Logarithmic Equation Research Articles

Fractional Choquard logarithmic equations with Stein-Weiss potential

In the present paper, we are concerned with the following fractional p-Laplacian Choquard logarithmic equation(−Δ)psu+V(x)|u|p−2u+(ln⁡|⋅|⁎|u|p)|u|p−2u=(∫RNF(y,u)|y|β|x−y|μdy)f(x,u)|x|βinRN, where N=sp≥2, s∈(0,1), 0<μ<N, β≥0, 2β+μ≤N and (−Δ)ps denotes the fractional p-Laplace operator, the potential V∈C(RN,[0,∞)), and f:RN×R→R is continuous. Under mild conditions and combining variational and topological methods, we obtain the existence of axially symmetric solutions both in the exponential subcritical case and in the exponential critical case. We point out that we take advantage of some refined analysis techniques to get over the difficulty carried by the competition of the Choquard logarithmic term and the Stein-Weiss nonlinearity. Moreover, in the exponential critical case, we extend the nonlinearities to more general cases compared with the existing results.

Evaluating gas breakthrough pressure and gas permeability in a landfill cover layer for mitigation of hazardous gas emissions

The construction of an engineered cover layer over landfills is a common method applied to reduce the emission of hazardous gases into the atmosphere. Landfill gas pressures can reach 50 kPa or even higher in some cases, thus posing a serious threat to nearby properties and human safety. As such, the evaluation of gas breakthrough pressure and gas permeability in a landfill cover layer is of great necessity. In this study, the loess soil that is often applied as a cover layer in landfills in northwestern China was used to conduct gas breakthrough, gas permeability, and mercury intrusion porosimetry (MIP) tests. Resultantly, the smaller the capillary tube diameter, the higher the capillary force, and the more significant the capillary effect. Gas breakthrough could be attained with no difficulty, provided that the capillary effect was minimal or approached zero. A good fit between the experimental gas breakthrough pressure–intrinsic permeability relationship and a logarithmic equation was found. The mechanical effect blew up the gas flow channel. In the worst-case scenario, the mechanical effect could lead to the overall failure of a loess cover layer in a landfill. A new gas flow channel was formed between the rubber membrane and the loess specimen as a result of the interfacial effect. Although both the mechanical and interfacial effects can elevate the gas emission rate, the latter did not play a role in the improvement of the gas permeability; therefore, misleading interference took place in the evaluation of the gas permeability, and an overall failure of the loess cover layer. To tackle this problem, the point at which the large- and small-effective stress asymptotes cross on the volumetric deformation–Peff diagram may be applied to give early warning signals of the potential overall failure of the loess cover layer in landfills in northwestern China.

Comparing environmental DNA with whole pond survey to estimate the total biomass of fish species in ponds

Abstract Environmental DNA (eDNA) has recently been developed for detecting the distribution of macroorganisms in aquatic systems and for estimating biomass/abundance. Although positive relationships between eDNA concentrations of species and biomass/abundance have been observed in previous studies, estimation of total biomass in a particular ecosystem based on eDNA concentration has not been investigated. To test this, we sampled eDNA from the littoral and pelagic zones of eight small ponds in Japan. Then, we drained ponds to capture all fish inhabiting each pond. We compared the eDNA copies measured by quantitative real‐time polymerase chain reaction analysis with the captured biomass of three dominant fish species (common carp, bluegill sunfish, and largemouth bass) obtained by direct capture. Differences in eDNA concentrations were compared between the littoral and pelagic zones of each pond. We found positive relationships between the eDNA copies and the whole captured biomass of all target species using Type II logarithmic equations, and the eDNA copies appeared to be saturated at a higher total biomass. We found positive correlations between the total number of eDNA copies and the total captured biomass of three fish species. This demonstrated the first step in estimating the population biomass of target fish species under natural conditions using eDNA measurement.

Flexural fatigue behavior and damage evolution analysis of aeolian sand concrete under freeze–thaw cycle

Aeolian sand is abundant and is widely distributed in western China and thus can reduce the project cost, alleviate the consumption of river sand resources, as well as protect the ecological environment in the application of concrete. Concrete structures, such as roads, bridges, and airport runways in northern cold regions, are exposed to the combined action of freeze–thaw cycle and fatigue load, and the damage mechanism is more serious and complex than that under a single factor. In the present study, the flexural fatigue behavior and damage evolution of aeolian sand concrete under freeze–thaw cycles are studied. Results show that freeze–thaw damage aggravates the flexural fatigue failure of concrete with aeolian sand, and the addition of aeolian sand enhances the average fatigue life of concrete after freeze–thaw cycle. The fatigue life of concrete that has aeolian sand content of 100% is improved more significantly than that of concrete with aeolian sand content of 50%. A double logarithmic fatigue equation is established for varying numbers of freeze–thaw cycles and different failure probabilities. Additionally, the course of flexural fatigue strain and fatigue modulus of aeolian sand concrete under freeze–thaw cycle action follows a typical three-stage development course. The effects of freezing and thawing action on fatigue behavior, such as fatigue strain growth, stiffness degradation, stress–strain relationship, and dissipated energy of 50% aeolian sand concrete (ASC50), are more significant, whereas the effects of 50 and 100 freeze–thaw cycles on the fatigue properties of 100% aeolian sand concrete (ASC100) are not evident. The research results are of great significance for the popularization and use of aeolian sand concrete in cold regions.

Growth and Development of Chicken Carcass in Different Sex and Age

This study was aimed to determine the growth and development of Broiler chicken carcass and components at different sex and age . The method used in this study is the Huxley Y = a x b allometric equation which is transformed into a logarithm equation Log Y = Log a + b Log X. This research was conducted in the Cage of the Animal Production Department of the Faculty of Agriculture, Halu Oleo University. The parameters observed in this study were the carcass and carcass parts. Based on the results of the study showed that the growth coefficient of male and female broiler carcasses were not significantly different (P&gt;0.05) which meant that they had a time of early development when compared to the growth of parts of the body as a whole. The growth and development patterns of the carcasses of both male and female broiler chickens at the age of 0 - 5 weeks is relatively the same.

Existence and uniqueness theorems for functional equations

In this paper we give simple extension and uniqueness theorems for restricted additive and logarithmic functional equations.

A detailed study of the CHN-BQ rock mass classification method and its correlations with RMR and Q system and Hoek-Brown criterion

A CHN-BQ method is firstly defined to present the system and connotation of the current Chinese national standard “Standards for Engineering Classification of Rock Mass” GB/T 50,218. The core idea and content of the CHN-BQ method are studied in detail. Based on a qualitative classification of rock masses at the sites of in situ rock deformation and shear strength tests with a total number of 1439 test sets in nearly 90 projects, the statistical characteristics of in situ test results of different grades of rock mass are analyzed; Thus, the mechanical parameters corresponding to each grade of rock mass basic quality are proposed. Based on 997 groups of various rock mass classification data in 10 projects, the comprehensive CHN-BQ method verifies, and some essential remarks are obtained. Firstly, the 312 sets of parallelly measured data show that the consistency of less than one-grade discrepancy of the qualitative and quantitative classification in the CHN-BQ method is up to 71% in percentage. Secondly, a highly linear correlativity of the CHN-BQ method with the RMR method is found, and a linear regression equation is proposed. Thirdly, a logarithmic regression equation between the CHN-BQ method and the Q system is obtained, and equivalent grading divisions of the Q system from 9-level intervals towards traditional 5-level intervals are proposed. Finally, the research shows that the GSI concept in the Hoek-Brown criterion and the qualitative classification of rock mass structure in the CHN-BQ method are essentially coincident. Based on the measured data in quartz mica schist rock of an arch dam project, a linear regression equation of the Kv index in the CHN-BQ method and GSI index in the Hoek-Brown criterion is obtained, and the comparison of the strength parameters of the CHN-BQ method and Hoek-Brown criterion is discussed.

Взаимодействие празеодима и диспрозия с кислородом в расплавах никеля при РAr = 0,1 MПa и анализ неметаллических включений

The interaction of Pr/Dy with oxygen in nickel melts at РAr = 0.1 MPa and constant temperature was studied by the method of instantaneous EMF fixation using a Mo[Cr/Cr2O3//ZrO2(MgO)//O(NiL)]Mo cell and certified sensors. Dependences a[O] = f[Pr/Dy, %] were expressed as logarithmic equations, which made it possible to compare them with each other in the concentration range of 0.001 – 0.2 % of each deoxidizer and determine that the deoxidizing capacity of Pr is 1.7 times higher in terms of compared to Dy. Calculated a[O] in Ni – O – Al – (Pr/Dy) melts compared to Al at a concentration of 0.05 % elements and showed a seven-fold decrease for the first deoxidizer and an eleven-fold decrease for the second. The morphology of non-metallic inclusions in metallographic sections of Ni – O – Pr/Dy alloys indicates that the inclusions are located along grain boundaries and have different configurations and complex heterophase composition. The analysis of non-metallic inclusions with the maximum content of the deoxidizing element proved the existence of Pr/Dy oxygen compounds, which confirms the data of thermodynamic and mass spectrometric studies. The average content of Zr in non-metallic inclusions during the deoxidation of Pr is twice as high as in the experiments during the deoxidation of Dy, which indicates the interaction of Pr/Dy with ZrO2 of the EMF sensor and the predominant interaction of Pr compared to Dy and correlates with the data on the determination of a[O].

Existence and exponential decay of a logarithmic wave equation with distributed delay

Existence and exponential decay of a logarithmic wave equation with distributed delay

The Effect of the Extrusion Ratio on Load and Die Wear in the Extrusion Process

In this paper, regarding the example of concurrent extrusion of a round bar, the results of a comparative analysis of the impact of the extrusion ratio R on process parameters such as die wear intensity, extrusion force, stress intensity in the die, and its elastic deformation during extrusion are presented. The tests were carried out in a wide range of the R (1.5-11), assuming a constant outer diameter of the die. A comparative analysis of the wear was carried out in the area of the die corner radius and the calibration strip since the wear is the highest there. Based on the result obtained, it was shown that the increase in the extrusion force was not proportional to the increase in the extrusion ratio, and this could be described by a logarithmic equation. On the other hand, the value of equivalent stresses in the area of the die corner depends, to a relatively small extent, on the value of the extrusion ratio. However, due to the deformation of the die, it is most advantageous to use an extrusion ratio greater than or equal to three. Based on the wear analysis, it was estimated that the increase in the extrusion ratio R from 3 to 7 caused a more than 2.5 times increase in tool wear. In turn, the implementation of the extrusion process with the use of R = 11 caused a 4.5-fold increase in the average depth of die wear compared to its wear using the extrusion ratio R = 3.

Modeling Tool for Studying the Influence of Operating Conditions on the Enzymatic Hydrolysis of Milk Proteins.

Systematic modeling of the enzymatic hydrolysis of milk proteins is needed to assist the study and production of partially hydrolyzed milk. The enzymatic hydrolysis of milk proteins was characterized and evaluated as a function of the temperature and protease concentration using Alcalase, Neutrase and Protamex. Modeling was based on the combination of two empirical models formed by a logarithmic and a polynomial equation to correlate the kinetic constants and the operating conditions. The logarithmic equation fitted with high accuracy to the experimental hydrolysis curves with the three proteases (R2 &gt; 0.99). The kinetic constants were correlated with the operating conditions (R2 &gt; 0.97) using polynomial equations. The temperature and protease concentration significantly affected the initial rate of hydrolysis, i.e., the kinetic constant a, while the kinetic constant b was not significantly affected. The values for the kinetic constant a were predicted according to the operating conditions and they were strongly correlated with the experimental data (R2 = 0.95). The model allowed for a high-quality prediction of the hydrolysis curves of milk proteins. This modeling tool can be used in future research to test the correlation between the degree of hydrolysis and the functional properties of milk hydrolysates.

Finite-time blowup and existence of global solutions for a logarithmic semilinear hyperbolic equation

In this paper we consider the semilinear wave equation with the product of logarithmic and polynomial nonlinearities and establish the global existence and finite-time blowup of solutions by using the potential well method.

Biology, digestive enzymes and organosomatic indices of Chrysichthys nigrodigitatus (Lacépède, 1803) from Oyan Dam, Southwestern Nigeria

Some aspects of the biology, digestive enzymes and organosomatic indices of Chrysichthys nigrodigitatus purchased from fishermen in Oyan dam was investigated. This study was aimed at providing information on the composition of food materials found in the gut and specific activities of selected enzymes as it affects the domestication of the species. 100 specimens of the species were examined for stomach contents, length-weight relationship, digestive enzyme assay and organosomatic indices, using standard methods. Food items observed were detritus (4%), fish part (12%), Insecta (13%). Sand was observed to be 11% of total stomach volume. The logarithmic equation for length-weight relationship lnW=2.68lnl-3.79 indicated that an increase in length led to a corresponding increase in weight with ‘R’ = 0.611, calculated ‘r’=0.78, ‘a’=2.68 and ‘b’=3.79 indicating positive allometric growth pattern. Amylase exhibited high activity in the stomach, while lipase and proteinase in the stomach and posterior intestine. Specific activities of digestive enzymes showed significant differences (p&lt;0.05). Viscerosomatic (2.92± 0.25), hepatosomatic (2.27± 0.22) indices and Fulton condition factor (1.93± 0.06) were recorded. Feed items present in the species confirm its overlapping feeding habit, indicating that the species is an omnivorous detritivore. This was also depicted in the activities of the different digestive enzymes.

Leaching NCM cathode materials of spent lithium-ion batteries with phosphate acid-based deep eutectic solvent

Deep eutectic solvents (DESs) play an important role in efficient recovery of spent lithium-ion batteries (LIBs). In this study, we proposed an efficient and safe method by using a choline chloride-phenylphosphinic acid DES as a lixiviant for the leaching of LiNixCoyMnzO2 (NCM) cathode active materials of spent LIBs. The leaching conditions were optimized based on the leaching time, liquid–solid ratio, and leaching temperature. Under optimal experimental conditions, the leaching efficiencies of Li, Co, Ni, and Mn reached 97.7 %, 97.0 %, 96.4 %, and 93.0 %, respectively. The kinetics of the leaching process were well-fitted using the logarithmic law equation. The apparent activation energies for Li, Co, Ni, and Mn have been reported to be 60.3 kJ/mol, 78.9 kJ/mol, 99.3 kJ/mol, and 82.1 kJ/mol, respectively. UV–visible spectroscopy and Fourier transform infrared analysis revealed that the coordination configurations of Ni and Co in the leaching solution were octahedral and tetrahedral, respectively. In addition, the PO bond in phenylphosphinic acid was involved in coordination during leaching. This finding may provide an effective and safe approach for leaching valuable metals from spent LIBs.

Local existence, global existence and decay results of a logarithmic wave equation with delay term

In this article, we deal with a logarithmic wave equation with strong damping and delay. Firstly, we prove the local existence by utilizing the semigroup theory. Later, we obtain the global existence of solutions by using the well‐depth method. Moreover, under appropriate assumptions on the weight of the delay and that of strong damping, we get the exponential decay.

Long-time behaviour of classical solutions to the relativistic Euler equations with logarithmic equation of state

Long-time behaviour of classical solutions to the relativistic Euler equations with logarithmic equation of state

New kinetic models for predicting the removal of oil and grease from food-processing industry wastewater

Biochemical accumulation rate of Serratia marcescens SA30 to remove the oil and grease (O&G) contaminant from a wastewater can be described using the kinetic models of packed-bed column reactor (PBCR) treatment system. The aim of this study was to develop the linear and logarithmic equations for predicting the efficiency of O&G removal from food-processing industry wastewater. The results showed that the performance of PBCR treatment process can achieve nearly 100% efficiency when the PBCR experiments were run using the initial O&G concentrations of 26.9 and 33.5 g L−1 with a volumetric flow rate of 0.18 L h−1. The applicability of the linear and logarithmic equations represented a quick and reliable method to monitor the trend of O&G removal by the PBCR treatment process contributes to strengthening the future biotechnological methods of treating oil-contaminated wastewater in the tropical environment.

Mathematical Model for 100- and 200-meter Olympic Games Running Championship Time Records

The 100 and 200- meter running championships for both males and females were first held in 1948 at London Olympic Games 1948. Some time records made by the running champions have been continuously improved. Thus, running championship is not only intended to win the gold medals but also to make new world records. The secondary data were in the form of running championships' time records used to formulate the mathematical models and determine the minimum time limits (fastest). This research used the time-record data of 100 and 200- meter running championships for both males and females winning the gold medals from the Olympic Games held in 1948 to 2020. The mathematical Model for 100 and 200- meter running championships was more appropriately formulated using a logarithmic regression equation. Meanwhile, the time records for running championships of 100 meters for females as well as those of 200 meters for both males and females used a simple linear regression. The world record for running 100 meters for males still belongs to Usain Bolt (9.63 seconds). By using an assumption that the time records are normally distributed, those time records can be broken/improved into 9.53 seconds. Moreover, if the analysis is made using a box-plot diagram, the fastest time can be 9.42 seconds. A similar conclusion was also obtained for the world records of running 100 meters for females and 200 meters for males and females mentioning that the recently achieved time records still can be broken/improved in the future.

Variability of phosphorus sorption properties in hydromorphic soils: Consequences for P losses in agricultural landscapes

AbstractIncreasing concerns over water eutrophication due to agricultural phosphorus (P) loss have led to the development of indicators to assess the risk of P release from agricultural soils. Recently, a logarithmic equation linking the degree of phosphorus saturation (DPS) to the simple water‐soluble P (WSP) content of soils has been proposed as a universal method to assess this risk based, however, mainly on the analysis of well‐drained soils. Here, we studied the P sorption properties and DPS values of 69 hydromorphic soils from cultivated and uncultivated wetland zones located in Brittany, Western France, to test whether the method could also apply to poorly‐drained soils. The bulk soil analysis showed that P contents of the studied hydromorphic soils were 30% to 80% higher than P contents normally found in Brittany soils, evidencing a possible P enrichment process. Adsorption isotherms revealed a surprisingly high variability in the P sorption properties as a function of the location of the soil (maximum P adsorption capacity ranging from 500 to 1850 mg kg−1), which is caused by variations in the phases controlling P sorption in soil (from clay to organic matter and/or iron and aluminium oxides, depending on the soil location). Distinct relationships between DPS and WSP values were also obtained depending on the location of the soils. The obtained DPS versus WSP relationships showed that the P saturation threshold above which the risk of dissolved P release increases markedly is 30% lower on average for hydromorphic soils than for well‐drained soils. Hydromorphic soils appear to be more at risk of releasing dissolved P at the same DPS values than well‐drained soils. The present study indicates an underestimation of the P release risk from hydromorphic soils by the existing method developed for well‐drained soils and calls for the development of specific risk assessment tools for hydromorphic soils, especially given on the strong spatial heterogeneity of their P sorption properties.Highlights Wetlands soils in western France show high variability in their phosphorus properties. There is a high‐risk of dissolved P losses from wetland soils to watercourses. The risk can remain high even in soils that are currently not cultivated Degree of phosphorus saturation method to assess dissolved P release risk could underestimate it.

A maturation scale for molecular simulation of kerogen thermal degradation

Correlating molecular dynamics (MD) simulations of thermal degradation of organic matter with natural processes in ancient sediments is vital for revealing the mechanisms of hydrocarbon generation at molecular and atomic levels. This study establishes a practical maturation scale for molecular modeling of the thermal degradation of kerogen. ReaxFF reactive MD simulations of the thermal evolution of typical kerogen types (Type I, II, and III) were carried out at temperatures ranging from 1200 K to 2600 K. The results demonstrate that the H/C and O/C atomic ratios of kerogen molecules decrease as simulation temperatures increase and that the evolutionary paths of the three kerogen types fit well with those shown in the van Krevelen diagram. The corrected activation energies of pyrolysis for these three typical kerogen types were found to be 61.1 kcal/mol, 53.5 kcal/mol and 52.8 kcal/mol, respectively, consistent with those obtained from both laboratory simulations and basin modeling. By comparison with the relationships between measured vitrinite reflectance values and the H/C atomic ratios of kerogen samples reported in the literature, a quantitative relationship was established between the vitrinite reflectance (%Ro) values of kerogen and simulation temperatures (T). The cross-plots of T vs %Ro for kerogen types II and III show that vitrinite reflectance exponentially increases with increasing simulation temperature. Two logarithmic correlation equations were constructed to match geological maturation levels to the corresponding levels in MD simulations of thermal degradation of types II and III kerogen. These maturation scales are generally in line with the simulation temperatures set for thermal degradation in kerogen pyrolysis in previous studies. This work has promising practical implications for research on the thermal evolution behavior of ancient sedimentary organic matter using ReaxFF MD simulation.