Law Exponent Research Articles

Solution of Fractional Differential Equations Involving Hilfer-Hadamard Fractional Derivatives

Objectives: The aim is to establish prerequisite properties for the Hilfer-Hadamard fractional derivatives and address boundary value problems related to fractional polar Laplace and fractional Sturm-Liouville equations involving Hilfer-Hadamard fractional derivatives. Methods: Existing definitions and findings are utilized to obtain the properties for fractional derivatives, and the Adomian decomposition method is employed to solve the fractional differential equations. Findings: Validity conditions for the law of exponents are determined, and the study investigates the fractional differential equations and their corresponding solutions, possessing the capacity to replace the traditional polar Laplace and Sturm-Liouville boundary value problems to effectively represent real-world phenomena. Novelty: The study introduces the substitution of two consecutively operated Hilfer-Hadamard fractional derivatives with a corresponding single Hilfer-Hadamard fractional derivative using the law of exponents. Additionally, the polar Laplace and Sturm-Liouville boundary value problems are extended to their respective fractional counterparts, expressed in a concise format using HilferHadamard fractional derivatives. Keywords: Adomian decomposition method, Hilfer-Hadamard fractional derivative, Fractional polar Laplace equation, Fractional Sturm-Liouville boundary value problem

Disorder-induced non-linear growth of fingers in immiscible two-phase flow in porous media

Immiscible two-phase flow in porous media produces different types of patterns depending on the capillary number Ca and viscosity ratio M. At high Ca, viscous instability of the fluid–fluid interface occurs when the displaced fluid is the more viscous, and leads to viscous fingering, which is believed to exhibit the same growth behavior as the viscously-unstable fingers observed in Hele–Shaw cells by Saffman and Taylor [“The penetration of a fluid into a porous medium or Hele–Shaw cell containing a more viscous liquid,” Proc. R. Soc. London 245, 312 (1958)], or as diffusion-limited aggregates (DLA). In such Laplacian growth processes, the interface velocity depends linearly on the local gradient of the physical field that drives the growth process (for two-phase flow, the pressure field). However, a non-linear power-law dependence between the flow rate and the global pressure drop, reminiscent of what has also been observed for steady-state two-phase flow in porous media, was evidenced experimentally for the growth of viscously-unstable drainage fingers in two-dimensional porous media, 20 years ago. Here, we revisit this flow regime using dynamic pore-network modeling and explore the non-linearity in the growth properties. We characterize the previously unstudied dependencies of the statistical finger width and non-linear growth law's exponent on Ca, and discuss quantitatively, based on theoretical arguments, how disorder in the capillary barriers controls the growth process' non-linearity, and why the flow regime crosses over to Laplacian growth at sufficiently high Ca. In addition, the statistical properties of the fingering patterns are compared to those of Saffman–Taylor fingers, DLA growth patterns, and the results from the aforementioned previous experimental study.

An Experimental Study on the Combustion Characteristics of a Methane Diffusion Flame within a Confined Space under Sub-Atmospheric Pressure

Gas-fired boilers, gas stoves, and wall-mounted gas boilers are the main consumers of gas fuel, but they generally encounter problems when operating at high altitudes, such as reduced thermal efficiency and increased pollutant emissions. Previous studies on gas combustion characteristics under sub-atmospheric pressure were mostly carried out in a large space, which is quite different from chamber combustion equipment. Therefore, it is insufficient to guide the design and operation optimization of plateau gas equipment. In this paper, experimentations were carried out to explore the characteristics of a methane diffusion flame under sub-atmospheric pressures. The mass flow rates of methane and air remain consistent under different pressure conditions. The centerline temperature (Tc) distribution, flame appearance, smoke point, CO emission, and NOx emission under different pressures (ranging from 61.66 to 97.75 kPa) were examined under both fuel rich and lean conditions. The results show that Tc at the rear and front of furnace variation with pressure is opposite under fuel-lean and -rich combustion. The Tc at the front of furnace decreases with decreasing pressure, whereas Tc at the rear of furnace increases with decreasing pressure. With decreasing pressure, flame length decreases under lean combustion, but increases under rich combustion. The smoke point fuel flow rate, flame length, and residence time increases with decreasing pressure, following the law of negative exponent. The CO emission decreases with decreasing pressure, which indicates that the reduced pressure makes methane combustion more complete. For NO emission, the reduced pressure results in an opposite tendency under fuel-lean and -rich combustion. With decreasing pressure, the NO emission decreases under fuel-lean combustion but increases under fuel-rich combustion.

High-order optimization of bicubic parametric convergent curves for carbon capture nozzles in hydrogen-rich fuel

Supersonic condensation separation is a potential environmentally-friendly hydrogen purification and CO2 capture technology. Among them, the nozzle contraction structure is essential in creating conditions conducive to CO2 condensation. In this study, the H2–CO2 binary condensate flow model based on the condensation dynamics theory was used to predict the influence of the contraction curve on the condensation parameters. Furthermore, this paper proposed a combined high-order curve correction method based on the standard Bicubic parametric curve and summarized the universal formula of the High-order correction curve (HOCC). Furthermore, the influence law of the correction exponent and quantile location on the hydrogen-rich fuel gas decarburization performance was further obtained through numerical calculation. After two-stage optimization, the HOCC n = 7 xm = 0.35 structure can improve the decarburization performance of the supersonic purification device by 7.44%. Moreover, the H2–CO2 condensate flow will not flow separation in the nozzle.

Robotic Tree-fruit harvesting with arrays of Cartesian Arms: A study of fruit pick cycle times

In some cases, it has been shown that fruits on trees with SNAP (Simple, Narrow, Accessible, and Productive) architectures are reachable by robot arms using three linear degrees of freedom; hence, high fruit-picking efficiencies can be achieved with simpler arms. This paper uses digitized fruit position data to compute the fruit pick cycle times (PCT) of robotic fruit harvesters with multiple arms arranged in grid configurations, i.e., operating in disjoint rectangular work cells independently of each other. The effects of the robot joints’ maximum linear acceleration and maximum linear velocity on PCTs were studied. As Vmax increased, the PCT followed a negative exponent power law (diminishing return) for any given Amax. Similarly, for a constant Vmax, the improvement of PCT as Amax increased slowed down at higher acceleration values.Also, the PCTs were computed when each arm work cell was designed based on equal fruit load or equal size criteria, using four workspace partitioning schemes (height split, length split, height split matrix, and length split matrix). Equal fruit-load configurations resulted in load balancing and exhibited lower PCTs; the height-split configuration was the best among the four different partitioning schemes, possibly because it compensated better for the fruit distribution's non-uniformity along the trees' height.Finally, the PCTs were computed while harvesting one side of an orchard row or both sides concurrently. Harvesting sides separately resulted in lower PCTs (greater speed) due to non-uniform fruit distributions on the different sides of trees. The insights gained in this paper can inform the design of harvesting robots utilizing arrays of 3-dof linear arms.

Global Spatial Distribution of Hack's Law Exponent on Mars Consistent With Early Arid Climate

AbstractWidespread valley networks (VNs) on Mars and other evidence point to an early warm and wet climate. However, ongoing debates still exist about VN's formation processes and associated climatic conditions. The power law relationship between basin length and area (Hack's Law) can be diagnostic of different fluvial processes related to climatic conditions. Past studies of Hack's Law on Mars at local sites have produced inconclusive results. Here we used a parameter‐free method to delineate watersheds globally on Mars based on mapped VNs and extracted their Hack's Law exponent (h). The majority of h values on Mars are similar to those in arid areas on Earth, suggesting similar runoff processes and arid conditions for VN formation on early Mars. Statistical analyses show that the spatial distribution of h on Mars is not random, but with a few clustered high and low values, likely controlled by local conditions (e.g., regional topographic slope).

Effectiveness of Blended Learning Techniques in Teaching Laws of Exponents

This study sought to ascertain whether blended learning techniques helped teach the laws of exponents to students in Grade 7 at Pulo National High School-Diezmo Ext., City of Cabuyao, Laguna for the SY 2021-2022. For this investigation, a quasi-experimental research approach was used. Sixty (60) seventh-grade students took part in this study. The pretest and posttest used in this study covered the fundamental idea of laws of exponents. The experimental group outperformed the comparison group on average, scoring a mean of 26.13 as opposed to 18.27 for the comparison group, according to the formative test results. The formative assessments for the two groups differed noticeably on p.01. Results from the posttest frequently improved over those from the pretest. The difference between the pretest and posttest mean scores was significant at p.01 for the experimental and comparison groups, indicating that it is very effective for both groups of students. The difference between the pretest and posttest scores for the experimental group produced a t-value of 9.525, which is significant at p.01. Based on the study's findings, the researchers concluded that blended learning is a successful intervention that may be used in the classroom to help kids get better at math.

Laws of Exponents by Differentiation

Laws of Exponents by Differentiation

Finite Amplitude Cellular Thermohaline Convection

Abstract In this paper, linear and nonlinear properties of thermohaline convection at the onset with the stress-free boundary conditions are investigated using perturbation analysis relevant to oceanic water and ground water. The nonlinear governing equations are expanded using linear stability solutions, related to the temperature, motion, and concentration in the series of nonhomogeneous linear equations. The proposed method yields finite amplitude steady solutions by means of successive approximations. The rate of heat transfer is studied up to sixth-order using an expansion of thermal Rayleigh number (Ra1) as proposed by Kuo (1961, “Solution of the Non-Linear Equations of the Cellular Convection and Heat Transport,” J. Fluid Mech., 10(04), pp. 611–630). The heat transfer rate of the system depends on Lewis number, Prandtl number, and thermal (Ra1) as well as solutal (Ra2) Rayleigh number. The result for the heat transport (N) in the system fits a power law of exponent of 0.288, i.e., N≃(Ra1/Ra1o)0.288 for Ra2=0, Pr = 10 and Ra1>4Ra1o. This power law of exponent value of 0.288 is close to the results (0.25) obtained by Kuo (1961, “Solution of the Non-Linear Equations of the Cellular Convection and Heat Transport,” J. Fluid Mech., 10(04), pp. 611–630), and also to the experimental results for laminar convection by Jakob (1949, Heat Transfer, Vol. 1, Wiley, New York.). The characteristics of heat transfer and flow field results are depicted by means of isotherms and streamlines, respectively. The path of convective heat transport and the comprehensive analysis of energy distribution by means of heatlines are explained using the concept of heat function. Laminar natural convection using entropy generation analysis due to fluid friction and heat transfer is also being studied. The numerical simulation for total entropy generation has been carried out for Prandtl numbers pertaining to ground water for different Ra1 and Ra2.

Large-scale frictionless jamming with power-law particle size distributions.

Due to significant computational expense, discrete element method simulations of jammed packings of size-dispersed spheres with size ratios greater than 1:10 have remained elusive, limiting the correspondence between simulations and real-world granular materials with large size dispersity. Invoking a recently developed neighbor binning algorithm, we generate mechanically stable jammed packings of frictionless spheres with power-law size distributions containing up to nearly 4000000 particles with size ratios up to 1:100. By systematically varying the width and exponent of the underlying power laws, we analyze the role of particle size distributions on the structure of jammed packings. The densest packings are obtained for size distributions that balance the relative abundance of large-large and small-small particle contacts. Although the proportion of rattler particles and mean coordination number strongly depend on the size distribution, the mean coordination of nonrattler particles attains the frictionless isostatic value of six in all cases. The size distribution of nonrattler particles that participate in the load-bearing network exhibits no dependence on the width of the total particle size distribution beyond a critical particle size for low-magnitude exponent power laws. This signifies that only particles with sizes greater than the critical particle size contribute to the mechanical stability. However, for high-magnitude exponent power laws, all particle sizes participate in the mechanical stability of the packing.

The Detection of Self-Similar/Branching Processes in Complex Biological Systems: Analysis of the Temporal Evolution of Impedance Measurements in Tulsi (Holy Basil) Leaves "Ocimumtenuiflorum".

In this paper, a theoretical model has been proposed for the first time to illustrate the living Tulsi (Holy Basil) leaf evolution in describing the branching systems in the intermediate range of frequencies. The proposed fitting function following the applied model portrays the whole stages of the temporal evolution of Tulsi (Holy Basil, "Ocimumtenuiflorum") leaf during the six days of impedance measurements with a high degree of accuracy (with fitting error less than 0.1 percent). The developed fitting parameter enables to identify the three stages of temporal evaluation of the Tulsi leaf as "vital leaf activity" (first stage), "quasi-chaotic behaviour" (second stage), and the "dying" (third stage). The theoretical model proposed in this work incorporated a fractal element having complex conjugated power law of exponents and a fractal element with a time lagged branching process. This novel approach introduces an additional degree of freedom over the previously proposed impedance models in terms of its imaginary part of complex conjugated power-law exponent. Further, it increases flexibility and versatility in accurately modeling the behavioral variations of complex branching systems, whose seemingly nondeterministic temporal nature had been considered earlier as counterintuitive and random. The incorporation of branching processes in comprehensively explaining the complex biological systems allows us to gain a deeper insight into the transfer of charge processes in the intermediate range of available frequency scales. Furthermore, the proposed model validates the presence of new fractal elements with a complex conjugated power law of exponents in naturally occurring biological processes. The experimental confirmation can play a key role in explaining a wide class of branching processes in complex systems and enrich the modern theory of fractional calculus.

The impact of the large-scale circulation anomalies in the northern hemisphere on air quality in Wuhan of China during the Spring Festival in 2020

During the 2020 Chinese Spring Festival extended holidays and the COVID-19 period in Wuhan, the “quasi-zero emissions increase” occurred due to the human activities have a considerably lower impact on air quality. A new scientific question of concern: Why has high of O3 and PM2.5 still been observed? The comparative analysis of the influences of anomalous atmospheric circulation and weather conditions during this special period in 2019–2020 with those in the historical years can be useful. The results in the study were as follows:(1) Even during this “quasi-zero emissions increase” period, the concentrations of O3 and PM2.5 are higher than normal, with O3 being significantly higher in Wuhan. Which is associated with a anomaly large-scale humid/warm water vapor transport in the Northern Hemisphere which is significantly different from that of the same period with historical more-haze years. (2) In this scenario, the large-scale humid/warm water vapor transport brings sustained high-humidity water vapor lifting conditions to north China. The daily cycling change of the sun's zenith angle to support the high-humidity and high-condensation condition are conducive to the elevation of high condensation (fc) and degree of super-saturation (S) going up to the peak of S >10%. Under the condition driven by high fc, it causes the power exponent law to be followed converting NO2 to O3, thereby resulting in an increase in O3 concentration. This additional contribution to O3 concentration is as high as 37% than those from pure chemical process. (3) An objective quantitative reversal method for calculating the monthly threshold in fc is given based on the power exponential conversion law by the observed data. It can be prospectively applied to assess the feedback on the impact of pollution on urban climate.

Metabolic shifts of oceans: Summoning bacterial interactions

Metabolic scaling can pinpoint monitoring priorities for anomalous communities and inform local eco-engineering restoration enhancing ecosystem function. We discovered an entropic Kleiber’s Law between bacterioplankton phylum total directed interactions and population and community abundane, with an average exponent Φ≃23 in striking accordance with theoretical expectations. The taxa-independent allometric relationship largely varies by habitat (shifting from 2/3 to 3/4 as power-exponent) and is better explained by the community abundance spectrum and its organization than by the envirome defined by environmental interactions. Yet, greater envirome disorganization meant higher environmental impacts causing larger bacteriome disorganization toward random interaction topologies and higher Kleiber’s Law exponents. Habitat-related bacteriome differences revealed that keystone phyla were highly interacting and rare (like Acidobacteria, SBR1093, and ZB3) maintaining weak but salient interactions with both rare (like Spirochaetes and Firmicutes) and dominant phyla. Consistent with their small divergence from main Kleiber’s law and abundance spectrum patterns cosmopolitan phyla, like Bacteroidetes and Proteobacteria, were dominant but weakly interacting (in magnitude) emphasizing the importance of weak ties in collective dynamics. The most salient interactions were always between phyla with high and low node centrality defined by their total directed interactions and abundance. Envirome network centrality shows temperature and nutrient concentrations (total and dissolved organic phosphorus) as central in estuarine habitats impacted by both global ocean factors and near-shore river effluxes; salinity and suspended particle matter were central in more off-shore habitats. Our results emphasize the importance of a probabilistic assessment of community interactions over individual bacterial abundance analyses when searching for collective patterns informative of ecosystem state and their environmental determinants.

Effect of laser power on the microstructure and properties of additive manufactured 17-4 PH stainless steel in different fabrication atmosphere

The effect of laser power (400, 600, 800 W) on the microstructure and properties, such as porosity, strength and microhardness, of directed energy deposition 17-4 PH stainless steel in different fabrication atmosphere (atmosphere air and atmosphere Ar) was investigated via the view of thermal process. With the increase of laser power, the porosity of the samples decreased, and the Ar atmosphere further enhanced the effect through a power exponent law. The submicron (∼0.5 μm) amorphous SiO2 particles and the nanoscale (∼35 nm) Cu-rich precipitates dispersed in the matrix strengthened the samples, and the dimensions changed when different atmosphere and heat treatment were employed which in turn influenced the strength and microhardness. The mechanical properties were also affected by content of austenite and the release of oxygen, which were dominated by thermal process and heat treatment.

Influence of Graded Surface Decarburization of Automobile Forging Front Axle on the Bending Behavior Based on a Third-Order Shear Deformation Beam Theory

During the forging process of automobile front axle, the steel near the surface is often decarburized for a certain depth. The mechanical properties at the decarburization layer are graded and different from the inner area, influencing the bending behavior of axles under heavy loads. In this paper, the decarburized forging of front axle is regarded as a rectangular thick sandwich beam, composed of a homogeneous core and the functionally graded layer coated on both bottom and top surface. A Third-order Shear Deformation Theory (TSDT) is employed to investigate the static bending behaviors under two point−loads. The properties of sandwich FG material are represented with a piecewise power−law function, and the displacement field governing equations are derived through the virtual work principle. The Navier analytical method and numerical DQM procedures are employed to obtain the bending responses under simply supported boundary conditions, and the results are validated through the comparison with an example in the literature. Then, the transverse deflection, rotation, axial stress, and shear stress are studied in terms of different power−law exponents, decarburization depth, unsymmetrical decarburization depth, unbalance loading, and beam sectional dimension. The study reveals the influence of surface decarburization on the bending behavior of forged automobile front axles, and contributes to the optimization of structure design.

Temperature-Dependent Viscoelastic Energy Dissipation and Fatigue Crack Growth in Filled Silicone Elastomers

There has always been a need to investigate the mechanical properties and fracture behaviors of materials that are widely used commercially, such as filled silicone rubbers. In this work, a thorough study was performed to evaluate the temperature dependencies of viscoelastic energy dissipation and fatigue crack propagation. Small strain viscoelastic behavior was examined by using dynamic mechanical analysis (DMA), and larger strain viscoelastic dissipation was quantified by defining an effective viscoelastic phase angle from large-amplitude cyclic deformation. Pure shear mode I fatigue tests were performed for quantification of material toughness and crack propagation resistance. In addition, natural rubber with different filler levels was tested to provide a direct comparison between two important classes of filled rubber elastomers. Major differences were identified for the two elastomers. However, for both types of elastomers, a common power law relationship with an exponent of 1.5 describes the dependence of the fatigue crack growth per cycle on a normalized driving force obtained from the stored elastic energy. We found that the silicone rubber deviated from the 1.5 exponent power law relation at elevated temperature, and its prefactor in this power law was correlated to the crack morphology, with rough cracks providing a higher fatigue resistance than smooth cracks.

Equivalent circuit properties of organic food extracts doped hydrogels and their applications in bioelectronics

Dielectric/electrical properties of organic food extracts doped hydrogels, which can be envisaged to lead the technology applications of biosensors, bio-imaging and wireless biological sensing schemes in bio-electronic circuits, were analysed at room temperature by means of impedance spectroscopy. Electrode and interfacial polarization mechanisms were determined to be dominant on the dielectric properties of pure and organic food extracts doped hydrogels in the low frequency region. Dielectric relaxation, viscoelastic mechanism, Brownian motion of food extract molecules and Maxwell-Wagner approach were used to explain the dielectric properties of the samples in the high frequency region. It was determined that the dielectric plane plots that provide information about the transmission mechanism of all samples are compatible with the Cole-Cole relaxation models. The conductivity characteristics of the all samples depending on the s parameter/power law exponent were explained for the three different regions by dc conductivity, the Correlated Barrier Hoping conductivity and Quantum Mechanical Tunnel conductivity models, respectively. From the experimental results obtained in this study, it was concluded that organic food extracts doped hydrogels are an inspirational material for applications such as biosensors, bio imaging and wireless bio-sensing device that can be used in bio-systems.

Vibration analysis of FG cylindrical shell: Evaluation of Ritz-polynomial mixed with ring terms

Here the Rayleigh - Ritz method has been applied to derive the shell vibration frequency equation. This equation has been formed as an eigenvalue problem form. MATLAB software package has been utilized for extracting shell frequency spectra. Nature of materials used for construction of cylindrical shells also has visible impact on shell vibration characteristics. For isotropic materials, the physical properties are same everywhere, the laminated and functionally graded materials vary from point to point. Here the shell material has been taken as functionally graded material. Moreover, the impact of ring supports around the shell circumferential has been examined for the various positions along the shell axial length. These shells are stiffened by rings in the tangential direction. These ring supports are located at various positions along the axial direction round the shell circumferential direction. These variations have been plotted against the locations of ring supports for three values of exponents of volume fraction law. For three conditions, frequency variations show different behavior with these values of exponent law. The influence of the positions of ring supports for simply supported end conditions is very visible. The frequency first increases and gain maximum value in the midway of the shell length and then lowers down. The comparisons of frequencies have been made for efficiency and robustness for the present numerical procedure.

Investigation on Non-Debye type relaxation and polaronic conduction mechanism in ZnCr2O4 ternary spinel oxide

Ternary spinel chromite ZnCr2O4 was synthesized by sol-gel auto-combustion method. Rietveld profile analysis of X-ray powder diffraction data revealedthe cubic spinel crystal structure with Fd-3 m space group. Thedielectric relaxation and transport behavior of ZnCr2O4were studied in detail by complex impedance spectroscopy in the frequency range of70-1.1x106Hzover the measured temperature range of293-523K. Nyquist plots modeled by the equivalent circuit RG‖QG+RGB‖QGBindicated the presence of two relaxation processes in the material. The AC conductivity followed Jonscher power law and variation of frequency exponent with temperature depicted non-overlapping small polaron tunneling as a dominant electrical transport mechanism in the sample. Moreover, the DC conductivity obeyed Mott and Davis law. The frequency and temperature dependence of dielectric constant and loss factor was discussed in terms of space charge polarization, Maxwell-Wagner type dielectric relaxation and hopping of charge carriers between Cr2+ and Cr3+networks in ZnCr2O4 system.

Superspreaders and lockdown timing explain the power-law dynamics of COVID-19.

Infectious disease outbreaks are expected to grow exponentially in time when left unchecked. Containment measures such as lockdown and social distancing can drastically alter the growth dynamics of the outbreak. This is the case for the 2019-2020 COVID-19 outbreak, which is characterized by a power-law growth. Strikingly however, the power-law exponent is different across countries. Here I illustrate the relationship between these two extreme scenarios, exponential and power-law growth, based on the impact of superspreaders and lockdown strategies to contain the outbreak. The theory predicts a relationship between the power- law exponent and the time interval between the first case and lockdown that is validated by the observed COVID-19 data across different countries.