In this work model of mirror elements equilibrium of the micromechanical components is developed, the behavior analysis of the mirror element of micromechanical mirrors in case changing of control voltages of electrostatic actuators is carried out, an expression for determining the maximum value of deflection voltage at which the snap-down effect will take the following form is obtained in case of the influence of the coefficient of the electrostatic rigidity of electrostatic actuators. The developed equilibrium model of mirror elements and the obtained results of modeling can be used at design of micromechanical mirrors with internal suspensions.

Background: Vestibular Neuritis (VN) induces Acute Vestibular Syndrome (AVS). This study aimed to identify predictive factors of VN recovery by healing and/or compensation. Methods: In this longitudinal and prospective study, 40 patients were included. The first assessment (T0) included medical history taking (general and specific), for cardiovascular risk factors (CVRFs), a battery of diagnostic tests, bithermal caloric test (BCT), video Head Impulse Test (VHIT) and skull vibration induced nystagmus (SVIN). All patients also completed a Dizziness Handicap Inventory (DHI). All assessments were repeated 90±15 days (T3) later. Subjective criteria compensation were based on DHI total score and objective criteria compensation on the instrumental test results. Four groups of patients (A, B, C, D) were delineated by combining patients with normal vs abnormal vestibular tests and patients with normal vs abnormal DH. Results: CVRFs were associated with a poorer recovery of symptoms but not the age or BMI. Vestibular tests mainly BCT (canal paresis %), VHIT (lateral semicircular canal gain) and SVINT (nystagmus slow phase velocity) recovered within normal values in 20%, 20%, 26% of cases respectively at T3 for the 40-patient group. Conclusions: Vascular risk factors (hypercholesterolemia) are correlated with patients who do not recover their symptoms either by healing or compensation. The evolution did not show any significant difference between high or low frequency vestibular tests in patients recovering from their symptoms. However, some patients with objective recovery may continue to show persisting subjective symptoms.

In this paper, we extend the KdVSKR equations in (1+1)-dimension and (2+1)-dimension to fractional KdVSKR equations with modified Riemann-Liouville derivative. The (2+1)-dimensional KdVSKR equation, which is a recent extension of (1+1)-dimensional KdVSKR equation, can model the resonances of solitons in shallow water. By means of the Hirota bilinear method, finite symmetry group method and consistent Riccati expansion method, many new interaction solutions have been derived. Soliton-cnoidal interaction solution of the (1+1)-dimensional fractional KdVSKR equation has been derived for the first time. For the (2+1)-dimensional frac-tional KdVSKR equation, two-wave interaction solutions and three-wave interaction solutions, including dark-soliton-sine interaction solution, bright-soliton-elliptic interaction solution, and lump-hyperbolic-sine interaction solution, have been derived. Impact of the fractional order on the shapes of these solutions has been illustrated by figures. Three-wave interaction solutions of fractional systems have not been reported in the existing references. The research idea in this paper can be applied to other fractional differential equations.

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.

In this paper, the time-resolved model of photoacoustic signal for samples with a complex inner structure is derived including local non-equilibrium of structural elements with multiple degrees of freedom, i.e. structural entropy of the system. The local non-equilibrium is taken into account through the fractional operator. By analyzing the model for two types of time-dependent excitation, a very short pulse and a very long pulse, it is shown that the rates of non-equilibrium relaxations in complex samples can be measured by applying the derived model and time-domain measurements. Limitations of the model and further directions of its development are discussed.

This article aims to explore the existence and stability of solutions to differential equations involving a regularized ψ−Hilfer fractional derivative, which, compared to standard ψ−Hilfer fractional derivatives, provide a clear physical interpretation when dealing with initial conditions. We discovered that the regularized ψ−Hilfer fractional derivative can be represented as the inverse operation of the ψ−Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a regularized ψ−Hilfer fractional derivative. Additionally, we applied the Mönch’s fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a regularized ψ−Hilfer fractional derivative, and further discussed Ulam-Hyers-Rassias stability and semi-Ulam-Hyers-Rassias stability of these solutions. Finally, we illustrated our results through case studies.

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.

In this paper, we study the semilinear pseudo-parabolic equations $\displaystyle u_{t} - \triangle_{\mathbb{B}}u - \triangle_{\mathbb{B}}u_{t} = \left|u\right|^{p-1}u$ on a manifold with conical singularity, where $\triangle _{\mathbb{B}}$ is Fuchsian type Laplace operator investigated with totally characteristic degeneracy on the boundary $x_{1} = 0$. Firstly, we discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence and nonexistence of global weak solution: for the low initial energy $J(u_{0})<d$, the solution is global in time with $I(u_{0}) >0$ or $\displaystyle\Vert \nabla_{\mathbb{B}}u_{0}\Vert_{L_{2}^{\frac{n}{2}}(\mathbb{B})} = 0$ and blows up in finite time with $I(u_{0}) < 0$; for the critical initial energy $J(u_{0}) = d$, the solution is global in time with $I(u_{0}) \geq0$ and blows up in finite time with $I(u_{0}) < 0$. The decay estimate of the energy functional for the global solution and the estimates of the lifespan of local solution are also given.

Local polynomial smoother (LPS) is a weighted local least-squares nonparametric method. It provides a local Taylor series fit of the data at any location and can be directly used in a differential equation to provide a numerical scheme. In this article, we introduce this new nonparametric idea based on local polynomial smoother, for acquiring the numerical solution of the Bagley-Torvik fractional-order differential equations. Furthermore, this paper will present a numerical comparison with some methods, such as legendre operational matrix and pseudo-spectral method. The efficiency and accuracy of the LPS method are demonstrated by two numerical examples.

The fundamental period is one of the most important parameters for the design of new structures , as well as the estimation of the capacity of existing ones. However, its calculation requires the solution of an eigenvalue problem. Especially for new buildings, this can involve material and geometric parameters that are not known beforehand. Thus, to estimate it, various design codes and researchers have adopted a several approximate analytical equations based on a number of key structural parameters. To this end, the present study introduces a novel methodology for the derivation of analytical equations for the fundamental period of Reinforced Concrete (RC) structures. The methodology is based on Machine Learning (ML) explainability techniques and specifically the so-called SHapley Additive exPlanations (SHAP) values. The novel methodology allows these equations to be constructed sequentially and in an informed manner, controlling the balance between accuracy and complexity. An extended dataset comprised of 4026 data points is employed. The results showed excellent accuracy, with a coefficient of determination R2≈0.95. This demonstrates the potential applicability of the proposed methodology in a wide array of similar engineering challenges.

This paperfocusesontheoscillatorypropertiesofthesemi-canonicalnonlineardelaydifferential equation ofthird-order (δ2(t)(δ1(t)ζ′(t))′)′ + η(t)ζα(τ(t))= 0, t ≥ t0, under theconditions ∞t01δ2(t)dt = ∞ andZ ∞t01δ1(t)dt < ∞. Using the new canonical transform method, we transformed the studied equation into a canonical type equation and then we find the conditions for the oscillation of the studied equation from the canonical equations. The obtained results are new and complement to the existing results mentioned in the literature. Examples are provided to illustrate the importance and novelty of the main results.

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.

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.

Background: Scientific literature on COVID-19 is still limited about long-term impacts on the all-body systems and they treatments. The aim of the study was to create a safe protocol-based in-tervention to improve functional and equilibrium abilities in older adults impacted by COVID-19. Methods: The study completed 46 people (intervention group: n=26; control group: n=20). Re-sistance training (RT) was held twice a week, 60 min per session for 8 weeks. The postural stability and quality of life questionnaire (WHQOOL) were conducted during pre- and post-testing. Results: Results indicated significant differences in overall stability index (OSI) with eyes open (EO), ante-rior-posterior stability index (APSI) EO, fall risk index 6-2 (FRI6-2) values in males (p

The digital age has ushered in unprecedented connectivity and convenience, but it has also given rise to escalating cyber threats and privacy concerns. This study examines the intricate balance between technological advancement and personal privacy in the face of evolving cyber risks. A comprehensive literature review was conducted, analyzing peer-reviewed articles, government reports, and industry white papers from the past decade. Additionally, case studies of major data breaches and privacy violations were examined. Surveys and interviews with cybersecurity experts, policymakers, and privacy advocates were conducted to gather diverse perspectives. The research reveals a significant increase in both the frequency and sophistication of cyber-attacks, coupled with a growing public awareness of privacy issues. However, there is a notable disconnect between individuals' stated privacy concerns and their online behaviors. Regulatory efforts to protect privacy often lag behind technological advancements and cyber threats. Maintaining equilibrium between technological progress and privacy protection requires a multifaceted approach involving enhanced cybersecurity measures, updated regulations, and improved digital literacy. The study underscores the need for ongoing collaboration between government, industry, and academia to address this complex challenge.

In this paper we introduce the Adomian decomposition method for a class of first order fuzzy dynamic equations on arbitrary time scales for existence of solutions. The results are provided with suitable numerical examples.

In this paper, we study the problem of controllability of impulsive neutral evolution integrodifferential equations with state dependent delay in Banach spaces. The main results are completely new and are obtained by using Sadovskii's fixed point theorem, theory of resolvent operators, and an abstract phase space. An example is given to illustrate the theory.

Objective: To develop and validate the short version of the JAEN Scale for use in women with Fibromyalgia Syndrome (FMS) to make the balance disorder measurement process more efficient. Methods: A Cross-sectional observational validation study was conducted. Fifty-six women with FMS and 44 healthy control were included. Certain items of the original tool were selected with the aim of 1) Improving internal consistency by reducing item redundancy 2) Obtaining a diagnostic capacity with an Area Under the ROC Curve (AUC) greater than 0.70 for discriminating FMS patients and fallers. The internal consistency, factorial validity, concurrent validity and diagnostic capacity of the new tool were analyzed. Results: Factorial analysis showed a two-factor structure that explained 72% of variance. Cronbach alpha coefficients of 0.904 were obtained for the total score of the JAEN-10 items. Concurrent validity analysis showed strong correlations of the JAEN-10 with other instruments that measured quality of life, postural balance or disability related to dizziness. The score of the JAEN-10 items showed an AUC of 0.858 with a sensitivity of 64.29 and a specificity of 95.45 for discriminating between FMS and healthy controls, and an AUC of 0.835 with a sensitivity of 90.48 and a specificity of 67.24 for discriminating between fallers and non-fallers.; Conclusions: The 10-item JAEN scale is a valid instrument for discriminating between subjects with or without FMS and between fallers and non-fallers. Its psychometric properties are good and similar to those of the original 20-item scale. Moreover, it is quicker to complete, which may be relevant in subjects with a tendency to experience fatigue.

In this paper, we investigate the equilibrium strategies for customers arriving at overtaking-free queueing networks and receiving partial information about the system’s state. In an overtaking-free network, customers cannot be overtaken by others arriving after them. We assume that customer arrivals follow a Poisson process and that service times at any queue are independent and exponentially distributed. Upon arrival, the received partial information consists in the total number of customers already in the network, however the distribution of these among the queues is left unknown. We prove the existence of equilibrium threshold strategies for such systems and provide methods to compute them.

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

Traditional population-based metaheuristic algorithms are effective in solving complex real-world problems but require careful strategy selection and parameter tuning. Parameter-less population-based optimization algorithms have gained importance due to their simplicity and efficiency. However, research on their applicability for solving large systems of nonlinear equations is still incipient. In this paper, a systematic review and detailed description of metaphor-less optimization algorithms is made, including the Jaya algorithm, the main Jaya variants for continuous optimization problems, with a focus on the Enhanced Jaya (EJAYA) algorithm, the three Rao algorithms, the best-worst-play (BWP) algorithm, and the very recent max-min-greedy-interaction (MaGI) algorithm. This paper also discusses a few previous parallelizations of the Jaya optimization algorithm, as well as recent GPU-based massively parallel implementations of the Jaya, Enhanced Jaya (EJAYA), Rao, and BWP algorithms developed by the authors. In addition, a novel GPU-accelerated version of the MaGI algorithm is proposed. The GPU-accelerated versions of the metaphor-less algorithms developed by the authors were implemented using the Julia programming language and tested primarily on GeForce RTX 3090 and NVIDIA A100 GPUs, as well as on Tesla V100S and Tesla T4 GPUs, using a set of difficult, large-scale nonlinear equation system problems. The computational experiments carried out produced quite significant speedups, which highlights the efficiency of the GPU-based versions of the metaphor-less algorithms developed for solving large-scale systems of nonlinear equations.

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.

Considered is “time as an interval” including time from the past and from the future, in contrast to time as a moment. Equilibrium as the basis for a description of changing properties in physics is understood to depend on the “mean velocity theorem”, while a “time” of equilibrium resembles a center of weight. This turns out to be a good method to derive properties for any function of time t including space coordinates q(t) and expressions for the time dependent Hamiltonian. Introduced are derivatives depending on time intervals instead of time moments and with these a new relation between the Lagrangian L and the Hamiltonian H. As an application introduced is a step by step method to integrate stationary state “local” time interval measurements to beyond “locality” in General Relativity. Because of limits on the measures of the resulting time intervals and their asymmetry, this allows for a probabilistic interpretation of quantities that have these intervals as time domain in QM. Their asymmetry also questions the time reversal symmetry of GR. Another application of time intervals is the discussion of the measurement of starlight radiation energy and QM wave packet collapse as an example of a time dependent Hamiltonian. Finally a relation between starlight frequency, metric and space- and time intervals is found. Discussed is how finite and asymmetric time intervals correspond to time dependent H and symmetric infinite time intervals to a time independent H. From there, in cosmological perspective, finite time intervals can help to describe how entropy change could relate to dark energy.

The second article in the series presents the application of the Smoothed Particle Hydrodynamics (SPH) method to modelling of batch crystallisation in stirred tanks. A methodology to integrate the population balance equations (PBE) in parallel and independently from the Navier-Stokes equations is demonstrated. The benefits of the proposed methodology in terms of computational requirements, accuracy and availability of the crystal size distribution are discussed. The specific formulation of the SPH equations where the resulting system of ordinary differential equations is solved using the weighted contributions rather than numerically by solving a linear system of equations allows for massive parallelisation and a very loose coupling of the population balance and the fluid dynamics. It has been demonstrated, that the population balance equations can be solved on a Shared Memory Architecture (SMA) system using the OpenMP interface while the fluid dynamics equations being computed independently on a General Purpose Graphics Processing Unit (GPGPU) using the NVidia CUDA technology. This way, a significant portion of the computational overhead due to the large number of additional transport equations resulting from the discretisation of the population balance was removed: the SPH simulation coupled with 200 population balance equations was only 40% slower compared to SPH-only simulation. Two methods for the solution of population balance equations that preserve full crystal size distribution were implemented: discretised population balance (DPB) and method of characteristics (MOCH). The DPB equations are solved using the high-resolution finite-volume method with flux limiter and the effect of a large number of different flux limiters have been investigated. Both methods were validated using the case studies from the literature where an analytical solution can be derived. The developed models were applied to a numerical solution of coupled computational fluid dynamics and population balance equations to model a batch crystallization process. The effect of the hydrodynamics on the local temperature/supersaturation and the resulting crystal size distribution was captured and compared to the ideal mixing case. The simulation results from the DPB and MOCH methods were compared in terms of computational requirements and accuracy and MOCH selected as computationally more efficient and accurate.

This paper aims to study and analyze the effect of the flow rate and amplitude of walling waves on the peristaltic flow of Jeffrey’s fluid through an irregular channel. In addition, it analyses streamlined patterns and their local and global bifurcation flow. The theory of dynamical systems is used to explore the position of critical points and their bifurcation. Numerical and analytical methods are presented to display effective changes in amplitude and rate of the flow on the transport of fluid. Three types of streamline patterns occur backward, trapping and augmented flow in the channel through movement due to changes in the value rate parameters.

The bentonite slurry trenches are becoming increasingly popular in the excavation process of trenches, especially for diaphragm wall construction. The problem that needs to be addressed is the stability of bentonite slurry trenches. This paper presents a stability analysis of trenches with temporary support of bentonite slurry, with unit weights ranging from 10.5 to 12.0 kN/m3, in the realistic stratum: C3 in the Hue city area. The analysis employs the shear strength reduction technique (SSR) with Mohr-Coulomb materials for the numerical method to evaluate the factor of safety (FS). The finite element method (FEM) software program (RS) and the finite difference method (FDM) FLAC software were applied. Additionally, the limit equilibrium method (LEM) of Bell-Washbourne and three-dimensional (3D) Bishop are used for calculating trench stability. The analysis results show a good agreement between RS2 and FLAC2D and RS3 and FLAC3D. Secondly, upon comparison, it is noted that the safety factors of the 3D software programs (RS3, FLAC3D) are higher than those of the 2D software programs (RS2, FLAC2D), ranging from 52.3 to 63.0% for trench lengths of 6m. However, for trench lengths of 54m, the safety factor values in 2D and 3D are nearly equal. Thirdly, the safety factor of the Bell-Washbourne method (LEM) is lower than the numerical analysis methods (FLAC and RS programs). Using the three-dimensional numerical method appears to be effective for estimating stability.

The objective of this study is to evaluate the effect of constitutive equations on the prediction accuracy for springback in cold stamping with various deformation modes. In this study, two types of yield functions—Hill’48 and Yld2000-2d—were considered to describe yield behavior. Isotropic and kinematic hardening models based on the Yoshida–Uemori model were also adopted to describe hardening behavior. Various material tests (such as uniaxial tension, tension- compression, loading-unloading, and hydraulic bulging tests) were carried out to determine the material parameters of the models. The obtained parameters were implemented in the finite element (FE) simulation to predict springback, and the results were compared with experimental data. U-bending and T-shape drawing were employed to evaluate the springback prediction accuracy. Obviously, the springback prediction accuracy was greatly influenced by constitutive equations. Therefore, it is important to choose appropriate constitutive equations for accurate description of material behaviors in FE simulation.

The Macroscopic Mechanical Equilibrium Condition can be shown to be the physical origin of the Equation of State of any material system, then can describe all thermodynamic quasi-equilibrium processes caused by the change of the external temperature and/or mechanical environment. If it cannot be satisfied, the system would be broken.

When discussing the Diophantine quintic equation $p(a^5 + b^5) = q(c^5 + d^5)$ where p is a prime and q is an integer, there is a clear gap between the mathematical literature and online forums. Because of the intrinsic complexity of parameterizing fifth-degree equations, this equation re- mains largely unexplored. In this work, we approach this quintic problem through algebraic methods with the goal of providing numerical solutions that illuminate its properties and behaviors.Our research uncovers in- triguing patterns and connections that shed light on the enigmatic nature of Diophantine quintic equations in this particular form.

This paper presents machine learning methods for approximate solutions of reaction-diffusion equations with multivalued interaction functions. This approach addresses the challenge of finding all possible solutions for such equations, which often lack uniqueness. The proposed method utilizes physics-informed neural networks (PINNs) to approximate generalized solutions.

The study of different forms of preconditioners for solving a system of nonlinear equations, by using Newton’s method, is presented. The preconditioners provide numerical stability and rapid convergence with reasonable computation cost, whenever chosen accurately. Different families of iterative methods can be constructed by using a different kind of preconditioners. The multi-step iterative method consists of a base method and multi-step part. The convergence order of base method is quadratic and each multi-step add an additive factor of one in the previously achieved convergence order. Hence the convergence of order of an m-step iterative method is m + 1. Numerical simulations confirm the claimed convergence order by calculating the computational order of convergence. Finally, the numerical results clearly show the benefit of preconditioning for solving system of nonlinear equations.

The General equilibrium theory tries to show how and why all free markets tend toward equilibrium in the long run. However, what is meant by equilibrium in this paper is more from a thermodynamical point of view. In order to understand the actual situation, it is necessary to study open systems which are complex. In physics, such a behavior in a complex system can be explain by using Non Equilibrium Thermodynamics. A system is able to self-organize and sustain itself away from equilibrium. Economic systems may fluctuate around a particular point. To sustain it far from the equilibrium state, it needs to degrade more energy and materials. In this study, the energy consumption patterns of Sri Lanka and USA are discussed. The pattern concerning Sri Lanka is close to the model proposed here, whereas the energy consumption pattern of USA is more complicated due to external factors.

Neural Fractional Differential Equations (Neural FDE) represent a neural network architecture specifically designed to fit the solution of a fractional differential equation to given data. This architecture combines an analytical component, represented by a fractional derivative, with a neural network component, forming an initial value problem. During the learning process, both the order of the derivative and the parameters of the neural network must be optimised. In this work, we investigate the non-uniqueness of the optimal order of the derivative and its interaction with the neural network component. Based on our findings, we perform a numerical analysis to examine how different initialisations and values of the order of the derivative (in the optimisation process) impact its final optimal value. Results show that the neural network on the right-hand side of the Neural FDE struggles to adjust its parameters to fit the FDE to the data dynamics for any given order of the fractional derivative. Consequently, Neural FDEs do not require a unique α value, instead, they can use a wide range of α values to fit data. This flexibility is beneficial when fitting to given data is required, and the underlying physics is not known.

This paper provides an alternative Slope-Deflection equation for performing the static analysis of the indeterminate structures. While the conventional Slope-Deflection method introduced in undergraduate structural analysis course involves the use of fixed end moments in the Slope-Deflection equations, the proposed method employs the slope of a simply supported beam to obtain the unknown moments at the joints. Three numerical examples of continuous beams with different load configurations are chosen and the efficacy of the proposed equations is demonstrated. From the perspective of undergraduate structural analysis course in Civil Engineering education, the approach proposed in this paper offers two advantages: (1) prior knowledge of fixed beams is not required and (2) the concepts gained from determinate structural analysis are directly applicable.

This study derives space travel pricing by Walrasian Equilibrium, which is logical reasoning from the general relativity theory (GRT), the accounting equation, and economic supply and demand functions. The Cobb-Douglas functions embed the endogenous space factor as new capital to form the space travel firm's production function, which is also transformed into the consumer's utility function. Thus, the market equilibrium occurs at the equivalence of supply and demand functions, like the GRT presents the equivalence between the spatial geometric tensor and the energy-momentum tensor, which explains the principles of gravity and the motion of space matter in the spacetime framework. The mathematical axiomatic set theory of the accounting equation explains the equity premium effect that causes a short-term accounting equation inequality, then reaches the equivalence by suppliers' incremental equity through the closing accounts process of the accounting cycle. On the demand side, the consumption of space travel can be assumed as a value at risk (VaR) investment to attain the specific spacetime curvature in an expected orbit. Spacetime market equilibrium is then achieved to construct the space travel pricing model. The methodology of econophysics and analogy method was applied to infer the space travel pricing by the model of profit maximization, single-minded, and envy-free pricing in unit-demand markets. A case study with simulation was conducted for empirical verification of the math models and algorithm. Results show that space travel pricing remains the principle of market equilibrium but needs to be extended to the spacetime tensor of GRT.

This article focuses on obtaining the analytical solutions for parabolic Volterra integro- differential equations in d-dimensional with different types frictional memory kernel. Based on theories of Laplace transform, Fourier transform, the properties of Fox-H function and convolution theorem, analytical solutions of the equations in the infinite domain are derived under three frictional memory kernel functions respectively. The analytical solutions are expressed by infinite series, the generalized multi-parameter Mittag-Leffler function, Fox-H function and convolution form of Fourier transform. In addition, the graphical representations of the analytical solution under different parameters are given for one-dimensional parabolic Volterra integro-differential equation with power-law memory kernel. It can be seen that the solution curves subject to Gaussian decay at any given moment.

Maxwell’s equations are valid only for a stationary observation point, therefore, to adequately describe real processes so far we have had to move to a moving reference frame. This paper presents the equations of electrodynamics for the moving observation point, it is shown that plane and spherical electromagnetic waves are their solutions, while the spherical wave propagates only outward, which cannot be said about Maxwell’s equations. The fields of uniformly moving charges are also solutions of the equations. Now there is no need to move to a moving reference frame, to use four-dimensional space and covariant form of equations. The question of finding a universal form of the equations that allows a solution in the form of the field of an arbitrarily moving charge remains open. This raises the question of the existence of a two-parameter group of transformations of electromagnetic fields along with the known one-parameter group has been posed. The phenomena derived from the equations, which make an additional contribution to the phase overrun in the Aharonov-Bohm effect are considered. The equation of motion of a charged particle in an electromagnetic field without simplifying approximations is considered, which allows us to take into account the radiation effects. It is shown that the fields in a moving observation point depend on its velocity and acceleration. In particular, although in a constant uniform electric field a force qE acts on a motionless charged particle, but on the same motionless but not fixed particle the force 4/3qE acts already, because it has a nonzero acceleration and the electric field at this point is larger. As the speed increases, the field decreases, and when it reaches the speed of light, when the particle stops accelerating, the force again becomes equal to qE The principle of operation of an unconventional alternator in a constant electric field and its corresponding engine, as well as new types of direct and impulse current generators, predicted by the equations, are described.

The exact solutions in the wave form are derived for the time fractional KdV and the time fractional Burgers' equations in conformable fractional derivative sense. The fractional variable change using the fundamental properties of the conformable derivative reduces both equations to some nonlinear ODEs. The predicted solution is assumed to be a finite series form of a function satisfying a particular first-order ODE whose solution contains an exponential function in the denominator. The solutions are represented in the explicit forms and illustrated by some choices of the parameters for various fractional orders of the equations.

The well-known equations for the process of powder compaction (PCP) in a rigid die published over the period from the beginning of the last century to the present are considered. Most of the considered equations are converted into the dependence of the densification pressure on the relative density. The equations were analyzed and their ability to describe PCP was assessed by determining the coefficient of determination when approximating experimental data on the compaction of various powders. It is shown that most of the equations contain two constants, the values of which are determined by fitting the mathematical dependence to the experimental curve. Such equations are able to describe PCP with high accuracy in the case of compaction of powders up to a relative density of 0.9-0.95. It is also shown that different equations can describe PCP in the range of density change from the initial to 0.9 with the same and sufficiently high accuracy, but when the process of compaction is extrapolated to higher values of density, the curves diverge. This indicates the importance of equations that can unambiguously describe PCP to a relative density equal to or close to 1.0. For an adequate description of PCP to a relative density greater than 0.95, equations containing three or four constants have proven themselves well.

We deal with Ulam stability of second order periodic linear differential equations with periodic damping. Necessary and sufficient condition for the Ulam stability of second order periodic linear differential equations is obtained by giving an Ulam stability theorem and an Ulam instability theorem. These results extend the conclusions in relevant literature. In addition, for the situation of constant coefficients, the minimum Ulam constant is given.

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

Abstract. Background: This paper investigates the propagation of behaviorally transmitted traits with negative effect on host fitness. Methods: We analyze equilibrium between genetically transmitted and behaviorally transmitted competing propagators and consider whether a behavioral propagator is linked to reproduction (e.g. vertical culture transmission), or not. We employ combined genetic and behavior-induced fitness components for hosts, while behavioral propagators have replication factors to distinguish from what’s good for the host (fitness). Results: A trait which spreads faster than its marginal host fitness contribution reduces population will establish itself. The often transient nature of laterally transmitted traits may be a defense against accumulation of deleterious traits. Laterally transmitted traits with high spreading rate often do not equalize with genetic traits, spreading outside natural selection of the hosts. Vertical transmission reduces replication rate and allows group selection against deleterious behaviorally transmitted traits. Competing mutually exclusive propagators contribute to inequality and altruism, but compete through adverse fitness since exclusivity assumes low conversion. Conclusion: Behaviorally transmitted traits, in some cases a tremendous advantage, may also be a significant problem in the development of societies.

The Rybczynski theorem describes the trade effect within production analyses between factor endowments and outputs. The Stolper-Samuelson theorem focuses on cost analyses between factor reward and commodity price. This paper examines the trade effect of changes of factor endowments on prices, based on general equilibrium. The study shows that changes of factor endowments cause domestic output changes (the Rybczynski effect), which affect output prices and factor prices (the Stolper-Samuelson effect). It is like a chain of effects that the Rybczynski’s trade effect triggers the Stolper-Samuelson’s trade effect. The analysis of this paper shows that a small increase of a factor endowment of any country rewards another factor and the commodity using the latter factor intensively. It displays a tuneful circle. Trade brings a well-balanced development to the world.

We present a normative methodology, for value-pricing B2B services, using a Nash Equilibrium mechanism. Value is fundamental to any service, yet it has defied a definition. The literature is conspicuously silent on how to define value, but abounds in richly descriptive characterizations. To overcome this deficit, we focus on the financial economics of service-value. We formulate an axiomatic definition and a differential equation that embodies the idea of service-value. We specify a set of value axioms and multidisciplinary postulates that coherently form our service-value constructs. Value-pricing is challenging. It is natural that providers desire a high price and customers want a low price. Realistically, both will agree on a win-win price. To uncover this price, we specify an algorithm to reveal the Nash Equilibrium. Once agreed, it validates providers’ and customers’ commitments, and payment obligations. Service value is cocreated by both provider and customer. For a reciprocal process, its treatment is remarkably asymmetric favoring the customer. We argue that one-sided descriptions of value-in-use and value-proposition are limiting mental models. We propose the additional ideas of value-from-use and value-supposition to strengthen the conceptual symmetry of cocreation. Our work reveals a critical gap in service science, metrology. Metrology, the science of measurements, is absent from service-science. Service-value is silent on the questions of quantities, units, scales, measurement principles and instruments. We argue for a call to action for Service Metrology. We sketch a roadmap of actionable suggestions to get started.

Meixner's historical remark in 1969 "... it can be shown that the concept of entropy in the absence of equilibrium is in fact not only questionable but that it cannot even be defined...." is investigated from today's insight. Several statements --such as the three laws of phenomenological thermodynamics, the embedding theorem and the adiabatical uniqueness-- are used to get rid of non-equilibrium entropy as a primitive concept. In this framework, Clausius inequality of open systems can be derived by use of the defining inequalities which establish the non-equilibrium quantities contact temperature and non-equilibrium molar entropy which allow to describe the interaction between the Schottky system and its controlling equilibrium environment.

Numerical models widely used for hydrocarbon phase behavior and compositional flow simulations are based on assumption of thermodynamic equilibrium. However, it is not uncommon for oil and gas-condensate reservoirs to exhibit essentially non-equilibrium phase behavior, e.g., in the processes of secondary recovery after pressure depletion below saturation pressure, or during gas injection, or for condensate evaporation at low pressures. In many cases the ability to match field data with equilibrium model depends on simulation scale. The only method to account for non-equilibrium phase behavior adopted by the majority of flow simulators is the option of limited rate of gas dissolution (condensate evaporation) in black oil models. For compositional simulations no practical yet thermodynamically consistent method has been presented so far except for some upscaling techniques in gas injection problems. Previously reported academic non-equilibrium formulations have a common drawback of doubling the number of flow equations and unknowns compared to the equilibrium formulation. In the paper a unified thermodynamically-consistent formulation for compositional flow simulations with non-equilibrium phase behavior model is presented. Same formulation and a special scale-up technique can be used for upscaling of an equilibrium or non-equilibrium model to a coarse-scale non-equilibrium model. A number of test cases for real oil and gas-condensate mixtures are given. Model implementation specifics in a flow simulator are discussed and illustrated with test simulations. A non-equilibrium constant volume depletion algorithm is presented to simulate condensate recovery at low pressures in gas-condensate reservoirs. Results of satisfactory model matching to field data are reported and discussed.

There is little doubt that life's origin followed from the known physical and chemical laws of Nature. The most general scientific framework incorporating the laws of Nature and applicable to most known processes to good approximation, is that of thermodynamics and its extensions to treat out-of-equilibrium phenomena. The event of the origin of life should therefore also be amenable to such an analysis. In this paper, I describe the non-equilibrium thermodynamic foundations of the origin of life for the non-expert. This ``Thermodynamic Dissipation Theory for the Origin of Life'' is founded on Classical Irreversible Thermodynamic theory developed by Lars Onsager, Ilya Prigogine, and coworkers.

The study aims to explain the different effects between the multiplication, addition, division and subtraction equations. In this sense, multiplication and addition equations develop a new element or fact, meanwhile, division equations express information within the element or fact, subtraction is the selection of element or fact. Furthermore, for a multiplication and addition equation results in the new element or fact, it is necessary to depend on specific physical concepts, without extreme alteration, as it would result in the alteration of the element or fact. It is worth highlighting the importance of fixed values ​​in the equation to describe the specific nature of the element or fact. The study makes it possible to understand open equations (without solutions) in mathematics and physics, in addition to understanding existing equations. The study makes reference to the theory of obligatory necessity and the theory of differences between elements.

The exact solutions of some conformable time fractional PDEs are presented explicitly. The modified Kudryashov method is applied to construct the solutions to the conformable time fractional Regularized Long Wave-Burgers (RLW-Burgers, potential Korteweg-de Vries (KdV) and clannish random walker's parabolic (CRWP) equations. Initially, the predicted solution in the finite series of a rational form of an exponential function is substituted to the ODE generated from the conformable time fractional PDE by using wave transformation. The coefficients used in the finite series are determined by solving the algebraic system derived from the coefficients of the powers of the predicted solution.

The structure of turbulent flow over non-flat surfaces is a topic of major interest in practical applications in both engineering and geophysical settings. A lot of work has been done in the fully rough regime at high Reynolds numbers where the effect on the outer layer turbulence structure and the resulting friction drag is well documented. It turns out that surface topology plays a significant role on the flow drag especially in the transitional roughness regime and therefore, hard to characterize. Survey of literature shows that roughness function depends on the interaction of roughness height, flow Reynolds number and topology shape. In addition, if the surface topology contains large enough scales then it can impact the outer layer dynamics and in turn modulate the total frictional force. Therefore, it is important to understand the mechanisms underlying drag increase from systematically varied surface undulations in order to better interpret quantifications based on mean statistics such as roughness function. In this study, we explore the mechanisms that modulate the turbulence structure over a two-dimensional (2D) sinusoidal wavy surface with a fixed amplitude, but varying slope. To accomplish this, we model the turbulent flow between two infinitely wide 2D wavy plates at a friction Reynolds number, $Re_{\tau}=180$. We pursue two different but related flavors of analysis. The first one adopts a roughness characterization flavor of such wavy surfaces. The second one focuses on understanding the non-equilibrium near surface turbulence structure and their impact on roughness characterization. Analysis of the different statistical quantifications show strong dependence on wave slope for the roughness function indicating drag increase due to enhanced turbulent stresses resulting from increased production of vertical velocity variance from the surface undulations.

The purpose of this study is to determine the equilibrium conditions for the formation of a mixture of propane and normal butane hydrates including temperature, pressure and mole fraction. In order to prevent the formation of hydrates in the cooling path, it is necessary to examine the conditions of hydrate formation and provide solutions. Modeling of hydrate formation conditions was performed using Hydoff software and compared with experimental results in this field, which obtained an acceptable error percentage. The range of temperature is between 267-276 °C and the molar percentage of propane is 0.7,0.8 and 0.9 and the mathematical equation was presented to predict hydrate formation. By analyzing the results, it was found that by increasing the concentration of ethane in the presence of other compounds, hydrate growth increased and hydrates formed more stable, also by increasing the concentration of propane and normal butane the amount of equilibrium pressure will decrease.

Good Boussinesq equations will be considered in this work. First we apply three combined compact schemes to approximate spatial derivatives of good Boussinesq equations. Then three fully discrete schemes are developed based on symplectic scheme in time direction, which are sympletic-structure preserving.Meanwhile, the convergence and conservation of the fully discrete schemes are analyzed. Finally, we present numerical experiments to confirm our theoretical analysis. Both our analysis and numerical test indicate that the fully discrete schemes are efficient in solving the spatial derivative mixed equation.

In this work, we introduce a new and generalized concept (herein referred to as Caputo fractional delta derivative and Caputo fractional delta dini derivative of order $\alpha\in(0,1)$) for Caputo fractional derivatives on an arbitrary time domain $\mathbb{T}$ which is a closed subset of $\mathbb{R}$. Combining the continuous and discrete time domains, we create a unified framework for stability analysis on time scales. Comparison results and stability criteria for the considered Caputo fractional dynamic equations are presented based on a new definition for Caputo fractional delta derivative of a Lyapunov function, contributing to the broader understanding of fractional calculus on time scales. The work also incorporates an illustrative example to demonstrate the relevance, effectiveness and applicability of the established stability results over that of the integer order.

The paper is considering of the basic differential equations of hydrodynamics: the equation of continuity and motion. On the simplest example, it is shown that the equation of continuity in a system with the equation of motion leads to contradictions and erroneous results of modeling. A more correct form of the continuity equation is described. It is shown that the equations of motion can be written in the form of complete differentials. Three possible integral forms of the equations of motion are presented. As a conclusion, the existence and smoothness of the solution of the Navier-Stokes equations are considered.

The paper considers the physical consequences of bimetric geodesically (pro jectively) invariant equations of gravitation. The correct expression for the energy- impulse tensor of the gravitational field is obtained and with its help the key role of the gravitational defect of mass-energy for the energy of the newborn universe is shown..

Most fully developed galaxies have vivid spiral structure, but the formation and evolution of spiral structure is still a mystery that is not fully understood in astrophysics. We find that the currently used equations of galactic dynamics contain some unreasonable components. In this paper, the following three working assumptions are introduced to simplify the galactic structural equations. 1. In the research of large-scale structure, the retarded potential of the gravitational field should be taken into account. The propagating time of the gravitational field from center to border is longer than the revolution periods of the stars near the center of galaxy. Newton's gravitational potential is unreasonable for such case, and the weak field and low velocity approximation of Einstein's field equation should be adopted. 2. The stars in a fully developed galaxy should be zero-pressure and inviscid fluid, and the equation of motion is different from that of ordinary continuum mechanics. Stars move along geodesics. 3. The structure of the galaxy is only related to the total mass density distribution. The equation of state of dark halo is different from that of ordinary luminous interstellar matter, so their trajectories are also very different. Dark halo and ordinary matter in galaxy are automatically separated. The total mass density distribution can be presupposed according to the observation data, and then it can be determined by comparing the solution of the equations with the observed data. These assumptions and treatments are supported by theory and observation. The variables of the equations of simplified galactic dynamics are separated from each other, and the equations are well-posed and can be solved according to a definite procedure. Therefore, this simplified dynamic equation system provides a more reasonable and practical framework for the further study of galactic structure, and can solve many practical problems. Besides, it is closely related to the study of dark energy and dark matter.

Exact solutions to conformable time fractional (3+1)- dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE. The resultant polynomial equation is solved by using algebraic operations. The method works for the Jimbo-Miwa, the Zakharov-Kuznetsov and the modified Zakharov-Kuznetsov equations in conformable time fractional forms. All the solutions are expressed in explicit forms.

Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. This involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We also state Abelian results which yield asymptotic formulas for weighted summatory function from that for the original summatory function

Recently, the properties of bouncing oil droplets, also known as `walkers', have attracted much attention because they are thought to offer a gateway to a better understanding of quantum behaviour. They indeed constitute a macroscopic realization of wave-particle duality, in the sense that their trajectories are guided by a self-generated surrounding wave. The aim of this paper is to try to describe walker phenomenology in terms of de Broglie-Bohm dynamics and of a stochastic version thereof. In particular, we first study how a stochastic modification of the de Broglie pilot-wave theory, à la Nelson, affects the process of relaxation to quantum equilibrium, and we prove an H-theorem for the relaxation to quantum equilibrium under Nelson-type dynamics. We then compare the onset of equilibrium in the stochastic and the de Broglie-Bohm approaches and we propose some simple experiments by which one can test the applicability of our theory to the context of bouncing oil droplets. Finally, we compare our theory to actual observations of walker behavior in a 2D harmonic potential well.

The general coupling between particle transport and ionization-recombination processes in hot plasma is considered on the key concept of equilibrium charge state (CS) transport. A theoretical interpretation of particle and CS transport is gained in terms of a two-dimensional (2D) Markovian stochastic (random) processes, a discrete 2D Fokker-Plank-Kolmogorov equation (in charge and space variables) and generalized 2D coronal equilibrium between atomic processes and particle transport. The basic tool for analysis of CS equilibrium and transport is the equilibrium cell (EC) (two states on charge and two on space), which presents (i) a unit phase volume, (ii) the characteristic scale of local equilibrium, (iii) a comprehensive solution for the simplest nonlinear relations between transport and atomic processes. The approach opens up new perspectives on transport studies: (i) the direct modelling of equilibrium and transport of impurity using the atomic data base, (ii) recovery of the complete recombination rate profile based on knowledge of density profiles and ionization rate profiles, (iii) the local transport analysis, based on the reduction of the equilibrium set to the single EC (in particular, central or edge), (iv) analysis of the reduced transport coefficients (diffusion and convection) on the density profile measurements.

This paper proposes a game-theoretic framework for quantifying effects of strategic bidding behavior of load aggregators (LAs) on wholesale electricity market equilibrium. The independent system operator (ISO) employs double-sided auction design to settle transactions in the wholesale electricity market. Generating companies (GENCOs) and LAs submit their bids to the ISO in the economic model of supply function equilibrium. Under the smart grid paradigm, a LA participates in the electricity market on behalf of end-consumers to minimize their total payment for purchasing electricity. In this paper, bi-level programming (BLP) method is employed to determine optimal bidding strategy of self-interested market participants. In the upper level, the profit of intended market participant is maximized and in the lower level, the ISO clears the market via a bid-based security constrained economic dispatch (SCED). Also, competition of GENCOs and LAs is modeled as a non-cooperative game. To examine the effects of strategic bidding behavior of LAs on market equilibrium, the proposed model is applied to the IEEE 9-bus and the IEEE 30-bus test systems. According to numerical results, to allow the LAs to behave strategically along with the GENCOs makes the former better off and the latter worse off, while the net effect of this on total social welfare turns out to be case-contingent.

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.

Nested structure is a structural feature that is conducive to system stability formed by the co-evolution of species in mutualistic ecosystems, and reflects the ability of ecosystem stability to be restored to a stable state again after being destroyed. The co-opetition relationship and value flow between industrial sectors in the global value chain are similar to the mutualistic ecosystem, and the pattern of the global economic system is always changing in dynamic equilibrium. Nestedness theory is used in this article to define the generalist and specialist sectors in the global value chain to analyze the changes in the global supply pattern. Then we study the mechanism of the global economic system to reach a stable equilibrium and the role of different sectors in the steady of the economic system, so as to provide countermeasures for enhancing the stability of the global economic system. At the end of the article, the domestic trade network, export trade network and import trade network of each country are extracted, and an econometric model is designed to analyze how the microstructure of the production system affects a country’s macroeconomic performance, thereby deriving the conclusion that the stability of the international trade network is crucial to a country's economic development.

In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power q=1m,m∈Z+, where the fractional derivative of order α=qγ for which γ∈Z+. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm's accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of problem. Under Caputo meaning the obtained results are illustrated numerically, and graphically. Geometrically, the behavior of the solution declares that the changing of the fractional derivative parameter values' in their domain alters the style of the attained solution in a symmetrical meaning and fully coinciding to the ordinary derivative value'. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for Integral, and integro-differental equations of fractional order.

A careful examination of the fundamentals of electromagnetic theory shows that due to the underlying mathematical assumptions required for Stokes' Theorem, global charge conservation cannot be guaranteed in topologically non-trivial spacetimes. However, in order to break the charge conservation mechanism we must also allow the electromagnetic excitation fields D, H to possess a gauge freedom, just as the electromagnetic scalar and vector potentials $\varphi$ and A do. This has implications for the treatment of electromagnetism in spacetimes where black holes both form and then evaporate, as well as extending the possibilities for treating vacuum polarisation. Using this gauge freedom of D, H we also propose an alternative to the accepted notion that a charge passing through a wormhole necessarily leads to an additional (effective) charge on the wormhole's mouth.

The absorption of medium-chain fatty acids (MCFA) depends on the solubility of these components in the gastric fluid. Parameters such as the total MCFA concentration, carboxyl ionization level, and carbon chain length affect the solubility of these molecules. Moreover, the enzymatic lipolysis of solubilized triacylglycerol (TAG) molecules may depend on the carbon chain length of the fatty acids (FAs) components and their positions on the glycerol backbone. This present study aimed at investigating the effect of electrolyte usually formed during the gastric digestion phase on the solubility of MCFA, and evaluating the influence of the FA carbon chain length on the lipolysis rate during the in vitro digestion simulation. The results obtained here showed that the increasing of electrolyte concentrations tend to decrease the mutual solubility of systems composed by the caproic and caprylic fatty acids + sodium chloride, sodium bicarbonate, and potassium chloride solutions. We also observed that a conventional version of the thermodynamic UNIQUAC model was able to correlate the liquid-liquid phase behavior of the electrolyte solutions. Regarding the in vitro digestion simulation, the experimental data indicated that the action of the pancreatic enzyme occurred preferentially in TAG molecules comprised of short and medium-chain fatty acids.

The three dimensional conformable time fractional Kadomtsev-Petviashvili and the conformable time fractional modified Kawahara equations are solved by implementing the Kudryashov's procedure. The corresponding wave transformation reduces both equations to some ODEs. Balancing the nonlinear and the highest order derivative terms gives the structure of the solutions in the finite series form. The useful symbolic tools are used to solve the resultant algebraic systems. The solutions are expressed in explicit forms.

An evolutionary game is introduced which considers game-theoretic strategies in the context of non-linear population matrix models. This game considers the states and actions of the organisms of the evolving population, and a notion of dynamic equilibrium between strategies is described. The game’s formalism is expounded and a proof about equilibrium is given; specifically that any stable equilibrium can be described by proportions of pure strategies; particularly when population matrices are not defective.

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.

In this paper, we investigate the equilibrium points of following a system of difference equations x_n+1 = x_n 2y_n−1, y _n+1 = y_n 2x_n−1. We also study the asymptotic stability of related system of difference equations. Further we examine the periodic solutions of related system with period two. Additionally, we find out the invariant interval and periodic cycles of related system of difference equations.

The aim of this paper is to obtain Karush-Kuhn-Tucker optimality conditions for weakly efficient solutions to vector equilibrium problems with the addition of constraints in the novel context of Hadamard manifolds as opposed to the classical examples of Banach, normed or Hausdorff spaces. More specifically, classical necessary and sufficient conditions for weakly efficient solutions to the constrained vector optimization problem are presented. As well as some examples. The results presented in this paper generalize results obtained by Gong (2008) and Wei and Gong (2010) and Feng and Qiu (2014) from Hausdorff topological vector spaces, real normed spaces and real Banach spaces to Hadamard manifolds, respectively.

Many systems in nature and laboratories are far from equilibrium and exhibit significant fluc- tuations, invalidating the key assumptions of small fluctuations and short memory time in or near equilibrium. A full knowledge of Probability Distribution Functions (PDFs), especially time- dependent PDFs, becomes essential in understanding far-from-equilibrium processes. We consider a stochastic logistic model with multiplicative noise, which has gamma distributions as stationary PDFs. We numerically solve the transient relaxation problem, and show that as the strength of the stochastic noise increases the time-dependent PDFs increasingly deviate from gamma distributions. For sufficiently strong noise a transition occurs whereby the PDF never reaches a stationary state, but instead forms a peak that becomes ever more narrowly concentrated at the origin. The addition of an arbitrarily small amount of additive noise regularizes these solutions, and re-establishes the existence of stationary solutions. In addition to diagnostic quantities such as mean value, standard deviation, skewness and kurtosis, the transitions between different solutions are analyzed in terms of entropy and information length, the total number of statistically distinguishable states that a system passes through in time.

The ordinary continuity equation relating the current and density of a system is extended to incorporate systems with dual (longitudinal and transverse) currents. Such a system of equations is found to have the same mathematical structure as that of Maxwell equations. The horizontal and transverse currents and the densities associated with them are found to be coupled to each other. Each of these quantities are found to obey a wave equation traveling at the velocity of light in vacuum. London's equations of super-conductivity are shown to emerge from some sort of continuity equations. The new London's equations are symmetric and are shown to be dual to each other. It is shown that London's equations are Maxwell's equations with massive electromagnetic field (photon). These equations preserve the gauge invariance that is broken in other massive electrodynamics. The duality invariance may allow magnetic monopoles to be present inside superconductors. The new duality is called the comprehensive duality transformation.

Where mining activities cause disturbance in catchments, streams are often impacted by heavy loads of fine eroded material. Since geomorphological processes are very slow, it is expected that during rehabilitation typically hundreds of years are required for a mine landform to return back to stability. A sensitive approach to analyzing post-mining landform stability in tropical regions is to assess the quantity of fine suspended sediments (FSS = silt + clay (0.45 µm < diameter < 63 µm)) leaving the catchment where the mine resides and entering the receiving streams in response to storm events. Continuous stream discharge and FSS quantities upstream and downstream of the catchment where the mine resides were modeled using HEC-HMS (Hydrologic Engineering Centre-Hydrologic Modeling System). Once calibrated, the model was run for a thousand years to predict continuous stream discharge and FSS quantities for various predicted rainfall scenarios. Short-term erosion and deposition across the mine catchment were also evaluated using a cali-brated landform evolution model, CAESAR-Lisflood. This paper reviews watershed soil erosion measurements and modeling research. It compares the previous research to approach above that assesses mine landform erosion and stability in terms of fine-suspended sediments. This approach can be used to determine mine landform erosion dynamics, predict achieving stability equilib-rium, and as a post-mining rehabilitation assessment tool.

Despite more than hundred years of research since Sir Sherrington’s studies on reflexes, his questions are still somehow unanswered. On what anatomical stage do the play of spinal reflex interaction take place? What are the physiological properties of this anatomical substrate? In this paper, we address these questions in light of the most advanced theory of motor control and the anatomical discoveries on the fascia that are changing how we think about control of action and perception. There are two sides of the problem: the neurological (reflex) connections that are at the base of movement, and the anatomical substrate that regulates and coordinates the movement. We recently advanced a hypothesis on how these two elements are connected and how they interplay. Here we further explain the concept of the somatic equilibrium point – SEP – and its central role in movement control and coordination. It is our belief that the concept of SEP explains how the neuro-mechanical control of movement is organized at peripheral level. At this level, intrafusal and extrafusal muscle fibres are combined in myofascial units, organized in anatomical directions. Myofascial units are closed systems whose behaviour can be affected by neural (voluntary) control or changes in external forces. SEPs represent the intrinsic equilibrium of the myofascial units, and are connected through the continuum of the fascia so that mechanical transfer of tension from segment to segment pre-adjust muscle fibers length and hence their excitation level. This is how coordination between segments is achieved. Finally, we suggest SEPs create the neurological representation of the referent configuration for action, and configurations are linked to the architecture of the fascial system.

Understanding the equilibrium saturation level is crucial to Binder Jetting (BJ). Saturation level influences dimensional accuracy, print time, green strength, and final material properties. Improved understanding of the saturation level can reduce development time for new materials and improve existing processes in BJ. Attempts have been made to predict saturation levels of parts with simple calculations from droplet primitives and capillary pressure. There is, however, limited experimental validation for these methods and they do not include the impact of drop velocity and droplet spacing. This study incorporates the influences of drop velocity and droplet spacing on the saturation level of the part. Drop primitives of varying droplet velocity and droplet spacing were compared. Results show that velocity impacts the feasible parameter space.

We prove general stability theorems for $n$-dimensional quartic-cubic-quadratic-additive type functional equations of the form \begin{eqnarray*} \sum_{i=1}^\ell c_i f \big( a_{i1}x_1 + a_{i2}x_2 + \cdots + a_{in}x_n \big) = 0 \end{eqnarray*} by applying the direct method. These stability theorems can save us much trouble of proving the stability of relevant solutions repeatedly appearing in the stability problems for various functional equations.

The use of biodiesel fuels in compression ignition engines leads to decrease CO, PM, HC and smoke opacity. However, NOx emissions increase importantly. Various methods to reduce NOx are used namely the EGR, the water injection method and the steam injection method. In this study, the steam injection method has been used instead of the other methods because of its benefits. This study examines the effects of steam injection on combustion products of diesel and biodiesel fuels by considering chemical equilibrium in order to determine the equilibrium combustion products involving 10 combustion products. A developed simulation code determing the equilibrium mole fractions and thermodynamic properties of combustion products is used for diesel and biodiesel fuels. It can be used for any fuel consisting of carbon, hydrogen or any oxygenated fuel. The results show that the mole fraction of CO2 and CO decrease with the steam injection ratios. NO mole fractions decrease with steam injection for lean mixtures but they increase slightly in the case of rich mixtures. The specific heat of combustion products increase with the steam injection ratios. Thus, the engine performance can be improved using the method. The model has been validated by comparing model results with the ones of NASA CEA software and GASEQ software using the methane as fuel. The relative errors of equilibrium mole fractions and thermodynamic properties of combustion products are less than 0.98 %.

This work presents significant results obtained by the application of global optimization techniques to the design of finite, normal form games with mixed strategies. To that end, the Fuzzy ASA global optimization method is applied to several design examples of strategic games, demonstrating its effectiveness in obtaining payoff functions whose corresponding games present a previously established Nash equilibrium. In other words, the game designer becomes able to choose a convenient Nash equilibrium for a generic finite state strategic game and the proposed method computes payoff functions that will realize the desired equilibrium, making it possible for the players to reach the favorable conditions represented by the chosen equilibrium. Considering that game theory is a very significant approach for modeling interactions between competing agents, and Nash equilibrium represents a powerful solution concept, portraying situations in which joint strategies are optimal in the sense that players cannot benefit from individually modifying their current strategies provided that other players do not change their strategies as well, it is natural to infer that the proposed method may be very useful for strategists in general. In summary, it is a genuine instance of artificial inference of payoff functions after a process of global machine learning, applied to their numerical components.

We derive time evolution equations, namely the Schrodinger-like equations and the Klein-Gordon equations for coherent fields and the Kadanoff-Baym (KB) equations for quantum fluctuations, in Quantum Electrodynamics (QED) with electric dipoles in 2 + 1 dimensions. Next we introduce a kinetic entropy current based on the KB equations in the 1st order of the gradient expansion. We show the H-theorem for the Leading-Order self-energy in the coupling expansion (the Hartree-Fock approximation). We show a conserved energy in the spatially homogeneous systems in the time evolution. We derive aspects of the super-radiance and the equilibration in our single Lagrangian. Our analysis can be applied to Quantum Brain Dynamics, that is QED with water electric dipoles. The total energy consumption to maintain super-radiant states in microtubules seems to be within the energy consumption to maintain the ordered systems in a brain.

Pairs of potential and flow variables have long been established for interfacing object-oriented component models of physical systems. Recently, however, the use of triplets has been discovered for the purpose of robust modeling. New and powerful Modelica libraries have been developed such as the DLR ThermoFluid Stream library or the introduction of the Dialectic Mechanics library. Their use of triplets is entangled with a special modeling approach that uses Linear Implicit Equilibrium Dynamics. In this paper, we study the basic motivation of this approach, its benefits and drawbacks before we finally demonstrate how it beneficially impacts the generation of corresponding simulation code.

Evolutionary theory suggests that the origin, persistence, and evolution of biology is driven by the “natural selection” of characteristics improving the differential reproductive success of the organism in the given environment. The theory, however, lacks physical foundation, and, therefore, at best, can only be considered a heuristic narrative, of some utility for assimilating the biological and paleontological data at the level of the organism. On deeper analysis, it becomes apparent that this narrative is plagued with problems and paradoxes. Alternatively, non-equilibrium thermodynamic theory, derived from physical law, provides a physical foundation for describing material interaction with its environment at all scales. Here we describe a “natural thermodynamic selection” of characteristics of structures (or processes), based stochastically on increases in the global rate of dissipation of the prevailing solar spectrum. The different mechanisms of thermodynamic selection are delineated for the different biotic-abiotic levels, from the molecular level at the origin of life, up to the level of the present biosphere with non-linear coupling of biotic and abiotic processes. At the levels of the organism and the biosphere, the non-equilibrium thermodynamic description of evolution resembles, respectively, Darwinian and Gaia descriptions, although the underlying mechanisms and the objective function of selection are fundamentally very different.

This works is related to the modeling of cellulose pyrolysis with a pseudo equilibrium approach. The objective is to model the kinetics of the cellulose pyrolysis with a semi-global mechanism obtained from the literature, in order to obtain the yield and the rate of formation of char. The pseudo equilibrium sense consists in the supposition that the solid phase devolatilization can be described kinetically - at finite rate - , preserving the competitive characteristic between the production of char and tar, while the gas phase can be described by means of chemical equilibrium. A set of ordinary, linear and non linear, differential equations was obtained and solved numerically with a simple but consistent scheme using the Totally Implicit Euler method. Chemical equilibrium was solved using CANTERA coupled with a code written in Matlab. Results showed that the scheme preserve the tar-gas competitive characteristic for cellulose pyrolysis. The gas phase, is defined as a mixture of CO₂, CO, H₂O, CH₄, H₂ and N₂ showed similarly composition compared with models from literature. Finally, the extension of the model to biomass in general, is straightforward, in order to include the hemicellulose and lignin.

The information entropy for Smoluchowski coagulation equation is proposed based on statistical mechanics. And the normalized particle size distribution is a lognormal function at equilibrium from the principle of maximum entropy and moment constraint. The geometric mean volume and standard deviation in the distribution function are determined as simple constant. The results reveal that the assumption that algebraic mean volume be unit in self-preserving hypothesis is reasonable in some sense. Based on the present definition of information entropy, the Cercignani’s conjecture holds naturally for Smoluchowski coagulation equation. Together with the proof that the conjecture is also true for Boltzmann equation, Cercignani’s conjecture will holds for any two-body collision systems, which will benefit the understanding of Brownian motion and molecule kinematic theory, such as the stability of the dissipative system, and the mathematical theory of convergence to thermodynamic equilibrium.

We investigated the dynamics of a simple master equation describing the evolution of a probability distribution between two states, \( p_1 \) and \( p_2 \) (with \( p_1 + p_2 = 1 \)). Our focus is on the behavior of the entropy \( S \), the distance \( D \) from the uniform distribution (\( p_1 = p_2 = 1/2 \)), and the free energy \( F \). We define two ratios, \( \eta_S = \frac{dS/dt}{dF/dt} \) and \( \eta_D = \frac{dD/dt}{dF/dt} \), and find that both diverge to plus-minus infinity as the system evolves towards the uniform distribution. This divergence marks a critical transition point where the master equation temporarily loses physical meaning. However, after this divergence, the system settles into a new, well-behaved regime where the solutions stabilize at finite values, representing a new equilibrium or steady state. This two-regime behavior underscores the intricate dynamics of simple probabilistic systems as they approach and surpass equilibrium, providing insights into the complex interplay of entropy, distance, and free energy in statistical mechanics.

The research paper provides a mathematical framework for understanding phase shift in wave phenomena, bridging theoretical foundations with real-world applications. It emphasizes the importance of phase shift in physics and engineering, particularly in fields like telecommunications and acoustics. Key equations are introduced to explain phase angle, time delay, frequency, and wavelength relationships. The study also introduces the concept of time distortion due to a 1° phase shift, crucial for precise time measurements in precision instruments. The research also addresses infinitesimal wave energy loss related to phase shift, enriching our understanding of wave behavior, and impacting scientific and engineering disciplines.

Electrically charged particles such as Electrons and Protons carry electric, E, and magnetic, B, fields. In addition to these fields, Quantum Mechanics (QM) endows these particles with an `arcane and spooky' field --- the wavefunction. This wavefunction of QM is not only assumed to be separate but distinct from the electromagnetic field. We herein upend this view by demonstrating otherwise. That is, we demonstrate that the four components of the Dirac wavefunction, can be shown to not only be an intimate, but, a direct function of the electromagnetic field carried by the particle in question. Insofar as unity, depth in our understanding and insight into both Dirac and Maxwell's equations as major pillars of Modern Physics, we believe that this work may very well inch us one-step-closer to the truth.

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.

We establish certain modular equations of degree three based on some results of Ramanujan's theta functions.

Autonomous differential equations of fractional order and non-singular kernel are solved. While solutions can be obtained through numerical, graphical, or analytical solutions, we seek an implicit analytical solution.

In this paper we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, we analyze the analytical solution for a class of fractional models associated with Logistic model, Von Foerster model and Bertalanffy model is presented graphically for various fractional orders and solution of corresponding classical model is recovered as a particular case.

In the FCC conversion of heavy petroleum fractions as atmospheric residues, the main challenge for refiners to achieve the quantity and quality of various commercial products depends essentially on the catalyst used in the process. A deep characterization of the catalyst at different steps of the process (fresh, regenerated, and spent catalyst) was investigated to study the catalyst's behavior including the physicochemical evolution, the deactivation factor, and kinetic-thermodynamic parameters. All samples have been characterized by various spectroscopy methods such as N2 adsorption-desorption, UV-visible spectroscopy, Raman spectroscopy, LECO carbon analysis, Scanning electron microscopy (SEM), X-ray diffraction (XRD), X‐ray fluorescence (XRF), nuclear magnetic resonance spectroscopy (NMR13C) analysis, and Thermogravimetric analysis. The results of N2 adsorption-desorption, UV-vis, Raman, LECO carbon, and SEM images showed that the main causes of catalyst deactivation and coking were the deposition of carbon species that covered the active sites and clogged the pores, and the attrition factor due to thermal conditions and poisonous metals. The XRD and XRF results showed the catalyst’s physicochemical evolution during the process and the different interlinks between catalyst and feedstock (Nickel, Vanadium, Sulfur, and Iron) elements which should be responsible for the coking and catalyst attrition factor. It has been found that, in addition to the temperature, the residence time of the catalyst in the process also has an influence on catalyst structure transformation. NMR13C analysis revealed that polyaromatic hydrocarbon is the main component in the deposited coke of the spent catalyst. Thermogravimetric analysis showed that the order of catalyst mass loss was fresh > regenerated > spent catalyst due to the progressive losses of the hydroxyl bonds (OH) and the structure change along the catalyst thermal treatment. Moreover, the kinetic and thermodynamic parameters showed that all zones are non-spontaneous endothermic reactions.

The paper proves the existence of a family of axisymmetric equilibrium figures as solutions of a stationary problem with unknown boundaries for the Navier–Stokes equations corresponding to the slow rotation of a viscous compressible two-layer liquid mass about some axis. It is assumed that the liquids are barotropic, capillary and have different viscosities, the internal fluid being bounded by a closed surface. This interface does not intersect with the external boundary of the cloud. The proof is based on implicit function theorem and carried out in the Hölder spaces.

To date, several different approaches are available to study sediment dynamics at reach or watershed scale, based on very different hypothesis. One of such assumptions, the so-called “morphodynamic equilibrium hypothesis” is becoming little unpopular for its embedded simplifications. The aim of this work is to demonstrate how this approach proves yet effective in modelling landscape morphodynamics at the watershed scale, for what concerns the longitudinal profile of a river and the sedimentary aspects. The application of a 1-D model based on the equilibrium hypothesis has been implemented for several large rivers worldwide. Geomorphological parameters have been analysed, which describe the evolution of longitudinal profile (concavity) and sediments characteristics (aggrading and fining), and the results show a reasonably good correspondence with qualitative estimation of the same parameters. At the scale of analysis and for the chosen systems, which show high inertia to geomorphological changes likely owing to their longitudinal extension, the model can detect where the present conditions reflect a big disturbance to the “natural equilibrium” thus allowing water managers to identify present issues to be addressed.

We explored an extension of the Cauchy-Riemann equations to the algebra of treons, recently described by Alejandro Bermejo, whose elements are ordered 3-tuples. We leveraged the isomorphism between the algebra of treons and algebra B, and deduced the Cauchy-Riemann equations for the algebra of treons, establishing the necessary conditions for analyticity in this algebraic structure. This work significantly broadened our horizons in complex analysis and introduced new possibilities for applications across various fields of advanced mathematics.

We present a plausible solution to the now forty seven year old paleoclimatology riddle of the so-called Faint Young Sun Paradox via the combined hypothesis of the conservation of the state of radiation balance between the Earth and Sun and that of an expanding Earth, where, in the face of a changing (increasing) Solar luminosity, the Earth would maintain steady temperatures by re-adjusting the height of its atmosphere. That is to say, depending on whether or not the radius of the solid Earth is changing, this re-adjustment of the height of the Earth’s atmosphere would mean two things — i.e.: (1) either the height increases — in which event the Earth accretes matter from its immediate surroundings (i.e., the obvious pool formed by the Solar wind) thereby increasing the mass of the Earth’s atmosphere, or: (2) the height decreases — in which event the Earth naturally expels matter from its atmosphere, thereby decreasing the effective mass of the Earth. We demonstrate that if –- as the current state of the art ITRF observations seem to indicate, namely that — the Earth’s landmass is steadily expanding globally at a paltry rate of ∼+0.45±0.05mm·yr−1, and, that the Earth’s atmosphere is to have a present radial vertical height of about one third of the Earth’s radius (∼2860km) from the Earth’s surface, then, one can (might) with relative ease, explain not only the presence of liquid water on the Earth’s surface some ∼3.20±0.70Gyr ago during the Archaean eon when the Sun was about 75% of its current luminosity, but also the present radial expansion rate of the Earth. When all is said and done, the Earth system is herein cast as an auto-self-regulating incubator where the auto-self-regulating mechanism is as a result of the Earth’s atmosphere responding by automatically re-adjusting its height.

Recently several papers have been published on the Liénard equation x'' +L x+m x^3+n x^5=0, where the authors studied their explicit solutions and their applications. Here we describe the complete dynamics of these differential equations in the Poincar\'e disc. The Poincar\'e disc is the closed disc centered at the origin of coordinates of radius one, the whole plane R^2 is identified with the interior of this disc, and its boundary, the circle $\sss^1$ is identified with the infinity of R^2.

The aim of this article is to propose a core game theory model of transaction costs wherein it is indicated how direct costs determine the probability of loss and subsequent transaction costs. The existence of optimum is proven, and the way in which exposure influences the location of the optimum is demonstrated. The decisions are described as a two-player game and it is discussed how the transaction cost sharing rule determines whether the optimum point of transaction costs is the same as the equilibrium of the game. The dispute between actors regarding changing the share of transaction costs to be paid by each party is modeled by a non-cooperative bargaining game. Requirements of efficient transaction cost sharing rules are defined, and it is posited that solution exists which is not unique. Policy conclusions are also devised based on principles of design of institutions to influence the nature of transaction costs.

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.

The financial equilibrium of the enterprise is an important company function ensuring by itself the maintenance of the enterprise of the competitive market. The financial equilibrium is analysed in numerous empirical studies. In this empirical study we highlight for the first time in a research the impact that taxation has on the financial equilibrium of some companies over ten years, period that includes the pre-crisis, the financial crisis and post-crisis periods.