The simplest elasticity model of foundation underlying a slender beam under flexure was conceived by Winkler, requiring local proportionality between soil reactions and beam deflection. Such an approach leads to well-posed elastostatic and elastodynamic problems, but, as highlighted by Wieghardt, it provides elastic responses which are not technically significant for a wide variety of engineering applications. Thus, Winkler's model was replaced by Wieghardt himself by assuming that the beam deflection is the convolution integral between soil reaction field and an averaging kernel. Due to conflict between constitutive and kinematic compatibility requirements, the corresponding elastic problem of an inflected beam resting on Wieghardt foundation results to be ill-posed. Modifications of the original Wieghardt model were proposed by introducing fictitious boundary concentrated forces of constitutive type, which are physically questionable, being significantly influenced on prescribed kinematic boundary conditions. Inherent difficulties and issues are overcome in the present research using a displacement-driven nonlocal integral strategy got by swapping input and output fields involved in Wieghardt's original formulation. That is, nonlocal soil reaction fields are output of integral convolutions of beam deflection fields with an averaging kernel. Equipping the displacement-driven nonlocal integral law with the bi-exponential averaging kernel, an equivalent nonlocal differential problem, supplemented with non-standard constitutive boundary conditions involving nonlocal soil reactions, is established. As a key implication, the integro-differential equations governing the elastostatic problem of an inflected elastic slender beam resting on displacement-driven nonlocal integral foundation are replaced with much simpler differential equations supplemented with kinematic, static and new constitutive boundary conditions. The proposed nonlocal approach is illustrated by examining and analytically solving exemplar problems of structural engineering. Benchmark solutions for numerical analyses are also detected.

This paper presents the simplified and quick way to obtain an experimental measure of the modulus of elasticity(E) of a piece of metal rod using the Smartphone microphone and through the experimental configuration in mode of unsupport beam Euler-Bernoulli(E-B). To understand it, it is required to have knowledge about flexural vibration of E-B beams and to know the analysis of the spectogram of the audio signal. The methodology consists in measuring the flexural resonance frequency through an app installed on a Smartphone and using the E formula deduced from the E-B theoretical framework. The results of the experimental measurements have a discrepancy of less than 5% with respect to the factory value, for a carbon steel rod with circular cross section using the impact excitation technique. It describes the precision and accuracy of the experimental measurement. It is useful, because it helps to classify the materials in the Faculty of Engineering and it helps to make a more efficient quality control.

This paper presents an approach to the vibration analysis of axially functionally graded (AFG) non-prismatic Euler-Bernoulli beams using the finite difference method (FDM). The characteristics (cross-sectional area, moment of inertia, elastic moduli, and mass density) of AFG beams vary along the longitudinal axis. The FDM is an approximate method for solving problems described with differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, differential equations were formulated with finite differences, and additional points were introduced at the beam’s ends and at positions of discontinuity (supports, hinges, springs, concentrated mass, spring-mass system, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. Vibration analysis of AFG non-prismatic Euler-Bernoulli beams was conducted with this model, and natural frequencies were determined. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of AFG non-prismatic Euler-Bernoulli beams, considering the damping. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was always increased through a grid refinement.

This paper presents an approach to the Euler-Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. Generally, the finite difference approximations are derived based on fourth-order polynomial hypothesis (FOPH) and second-order polynomial hypothesis (SOPH) for the deflection curve; the FOPH is made for the fourth and third derivative of the deflection curve while the SOPH is made for its second and first derivative. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, the FOPH was made for all of the derivatives of the deflection curve, and additional points were introduced at the beam’s ends and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Furthermore, tapered beams were analyzed (element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was increased through a grid refinement. Especially in the first-order analysis of uniform beams, the results were exact for uniformly distributed and concentrated loads regardless of the grid. Further research will be needed to investigate polynomial refinements (higher-order polynomials such as fifth-order, sixth-order…) of the deflection curve; the polynomial refinements aimed to increase the accuracy, whereby non-centered finite difference approximations at beam’s ends and positions of discontinuity would be used.

Metastases cause high mortality in several cancers and immunotherapies are expected to be effective in the prevention and treatment of metastatic disease. However, only a minority of patients benefit from immunotherapies. This creates a need for novel therapies that are efficacious regardless of the cancer types and metastatic environments they are growing in. Preclinical immuno-oncology models for studying metastases have long been limited to syngeneic or carcinogenesis-inducible models that have murine cancer and immune cells. However, the translational power of these models has been questioned. Interactions between tumor and immune cells are often species-specific and regulated by different cytokines in mice and humans. For increased translational power, mice engrafted with functional parts of human immune system have been developed. These humanized mice are utilized to advance understanding the role of immune cells in the metastatic process, but increasingly also to study the efficacy and safety of novel immunotherapies. From these aspects, this review will discuss the role of immune cells in the metastatic process and the utility of humanized mouse models in immuno-oncology research for metastatic cancers, covering several models from the perspective of efficacy and safety of immunotherapies.

Two-photon imaging of light stimulus-evoked neuronal activity has been used to study all neuron classes in the vertebrate retina, from the photoreceptors to the retinal ganglion cells. Clearly, the ability to study retinal circuits down to the level of single synapses or zoomed out at the level of complete populations of neurons, has been a major asset in our understanding of this beautiful circuit. In this chapter, we discuss the possibilities and pitfalls of using an all-optical approach in this highly light-sensitive part of the brain.

In the paper, by a very simple approach, the author establishes an expression in terms of a lower Hessenberg determinant for the Euler polynomials. By the determinantal expression, the author finds a recurrence relation for the Euler polynomials. By the way, the author derives the corresponding expression and recurrence relation for the Euler numbers.

Euler sums are alternating (or level two) extension of multiple zeta values (MZVs). Kaneko and Tsumura initiated the study of multiple T-values (MTVs), another level two generalization, by restricting the summation indices in the definition of MZVs to a fixed parity pattern. In this paper, we shall study finite MTVs and their alternating versions which are level two and level four variations of finite MZVs, respectively. We conjecture that all finite MZVs are in the Q-span of finite MTVs which in turn apparently lie in the span of finite Euler sums, and the inclusions are both proper. We shall first provide some structural results for Euler sums of small weights, guided by the author's previous conjecture that the finite Euler sum space of weight w is isomorphic to a quotient Euler sum space of weight w. Then, by utilizing some well-known properties of the classical alternating MTVs, we shall derive a few important Q-linear relations among the finite alternating MTVs, including the reversal, linear shuffle and sum relations. We then compute the upper bound for the dimension of the Q-span of weight w finite (alternating) MTVs for w<9, both rigorously using the newly discovered relations and numerically aided by computers.

In this paper, we investigate the recently introduced degenerate Bernstein polynomials and operators and derive some of their properties. Also, we give some properties of the degenerate Euler numbers and polynomials and their connection with the degenerate Euler polynomials.

In the present paper, we introduce (p, q)-extension of Apostol type Frobenius-Euler polynomials and numbers and investigate some basic identities and properties for these polynomials and numbers, including addition theorems, difference equations, derivative properties, recurrence relations and so on. Then, we provide integral representations, explicit formulas and relations for these polynomials and numbers. Moreover, we discover (p, q)-extensions of Carlitz's result [L. Carlitz, Mat. Mag., 32 (1959), 247–260] and Srivastava and Pintér addition theorems in [H. M. Srivastava, A. Pinter, Appl. Math. Lett., 17 (2004), 375–380].

Motivated by Kim-Kim [19] introduced the new type of degenerate poly- Bernoulli polynomials by means of the degenerate polylogarithm function. In this paper, we define the degenerate poly-Frobenius-Euler polynomials, called the new type of degenerate poly-Frobenius-Euler polynomials, by means of the degenerate polylogarithm function. Then, we derive explicit expressions and some identities of those numbers and polynomials.

We introduce the Hermite based poly-Bernoulli polynomials with a q parameter and give someof their basic properties including not only addition property, but also derivative properties and integralrepresentations. We also de.ne the Hermite based λ -Stirling polynomials of the second kind, and thenprovide some relations. Moreover, we derive several correlations and identities including the Hermite-Kampéde Fériet (or Gould-Hopper) family of polynomials, the Hermite based poly-Bernoulli polynomials with a qparameter and the Hermite based λ -Stirling polynomials of the second kind.

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.

Inspired by the definition of degenerate multi-poly-Genocchi polynomials given by using the degenerate multi-polyexponential functions. In this paper, we consider a class of new generating function for the degenerate multi-poly-Bernoulli polynomials, called the type 2 degenerate multi-poly-Bernoulli polynomials by means of the degenerate multiple polyexponential functions. Then, we investigate their some properties and relations. We show that the type 2 degenerate multi-poly-Bernoulli polynomials equals a linear combination of the weighted degenerate Bernoulli polynomials and Stirling numbers of the first kind. Moreover, we provide an addition formula and a derivative formula. Furthermore, in a special case, we acquire a correlation between the type 2 degenerate multi-poly-Bernoulli numbers and degenerate Whitney numbers.

The main goal of the present paper is to construct some families of the Carlitz’s q-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz’s q-Bernoulli polynomials and numbers with weight (α, β) and investigate their some explicit properties and identities arising from the bosonic q-Volkenborn integral on ℤ_p. We then define the modified degenerate Carlitz’s q-Bernoulli polynomials and numbers with weight (α, β) and obtain some recurrence relations and other identities. Moreover, we derive some correlations with the modified Carlitz’s q-Bernoulli polynomials with weight (α, β), the modified degenerate Carlitz’s q-Bernoulli polynomials with weight (α, β), the Stirling numbers of the first kind and second kind.

In this paper, we consider a new class of polynomials which is called the multi-poly-Euler polynomials. Then, we investigate their some properties and relations. We provide that the type 2 degenerate multi-poly-Euler polynomials equals a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind. Moreover, we provide an addition formula and a derivative formula. Furthermore, in a special case, we acquire a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers.

In this work, an improved collocation method based on the Bernoulli polynomials is presented to solve the Volterra integral equation (VIE) of the second kind. The main idea of the proposed method is to improve the results of the classic Bernoulli collocation method (BCM) by dividing the interval into some sub-intervals and considering the collocation points on each of them. Here, the zeros of the shifted Chebyshev polynomials (SCPs) are considered as collocation points. Then, BCM is applied step by step from the first sub-interval to the last one. By this process, a system of algebraic equations is attained for each sub-interval that could be easily solved. Convergence of the scheme is analyzed. For the purpose of demonstrating the validity, applicability, and efficiency of the suggested scheme several numerical examples are provided. Numerical results illustrate that the accuracy of the improved Bernoulli collocation method (IBCM) is more than BCM.

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.

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.

In this paper, we study nonlocal problems for a fractional diffusion equation and a degenerate hyperbolic equation with singular coefficients in the lower terms. The uniqueness of the solution to the problem is proved by the method of energy integrals. The existence of a solution is equivalently reduced to the question of the solvability of Volterra integral equations of the second kind and a fractional differential equation. For a particular solution of the proposed problem, its visualization is carried out for various values of the order of the fractional derivative. It is shown that the order of the derivative affects the intensity of the diffusion process (subdiffusion), as well as the shape of the wave front.

In this paper, we construct the degenerate poly-Frobenius-Genocchi polynomials, called the type 2 degenerate poly-Frobenius-Euler polynomials, by means of polyexponential function. We derive explicit expressions and some identities of those polynomials. In the last section, we introduce type 2 degenerate unipoly-Frobenius-Genocchi polynomials by means of unipoly function and derive explicit multifarious properties.

Kim-Kim [12] introduced the new type of degenerate Bernoulli numbers and polynomials arising from the degenerate logarithm function. In this paper, we introduce a new type of degenerate poly-Euler polynomials and numbers, are called degenerate poly-Euler polynomials and numbers, by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Euler polynomials and numbers. In the last section, we also consider the degenerate unipoly-Euler polynomials attached to an arithmetic function, by using the degenerate polylogarithm function and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

By introducing environmental mediator, for avoiding randomness in selecting indicators and subjectivity in deciding interaction path coefficients to reveal interaction mechanism among forest industry, ecology and environmental capacity, structural Pressure–State–Impact–Response (PSIR) and quantitative Structural Equation Modeling (SEM) are utilized with data on thirty–one provinces of China. We can find that: (i) forest industry has a negative influence on ecology, whereas ecology has a positive influence on industry; (ii) the destructive influence that industry has on ecology is much greater than the positive effect ecology has on industry; and (iii) environmental capacity plays a partial mediating role between industry and ecology. According to these results we can conclude that: (i) forest industry and ecology are not in a symbiotic relationship and have not reached a level of ecological security; (ii) the interaction between industry and ecology is in a period of transition from antagonistic to beneficial; and (iii) industry dominates ecology. However, we prospect the following advice: (i) methods of forest industry should be changed; (ii) new modes of industrial integration and circular economy should be developed; and (iii) the ability of the environment to sustain itself should be enhanced. The level of forest ecological security to discuss the symbiotic mechanism of industry and ecology will propose to be measured in future.

In the paper, by virtue of some properties for the Riemann zeta function, the author finds a double inequality for the ratio of two consecutive Bernoulli numbers with even indexes and analyzes the approximating accuracy of the double inequality.

The resilience of towns and cities refers to their ability to adapt and recover when confronted with pressures or challenges. This study employs ten variables as indicators to assess the resilience of urban areas by analyzing official data and constructing a multilevel mixed-effects structural equation model. The empirical findings demonstrate that enhancing resilience significantly influ-ences urban service functions, such as strengthening healthcare systems, stabilizing the real estate market, and addressing socio-economic challenges. Specifically, factors such as healthcare re-sources, housing demand, population structure, and social security exhibit varying levels of resil-ience, which profoundly impact the spatial distribution of these factors and socio-economic de-velopment. Disparities in healthcare resource allocation, demographic aging trends, and shifts in the housing market closely correlate with resilience levels in different regions. Additionally, the study emphasizes that improving resilience not only boosts the reliability and security of urban services but also ensures that communities receive necessary support and protection during times of adversity. Consequently, policymakers should recognize the pivotal role of urban resilience in resource allocation and social equity, devising corresponding strategies to enhance the resilience of these areas.

The objective of this study was to analyze the in vitro stability and toxicity of liposomes containing guarana in skin cell lines. The liposomes were produced by the reverse phase evaporation method containing 1 mg/mL guarana. The stability of the liposomes was evaluated by physical-chemical parameters for up to 90 d using three different storage conditions. The cytotoxicity of guarana (GL), liposomes (B-Lip), and guarana-loaded liposomes (G-Lip) was evaluated on spontaneously immortalized human keratinocyte cell lines (HaCaT), murine swiss albino fibroblasts (3T3), and human fibroblast (1BR.3.G). The evaluation was performed using cellular viability analysis. The techniques used were 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT) and neutral red capturing (NRU) and the analyses were conducted after 24, 48, and 72 h exposure of these cells to the different treatments. The G-Lip exhibited physical-chemical stability for 60 d when the samples were stored in a refrigerator. The GL, B-Lip, and G-Lip demonstrated low cytotoxicity in the three cell cultures tested since a low reduction in cell viability was only observed at the highest concentrations. In addition, greater cell damage was observed for B-Lip; however, guarana protects cells from this damage. Thus, G-Lip structures can be considered as promising systems for topical applications.

Inverse Sumudu transform multiple shifting properties are used to design methodology for solving ordinary differential equations. Then algorithm applied to solve Whittaker and Zettl equations to get their new exact solutions and profiles which shown through Maple complex graphicals. Table of inverse Sumudu transforms for elementary functions given for supporting the differential equations solving using inverse Sumudu transform.

Chaos theory is widely used in the design of Pseudorandom Number Generators (PRNG), which produce number sequences with statistically uniform distributions and random appearance. However, certain methods for implementing chaotic maps can lead to dynamic degradation of the generated number sequences. To solve such problem, we develop a method for generating pseudo-random number sequences based on multiple one-dimensional chaotic maps. In particular, we introduce a Bernoulli chaotic map that utilizes function transformations and constraints on its control parameter, covering complementary regions of the phase space. This approach enables the generation of chaotic number sequences with a wide coverage of phase space, increasing uncertainty in the number sequence generation process. Moreover, by incorporating a scaling factor and a sinusoidal function, we develop a robust chaotic map, referred to as the Sine-Multiple Modified Bernoulli Chaotic Map (SM-MBCM), which ensures a high level of randomness, validated through statistical mechanics analysis tools. Using the SM-MBCM, we propose a Chaotic PRNG (CPRNG) and evaluate its quality through correlation coefficient analysis, key sensitivity tests, statistical and entropy analysis, key space evaluation, linear complexity analysis, and performance tests. Furthermore, we present an FPGA-based implementation scheme that leverages equivalent MBCM variants to optimize the electronic implementation process. Finally, we compare the proposed system with existing designs in terms of throughput and key space.

We develop an interactive likelihood (ILH) for sequential Monte-Carlo (SMC) methods for image-based multiple target tracking applications. The purpose of the ILH is to improve tracking accuracy by reducing the need for data association. In addition, we integrate a recently developed deep neural network for pedestrian detection along with the ILH with a multi-Bernoulli filter. We evaluate the performance of the multi-Bernoulli filter with the ILH and the pedestrian detector in a number of publicly available datasets (2003 PETS INMOVE, AFL, and TUD-Stadtmitte) using standard, well-known multi-target tracking metrics (OSPA and CLEAR MOT). In all datasets, the ILH term increases the tracking accuracy of the multi-Bernoulli filter.

The fractional advection-dispersion equation FADE plays an important role, especially in explaining the process of solute-transport in natural porous media with different degrees of heterogeneity. However, studies of the flow and transport of contaminants in the underground environment are based on the knowledge of many functional parameters of the subsoil. The delay factor caused by nonlinear absorption is one of these parameters inheriting significant ambiguities in its estimation in heterogeneous media. In the study, a fractional advection-dispersion FADE model was developed by the implicit Euler method, based on a Grunwald and Caputo-Liousville approximation to explicitly evaluate and simulate the transport mechanism of contaminants in media with unsaturated porous materials subjected to a nonlinear sorption model. Better solute mass retention was observed for fractional advection-dispersion FADE model. Moreover, the absorption parameter of the retardation coefficient has a better impact on the motion of the solute in a saturated porous medium. In other words, it has a significant effect on the dispersion of pollutants in aquifers. This study highlights the need to characterize the aquifer to predict solute transport in porous media.

In the present paper, the (p,q)-Hermite based Apostol type Frobenius-Euler polynomials and numbers are firstly considered and then diverse basic identities and properties for the mentioned polynomials and numbers, including addition theorems, difference equations, integral representations, derivative properties, recurrence relations. Moreover, we provide summation formulas and relations associated with the Stirling numbers of the second kind.

Industrial manipulators and parallel robots are often used for tasks like drilling or milling, that require three translational, but only two rotational degrees of freedom (“3T2R”). While kinematic models for specific mechanisms for these tasks exist, a general kinematic model for parallel robots is still missing. This paper presents the definition of the rotational component of kinematic constraints equations for parallel robots based on two reciprocal sets of Euler angles for the end-effector orientation and the orientation residual. The method allows to completely remove the redundant coordinate in 3T2R tasks and to solve the inverse kinematics for general serial and parallel robots with the gradient-descent algorithm. The functional redundancy of robots with full mobility is exploited using nullspace projection.

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.

This article aims to identify the generating function of modi…ed Apostol type q-Bernoulli polynomials. With the aid of this generating function, some properties of modi…ed Apostol type q-Bernoulli polynomials are given. It is shown that aforementioned polynomials are q-Appell. Hence, we make use of these polynomials to have applications on q-Umbral calculus. From those applications, we derive some theorems in order to get Apostol type modi…ed q-Bernoulli polynomials as a linear combination of some known polynomials which we stated in the paper.

We investigate solutions of the classical Mathieu-Hill (MH) equation which characterizes the dynamics of trapped ions, based on the Floquet theory and starting from a well known analytical model. The analytical model we introduce demonstrates that the equations of motion are equivalent to those of the harmonic oscillator (HO). Two independent approaches are used, based on two classes of complex solutions of the MH equation. The paper addresses both the damped HO and parametric oscillator (PO) for an ion confined in an electrodynamic (Paul) trap. Stability and instability regions for the periodic orbits associated to the MH equation which describes a HO (PO) confined in an electrodynamic trap are also discussed, depending on the values of the Floquet exponent, for very small values of the $q$ parameter. The main objective of the paper is to identify stable and bounded solutions for the dynamics of a HO levitated in an electrodynamic trap. This paper represents a follow-up of a recently published article in Photonics, 11 (6) 551.

Modern physics is situated on the fine line between scientific research and an attempt to answer metaphysical questions. Contemporary scholars such as Henry Stapp and Roger Penrose attempt to create an ontological interpretation of Schrödinger's equation in superposition state and its reduction or collapse to classical state. In this short essay I will try to explore how Immanuel Kant's metaphysics and the concept of noumena can be possibly integrated with quantum physics and Schrödinger's equation.

The stability properties of the Hill equation are discussed, and especially those of the Mathieu equation that characterize ion motion in electrodynamic traps. The solutions of the Mathieu equation for a trapped ion are characterized by using the Floquet theory and Hill's Method solution, which yields an infinite system of linear and homogeneous equations whose coefficients are recursively determined. Stability is discussed for parameters $a$ and $q$ that are real. Characteristic curves are introduced naturally by the Sturm-Liouville problem for the well known even and odd Mathieu equations $ce_m(z,q)$ and $se_m(z,q)$. We illustrate the stability diagram for a combined (Paul and Penning) trap and represent the frontiers of the stability domains for axial and radial motion. In case of a Paul trap the stable solution corresponds to a superposition of harmonic motions. The problem of evaluating the maximum amplitudes of stable oscillations for the ideal conditions (taken into consideration) is also approached. Anharmonic corrections are discussed within the frame of the perturbation theory, while the frontiers of the modified stability domains are determined as a function of the chosen perturbation parameter. The results apply to 2D and 3D ion traps used for different applications in quantum engineering, among which optical clocks, quantum logic and quantum metrology, but not restricted only to these.

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.

In the paper, by virtue of the Fa`a di Bruno formula, some properties of the Bell polynomials of the second kind, and the inversion formulas of binomial numbers and the Stirling numbers of the first and second kinds, the authors simplify meaningfully and significantly coefficients in two families of ordinary differential equations associated with higher order Frobenius–Euler numbers.

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 apply utilized the extended form of the auxiliary equation method to obtain extensively reliable exact travelling wave solutions of perturbed Gerdjikov–Ivanov equation (GIE) that is widely used as a model in the field theory of quanta and non-linear optics. The method is based on a simple first order second degree ODE. The new form of the approach gives more solutions to the governing equation efficiently.

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.

A proof is given showing the Collatz Conjecture is true for all positive integers. The proof was possible once the perspective of the proof was changed from looking at the pattern of the positive integers to looking at the conjecture rules. The conjecture rule for even numbers organizes all positive integers into unique sets and the rule for odd numbers interconnects the unique sets into dendritic pathways to “1.” Infinite loops, other than the minor 4-2-1 loop after reaching “1,” and pathways to infinity are shown to be mathematically impossible. The proof predicted a general equation that shows all positive integers ultimately reach a final value of “1," and calculates the values and locations of the odd positive integers during the iterations for each tested positive integer.

In the framework of the self-consistent Maxwell-Pauli theory, the non-linear Pauli equation is obtained. Stationary and nonstationary solutions of the nonlinear Pauli equation for the hydrogen atom are studied. We show that spontaneous emission and the related rearrangement of the internal structure of an atom, which is traditionally called a spontaneous transition, have a simple and natural description in the framework of classical field theory without any quantization and additional hypotheses. The behavior of the intrinsic magnetic moment (spin) of an electron wave in an external magnetic field is considered. We show that, according to the self-consistent Maxwell-Pauli theory, in a weak magnetic field, the intrinsic magnetic moment of an electron wave is always oriented parallel to the magnetic field strength vector, while in a strong magnetic field, depending on the initial orientation of the intrinsic magnetic moment, two orientations are realized: either parallel or antiparallel to the magnetic field strength vector.

Traditional discrimination techniques for Euler deconvolution use only the color spectrums of structural indexes, without considering the spatial distribution characteristics and inherent relationships among the Euler solutions to separate adjacent causative sources. In the present study, a new approach was developed for discriminating uncorrelated Euler solutions from coherent solutions based on the focusing levels indicated by the probability density distributions generated using multivariate kernel density estimations (KDE). A novel multiple coverage technique was proposed by using a series of different sized moving windows over gridded gravity data, which formed tight clusters of Euler solutions for different sized causative sources. The results of the probability density distributions were obtained using a 3-D KDE method for the Euler solution subsets {x, y, z} of synthetical models, and real data from a survey conducted in British Columbia (Canada) which had successfully established more credible and meaningful geological models when compared with three other subsets.

In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.

Current physiology attributes the mechanism of membrane potential generation to transmembrane ion transport, but ion adsorption could just as well play this fundamental role. The evidence shows that the ion adsorption mechanism accurately reproduces the experimentally measured membrane potential. The Goldman-Hodgkin-Katz equation (GHK eq.) and the Nernst equation (Nernst eq.) are the typical mathematical formulas representing the membrane potential in current physiology. However, the authors were able to show that the potential formulas by ion adsorption mechanism give identical results to GHK eq. and Nernst. eq. Our experimental and theoretical analyses suggest that there is a special relationship between the membrane potential and the membrane surface charge density, and this unique equation inevitably leads to the establishment of a GHK eq and/or a Nernst eq. The authors found that the unique equation is the foundation of thermodynamics “Boltzmann distribution”. Thus, the GHK eq. and the Nernst eq. are simply the natural consequence of thermodynamics from the view of the ion adsorption mechanism.

In this paper, we revisit the two theoretical approaches for the formulation of the tachyonic Dirac equation. The first approach works within the theory of restricted relativity, starting from a Lorentz invariant Lagrangian consistent with a spacelike four-momentum. The second approach uses the theory of relativity extended to superluminal motions and works directly on the ordinary Dirac equation through superluminal Lorentz transformations. The equations resulting from the two approaches show mostly different, if not opposite, properties. In particular, the first equation violates the invariance under the action of the parity and charge conjugation operations. Although it is a good mathematical tool to describe the dynamics of a space-like particle, it also shows that the mean particle velocity is subluminal. In contrast, the second equation is invariant under the action of parity and charge conjugation symmetries, but the particle it describes is consistent with the classical dynamics of a tachyon. This study shows that it is not possible with the currently available theories to formulate a covariant equation that coherently describes the neutrino in the framework of the physics of tachyons, and depending on the experiment, one equation rather than the other should be used.

The effects on Raychaudhuri's equation of an intrinsically discrete or particle nature of spacetime are investigated, through consideration of null congruences emerging from, or converging to, a generic point of spacetime, i.e. in geometric circumstances somehow prototypical of singularity issues. This is done from an effective point of view, meaning through a (continuous) description of spacetime modified to embody the existence of an intrinsic discreteness on the small scale, adding to previous results for non-null congruences. Various expressions for the effective rate of change of expansion are derived. They in particular provide finite values for the limiting effective expansion and its rate of variation when approaching the focal point; this on top of resulting non vanishing the limiting cross-sectional area itself of the congruence.

In this paper, we study the p-adic integral representation on Z_p of q-Bernoulli numbers arising from two variable q-Bernstein polynomials and investigate some properties for the q-Bernoulli numbers. In addition, we give some new identities of q-Bernoulli numbers.

The study aimed to analyze thoracic and lumbar erector spinae muscle activations during 3D elongation exercises in individuals with adolescent idiopathic scoliosis (AIS). Participants included 24 adolescents, meeting specific criteria such as a Cobb angle between 10° and 20°, and not having undergone surgical or brace treatments. EMG data were collected to evaluate muscle activations. The study found no significant interaction between exercise type and the side of the curve, but a significant main effect of exercise type on muscle activation, irrespective of Cobb angle. Different exercise types significantly affected muscle activity, with the highest activations in the TheraBand exercise, followed by manual and without equipment conditions. This suggests that exercise type significantly impacts muscle engagement in AIS patients, providing valuable insights for targeted exercise program design. The study, however, focused on short-term muscle activation, not long-term Cobb angle progression, pointing to the need for further research in this area.

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.

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

Defining integrals over volume element in curved spacetime together with the mass and energy densities within that region of spacetime is apparently equivalent to the given energy and mass of that body or particle. With this in mind equations involving energies and mass can be rewritten in terms of their densities and the integral over the volume element. The idea find its place in the Hamiltonian formulations, and can be carried over into quantum mechanical frameworks systematically accounting for the effect of spacetime volume structures and gravity on dynamics of quantum systems. This article briefly explores this concept and runs through various frameworks and equations relevant to this idea, particularly the schrodinger-Newton equation and discusses the cosmological constant as a vacuum energy density.

In this paper, we introduce a way of representing a given mapping as the sum of odd and even mappings. Then, by using this representation, we investigate the stability of various forms for the general nonic functional equation.

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.

In this short note we present a technique using which one attributes frequency and wavevector to (almost) arbitrary scalar fields. Our proposed definition is then applied to the classical wave equation to yield a novel nonlinear PDE.

Maxwell equation for the electric field has been solved for any medium by suggesting the wavenumber and angular frequency be complex quantities. This accounts for the field decay by interaction with the medium.This expression for the time and special decay of the electric field in a medium is used to construct a new wave function sensitive to the medium physical properties. This new wave function, unlike the conventional one, differentiates between a beam of particles in a vacuum and that enters a medium which is an attenuated due to the scattering effect.Another expression for time decaying electric field was obtained using Newton’s laws for frictional medium. This expression shows that the electric field diminishes due to friction.Fortunately, this time decaying part of the electric field is typical to that derived from Maxwell’s equations. Finally, a new Schrodinger equation sensitive to the medium properties was derived. This equation, fortunately, describes some scattering processes for Protein scattering, scattering of x-rays, opto-acoustic phonons, and Raman scattering for some materials successfully .

An adequate representation of the water infiltration process in the soil allows improving the efficiency in application and the uniformity in surface irrigation. The Green and Ampt model has shown a good representation of the process, and researchers from the United States Department of Agriculture (USDA) determined the values of their parameters for soils of that country, which are shown in tables or through functional relationships and this information is used as reference in several parts of the world, although there is no certainty that they are representative of the soils in Mexico. In this study, the parameters of the Green & Ampt equation were determined and evaluated in some soils of agricultural importance in Mexico. The parameters were obtained in four ways: one of them applied a methodology adapted from Brooks and Corey to quantify the wetting front capillary pressure head and used an permeameter under constant hydraulic head to determine the saturated hydraulic conductivity, and the other three consisted in taking them from three studies reported by the USDA. The values of the parameters suggested in Mexico drastically underestimated the results with relative errors (RE) in a range of -49.0 to -94.0% and the most representative were those obtained with the methodology proposed in this research with RE of -15.0 to 6.0%.

Proposed in 1889 by Svante Arrhenius, the Arrhenius equation is an important result in physical chemistry which aims to find the relationship between temperature and reaction rates. It is also an essential equation for analytic chemistry which helps to determine the energy barrier for a given reaction. In this article, we will go through a detailed derivation of the exponential factor in the Arrhenius Equation based on the Boltzmann distribution of particle energy probability. Then, we will attempt to model the pre-exponential factor in accordance with the principles of collision theory for bi-molecular reaction with gaseous reactants. Understanding the basic principles of chemical reaction provides valuable tools to analyze complex reaction mechanisms in the perspective of optimizing existing industrial processes.

Recently quantum cosmology in tomographic representation was considered. as an important tool to analyze the quantum states of the early universe in relation with the subsequent classical evolution. Given a wave function the correspondent tomogram is defined proportional to the square modulus of its fractional Fourier transform. The classical limit obtained by taking the limit ℏ→0 can be compared with the classical tomogram obtained from the classical hamiltonian formalism. In this paper we show that a set of tomograms can be derived as exact solutions of a third order equation when a de Sitter quantum universe is considered. We finally discuss the limits of this approach to quantum cosmology because the extension to equations with arbitrary potentials can become extremely complicated for potentials given by algebraic polynomials of degree greater than two and therefore also for potentials expressible by arbitrary functions.

This article is devoted to Feller's diffusion equation which arises naturally in probabilities and physics (e.g. wave turbulence theory). If discretized naively, this equation may represent serious numerical difficulties since the diffusion coefficient is practically unbounded and most of its solutions are weakly divergent at the origin. In order to overcome these difficulties we reformulate this equation using some ideas from the Lagrangian fluid mechanics. This allows us to obtain a numerical scheme with a rather generous stability condition. Finally, the algorithm admits an elegant implementation and the corresponding Matlab code is provided with this article under an open source license.

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.

We still do not have satisfactory answers about the inflationary and accelerating expansions of the universe, the missing mass problem, and the origins of the four fundamental forces and the quantum phenomena. Here, we introduce a new model of matter and space and a postulate on a constant of motion, and then show that an equation of motion derived from the constant of motion could provide new understandings of the above problems. For example, we show that there is a natural inflation scenario and that the co-moving distance to the closest accelerating supernova is clearly given as $c/(\sqrt{3} H)$ ($\sim$8.1 billion ly). In addition, we show that, in a system including a proton and an electron, the equation of motion has eight acceleration terms, some of which are suspected to be responsible for the four fundamental forces and the galaxy rotation anomaly. From further discussion, we conjecture that in order to understand the gravitational and strong interactions and the quantum phenomena, it is necessary to consider the expansion of the universe. The lensing mass distribution is explained, based on another postulate on a method of representing superposition.

Rough volatility models are popularized by \cite{gatheral2018volatility}, where they have shown that the empirical volatility in the financial market is extremely consistent with rough volatility. Fractional Riccati equation as a part of computation for the characteristic function of rough Heston model is not known in explicit form as of now and therefore, we must rely on numerical methods to obtain a solution. In this paper, we give a short introduction to option pricing theory and an overview of the current advancements on the rough Heston model.

This article aims to discuss, the stability and boundedness character of the solutions of the rational equation of the form \begin{equation}\label{eql21.1} y_{t+1}=\frac{\nu\epsilon^{-y_t}+\delta\epsilon^{-y_{t-1}}}{\mu+\nu y_t+\delta y_{t-1}},\quad t\in N(0). \end{equation} Here, $\epsilon>1, \nu,\delta,\mu\in (0,\infty)$ and $y_0, y_1$ are taken as arbitrary non-negative reals and $N(a)=\{a,a+1,a+2,\cdots \}$. Relevant examples are provided to validate our results. The exactness is tested using MATLAB.

This paper secures the analytical soliton solutions of the perturbed NLSE gives to (1). The existence criteria have been demonstrated and the wave speed of the soliton befall out. To achieve all the results obtained we used the mathematical technique. In view of the results obtained, some new additional one have been reported in left-handed metamaterials.

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.

We show how the Poincar\'e--Riemann--Hilbert boundary-value problem enables us to construct effective estimates of the potential in the Schr\"odinger equation. The apparatus of the three-dimensional inverse problem of quantum scattering theory is developed for this. It is shown that the unitary scattering operator can be studied as a solution of the Poincar\'e--Riemann--Hilbert boundary-value problem. This allows us to go on to study the potential in the Schr\"odinger equation

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

The ordinary 3D wave equation for nondissipative, homogeneous, isotropic media admits solutions where the point sources are permitted to move, but it does not admit solutions where the receiver is allowed to move. To overcome this limitation, the new wave equation is derived, permitting both the receiver and the source to move. This new wave equation is a generalization of the standard wave equation, and it reduces to the standard wave equation when the receiver is at rest. To derive this new wave equation, we first mathematically define a diverging spherical wave caused by the stationary point source. From this purely mathematical definition, the wave equation for the stationary source and the moving receiver is derived together with the corresponding free-space Green's function. Utilizing the derived Green's function, it was shown that this new wave equation also permits the solutions where both the receiver and the source are permitted to move. In conclusion, this paper demonstrates that, instead of an ordinary wave equation, the wave equation for the moving source and the moving receiver governs the waves emitted by the moving point sources and received by the moving receivers.

Let $S=\left(s_1, \ldots, s_n\right)$ be a finite sequence of integers. Then $S$ is a Gilbreath sequence of length $n$, $S\in\mathbb{G}_n$, iff $s_1$ is even or odd and $s_2, \ldots, s_n$ are respectively odd or even and $\min\mathbb{K}_{\left(s_1, \ldots, s_m\right)}\leq s_{m+1}\leq\max\mathbb{K}_{\left(s_1, \ldots, s_m\right)}\forall m\in\left[\left.1, n\right)\right.$. This, applied to the order sequence of prime number $P$, defines Gilbreath polynomials and two integer sequences A347924 \cite{oeisA347924} and A347925 \cite{oeisA347925} which are used to prove that Gilbreath conjecture $GC$ is implied by $p_n-2^{n-1}\leqslant\mathcal{P}_{n-1}\left(1\right)$ where $\mathcal{P}_{n-1}\left(1\right)$ is the $n-1$-th Gilbreath polynomial at 1.

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

New proposals for the equation of state of four- and five-dimensional hard-hypersphere mixtures in terms of the equation of state of the corresponding monocomponent hard-hypersphere fluid are introduced. Such proposals (which are constructed in such a way so as to yield the exact third virial coefficient) extend, on the one hand, recent similar formulations for hard-disk and (three-dimensional) hard-sphere mixtures and, on the other hand, two of our previous proposals also linking the mixture equation of state and the one of the monocomponent fluid but unable to reproduce the exact third virial coefficient. The old and new proposals are tested by comparison with published molecular dynamics and Monte Carlo simulation results and their relative merit is evaluated.

DGJ (Daftardar-Gejii-Jafaris) method is used to obtain numerical solution of the third order fractional differential equation. Providing the DGJ method converges, the approximate solution is a good and effective numerical result which is close to the exact solution or the exact solution. For this,the examples of the explaning the method are presented. The proposed method is implemented for the approximation solution of the third order nonlinear fractional partial differential equations. The method was shown to be unsuitable and inconsistent for an example of a nonlinear fractional partial differential equation depend on initial-boundary value conditions. The fact that these numerical results are not consistent can be explained by the fact that the method is not convergent.

Soil erosion and sediment transport are quite complex processes as they depend on physical, biological, mechanical, and chemical processes within a particular catchment. Therefore, it is highly essential to better explain engaged physical processes and means of accounting for site-specific conditions, for soil loss and sediment yield estimation. This paper mainly focuses on physical explanations behind soil erosion and common soil erosion models like Universal or Revised Universal Soil Loss Equation(USLE/RUSLE) and Modified Universal Soil Loss Equation(MUSLE). Based on the physical explanations and overall limitations, the MUSLE is selected for the application of sediment yield estimation. The main objective of this paper is to estimate the best exponent of the MUSLE, and to estimate the best combination of the exponent and topographic factor of the MUSLE under hydro-climatic conditions of Ethiopia. For the sake of calibration procedure, the main parameters of the MUSLE which directly affect soil erosion process such as cover, conservation practice, soil erodibility, and topographic factors are estimated based on the past experiences from literature and comparative approaches, whereas the other parameters which do not directly affect the erosion process or which have no any physical meaning (i.e coefficient a and exponent b) are estimated through calibration. It is verified that the best exponent of the MUSLE is 1 irrespective of the topographic factor, which results in the maximum performance of the MUSLE (i.e approximately 100\%). For the best combination of the exponent and topographic factor, the performance of the MUSLE is greater than or equal to 80\% for all four watersheds under our consideration, we expect the same for other watersheds of Ethiopia.

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.

This paper presents the perturbation theory for the double–sine–Gordon equation. We obtain a system of differential equations that shows the amplitude and phase modulation of the approximate solution.. In the particular case λ = 0 we get the well-known perturbation theory for the sine–Gordon equation. For a special value λ=-1/8, we derive a phase-locked solution with the same frequency of the linear case. In general we obtain both coherent (solitary waves, lumps and so on) solutions as well as fractal solutions. We can demonstrate the existence of envelope wobbling solitary waves, because of the phase modulation depending on the solution amplitude and on the position. The main conclusion is that it is too reductive focus only on coherent solutions for the double sine-Gordon equation, because of the very rich behavior for the DSG nonlinear equation, including wobbling chaotic and fractal solutions.

Although there is a common physiological notion that the origin of the membrane potential is attributed to transmembrane ion transport, it is theoretically possible to explain its generation by the mechanism of ion adsorption. It was previously suggested that the ion adsorption mechanism led even to the potential formulas which are even identical to either the famous Nernst equation or Goldman-Hodgkin-Katz equation. Our further analysis shown in this paper indicates that the potential formula based on the ion adsorption mechanism leads to one equation which is the function of material surface charge density and the material surface potential. Furthermore, we confirmed that the equation holds in all the different experimental systems we studied. Although we have not succeeded in elucidating why such an equation is established, the equation appears to be the key equation governing the characteristics of the membrane potential regardless of the systems in question.

HU is the Hypergeometrical Universe Theory (HU)[1-8], proposed in 2006, where the Universe is a Lightspeed Expanding Hyperspherical Hypersurface and Gravitation is an absolute-velocity-dependent, epoch-dependent force. Here we introduce the Big Pop Cosmogenesis and show our calculations associated with the Equation of State of the Universe. This article is the first in a series of articles[9-22] supporting the paradigm shift.

A solution for the Prandtl-Blasius equation is essential to all kinds of boundary layer problems. This paper revisits this classic problem and presents a general Maple code as its numerical solution. The solutions were obtained from the Maple code, using the Runge-Kutta method. The study also considers convergence radius expanding and an approximate analytic solution is proposed by curve fitting. Similarly, the study resolves some boundary layer related problems and provide relevant Maple codes for these.

Recent astronomical observations with respect to measurements in distant supernovas, cosmic microwave background and weak gravitational lensing confirm that the Universe is undergoing a phase of accelerated expansion and it has been proposed that this cosmological behavior is caused by a hypothetical dark energy which has a strong negative pressure that allows explain the expanding universe. Several theoretical ideas and models related dark the energy includes the cosmological constant, quintessence, Chaplygin gas, braneworld and tachyonic scalar fields. In this paper, we have obtained new relativistic stellar configurations considering an anisotropic fluid distribution with a charge distribution which could represents a potential model of a dark energy star. In order to investigate the effect of a quadratic equation of state in this anisotropic model we specify particular forms for the gravitational potential that allow solving the Einstein-Maxwell field equations. For these new solutions we checked that the radial pressure, metric coefficients, energy density, anisotropy factor, charge density , mass function are well defined and are regular in the interior of the star. The solutions found can be used in the development of dark energy stars models satisfying all physical acceptability conditions but the causality condition and strong energy condition are violated. We expect that these models have multiple applications in astrophysics and cosmology.

In this paper we study the initial boundary value problem for logarithmic Higher Order Wave equation. Introducing the Logarithmic Sobolev inequality and using the combination of Galerkin method, we consider the theorem of existence of a global weak solution to problem for the initial boundary value problem of the logarithmic wave equation. By constructing an appropriate Lyapunov function, we obtain the decay estimates of energy for logarithmic Higher Order Wave equation. The proof of the main theorem is given.

The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender-Brody-Müller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta function, and provide an analytical expression for the eigenvalues of the results. The holomorphicity of the resulting eigenvalues is examined. Moreover, a second quantization of the resulting Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function converges at σ = 1/2.

This study delves into the Schrödinger equation's new invariants, shedding light on their crucial implications for theoretical physics and beyond. Specifically, it explores the behavior of scattering amplitudes and bound states concerning the choice of coordinate systems. The research unveils that while the energy of bound states remains invariant under coordinate transformations, the phase of scattering amplitudes undergoes variations, underscoring its pivotal role in theories reliant on phase normalization. Moreover, these findings have played a pivotal role in constructing estimates for three-dimensional Navier-Stokes equations, enhancing our ability to model complex fluid dynamics with greater precision and reliability. Additionally, in seismic exploration, tomography, and ultrasound imaging, these properties allow researchers to optimize phase selection, facilitating the most effective interpretation of measurement results. This strategic phase manipulation ensures clearer insights into subsurface structures, tissue composition, and fluid dynamics, with the flexibility to seamlessly transition back to the original coordinate system post-interpretation. This interdisciplinary synergy underscores the profound impact of fundamental theoretical research on advancing practical applications across diverse scientific domains.

The mind-body problem is the central issue in both of philosophy and life science for several centuries. To date, there is still no conclusive theory to interpret the relation between the mind and the body, even just the mind alone. Here, we promote a novel model, a derived mathematic equation called the entropic system equation, to describe the innate characters of the mind and the mechanism of the mind-body coupling phenomenon. As the semi-open thermodynamic systems far from equilibrium, the living organisms could be logically considered as an entropic system. In the living organisms or the entropic systems, there also are three essential existing elements including mass, energy and information, in which the mind and the body are hypothetically coupled by free energy and entropic force.

An extension of the Stoney formula for the case of a back side metallized 8” silicon taiko wafer has been developed, in the elastic regime, within the frame of the theory of elasticity. A good correlation between the calculated warpage, determined by the stress released by a given back side metallization (BSM), and the corresponding experimental warpages of the same thick metal layers deposited on an 8” silicon taiko wafer provides evidences of the correctness of the developed theory. This development suggests the possibility to extend this approach to the case of 8” taiko wafers based on a wide band gap semiconductor such as silicon carbide (SiC).

This article deals with the conformable double Laplace transforms and their some properties with examples and also the existence Condition for the conformable double Laplace transform is studied. Finally, in order to obtain the solution of nonlinear fractional problems, we present a modified conformable double Laplace that we call conformable double Laplace decomposition methods (CDLDM). Then, we apply it to solve, Regular and singular conformable fractional coupled burgers equation illustrate the effectiveness of our method some examples are given.

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 this paper, we analyze the generalized Ulam-Hyers stability of affine type AQ Functional Equation in various Banach Spaces using Direct and Fixed Methods

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.

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

We present set-theoretical solutions of the Yang-Baxter equation in BCK–algebras. Some solutions in BCK−algebras are not solutions in other structures (such as MV −algebras). Related to our investigations we also consider some new structures: Boolean coalgebras and a unified braid condition – quantum Yang-Baxter equation. Finally, we will see how poetry has accompanied the development / history of the Yang–Baxter equation.

We give an operator solution to an advanced-retarded differential equation. The application of the operators involved produces a solution in terms of Bessel functions.

The generalized Kuramoto–Sivashinsky equation is investigated using the modified Kudrayshov method for the exact analytical solution. The modified Kudrayshov method converts the nonlinear partial differential equation to algebraic equations, as a result of various steps, which on solving the so obtained equation systems yields the analytical solution. By this way various exact solutions including complex structures are found and drawn their behaviour in complex plane by Maple to compare the uniqueness of solutions.

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 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.

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.

We propose a generalization of Nambu-Hamilton equation in superspace $\mathbb R^{3|2}$ with three real and two Grassmann coordinates. We construct the even degree vector field in the superspace $\mathbb R^{3|2}$ by means of the right-hand sides of proposed generalization of Nambu-Hamilton equation and show that this vector field is divergenceless in superspace. Then we show that our generalization of Nambu-Hamilton equation in superspace leads to family of ternary brackets of even degree functions defined with the help of Berezinian. This family of ternary brackets is parametrized by the infinite dimensional group of invertible second order matrices, whose entries are differentiable functions on the space $\mathbb R^{3}$. We study the structure of ternary bracket in a more general case of a superspace $\mathbb R^{n|2}$ with $n$ real and two Grassmann coordinates and show that for any invertible second order functional matrix it splits into the sum of two ternary brackets, where one is usual Nambu-Poisson bracket, extended in a natural way to even degree functions in a superspace $\mathbb R^{n|2}$, and the second is a new ternary bracket, which we call $\Psi$-bracket, where $\Psi$ can be identified with invertible second order functional matrix. We prove that ternary $\Psi$-bracket as well as the whole ternary bracket (the sum of $\Psi$-bracket with usual Nambu-Poisson bracket) is totally skew-symmetric, satisfies the Leibniz rule and the Filippov-Jacobi identity (Fundamental Identity).

In this paper, we gave an attack on RSA when Euler function has small multiplicative inverse modulo "e" and the prime sum p+q is of the form p+q=2^nk_0+k_1 where n is a given positive integer and k_0 and k_1 are two suitably small unknown integers using sublattice reduction techniques and Coppersmith's methods for finding small roots of modular polynomial equations. When we compare this method with an approach using lattice based techniques, this procedure slightly improves the bound and reduces the lattice dimension.