General Equation Research Articles

Mathematical modeling of insulating coating of thermal conductivity including body’s own radiation and non-local spatial effects

Abstract In this paper thermal conductivity analysis of insulating coating is discussed. The general partial differential thermal conductivity equation is used with consideration to nonlocal boundary effects. We relying on the Eringen’s approach involves heat-flow vector transformation using nonlocal influence function and weight coefficients. Also presented model includes own surface radiation of the coating.

Wigner Function Method for Describing Electromagnetic Field in Plasma-Like Media with Spatial Dispersion and Resonant Dissipation

The paper gives a systematic presentation of the Wigner function method (Weyl formalism) for modeling the propagation and absorption of electromagnetic waves in anisotropic and gyrotropic dissipative media with spatial dispersion. A general kinetic equation for the Wigner function (tensor) is formulated, and its asymptotic expansion up to the second order for smoothly inhomogeneous and weakly dissipative media is constructed. As a result, a modification of the method of the kinetic equation for rays is proposed, based on the stochastic description of rays, making it possible to increase the accuracy of numerical modeling of wave problems with strong transverse inhomogeneity of the absorption coefficient without increasing the amount of calculations. The technique developed can be used to describe the propagation, absorption, and scattering of electron-cyclotron waves in high-temperature plasma of magnetic traps for controlled fusion in cases where standard modeling methods do not provide the necessary accuracy.

Induced gravitational wave interpretation of PTA data: a complete study for general equation of state

We thoroughly study the induced gravitational wave interpretation of the possible gravitational wave background reported by PTA collaborations, considering the unknown equation of state w of the early universe. We perform a Bayesian analysis of the NANOGrav data using the publicly available PTArcade code together with SIGWfast for the numerical integration of the induced gravitational wave spectrum. We focus on two cases: a monochromatic and a log-normal primordial spectrum of fluctuations. For the log-normal spectrum, we show that, while the results are not very sensitive to w when the GW peak is close to the PTA window, radiation domination is out of the 2σ contours when only the infra-red power-law tail contributes. For the monochromatic spectrum, the 2σ bounds yield 0.1 ≲ w ≲ 0.9 so that radiation domination is close to the central value. We also investigate the primordial black hole (PBH) abundance for both monochromatic and log-normal power spectrum. We show that, in general terms, a larger width and stiffer equation of state alleviates the overproduction of PBHs. No PBH overproduction requires w ≲ 0.57 up to 2-σ level for the monochromatic spectrum. Furthermore, including bounds from the cosmic microwave background, we find in general that the mass range of the PBH counterpart is bounded by 10-5 M ⊙ ≲ M PBH ≲ 10-1 M ⊙. Lastly, we find that the PTA signal can explain the microlensing events reported by OGLE for w ~ 0.7. Our work showcases a complete treatment of induced gravitational waves and primordial black holes for general w for future data analysis.

Evidence for Nightside Water Emission Found in Transit of Ultra-hot Jupiter WASP-33 b

To date, the dayside thermal structure of ultra-hot Jupiters (UHJs) is generally considered to be inverted, but their nightside thermal structure has been less explored. Here, we explore the impact of nightside thermal emission on high-resolution infrared transmission spectroscopy, which should not be neglected, especially for UHJs. We present a general equation for the high-resolution transmission spectrum that includes planetary nightside thermal emission. This provides a new way to infer the thermal structure of the planetary nightside with high-resolution transmission spectroscopy. Using the cross-correlation technique, we find evidence for the presence of an H2O emission signature on the UHJ WASP-33 b during the transit, indicating an inverted temperature structure on its nightside. Such a result suggests a stronger heat transport through the circulation than currently expected. An alternative explanation is that the rotating visible hemisphere during transit leads to the potential contribution of the limb and dayside atmospheres to the detected emission signature. In the future, the combination of high-resolution, full-phase-curve spectroscopic observations and general circulation models will hopefully solve this puzzle and provide a complete picture of the three-dimensional nature of the chemistry, circulation, and thermal structure of UHJs.

Nonlinear elastic general integral equations in micromechanics of random structure composites

We consider static problems for composite materials (CMs) with either locally elastic or peridynamic constitutive properties. The general integral equation (GIE) is the exact integral equationWe consider static problems for composite materials connecting the random fields at the point being considered and the surrounding points. There is a very long and colored history of the development of GIE which goes back to Lord Rayleigh. Owing to the new GIE (forming the second background of micromechanics also called the computational analytical micromechanics, CAM), one proved that local micromechanics (LM) and peridynamic micromechanics (PM) are formally similar to each other for CM of random structures. By now, the GIEs are generalized to CMs (of statistically homogeneous and inhomogeneous structures) with the phases described by local models, strongly nonlocal models (strain type and displacement type, peridynamics), and weakly nonlocal models (strain-gradient theories, stress-gradient theories, and higher-order models). However, a fundamental restriction of all mentioned GIEs is their linearity with respect to a primary unknown variable. The goal of this study is obtaining nonlinear GIEs for PM, and, in a particular case, for LM. For the presentation of PM as a unified theory, we describe PM as the formalized schemes of blocked (or modular) structures so that the experts developing one block need not be experts in the underlying another block (this is a good background for effective collaborations of different teams in so many multidisciplinary areas as PM). The opportunity for the creation of this blocked structure of the PM is supported by a critical generalization of CAM which is extremely flexible, robust, and general.

New solutions of the general elliptic equation and its applications to the new (3 + 1)-dimensional integrable Kadomtsev-Petviashvili equation

In this article, we introduce many new Jacobi elliptic function solutions to the general elliptic equation. Consequently, the Jacobi elliptic function expansion method is improved to accommodate the general elliptic equation and its new solutions for constructing exact traveling wave solutions of nonlinear partial differential equations (NLPDEs). Moreover, the improved method is used to obtain new explicit solutions for the (3+1)-dimensional integrable Kadomtsev-Petviashvili (KP) equation. This method can be applied to many other NLPDEs as well for obtaining new exact solutions.

On computation of solution for [formula omitted] dimensional fractional order general wave equation

In this research article, we handle a class of (2+1) dimensional wave equations under fractional-order derivatives using an iterative integral transform due to Laplace. The concerned derivative is taken in the Caputo’s sense. The proposed method is a purely algebraic manipulation approach to compute solutions without a priori knowledge of geometry and physical meaning related to the proposed problem. In fact, in this procedure, we combine two novel techniques, Laplace transforms (LT) and iterative procedures to form a hybrid technique for computation of the solution to the proposed problem. The method is rapidly convergent. Here, we give various examples for the validation of our proposed method. The superiority of the method over the existing numerical method is that it does not require any prior discretization or collocation of functions. Also, the method is independent of axillary parameters as needed in the homotopy methods, because such auxiliary parameters control the efficiency of the mentioned methods. The proposed procedure is simple and straightforward. In addition, some comparison between exact and approximate solutions is also given. The proposed method is compared with the homotopy perturbation method (HPM) which shows that the proposed technique is more efficient and easy to implement.

Global Well-Posedness and Long-Time Asymptotics of a General Nonlinear Non-local Burgers Equation

Global Well-Posedness and Long-Time Asymptotics of a General Nonlinear Non-local Burgers Equation

A novel model for count data: zero-inflated Probit Bell model with applications

A novel model for count data is proposed in this work. This new model is briefly called the Zero-Inflated Probit Bell (ZIPBell) model. The ZIPBell model can be used to simulate and analyze traditional count data and can be used well in the simulation and analysis of count data with the phenomenon that the frequency for zero occupies a very high percentage. In the scope of this study, we first provide the general formula, functions or related equations for the ZIPBell model. The results of asymptotic inferences for the ZIPBell model are also given in this work. Maximum likelihood estimator (MLE) is used to check its performance of the proposed model. According to the numerical illustrations via several simulation researches and the analysis of a practical data set, it has been seen that the MLE method has offered correct and reliable inferences. Finally, the inferences and conclusions of this study are mentioned.

Analysis of Collatz Conjecture Rules

Abstract A proof of the Collatz Conjecture is presented. Changing the perspective of the problem from looking at the pattern of the positive integers to looking at the conjecture rules made the proof possible. 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 values continually increasing to infinity are shown to be mathematically impossible. The proof predicted a general equation that shows all positive integers 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. JEL classification numbers: CO2; C65. Keywords: Collatz Conjecture, Proof, Rules, Dendritic pathway.

Postoperative body weight change and its influencing factors in patients with gastric cancer.

The high incidence and mortality of gastric cancer (GC) pose a significant threat to human life and health, and it has become an important public health challenge in China. Body weight loss is a common complication after surgical treatment in patients with GC and is associated with poor prognosis and GC recurrence. However, current attention to postoperative weight change in GC patients remains insufficient, and the descriptions of postoperative weight change and its influencing factors are also different. To investigate body weight changes in patients with GC within 6 mo after gastrectomy and identify factors that influence dynamic body weight changes. We conducted a prospective longitudinal study of 121 patients with GC and collected data before (T0) and 1 (T1), 3 (T2), and 6 (T3) mo after gastrectomy using a general data questionnaire, psychological distress thermometer, and body weight measurements. The general estimation equation (GEE) was used to analyze the dynamic trends of body weight changes and factors that influence body weight changes in patients with GC within 6 mo of gastrectomy. The median weight loss at T1, T2, and T3 was 7.29% (2.84%, 9.40%), 11.11% (7.64%, 14.91%), and 14.75% (8.80%, 19.84%), respectively. The GEE results showed that preoperative body mass index (BMI), significant psychological distress, religious beliefs, and sex were risk factors for weight loss in patients with GC within 6 mo after gastrectomy (P < 0.05). Compared with preoperative low-weight patients, preoperative obese patients were more likely to have weight loss (β = 14.685, P < 0.001). Furthermore, patients with significant psychological distress were more likely to lose weight than those without (β = 2.490, P < 0.001), and religious patients were less likely to lose weight 6 mo after gastrectomy than those without religious beliefs (β = -6.844, P = 0.001). Compared to female patients, male patients were more likely to experience weight loss 6 mo after gastrectomy (β = 4.262, P = 0.038). Male patients with GC with high preoperative BMI, significant psychological distress, and no religious beliefs are more likely to lose weight after gastrectomy.

Existence and stability of weak solutions of the Vlasov–Poisson system in localised Yudovich spaces

We consider the Vlasov–Poisson system both in the repulsive (electrostatic potential) and in the attractive (gravitational potential) cases. Our first main theorem yields the analog for the Vlasov–Poisson system of Yudovich’s celebrated well-posedness theorem for the Euler equations: we prove the uniqueness and the quantitative stability of Lagrangian solutions f=f(t,x,v) whose associated spatial density ρf=ρf(t,x) is potentially unbounded but belongs to suitable uniformly-localised Yudovich spaces. This requirement imposes a condition of slow growth on the function p↦‖ρf(t,⋅)‖Lp uniformly in time. Previous works by Loeper, Miot and Holding–Miot have addressed the cases of bounded spatial density, i.e. ‖ρf(t,⋅)‖Lp≲1 , and spatial density such that ‖ρf(t,⋅)‖Lp∼p1/α for α∈[1,+∞) . Our approach is Lagrangian and relies on an explicit estimate of the modulus of continuity of the electric field and on a second-order Osgood lemma. It also allows for iterated-logarithmic perturbations of the linear growth condition. In our second main theorem, we complement the aforementioned result by constructing solutions whose spatial density sharply satisfies such iterated-logarithmic growth. Our approach relies on real-variable techniques and extends the strategy developed for the Euler equations by the first and fourth-named authors. It also allows for the treatment of more general equations that share the same structure as the Vlasov–Poisson system. Notably, the uniqueness result and the stability estimates hold for both the classical and the relativistic Vlasov–Poisson systems.

The characteristics and mechanism NO formation during hydroxylated fuels/NH3 oxidation in oxy-fuel atmospheres

Oxy-fuel combustion, as an important technology to realize CCUS in the field of combustion, needs to control the generation of pollutants such as NOx while realizing CO2 capture, and the conversion of fuel-N to N2 is crucial. Previous experiments have provided insights into the influence of different atmospheres on NO during oxy-fuel combustion of alcohols. However, the specific reaction mechanism underlying how the atmosphere and hydroxyl fuels impact NO generation through the free radical pool remains unclear. In this study, the formation and reduction of NO during the oxidation of CH3OH/NH3 and C2H5OH/NH3 in O2/CO2, O2/H2O and O2/CO2/H2O atmospheres with oxy-fuel condition were investigated by simulation. In the NO generation stage, different H2O, CO2, O2 and fuel types have little effect on the peak of NO although they change the NH3→NO reaction path. In the process of NO reduction, stage 1 is the main reaction phase of NO→N2, and both CO2 and H2O are beneficial for the reduction of NO. But the reduction of NO by H2O was more pronounced. In the case of simultaneous addition of H2O and CO2, the reducing effect of H2O and CO2 alone on NO was attenuated. On the other hand, the reduction effect exerted by H2O and CO2 was stronger in high than in low oxygen-fuel ratio, stronger in C2H5OH than in CH3OH. The reduction of NO by H2O and CO2 is based on the general equation NO→N2+O2. H2O is used to promote the reaction in the positive direction by consuming O2 to convert it to H and OH, so that more of the element O is converted to OH than O2 to inhibit the production of NO. The CO2 contributes to the formation of less O2 during the reduction of N2 from NO by promoting reaction NO + N→O + N2, while the additional OH increases the "capacity" of O2 in the NO→N2+O2 reaction.

The MEMENTO code for modeling of macroscopic melt motion in fusion devices

The MEMENTO (MEtallic Melt Evolution in Next-step TOkamaks) code is a new numerical implementation of the physics model originally developed for the MEMOS-U code with the objective to self-consistently describe the generation of melt and its subsequent large scale dynamics in fusion devices and to assess the damage of metallic reactor armor under powerful normal and off-normal plasma events. The model has been validated in multiple dedicated EUROfusion experiments. MEMENTO solves the heat and phase transfer problem coupled with the incompressible Navier–Stokes equations in the shallow water approximation for the thin liquid film over the solid metal and with the current propagation equations on a domain that features a time-evolving deforming metal-plasma interface. The code utilizes non-uniform and adaptive meshing along with sub-cycling in time facilitated by the AMReX open-source framework as well as AMReX’s built-in parallelization capabilities.

Re-evaluating the cosmological redshift: Insights into inhomogeneities and irreversible processes

Understanding the expansion of the Universe remains a profound challenge in fundamental physics. The complexity of solving general relativity equations in the presence of intricate, inhomogeneous flows has compelled cosmological models to rely on perturbation theory in a homogeneous Friedmann-Lemaître-Robertson-Walker background. This approach accounts for a redshift of light encompassing contributions from both the cosmological background expansion along the photon's trajectory and Doppler effects at emission due to peculiar motions. However, this computation of the redshift is not covariant, as it hinges on specific coordinate choices that may distort physical interpretations of the relativity of motion. In this study we show that peculiar motions, when tracing the dynamics along time-like geodesics, must contribute to the redshift of light through a local volume expansion factor, in addition to the background expansion. By employing a covariant approach to redshift calculation, we address the central question of whether the cosmological principle alone guarantees that the averaged local volume expansion factor matches the background expansion. We establish that this holds true only in scenarios characterised by a reversible evolution of the Universe, where inhomogeneous expansion and compression modes compensate for one another. In the presence of irreversible processes, such as the dissipation of large-scale compression modes through matter virialisation and associated entropy production, the averaged expansion factor becomes dominated by expansion in voids that can no longer be compensated for by compression in virialised structures. Furthermore, for a universe in which a substantial portion of its mass has undergone virialisation, adhering to the background evolution on average leads to significant violations of the second law of thermodynamics. Our approach shows that entropy production due to irreversible processes during the formation of structures plays the same role as an effective, time-dependent cosmological constant (i.e. dynamical dark energy) without the need to invoke new unknown physics. Our findings underscore the imperative need to re-evaluate the influence of inhomogeneities and irreversible processes on cosmological models, shedding new light on the intricate dynamics of our Universe.

Optimal control technique applied to the minimization of uncertainty measurements in surveying instruments

The objective of this study was to develop an optimal control approach by numerical calculus to predict how to reduce the overall uncertainty of survey instruments unable to directly measure inaccessible points. To reach our goal, two approaches were used to attain the objective. The first was inspired by mathematical models related to three methods appropriately selected and contained in Zhuo’s work proposed in 2012. These were Remote Elevation Measurement (REM), Remote Elevation Dual Measurement (REDM), and Front-to-Back Measurement (FBM) methods whose uncertainties on the measurements of points were deduced using error propagation equations. Optimal control technique helps us to show that for the REM, the height h of the prism contributed more than 70% compared to the global uncertainty for ranges [Formula: see text] from the prism. For the REDM, when the distance between two consecutive stations increases, the weight of the contribution of the two zenith angles [Formula: see text] and [Formula: see text] tends to 50% each for [Formula: see text] close to [Formula: see text], which is to be avoided. For the FBM, the weight of the contribution during the front measurement process before is negligible. The second approach used the Swedish regulation of SIS-TS 21143:2009 which classified total stations according to types of uncertainty to compare the results given by the total station of class T3 unable to directly measure inaccessible points with the more sophisticated class T1 station with direct measurements. Thus, for small spans at the rear measurements [Formula: see text], the height [Formula: see text] of the front prism has the greatest relative contribution more than 90% for zenithal differences [Formula: see text]. This results of our analysis were convincing and provided designers with the data to minimize the overall uncertainties essential in the conception of total stations.

Quintessential f(G) gravity with statistically fitting of H(z)

Inspired by high-energy physics and by considering standard gravity theory as the low-energy limit of quantum theory, the modification of general relativity as [Formula: see text] has been proposed. This paper is dedicated to investigating the cosmological model of modified Gauss–Bonnet gravity in four dimensions. General field equations for the four-dimensional Friedmann– Robertson–Walker metric are given in detail. Also, it is well known that Hubble parameter [Formula: see text] measurements are helpful for the inference of cosmological model parameters. We try to explain the MCMC statistical approach used to calculate Hubble’s parameter. The resulting form of [Formula: see text] was limited using Cosmic Chronometric data, Standard Candles SN Ia from Pantheon, and Baryon Acoustic Oscillations to obtain the best match values by using the regression equation. Finally, to examine the late-time dynamics of our Universe, the behavior of physical and kinematic variables like the deceleration parameter [Formula: see text], State-finder parameters [Formula: see text], diagnostic parameter ([Formula: see text]), equation of state parameter as well as the behavior of energy conditions versus redshift with the parameter values are presented.

Free fermions in the repeated interactions scheme

We discuss collision models made of fermions colliding repeatedly with an extended fermionic system whose hamiltonian is quadratic. After a review of the general theory of collision models, we derive a general Lyapunov dynamical equation that governs the covariance matrix of the fermionic extended system. We solve explicitly, in a wide range of physical parameters, several situations of interest such as the case of a boundary driven chain or the replication of entanglement leading to a rainbow like state.

Hyperstability Results for the General Linear Functional Equation in Non-Archimedean 2-Banach Spaces

Hyperstability Results for the General Linear Functional Equation in Non-Archimedean 2-Banach Spaces

Equation of spin motion for a particle with electric and magnetic charges and dipole moments

The general classical equation of spin motion is rigorously derived for a particle with electric and magnetic charges and electric and magnetic dipole moments in electromagnetic fields. The equation describing the spin motion relative to the momentum direction in storage rings is also obtained. The importance of the derivation follows from the potential presence of magnetic charges and EDMs due to the pseudoscalar CP-violating electromagnetic interaction caused by dark matter axions.