General Equation Research Articles

SDEs with two reflecting barriers driven by optional processes with regulated trajectories

We study the existence, uniqueness, and approximation of solutions of general stochastic differential equations (SDEs) with two time-dependent reflecting barriers driven by optional semimartingales. We do not assume that the probability space has to satisfy the usual conditions. We define and solve an appropriate version of the deterministic Skorokhod problem for regulated functions. Applications to currency option pricing in financial models are given.

General relativistic gravitational induction and causal temperatures

Abstract In this paper, we describe the process of general relativistic gravitational induction in spherically symmetric spacetimes by defining an energy momentum tensor for the induction process, which is divergence-free and hence conserved. The aforementioned tensor explicitly describes how the matter-free gravity, as measured by the geometrical Weyl curvature, interacts with the matter. This tensor is clearly different from the energy momentum tensor of the standard matter and we transparently show that in spherical symmetry, the Bianchi identities reduce to the conservation laws for these two such energy momentum tensors. Working with a semitetrad covariant formalism in spherically symmetric spacetimes, we then demonstrate the process of constructing a consistent causal thermodynamical picture for the free gravity and matter interaction via the general non-truncated Israel-Stewart heat transport equation. As an illustrative example, we consider the Lemaître-Tolman-Bondi spacetime to highlight the relationship between the shear and the Weyl curvature in determining the inductive heat flux.

Frequency-resolved measurement of two-color air plasma terahertz emission

We investigated the far-field terahertz beam profile generated from an air plasma induced by two-color femtosecond laser pulses. Under our experimental conditions (filament length shorter than the dephasing length between the two-color pulses), using electro-optic sampling in both ZnTe (0.2-2.2 THz) and GaP (0.4-6.8 THz) crystals, and ultra-broadband ABCD technique (1-17.5 THz), we determined that the THz beam exhibits a unimodal beam pattern below 4 THz and a conical one above 6 THz. This experimental finding is consistent with theoretical studies based on the unidirectional pulse propagation equation, which predict the transition of THz emission from a flat-top profile to a conical one due to the destructive interference of THz waves emitted from the plasma filament. Our results also underscore the importance of accounting for experimental artifacts, such as photo-excited losses in silicon resulting in on-axis THz absorption along with the influence of drilled mirrors, in characterizing the complex spatial and frequency-dependent behavior of two-color plasma-induced terahertz emission.

Stability of a General Quadratic-Cubic Functional Equation in Non-Archimedean 2-Normed Spaces

This research aims to investigate the Hyers-Ulam stability of the mixed-type quadratic-cubic functional equation in non-Archimedean 2-normed spaces using the fixed-point method. Additionally, some counter-examples are illustrated for instability. The exciting possibilities of this cutting-edge research and unlocking new frontiers in mathematical analysis are explored.

Impact of Generalized Versus Individualized Load-Velocity Equations on Velocity-Loss Magnitude in Bench-Press Exercise: Mixed-Model and Equivalence Analysis.

This study analyzed the influence of 2 velocity-based training-load prescription strategies (general vs individual load-velocity equations) on the relationship between the magnitude of velocity loss (VL) and the percentage of repetitions completed in the bench-press exercise. Thirty-five subjects completed 6 sessions consisting of performing the maximum number of repetitions to failure against their 40%, 60%, and 80% of 1-repetition maximum (1RM) in the Smith machine bench-press exercise using generalized and individualized equations to adjust the training load. A close relationship and acceptable error were observed between percentage of repetitions completed and the percentage of VL reached for the 3 loading magnitudes and the 2 load-prescription strategies studied (R2 from .83 to .94; standard error of the estimate from 7% to 10%). A simple main effect was observed for load and VL thresholds but not for load-prescription strategies. No significant interaction effects were revealed. The 40% and 60% 1RM showed equivalence on data sets and the most regular variation, whereas the 80% 1-repetition maximum load showed no equivalence and more irregular variation. These results suggest that VL is a useful variable to predict percentage of repetitions completed in the bench-press exercise, regardless of the strategy selected to adjust the relative load. However, caution should be taken when using heavy loads.

A Semi‐Analytical Method for Simulating Near‐Field Antiplane Wave Propagation in Layered Fluid‐Saturated Porous Media

ABSTRACTA semi‐analytical method for the near‐field antiplane wave propagation analysis in the layered fluid‐saturated porous media (FSPM) is proposed based on the Biot u–U dynamic formulation. The wave propagation equations of the FSPM are decoupled by the variable‐separating method. The thin‐layer element method (TLEM) is applied to discretize the infinite domain and construct the consistent artificial boundary condition. The finite element method (FEM) is adopted for the space discretization of the finite domain and the numerical solution of the dynamic response. The proposed method is validated by the comparison of the numerical results of this method with those in the published references and acquired from the remote artificial boundary. Subsequently, this method is applied to investigate typical near‐field antiplane wave propagation problems in the FSPM. Parametric sensitivity investigations are also executed to explore the impact of mechanical parameters, including permeability coefficients, porosity, and shear modulus of the solid phase, on the dynamic response of the FSPM. The study results confirm the efficacy and efficiency of the proposed method in the near‐field antiplane wave propagation analysis in the FSPM.

Road traffic noise prediction model "ASJ RTN-Model 2023" proposed by the Acoustical Society of Japan - Part 1: Framework of the calculation model

The Acoustical Society of Japan (ASJ) has had the Research Committee on Road Traffic Noise to develop a prediction model for road traffic noise for more than 50 years. As a result of the continuous research activities, a new version of model, "ASJ RTN-Model 2023", was published in this April. It is an upgrade version of the previous model, "ASJ RTN-Model 2018", proposed in 2019. In developing the latest version, existing knowledge was widely taken into account, in particular, the sound power levels of road vehicles under acceleration running conditions and calculation methods for sound propagation were improved in wide range. This new model is introduced in three papers with the same main title in this congress. In this paper, the framework of the model, the general procedure of prediction calculation, and the outlines of sound emissions and the calculation equations of sound propagation are presented.

Evaluating the accuracy of population-specific versus generic stature estimation regression equations in a South African sample.

Accurate estimates of stature play an important role in the personal identification of unknown decedents, however a drawback in the application of many stature estimation equations is the need for known sex and population, the assignment of which can be challenging. Researchers have formulated equations for stature estimation that are neither population- nor sex-specific and thereof the aim of this study was to assess the applicability of these stature estimation equations proposed by Albanese et al.. (2016) (Albanese J, Tuck A, Gomes J, Cardoso HFV (2016) An alternative approach for estimating stature for long bones that is not population- or group-specific. Forensic Sci Int 259:59-68). The physiological length of the femur, condylar malleolar length of the tibia and a combination of these measurements, collected from Magnetic Resonance Imaging Scanograms of adult (20-60 years) White South African males (n = 30) and females (n = 44) were used to assess the accuracy of the Albanese et al. (Albanese J, Tuck A, Gomes J, Cardoso HFV (2016) An alternative approach for estimating stature for long bones that is not population- or group-specific. Forensic Sci Int 259:59-68). sex-specific and generic stature estimation equations. The stature estimates were compared with measured living stature (LSM), using paired t-tests. Results indicated that the Albanese et al. (Albanese J, Tuck A, Gomes J, Cardoso HFV (2016) An alternative approach for estimating stature for long bones that is not population- or group-specific. Forensic Sci Int 259:59-68). equations underestimated living stature by between 1.1 and 5.0cm. These underestimations were significantly different between the LSM and the sex-specific estimates for females and the LSM and the generic estimates for males and the tibia for sex-specific equation. All stature estimates however fell in between two standard error of estimates for the sex-specific equations for males and the generic equations for the females. Although, the equations by Albanese et al. (Albanese J, Tuck A, Gomes J, Cardoso HFV (2016) An alternative approach for estimating stature for long bones that is not population- or group-specific. Forensic Sci Int 259:59-68). can be used to estimate stature in White South Africans in certain cases, the use of sex/population-specific equations remains the method of choice.

A derivation of the Schrödinger equation from Feynman’s path-integral formulation of quantum mechanics

The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.

Two‐Streams Revisited: General Equations, Exact Coefficients, and Optimized Closures

AbstractTwo‐Stream Equations are the most parsimonious general models for radiative flux transfer with one equation to model each of upward and downward fluxes; these are coupled due to the transfer of fluxes between hemispheres. Standard two‐stream approximation of the Radiative Transfer Equation assumes that the ratios of flux transferred (coupling coefficients) are both invariant with optical depth and symmetric with respect to upwelling and downwelling radiation. Two‐stream closures are derived by making additional assumptions about the angular distribution of the intensity field, but none currently works well for all parts of the optical parameter space. We determine the exact values of the two‐stream coupling coefficients from multi‐stream numerical solutions to the Radiative Transfer Equation for shortwave radiation. The resulting unique coefficients accurately reconstruct entire flux profiles but depend on optical depth. More importantly, they generally take on unphysical values when symmetry is assumed. We derive a general form of the Two‐Stream Equations for which the four coupling coefficients are guaranteed to be physically explicable. While non‐constant coupling coefficients are required to reconstruct entire flux profiles, numerically optimized constant coupling coefficients (which admit analytic solutions) reproduced shortwave reflectance and transmittance with relative errors no greater than over a large range of optical parameters. The optimized coefficients show a dependence on solar zenith angle and total optical depth that diminishes as the latter increases. This explains why existing coupling coefficients, which often omit the former and mostly neglect the latter, tend to work well for only thin or only thick atmospheres.

Effect of the particle shape on the shear mechanical behavior of coral sand

Abstract Coral sand particles exhibit a wide range of shapes, which can be divided into four shapes, e.g., blocky, dendritic and rodlike, flaky, and shell debris. The particle shape of these mixtures is defined by the sphericity, concavity, aspect ratio, flatness and overall regularity, which ranges from 0 to 1. The effect of particle shape on the strength, crushing characteristics, and critical state parameter is systematically investigated through a series of triaxial drainage shear tests under different confining pressures. And the relationship between critical state parameters and mechanical parameters is established. The test results demonstrate the existence of an evident strain-hardening phenomenon in the stress–strain curve of coral sand, accompanied by a strain-softening phenomenon when the bias stress reaches its peak value. The sample is initially subjected to shear shrinkage, followed by shear expansion. The volumetric deformation of the coral sand decreased with increasing peripheral pressure. The particles are transformed from rough irregular shapes to smooth spheres as evidenced by an increase in the shape parameter. The greater the degree of irregularity in the shape of the particles, the more pronounced the resulting change in size reduction. In addition, the critical state parameter was found to be influenced by the shape of the coral sand particles and the mode of particle accumulation. The overall shear resistance of coral sand particles was found to depend on particle rearrangement in addition to particle surface roughness and interparticle friction. It is proposed that the general regularity critical state parameter equation relates the particle shape of coral sand to its critical state mechanical properties, which is of great importance to the practical application and research of coral sand in engineering, and provides an effective means of predicting mechanical properties granular materials.

Examining the Mathematica algorithm for general Heun function calculation: a comparative analysis

Abstract We investigate the numerical calculation of the general Heun equation using Wolfram Mathematica’s functions, comparing the numerical solutions with hypergeometric and explicit solutions. This exploration sheds light on the efficacy and accuracy of the numerical algorithm implemented in Mathematica for computing Heun functions.

A bidirectional fifth-order partial differential equation: Lax representation and soliton solutions

In this paper, we study the bidirectional fifth-order partial differential equation uttt−uxxxxt−4(uxut)xx−4(uxuxt)x=0, which was proposed as a new integrable equation by Wazwaz [Phys. Scr. 83, 015012 (2011)]. By means of the prolongation structure method of Wahlquist and Estabrook [J. Math. Phys. 16, 1–7 (1975)], we construct a Lax representation for this equation. This enables us to confirm its integrability and identify it as the potential second-order flow in a Gelfand-Dickey hierarchy. We also use the Darboux transformation and present the multi-soliton solutions in the Wronskian form. Finally, we comment on a more general bidirectional partial differential equation.

Synchronous consistent integration for superconvergent isogeometric analysis of structural vibrations

A commonly used procedure to improve the frequency accuracy of isogeometric structural vibration analysis is the design of special superconvergent quadrature rules in accordance with the frequency error measure derived from the generic stencil equations. However, it is also numerically observed that such superconvergent quadrature rules may not consistently lead to the expected accuracy. The reason underlying this inconsistency is related to the failure of quadrature rules to meet the condition of variational integration consistency. Meanwhile, by means of the tensor product nature of isogeometric basis functions and quadrature rules, it is theoretically proved that in one- as well as multi-dimensional scenarios, the variational integration consistency of the isogeometric weak formulation necessitates the quadrature rules be exact through 2(p-m)th degree polynomials, where p is the complete degree of multi-dimensional basis functions or the degree of their correspondent one-dimensional basis functions, m is the order of differentiation in Galerkin weak form. A synchronous consistent superconvergent quadrature rule ensuring frequency superconvergence should simultaneously fulfill both the constraint of frequency error measure and the condition of variational integration consistency. Subsequently, the issues associated with certain quadrature rules that violate the variational integration consistency are examined for both second- and fourth-order problems, and the corresponding synchronous consistent quadrature rules are particularly elaborated.

Correctness of the mixed problem for degenerate multidimensional elliptic-parabolic equations

The initial-boundary value problem (Dirichlet problem) for general elliptic-parabolic equations of second order was first posed by G. Fichera. Further investigation of this problem was carried out in the monograph by O.A. Oleinik and E.V. Radkevich and the works by V.N. Vragov. In these works, the authors examined mixed problems for degenerate multidimensional elliptic equations. The articles by S.A. Aldashev focused on the correctness (in the sense of uniqueness of solvability) of the Dirichlet problem in a cylindrical domain for multidimensional elliptic-parabolic equations. A mixed problem for these equations has not been studied. In this paper, the authors demonstrate the uniqueness of solvability and obtain an explicit representation of the classical solution of the mixed problem for degenerate multidimensional elliptic-parabolic equations. The proposed method allows reducing the problem under study to a mixed problem for a degenerate multidimensional elliptic equation examined by S.A. Aldashev.

A Quantization for the Toy Model of Black Hole and a Suggestion to the Unification Theory

In my previous paper about the correlations between quantum mechanics and the four different natural forces, I suggested that there may be a fifth force or even the existence of a new particle. However, it remains a mystery to physicists, even after nearly a hundred years, that they cannot unify these four natural forces. This may be because there are indeed too many variables for them to consider. Also, there is a possibility that quantum mechanics may coexist with quantum field theory. In the present paper, this author proposes that there may be a misconception in the computational equation for relative time or gaps in the system for measuring time between quantum mechanics and general relativity. Hence, one may still not be able to unify these natural forces. This author suggests that we may need to rewrite parts of quantum mechanics – the Schrödinger equation – or even the general relativity equation. Additionally, this author proposes that there may be a bridge equation converting between quantum mechanics and general relativity. Moreover, this author has employed the non-trivial zeta zeros to simulate the black hole or the so-called black hole toy model. In practice, there may be an electromagnetic field surrounding the boundary of the black hole, as well as the existence of a continuum along the boundary contour. This author hopes that, in such a case, we may decode those high-frequency electromagnetic waves into useful information and take a further step toward verifying Stephen Hawking’s famous theory on black hole radiation and information entropy. In fact, this author has used the HKLam statistical model theory to express the electromagnetic field energy-stress tensor (with the possibility of quantization) to analogically establish a quantized model for the Einstein Gravitational Field Equation. Hence, the problem of quantum gravity may then be solved. Last but not least, this author also expects that humanity may finally find a way to unify quantum mechanics and general relativity through modifications to the current quantum gravity theory, such as my proposed bridge-converting equation, etc.

Semi-analytic approximate time-optimal asteroid landing with dimensionality reduction shooting

Pinpoint landing on asteroid is very challenging due to the uncertainties of the gravitational field, which highlights the urgent need for autonomous landing algorithms under such uncertainties. Some well-known guidance algorithms, such as APDG and real-time convex programming, either ignore thrust boundary constraints or are severely time consuming. In addition, these algorithms have the same difficulty in analyzing landing errors due to model uncertainty propagation, making the reliability of the algorithms in actual landings questionable. To address these challenges, we propose a landing framework that combines model identification, trajectory analytical solution and closed-loop corrections to achieve near-optimal real-time landing control. Firstly, we simplified the landing scenario, focusing on reducing the dimension of shooting variables to enable rapid trajectory calculation. Secondly, we derived error propagation equations and established criteria for trajectory replanning based on landing error prediction. Finally, we validated our approach with practical examples of the Shoemaker probe landing on Eros 433. Through real-time landing error estimation, our proposed framework enables spacecraft to achieve near time-optimal land within a given error threshold.

Levinson–Smith Dissipative Equations and Geometry of GENERIC Formalism and Contact Hamiltonian Mechanics

We apply Jacobi’s Last Multiplier theory to construct the non-standard Lagrangian and Hamiltonian structures for the Levinson–Smith equations satisfying the Chiellini integrability condition. Then after a brief exposition of the contact geometry, we explore its connection with the non-standard Hamiltonian structures. We present the formulation of the Levinson–Smith equation in terms of General Equation for the Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) method and also study the gradient-type flow. We give a geometric formulation of GENERIC and apply this to general Levinson–Smith equations.

Pseudo Master Curve Analysis of an Infinite Number of Parallel First-Order Reactions: Improved Distributed Activation Energy Model.

The so-called Distributed Activation Energy Model (DAEM) has been used extensively, mainly to analyze pyrolysis reactions of solid reactants. The model expresses many parallel first-order reactions using the distributions of activation energy f(E) and frequency factor k 0(E). Miura and Maki presented a method to estimate both f(E) and k 0(E) in the DAEM in 1998. This model has been used successfully by many researchers. In this paper more general basic equations are derived for describing an infinite number of parallel first-order reactions by extending the basic equations for the finite number of parallel first-order reactions. Revisiting the Miura-Maki method based on the general basic equations, a graphical analysis method that may be called "Pseudo Master Curve Analysis" is presented. The method not only supplements the Miura-Maki method but gives the underlying concept of the Miura-Maki method clearly. It is also shown that the graphical method can be applicable to analyze single reactions and the experimental data obtained using isothermal reaction techniques. Next, a method that improves the estimation accuracy of k 0(E) is presented. Practical examples analyzing several experimental data are also given to show the usefulness and validity of the Miura-Maki method and the graphical method. Through the examination, it is proposed that the DAEM should be renamed, for example, as the Distributed Rate Constant Model (DRCM).

On formal series solutions to 4th-order quadratic homogeneous differential equations and their convergence

Abstract It is known that all τ functions of the Painlevé equations satisfy the fourth-order quadratic differential equation. Among them, for the III, V, and VI equations, it is possible to express the formal series solutions combinatorially by using conformal blocks. In this paper, we show the convergence of the formal series, including the solutions of more general equations. The convergence of the conformal block function also follows in the case c = 1 by the absolute convergence of τ series, since it is a partial sum of the τ series. We also characterized the form of a homogeneous quadratic equation with a series solution similar to the tau functions of the Painlevé equations.