For crystal periodic structure prediction, a new and concise approach based on the principles of statistical physics was employed to derive a new form of the equation to determine their period vectors (cell edge vectors), under general external stress. Then the new form is applicable to both classical physics and quantum physics. It also turned out to be the equation of state and the mechanical equilibrium condition for crystals under external stress and temperature. It should be helpful in piezoelectric and piezomagnetic studies, as the period vectors were changed by the external stress. For linear elastic crystals, it is actually also the microscopic but temperature-dependent form of the generalized Hooke's law, then can be used to calculate the corresponding elastic constants of the law, for given temperatures.

Since crystals are made of periodic structures in space, predicting their three period vectors starting from any values based on the inside interactions is a basic theoretical physics problem. For the general situation where crystals are under constant external stress, we derived dynamical equations of the period vectors in the framework of Newtonian dynamics, for pair potentials recently (doi:/10.1139/cjp-2014-0518). The derived dynamical equations show that the period vectors are driven by the imbalance between the internal and external stresses. This presents a physical process where when the external stress changes, the crystal structure changes accordingly, since the original internal stress can not balance the external stress. The internal stress has both a full kinetic energy term and a full interaction term. It is extended to many-body interactions in this paper. As a result, all conclusions in the pair-potential case also apply for many-body potentials.

In crystal structure prediction, Newton's second law can always be applied to particles (atoms or ions) in a cell to determine their positions. However the values of the period vectors (crystal cell edge vectors) should be determined as well. Here we applied the dynamical equations of period vectors derived recently based on Newton's laws (doi:/10.1139/cjp-2014-0518) for that purpose, where the period vectors are driven by the imbalance between the internal and external pressures/stresses. For equilibrium states, they became equations of state, which essentially turn out the equilibrium conditions of crystals from mechanical point of view. For external pressure, by ignoring some effects, they reduced to the Mie-Gruneisen equation. Since the internal stress has both a full kinetic energy term and a full interaction term, the influences of both external temperature and stress/pressure on crystal structures can be calculated, then thermodynamical properties and processes were presented. Contrary to usual ideas, the equations show that pure harmonic vibrating phonons can result in thermal expansion in crystals when the external temperature is changed. Finally, crystal system collapse due to temperature and/or stress/pressure change was discussed.

A reduced-dimension (RD) method based on the proper orthogonal decomposition (POD) technology and the linearized Crank-Nicolson mixed finite element (CNMFE) scheme for solving the 2D nonlinear Extended Fisher-Kolmogorov (EFK) equation is proposed. The method reduces CPU runtime and error accumulation by reducing the dimension of the unknown CNMFE solution coefficient vectors. For this purpose, The CNMFE scheme of the EFK equation is established and the uniqueness, stability and convergence of the CNMFE solutions are discussed. Subsequently, the matrix-based RDCNMFE scheme is derived by applying the POD method. Furthermore, the uniqueness, stability and error estimates of the linearized RDCNMFE solution are proved. Finally, numerical experiments are carried out to validate the theoretical findings. In addition, we contrast the RDCNMFE method with the CNMFE method, highlighting the advantages of the dimensionality reduction method.

By using Modified simple equation method, we study the Cahn Allen equation which arises in many scientific applications such as mathematical biology, quantum mechanics and plasma physics. As a result, the existence of solitary wave solutions of the Cahn Allen equation is obtained. Exact explicit solutions interms of hyperbolic solutions of the associated Cahn Allen equation are characterized with some free parameters. Finally, the variety of structure and graphical representation make the dynamics of the equations visible and provides the mathematical foundation in mathematical biology, quantum mechanics and plasma physics.

Under suitable continuity and uniqueness conditions, solutions of an α order Caputo fractional boundary value problem are differentiated with respect to boundary values and boundary points. This extends well-known results for nth order boundary value problems. The approach used is a standard technique and makes heavy use of recent results for differentiation of solutions of Caputo fractional intial value problems with respect to initial conditions and continuous dependence for Caputo fractional boundary value problems.

We investigate a singularly perturbed q-difference differential Cauchy problem with polynomial coefficients in complex time and space with quadratic nonlinearity. We construct local holomorphic solutions on sectors in the complex plane with respect to the perturbation parameter with values in some Banach space of formal power series in space with analytic coefficients on shrinking domains in time. Two aspects of these solutions are addressed. One feature concerns asymptotic expansions in the parameter for which a Gevrey type structure is unveiled. The other fact deals with confluence properties as q tends to 1. In particular the built up Banach valued solutions are shown to merge in norm to a fully bounded holomorphic map in all its arguments that solves a nonlinear partial differential Cauchy problem.

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.

Hamiltonians describing the relativistic quantum dynamics of a particle with an arbitrary spin are shown to exhibit a supersymmetric structure when the even and odd elements of the Hamiltonian commute. For such supersymmetric Hamiltonians an exact Foldy-Wouthuysen transformation exits which brings it into a block-diagonal form separating the positive and negative energy subspaces. Here the supercharges transform between energy eigenstates of positive and negative energy. The relativistic dynamics of a charged particle in a magnetic field is considered for the case of a scalar (spin-zero) boson obeying the Klein-Gordan equation, a Dirac (spin one-half) fermion and a vector (spin-one) boson characterised by the Proca equation.

Additive manufacturing (AM) techniques are widely investigated for the cost-effective use of titanium (Ti) alloys in various aerospace applications. One of the AM techniques developed for such applications is plasma transferred arc solid free-form fabrication (PTA-SFFF). Materials manufactured through AM techniques often exhibit anisotropies in mechanical properties due to the layer-by-layer material build. In this regard, the present study investigates the isothermal directional fatigue of Ti-TiB metal matrix composite (MMC) manufactured by PTA-SFFF. This investigation includes rotating beam fatigue test, electron microscopy, and fatigue calculations. The fatigue experiments were performed at 350 ºC using specimen with the test axis oriented diagonally (45º) and parallel (90º) to the AM builds directions. The fatigue values from the current experiments along with literature data find that Ti MMC manufactured via PTA-SFFF exhibit fatigue anisotropy reporting highest strength in 90º and lowest in perpendicular (0º) AM build directions. Further, the electron microscopy investigations on 0º, 45º, and 90º AM build specimens reveal frequent TiB clusters in all three AM build directions and suggest that the spread of these TiB clusters play a role in fatigue anisotropy. Moreover, the fatigue calculations were performed using both the Paris’ and recently reported modified Paris’ equations. Comparison of R^2 values for these calculations show that modified Paris’ equation predicts the fatigue life of AM Ti-TiB MMC more accurately than the Paris’ equation.

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.

In this paper, the transmutation operator method is used to construct an exact solution to the initial boundary value problem for the equation of multidimensional free transverse vibration of a thin elastic plate with a singular Bessel operator acting on geometric variables. We emphasize that multidimensional Erdélyi–Kober operators of fractional order have the property of a transmutation operator, allowing one to transform more complex multidimensional partial differential equations with singular coefficients acting over all variables into simpler ones. If formulas for solutions are known for a simple equation, then we also obtain representations for solutions to the first complex partial differential equation with singular coefficients. In particular, it is successfully applied to the singular differential equations, especially with Bessel-type operators. Using this operator, the considered problem is reduced to a similar problem without the Bessel operator. Based on the solution to the auxiliary problem, an exact solution to the original problem is constructed and analyzed.

For almost 75 years, the general solution for the Schrödinger equation was assumed to be generated by an exponential or a time-ordered exponential known as the Dyson series. We study the unitarity of this solution in case of singular Hamiltonian and provide a new methodology that is not based on the assumption that the underlying space is L2(R). Then, an alternative operator for generating the time evolution is suggested that is manifestly unitary, regardless of the choice of the Hamiltonian. The new construction involves an additional positive operator that normalizes the wave-function locally and allows us to preserve unitary even on indefinite norm spaces. Our considerations show that Schrödinger's and Liouville's equations are, in fact, two sides of the same coin, and together they become the unified description of quantum systems.

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 objective of the research was to examine the determinants that determine farmer participation and shallot farming development in search of effective farm management practices. The study used Structural Equation Modeling data analysis. The primary data was collected from direct-structural interviews with 150 randomly chosen shallot farmers in Bantaeng Regency, Indonesia. It was found that the physical attributes of the land, the economic framework of the community, and the political system of the agricultural community were fundamental factors that exerted a positive and significant influence on farmer participation. Therefore, improvements in the physical aspects of land, the economic framework, and the political structure of agricultural communities could promote farmer participation. Furthermore, the farmers’ participation and the political structure of agricultural communities have a positive and significant impact on the development of shallot farming. Thus, by increasing the influence of government officials and community leaders, shallot farming can be promoted. The farmers can then enhance their participation in the shallot farming plan's formulation and implementation, providing the continued development of shallot farming. The findings of this study contribute significantly to the body of knowledge by validating previous research and proposing different ways to improve effective farm management practices in shallot farming.

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 study, we propose a conservative and compact finite difference scheme designed to preserve both the mass change rate and energy for solving the sixth-order Boussinesq equation with the surface tension. Theoretical analysis confirms that the proposed scheme achieves second-order accuracy in temporal discretization and fourth-order accuracy in spatial discretization. The solvability, convergence, and stability of the difference scheme are rigorously established through the application of the discrete energy method. Additionally, a series of numerical experiments are conducted to illustrate the effectiveness and reliability of the conservative scheme for long-time simulations.

We will demonstrate that the photon energy density parameter Ωγ of the universe can be derived exactly in Rh = ct cosmology. We find that it must be Ωγ = 1/5760π ≈ 5.52621330180192× 10^(−5). This indicates that there is no uncertainty in the radiation density within at least somesub-classes of Rh = ct cosmology.

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.

The Diophantine equation for which p is prime 2^n = p^2 + 7 and n is integers with p > 0, is examined in this work. It is demonstrated by examining the characteristics of the equation that the only solutions meeting these requirements are p = 3 with n = 5 and p = 5 with n = 7. In order to analyze the above equation, it is necessary to look at the properties of primes and how they relate to powers of two. These results advance our knowledge of prime numbers and how they behave in certain mathematical problems.

Up to now, Schrödinger equation, Klein-Gordon equation (KGE) and Dirac equation are believed the fundamental equations of quantum mechanics. Schrödinger equation has a defect that there is no NKE solutions. Dirac equation has positive kinetic energy (PKE) and negative kinetic energy (NKE) branches. Both branches should have low momentum, or nonrelativistic, approximations: one is Schrödinger equation and the other is NKE Schrödinger equation. KGE has two problems: it is an equation of second time derivative, and calculated density is not definitely positive. To overcome the problems, it should be revised as PKE and NKE decoupled KGEs. The fundamental equations of quantum mechanics after the modification have at least two merits. They are of unitary in that everyone contains the first time derivative and are symmetric with respect to PKE and NKE. This reflects the symmetry of the PKE and NKE matters, as well as matter and dark matter, of our universe. The problems of one-dimensional step potentials are resolved by means of the modified fundamental equations for a nonrelativistic particle.

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.

Einstein equation approximated to second order in Newton constant G is quantized and applied to a large region of the universe. The two-point correlations of the vacuum energy density and pressure are included and the resulting metric agrees with the one derived from the $\Lambda CDM$ cosmological model plus an additional contribution of the quantum vacuum. The conclusion is that at least a part of the acceleration in the expansion of the universe is due to the vacuum fluctuations.

The current study compared the goodness of fit and estimation precision of Bayes and maximum likelihood (ML) structural equation modeling across sample sizes. Data consisted of Markov Chain Monte Carlo generated samples of 50, 75, 100, 125, 150, 200, 250, 500, 750, 1000, and 1500 observations. The Bayes and ML methods were used to estimate a structural equation model with 40 parameters, including 12 continuous observed indicators, four latent variables, and three structural paths. Analyses were conducted with the Mplus 8.0 statistical software and were replicated 500 times with each of the 22 conditions (11 sample sizes x 2 estimation methods). Indices of model fit were the Bayes information criterion, the Tucker-Lewis index, the comparative fit index, the root mean square error of approximation, the posterior predictive p-value (for Bayes models), and the ꭓ2 test (for ML models). Results showed that as the sample size increased, both estimation methods exhibited improved model fit and enhanced estimation precision, and both methods performed well with samples of 200 or larger; however, ML demonstrated a marginally superior fit with smaller samples. These findings were not consistent with previous research and suggest that ML remains a better choice with continuous data and small samples.

Recently, there has been a growing interest in the production of electricity from renewable energy sources (RES). The RES investment is characterized by uncertainty, which is long-term, costly, depend on feed-in-tariff and support schemes. In this paper, we address the real option valuation (ROV) of a solar power plant investment. The real option framework is investigated. This framework considers the renewable certificate price, furthermore the cost of delay between establishing and operating the solar power plant. The optimal time of launching the project and assess the value of deferred option are discussed. The new three stage numerical methods are constructed, the Lobatto3C-Milstein (L3CM) methods. The numerical methods are integrated with concept of Black-Scholes option pricing theory, and applied in option valuation for solar energy investment with uncertainty. The numerical results of L3CM, finite difference and Monte Carlo methods are compared to show the efficiency of our methods. Our data set refers to the Arab Republic of Egypt.

A one-dimensional model for fluid and solute transport in poroelastic materials (PEM) is studied. Although the model was recently derived and some exact solutions, in particular steady-state solutions and their applications, were studied, special cases occurring when some parameters vanish were not analysed earlier. Since the governing equations are nonintegrable in nonstationary case, the Lie symmetry method and modern tools for solving ODE systems are applied in order to construct time-dependent exact solutions. Depending on parameters arising in the governing equations, several special cases with new Lie symmetries are identified. Some of them have a highly nontrivial structure that cannot be predicted from a physical point of view or using Lie symmetries of other real-world models. Applying the symmetries obtained, multiparameter families of exact solutions are constructed, including those in terms of elementary and special functions (hypergeometric, Whittaker, Bessel and modified Bessel functions). A possible application of the solutions obtained is demonstrated and it is shown that some exact solutions can describe (at least qualitatively) the solute transport in PEM. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems.

The authors undertook pilot studies to assess the accuracy of the reverse engineering process of worn nonstandard spur gears (non-standardized modulus value, unusual profile/pressure angle, high or low teeth), using conventional techniques and measuring instruments. Eight gears were tested, the module of which ranges from 1.020 to 4.98 mm and the number of teeth from 13 to 42. The key parameter, the basic pitch, was selected to estimate the accuracy of the process. The goal is to determine the value of the profile angle. Eleven models were proposed to estimate the nominal tolerance field, using various types of random data distribution. The tested gears were made in IT grade: 6, 7, 8 and 9 according to DIN 3961. Vernier disc micrometers with various measurement ranges were used for research. It has been shown that the nominal module does not have to be treated as a random variable in the population. EIG (Equation of Identity) was developed, allowing conversion of any gear with specific values of geometric parameters into an identical gear with alternative values of these parameters. The directions and degrees of shifts of the estimated tolerance fields relative to the nominal field were determined. The most effective estimating model was selected in the context of the percentage filling of the nominal tolerance field, taking into account the symmetric Student-Fisher distribution with a confidence level of 60%. Further directions of the research activities were described.

Rheometric investigations facilitate the rheological analysis of cementitious suspensions. However, the choice of the rheometric device and raw data conversion affect the results. Contrary to small gap rheometry, where the flow data shear stress

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.

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.

In this paper, we develop an effective iterative algorithm to solve a generalized Sylvester tensor equation over quaternions which includes several well-studied matrix/tensor equations as special cases. We discuss the convergence of this algorithm within a finite number of iterations, assuming negligible round-off errors for any initial tensor. Moreover, we demonstrate the unique minimal Frobenius norm solution achievable by selecting specific types of initial tensors. Additionally, numerical examples are presented to illustrate the practicality and validity of our proposed algorithm. These examples include demonstrating the algorithm’s effectiveness in addressing three-dimensional microscopic heat transport and color video restoration problems.

Uncertain fractional differential equation (UFDE) is an excellent tool for describing complicated dynamic systems. This paper analyzes the valuation problems of currency option based on UFDE under optimistic value criterion. Firstly, an uncertain fractional currency model is formulated to describe the dynamics of foreign exchange rate. Then, the pricing formulae of European, American and Asian currency option are obtained under optimistic value criterion, respectively. Finally, numerical simulations are provided to illustrate the validity of our theoretical results.

In this paper, we consider the Cauchy problem of a time fractional nonlinear diffusion equation. According to the Kaplan’s first eigenvalue method, we first prove the blow-up of the solutions in finite time for some sufficient conditions. We next give sufficient conditions for the existence of global solutions by using the result of Zhang and Sun. In conclusions, we find the second critical exponent for the existence of global and non-global solutions via the decay rates of the initial data at spatial infinity.

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.

We propose a new hierarchy of the vector derivative nonlinear Schr{\"o}dinger equations and consider the simplest multiphase solutions of this hierarchy. The study of the simplest solutions of these equations led to the following results. First, the three-leaf spectral curves $\Gamma=\{(\mu,\lambda)\}$ of the simplest multiphase solutions have a quite simple symmetry. They are invariant with respect to holomorphic involution $\tau$. The type of this involution depends on the genus of the spectral curve. Or the involution has the form $\tau:(\mu,\lambda)\to(\mu,-\lambda)$, or $\tau:(\mu,\lambda)\to(-\mu,-\lambda)$. The presence of symmetry leads to the fact that the dynamics of the solution is determined not by the entire spectral curve $\Gamma$, but by its factor $\Gamma/\tau$, which has a smaller genus. Secondly, it turned out that the dynamics of the two-component vector $\bp=(p_1,p_2)^t$ is determined, first of all, by the dynamics of its length $\abs{\bp}$. Independent equations determine the dependence of the direction of the vector $\bp$ from its length. In cases where the direction of the vector $\bp$ is fixed, the corresponding spectral curve splits into separate components. In conclusion, we note that, as in the case of the Manakov system, the equation of the spectral curve is invariant with respect to the orthogonal transformation of the vector solutions. I.e., the solution can be found from the spectral curve up to the orthogonal transformation. This fact indicates that the spectral curve does not depend on the individual components of the solution, but on their symmetric functions. Thus, the spectral data of multiphase solutions have two symmetries. These symmetries make it difficult to reconstruct signals from their spectral data. The work contains examples illustrating these statements.

Using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations thereby establishing their complete integrability. The Gardner equation is chosen as the key example for it comprises both the Korteweg-de Vries and modified Korteweg de Vries (mKdV) equations. The Gardner and Miura transformations which connect these equations are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota’s method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case) whereas the focusing Gardner equation has the standard elastically colliding solitons. The paper’s aim is to provide a review of integrability properties and solutions of the Gardner equation and illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica but can be adapted for major computer algebra systems.

Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by spinors - coordinate spinors, momentum spinors, spinor wave functions. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Schrödinger equation the wave function is represented by a single complex quantity, in the Dirac equation a step forward is taken and the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Using the example of the interaction between an electron and an electromagnetic field, we can see that the spinor equation contains more detailed information about the interaction than the vector equations. This is not new for quantum mechanics, since it describes interactions using complex wave functions, which cannot be observed directly, and only when measured goes to probabilities in the form of squares of the moduli of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors.In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In section 3 of the paper this quadratic form is expressed through momentum spinors, which makes it possible to obtain an equation for the spinor wave function in spinor coordinate space by replacing the momentum spinor components by partial derivative operators on the corresponding coordinate spinor component. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. The Lagrangian densities for the scalar field and for the electron field, along with their corresponding propagators, are presented. An equation of motion for the electron is proposed that is relativistically invariant, in contrast to the Dirac equation, which lacks this invariance. This novel equation permitted the construction of a relativistically invariant procedure for the second quantization of the fermion field in spinor coordinate space.

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.

In the paper the new, fractional order, state space model of a heat transfer in a one- 1 dimensional metallic rod is addressed. The fractional orders of derivative along space and time 2 are not exactly known and they are described by intervals. The proposed model is the interval 3 expanding of the state space fractional model of the one dimensional heat transfer. It is expected to 4 better describe the reality because interval order of each real process is hard to estimate. Elementary 5 properties of the proposed model are analysed. Theoretical results are experimentally validated using 6 data from real experimental laboratory system.

The main object of this work is to establish the existence and uniqueness of a solution to the 3D Navier-Stokes (NS) system for an incompressible fluid with viscosity. The nonlinearity of the NS system, as well as the need to estimate velocity and pressure for every value of the viscosity parameter [1] make them challenging to solve. In this regard, in the present work, we study the Navier-Stokes system, which describe the flow of a viscous incompressible fluid and the solution was obtained for velocity and pressure in an analytical form. In addition, the found pressure distribution law, which is described by a Poisson type equation and plays a fundamental role in the theory of Navier-Stokes systems in constructing analytic smooth (conditionally smooth) solutions.

The differential-equation of second order for space-time oscillation in flat tangential-spacetime of Minkowski-type can be written as an energy-equation. For this energy-equation the analogon of damping energy in two coupled Minkoswki-spacetimes with ftl can be calculated and formulated over Lamberts W-function or via Euler-equation.

In this paper, we establish the necessary and sufficient conditions for solving a dual split quaternion matrix equation AXB=C, and present the general solution expression when solvability is achieved. As an application, we delve into the necessary and sufficient condition for the existence of Hermitian solution to this equation by using a newly defined real representation method. Furthermore, we obtain the solutions for the dual split quaternion matrix equations AX=C and XB=C. Finally, we provide a numerical example to demonstrate the findings of this paper.

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.

This work focused on the solubility of ethane in three promising ionic liquids {1-Hexyl-3-methylimidazolium bis(trifluormethylsulfonyl) imide [HMIM][Tf2N], 1-Butyl-3-methyl-imidazolium dimethyl-phosphate [BMIM][DMP] and 1-Propyl-3- methylimidazolium bis(trifluoromethyl-sulfonyl)-imide [PMIM][Tf2N]}. The solubilities were measured at 303.15 K to 343.15 K and pressures up to 1.4 MPa using a gravimetric microbalance. The overall ranking of ethane solubility in the ionic liquids from highest to lowest is the following: [HMIM][Tf2N] > [PMIM][Tf2N] > [BMIM][DMP]. The Peng- Robinson equation of state was used to model the experimental data using three different mixing rules: van der Waals one, van der Waals two and Wong-Sandler mixing rules combined with the Non-Random Two-Liquid model. The average absolute deviations for the three mixing rules for the ionic liquids at the three temperatures were (4.39, 2.45 and 2.45) %, respectively. Henry’s Law constants for ethane in [BMIM] [DMP] were the highest (lowest solubility) amongst other ionic liquids studied in this work. The selectivity of CO2 over C2H6 was estimated at the three temperatures and the overall ranking of the selectivity was in the following order: [PMIM][Tf2N] > [BMIM][DMP] > [HMIM][Tf2N]> Selexol. Selexol, a widely used physical solvent in gas sweetening has better selectivity than the three ionic liquids studied. [PMIM][Tf2N], a promising solvent, has the highest selectivity among the three ILs studied, and would therefore be the best choice if in addition to carbon dioxide capture, ethane co-absorption was to be avoided. The enthalpy and entropy of solvation at infinite dilution were also estimated.

In this paper we have investigated the driven quintic Duffing equa- tion. Using some analysis technique, we are able to predict the number of limit cycles around the equilibrium and to develop a theoretical ap- proach to chaos transition in damped driven systems.

The present work is concerned with the study of the generalized Langevin equation and its link to the physical theories of statistical mechanics and scale relativity. It is demonstrated that the form of the coefficients of Langevin equation depend critically on the assumption of continuity of the reconstructed trajectory. This in turn demands for the fluctuations of the diffusion term to be discontinuous in time. This paper further investigates the connection between the scale-relativistic and stochastic mechanics approaches, respectively, with the study of the Burgers equation, which in this case appears as a stochastic geodesic equation. By further demanding time reversibility of the drift the Langevin equation can also describe equivalent quantum-mechanical systems in a path-wise manner. The resulting statistical description obeys the Fokker-Plank equation of the probability density of the differential system, which can be readily estimated from Monte Carlo simulations of the random paths. Based on the Fokker-Plank formalism a new derivation of the transient probability densities is presented. Finally, stochastic simulations are compared to the theoretical results.

Reservoir routing has been a routine procedure in hydrology, hydraulics and water management. It is typically based on the mass balance (continuity equation) and a conceptual equation relating storage and outflow. If the latter is linear, then there exists an analytical solution of the resulting differential equation, which can directly be utilized to find the outflow from known inflow, and to obtain macroscopic characteristics of the process, such as response and residence times, and their distribution functions. Here we streamline the reservoir routing framework and extend it to find approximate solutions for nonlinear cases. The proposed framework can also be useful for climatic tasks, such as describing the mass balance of atmospheric carbon dioxide and determining characteristic residence times, which have been an issue of controversy. Application of the theoretical framework results in excellent agreement with real world data. In this manner, we easily quantify the atmospheric carbon exchanges, and obtain reliable and intuitive results, without the need to resort to complex climate models. The mean residence time of atmospheric carbon dioxide turns out to be no more than four years and the response time is smaller than that, thus opposing the much longer mainstream estimates.

Motivated by the well-known contradiction of special relativity and the heat equation, a wave equation for temperature scalar field is presented that also resolves the old controversy of (Lorentz) transformation of temperature and entropy. After showing that the current dogma of temperature and entropy being emergent concepts is based on but a logical fallacy, it is proposed that single particles posses entropy. This principle of fundamentality of entropy is then shown to be compatible with the equipartition theorem by yielding corrections in the quantum gravity regime.

There are several versions of the damped form of the celebrated Pinney equation, which is the natural partner of the undamped time-dependent harmonic oscillator. In this work these dissipative versions of the Pinney equation are briefly reviewed. We show that Noether’s theorem for the usual time-dependent harmonic oscillator as a guiding principle for derivation of the Pinney equation also works in the damped case, selecting a Noether symmetry based damped Pinney equation. The results are extended to general nonlinear damped Ermakov systems. A certain time-rescaling always allows to remove the damping from the final equations.

In order to study the thermal deformation behavior of uranium-50wt.%zirconium, used Gleeble 3800 thermal simulation testing machine to carry out compression deformation experiments at different temperatures for this material. The effects of deformation temperature and strain rate on the alloy were studied. The results show that the peak stress of U-50wt.%Zr metallic fuel decreases with the increase of deformation temperature after thermal deformation at 500°C~600°C, especially at 600°C compared with that at 500°C and 550°C. At the same temperature, the peak stress increases with the increase of strain rate, especially the peak stress at strain rate 0.1s-1 and 1s-1 is significantly higher than that at strain rate 0.01s^-1, and then the microstructure and diffraction results of U-50wt.%Zr metal-type fuel after thermal deformation are compared and analyzed, and the Arrhenius-type constitutive equation and thermal proces550℃,έ=1s^-1 diagram of U-50wt.%Zr metal-type fuel are established based on the peak stress.

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.

“Dark Web” usually describes a new world of e-living space perceived as a place, free from con-straints, in which its main actors generally are criminal, unstable, and mentally disturbed individ-uals. Nowadays, advancements in information and internet technologies have made the Dark Web more accessible in recent decades. The Dark Web is a virtual world component; therefore, as-sessing users’ behavior may be crucial in understanding motivation to use it. This study aims to discuss the behavior patterns of Dark Web users within the theoretical framework of the technol-ogy acceptance model via a structural equation model. The resulting structural equation model showed that considered causal relationships were statistically significant.

Homotopy perturbation and analysis methods have been widely used to obtain both approximate and exact so- lutions to nonlinear problems. In general, these two methods are based on the Taylor series with respect to an embedding parameter. Many researchers have compared the two methods and raised more concerns on the homo- topy perturbation method (HPM) because the homotopy analysis method (HAM) contains a convergence-control parameter ~: For this reason, in this article, a more general form of HPM is introduced as the -homotopy per- turbation method (-HPM), which contains a control parameter : The introduction of parameter  in this new modication gives a better way to adjust and control the convergence region and the rate of the series solution. We conrm through the given examples in this study that the HPM is a special case of the -HPM. The error and convergence analysis of this proposed method are also presented

The wave energy converter (WEC) technology constantly develops new designs and models. To encourage its development within the energy sector, research is required to improve current designs and create durable and effective devices. However, it is also crucial to investigate the suitability of a WEC to a site-specific sea condition as this can impact its optimal power absorption and constructability. This study evaluates and compares the efficiency of a heaving buoy with varied geometries in one of the most energetic places along the Mexican coast. A statistical analysis of the wave climate on the coast of Ensenada during the last 42 years is performed to define the conditions to which the device is subjected. The location of the WEC in shallow waters is chosen using a computational model which solves the modified mild slope equation in its elliptic shape. The heaving buoy floater is studied using three distinct geometries: a semi-sphere, a cylinder, and a proposed rounded semi-rectangle. The hydrodynamic response of the three geometries is then analysed in the frequency and time domain using ANSYS AQWA. The hydrodynamic study involves the assessment of the floating body dynamics, exerted forces, the power absorbed as well as the suitability of the proposed power take-off (PTO) system. Findings reveal that the proposed geometry absorbs the most energy, with an annual power of 135.11 MW, and that the transmission PTO design is appropriate for this type of technology.

In 2018, Iran witnessed approximately 21,000 fatalities resulting from road accidents, ranking it as the fifth leading cause of death across all age groups. These incidents carry substantial economic and social burden for the nation, claiming more than 5% of its GDP. Human behavior emerges as a significant contributing factor to high-risk driving offenses. It has been proposed that by influencing general deterrence, a rigorous defense can be established against such violations. Various theories posit that by reinforcing or diminishing influential factors, general deterrence perception can be molded, presenting a potential avenue for an effective deterrent against traffic infractions. This study examines deterrence perceptions prevailing among the public in Iran regarding the enforcement of traffic laws. Employing Structural Equation Modeling (SEM), we delve into the intricate relationships between variables, allowing for a comprehensive exploration of the factors influencing general deterrence. The study draws on the data derived from a survey comprising 548 questionnaires for modeling purposes. An innovative concept, private sector enforcement, is introduced and incorporated into the questioning and modeling process. The findings reveal that socioeconomic status (p-value&lt;0.01), police enforcement (p-value&lt;0.05), private sector enforcement (p-value&lt;0.05), specific deterrence (p-value&lt;0.01), and the use of technology (p-value&lt;0.01) have direct and indirect impacts on the general deterrence perception of drivers.

We investigate the behavior of the solutions of the ray equation in isotropic media. We discuss local and global transversals and wavefronts and give examples of rays without wavefronts.

Food security is the cornerstone of national security, and exploring the resilience of the grain industry chain is particularly important under the current complex and changing external environment. The study is to evaluate the grain industry chain resilience and identify its influencing factors. Based on the panel data of 30 provinces (cities) in China from 2011 to 2022, this paper adopted Partial Least Squares structural equation modeling method to carry out comprehensive evaluation, applied Dagum Gini coefficient to analyze spatial differences, and used geo-detectors to identify the influencing factors. The results showed that (1) from an overall perspective, the resilience of China's grain industry chain showed an overall upward trend, and each region had been greatly improved; (2) from the perspective of spatial differences, the overall spatial variability of China's grain industry chain is relatively small, and the gap mainly originates from inter-regional differences; (3) from the perspective of influencing factors, the level of grain foreign trade, regional market size and agricultural innovation level were important factors. In order to enhance the resilience of China's grain industry chain, based on the heterogeneous characteristics, the coordinated development of each region should be integrated, and the endogenous power should be given full play to enhance the resilience of the industry chain.

In this paper, we consider a fourth-order suspension bridge equation with nonlinear damping term |u_t|^m-2u_t and source term |u|^p-2u. We give necessary and sufficient condition for global existence and energy decay results without considering the relation between m and p. Moreover, when p>m, we give sufficient condition for finite time blow-up of solutions. The lower bound of the blow-up time T_max is also established. It worth to mention that our obtained results extend the recent results of Wang (J. Math. Anal. Appl., 2014) to the nonlinear damping case.

Dual generalized commutative quaternions have broad application prospects in many fields. Additionally, the matrix equation AXB = C has important applications in mathematics and engineering, especially in control systems, economics, computer science, and other disciplines. However, research on the matrix equation AXB = C over the dual generalized commutative quaternions remains relatively insufficient. In this paper, we derive the necessary and sufficient conditions for the solvability of the dual generalized commutative quaternion matrix equation AXB = C. Furthermore, we provide the general solution expression for this matrix equation, when it is solvable. Finally, a numerical algorithm and an example are provided to confirm the reliability of the main conclusions.

The microbial communities are closely related to the overall health and quality of soil, but studies on microbial ecology in apple-pear orchard soils are limited. In the current study, 28 soil samples were collected from three apple-pear orchards in northeastern China, and the composition and structure of fungal and bacterial communities were investigated by high-throughput sequencing. Molecular ecological network showed that the keystone taxa of bacterial communities were Actinobacteria, Proteobacteria, Gemmatimonadetes, Acidobacteria, Nitrospirae, and Chloroflexi, and the keystone taxa of fungal communities was Ascomycota. Mantel tests showed that soil texture and pH were important factors shaping soil bacterial and fungal communities, and soil water soluble organic carbon (WSOC) and nitrate nitrogen (NO3--N) were also closely related to soil bacterial communities. Canonical correspondence analysis (CCA) and variation partition analysis (VPA) analysis revealed that geographic distance, soil texture, pH, and other soil properties could explain 10.55%, 13.5%, and 19.03% of the overall variation of bacterial communities, and they explained 11.61%, 13.03%, and 20.26% of the overall variation of fungal communities, respectively. The key stone taxa of bacterial and fungal communities in apple-pear orchard soils and their strong correlation with soil properties could provide useful clues toward sustainable management of orchards.

The Polynomial coefficient function, from the complex hyperspace Cⁿ in itself, is the function Fⁿ that provides the coefficients of the polynomial (x - a₁)(x - a₂)... (x-a_n). A proof of the fundamental theorem of algebra is given where the surjectivity of this function is obtained from a Recurrent Coefficient Equation.

P-band radar remote sensing applied during the Airborne Microwave Observatory of Subcanopy and Subsurface (AirMOSS) mission has shown great potential for estimation of root zone soil moisture. When retrieving the soil moisture profile (SMP) from P-band radar, a mathematical function describing the vertical moisture distribution is required. Because only a limited number of observations are available, the number of free parameters of the mathematical model must not exceed the number of observed data. For example, a second order polynomial that contains 3 free parameters was presumed based on in-situ SMP data. The polynomial is currently parameterized based on 3 backscatter observations provided by AirMOSS (i.e. one frequency at three polarizations of HH, VV and HV). In this paper, a more realistic, physically-based SMP model containing 3 free parameters is derived based on a solution to Richards’ equation for unsaturated flow in soils. Evaluation of the new SMP model based on both numerical simulations and measured data revealed that it exhibits greater flexibility for fitting measured and simulated SMPs than the currently applied polynomial. It is also demonstrated that the new SMP model can be reduced to a second order polynomial at the expense of fitting accuracy.

The paper considers two models of queuing with a varying structure based on the introduction of additional transient intensities into known models or their combinations, which create stationary distributions convenient for calculation. In the first model, it is a probabilistic mixture of known stationary distributions with given weights. In the second model, this uniform distribution is repeatedly used in physical statistics. Both models are based on the selection of states, between which additional transient intensities are introduced. The algorithms used in the paper for introducing new transient intensities are closely related to the concept of flow in a deterministic transport network. The introduced controls are selected so that the marginal distribution of the combined system is a mixture of the marginal distributions of the combined systems with different weights determined by the introduced transient intensities. As a result, the process of functioning of the combined system is obtained by switching processes corresponding to different combined systems at certain points in time.

In this article, the general solution of the tachyonic Klein-Gordon equation is obtained as a Fourier integral performed on a suitable path in the complex \omega-plane. In particular, it is proved that under given boundary conditions this solution does not contain any superluminal components. On the basis of this result, we infer that all possible spacelike wave equations describe the dynamics of subluminal particles endowed with imaginary mass. This result is validated for the Chodos equation, used to describe the hypothetical superluminal behaviour of neutrino. In this specific framework, it is proved that the wave packet propagates in spacetime with subluminal group velocities and that for enough small energies it behaves as a localized wave.

Currently, two-component integrable nonlinear equations from the hierarchies of the vector nonlinear Schrodinger equation and the vector derivative nonlinear Schrödinger equation are being actively investigated. Usually, such two-component integrable nonlinear evolutionary equations are considered, the form of which does not depend on the replacement of one component with another. In this paper, we propose for consideration and usage a new hierarchy of two-component integrable nonlinear equations, which have an important difference from the already known equations. Among the hierarchy equations there are analogues of the two-component nonlinear Schrödinger equation (second equation from hierarchy) and the two-component modified Korteweg-de Vries equation (fourth equation from hierarchy). The third equation of the hierarchy is a combination of the nonlinear Schrödinger equation for one component and the modified Korteweg-de Vries equation for the second component. The equations for the individual components are very different from each other, even if they have the same order. Let us note that even hierarchy equations can be reduced to well-known variants of the scalar derivative nonlinear Schrödinger equations, and odd equations can’t be reduced. To construct the hierarchy equations, we use the monodromy matrix method, first proposed by B.A. Dubrovin. Knowledge of the monodromy matrix makes it possible to construct spectral curves of multiphase solutions, as well as to find stationary equations that these solutions satisfy. The last section presents the simplest solutions in the form of solitons and periodic one-phase waves, as well as spectral curves corresponding to these solutions.

This study investigates how customers’ technological readiness affects service quality and the adoption of self-service banking by developing an empirical model. The model was proposed to explore the relationships between technology readiness, self-service quality, and customer satisfaction. The hypotheses and stability of the model were tested using structural equation modeling. The findings revealed that technology readiness, including the optimism, innovation, and insecurity dimensions, had a significant positive impact on self-service quality. In addition, service quality fully mediated the relationship between technological readiness and customer satisfaction. These results suggest that when customers are ready to embrace and use new technology, their evaluation of service quality is high. Thus, bank managers should drive customer perceptions of the quality of self-service banking by improving their technological readiness through various measures. Furthermore, based on the mediating effect of self-service quality, banks must be aware that improving service quality can enhance customer satisfaction and maintain a competitive advantage.

Finite-time control theory has been widely used as a mathematical tool to design robust controllers. By manipulating the finite-time convergence proof of this theory, we developed a new control design appropriately tuned for the finite-time control of the chaotic logistics system. In our experimental setup, the logistic equation is programmed into a PIC microcontroller, and a part of the controller was conceived using analog electronics. Because the system to be controlled is in the discrete-time domain, and the finite-time stability proof is stated in the continuous-time representation, our finite-time control approach is a good example for designing control algorithms in both time domain schemes. Hence, our experimental results support our main contribution. Pulse Width Modulation (PWM) is the signal format used to translate digital signals into continuous-time fields.

In a multivariable system there are usually a number of relaxation times. When some of the relaxation times are shorter than others, the corresponding variables will decay to their equilibrium value faster than the others. After the fast variables have decayed, the system can be described with a smaller number of variables. From the theory of nonequilibrium thermodynamics, as formulated by Onsager, we know that the coefficients in the linear flux-force relations satisfy the so-called Onsager symmetry relations. The question we will address in this paper is how to eliminate the fast variables such that the coefficients in the reduced description for the slow variables still satisfy the Onsager relations. As the proof Onsager gave of the symmetry relations does not depend on the choice of the variables, it is equally valid for the subset of slow variables. Elimination procedures that lead to symmetry breaking are possible, but do not consider systems that satisfy the laws of nonequilibrium thermodynamics.

As currently understood, the Dirac theory employs a 4 x1 type wavefunction. This 4x1 Dirac wavefunction is acted upon by a 4x4 Dirac Hamiltonian operator, in which process, four independent particle solutions result. Insofar as the real physical meaning and distinction of these four solutions, it is not clear what these solutions really mean. We demonstrate herein that these four independent particle solutions can be brought together under a single roof wherein the Dirac wavefunction takes a new form as a 4x4 wavefunction. In this new formation of the Dirac wavefunction, these four particle solutions precipitate into three distinct and mutuality dependent particles that are eternally bound in the same region of space. Given that Quarks are readily found in a mysterious threesome cohabitation-state eternally bound inside the Proton and Neutron, we make the suggestion that these Dirac particles might be Quarks. For the avoidance of speculation, we do not herein explore this idea further but merely present it as a very interesting idea worthy of further investigation. We however must say that, in the meantime, we are looking further into this very interesting idea, with the hope of making inroads in the immediate future.

Here, we consider a closed composite system consisting of a primary system where the chemical reaction occurs and is located in a p, T reservoir. In the ideal isentropic case (thermodynamic process), the change in the internal energy of the composite system (due to a chemical reaction) is wholly converted into useful work - maximum useful work. Irreversibility - a measure for the degradation of useful work - in a closed system generates entropy, i.e., the thermodynamic process is no longer isentropic. Applying the principle of maximum entropy and minimum internal energy to a part of the system's total volume and applying the elements of the system's volume for which the law of local equilibrium applies useful work and irreversibility can be defined. Suppose the primary system is viewed as a set of black box volume elements in which a chemical reaction occurs. In that case, the criterion of an isentropic process or a process with entropy generation becomes the Gibbs-Duham equation. It has been proven that De Donder's equation of entropy production of a chemical reaction is a particular form of Calen's equation –TdST = dUT where dST is the differential change of total entropy and dUT is the differential change of total internal energy of the composite system.

This paper is a study of a generalization of the quantum Riemannian Hamiltonian evolution, previously analyzed by us in our work [6], in the geometrization of quantum mechanical evolution in a Finsler geometry. We find results with dynamical equations governing the evolution of the trajectories defined by the expectation values of position. The analysis appears to provide an underlying geometry described by a geodesic equation, with connection form with a second term which is an essentially quantum effect. These dynamical equations provide a new geometric approach to the quantum evolution where we suggest a definition for "local instability" in the quantum theory.

Body-centered cubic bismuth (Bi) is considered an enticing pressure marker, and therefore it is highly desirable to command its accurate equation of state (EOS). However, significant discrepancies are noted among the previous experimental EOSs. In the present work, the EOS up to 300 GPa is theoretically obtained by direct solving the partition function via a direct integral approach (DIA). The calculated results nearly reproduce the hydrostatic experimental measurements below 75 GPa, and the deviations from the measurements gradually get larger with increasing pressure. Based on the ensemble theory of equilibrium state, DIA works with high precision particularly in high-pressure conditions, so the hydrostatic EOS presented in this work is expected to be a reliable pressure standard.

In the present study, we derive a new B-spline technique namely trigonometric B-spline collocation algorithm to solve some initial boundary value problems for the nonlinear Klein-Gordon equation. In order to carry out the time integration with Crank-Nicolson implicit method, the order of the equation is reduced to give a coupled system of nonlinear partial differential equations. The collocation approximation based on trigonometric cubic B-splines for spatial discretization is followed by the linearization of the nonlinear term. The efficiency and accuracy of the present method are validated by measuring the error between the numerical and analytical solutions when exist. The conservation laws representing momentum and energy are also computed for all problems.

This paper employs the Finite Element Method (FEM) to simulate and analyze the effects of corrosion defect parameters on the stress and failure pressure of pipelines. It investigates how the boundary conditions of the pipeline model influence stress and examines the sensitivity of failure pressure to corrosion defect parameters. A nonlinear regression equation has been developed from a dataset obtained through simulation experiments to predict the failure pressure of corroded pipelines. To validate the effects of corrosion defect parameters on failure pressure, a hydrostatic test platform for an API 5L X65 pipeline with corrosion defects was established to measure stress levels and failure pressures across varying corrosion defects. The study reveals that failure pressure is negatively correlated with corrosion length and depth, while positively correlated with corrosion width. Among these parameters, corrosion depth exerts a more significant influence on the pipeline's failure pressure than corrosion length and width. Within the range of corrosion defect parameters examined, the maximum deviation of the prediction equation's results from the simulation results is 8.71%, with an average deviation of 5.81%. The standard deviation of the fitted residuals is 0.01837. Additionally, the maximum deviation between the predicted results and experimental measurements is 8.39%, with an average deviation of 7.77%. The strong agreement between the predicted results from the equation and the actual measured data underscores the effectiveness of the nonlinear regression equation.

Background: This study aimed to identify the association between occupational stress and depression-well-being by proposing a comprehensive and flexible job burden-capital model with its corresponding hypotheses. Methods: For this research, 1618 valid samples were gathered from the electronic manufacturing service industry in Hunan Province, China; self-rated questionnaires were administered to participants for data collection after obtaining their written consent. The proposed model was fitted and tested through structural equation model analysis. Results: Single-factor correlation analysis results indicated that coefficients between all items and dimensions had statistical significance. The final model demonstrated satisfactory global goodness of fit (CMIN/DF=5.37, AGFI=0.915, NNFI=0.945, IFI=0.952, RMSEA=0.052). Both the measurement and structural models showed acceptable path loadings. Job burden and capital were directly associated with depression and well-being or indirectly related to them through personality. Multi-group structural equation model analyses indicated general applicability of the proposed model to basic features of such a population. Gender, marriage and education led to differences in the relation between occupational stress and health outcomes. Conclusions: The job burden-capital model of occupational stress-depression and well-being was found to be more systematic and comprehensive than previous models.

The purpose of the research is to investigate the unique solvability of a nonlocal boundary value problem for a loaded equation of parabolic-hyperbolic type in a special domain. Methods: Using representations of a general regular solution, the existence and uniqueness of the problem posed is proved. Results: A new method for proving the unique solvability of nonlocal boundary value problems for one loaded equation of mixed type in a special domain has been formed and developed. Conclusions: The obtained results make it possible to solve some problems of genetics, immunology, transonic gas dynamics and thermal physics, which are associated with displacement.

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.

By expressing the Schrödinger wave function in the form $\psi =R{e}^{iS/\hslash }$ , where R and S are real functions, we have shown that the expectation value of S is conserved. The amplitude of the wave (R) is found to satisfy the Schrödinger equation while the phase (S) is related to the energy conservation. Besides the quantum potential that depends on R, viz., $V_Q=-\frac{{\hslash }^2}{2m}\frac{{\nabla }^{2}R}{R}$ , we have obtained a spin potential $V_S=-\frac{S{\nabla }^{2}S}{m}$ that depends on S which is attributed to the particle spin. The spin force is found to give rise to dissipative viscous force. The quantum potential may be attributed to the interaction between the two subfields S and R comprising the quantum particle. This results in splitting (creation/annihilation) of these subfields, each having a mass $m{c}^2$ with an internal frequency of $2m{c}^2/\hslash$ , satisfying the original wave equation and endowing the particle its quantum nature. The mass of one subfield reflects the interaction with the other subfield. If in Bohmian ansatz R satisfies the Klein-Gordon equation, then S must satisfies the wave equation. Conversely, if R satisfies the wave equation, then S yields the Einstein relativistic energy momentum equation.

In this paper, we investigate the Hawking radiation process by using the quantum tunneling phenomenon of massive spin-1 (W-bosons) and spin-0 particles from (2+1) dimensional Black Hoke with quintessential and magnetic field. For this purpose, using Hamilton-Jacobi ansatz, we apply the WKB approximation to the field equations of massive charged vector particles. We get the required tunneling rate of radiated particles and obtain their corresponding Hawking temperature $T_h$. In order to study the quantum gravity effects, we utilize the generalized Proca and Klein-Gordan equations incorporating the generalized uncertainty principle (GUP) and recover the accompanying quantum corrected temperature $T'_{h}$.

This article derives the closed form solution to finding the roots of a quadratic equation in SU3.

The present study discusses how adopting cryptos affects financial inclusion in developing economies. Primary constructs like Financial Inclusion (FI), Perceived Economic Empowerment (PEE), Trust in Financial Institutions (TFI), User Satisfaction (US), and Cryptocurrency Adoption (CA) were tested through Structural Equation Modeling (SEM). The results indicated that CA significantly and positively influenced FI, US, TFI, and PEE. These relationships extend to the interaction effects: US, TFI, and PEE all positively related to FI. It is a reflection of cryptocurrencies as an opportunity to redress most of the afflictions characteristic of traditional finance systems and to promote financial inclusion and economic empowerment in developing countries. Future research should also investigate whether digital literacy and regulatory environments support cryptocurrency access.

Inertinite, a product of wildfire, holds important information on global temperature change. The relationship between its reflectance and temperature has been widely used to identify wildfire events in paleo-sedimentary environments, but the currently used equations relating inertinite reflectance and combustion temperature are subject to large errors. Therefore, to clarify the relationship between inertinite reflectance and combustion temperature further, we systematically analyzed the changes in inertinite reflectance under different combustion durations based on literature data. The results confirmed that inertinite reflectance is related to combustion duration. Disregarding combustion duration, the combustion equation is T=267.52+110.19*Ro(R²=0.91), where T is the combustion temperature, Ro% is the measured inertinite reflectance, and R² is the correlation coefficient. Under a combustion duration of 1 hour, the equation is T=273.57+113.89*Ro(R²=0.91), and under a combustion duration longer than 4 hours, the equation is T=232.91+110.6*Ro(R²=0.94). These three equations not only account for the temporal factor but are also more precise than the commonly used formula. This study provides a scientific basis for research on paleo-wildfire.

We provide representations for the generalized Drazin inverse of an anti-triangular matrix of the form $\left( \begin{array}{cc} a&b\\ 1&0 \end{array} \right)$ in a Banach algebra $\mathcal{A}$, under the condition that $ab=ba$. Specifically, we present the representation of Drazin inverse for these types of anti-triangular matrices in Banach algebras.

In this paper, we establish the existence and uniqueness results of the fractional Navier-Stokes evolution equation using the Banach fixed point theorem, where the fractional order β is in the form of the Atangana-Baleanu Caputo fractional order. The iterative method combined with the Laplace transform and Sumudu transform are employed to find the exact and approximate solutions of the fractional Navier-Stokes equation of a one-dimensional problem of unsteady flow of a viscous fluid in a tube. Mathematica software is used to present the solutions graphically.

In the context of the minimally extended varying speed of light (meVSL) model, both the absolute magnitude and the luminosity distance of type Ia supernovae (SNe Ia) differ from those predicted by general relativity (GR). Utilizing the Pantheon+ data, we examine the plausibility of various dark energy models under meVSL. Both ωCDM and CPL parameterization dark energy (DE) models exhibit indications of cosmic variations in the speed of light at the 1-σ level. For Ωm0=0.30,0.31, and 0.32 with (ω0,ωa)=(−1,0), the 1-σ range of c˜˙0/c˜0(10−13yr−1) is (-8.76 , -0.89), (-11.8 , 3.93), and (-14.8 , -6.98), respectively. Meanwhile, the 1-σ range of c˜˙0/c˜0(10−12yr−1) for CPL dark energy models with −1.05≤ω0≤−0.95 and 0.28≤Ωm0≤0.32, is (-6.31 , -2.98). The value of c˜ at z=3 can exceed that of the present by 0.2∼3 % for ωCDM models and 5∼13 % for CPL models. Additionally, for viable models except for the CPL model with Ωm0=0.28, we find −25.6≤G˜˙0/G˜0(10−12yr−1)≤−0.36. For this particular model, we obtain an increasing rate of the gravitational constant within the range 1.65≤G˜˙0/G˜0(10−12yr−1)≤3.79.

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.

Inherent in every Active Magnetic Bearing (AMB) are differences between the expected geometric axes and the actual magnetic axes due to a combination of discrepancies, including physical variation from manufacturing tolerances and misalignment from mechanical assembly, fringing and leakage effects, as well as variations in magnetic material properties within a single AMB. A method is presented here for locating the magnetic axes of an AMB that will facilitate the accurate characterization of the bearing air gaps for potential improvement in field tuning, performance analyses and certain shaft force measurement techniques. This paper presents an extension of the application of the bias current perturbation method for the determination of the magnetic center [4] to the determination of magnetic axes for the further development of accurate current-based force measurement techniques [1].

Examining the energy consumer’s behavioral model is a critical process for both national governments and academia. This endeavor seeks to uncover effective solutions amid the energy crisis and climate change challenges. This article delves into legislative developments within the energy sector, European Commission recommendations for reducing energy consumption, and existing constraints impacting individual consumers. By scrutinizing relevant literature, we aim to identify and analyze factors that can enhance individual benefits derived from energy savings. To model the behavior of final consumers, a comprehensive set of variables is formulated. Data collection involved administering a questionnaire to individual consumers, consumer associations, and energy micro-enterprises in Romania. The gathered data was meticulously analyzed using the Smart-Pls 4 statistical software. Building upon insights from specialized literature, this paper pinpoints the behavioral determinants influencing the reduction of energy consumption. These determinants serve as independent variables shaping the voluntary adoption of measures in lifestyle and behavior among various types of energy users. The study’s findings validate the assumptions presented in this article, highlighting that the reduction of energy consumption is a direct and intrinsic outcome achieved by cumulatively addressing several factors. These factors encompass investments in the energy sector, budget allocation for energy consumption expenditure, adherence to social behavior norms, access to pertinent information about the consequences of the energy crisis, and individual responsibility. Notably, the perception of energy-saving opportunities emerges as a mediator with a significant effect between independent variables and energy savings. This aspect, developed for the first time in this article, draws inspiration from the prospect theory introduced by Kahneman and Tversky [1].

Our objective in this work is to demonstrate how Physics-Informed Neural Networks, a type of deep learning technology, can be utilized to examine the mechanical properties of a helicopter blade. The blade is regarded as a prismatic cantilever beam that is exposed to triangular loading, and comprehending its mechanical behavior is of utmost importance in the aerospace field. PINNs utilize the physical information, including differential equations and boundary conditions, within the loss function of the neural network to approximate the solution. Our approach determines the overall loss by aggregating the losses from the differential equation, boundary conditions, and data. We employed a Physics-Informed Neural Network (PINN) and an Artificial Neural Network (ANN) with equivalent hyperparameters to solve a fourth-order differential equation. By comparing the performance of the PINN model against the analytical solution of the equation and the results obtained from the ANN model, we have conclusively shown that the PINN model exhibits superior accuracy, robustness, and computational efficiency when addressing high-order differential equations that govern physics-based problems. In conclusion, the study demonstrates that PINN offers a superior alternative for addressing solid mechanics problems with applications in the aerospace industry.

The Dirac and Klein-Gordon equations have proven to be some of the most accurate theories for describing particle behaviour and have laid the foundations of Relatavistic Quantum Mechanics. We extend the formulations to consider a 5 Dimensional spacetime with two time dimensions (as first conceptualised by Sajjad Zahir) and extend the notions to Quantum Field Theory to arrive at the behaviour of particles through the Dirac and Klein-Gordon counterparts of the 5 Dimensional spacetime. The implications of this to anti-matter are discussed.

Reviewed the research progress of Newton’s iterative methods for nonlinear equation. Convergence with second order, third order, fourth order, fifth order, sixth order, seventh order, eighth order and ninth order Newton’s iterative methods are given in turn, and the efficiency index of these iterative methods are analyzed. Numerical experiments show that the convergence process of the three Newton iterative methods, and the results indicate that the convergence of the higher-order Newton’s method can be well demonstrated only when the initial point is close to the root.

We extended the usage of the expansion method based on Sine-Gordon equation to the two dimensional Fisher equation. The relation between the trigonometric and hyperbolic functions are derived from the Sine-Gordon equation dened in two space dimension. The complex-valued traveling wave solutions to the two dimensional Fisher and Nagumo equations are set in forms a nite series of multiplications of powers of sech(.) and tanh(.) functions.

Objective: This study aims to investigate the relationship between sports psychological capital, coach leadership behavior, and satisfaction among middle school students in the 2023 sports college entrance examination in Chongqing. Methods: Using the structural questionnaire design, this paper carries out a questionnaire survey on the candidates who come to participate in the volleyball specialty college entrance examination in Chongqing, the corresponding database was established and the data were statistically processed by spss21.0 and amos18.0. Results: 1) there was a significant positive correlation between sports psychological capital, coach leadership behavior and examinees satisfaction; 2) sports psychological capital and coach leadership behavior had a certain influence on five dimensions of examinees' personal treatment, personal performance, team integration, team interaction and support from relatives and friends; 3) The sense of self-efficacy in sports psychological capital and coach's leading behavior are the most important to examinees' satisfaction; 4) Both coaches' leadership behavior and sports psychological capital can directly affect examinees' satisfaction, but sports psychological capital is the most important. It can not only directly affect examinees' satisfaction, but also act as an intermediary between coaches' leadership behavior and examinees' satisfaction. Conclusion: Sports College Entrance Examination Students can improve training satisfaction and sports performance by strengthening sports psychological capital training and perceiving coach's leadership behavior; In the leadership behavior of coaches, we should emphasize training and appropriately weaken authoritative teaching, while the training of sense of hope and toughness should be highlighted in sports psychological capital.

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

According to our current understanding of quantum mechanics, a `measurement' violates unitarity. In other words as the act of measurement `resets' the probabilities described by the Schrödinger equation, in the very `moment' of the collapse of the wavefunction, conservation of probability does not hold. To make progress in our understanding of the measurement problem it is thus expected that one must encompass in a single equation both otherwise mutually-exclusive processes of measurement and unitary evolution. In this light, turning to the continuity equation, we realize the possibility that if we allow for existence of sources and sinks in the `fluid of probability' we get closer to such a theory by arriving at nonlinear generalizations of Schrödinger and Klein-Gordon equations. The generalized equations derived are then shown to satisfy all conditions that are expected from a proper generalization: simplification to their linear counterparts by a well-defined dynamical condition, and Galilean and Lorentz invariance.

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.

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.

New analytical solutions of the heat conduction equation are presented in cylindrical and spherical coordinates. Then these solutions are reproduced with high accuracy by recent explicit and unconditionally stable finite difference methods. After this, real experimental data from the literature regarding a heated cylinder are reproduced by the explicit numerical methods as well as by Finite Element Methods (FEM) ANSYS workbench. Convection and nonlinear radiation are also considered on the boundary of the cylinder, and it is shown that the explicit methods are much more accurate than the commercial software.

In this paper, I investigate Hyers-Ulam-Rassias stabilities of the general quintic functional equation and the general sextic functional equation.