The Probability Density Function Related to Shallow Cumulus Entrainment Rate and Its Influencing Factors in a Large-Eddy Simulation

The process of entrainment-mixing between cumulus clouds and the ambient air is important for the development of cumulus clouds.Accurately obtaining the entrainment rate(λ)is particularly important for its parameterization within the overall cumulus parameterization scheme.In this study,an improved bulk-plume method is proposed by solving the equations of two conserved variables simultaneously to calculateλof cumulus clouds in a large-eddy simulation.The results demonstrate that the improved bulk-plume method is more reliable than the traditional bulk-plume method,becauseλ,as calculated from the improved method,falls within the range ofλvalues obtained from the traditional method using different conserved variables.The probability density functions ofλfor all data,different times,and different heights can be well-fitted by a log-normal distribution,which supports the assumed stochastic entrainment process in previous studies.Further analysis demonstrate that the relationship betweenλand the vertical velocity is better than other thermodynamic/dynamical properties;thus,the vertical velocity is recommended as the primary influencing factor for the parameterization ofλin the future.The results of this study enhance the theoretical understanding ofλand its influencing factors and shed new light on the development ofλparameterization.

Prediction and Output Estimation of Pattern Moving in Non-Newtonian Mechanical Systems Based on Probability Density Evolution

A prediction framework based on the evolution of pattern motion probability density is proposed for the output prediction and estimation problem of non-Newtonian mechanical systems,assuming that the system satisfies the generalized Lipschitz condition.As a complex nonlinear system primarily governed by statistical laws rather than Newtonian mechanics,the output of non-Newtonian mechanics systems is difficult to describe through deterministic variables such as state variables,which poses difficulties in predicting and estimating the system’s output.In this article,the temporal variation of the system is described by constructing pattern category variables,which are non-deterministic variables.Since pattern category variables have statistical attributes but not operational attributes,operational attributes are assigned to them by posterior probability density,and a method for analyzing their motion laws using probability density evolution is proposed.Furthermore,a data-driven form of pattern motion probabilistic density evolution prediction method is designed by combining pseudo partial derivative(PPD),achieving prediction of the probability density satisfying the system’s output uncertainty.Based on this,the final prediction estimation of the system’s output value is realized by minimum variance unbiased estimation.Finally,a corresponding PPD estimation algorithm is designed using an extended state observer(ESO)to estimate the parameters to be estimated in the proposed prediction method.The effectiveness of the parameter estimation algorithm and prediction method is demonstrated through theoretical analysis,and the accuracy of the algorithm is verified by two numerical simulation examples.

High-dimensional uncertainty quantification of projectile motion in the barrel of a truck-mounted howitzer based on probability density evolution method

This paper proposed an efficient research method for high-dimensional uncertainty quantification of projectile motion in the barrel of a truck-mounted howitzer.Firstly,the dynamic model of projectile motion is established considering the flexible deformation of the barrel and the interaction between the projectile and the barrel.Subsequently,the accuracy of the dynamic model is verified based on the external ballistic projectile attitude test platform.Furthermore,the probability density evolution method(PDEM)is developed to high-dimensional uncertainty quantification of projectile motion.The engineering example highlights the results of the proposed method are consistent with the results obtained by the Monte Carlo Simulation(MCS).Finally,the influence of parameter uncertainty on the projectile disturbance at muzzle under different working conditions is analyzed.The results show that the disturbance of the pitch angular,pitch angular velocity and pitch angular of velocity decreases with the increase of launching angle,and the random parameter ranges of both the projectile and coupling model have similar influence on the disturbance of projectile angular motion at muzzle.

Probability Density Analysis of Nonlinear Random Ship Rolling

Ship rolling in random waves is a complicated nonlinear motion that contributes substantially to ship instability and capsizing.The finite element method(FEM)is employed in this paper to solve the Fokker Planck(FP)equations numerically for homoclinic and heteroclinic ship rolling under random waves described as periodic and Gaussian white noise excitations.The transient joint probability density functions(PDFs)and marginal PDFs of the rolling responses are also obtained.The effects of stimulation strength on ship rolling are further investigated from a probabilistic standpoint.The homoclinic ship rolling has two rolling states,the connection between the two peaks of the PDF is observed when the periodic excitation amplitude or the noise intensity is large,and the PDF is remarkably distributed in phase space.These phenomena increase the possibility of a random jump in ship motion states and the uncertainty of ship rolling,and the ship may lose stability due to unforeseeable facts or conditions.Meanwhile,only one rolling state is observed when the ship is in heteroclinic rolling.As the periodic excitation amplitude grows,the PDF concentration increases and drifts away from the beginning location,suggesting that the ship rolling substantially changes in a cycle and its stability is low.The PDF becomes increasingly uniform and covers a large region as the noise intensity increases,reducing the certainty of ship rolling and navigation safety.The current numerical solutions and analyses may be applied to evaluate the stability of a rolling ship in irregular waves and capsize mechanisms.

Explanation of Relation between Wave Function and Probability Density Based on Quantum Mechanics in Phase Space

The main problem of quantum mechanics is to elucidate why the probability density is the modulus square of wave function. For the purpose of solving this problem, we explored the possibility of deducing the fundamental equation of quantum mechanics by starting with the probability density. To do so, it is necessary to formulate a new theory of quantum mechanics distinguished from the previous ones. Our investigation shows that it is possible to construct quantum mechanics in phase space as an alternative autonomous formulation and such a possibility enables us to study quantum mechanics by starting with the probability density rather than the wave function. This direction of research is contrary to configuration-space formulation of quantum mechanics starting with the wave function. Our work leads to a full understanding of the wave function as the both mathematically and physically sufficient representation of quantum-mechanical state which supplements information on quantum state given solely by the probability density with phase information on quantum state. The final result of our work is that quantum mechanics in phase space satisfactorily elucidates the relation between the wave function and the probability density by using the consistent procedure starting with the probability density, thus corroborating the ontological interpretation of the wave function and withdrawing a main assumption of quantum mechanics.

Grain yield and N uptake of maize in response to increased plant density under reduced water and nitrogen supply conditions

The development of modern agriculture requires the reduction of water and chemical N fertilizer inputs.Increasing the planting density can maintain higher yields,but also consumes more of these restrictive resources.However,whether an increased maize density can compensate for the negative effects of reduced water and N supply on grain yield and N uptake in the arid irrigated areas remains unknown.This study is part of a long-term positioning trial that started in 2016.A split-split plot field experiment of maize was implemented in the arid irrigated area of northwestern China in 2020 to 2021.The treatments included two irrigation levels:local conventional irrigation reduced by 20%(W1,3,240 m^(3)ha^(-1))and local conventional irrigation(W2,4,050 m^(3)ha^(-1));two N application rates:local conventional N reduced by 25%(N1,270 kg ha^(-1))and local conventional N(360 kg ha^(-1));and three planting densities:local conventional density(D1,75,000 plants ha^(-1)),density increased by 30%(D2,97,500 plants ha-1),and density increased by 60%(D3,120,000 plants ha^(-1)).Our results showed that the grain yield and aboveground N accumulation of maize were lower under the reduced water and N inputs,but increasing the maize density by 30% can compensate for the reductions of grain yield and aboveground N accumulation caused by the reduced water and N supply.When water was reduced while the N application rate remained unchanged,increasing the planting density by 30% enhanced grain yield by 13.9% and aboveground N accumulation by 15.3%.Under reduced water and N inputs,increasing the maize density by 30% enhanced N uptake efficiency and N partial factor productivity,and it also compensated for the N harvest index and N metabolic related enzyme activities.Compared with W2N2D1,the N uptake efficiency and N partial factor productivity increased by 28.6 and 17.6%under W1N1D2.W1N2D2 had 8.4% higher N uptake efficiency and 13.9% higher N partial factor productivity than W2N2D1.W1N2D2 improved urease activity and nitrate reductase activity by 5.4% at the R2(blister)stage and 19.6% at the V6(6th leaf)stage,and increased net income and the benefit:cost ratio by 22.1 and 16.7%,respectively.W1N1D2 and W1N2D2 reduced the nitrate nitrogen and ammoniacal nitrogen contents at the R6 stage in the 40-100 cm soil layer,compared with W2N2D1.In summary,increasing the planting density by 30% can compensate for the loss of grain yield and aboveground N accumulation under reduced water and N inputs.Meanwhile,increasing the maize density by 30% improved grain yield and aboveground N accumulation when water was reduced by 20% while the N application rate remained constant in arid irrigation areas.

Evolutionary history shapes variation of wood density of tree species across the world

The effect of evolutionary history on wood density variation may play an important role in shaping variation in wood density,but this has largely not been tested.Using a comprehensive global dataset including 27,297 measurements of wood density from 2621 tree species worldwide,we test the hypothesis that the legacy of evolutionary history plays an important role in driving the variation of wood density among tree species.We assessed phylogenetic signal in different taxonomic(e.g.,angiosperms and gymnosperms)and ecological(e.g.,tropical,temperate,and boreal)groups of tree species,explored the biogeographical and phylogenetic patterns of wood density,and quantified the relative importance of current environmental factors(e.g.,climatic and soil variables)and evolutionary history(i.e.,phylogenetic relatedness among species and lineages)in driving global wood density variation.We found that wood density displayed a significant phylogenetic signal.Wood density differed among different biomes and climatic zones,with higher mean values of wood density in relatively drier regions(highest in subtropical desert).Our study revealed that at a global scale,for angiosperms and gymnosperms combined,phylogeny and species(representing the variance explained by taxonomy and not direct explained by long-term evolution process)explained 84.3%and 7.7%of total wood density variation,respectively,whereas current environment explained 2.7%of total wood density variation when phylogeny and species were taken into account.When angiosperms and gymnosperms were considered separately,the three proportions of explained variation are,respectively,84.2%,7.5%and 6.7%for angiosperms,and 45.7%,21.3%and 18.6%for gymnosperms.Our study shows that evolutionary history outpaced current environmental factors in shaping global variation in wood density.

A facile strategy for tuning the density of surface-grafted biomolecules for melt extrusion-based additive manufacturing applications

Melt extrusion-based additive manufacturing(ME-AM)is a promising technique to fabricate porous scaffolds for tissue engi-neering applications.However,most synthetic semicrystalline polymers do not possess the intrinsic biological activity required to control cell fate.Grafting of biomolecules on polymeric surfaces of AM scaffolds enhances the bioactivity of a construct;however,there are limited strategies available to control the surface density.Here,we report a strategy to tune the surface density of bioactive groups by blending a low molecular weight poly(ε-caprolactone)5k(PCL5k)containing orthogonally reactive azide groups with an unfunctionalized high molecular weight PCL75k at different ratios.Stable porous three-dimensional(3D)scaf-folds were then fabricated using a high weight percentage(75 wt.%)of the low molecular weight PCL 5k.As a proof-of-concept test,we prepared films of three different mass ratios of low and high molecular weight polymers with a thermopress and reacted with an alkynated fluorescent model compound on the surface,yielding a density of 201-561 pmol/cm^(2).Subsequently,a bone morphogenetic protein 2(BMP-2)-derived peptide was grafted onto the films comprising different blend compositions,and the effect of peptide surface density on the osteogenic differentiation of human mesenchymal stromal cells(hMSCs)was assessed.After two weeks of culturing in a basic medium,cells expressed higher levels of BMP receptor II(BMPRII)on films with the conjugated peptide.In addition,we found that alkaline phosphatase activity was only significantly enhanced on films contain-ing the highest peptide density(i.e.,561 pmol/cm^(2)),indicating the importance of the surface density.Taken together,these results emphasize that the density of surface peptides on cell differentiation must be considered at the cell-material interface.Moreover,we have presented a viable strategy for ME-AM community that desires to tune the bulk and surface functionality via blending of(modified)polymers.Furthermore,the use of alkyne-azide“click”chemistry enables spatial control over bioconjugation of many tissue-specific moieties,making this approach a versatile strategy for tissue engineering applications.

Flexible,high-density,laminated ECoG electrode array for high spatiotemporal resolution foci diagnostic localization of refractory epilepsy

High spatiotemporal resolution brain electrical signals are critical for basic neuroscience research and high-precision focus diagnostic localization,as the spatial scale of some pathologic signals is at the submillimeter or micrometer level.This entails connecting hundreds or thousands of electrode wires on a limited surface.This study reported a class of flexible,ultrathin,highdensity electrocorticogram(ECoG)electrode arrays.The challenge of a large number of wiring arrangements was overcome by a laminated structure design and processing technology improvement.The flexible,ultrathin,high-density ECoG electrode array was conformably attached to the cortex for reliable,high spatial resolution electrophysiologic recordings.The minimum spacing between electrodes was 15μm,comparable to the diameter of a single neuron.Eight hundred electrodes were prepared with an electrode density of 4444 mm^(-2).In focal epilepsy surgery,the flexible,high-density,laminated ECoG electrode array with 36 electrodes was applied to collect epileptic spike waves inrabbits,improving the positioning accuracy of epilepsy lesions from the centimeter to the submillimeter level.The flexible,high-density,laminated ECoG electrode array has potential clinical applications in intractable epilepsy and other neurologic diseases requiring high-precision electroencephalogram acquisition.

Method to measure tree-ring width,density,elemental composition,and stable carbon and oxygen isotopes using one sample

Tree-ring width(RW),density,elemental com-position,and stable carbon and oxygen isotope(δ^(13)C,δ^(18)O)are widely used as proxies to assess climate change,ecology,and environmental pollution;however,a specific pretreat-ment has been needed for each proxy.Here,we developed a method by which each proxy can be measured in the same sample.First,the sample is polished for ring width meas-urement.After obtaining the ring width data,the sample is cut to form a 1-mm-thick wood plate.The sample is then mounted in a vertical sample holder,and gradually scanned by an X-ray beam.Simultaneously,the count rates of the fluorescent photons of elements(for chemical characteriza-tion)and a radiographic grayscale image(for wood density)are obtained,i.e.the density and the element content are obtained.Then,cellulose is isolated from the 1-mm wood plate by removal of lignin,and hemicellulose.After producing this cellulose plate,cellulose subsamples are separated by knife under the microscope for inter-annual and intra-annual stable carbon and oxygen isotope(δ^(13)C,δ^(18)O)analysis.Based on this method,RW,density,elemental composition,δ^(13)C,and δ^(18)O can be measured from the same sample,which reduces sample amount and treatment time,and is helpful for multi-proxy comparison and combination research.

Multilevel carbon architecture of subnanoscopic silicon for fast‐charging high‐energy‐density lithium‐ion batteries

Silicon(Si)is widely used as a lithium‐ion‐battery anode owing to its high capacity and abundant crustal reserves.However,large volume change upon cycling and poor conductivity of Si cause rapid capacity decay and poor fast‐charging capability limiting its commercial applications.Here,we propose a multilevel carbon architecture with vertical graphene sheets(VGSs)grown on surfaces of subnanoscopically and homogeneously dispersed Si–C composite nanospheres,which are subsequently embedded into a carbon matrix(C/VGSs@Si–C).Subnanoscopic C in the Si–C nanospheres,VGSs,and carbon matrix form a three‐dimensional conductive and robust network,which significantly improves the conductivity and suppresses the volume expansion of Si,thereby boosting charge transport and improving electrode stability.The VGSs with vast exposed edges considerably increase the contact area with the carbon matrix and supply directional transport channels through the entire material,which boosts charge transport.The carbon matrix encapsulates VGSs@Si–C to decrease the specific surface area and increase tap density,thus yielding high first Coulombic efficiency and electrode compaction density.Consequently,C/VGSs@Si–C delivers excellent Li‐ion storage performances under industrial electrode conditions.In particular,the full cells show high energy densities of 603.5 Wh kg^(−1)and 1685.5 Wh L^(−1)at 0.1 C and maintain 80.7%of the energy density at 3 C.

11.2 W/mm power density AlGaN/GaN high electron-mobility transistors on a GaN substrate

In this letter,high power density AlGaN/GaN high electron-mobility transistors(HEMTs)on a freestanding GaN substrate are reported.An asymmetricΓ-shaped 500-nm gate with a field plate of 650 nm is introduced to improve microwave power performance.The breakdown voltage(BV)is increased to more than 200 V for the fabricated device with gate-to-source and gate-to-drain distances of 1.08 and 2.92μm.A record continuous-wave power density of 11.2 W/mm@10 GHz is realized with a drain bias of 70 V.The maximum oscillation frequency(f_(max))and unity current gain cut-off frequency(f_(t))of the AlGaN/GaN HEMTs exceed 30 and 20 GHz,respectively.The results demonstrate the potential of AlGaN/GaN HEMTs on freestanding GaN substrates for microwave power applications.

Robustness of the octupole collectivity in 144Ba within the cranking covariant density functional theory in 3D lattice

The octupole deformation and collectivity in octupole double-magic nucleus 144Ba are investigated using the Cranking covariant density functional theory in a three-dimensional lattice space.The reduced B(E3)transition probability is implemented for the first time in semiclassical approximation based on the microscopically calculated electric octupole moments.The available data,including the I-ωrelation and electric transitional probabilities B(E2)and B(E3)are well reproduced.Furthermore,it is shown that the ground state of 144Ba exhibits axial octupole and quadrupole deformations that persist up to high spins(I≈24h).

Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

Stochastic fractional differential systems are important and useful in the mathematics,physics,and engineering fields.However,the determination of their probabilistic responses is difficult due to their non-Markovian property.The recently developed globally-evolving-based generalized density evolution equation(GE-GDEE),which is a unified partial differential equation(PDE)governing the transient probability density function(PDF)of a generic path-continuous process,including non-Markovian ones,provides a feasible tool to solve this problem.In the paper,the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established.In particular,it is proved that in the GE-GDEE corresponding to the state-quantities of interest,the intrinsic drift coefficient is a time-varying linear function,and can be analytically determined.In this sense,an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original highdimensional linear fractional differential system can be constructed such that their transient PDFs are identical.Specifically,for a multi-dimensional linear fractional differential system,if only one or two quantities are of interest,GE-GDEE is only in one or two dimensions,and the surrogate system would be a one-or two-dimensional linear integer-order system.Several examples are studied to assess the merit of the proposed method.Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems,the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian,and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.

Probability Distribution Characteristics of Strong Nonlinear Waves Under Typhoon Conditions in the Northern South China Sea

The generation and propagation mechanism of strong nonlinear waves in the South China Sea is an essential research area. In this study, the third-generation wave model WAVEWATCH Ⅲ is employed to simulate wave fields under extreme sea states. The model, integrating the ST6 source term, is validated against observed data, demonstrating its credibility. The spatial distribution of the occurrence probability of strong nonlinear waves during typhoons is shown, and the waves in the straits and the northeastern part of the South China Sea show strong nonlinear characteristics. The high-order spectral model HOS-ocean is employed to simulate the random wave surface series beneath five different platform areas. The waves during the typhoon exhibit strong nonlinear characteristics, and freak waves exist. The space-varying probability model is established to describe the short-term probability distribution of nonlinear wave series. The exceedance probability distributions of the wave surface beneath different platform areas are compared and analyzed. The results show that with an increase in the platform area, the probability of a strong nonlinear wave beneath the platform increases.

Probability Laws Derived from the Gamma Function

Several densities or probability laws of continuous random variables derive from the Euler Gamma function. These laws form the basis of sampling theory, namely hypothesis testing and estimation. Namely the gamma, beta, and Student law, through the chi-square law and the normal law are all distributions resulting from applications of Euleur functions.

Electron density measurement by the three boundary channels of HCOOH laser interferometer on the HL-3 tokamak

Far-infrared(FIR)interferometer is widely used to measure the electron density in the magnetically confined fusion plasma devices.A new FIR laser interferometer with a total of 13 channels(8 horizontal channels and 5 oblique channels)is under development on the HL-3tokamak by using the formic-acid laser(HCOOH,f=694 GHz).In order to investigate the boundary electron density activity during the divertor discharge,three horizontal interferometry channels located at Z=-97,-76,76.5 cm have been successfully developed on HL-3 in 2023,and put into operation in recent experimental campaign,with a time resolution of<1.0μs and lineintegrated electron density resolution of~7.0×10^(16) m^(-2).This paper mainly focuses on the optical design of the three-channel interferometry system,as well as optical elements and recent experimental result on HL-3.

The effect of topographic density variations on the geoid and orthometric heights in Hong Kong

Utilizing the adopted average topographic density of 2670 kg/m^(3)in the reduction of gravity anomalies introduces errors attributed to topographic density variations,which consequently affect geoid modeling accuracy.Furthermore,the mean gravity along the plumbline within the topography in the definition of Helmert orthometric heights is computed approximately by applying the Poincar e-Prey gravity reduction where the topographic density variations are disregarded.The Helmert orthometric heights of benchmarks are then affected by errors.These errors could be random or systematic depending on the specific geological setting of the region where the leveling network is physically established and/or the geoid model is determined.An example of systematic errors in orthometric heights can be given for large regions characterized by sediment or volcanic deposits,the density of which is substantially lower than the adopted topographic density used in Helmert's definition of heights.The same applies to geoid modeling errors.In this study,we investigate these errors in the Hong Kong territory,where topographic density is about 20%lower than the density of 2670 kg/m^(3).We use the digital rock density model to estimate the effect of topographic density variations on the geoid and orthometric heights.Our results show that this effect on the geoid and Helmert orthometric heights reach maxima of about 2.1 and 0.5 cm,respectively.Both results provide clear evidence that rock density models are essential in physical geodesy applications involving gravimetric geoid modeling and orthometric height determination despite some criticism that could be raised regarding the reliability of these density models.However,in regions dominated by sedimentary and igneous rocks,the geological information is essential in these applications because topographic densities are substantially lower than the average density of 2670 kg/m^(3),thus introducing large systematic errors in geoid and orthometric heights.

On the Application of Mixed Models of Probability and Convex Set for Time-Variant Reliability Analysis

In time-variant reliability problems,there are a lot of uncertain variables from different sources.Therefore,it is important to consider these uncertainties in engineering.In addition,time-variant reliability problems typically involve a complexmultilevel nested optimization problem,which can result in an enormous amount of computation.To this end,this paper studies the time-variant reliability evaluation of structures with stochastic and bounded uncertainties using a mixed probability and convex set model.In this method,the stochastic process of a limit-state function with mixed uncertain parameters is first discretized and then converted into a timeindependent reliability problem.Further,to solve the double nested optimization problem in hybrid reliability calculation,an efficient iterative scheme is designed in standard uncertainty space to determine the most probable point(MPP).The limit state function is linearized at these points,and an innovative random variable is defined to solve the equivalent static reliability analysis model.The effectiveness of the proposed method is verified by two benchmark numerical examples and a practical engineering problem.

Adaptive Update Distribution Estimation under Probability Byzantine Attack

The secure and normal operation of distributed networks is crucial for accurate parameter estimation.However,distributed networks are frequently susceptible to Byzantine attacks.Considering real-life scenarios,this paper investigates a probability Byzantine(PB)attack,utilizing a Bernoulli distribution to simulate the attack probability.Historically,additional detection mechanisms are used to mitigate such attacks,leading to increased energy consumption and burdens on distributed nodes,consequently diminishing operational efficiency.Differing from these approaches,an adaptive updating distributed estimation algorithm is proposed to mitigate the impact of PB attacks.In the proposed algorithm,a penalty strategy is initially incorporated during data updates to weaken the influence of the attack.Subsequently,an adaptive fusion weight is employed during data fusion to merge the estimations.Additionally,the reason why this penalty term weakens the attack has been analyzed,and the performance of the proposed algorithm is validated through simulation experiments.