Exponential Form Research Articles

Junction formation rates, residence times, and the rate of plastic flow in FCC metals

During plastic flow in metals, dislocations from slip systems with different glide planes collide to form junctions. After being in-residence within the dislocation network for some period of time, these junctions then break, thereby liberating the attached dislocation lines. In this work we use random forest discrete dislocation dynamics simulations to quantify the junction formation rate and junction residence time as a function of stress for all junction types in face-centered cubic metals. We then relate these quantities to the dislocation link-length distribution, which is found to exhibit an exponential form. This enables us to quantify the mean junction strength and also the slip system interaction coefficients. Finally, using the link-length model we obtain a flow rule for our systems which is physics-based with all parameters determined from DDD simulations. The insights here provide a path forward for a dislocation network theory of plastic flow based on the link-length distribution.

Deep bed filtration and formation damage by particles with distributed properties

Current models for deep bed filtration describe particles with uniform properties. Yet, the sizes, densities, and mineral composition of particles vary significantly in the same injection well. The aim of this work is to provide an effective mathematical model for water injection of particles with distributed properties and formation damage prediction. We average the set of traditional population balance equations for single-property particles and obtain one upscaled equation. The upscaled equation for particle retention rate contains a non-linear function of suspended concentration, which we call the 'suspension function'. We derive analytical solutions for the upscaled equation for linear (coreflood) and radial (well injectivity) flows. Then we treat lab coreflood data to determine the model suspension function and provide a model for well injectivity prediction. The retention profile for the flow of uniform particles has an exponential form. Frequently reported in the literature, hyper-exponential forms have been hypothetically explained by multiple particle properties. The inverse solution allows revealing the individual filtration coefficients for binary mixtures from total breakthrough concentrations during coreflood. Treatment of the data from lab experiments reveals individual filtration coefficients that belong to common intervals. For the first time, deep bed filtration of particles with distributed properties is upscaled and presented using a single equation that reflects the particle property distribution. This equation provides an effective mathematical model for tuning lab coreflood data, determines the model function, and uses it for injectivity decline prediction.

Efficient simulation of potential energy operators on quantum hardware: a study on sodium iodide (NaI)

This study introduces a conceptually novel polynomial encoding algorithm for simulating potential energy operators encoded in diagonal unitary forms in a quantum computing machine. The current trend in quantum computational chemistry is effective experimentation to achieve high-precision quantum computational advantage. However, high computational gate complexity and fidelity loss are some of the impediments to the realization of this advantage in a real quantum hardware. In this study, we address the challenges of building a diagonal Hamiltonian operator having exponential functional form, and its implementation in the context of the time evolution problem (Hamiltonian simulation and encoding). Potential energy operators when represented in the first quantization form is an example of such types of operators. Through systematic decomposition and construction, we demonstrate the efficacy of the proposed polynomial encoding method in reducing gate complexity from O(2n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(2^n)$$\\end{document} to O∑i=1rnCr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( \\sum _{i=1}^{r} {} ^nC_r \\right)$$\\end{document} (for some r≪n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ll n$$\\end{document}). This offers a solution with lower complexity in comparison to the conventional Hadamard basis encoding approach. The effectiveness of the proposed algorithm was validated with its implementation in the IBM quantum simulator and IBM quantum hardware. This study demonstrates the proposed approach by taking the example of the potential energy operator of the sodium iodide molecule (NaI) in the first quantization form. The numerical results demonstrate the potential applicability of the proposed method in quantum chemistry problems, while the analytical bound for error analysis and computational gate complexity discussed, throw light on issues regarding its implementation.

An impact of Richardson number on the inclined MHD mixed convective flow with heat and mass transfer

AbstractThe two‐dimensional mixed convective MHD flow with heat and mass transfer is investigated for its behavior with Dufour and Soret mechanisms over the porous sheet. The copper–alumina (Cu–Al2O3) hybrid nanoparticles are used in the base fluid water. The governing system of partial differential equations is converted into a system of ordinary differential equations via similarity transformations, obtaining the solution for velocity, temperature, and concentration fields in exponential form. The problem is demonstrated in the Darcy–Brinkman model, the impact of included parameters such as Richardson number, magnetic field, and Dufour numbers are studied for the obtained solution with the help of graphs. Increasing the magnetic field decreases both transverse and axial velocity profiles. Increasing the magnetic field and Richardson's number decreases the solution (Al2O3–H2O). Increasing the values magnetic field and Richardson's number decreases both transverse and axial velocity profiles. Increasing the values of the Dufour effect increases the axial and transverse velocity boundary layer. The magnetohydrodynamic hybrid nanofluid flow over porous media works efficiently in liquid cooling and, therefore, has significant applications in industrial heating and cooling systems, solar energy, magnetohydrodynamic flow meters and pumps, manufacturing, regenerative heat exchange, thermal energy storage, solar power collectors, geothermal recovery, and chemical catalytic reactors.

Exact solitary wave propagations for the stochastic Burgers’ equation under the influence of white noise and its comparison with computational scheme

In this manuscript, the well-known stochastic Burgers’ equation in under investigation numerically and analytically. The stochastic Burgers’ equation plays an important role in the fields of applied mathematics such as fluid dynamics, gas dynamics, traffic flow, and nonlinear acoustics. This study is presented the existence, approximate, and exact stochastic solitary wave results. The existence of results is shown by the help of Schauder fixed point theorem. For the approximate results the proposed stochastic finite difference scheme is constructed. The analysis of the proposed scheme is analyzed by presented the consistency and stability of scheme. The consistency is checked under the mean square sense while the stability condition is gained by the help of Von-Neumann criteria. Meanwhile, the stochastic exact solutions are constructed by using the generalized exponential rational function method. These exact stochastic solutions are obtained in the form of hyperbolic, trigonometric and exponential functions. Mainly, the comparison of both numerical and exact solutions are analyzed via simulations. The unique physical problems are constructed from the newly constructed soliton solutions to compare the numerical results with exact solutions under the presence of randomness. The 3D and line plots are dispatched that are shown the similar behavior by choosing the different values of parameters. These results are the main innovation of this study under the noise effects.

INFLUENCE OF ELECTROMAGNETIC TREATMENT ON HYDRAULIC CHARACTERISTICS OF HOT PIPELINES DURING NEWTONIAN AND NON-NEWTONIAN PETROLEUM SYSTEMS TRANSPORTATION

This work examines the influence of radio-frequency (RF) electromagnetic (EM) influence on the hydraulic characteristics of hot pipelines for the transport of petroleum Newtonian and non-Newtonian systems.Newtonian and non-Newtonian petroleum and their rheological properties before and after RF EM exposure, as well as the hydraulic characteristics of hot petroleum pipelines for pumping selected petroleum, were selected as objects of study.As a result: a methodology for petroleum processing has been developed that corresponds to industrial practice in the operation of petroleum pipelines; empirical dependences of petroleum viscosity on temperature were obtained in the form of piecewise exponential functions. The influence of RF EM exposure on the thixotropic properties of petroleum and the values of ultimate shear stresses was assessed. A decrease or complete disappearance of hysteresis loops after RF EM exposure to petroleum and a decrease in the maximum shear stress for non-Newtonian petroleum were revealed.By using the software developed by the authors, «Hydraulic calculation of isothermal and non-isothermal petroleum pipelines (GRINT)», the processes of pipeline transport of petroleum treated with EM fields were modeled, taking into account the obtained empirical dependencies. An assessment of the influence of RF EM exposure on the hydraulic characteristics of «hot» pipelines was carried out. It has been established that after RF EM exposure to petroleum, the zone of stable operation of a «hot» petroleum pipeline increases and the required pressure for pumping the same volume of petroleum decreases.

Causal emergent principles and relations for mechanical properties of covalent and ionic crystals

A knowledge and data-synergized intelligent computation architecture for materials was proposed within the data science paradigm. As a vital operation, two digital ensemble descriptors implying chemical composition and structural trend for crystals were created using the features contained in the Periodic Table of elements without a priori assumption, which affords causal emergence and regulation principles for the mechanical response of covalent and ionic solids. In addition to a linear correlation among structural state/mechanical response parameters, causal analytic relations in an exponential form between structural and thermodynamic state/mechanical response parameters and a digital ensemble descriptor were unveiled through least squares regression, in which the coefficients are classified in accordance with symmetry principles on the atom and lattice. Thereafter, the underlying physicochemical mechanisms of chemical pressure and chemical bonding are found responsible for the mechanical responses of bulk modulus and hardness of solids. At last, a physical prediction model was established for crystalline solids and demonstrated the feasibility of the predictive design of novel superhard materials. It is believed that by constructing suitable digital ensemble descriptors, this intelligent computation architecture and consequent physical prediction models on the basis of causal analytic relations are able to generalize by depicting crystalline solids with covalent and ionic bonds in other crystallographic structures.

A theoretical and numerical analysis of solar chimney in multi-storey atrium building

Current research efforts on solar chimney are much on those configurations with single space or single vent, hampering its applications in complex buildings. To expand its implementation, the viability of solar chimney in multi-storey atrium buildings was explored numerically and theoretically. It was known that its natural ventilation performance can fulfil the WHO requirements (i.e., 6 air change per hour, ACH) even under low solar radiation of 200 W/m2 if it is appropriately designed, such as cavity gap (d), cavity height (Hc), solar radiation (Q), and window width (W). Its natural ventilation rate was found to be positively linear to the exponential form of the analyzed factors in this study, including cavity gap (d0.63∼0.69), cavity height (Hc0.59∼0.60), solar radiation (Q0.42∼0.43), and window width (W0.14∼0.20). The impacts of cavity gap and solar radiation are relatively weaker for big space such as atrium than those scenarios with single-space ventilation in the literature. The volume of the connected space was found to affect its natural ventilation performance, where the obtained ventilation rate of 4-storey is averagely 5.1 % lower than those of 3-storey buildings. A theoretical model was developed and experimentally validated, for the first time, to predict the natural ventilation rate of solar chimney when it is applied to the atrium with multiple air inlets. It was also noticed that the same theoretical model can be applied to the atrium buildings with odd and even storeys, but the area coefficients are different. The research outcomes of this study confirm the viability of solar chimney in complex buildings and offer a theoretical foundation for its implementation.

Course genres classification of music e-learning platform based on deep learning big data intelligent processing algorithm

At present, the growth trend of digital music information is showing an exponential form. Various songs can be classified according to their unique genres and styles, forming various “genres”. So, how to effectively manage the massive amount of music data has become a problem that needs to be solved to promote the balanced development of different types and genres of music. On this basis, this article adopts deep learning technology to study the classification methods of music genres oriented towards big data. Firstly, this article studies LSTM based on long and short term memory networks and compares its advantages and disadvantages with convolutional neural network CNN. Secondly, this article calculates the drift patterns of big data intelligent processing algorithms and conducts relevant performance tests on big data intelligent processing algorithms to verify the effectiveness of the proposed algorithms. Finally, this article addresses the problems in current music genre classification research by introducing Mel filters into the extraction of sound features to further improve the efficiency and speed of the model. And deep learning technology is adopted to construct a music genre classification model based on big data intelligent analysis, further promoting the effective improvement of music genre classification effectiveness.

A 3D multi-phase-field model for orientation-dependent complex crack interaction in fiber-reinforced composite laminates

We formulate a 3D multi-phase field model to simulate bulk and interfacial fracture in fiber-reinforced composites. The purpose of this study is to predict the complex crack interaction between the interlaminar and intralaminar fracture in FRPCs. For that, the discrete crack and the sharp interface are both smeared using the phase-field approach. A second-order anisotropic tensor is incorporated in the elastic as well as the fracture surface energy to promote direction-dependent crack growth. The interface constitutive behavior is represented by the uncoupled 3D traction–separation laws—in exponential and bi-linear form. The proposed model captures the complicated failure phenomena in FRPCs such as matrix damage, interfacial delamination, and their interactions for different configurations of lamina with different fiber orientations and fracture energies. This model’s capability is further illustrated to study the three common modes of fracture i.e. modes I, II, and III, in 3D FRPC laminates where the crack impinges on the interface. Finally, the obtained framework is validated by two benchmark problems including the mode I and mode II delamination test, and is compared against experimental data reported in the literature.

Study on the Macro-Fine Mechanical Behavior of Ore Flow Based on the Discrete Element Method

The mechanical behavior associated with the flow of ore-rock bulk materials is an important factor leading to the instability and failure of the shaft wall of the ore storage section in ore passes. It is of great significance for accurately understanding the stability failure characteristics of the shaft wall in the ore storage section in the ore-drawing process to understand the flow characteristics and internal mechanical transfer mechanism of ore-rock bulk. The flow characteristics, contact compactness, stress distribution characteristics, and contact force probability distribution of the ore-rock bulk are analyzed by the discrete element method, which realizes the quantitative characterization of the damage degree of ore-rock flow and reveals the damage mechanism of the shaft wall in the storage section of the ore pass. The results show that (1) in the process of ore-rock particle flow in the ore pass storage section, the macroscopic flow pattern of ore-rock particles changes from a “—” shape to a “V” shape, and the friction between ore-rock particles, particles, and the ore-pass wall is an important reason for the change of the macroscopic flow pattern; (2) the probability distribution of contact force strength between the particles decreases exponentially in the whole ore-drawing process, in which the strong force chains play a major role in the stability of the bulk system; and (3) the overpressure frequency and overpressure coefficient could be used to quantitatively characterize the wall damage degree under the action of ore-rock flow. The dynamic lateral pressure fluctuates periodically in exponential form and decreases, and the dynamic load formed by the ore-rock flow mainly acts on the lower part of the ore storage section.

Can the late dark energy parameterization reconcile the Hubble tension?* *Ming-Jian Zhang is Supported by the Natural Science Foundation of Shandong Province, China (ZR2021MA075), and the Talent Research Project of Qilu University of Technology (2023RCKY029). Li Chen is Supported by the Natural Science Foundation of Shandong Province, China (ZR2019MA033)

In this study, we constructed ten dark energy models to test whether they can reconcile the Hubble tension and how much it is affected by parameterization. To establish a fair test, the models are diverse, encompassing fractional, logarithmic, exponential, and inverse exponential forms as well as several non-parameterized models. The dataset we used includes the NPIPE pipeline of cosmic microwave background (CMB) power-spectrum data from Planck2020, Pantheon+ samples from Supernovae Type Ia, and baryon acoustic oscillations. The MCMC calculations imply dark energy transferring from to for the four parameterized dark energy models. However, these models cannot adequately reconcile the Hubble tension. Notably, we found that phantom-like dark energy with can achieve the greatest reduction in the Hubble tension to . However, AIC analysis indicates that this alleviation is at the cost of high AIC. We also investigated the effect of constructions on the derivative of the equation of state , cosmic density parameter, CMB power spectrum , and matter spectra . We also found that the Hubble tension may be related to the reionization process.

Effects of Adding the Antiparallel Diodes in a Model of Solar Photovoltaic Cell: Theory and Pspice Simulations

Background: The photovoltaic (PV) array consisting of modules is considered as the fundamental electrical power conversion unit of the PV generator system. The PV array has nonlinear characteristics with the current-voltage (I-V) characteristic in the exponential form introduced by diodes. In this context, the key objective of the research carried out in the present study is to propose a novel PV cell, with anti-parallel diodes introduced in the circuit model in place of the single diode. Objective: The objective of this research is to compare the effects, magnitudes of current and power generated by the PV cell, with antiparallel diodes to those with single diodes. Methods: The model circuit containing two antiparallel diodes is first designed, and then analyzed through the circuit Equationuations found using the Kirchhoff’s laws. The I-V characteristics are modified in the hyperbolic sine form, and the model Equationuation is solved analytically and numerically using the MATLAB software. The analytical and numerical investigations are then confirmed by Pspice simulation, which is made possible with the introduction of additional bloc necessary to make the output signal voltage tunable in the opened circuit. Results: The I-V characteristic is plotted showing that when the voltage is lower than a particular, the model containing the antiparallel diode has the increasing of current, while that with single diode has constant current. Additionally the current generated by the PV cell oscillates increasingly when the supplied signal voltage is negative. We also found that the maximum power increases with the temperature. The expression of the cutoff signal voltage leading to zero current was also found and plotted, the increasing function of the number Ns of series diodes. As application, the PV circuit designed is used in the charging of the battery of electrical vehicle. Conclusion: The PV cell with antiparallel diodes generates currents for large band of signal voltage, and the power generated increases with temperature.

Extracting the Ultimate New Soliton Solutions of Some Nonlinear Time Fractional PDEs via the Conformable Fractional Derivative

Nonlinear fractional-order differential equations have an important role in various branches of applied science and fractional engineering. This research paper shows the practical application of three such fractional mathematical models, which are the time-fractional Klein–Gordon equation (KGE), the time-fractional Sharma–Tasso–Olever equation (STOE), and the time-fractional Clannish Random Walker’s Parabolic equation (CRWPE). These models were investigated by using an expansion method for extracting new soliton solutions. Two types of results were found: one was trigonometric and the other one was an exponential form. For a profound explanation of the physical phenomena of the studied fractional models, some results were graphed in 2D, 3D, and contour plots by imposing the distinctive results for some parameters under the oblige conditions. From the numerical investigation, it was noticed that the obtained results referred smooth kink-shaped soliton, ant-kink-shaped soliton, bright kink-shaped soliton, singular periodic solution, and multiple singular periodic solutions. The results also showed that the amplitude of the wave augmented with the pulsation in time, which derived the order of time fractional coefficient, remarkably enhanced the wave propagation, and influenced the nonlinearity impacts.

Institutional quality and emigration nexus: Empirical evidence from Türkiye

AbstractThis study attempts to fill an important gap in the empirical literature by investigating the impact of institutional quality in destination countries on emigrants from Türkiye using bilateral migration data from 2010 through 2020. For this purpose, the study builds an augmented gravity model, including economic, geographic, and cultural variables in an exponential form and estimates it by the Poisson Pseudo Maximum Likelihood (PPML) method. Empirical results significantly confirm the pulling role of strong institutions in destination countries for emigrants from Türkiye since all institutional quality indicators are positively associated with emigrant movements, albeit just one of them, regulatory quality, is statistically insignificant. Among indicators of institutional quality, the most pulling ones for emigrants are voice and accountability and the rule of law, followed by government effectiveness, control of corruption, and political stability and absence of violence, respectively. Overall results indicate that institutions are front‐line players in the emigrants' migration decision and destination choice process. In this context, policy‐makers in both Türkiye and the destination country may implement an institutional policy considering the outcomes stemming from migration movements.

Phase change in a one-dimensional functionally graded material

Functionally graded materials (FGMs) are a class of materials with engineered microstructure to achieve spatial variation in properties so as to serve a desired function. While FGMs have been well-researched for their structural properties, there is a lack of investigation of FGMs in solid-liquid phase change heat transfer processes. A well-designed functionally graded phase change material (FG-PCM) may overcome some of the well-known limitations of the phase change process. In order to investigate this possibility, this work presents a theoretical model to predict phase change propagation in a one-dimensional FG-PCM containing an arbitrary spatial variation in thermal conductivity, thermal diffusivity and latent heat. The problem is solved using the method of eigenfunction expansion to determine the temperature distribution, wherein, spatial variation in thermophysical properties listed above is accounted for by deriving a first-order matrix ordinary differential equation in the coefficients of the series solution. Results are found to correctly reduce to the well-known Stefan solution under special conditions. The impact of variation in thermophysical properties is investigated in detail. It is found that while spatial variation in thermal conductivity and latent heat has a significant effect on the rate of phase change, the impact of the thermal diffusivity function is negligible for realistic values of the Stefan number. The effect of the Stefan number governing this problem on the rate of melting is investigated. The nature of phase change under linear (1+aξ), polynomial (1+aξn) and exponential (eaξ) functional forms of thermophysical properties is investigated. It is shown that an appropriate choice of these spatial functions may help overcome the well-known self-limiting nature of phase change and produce a nearly-linear phase change propagation curve. For example, results show 33% increase in phase change propagation for a linear (1+5ξ) variation in thermal conductivity compared to baseline. The theoretical tools developed here may help design and optimize functionally graded PCMs towards improved phase change heat transfer in applications of practical interest.

Hysteresis damping ratio for moment steel frame evaluation using DDBD

AbstractThe main purpose of this paper is to propose a model for estimating the hysteresis damping ratio of moment steel frames and compare them with the results obtained using direct displacement-based design. In this study, different models of steel bending frames were examined by using several type of analysis to obtain this ratio. Thirty-steel frame models with 3, 6, 9, 12, and 15 floors and 3 and 6 bay with 6 m span length were analyzed and evaluated. The results showed that the equivalent damping ratio at the level of safety based on Jacobsen and Jennings, and the values of the equivalent damping relationships according to FEMA-440 vary from the empirical formula proposed in displacement-based design-2012. Therefore, in the ductility domain the exponential relationship form was more consistent with the results used. A new formula was proposed to calculate the hysteresis damping ratio, and it is more accurate than the empirical formula proposed in displacement-based design-2012.

Unveiling new exact solutions of the unstable nonlinear Schrödinger equation using the improved modified Sardar sub-equation method

The improved modified Sardar sub-equation method is used in this study to look into the complex domain of the unstable nonlinear Schrödinger equation, which is a basic idea in quantum physics. Our main goal is to obtain new and exact solutions for the unstable nonlinear Schrödinger equation, which includes various mathematical structures such as rational, exponential, trigonometric, and hyperbolic trigonometric forms. These solutions provide insight into the complex mechanics that control wave events and their nonlinear behavior. By employing MATLAB, we display visual depictions of these solutions in different dimensions, improving the direct examination of the occurrences. This study enhances the existing knowledge of mathematical methods for solving nonlinear differential equations, which can be applied to other nonlinear wave equations in other scientific fields. The findings from this work not only enhance our comprehension of the unstable nonlinear Schrödinger equation but also open up new possibilities for studying nonlinear phenomena in several physical systems, including optics, condensed matter physics, and quantum information science.

Aufbau Suppressed Coupled Cluster Theory for Electronically Excited States.

We introduce an approach to improve single-reference coupled cluster theory in settings where the Aufbau determinant is absent from or plays only a small role in the true wave function. Using a de-excitation operator that can be efficiently hidden within a similarity transform, we create a coupled cluster wave function in which de-excitations work to suppress the Aufbau determinant and produce wave functions dominated by other determinants. Thanks to an invertible and fully exponential form, the approach is systematically improvable, size consistent, size extensive, and, interestingly, size intensive in a granular way that should make the adoption of some ground state techniques, such as local correlation, relatively straightforward. In this initial study, we apply the general formalism to create a state-specific method for orbital-relaxed, singly excited states. We find that this approach matches the accuracy of similar-cost equation-of-motion methods in valence excitations while offering improved accuracy for charge transfer states. We also find the approach to be more accurate than excited-state-specific perturbation theory in both types of states.

Eternal solutions to a porous medium equation with strong non-homogeneous absorption. Part I: radially non-increasing profiles

Existence of specific eternal solutions in exponential self-similar form to the following quasilinear diffusion equation with strong absorption \[ \partial_t u=\Delta u^m-|x|^{\sigma}u^q, \] posed for $(t,\,x)\in (0,\,\infty )\times \mathbb {R}^N$ , with $m>1$ , $q\in (0,\,1)$ and $\sigma =\sigma _c:=2(1-q)/ (m-1)$ is proved. Looking for radially symmetric solutions of the form \[ u(t,x)={\rm e}^{-\alpha t}f(|x|\,{\rm e}^{\beta t}), \quad \alpha=\frac{2}{m-1}\beta, \] we show that there exists a unique exponent $\beta ^*\in (0,\,\infty )$ for which there exists a one-parameter family $(u_A)_{A>0}$ of solutions with compactly supported and non-increasing profiles $(f_A)_{A>0}$ satisfying $f_A(0)=A$ and $f_A'(0)=0$ . An important feature of these solutions is that they are bounded and do not vanish in finite time, a phenomenon which is known to take place for all non-negative bounded solutions when $\sigma \in (0,\,\sigma _c)$ .