of the metalinear double-covering group SL¯(5,R) with respect to the general linear double-covering group GL¯(4,R) one can easily obtain the enlargements of the fields while remaining consistent with the BRST algebra. This leads to the descent equations allowing us to build the observables of the theory by means of the BRST algebra constructed using a sa¯(5,R) algebra-valued superconnection. In particular, we discuss the construction of topological invariants with torsion.

Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics and others. Due to its application in control theory, the matrix equation AXB=C has been extensively studied. However, there is currently limited information on matrix equation AXB=C over the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of the dual quaternion matrix equation AXB=C, and present the expression for the general solution when it is solvable. As an application, we derive the ϕ-Hermitian solutions for the dual quaternion matrix equation AXAϕ=C, where the ϕ-Hermitian extends the concepts of Hermiticity and η-Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper.

The goal of the current study is to analyse several nonlinear two-dimensional time-fractional Rosenau Hymanequations. The two-dimensional fractional Rosenau-Hyman equation has extensive use in engineering and applied sciences. The fractional view analysis of two-dimensional time-fractional Rosenau-Hyman equations is discussed using the homotopy perturbation approach, adomian decomposition method, and Yang transformation. Some examples involving two-dimensional time-fractional Rosenau-Hyman equations are provided in order to better understand the suggested approaches. The solutions appear as infinite series. We offer a comparison between the accurate solutions and those that are generated employing the proposed approaches in order to demonstrate the effectiveness and applicability of the proposed techniques. The results are graphically illustrated using 2D and 3D graphs. It has been noted that the obtained results and the targeted problems real solutions are quite similar. Calculated solutions at various fractional levels describe some of the problems useful dynamics. Other fractional problems that arise in other fields of science and engineering can be solved using a modified version of the current techniques.

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.

Monge-Ampere equations have important research significance in many fields such as geometry, convex geometry and mathematical physics. In this paper, under some superlinear and sublinear conditions, the existence of nontrivial solutions for a system arising from Monge-Ampere equations with two parameters is investigated based on Guo-Krasnosel'skii fixed point theorem. In the end, two examples are given to illustrate our main results.

In this paper we address the approximation of the coupling problem for the wave equation and Maxwell's equations of electromagnetism in the time domain in terms of the electric field, by means of a nodal linear finite-element discretization in space, combined with a classical'explicit finite-difference scheme for the time discretization. Our study applies to the particular case where the dielectric permittivity has a constant value outside a sub-domain, whose closure does not intersect the boundary of the domain where the problem is defined. Inside this sub-domain Maxwell's equations hold. Outside this sub-domain the wave equation holds, which may correspond to the Maxwell's equations with a constant permittivity under certain conditions. We consider as a model the case of first order absorbing boundary conditions. Optimal error estimates that hold in natural norms under reasonable assumptions are given, among which lies a typical CFL condition for hyperbolic equations.

This paper discusses how mathematical models are used to predict environmental conditions. While these models are successful for larger scale phenomena, it becomes more complex for local weather or wave processes. Several approaches have been proposed to address this problem, including neural networks and statistical and mathematical models. The novelty in this approach is the use of statistical manifolds, which provide a more accurate way to estimate distances and cost functions. This work focuses on advancements in differential and information geometry for creating different types of models that can be used for environmental simulation and forecasting. These fields are vital for many institutes and centres worldwide, as they help solve challenges in areas such as meteorology, renewable energy, and aviation safety. Some potential IG applications to advance energy were given. Closing remarks with new open problems combined with the next research phase are provided.

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.

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 .

Information geometry (IG) is a fascinating combination of differential geometry and statistics where a Riemannian manifold's structure is applied to a statistical model. It may find numerous fascinating uses in the domains of theoretical neurology, machine learning, complexity, and (quantum) information theory, among others. (IG) aims to provide a differential-geometric perspective on statistical geodesic models' structure. In this case, IG, KD, and J-divergence (JD) are used to define the manifold of the Generalised Brownian Motion (GBM). Consequently, the geodesic equations (GEs) are devised, and GB information matrix exponential (IME) is presented. Moreover, for first time ever, the necessary and sufficient mathematical requirement that characterizes the developability of Generalized Brownian Motion(GBM) manifold is devised. Also, a novel sufficient and necessary conditions which characterizes the regions where the surface describing GBM is minimal is determined. Also, it is shown that GBM has a non-zero 0-Gaussian and Ricci curvatures. Consequently, this advances the establishment of a unified GBM- Relativistic Info-Geometric based analysis.

In the evolving landscape of vibration analysis, the integration of machine learning technologies for semi-automated analytical processes presents challenges in establishing clear cause-and-effect relationships in decision-making formulas. This research addresses this gap by exploring analytical diagnosis through residual signals, enabling the identification of anomalies and underperformance in machinery. Vibration signature analysis, focusing on distinctive vibration patterns, proves instrumental in detecting bearing defects and optimizing maintenance for enhanced operational efficiency and safety. Mathematical modeling, particularly worst-case analysis, provides a structured framework for decision-making under uncertainty. The research employs descriptive methods to analyze bearing wear data, transforming it into quantitative data for multiple regression equations. The derived equations reveal the dominance of harmonics in determining bearing wear stages, with PeakVue and overall vibration further indicating damage progression. The findings emphasize the significance of harmonic detection and high-frequency vibration analysis in predicting bearing health, offering a comprehensive approach to enhance machinery reliability. The study concludes with practical recommendations for bearing damage diagnosis based on the developed mathematical models, providing valuable insights for predictive maintenance strategies.

Coupling the Maxwell tensor to the Riemann-Christoffel curvature tensor is shown to lead to a geometricized theory of electrodynamics. While this geometricized theory leads directly to the classical Maxwell equations, it also extends their physical interpretation by giving charge density, mass density and the four-velocity that describes their motions geometric definitions. Assuming the existence of a conserved energy-momentum tensor, all solutions to the geometricized theory of electrodynamics developed here are shown to be consistent with the emergence of gravity obeying the General Relativity field equation augmented by a term that mimics the properties of dark matter and/or dark energy. Finally, due to the symmetries of the theory, the properties and phenomenology of antimatter emerge in solutions.

This paper presents a new method for parameterizing the Lorentz group based on coupled parameters. From the Euler–Hamilton equations, an additional angular rapidity and perpendicular rapidity are obtained, and the Hamiltonian and Lagrangian of a relativistic particle are expanded into rapidity spectra. A so-called passage to the limit is introduced that makes it possible to decompose physical quantities into spectra in terms of elementary functions when explicit decomposition is difficult. New rapidity-dependent Lorentz-invariant coordinates are obtained, and the descriptions of particle motion using the old and new Lorentz-invariant forms as applied to plane waves are compared. Based on a classical model of particle motion in the field of a plane monochromatic electromagnetic wave and in that of a plane laser pulse, the rapidity-dependent spectral decompositions into elementary functions are presented, and the Euler–Hamilton equations are derived as rapidity functions in 3+1 dimensions.

This article presents a concise survey of basic discrete and semi-discrete nonlinear models which produce two- and three-dimensional (2D and 3D) solitons, and a summary of main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross-Pitaevskii (GP) equations with the Lee-Huang-Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole-dipole and quadrupole-quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ⁽²⁾) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin-orbit-coupled (SOC) binary Bose-Einstein condensate, and others. The article presents a review of basic species of multidimensional discrete modes, including fundamental and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.

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.

This experimental research presents an easier model for fitting experimental results that can be simulate the dynamical behaviour of a single degree freedom system, SDFS, subjected in a forced vibration in a shaking table. The fractional damping using a Kelvin-Voigt fractional derivative rheological model, KVFRdm, to represent a fractional oscillator and compare with the experimental model.

To accurately predict the matching relationships between the various components and the engine performance in the whole aero-engine environment, this study introduces a two-dimensional throughflow simulation method for the whole aero-engine. This method is based on individual throughflow solvers for the turbo-machinery and the combustor. It establishes a throughflow simulation model for the whole engine by integrating with the compressor-turbine co-operating equations and boundary conditions. The turbo-machinery throughflow solver employs a circumferentially averaged form of the time-dependent Navier-Stokes equations (N-S) as the governing equation. The combustor solver uses the Reynolds Average Navier-Stokes (RANS) method to solve flow and chemical reaction processes by constructing turbulence, combustion, and radiation models. The accuracy of the component solver is validated using Pratt & Whitney’s three-stage axial compressor (P&W3S1) and General Electric’s high-pressure turbine (GE-EEE HPT), and the predicted results are consistent with the experimental data. Finally, the developed throughflow method is applied to simulate the throttling characteristics of the WZ-X turboshaft engine. The results predicted by the throughflow program are consistent with the GasTurb calculations, including the trends of shaft power delivered, specific fuel consumption (SFC), inlet airflow, and total pressure ratio of the compressor. The developed method to perform throughflow simulation of the whole aero-engine eliminates the dependence on a general component map. It can quickly obtain the meridian flow field parameters and overall engine characteristics, which is expected to guide the design and modification of the engine in the future.

Unsupervised physics-informed deep learning can be used to solve computational physics problems by training neural networks to satisfy the underlying equations and boundary conditions without labeled data. Parameters such as network architecture and training method determine the training success. However, the best choice is unknown a priori as it is case specific. Here, we investigated network shapes, sizes, and types for unsupervised physics-informed deep learning of the two-dimensional Reynolds-averaged flow around cylinders. We trained mixed-variable networks and compared them to traditional models. Several network architectures with different shape factors and sizes were evaluated. The models were trained to solve the Reynolds-averaged Navier-Stokes equations incorporating Prandtl's mixing length turbulence model. No training data were deployed to train the models. The superiority of the mixed-variable approach was confirmed for the investigated high Reynolds number flow. The mixed-variable models were sensitive to the network shape. For the two cylinders, differently deep networks showed superior performance. The best fitting models were able to capture important flow phenomena such as stagnation regions, boundary layers, flow separation, and recirculation. We also encountered difficulties when predicting high Reynolds number flows without training data

This paper provides the numerical investigation of two-dimensional stochastic Navier-Stokes equation driven by additive white noises. The vorticity-stream function method is presented to calculate the incompressible fluid flow. In order to solve the formulation of steam function equation, we propose a Crank-Nicolson Fourier pseudo-spectral method. Numerical results obviously show that the presence of stochastic forcing can have an impact upon the shapes of vortex in fluid flow. To a certain extent, the phenomenon of eddy shedding and the formation of some new vortex in a random medium can be observed, which make it possible for us to conduct optimal control experiments in the development of vortex structures.

Multi-parameter families of Lax pairs for the modified Korteweg-de Vries (mKdV) equation are defined by applying a direct method developed in the present study. The gauge transformations converting the defined Lax pairs to some simpler forms and reducing the variety of the Lax pairs this way are found. The direct method and its possible applications to other types of evolution equations are discussed.

This is the second paper in a cycle investigating the exact solution of loop equations in decaying turbulence. We perform numerical simulations of the Euler ensemble, suggested in the previous work, as a solution to the loop equations. We designed novel algorithms for simulation, which take a small amount of computer RAM so that only the CPU time grows linearly with the size of the system and its statistics. This algorithm allows us to simulate systems with billions of discontinuity points on our fractal curve, dual to the decaying turbulence. For the vorticity correlation function, we obtain \textbf{quantum fractal laws} with regime changes between universal discrete values of indexes. The traditional description of turbulence with fractal or multifractal scaling laws does not apply here. The measured conditional probabilities of fluctuating variables are smooth functions of the logarithm of scale, with statistical errors being negligible at our large number of random samples $T = 167,772,160$ with $N = 200,000,000$ points on a fractal curve. The quantum jumps arise from the analytical distribution of scaling variable $X = \frac{\cot^2(\pi p/q)}{q^2}$ where $\frac{p}{q}$ is a random fraction. This distribution is related to the totient summatory function and is a discontinuous function of $X$. In particular, the energy spectrum is a devil's staircase with tilted uneven steps with slopes between $-1$ and $-2.03$. The logarithmic derivative of energy decay $n(t)$ as a function of time is jumping down the stairs of universal levels between $1.50$ and $1.53$. The quantitative verification of this quantization would require more precise experimental data.

Real and complex analytic functions are largely studied in the field of complex analysis and are seen as very useful tools in solving problems in mathematics, physics and engineering. Real and complex numbers are a subset of semi-structured complex numbers (a new number set created to algebraically solve division by zero). Nevertheless, the properties of analytic functions made up of semi-structured complex variables (called semi-structured complex analytic functions) is yet to be explored. This limits the range of possible problems that can be resolved using analytic functions). In this regard, the aim of this paper was to expound upon the properties of semi-structured complex analytic functions and show their application in solving engineering problems. The results of this paper included (1) developing a full set of Cauchy–Riemann Equations for the semi-structured complex -space; (2) define sufficient and necessary conditions for a semi-structured complex function to be analytic; (3) determine the relationship between semi-structured complex analytic functions, Laplace’s Equations and Poisson’s Equations; and (5) provide an example of the use of these functions.

Antiangiogenic therapy (AAT) is an indirect oncological modality which is aimed at disruption of cancer cells nutrient supply. Invasive tumors have been shown to possess inherent resistant to this treatment, while compactly growing benign tumors react to it by shrinkage. It is generally accepted that AAT by itself is not curative. This study presents a mathematical model of non-invasive tumor growth with physiologically justified account of alteration of microvasculature and biomechanical aspects during tumor growth and AAT. In untreated setting the model reproduces tumor growth with saturation, where the maximum tumor volume depends on the level of angiogenesis. The outcomes of simulations of AAT depend on the tumor size at the moment of treatment initiation. If it is close to the stable size of avascular tumor grown in absence of angiogenesis, then the tumor is rapidly stabilized by AAT. Treatment of large tumors is accompanied by displacement of normal tissue due to the tumor shrinkage. During it, microvasculature undergoes distortion which degree depends on the displacement distance. As it affects tumor nutrient supply, the stable size of a tumor that undergoes AAT negatively correlates with its size at the beginning of treatment. For sufficiently large initial tumors, the long-term survival of tumor cells is compromised by competition with normal cells for severely limited inflow of nutrients, which makes AAT effectively curative.

The article proposes asymptotic methods for calculating Sommerfeld integrals, which enable us to calculate the integral using the expansion of a function into an infinite power series at the saddle point, where the role of a rapidly oscillating function under the integral can be fulfilled either by an exponential or by its product by the Hankel function. The proposed types of Sommerfeld integrals are generalized on the basis of integral representations of the Hertz radiator fields in the form of the inverse Hankel transform with the subsequent replacement of the Bessel function by the Hankel function. It is shown that the numerical values of the saddle point are complex. During integration, auxiliary or so-called standard integrals, which contain the main features of the integrand function, were used. As a demonstration of the accuracy of the technique, a previously known asymptotic formula for the Hankel functions was obtained in the form of an infinite series. The proposed method for calculating Sommerfeld integrals can be useful in solving the half-space Sommerfeld problem. An example is the expression in the form of an infinite series for the magnetic field of reflected waves, obtained directly through the Sommerfeld integral (SI).

We consider two abstract systems of parabolic-hyperbolic type that model thermoelastic problems. We study the influence of the physical constants and the initial data on the nonexistence of global solutions that in our framework are produced by the blow-up in finite time of the norm of the solution in the phase space. We employ a differential inequality to find sufficient conditions that produce the blow-up. To that end, we construct a set that is positive invariant for any positive value of the initial energy. As a result we found that the coupling with the parabolic equation stabilizes the system, as well as the damping term in the hyperbolic equation. Moreover, for any pair of positive values $(\xi,\epsilon)$, there exist initial data such that the corresponding solution with initial energy $\xi$ blows-up at a finite time less than $\epsilon$. Our purpose is to improve results previously published in the literature.

The work is devoted to the development of methods for constructing asymptotic formulas as x→∞ of a fundamental system of solutions of linear differential equations generated by a symmetric two-term differential expression of odd order. The coefficients of the differential expression belong to classes of functions that allow oscillation (for example, those that do not satisfy the classical Titchmarsh-Levitan regularity conditions). As a model equation, the 5th order equation i2p(x)y′′′′′+p(x)y′′′′′+q(x)y=λy, for which various cases of behavior of the coefficients p(x),q(x), is investigated. New asymptotic formulas are obtained for the case when the function h(x)=−1+p−1/2(x)∉L1[1,∞) significantly influences the asymptotics of solutions to the equation. The case when the equation contains a nontrivial bifurcation parameter is studied.

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.

Exercising surds to represent figures is a common practice in scientific and Engineering fields, especially in scripts where calculators are banned or unapproachable. Peter Chew Theorem make result becomes simple when dealing with converting Quadratic Surds. The substance of the Peter Chew Theorem lies in enabling the forthcoming generation to simple break problems related to Quadratic Surds more effectively, easing a direct comparison with contemporary results. By employing the Peter Chew Theorem, one can streamline the tutoring and literacy of math, particularly concerning second- order direct equations with constant portions. This theorem's objective aligns with Albert Einstein's famed quotation Everything should be made as simple as possible, but not simpler.

Empirical equations representing, interpolating and smoothing groups of measurement results by regression methods are in widespread use in metrology and other fields of science and engineering. Standard equations for the propagation of measurement uncertainties to values computed from such equations are available but suffer from a lack of general acceptance and are only infrequently applied in practice. One reason for the slow uptake is the lack of clear methods to account for systematic error. In this paper, uncertainty propagation equations in terms of covariance matrices are generalized to allow for systematic errors in least-squares regression, and effects of the resulting uncertainties are investigated analytically. A stochastic ensemble model of systematic error is proposed for the computation of the non-diagonal elements of the weight matrix of the Generalized Least-Squares method (GLS) from the measurement uncertainty. A GLS projector formalism is described which separates the effects of measurement scatter on values calculated by the equation from those on the related “residuals”, i.e., the residual errors in the fitted data. The same projectors act similarly on the associated uncertainties and covariance matrices and permit the effects of systematic errors on the simulation covariance to be quantified. It is demonstrated that, in order to include systematic errors in the uncertainty estimates of GLS, covariance matrices may be substituted by novel, specifically defined dispersion matrices that are not specified yet by the GUM[1]. Systematic measurement errors may be estimated from various sources; a particularly easy way suggested here involves analyzing the structure in the regression residuals. It is demonstrated that GLS projectors exhibit inherent features of “error blindness” also with respect to systematic errors. [1] GUM: Guide to the Expression of Uncertainty in Measurement, http://www.bipm.org/en/publications/guides

New possibilities of Gramian computation by using canonical transformations into diagonal, controllable and observable canonical forms are shown. With the help of such a technique the Gramian matrices can be represented in the form of products of Hadamard matrices of multipliers and matrices of the transformed right-hand side of Lyapunov equations. It is shown that the multiplier matrices are invariant under various canonical transformations of linear continuous systems. The modal Lyapunov equations for continuous SISO LTI systems in diagonal form are obtained and their new solutions based on Hadamard decomposition are proposed. New algorithms for element-by-element computation of Gramian matrices for stable continuous MIMO LTI systems are developed. For continuous SISO LTI systems given by equations of state in controllable and observable canonical forms, new algorithms for the computation of controllability Gramians and their traces in the form of Hadamard products in the form of Xiao matrices are developed. The application of transformations to the canonical forms of controllability and observability allowed to simplify the formulas of spectral decompositions in the form of Xiao matrices. In the paper new spectral decompositions in the form of Hadamard’s products for solutions of Sylvester algebraic and differential equations of MIMO LTI systems, including spectral decompositions of finite and infinite cross-Gramians of continuous MIMO LTI systems. Recommendations on the use of the obtained results are given.

A system of simultaneous multi-variable nonlinear equations can be solved by the Newton’s method with local q-quadratic convergence if the Jacobian is analytically available. If this is not the case, then quasi-Newton methods with local q-superlinear convergence give solutions by approximating the Jacobian in some way. Unfortunately, the quasi-Newton condition (secant equation) doesn’t completely specify the Jacobian approximate in multi-dimensional case, so its full-rank update is not possible with classic variants of methods. The suggested new iteration strategy (“T-Secant”) allows full-rank update of the Jacobian approximate in each iteration by determining two independent approximates for the solution. They are used to generate a set of new independent trial approximates, then the Jacobian approximate can fully be updated. It is shown, that the T-Secant approximate is in the vicinity of the classic quasi-Newton approximate, providing that the solution is evenly surrounded by the new trial approximates. The suggested procedure increases the super-linear convergence of the secant method (φS = 1.618...) to super-quadratic (φT = φS + 1 = 2.618...) and the quadratic convergence of the Newton-method (φN = 2) to cubic (φT = φN + 1 = 3) in one dimensional case. The Broyden-type efficiency (mean convergence rate) of the suggested method in multi-dimensional case is an order higher than the efficiency of other classic low-rank update quasi-Newton methods as shown by numerical examples on a Rosenbrock-type test-function with up to 1000 variables. The geometrical representation (hyperbolic approximation) in single variable case helps explaining the basic operations and a vector-space description is also given in multi-variable case.

The article is devoted to the analytical and numerical study of the pattern of propagation and attenuation, due to Coulomb friction, of shear waves in an infinite thin elastic plate with a circular orifice of radius lying on a rough base. In the field of motion, an exact analytical solution of a nonlinear boundary value problem for tangential stresses and transversal velocities is obtained in quadratures by the method of Laplace transformations. It turned out that the complete exhaustion of the wave front of a strong rupture occurs at a finite distance from the center of the hole and an elementary formula is given for this distance (the case of tangential forces instantly applied to the orifice boundary, and then constant in time, is considered). For various ratios of the magnitude of the limiting friction force to the amplitude of the applied load, the trailing wave fronts are obtained, after which a state of static equilibrium between the elastic and friction forces with a nonlinear distribution of residual deformations is established in the region .

We present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection-reaction-diffusion equations on surfaces. The temporal direction adopted backward-Euler method, while the spatial direction employed surface finite element method. In contrast to regular domains, it is observed that the point in the characteristic direction traverses the surface only once within a brief time. Thus, the good approximation of the solution in the characteristic direction holds significant importance for the numerical scheme. In this regard, Taylor expansion is employed to reconstruct the solution beyond the surface in the characteristic direction. The stability of our scheme is then proved. A comparison is made with an existing characteristic finite element method based on face mesh. Numerical examples are provided to validate the effectiveness of our proposed method.

Ambient vibration energy is widely being harnessed as a source of electrical energy to drive low-power devices. The vibration energy harvester (VEH) of interest employs an electromagnetic transduction mechanism, whereby ambient mechanical vibration is converted to electrical energy. The limitations affecting the performance of VEHs, with an electromagnetic transduction structure, include its operational bandwidth as well as the enclosure-size constraint. In this study, an analysis and design of a nonlinear VEH system is conducted, using the Output Frequency Response Function (OFRF) representations of the actual system model. However, the OFRF representations are determined from the Generalised Associated Linear Equation (GALE) decompositions of the system of interest. The effect of both nonlinear damping and stiffness characteristics, to respectively extend the average power and operational bandwidth of the VEH device, is demonstrated.

This paper describes developing and analyzing an analytical model of Microelectromechanical Systems (MEMS) internal vibrating ring gyroscope. The internal ring structure consists of eight semicircular beams attached to the externally placed anchors. The paper analyses the vibrating ring gyroscope inplane displacement behavior and the resulting elliptical vibrational modes. The elliptical vibrational modes appear as pairs with the same resonance frequency due to the symmetric nature of the design. The analysis commences by conceptualizing the ring as a geometric structure with a circular shape, possessing specific dimensions such as thickness, height, and radius. We construct a linear model that characterizes the vibrational dynamics of the internal vibrating ring. The analysis encompasses comprehensive mathematical formulations for the radial and tangential displacements in local polar coordinates, considering the inextensional displacement of the ring structure. By utilizing the derived equations, we highlight the underlying relationships driving the vibrational characteristics of the MEMS vibrating ring gyroscope. These dynamic vibrational relationships are essential in enabling the vibrating ring gyroscope's future utilization in accurate navigation and motion sensing technologies.

Partial differential equations (PDEs) are used to describe a wide range of phenomena, such as heat transfer, fluid dynamics, and quantum mechanics. By solving PDEs, we can ob- tain insights into the behavior of the system and make predictions about its future evolution. Conventional numerical methods for obtaining the approximate solutions of PDEs may re- quire extensive computational resources and time, especially for complex PDEs and large- scale problems. The recently developed physics-informed neural network (PINN) has shown promise in a variety of scientific and engineering fields by incorporating physical laws into the loss functions of the neural network (NN). In addition, the training of PINN does not require ground truth data, but it has poor generalization ability to unseen domains. On the other hand, a convolutional neural network (CNN) has fast inference and better generalization ability, but it requires a large amount of training data. Taking the advantages of PINN and CNN by using Legendre multiwavelets (LMWs) as basis functions, we introduce a new method to approach the PDEs in this paper, namely Physics-Informed Legendre Multiwavelets CNN (PiLMWs-CNN), in order to continuously approximate a grid-based state representation that can be handled by a CNN. PiLMWs-CNN enable us to train our models using only physics-informed loss functions without any pre- computed training data, simultaneously providing fast and continuous solutions that gener- alize to previously unknown domains. In particular, the LMWs can simultaneously possess compact support, orthogonality, symmetry, high smoothness, and high approximation order. Compared to orthonormal polynomial (OP) bases, the approximation accuracy can be greatly increased and computation costs can be significantly reduced by using LMWs. We applied PiLMWs-CNN to approximate the damped wave equation, incompressible Navier-Stokes (N- S) equation, and two-dimensional heat conduction equation. The experimental results show that this method provides more accurate, efficient, and fast convergence with better stability when approximating the solution of PDEs.

Applications of gradient method for nonlinear optimization in development of Gradient Neural Network (GNN) and Zhang Neural Network (ZNN) are investigated. Particularly, the solution of the matrix equation AXB=D which changes over time is studied using the novel GNN model, termed as GGNN(A,B,D). The GGNN model is developed applying GNN dynamics on the gradient of the error matrix used in the development of the GNN model. The convergence analysis shows that the neural state matrix of the GGNN(A,B,D) design converges asymptotically to the solution of the matrix equation AXB=D, for any initial state matrix. It is also shown that the convergence result is the least square solution which is defined depending on the selected initial matrix. A hybridization of GGNN with analogous modification GZNN of the ZNN dynamics is considered. The Simulink implementation of presented GGNN models is carried out on the set of real matrices.

In this paper, existence and stability results of non-local integro-differential equation with Hilfer derivative of order ω∈(1,2) have been investigated. In this derivative is applied in RLC circuit equation of various fractional order. To proposed problem the existence solutions are derived using Schaefer’s fixed point theorem and the Banach contraction principle is using for uniqueness results. To demonstrate the effectiveness and applicability of our theoretical conclusions and two-step Lagrange polynomial interpolation were used to solve four numerical results.

The coupled Dirac equations for solar e- and μ-neutrinos are solved in the presence of a small uniform interaction of the e-neutrinos with matter (the Sun). It is shown that the interaction leads to an increase of the amplitude of the neutrinos oscillations and a decrease of the lo- calization probability of the e-neutrinos. However, the strength of this interaction is insufficient to explain the magnitude of the discrepancy between the solar neutrino flux and the neutrino flux measured on the Earth. An alternative interpretation related to a possible superfluid neutrino state is advanced, which may throw some light on the solar neutrino problem.

Recently, a paper was written to establish semi-structured complex numbers ℍ (a new number set invented to enable division by zero) as a useful mathematical tool providing novel results from its application in engineering and science. Whilst the paper was a milestone, the application of this number set to quantum and classical physics had not been fully explored. Consequently, the aim of this research was to use semi-structured complex numbers to develop a new mathematical model for the mass of an object and in course unite classical and quantum physics in a simple yet effective manner. As its major contributions this paper: (1) develops a new mathematical model for the mass of an object called the wave-mass function using De Broglie’s mass and frequency relation and semi-structured complex numbers; (2) uses the new wave-mass function to determine what happens to an object traveling at the speed of light; (3) derives a relation between Schrödinger’s wave function and the new wave-mass function; (4) derives a relation between Einstein’s Field Equation and the new wave-mass function; (5) uses the wave-mass function to combine Einstein’s Field Equation with Schrödinger’s wave function to form a new quantum gravity equation. These results and their applications provide a firm foundation to advance the number set ℍ as a useful mathematical tool.

The Suter curves for Wh and Wm characteristics of collected and the first time presented Four-quadrant (4Q) diagrams of 11 radial pump-turbine models with different specific speeds (nq=24.34, nq=24.8, nq=27, nq=28.6, nq=38, nq=41.6, nq=41.9, nq=43.83, nq=50, nq=56, and nq=64.04) are given in the paper, as well as the Suter curves for 2 pump models (nq=25, and nq=41.8) which were already published in the literature. These 11+2 curves were analyzed to establish a certain universal law of behavior in dependence on the specific speed. For the determination of existing laws in behavior, the fitting procedure using the Regression and Spline methods was used. The paper presents a research plan and structures that include: data collection of four-quadrant diagrams for models of pump-turbines and pumps with different specific speeds (nq), the procedure of recalculation of Four-quadrant diagrams of the models into Suter curves for Wh and Wm characteristics, the definition of the optimal point for the pump and turbine operating mode in pump-turbine models for different specific speeds, and developing numerical models in the MATLAB program to obtain the Universal Equation for the Wh and Wm characteristics. The scientific contribution of this paper is the firstly published original mathematical curve named Universal Equation for the Wh and Wm characteristics for radial pumps and pump-turbines. Application of the equation has been demonstrated at the pumping station where two radial pumps are installed. The calculation of transient processes was performed using the numerical model developed by the authors in the mathematical program MATLAB. Comparison of the transition process results has been made for the two cases - when input data in the numerical model have been firstly used with the values of Suter curves for Wh and Wm characteristics obtained by recalculation from the Four-quadrant operating characteristics (Q11, n11, M11) of model’s specific speed, and then secondly with the values of the Suter curves for Wh and Wm characteristics obtained from the Universal Equation.

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.

It is universally accepted that Maxwell equations do not remain invariant under the Galilean transformation. This conflicts the principle of relativity which states that the physical law must remain invariant in the mathematical form in all inertial frames of reference. For this reason, the Lorentz transformation is invented, and the Galilean transformation is nowadays superseded by the Lorentz transformation. However, this paper challenges this widely held belief that the Maxwell equations are not invariant under the Galilean transformation. By applying the Galilean transformation to Lienard-Wiechert electromagnetic fields it is mathematically proven that the Maxwell equations indeed remain invariant under the Galilean transformation. In addition, the critical error in Lorentz's proof of Galilean non-invariance of Maxwell equations is pointed out, and it turns out that Lorentz's conclusion that Maxwell equations are not invariant under the Galilean transformation is the result of a mathematical error.

In this article, an iterative method of two step is proposed for finding all roots simultaneously of polynomial equations. The order of convergence of the proposed algorithm is 2m, by using any iterative scheme of order m. Numerical tests are performed to confirm the theoretical results and to compare the proposed scheme with existing methods for finding all roots simultaneously.

Recently, Li et al [1] investigated the nonlinear Langevin equations existence including two fractional orders with infinite-point boundary conditions. It has shown that the outcomes given it does count on the solution form for which it has boundary values. It means that their solution needs more to be in the final step. In current work, we will have discussion about the existence and uniqueness for the same boundary conditions by investigating the closely explicit solution form that has no boundary values. Numerical example is supported to have more vision to compare the between new and previous results. It turns out that our results are superior.

In recent years, applying deep learning to solve physics problems has attracted much attention. Data-driven deep learning methods produce operators that can learn solutions to the whole system of partial differential equations. However, the existing methods are only evaluated on simple flow equations (e.g., Burger's equation), and only consider the generalization ability on different initial conditions. In this paper, we construct CFDBench, a benchmark with four classic problems in computational fluid dynamics (CFD): lid-driven cavity flow, laminar boundary layer flow in circular tubes, dam flows through the steps, and periodic Karman vortex street. Each flow problem includes data with various boundary conditions, fluid physical properties, and domain geometries. Compared to existing datasets, the advantages of CFDBench are (1) comprehensive. It contains common physical parameters such as velocity, pressure, and cavity fraction. (2) realistic. It is very suitable for deep learning solutions of fluid mechanics equations. (3) challenging. It has a certain learning difficulty, prompting to find models with strong learning ability. (4) standardized. CFDBench facilitates a comprehensive and fair comparison of different deep learning methods for CFD. We make appropriate modifications to popular deep neural networks to apply them to CFDBench and enable the accommodation of more changing inputs. The evaluation on CFDBench reveals some new shortcomings of existing works and we propose possible directions for solving such problems.

RANS simulations have been broadly used to investigate turbulence in the oceans and atmosphere. Within these environments there are a multitude of tracers undergoing reactions (eg. phytoplankton growth, chemical reactions). The distribution of these reactive tracers is strongly influenced by turbulent mixing. With a 50 member ensemble of two dimensional Rayleigh-Taylor induced turbulent mixing, we show that the dynamics of a reactive tracer growing according to Fisher’s equation are poorly captured by the mean. A fluctuation-dependent sink introduced by Reynolds averaging Fisher’s equation transfers concentration from the mean to the fluctuations. We compare the dynamics of the reactive tracer with those of a passive tracer. The reaction causes the concentration of reactive tracer to increase thereby increasing Fickian diffusion and allowing the reactive tracer to diffuse into turbulent structures that the passive tracer cannot reach. A positive feedback between turbulent mixing and fluctuation growth is identified. We show that eddy viscosity and diffusivity parameterizations fail to capture the bulk trends of the system and identify a need for negative eddy diffusivities. One must therefore be cautious when interpreting RANS results for reactive tracers.

This study delves into the intricate dynamics of sustainable fisheries through the lens of mathematical modeling, specifically employing the Lotka-Volterra equations. The Lotka-Volterra equations, originally conceived to understand predator-prey interactions, offer a potent framework to model and predict fish population dynamics within aquatic ecosystems. The aim of this research is to demonstrate the applicability and efficacy of the Lotka-Volterra equations in understanding the delicate balance of fish populations and aiding sustainable fisheries management. Through this mathematical approach, we seek to provide a structured methodology and present results that showcase the potential of this model in predicting and managing fish populations. In our analysis, we adjust the parameters of the Lotka-Volterra equations to simulate various scenarios, enabling a comprehensive understanding of sustainable fishing practices. Our findings underscore the significance of mathematical models, particularly the Lotka-Volterra equations, in informing strategies to maintain fish populations in a sustainable and ecologically balanced manner, thereby ensuring the longevity of marine ecosystems and the fisheries thatrely upon them.

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This solution family is equivalent to a fractal curve in complex space $\mathbb C^d$ with random steps parametrized by $N$ Ising variables $\sigma_i=\pm 1$, in addition to a rational number $\frac{p}{q}$ and an integer winding number $r$, related by $\sum \sigma_i = q r$. This equivalence provides a \textbf{dual} theory describing a strong turbulent phase of the Navier-Stokes flow in $\mathbb R_d$ space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a \textbf{quantum} statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size $N \rightarrow \infty$ or its chemical potential $\mu \rightarrow 0$. The system with fixed $N$ has different asymptotics at odd and even $N\rightarrow \infty$, but the limit $\mu \rightarrow 0$ is well defined. The energy dissipation rate is analytically calculated as a function of $\mu$ using methods of number theory. It grows as $\nu/\mu^2$ in the continuum limit $\mu \rightarrow 0$, leading to anomalous dissipation at $\mu \propto \sqrt{\nu} \to 0$. The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as $t^{-\lambda}$, where the anomalous dimensions $\lambda$ satisfy the spectral equation \eqref{spectralEq}. The spectrum becomes a continuum in the statistical limit $N \to \infty$, presumably leading to multifractal phenomena.

The fractional diffusion equations with integrals play a significant role in describing anomalous diffusion phenomena. We first construct an appropriate resolvent family, and use the convolution theorem of Laplace transform, the probability density function, Cauchy integral formula and Fubini theorem to study its related equicontinuity, strongly continuity, compactness, etc. Then, a reasonable mild solution is defined. Finally, by combining some fixed point theorems, we obtain some conclusions on the existence and uniqueness of mild solutions.

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.

The standard approach to integrable nonlinear evolution equations (NLEE) usually uses the following steps: 1) Lax representation $[L,M]=0$; 2)~construction of fundamental analytic solutions (FAS); 3) reducing the inverse scattering problem (ISP) to a Riemann-Hilbert problem (RHP) $\xi^+(x,t,\lambda) =\xi^-(x,t,\lambda) G(x,t\lambda)$ on a contour $\Gamma$ with sewing function $G(x,t,\lambda)$; 4) soliton solutions and possible applications.Step 1) involves several assumptions: the choice of the Lie algebra $\mathfrak{g}$ underlying $L$, as well as its dependence on the spectral parameter, typically linear or quadratic in $\lambda$. In the present paper we propose another approach which substantially extends the classes of integrable NLEE.Its first advantage is that one can effectively use any polynomial dependence in both $L$ and $M$. We use the following steps: A) Start with canonically normalized RHP with predefined contour $\Gamma$; B) Specify the $x$ and $t$ dependence of the sewing function defined on $\Gamma$; C) Introduce convenient parametrization for the solutions $ \xi^\pm(x,t,\lambda)$ of the RHP and formulate the Lax pair and the nonlinear evolution equations (NLEE); D) use Zakharov-Shabat dressing method to derive their soliton solutions. This needs correctly taking into account the symmetries of the RHP. E) Define the resolvent of the Lax operator and use it analyze its spectral properties.

Maxwell equations governing electromagnetic effects are being shown to be equivalent to the compressible inviscid Navier-Stokes equations applicable in fluid dynamics and representing conservation of mass and linear momentum. The latter applies subject to a generalized Beltrami condition to be satisfied by the magnetic field. This equivalence indicates that the compressible inviscid Navier-Stokes equations are Lorentz invariant as they derive directly from the Lorentz invariant Maxwell equations subject to the same Beltrami condition. In addition, the derivation results provide support for the claim that electromagnetic potentials have physical significance as demonstrated by Aharonov-Bohm effect, and are not only a convenient mathematical formulation.

In this paper, we derive Jeffreys divergence, generalized Fisher divergence and corresponding De Bruijn identities on space-time random field. First, we determine the relation between Fisher information on the space-time random field in one of the space-time points and the ratio of Jeffreys divergence on a space-time random field at distinct space-time positions to the square of coordinate difference. In addition, we also find identities between the partial derivative of the Jeffreys divergence and the generalized Fisher divergence with respect to space-time variables, i.e. the De Bruijn identities, between two space-time random fields obtained by different parameters under the same Fokker-Planck equations. At the end of this paper, we present three examples of the Fokker-Planck equations on space-time random fields, identify their density functions, and derive the Jeffreys divergence, generalized Fisher information, generalized Fisher divergence, and accompanying De Bruijn identities.

We present three reduced integrable hierarchies of nonlocal integrable NLS type equations from a vector integrable hierarchy associated with a matrix Lie algebra, not being A type. Three similarity transformations are taken to keep the invariance of the transformed zero curvature equations. The key step is to formulate a solution to a reduced stationary zero curvature equation so that the zero curvature formulation works for a reduced case.

The article introduces the reader to the new thermodynamic potential of classical thermodynamics, called the J-potential. For the case of a homogeneous system, it vanishes, but it perfectly serves multiphase (heterogeneous) systems. The J-potential is especially important for the thermodynamics of capillary systems in colloidal science. The J-potential simplifies the derivation of the Neumann and Young equations, and has found application in the thermodynamics of thin films and the theory of strength of materials. In this work, for example, it was used to study the dependence of the chemical potential of a substance of a small bubble on the bubble size.

In this paper, a short survey of the existed results concerning the Jacobian Conjecture is first given. Then the 3-fold linear type polynomial map will be analyzed in detail. The expansion of the Jacobian condition is deduced to obtain its equivalent algebraic equations, and the Jacobian condition will be analyzed to derive two coordinate transformations that can maintain the invariance of the Jacobian condition. Finally, it is proved by mathematical induction method that one general chain expression presented in this paper is just the inverse polynomial map of 3-fold linear type polynomial map, i.e. LJC(n,[3]) holds such that the Jacobian Conjecture holds.

This paper presents a numerical method for solving a two-dimensional subdiffusion equation with a Caputo fractional derivative. The problem considered assumes symmetry in both the equation’s solution domain and the boundary conditions, allowing for a reduction of the two-dimensional equation to a one-dimensional one. The proposed method is an extension of the fractional Crank-Nicolson method, based on the discretization of the equivalent integral-differential equation. To validate the method, the obtained results were compared with a solution obtained through Laplace transform. The analytical solution in the image of the Laplace transform was inverted using the Gaver-Wynn-Rho algorithm implemented in the specialized mathematical computing environment, Wolfram Mathematica. The results clearly show the mutual convergence of the solutions obtained by the two methods.

For a linear ordinary differential equation (ODE in short) of the third order, results are presented that supplement the theory of conjugate points and extremal solutions by W. Leighton, Z. Nehari, M. Hanan. It is especially noted the sensitivity of solutions to the initial data, which makes their numerical study difficult. Similar results were obtained for the third-order nonlinear equations of the Emden-Fowler type.

Novel approaches under the extreme conditions of ultra-high gravity of a super massive black hole allow us to assume that the spacetime degrades into the single dimension of time. A unique interpretation of the non-relativistic Schrödinger equation together with the relativistic Klein-Gordon and Dirac equations is presented by swapping the three dimensional space derivatives with the singular time derivative in the D’Alembertian operator. This apparatus reveals a unique time dependent wave equation that results in some implications such as possible time quantization with the intervals of around 3.36 × 10−21 ? which is in the order of uncertainty in time, ∆? = 1.3 × 10−21 ?, calculated from the energy-time version of the Heisenberg uncertainty principle for electrons. As a result of this, an antisymmetric wave function around the edge of the event-horizon zone appears and indicates that the virtual particle-antiparticle pairs of the electron field split into two real components with the spin-up and down states, reconfirming the Hawking radiation. Energy eigenvalues calculated from the application of the Dirac equation satisfies the preservation of energy and charge. The wave equation invoked in this study reduces into the time dependent Dirac equation within the neighborhood of zero-point, ? = 0, insuring the relevance of the evaluations.

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.

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.

We develop a quantitative microscopic theory of decaying Turbulence by studying the dimensional reduction of the Navier-Stokes loop equation for the velocity circulation. We have found an infinite dimensional manifold of solutions of the Navier-Stokes loop equation\cite{M93, M23PR} for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This family of solutions corresponds to a fractal curve in complex space $\mathbb C^d$, described by an algebraic equation between consecutive positions plus a nonlinear periodicity condition. We derive the constrained SDE for the evolution of the fractal curve at a fixed moment of physical time as a function of an auxiliary stochastic time. We expect this stochastic process to cover our fixed manifold of the solutions of the decaying Turbulence. The energy density of the fluid decays as $\mathcal E_0/t$, where $\mathcal E_0$ is an initial dissipation rate. Presumably, we have found a new phase of extreme Turbulence yet to be observed in real or numerical experiments.

Reference evapotranspiration (ET0) is the first step in calculating crop irrigation demand, and numerous methods have been proposed to estimate this parameter. FAO-56 Penman-Monteith (PM) is the only standard method for defining and calculating ET0. However, it requires radiation, air temperature, atmospheric humidity, and wind speed data, limiting its application in regions where these data are unavailable; therefore, new alternatives are required. This study compared the accuracy of ET0 calculated with the Blaney-Criddle (BC) and Hargreaves-Samani (HS) methods versus PM using information from an automated weather station (AWS) and the NASA-POWER platform (NP) for different periods of time. The information collected corresponding at Module XII of the Lagunera Region Irrigation District 017, a semi-arid region in the North of Mexico. The HS method underestimated by 5.5 % the reference evapotranspiration (ET0) compared to the PM method during the period from March to August, and yielded the best fit in the different evaluation periods: daily, average, and 5-day cumulative; the latter showed the best values of inferential parameters. The information about maximum and minimum temperatures from the NP platform was suitable for estimating ET0 using the HS equation. This data source is a timely alternative, particularly in semi-arid regions where no data from weather stations are available.

In this paper, we introduce as a pedagogical strategy an internal division-free, straightforward, and symmetrically progressing algorithm in manually computing matrix inverse and solving systems of linear equations by revisiting the application of elementary row operations in the Gauss-Jordan reduction method and connecting it to the determinant method. The proposed cross-multiplication-flip (CMF) algorithm employs cross-multiplication similar to the butterfly movement in computing determinants as a strategic application of elementary row operations to efficiently reduce the rows and then applies flipping of rows and entries to put an upper triangular matrix into lower triangular form to continue the reduction process.

Volumetric modification of transparent materials by femtosecond laser pulses is successfully used in a wide range of practical applications. The level of modification is determined by the locally absorbed energy density, which depends on numerous factors. In this work, it is shown experimentally and theoretically that, in a certain range of laser pulse energies, the peak of absorption of laser radiation for doughnut-shaped (DS) pulses is several times higher than for Gaussian ones. This makes the DS pulses very attractive for material modification and direct laser writing applications. Details of the interaction of laser pulses of Gaussian and doughnut shapes with fused silica obtained by numerical simulations are presented for different pulse energies and compared with the experimentally obtained data. The effect of absorbed energy delocalization with increasing laser pulse energy is demonstrated for both beam shapes while, at relatively low pulse energies, the DS beam geometry provides a stronger local absorption compared to Gaussian one. Implications of a DS pulse action on post-irradiation material evolution are discussed based on thermoelastoplastic modeling.

The correspondence between classical and quaternion quantum equations, permits considering the universe (vacuum) as an ideal elastic solid. Elementary particles would have to be standing or soliton-like waves. Tension induced by the compression and twisting of the elastic medium would increase the energy density, consequently generate a gravity forcing and affect the wave speed. Consequently the gravity could be described by an index of refraction. Theory was created by combining the Cauchy model of the elastic continuum with the Planck-Kleinert crystal hypothesis. The quaternion-imaginary Lagrange’an, the quaternion motion equation and the quaternionic oscillator allowed deriving: - The Schrödinger equation from the functional integral, which identifies the quaternion-imaginary quantum Hamiltonian. - The 2nd order wave equation system describing both the bosons and the gravity in terms of quaternionic Poisson equation. - The first order quaternionic wave equation system. - The family of the second order wave equation systems describing both the particles and the generated quaternionic force-fields (four-potentials). - The fundamental Planck and gravity constants. - The quaternionic continuity equation in an ideal elastic solid.

The (numerical) solution of Ordinary Differential Equations (ODEs) problems is of paramount relevance, ODEs system being an ubiquitous mathematical formulation of many physical phenomena (such as those involved in fluid dynamics, chemistry, biology, evolutionary-anthropology, ...): almost every dynamici phenomenon can be modeled by means of an ODEs system. The present paper is the first manifesto of FOODIE, a library aimed to numerically solve ODEs problems by means of a clear, concise and efficient abstract interface. FOODIE, meaning Fortran Object oriented Ordinary Differential Equations integration library, has manifolds aims: to provide a set to built-in numerical schemes that are accurate, robust, validated and efficient and to allow easy application of these schemes to (almost) all ODEs problems by means of an effective Abstract Calculus Pattern. The key idea is to allow the same solver-implementation to be applied to all ODEs problems thus avoiding the re-implementation of the ODEs solver for each different ODEs problem: code re-usability is consequently maximized, FOODIE being a general robust framework. Besides, the same framework also allows rapid development of new ODEs solvers due to the high abstraction level of the library itself. The present paper is the first announcement of FOODIE project: the current implementation is extensively discussed and its capabilities are proved by means of tests and examples.

In the last years, a recent growing line of research has been developed with fruitful results by using fractional and conformable derivatives in the iterative procedures of classical methods for solving nonlinear equations. In that sense, the use of conformable derivatives has shown better behavior than fractional ones, not only in the theory, but also in the practice. In this work, we adapt the approximation of conformable derivatives in order to design the first conformable derivative-free iterative schemes to solve nonlinear equations: a Steffensen’s type method and a Secant type method; the latter with memory. Convergence analysis is made, preserving the order of classical cases, and the numerical performance is studied in order to confirm the results in the theory. It is shown that these methods can present some numerical advantages versus their classical partners, with wide sets of converging initial estimations.

In this paper, we report orthogonal fuzzy versions of some celebrated iterative mappings. We provide various concrete conditions on the real valued functions J,S:(0,1]→(−∞,∞) for the existence of fixed-points of (J,S)-fuzzy iterative mappings. We obtain many fixed point theorems in orthogonal fuzzy metric spaces. We apply (J,S)-fuzzy version of Banach fixed point theorem to show the existence and uniqueness of the solution. We support these results with several non-trivial examples and applications to Volterra-type integral equations and fractional differential equations.

The creep response of commercially pure titanium was investigated in air at 550, 600 and 650°C, to assess the effect of oxidation on the mechanical response. Experiments demonstrated that exposure at high temperature produced a marked reduction of the minimum creep rate under a given applied stress. Microhardness measurements showed that a hardened zone forms in proximity of the surface, due to oxygen penetration into metal. A simplified composite-model was then used to describe the creep response. In this model, the sample consisted of two zones, the hard case, enriched in oxygen, and the soft pure-titanium core, both creeping with similar strain rates. Calculation led to an estimation of the dependence of the minimum creep rate on stress and temperature for the hard high-oxygen zone. The simplified composite-model here presented gave an excellent description of experimental creep data for pure titanium tested in air and also provided a reliable picture of the effect of oxidation on complex Ti-alloys.

The dynamics of the main photosynthetic structures is studied by mathematical modeling methods in this article. Chlorophyll portion variability in phytoplankton and the formation of energy-intensive substances in the process of photosynthesis underlie the models. These cellular components are considered in terms of their participation in the growth of specific biomass. Computational experiments are presented that simulate various degrees of environmental friendliness. The corresponding functions are built in accordance with seasonal fluctuations throughout the year in the Far East region of Russia. The stability of model solutions in long-term dynamics is also investigated. The models were tested for biological adequacy and their effectiveness was compared. For the model selected as a result of the comparison, the optimal control problem has been formulated and solved. It is shown that in this way it is possible to reduce the space of the initial components of the model system.

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.