General Equation Research Articles

The role of κ‐deformed Kaniadakis distributed electrons on the dust ion‐acoustic waves in charge‐varying dusty plasma

AbstractThe one‐dimensional dynamics of nonlinear electrostatic dust‐ion acoustic waves (DIAWs), in a plasma containing Kaniadakis‐distributed electrons, fluid ions, and variable‐charge dust grains has been investigated. The κ−generalized electron charging current is derived based on the orbit‐motion limited approach. The reductive perturbation technique has been used to derive a KdV Burger equation of propagation. We have highlighted the dependence between the values of the dust particles charge Qd0 and the validity domain of the Kaniadakis parameter κ, where the Burger term C has positive. By integrating the KdV Burger equation numerically and assuming a laboratory dusty plasma, a transition from oscillatory to monotonic shock waves is observed with increasing values of κ. We have also found that the phase velocity of DIA shock waves decreases, and the dissipation can prevail over that of dispersion as the electrons move further away from thermal equilibrium (κ is increased). The coexistence of compressive and rarefactive of monotonic shock profile is observed. It is found that an increase in κ leads to an increase of the compressive monotonic shock wave and decrease the rarefactive monotonic shock wave.

Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations

The equation G(x,x)=y˜, where G:X×X→Y, and X,Y are vector metric spaces (meaning that the values of a distance between the points in these spaces belong to some cones E+,M+ of a Banach space E and a linear space M, respectively), is considered. This operator equation is compared with a “model” equation, namely, g(t,t)=0, where a continuous map g:E+×E+→M+ is orderly covering in the first argument and antitone in the second one. The idea to study equations comparing them with “simpler” ones goes back to the Kantorovich fixed-point theorem for an operator acting in a Banach space. In this paper, the conditions under which the solvability of the “model” equation guarantees the existence of solutions to the operator equation are obtained. The statement proved extends the recent results about fixed points and coincidence points to more general equations in more general vector metric spaces. The results obtained for the operator equation are then applied to the study of the solvability, as well as to finding solution estimates, of the Cauchy problem for a functional differential equation.

Theoretical computation of the band broadening in micro-pillar array columns

We have investigated the possibility to establish a theoretical plate height expression for the band broadening in the most widely used micro-pillar array column format, i.e., a cylindrical pillar array wherein the pillar walls and the channel bottom are coated with a thin layer of meso‑porous material.Assuming isotropic diffusion in the shell-layer, it was found that the vertical diffusive transport along the porous shell-layer covering the pillar walls significantly suppresses the band broadening originating from the vertical migration velocity gradients. As the vertical transport in the shell-layer increases linearly with the retention equilibrium constant K, this leads to an anomalous dependency on the retention factor. Indeed, instead of increasing with k’’ and following the classic (1+ak’’+bk’’2)/(1 + k’’)2-dependency governing a classic Taylor-Aris system, the variation of the mobile zone mass transfer resistance term hCm in a 3D pillar array with bottom-wall retention goes through a maximum (resp. factor 1.5 (k’’=4) and 2 (k’’=16) difference between observed and classic Taylor-Aris behaviour). This effect increases with increasing pillar heights and increasing reduced velocities. Because of this complex k’’-dependency, it proves very cumbersome to establish a general plate height equation covering all conditions. Instead, a plate height expression was established that is limited up to k’’=4, but remains accurate for higher k’’-values for cases where the ratio of pillar height over inter-pillar distance remains below 5. It can however be anticipated the proposed analytical model is only valid in a rather limited range around the presently considered external porosity of ε=0.5.

Mixed approximation of nonlinear acoustic equations: Well-posedness and a priori error analysis

Accurate simulation of nonlinear acoustic waves is essential for the continued development of a wide range of (high-intensity) focused ultrasound applications. In this article, we explore mixed finite element formulations of classical strongly damped quasilinear models of ultrasonic wave propagation, the Kuznetsov and Westervelt equations. Such formulations allow simultaneous retrieval of the acoustic particle velocity and either the pressure or acoustic velocity potential, thus characterizing the complete ultrasonic field at once. Using non-standard energy analysis and a fixed-point technique, we establish sufficient conditions for the well-posedness, stability, and optimal a priori errors in the energy norm for the semi-discrete equations. For the Westervelt equation, we also determine the conditions under which the error bounds can be made uniform with respect to the involved dissipation parameter. Additionally, we discuss convergence in the maximum error norm of the involved scalar quantities. Finally, computer experiments for Raviart–Thomas (RT) and Brezzi–Douglas–Marini (BDM) elements are performed to confirm the theoretical findings.

Superoperator Master Equations and Effective Dynamics.

We developed the projection method to derive an analog of the quantum master equation for propagators rather than density matrices themselves. As these propagators are superoperators, we call them superoperator master equations. Furthermore, as the projector maps superoperators to superoperators, we call it a hyperprojector. We gave general perturbative expansions for generators of the weak coupling superoperator master equation and the stroboscopic limit superoperator master equation. After that, we considered a particular example of a hyperprojector that is the infinite time average of unitary dynamics. We call it the averaging hyperprojector. We discussed the properties of this hyperprojector and its physical meaning. Then, we illustrated our general second order superoperator master equations arising in the weak coupling limit and the stroboscopic limit, taking the averaging hyperprojector as an example, which we call effective dynamics. We discussed some properties of these superoperator master equations, in particular, the entropy increase.

Analysis of high-contrast all-optical dual-wavelength switching in asymmetric dual-core fibers.

We systematically present experimental and theoretical results for the dual-wavelength switching of 1560 nm, 75 fs signal pulses (SPs) driven by 1030 nm, and 270 fs control pulses (CPs) in a dual-core fiber (DCF). We demonstrate a switching contrast of 31.9 dB, corresponding to a propagation distance of 14 mm, achieved by launching temporally synchronized SP-CP pairs into the fast core of the DCF with moderate inter-core asymmetry. Our analysis employs a system of three coupled propagation equations to identify the compensation of the asymmetry by nonlinearity as the physical mechanism behind the efficient switching performance.

Nonlinear transmission dynamics of mutual transformation between array modes and hollow modes in elliptical sine-Gaussian cross-phase beams

In this paper, based on the nonlocal nonlinear Schrödinger equation, the nonlinear transmission characteristics of elliptical sine-Gaussian cross-phase (ESGCP) beams in strongly nonlocal nonlinear media are researched, and propagation expression of ESGCP beams in transmission process is given. One of the characteristics of beams is that the array mode and the hollow mode can be transformed into each other during the transmission process, which can produce more abundant light intensity modes. Numerical simulation is used to determine how the parameters in the initial propagation equation affect the beam width. In addition, evolution laws of light intensity, phase, and critical power are explored. The results show that the evolution of the beams is controlled by the parameters of the sine term and the cross-phase term in the initial propagation expression. The relationship between initial incident power and evolution period is introduced, and the reason is explained. The beams have great potential for development in the fields of optical communication and light capture.

The Cauchy Problem for the General Telegraph Equation with Variable Coefficients under the Cauchy Conditions on a Curved Line in the Plane

The Riemann method is used to prove the global correctness theorem to Cauchy problem for a general telegraph equation with variable coefficients under Cauchy conditions on a curved line in the plane. The global correctness theorem consists of an explicit Riemann formula for a unique and stable classical solution and a Hadamard correctness criterion for this Cauchy problem. From the formulation of the Cauchy problem, the definition of its classical solutions and the established smoothness criterion of the right-hand side of the equation, its correctness criterion is derived. These results are obtained by Lomovtsev’s new implicit characteristics method which uses only two differential characteristics equations and twelve inversion identities of six implicit mappings. If the righthand side of general telegraph equation depends only on one of two independent variables, then it is necessary and sufficient that it be continuous with respect to this variable. If the right-hand side of this equation depends on two variables and is continuous, then in its integral smoothness requirements it is necessary and sufficient the continuity in one and continuous differentiability in the other variable. The correctness criterion represents the necessary and sufficient smoothness requirements of the right-hand side of the equation and the Cauchy data. From the established global correctness theorem, the well-known Riemann formulas for classical solutions and correctness criteria to Cauchy problems for the general and model telegraph equations in the upper half-plane are derived. In the works of other authors, there is no necessary (minimally sufficient) smoothness on the right-hand sides of the hyperbolic equations of real Cauchy problems for the set of classical (twice continuously differentiable) solutions.

Revisiting generic allometric equations for estimating forest aboveground biomass in Japan: Importance of incorporating plant functional types and origins

Forest aboveground biomass serves as an important indicator of carbon resources and carbon sequestration in terrestrial systems, and its accurate estimation is crucial for carbon evaluation. Allometric equations are among the most commonly used approaches, but the frequent misuse of locally developed models at a different scale has often led to biased results. Despite many attempts to construct generally applicable allometric equations, two crucial factors of plant functional types and origins have rarely been incorporated into the allometric equation models. In this study, the effectiveness of incorporating both plant functional types and origins into generic models was investigated using a newly compiled data set in Japan. The results showed that incorporating both plant functional type and origin into models significantly improved model performance. The results of this study suggest that the inclusion of both plant functional type and plant origin in generic allometric equations is essential, and highlight the possibilities of including both plant functional types and origins in models for accurate estimation of forest AGB in Japan.

Investigation of guided wave propagation in nanoscale layered periodic piezoelectric plates based on Eringen's nonlocal and strain gradient theory

This paper investigates the dynamic propagation behavior of guided waves in nanoscale media composed of alternating layers of piezoelectric phases. The study employs Eringen's nonlocal and strain gradient theories to derive equations for symmetric wave propagation and the scattering relations of guided waves perpendicular to these structures. Dispersion curves are calculated and plotted to compare these theories, considering the influence of nanoscale size-effect, volume ratio of constituent layers, and material length scale. Using the strain gradient theory, the paper explores the impact of length scale parameters on the propagation behavior of horizontally polarized guided waves in an alternating layer structure of nanoscale piezoelectric material, specifically nanoscale piezoelectric laminates. The analysis involves calculating localization factors and dispersion curves to analyze the effects of size-effect and length scale parameters on wave propagation and band structures. Additionally, the study plots and analyzes changes in mechanical displacement and electrical potential, followed by an investigation and discussion of nanoscale size effects and the effects of the material's length scale parameter on dispersion curves and conversion modes within specific regions. The results highlight that the length scale effect parameter significantly impacts the mode conversion zones and band gap structures, more so than the NLPE continuum theory suggests. As the strain gradient parameter increases, these zones shift towards higher wavenumbers and to the right . Furthermore, introducing the length scale factor results in a substantial decrease in both the formation of conversion zones and the density of dispersion curves.

Corrosion evolution and axial mechanical performance degradation of corroded M24 high-strength bolts

To investigate the axial static performance degradation of high-strength bolts subjected to long-term atmospheric corrosion in coastal environments, a neutral salt spray accelerated corrosion test was performed on M24 high-strength bolts over a period ranging from 0 to 330 days. Microscopic morphology observations and axial tensile tests were conducted after corrosion. A linear relationship between mass loss rate and corrosion period was established, and the evolution of pit depth and aspect ratio was proposed. Based on the tensile test, the failure modes of corroded high-strength bolts were categorized as bolt breakage and thread shear-out. The critical mass loss rate for the transition between these two modes was found to be 7.05% (corrosion period of 180 days). Equations characterizing the degradation of residual ultimate strength and relative stiffness as a function of mass loss rate were derived. Furthermore, based on average thickness loss and pit loss, a theoretical analysis on ultimate strength degradation was conducted, and its results were compared to the refined numerical simulations. Both methods were found to provide acceptable outcomes, and numerical results demonstrated more precise and conservative. With numerical method, parametric analysis was conducted, involving 5 groups and 222 models, leading to the formulation of a general ultimate bearing capacity assessment equation concerning bolt diameter and thread pitch.

Single-shot observation of nonlinear pulse splitting in a Kerr medium.

We report single-shot, time-resolved observation of self-steepening and temporal splitting of near-infrared, 50 fs, micro-joule pulses propagating nonlinearly in flint (SF11) glass. A coherent, smooth-profiled, 60-nm-bandwidth probe pulse that propagated obliquely to the main pulse through the Kerr medium recorded a time sequence of longitudinal projections of the main pulse's induced refractive index profile in the form of a phase-shift "streak," in which frequency-domain interferometry recovered with ∼10 fs temporal resolution. A three-dimensional simulation based on a unidirectional pulse propagation equation reproduced observed pulse profiles.

Modular differential equations with movable poles and admissible RCFT characters

Studies of modular linear differential equations (MLDE) for the classification of rational CFT characters have been limited to the case where the coefficient functions (in monic form) have no poles, or poles at special points of moduli space. Here we initiate an exploration of the vast territory of MLDEs with two characters and any number of poles at arbitrary points of moduli space. We show how to parametrise the most general equation precisely and count its parameters. Eliminating logarithmic singularities at all the poles provides constraint equations for the accessory parameters. By taking suitable limits, we find recursion relations between solutions for different numbers of poles. The cases of one and two movable poles are examined in detail and compared with predictions based on quasi-characters to find complete agreement. We also comment on the limit of coincident poles. Finally we show that there exist genuine CFT corresponding to many of the newly-studied cases. We emphasise that the modular data is an output, rather than an input, of our approach.

Allometric relationships between sapwood area and shrub dimensions for six common Southern African savanna bush encroacher species: Universal or species‐specific?

AbstractSouthern African savanna rangelands are facing a widespread degradation pattern called bush encroachment. This is associated with implications for various aspects of the water cycle and in particular canopy transpiration. At the individual‐tree scale, it is estimated by scaling sap‐flux density by sapwood area. However, the direct measurement of sapwood area is impracticable at landscape scale and general allometric equations of the West‐Brown‐Enquist (WBE) model relating sapwood area to primary size measures seem to fail for some species and climates. Therefore, we conducted intensive field measurements to establish species‐specific allometric relationships between sapwood area and sizes (stem diameter, crown area) in six dominant shrub species involved in bush encroachment in Namibia (Colophospermum mopane, Senegalia mellifera, Vachellia reficiens, Dichrostachys cinerea, Vachellia nebrownii, Catophractes alexandri). We found strong allometric relationships between sapwood area and stem diameter as well as between sapwood area and crown area for all six species. These relations are largely in line with the WBE theory but still provide estimates that are more accurate. Only in D. cinerea, the sapwood area was significantly smaller than predicted by the WBE theory, which might be caused by a larger need for stabilizing heartwood. Our results are useful to estimate water loss via transpiration at a large scale using remote sensing techniques and can promote our understanding of the ecohydrological conditions that drive species‐specific bush encroachment in savannas. This is particularly important in the light of climate change, which is considered to have major implications on ecohydrological processes in savannas.

Application and performance of heart-rate-based methods to estimate oxygen consumption at different exercise intensities in postmenopausal women.

Heart rate (HR) is a widespread method to estimate oxygen consumption ( O2), exercise intensity, volume, and energy expenditure. Still, accuracy depends on lab tests or using indexes like HRnet and HRindex. This study addresses HR indexes' applicability in postmenopausal women (PMW), who constitute over 50% of the aging population and may have unique characteristics (e.g., heart size) affecting HR use. Fourteen PMW underwent a cycling ramp incremental test to establish the relationships between O2 (in MET) and absolute HR, HRnet, and HRindex. In a second group of ten PMW, population-specific and general equations were tested to predict MET and energy expenditure during six constant work exercises at various intensities. Pulmonary gas exchange and HR were continuously measured using a metabolic cart. Correlations, Bland-Altman analysis, and two-way RM-ANOVA were used to compare estimated and measured values. Strong linear relationships between the three HR indexes and MET were found in Group 1. In Group 2, population-specific equations showed medium-to-high correlations, precision, and no significant biases when estimating MET and energy expenditure. HRnet and HRindex outperformed absolute HR in accuracy. General HR equations had similar correlations but exhibited larger biases and imprecision. Statistical differences between measured and estimated values were observed at all intensities with general equations. This investigation confirms the suitability of HR for estimating aerobic metabolism in one of the most significant aging populations. However, it emphasizes the importance of considering individual variability and developing population-specific models when utilizing HR to infer metabolism.

A method for estimating the transverse forces developed by the propulsion and steering system, based on the results of vessel circulation tests

In the article the most useful propulsion complexes are described. General equation for transverse force on propulsion complex is shown. Authors discussed values of flow transformation coefficients discovered by A. Gofman. Disadvantages of general method for estimation of transvers propulsive forces are described. In the article the method for numerical evaluation of transverse propulsion complex forces is presented. This method is based on calculation of total force acting on hull-screw-rudder complex using full scale trials data. Values of coefficients which describe total force acting on propulsion complex are calculated using mean least squares method. Values were calculated for two types of cargo vessels. Verification of calculation was done.

Numerical simulation of the permeable wedge-shaped geometry incorporating electroosmosis, solar radiation and heat source-sink effects

The present work deals with the convective double diffusive magneto-cross-nanofluid flow across the permeable wedge. Inclined magnetic field, electro-osmosis, heat source-sink parameter, and solar radiation process are the manuscript's standout characteristics. The numerical simulation of the given flow results in a system of generic nonlinear differential equations, which are then exactly transformed into ordinary differential equations. The visual results are drawn using the bvp4c approach to see the effects of relevant changing elements across the porous wedge. The velocity field generally behaves as the increasing function of thermal and solutal Grashof number but surges along the nanofluid Grashof number. The Brownian diffusion coefficient surges the temperature and double-diffusivity, while minors the concentration. The radiation increases the temperature while lowering the concentration rate and the double diffusivity due to the relative importance of thermal radiation and conduction heat transfer. The nanofluid Lewis number particularly slows down the double-diffusivity profile. Electro-osmotic flow is a typical technique in chemical analysis, soil analysis and processing, and microfluidic devices employing structures with high-voltage surfaces. In the limiting case, current results are also compared to previously published literature. The graphical sketches of local skin friction, Nusselt number, and Sherwood number contour elaborated our analyses more precisely.

Accuracy of Bioelectric Impedance Analysis Devices to Estimate Body Fat and Fat-Free Mass in College Women Athletes

Background: Body composition is frequently measured in women athletes to evaluate training changes, assist in dietary planning, and avoid the female athlete triad. Measurements to monitor %fat and fat-free mass (FFM) can provide valuable information for coaches and athletes throughout the training process. However, questions remain concerning the accuracy of various methods used to measure %fat. The purpose of this study was to assess the accuracy of bioelectric impedance analysis (BIA) devices to estimate %fat and FFM compared to dual-energy X-ray absorptiometry (DXA) in college women athletes.
 Methods: A cross-section design was employed to assess %fat and FFM among college women athletes. Fiftyseven athletes (age = 20.0 1.4 yrs, height = 179.2 6.0 cm, weight = 74.3 4.4 kg) from soccer (n = 29), basketball (n = 15), and swimming (n = 13) had %fat estimated from four single-frequency BIA devices. Two BIA devices had general population equations (BIA1 and BIA2) and two had athletespecific equations (BIA3 and BIA4). Each device had proprietary equations for estimating %fat and was not capable of being updated. Each device had a 2-point electrode contact with either hands or feet. DXA %fat served as the criterion measurement. Percent fat was estimated directly by each device, and FFM was calculated as body mass minus fat mass. All measures were completed in single sessions for each athletic group with different sports groups being measured at the onset of their competitive season. Athletes were measured between 1400 and 1600 hours in a rested stated with hydration assumed and after voiding the bladder. A repeated-measures one-way analysis of variance (ANOVA) with Bonferroni post hoc testing was used to evaluate differences among measurement techniques with significance set at p<0.05.
 Results: Three arm-to-arm BIA devices (BIA1, BIA2, and BIA3) were not significantly different in %fat estimates (23.1 ± 5.0%, 23.7 ± 4.7%, and 23.6 ± 4.3%, respectively) but were significantly lower than DXA (29.5 ± 5.1%). The leg-to-leg athletic BIA (BIA4) had a significantly higher %fat estimate (24.6 ± 5.7%) than BIA1 but was not significantly different from BIA2 and BIA3. The correlation of DXA %fat with BIA1 (r = 0.84), BIA2 (r = 0.85), and BIA3 (r = 0.85) were significant but not statistically different across the 3 devices. BIA4 had a significantly lower correlation (r = 0.66) with DXA %fat. The lower estimates in %fat resulted in significantly higher calculated FFM values for BIA1 (51.1 ± 5.5 kg), BIA2 (50.8 ± 59 kg), BIA3 (50.9 ± 6.9 kg), and BIA4 (50.1 ± 5.8 kg) than for DXA (47.5 ± 5.9 kg). However, all BIA estimates of FFM were highly correlated with DXA FFM (r = 0.90-0.93). Limits of agreement analysis indicatedthe average bias ranged from 2.2 kg (BIA4) to 3.4 kg (BIA1).
 Conclusion: Single-frequency BIA devices utilized in this study tend to underestimate %fat and overestimate FFM compared to DXA in college women athletes. However, high correlations between predicted and actual FFM values indicate that single-frequency BIA devices may be useful for tracking changes in women athletes across seasons.

Existence and Uniqueness of Solutions of Hammerstein-Type Functional Integral Equations

The authors deal with nonlinear and general Hammerstein-type functional integral equations (HTFIEs). The first objective of this work is to apply and extend Burton’s method to general and nonlinear HTFIEs in a Banach space via the Chebyshev norm and complete metric. The second objective of the paper is to extend and improve some earlier results to nonlinear HTFIEs. The authors prove two new theorems with regard to the existence and uniqueness of solutions (EUSs) of HTFIEs via a technique called progressive contractions, which belongs to T. A. Burton, and the Chebyshev norm and complete metric.

The adjoint map of Euclidean plane curves and curvature problems

The adjoint map of a pair of naturally parametrized curves in the Euclidean plane is studied from the point of view of the curvature. A main interest is when the given curve and its adjoint curve share the same natural parameter and the same curvature. For the general linear second order differential equation we introduce a function expressing the deformation of curvatures induced by the adjoint map.