Representation of three-dimensional mass distribution of the Earth's interior by biorthogonal series and its use for studying internal structure of the planet

This paper utilizes the mathematical concept of approximation within an ellipsoid from a single viewpoint to present the spatial mass distribution function of the Earth's interior and its internal potential.The primary focus lies in constructing the volume distribution of masses in the planet's interior, with the expansion coefficients being linear combinations of the Stokes constants. Several possible approaches are suggested for determining accurately these coefficients employing three-dimensional(biorthogonal)polynomials. By expressing the mass distribution function of the Earth's interior and its internal potential as a series, an algorithm is introduced for the calculation of gravitational energy. It allows us to estimate fluctuations in gravitational energy. The implementation of this algorithm offers the means of establishing the extent to which the Earth deviates from a state of hydrostatic equilibrium as a celestial body.Due to the aforementioned method, calculations have been conducted to validate its effectiveness and reliability. This example is given as an illustration of a given method for studying the internal structure of planets.

Pairing effects on the fragment mass distribution of Th,U,Pu,and Cm isotopes

In this article,a comprehensive study of the fission process of Th,U,Pu,and Cm isotopes using a Yukawa-folded meanfield plus standard pairing model is presented.The study focused on analyzing the effects of the pairing interaction on the fragment mass distribution and its dependence on nuclear elongation.The significant role of pairing interactions in the fragment mass distributions of^(230)Th,^(234)U,^(240)Pu,and^(246)Cm was demonstrated.Numerical analysis revealed that increasing the pairing interaction strength decreased the asymmetric fragment mass distribution and increased the symmetric distribution.Furthermore,the odd-even mass differences at symmetric and asymmetric fission points were examined,highlighting their sensitivity to changes in the pairing interaction strength.Systematic analysis of the Th,U,Pu,and Cm isotope fragment mass distributions demonstrated the effectiveness of the model in reproducing the experimental data.In addition,the effects of the zero-point energy and half-width parameter on the fragment mass distribution for^(240)Pu were explored.Thus,this study provides valuable insights into the fission process by emphasizing the importance of pairing interactions and their relationship with nuclear elongation.

Measuring the flexibility matrix of an eagle’s flight feather and a method to estimate the stiffness distribution

Flight feathers stand out with extraordinary mechanical properties for flight because they are lightweight but stiff enough.Their elasticity has great effects on the aerodynamics, resulting in aeroelasticity.Our primary task is to figure out the stiffness distribution of the feather to study the aeroelastic effects.The feather shaft is simplified as a beam, and the flexibility matrix of an eagle flight feather is tested.A numerical method is proposed to estimate the stiffness distributions along the shaft length based on an optimal Broyden–Fletcher–Goldfarb–Shanno(BFGS) method with global convergence.An analysis of the compressive behavior of the shaft based on the beam model shows a good fit with experimental results.The stiffness distribution of the shaft is finally presented using a 5 th order polynomial.

Force Analysis of the Overconstrained Mechanisms Based on Equivalent Stiffness Considering Limb Axial Deformation

Overconstrained mechanism has the advantages of large bearing capacity and high motion reliability,but its force analysis is complex and difficult because the mechanism system contains overconstraints.Considering the limb axial deformation,taking typical 2SS+P and 7-SS passive overconstrained mechanisms,2SPS+P and 7-SPS active overconstrained mechanisms,and 2SPS+P and 7-SPS passive-input overconstrained mechanisms as examples,a new force analysis method based on the idea of equivalent stiffness is proposed.The equivalent stiffness matrix of passive overconstrained mechanism is derived by combining the force balance and deformation compatibility equations with consideration of axial elastic limb deformations.The relationship between the constraint wrench magnitudes and the external force,limb stiffness is established.The equivalent stiffness matrix of active overconstrained mechanism is derived by combining the force balance and displacement compatibility equations.Here,the relationship between the magnitudes of the actuated wrenches and the external force,limb stiffness is investigated.Combining with the equivalent stiffness of the passive overconstrained mechanism,an analytical relationship between the actuated forces of passive-input overconstrained mechanism and the output displacement,limb stiffness is explored.Finally,adaptability of the equivalent stiffness to overconstrained mechanisms is discussed,and the effect of the limb stiffness on overconstrained mechanisms force distribution is revealed.The research results provide a theoretical reference for the design,research and practical application of overconstrained mechanism.

Seasonal Variations of Number Size Distributions and Mass Concentrations of Atmospheric Particles in Beijing

Particle number and mass concentrations were measured in Beijing during the winter and summer periods in 2003, together with some other parameters including black carbon (BC) and meteorological conditions. Particle mass concentrations exhibited low seasonality, and the ratio of PM2.5/PM10 in winter was higher than that in summer. Particle number size distribution (PSD) was characterized by four modes and exhibited low seasonality. BC was well correlated with the number and mass concentrations of accumulation and coarse particles, indicating these size particles are related to anthropogenic activities. Particle mass and number concentrations (except ultra-fine and nucleation particles) followed well the trends of BC concentration for the majority of the day, indicating that most particles were associated with primary emissions. The diurnal number distributions of accumulation and coarse mode particles were characterized by two peaks.

Sand particle lift-off velocity measurements and numerical simulation of mass flux distributions in a wind tunnel

Lift-off velocity of saltating sand particles in wind-blown sand located at 1.0 mm above the sand bed surface was measured using a phase Doppler particle analyzer in a wind tunnel. The results show that the probability distribution of lift-off velocity can be expressed as a lognormal function, while that of lift-off angle follows an exponential function. The probability distribution of lift-off angle conditioned for each lift-off velocity also follows an exponential function, with a slope that becomes steeper with increasing lift-off velocity. This implies that the probability distribution of lift-off velocity is strongly dependent on the lift-off angle. However, these lift-off parameters are generally treated as an independent joint probability distribution in the literature. Numerical simulations were carried out to investigate the effects of conditional versus independent joint probability distributions on the vertical sand mass flux distribution. The simulation results derived from the conditional joint probability distribution agree much better with experimental data than those from the independent ones. Thus, it is better to describe the lift-off velocity of saltating sand particles using the conditional joint probability distribution. These results improve our understanding of saltation processes in wind-blown sand.

Vertical distribution characteristics of dust aerosol mass concentration in the Taklimakan Desert hinterland

The different height mass concentrations of dust aerosol data from the atmosphere environment observation station （Ta- zhong Station） was continuously observed by instruments of Grimm 1.108, Thermo RP 1400a and TSP from January of 2009 to February of 2010 in the Taklimakan Desert hinterland. Results show that： （1） The mass concentration value of 80 m PMl0 was higher, but PM2.5 and PM1.0 concentrations at 80 m was obviously lower than 4 m PMl0, and the value of 80 m PM1.0 mass concentration was the lowest. （2） The PM mass concentrations gradually decreased from night to sunrise, with the lowest concentration at 08：00, with the mass concentration gradually increased, up to the highest concentration around 18：00, and then decreased again. It was exactly the same with the changes of wind speed. （3） The high monthly average mass concentration of TSP mainly appeared from March to September, and the highest concentration was in April and May, subsequently gradually decreased. Also, March-September was a period with high value area of PM monthly average mass concentration, with the highest monthly average mass concentration of 846.0 p.g/m3 for 4 m PM^0 appeared in May. The concentration of PM10 was much higher than those of PM2.5 and PM1.0 at 80 m. There is a small difference between the concentration of PM2.5 and PM~ 0. Dust weather was the main factor which influenced the concentration content of the different diameter dust aerosol, and the more dust weather days, the higher content of coarse particle, conversely, fine particle was more. （4） The mass concentration of different diameter aerosols had the following sequence during dust weather： clear day 〈 blowing dust 〈 floating and blowing dust 〈 sandstorm. In different dust weather, the value of PM^o/TSP in fine weather was higher than that in floating weather, and much higher than those in blowing dust and sandstorm weather. （5） During the dust weather process, dust aerosol concentration gradually decreased with particle size decreasing. The dust aerosol mass concentration at different heights and diameter would have a peak value area every 3-4 days according to the strengthening process of dust weather.

Molar Mass Distribution and Chain Conformation of Polysaccharides from Fortunella margarita(Lour.) Swingle

In order to study the effects of different extraction methods on the molar mass distribution and chain conformation of Fortunella margarita （Lour.） Swingle polysaccharides （FP）, we used extraction by hot water （WFP）, ultrasonic-assisted treatment （UFP）, microwave-assisted treatment （MFP） and ultrasonic/microwave-assisted treatment （UMFP）, and then Fourier transform infrared （FT-IR） spectroscopy as well as 1H and 13C nuclear magnetic resonance （NMR） spectroscopy to characterize the structural properties of FP extracted. The molar weight （Mw）, polydispersity index （Mw/Mn）, root-mean-square （RMS） turning radius （Rg）, molar mass distribution and chain conformation of FP were studied systematically using size-exclusion chromatography （SEC）, multi-angle laser light-scattering （MALLS） and refractive index （RI）. WFP, UFP, MFP and UMFP are all typical carbohydrates according to 1H NMR, laC NMR and FT-IR measurements. The type of glycosidic linkage is mainly a fl-glycosidic bond with a small amount of a-glycosidic bond. The results obtained by the SEC-MALLS-RI system showed the molar masses of WFP and UMFP were distributed mainly in the range of 5.0x10^6-1.0x10^7 g-mo1-1 and they accounted for 57.80% and 56.84% of total FP, respectively. The molar masses of UFP and MFP were distributed mainly in the 1.0x10^6-5.0x10^6 g.mol-1 range, which accounted for 38.24% and 52.39% of FP, respectively. WFP and UMFP in water were uniform spherical polymers; UFP and MFP were typical highly branched polymers and the degree of branching for MFP was higher compared to UFP. These results indicated the ultrasonic- and microwave-assisted extraction methods caused a significant decrease of the molar mass of FP but the ultrasonic/microwave synergistic extraction method had no effect.

Connection Between Liquid Distribution and Gas-Liquid Mass Transfer in Monolithic Bed

With a particular focus on the connection between liquid flow distribution and gas-liquid mass transfer in monolithic beds in the Taylor flow regime, hydrodynamic and gas-liquid mass transfer experiments were carriedout in a column with a monolithic bed of cell density of 50 cpsi with trio different distributors （nozzle and packed bed distributors）. Liquid saturation in individual channels was measured by using self-made micro-conductivity probes. A mal-distribution factor was used to evaluate uniform degree of phase distribution in monoliths. Overall bed pressure drop and mass transfer coefficients were measured. For liquid flow distribution and gas-liquid masstransfer, it is found that the superficial liquid velocity is a crucial factor and the packed bed distributor is better than the nozzle distributor. A semi-theoretical analysis using single channel models shows that the packed bed distributor always yields shorter and uniformly distributed liquid slugs compared to the nozzle distributor, which in turn ensures a better mass transfer performance. A bed scale mass transfer model is proposed by employing the single channel models in individual channels and incorporating effects of non-uniform liquid distribution along the bedcross-section. The model predicts the overall gas-liquid mass transfer coefficient wig a relative error within ＋30%.

Configurational information entropy analysis of fragment mass cross distributions to determine the neutron skin thickness of projectile nuclei

Configurational information entropy(CIE)analysis has been shown to be applicable for determining the neutron skin thickness(δnp)of neutron-rich nuclei from fragment production in projectile fragmentation reactions.The BNN+FRACS machine learning model was adopted to predict the fragment mass cross-sections(σ_(A))of the projectile fragmentation reactions induced by calcium isotopes from ^(36)Ca to ^(56)Ca on a ^(9)Be target at 140MeV/u.The fast Fourier transform was adopted to decompose the possible information compositions inσA distributions and determine the quantity of CIE(S_(A)[f]).It was found that the range of fragments significantly influences the quantity of S_(A)[f],which results in different trends of S_(A)[f]~δnp correlation.The linear S_(A)[f]~δnp correlation in a previous study[Nucl.Sci.Tech.33,6(2022)]could be reproduced using fragments with relatively large mass fragments,which verifies that S_(A)[f]determined from fragmentσAis sensitive to the neutron skin thickness of neutron-rich isotopes.

Micro sliding friction model considering periodic variation stress distribution of contact surface and experimental verification

Micro sliding phenomenon widely exists in the operation process of mechanical systems,and the micro sliding friction mechanism is always a research hotspot.In this work,based on the total reflection method,a measuring device for interface contact behavior under two-dimensional(2D)vibration is built.The stress distribution is characterized by the light intensity distribution of the contact image,and the interface contact behavior in the 2D vibration process is studied.It is found that the vibration angle of the normal direction of the contact surface and its fluctuation affect the interface friction coefficient,the tangential stiffness,and the fluctuation amplitude of the stress distribution.Then they will affect the change of friction state and energy dissipation in the process of micro sliding.Further,an improved micro sliding friction model is proposed based on the experimental analysis,with the nonlinear change of contact parameters caused by the normal contact stress distribution fluctuation taken into account.This model considers the interface tangential stiffness fluctuation,friction coefficient hysteresis,and stress distribution fluctuation,whose simulation results are consistent well with the experimental results.It is found that considering the nonlinear effect of a certain contact parameter alone may bring a greater error to the prediction of friction behavior.Only by integrating multiple contact parameters can the accuracy of friction prediction is improved.

Hospital-Based Phenotypic Features and Treatment Outcomes of Chinese Women with Polycystic Ovary Syndrome: The Effect of Body Mass Index and Geographic Distribution

Genetic,lifestyle,and environmental factors contribute to the etiology of polycystic ovary syndrome(PCOS).Increased body mass index(BMI)exacerbates the reproductive and metabolic parameters and reduces the fecundity of women with PCOS.This is a secondary analysis of a large-sample,multicenter,randomized controlled trial conducted at 21 sites in Chinese mainland.A total of 1000 women diagnosed with PCOS were enrolled in this trial.Of these,998 women with PCOS were included in the analysis.Increased BMI was associated with more severe menstrual irregularities,elevated testosterone level,higher prevalence of metabolic syndrome,and poorer quality of life.The rates of ovulation per woman for the normal,overweight,and obese BMI groups were 83.0%,78.2%,and 63.6%,respectively(P<0.001),and the rates of live birth were 23.6%,18.1%,and 15.3%(P=0.030).Northern PCOS patients showed more severe reproductive,glucose,and lipid profiles;less exercise;and lower total ovulation rates compared with PCOS patients from Southern China(74.8%vs 81.2%,absolute difference 6.4%,95%confidence interval 1.2%–11.5%).The results show the typical phenotypic features of Han women with PCOS in Northern and Southern China.The women living in Northern China showed a higher BMI,more severe glycolipid metabolism profiles,and subsequently worse clinical outcomes by the same interventions than those living in Southern China.The difference in phenotypic features can be explained mostly by differences in BMI and the resulting difference in ovulation.

Applicability of law of mass action to distribution of manganese between slag melts and liquid iron

According to the law of mass action and the coexistence theory of slag structure, the distribution of manganese between MnO FeO SiO 2 and MgO MnO FeO SiO 2 slag melts as well as liquid iron was analyzed. It is shown that K ′ Mn and K MnO are only dependent on temperature and don’t change with basicities and compositions of slag melts. So the distribution of manganese between the above mentioned slag melts and molten iron obeys the law of mass action. But analysis of experimental results from other sources shows that K ′ Mn and K MnO really change with basicities of slag, which is probably arisen from not approaching equilibrium under low basicity slag melts.

Preliminary analysis of distribution and variation of perennialmonthly mean water masses in the Bohai Sea,the Huanghai Sea and the East China Sea

On the basis of perennial monthly mean temperature and salinity data, the classification of monthly water masses at the surface and the bottom in the Bohai Sea, the Huanghai Sea and the East China Sea, has been made by using the method of fuzzy cluster from the modified characteristic of water masses in the shallow water area. In this paper, the basic features, growth and decline patterns of water masses in relation to fishing grounds in the whole shelves of the Bohai Sea, the Huanghai Sea and the East China Sea are discussed with emphasis.

Exact Statistical Distribution of the Body Mass Index (BMI): Analysis and Experimental Confirmation

Body Mass Index (BMI), defined as the ratio of individual mass (in kilograms) to the square of the associated height (in meters), is one of the most widely discussed and utilized risk factors in medicine and public health, given the increasing obesity worldwide and its relation to metabolic disease. Statistically, BMI is a composite random variable, since human weight (converted to mass) and height are themselves random variables. Much effort over the years has gone into attempts to model or approximate the BMI distribution function. This paper derives the mathematically exact BMI probability density function (PDF), as well as the exact bivariate PDF for human weight and height. Taken together, weight and height are shown to be correlated bivariate lognormal variables whose marginal distributions are each lognormal in form. The mean and variance of each marginal distribution, together with the linear correlation coefficient of the two distributions, provide 5 nonadjustable parameters for a given population that uniquely determine the corresponding BMI distribution, which is also shown to be lognormal in form. The theoretical analysis is tested experimentally by gender against a large anthropometric data base, and found to predict with near perfection the profile of the empirical BMI distribution and, to great accuracy, individual statistics including mean, variance, skewness, kurtosis, and correlation. Beyond solving a longstanding statistical problem, the significance of these findings is that, with knowledge of the exact BMI distribution functions for diverse populations, medical and public health professionals can then make better informed statistical inferences regarding BMI and public health policies to reduce obesity.

CORRELATION BETWEEN HYSTERESIS OF GAS-LIQUID MASS TRANSFER AND LIQUID DISTRIBUTION IN A TRICKLE BED

The hysteresis of gas-liquid mass transfer rate and the corresponding radial liquiddistribution in a trickle bed reactor are measured to provide evidence for the correlation between thesetwo behaviors.Experimental results indicate that the hysteresis of gas-liquid mass transfer originatesfrom the nonuniformity of the hydrodynamic state of gas-liquid flow and the radial maldistributionof local kgia corresponds very well to the radial maldistribution of liquid flow in the bed.The localliquid flow rate is also found to be nonuniform in the azimuthal direction.In view of maldistributedliquid flow even in the pulsing flow regime,the conventional plug flow model seems oversimplifiedfor describing the behavior of a trickle bed.

ELASTO-PLASTIC STIFFNESS MATRIX OF ROCK MASS ELEMENT WITH SOFT CLAY STRATA AND ITS APPLICATION

The fault element is used to handle soft clay strata in a rock mass.The formulas or clasto-plastic stiffnessmatrix for the fault element are derived using the constitutive relationship between plastic increment stress andstrain.A numerical example of a circular tunnel with soft clay strata in the rock medium are examined.

Mass-Stiffness Templates for Cubic Structural Elements Dedicated to Professor Karl Stark Pister for his 95th birthday

1 This paper considers Lagrangian finite elements for structural dynamics constructed with cubic displacement shape functions.The method of templates is used to investigate the construction of accurate mass-stiffness pairs.This method introduces free parameters that can be adjusted to customize elements according to accuracy and rank-sufficiency criteria.One-and two-dimensional Lagrangian cubic elements with only translational degrees of freedom(DOF)carry two additional nodes on each side,herein called side nodes or SN.Although usually placed at the third-points,the SN location may be adjusted within geometric limits.The adjustment effect is studied in detail using symbolic computations for a bar element.The best SN location is taken to be that producing accurate approximation to the lowest natural frequencies of the continuum model.Optimality is investigated through Fourier analysis of the propagation of plane waves over a regular infinite lattice of bar elements.Focus is placed on the acoustic branch of the frequency-vs.-wavenumber dispersion diagram.It is found that dispersion results using the fully integrated consistent mass matrix(CMM)are independent of the SN location whereas its lowfrequency accuracy order is O(κ8),whereκis the dimensionless wave number.For the diagonally lumped mass matrix(DLMM)constructed through the HRZ scheme,two optimal SN locations are identified,both away from third-points and of accuracy order O(κ8).That with the smallest error coefficient corresponds to the Lobatto 4-point integration rule.A special linear combination of CMM and DLMM with nodes at the Lobatto points yields an accuracy of O(κ10)without any increase in the computational effort over CMM.The effect of reduced integration(RI)on both mass and stiffness matrices is also studied.It is shown that singular mass matrices can be constructed with 2-and 3-point RI rules that display the same optimal accuracy of the exactly integrated case,at the cost of introducing spurious modes.The optimal SN location in two-dimensional,bicubic,isoparametric plane stress quadrilateral elements is briefly investigated by numerical experiments.The frequency accuracy of flexural modes is found to be fairly insensitive to that position,whereas for bar-like modes it agrees with the one-dimensional results.

Contrasting vertical distribution between prokaryotes and fungi in different water masses on the Ninety-East Ridge, Southern Indian Ocean

Although the microbial diversity of the Indian Ocean has been extensively investigated,little is known about the community composition of microbes in the Southern Indian Ocean.In the present study,we divided 60 water column samples on the Ninety-East Ridge(NER)into fi ve water masses according to the temperature-salinity curves.We presented,for the fi rst time,a full description of the microbial biodiversity on NER through high-throughput amplicon sequencing approach,including bacteria,archaea,and fungi.We found that bacteria exhibited higher richness and diversity than archaea and fungi across the water masses on NER.More importantly,each water mass on NER featured distinct prokaryotic microbial communities,as indicated by the results of non-metric multidimensional scaling.In contrast,fungi were eurybathic across the water masses.Redundancy analysis results demonstrated that environmental factors might play a pivotal role in the formation and stability of prokaryotic communities in each water mass,especially that of archaea.In addition,indicator species might be used as fi ngerprints to identify corresponding water masses on NER.These results provide new insights into the vertical distribution,structure,and diversity of microorganisms on NER from the perspective of water mass.

Distribution of Mass and Energy in Five General Cosmic Models

Distributions of the universe horizon distance and universe horizon volume were investigated in the light of five general cosmic models which were constructed in a previous study. Both distributions increase so slowly up to t ≈ 21.5444 Myr, then they start raising very fast up to t ≈ 60 Gyr. Afterwards, they increase again very slowly until t ≈ 124 Gyr. Distributions of mass of radiation, matter and dark energy within the horizon volume of the universe were also studied in the five general cosmic models. The masses of both radiation and matter decrease gradually with time while the mass of dark energy increases. The mass of radiation prevailed in the early universe up to t ≈ 34627.5 - 55916.2 yr, where it becomes equal to the mass of matter. Then the mass of matter dominated until t ≈ 9.4525 - 10.0632 Gyr, where it becomes equal to the mass of dark energy. Thenceforward, the mass of dark energy prevails the universe. The cosmic space becomes approximately matter empty in the so far future of the universe.