MULTIFRACTAL ANALYSIS OF CONVERGENCE EXPONENTS FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

For each real number x∈(0,1),let[a_(1)(x),a_(2)(x),…,a_n(x),…]denote its continued fraction expansion.We study the convergence exponent defined byτ(x)=inf{s≥0:∞∑n=1(a_(n)(x)a_(n+1)(x))^(-s)<∞},which reflects the growth rate of the product of two consecutive partial quotients.As a main result,the Hausdorff dimensions of the level sets ofτ(x)are determined.

Effects of long-term partial substitution of inorganic fertilizer with pig manure and/or straw on nitrogen fractions and microbiological properties in greenhouse vegetable soils

Partial substitution of inorganic fertilizers with organic amendments is an important agricultural management practice.An 11-year field experiment(22 cropping periods)was carried out to analyze the impacts of different partial substitution treatments on crop yields and the transformation of nitrogen fractions in greenhouse vegetable soil.Four treatments with equal N,P_(2)O_(5),and K_(2)O inputs were selected,including complete inorganic fertilizer N(CN),50%inorganic fertilizer N plus 50%pig manure N(CPN),50%inorganic fertilizer N plus 25%pig manure N and 25%corn straw N(CPSN),and 50%inorganic fertilizer N plus 50%corn straw N(CSN).Organic substitution treatments tended to increase crop yields since the 6th cropping period compared to the CN treatment.From the 8th to the 22nd cropping periods,the highest yields were observed in the CPSN treatment where yields were 7.5-11.1%greater than in CN treatment.After 11-year fertilization,compared to CN,organic substitution treatments significantly increased the concentrations of NO_(3)^(-)-N,NH_(4)^(+)-N,acid hydrolysis ammonium-N(AHAN),amino acid-N(AAN),amino sugar-N(ASN),and acid hydrolysis unknown-N(AHUN)in soil by 45.0-69.4,32.8-58.1,49.3-66.6,62.0-69.5,34.5-100.3,and 109.2-172.9%,respectively.Redundancy analysis indicated that soil C/N and OC concentration significantly affected the distribution of N fractions.The highest concentrations of NO_(3)^(-)-N,AHAN,AAN,AHUN were found in the CPSN treatment.Organic substitution treatments increased the activities ofβ-glucosidase,β-cellobiosidase,N-acetyl-glucosamidase,L-aminopeptidase,and phosphatase in the soil.Organic substitution treatments reduced vector length and increased vector angle,indicating alleviation of constraints of C and N on soil microorganisms.Organic substitution treatments increased the total concentrations of phospholipid fatty acids(PLFAs)in the soil by 109.9-205.3%,and increased the relative abundance of G^(+)bacteria and fungi taxa,but decreased the relative abundance of G-bacteria,total bacteria,and actinomycetes.Overall,long-term organic substitution management increased soil OC concentration,C/N,and the microbial population,the latter in turn positively influenced soil enzyme activity.Enhanced microorganism numbers and enzyme activity enhanced soil N sequestration by transforming inorganic N to acid hydrolysis-N(AHN),and enhanced soil N supply capacity by activating non-acid hydrolysis-N(NAHN)to AHN,thus improving vegetable yield.Application of inorganic fertilizer,manure,and straw was a more effective fertilization model for achieving sustainable greenhouse vegetable production than application of inorganic fertilizer alone.

Key Physical Factors Affecting Spatial-temporal Variation of Labile Organic Carbon Fractions by Biochar Driven in Mollisols Region of Northeast China

Biochar is widely used to improve soil physical properties and carbon sequestration. However, few studies focuse on the impact of maize stalk biochar on labile organic carbon(LOC) pool and the relationship between physical properties and LOC fractions. A field positioning experiment was performed in Mollisols region of Northeast China to evaluate the influence of maize stalk biochar on the spatial distribution and temporal changes of physical properties and LOC fractions. Maize stalk biochar treatments included C1(1.5 kg·hm^(-2)), C2(3 kg·hm^(-2)), C3(15 kg·hm^(-2)), C4(30 kg·hm^(-2)), and CK(0). The results showed that maize stalk biochar increased soil water contents(SWC) and soil porosity(SP), but reduced bulk density(BD). Maize stalk biochar reduced dissolved organic carbon(DOC) contents in the 0-20 cm soil layer, ranging from 0.25 g·kg^(-1) to 0.31 g·kg^(-1) in harvest period, while increased in the 20-40 cm soil layer. In addition, the application of biochar had a significant impact on the spatial distribution and temporal change of SWC, BD, SP, DOC, hot-water extractable carbon(HWC), acid hydrolyzed organic carbon(AHC Ⅰ, Ⅱ), and readily oxidized organic carbon(ROC). High amounts of maize stalk biochar up-regulated the contents of soil organic carbon SOC, HWC, AHC Ⅰ, AHC Ⅱ, and ROC. In addition, SWC and SP were the key physical factors to affect LOC fractions. In conclusions, maize stalk biochar could improve physical properties, and then influence LOC fractions, and maize stalk biochar could be used as an organic amendment for restoring degraded soils governed by their rates of addition.

Deadwood affects the soil organic matter fractions and enzyme activity of soils in altitude gradient of temperate forests

The main objective of our study has been to determine the role of deadwood in the shaping of the amount of soil organic matter fractions in mountain forest soils.For this purpose,a climosequence approach comprising north(N)and south(S)exposure along the altitudinal gradient(600,800,1000 and 1200 m a.s.l.)was set up.By comparing the properties of decomposing deadwood and those of the soils located directly beneath the decaying wood we drew conclusions about the role of deadwood in the shaping of soil organic matter fractions and soil carbon storage in different climate conditions.The basic properties,enzymatic activity and fractions of soil organic matter(SOM)were determined in deadwood and affected directly by the components released from decaying wood.Heavily decomposed deadwood impacts soil organic matter stabilization more strongly than the less decayed deadwood and the light fraction of SOM is more sensitive to deadwood effects than the heavy fraction regardless of the location in the altitude gradient.Increase in SOM mineral-associated fraction C content is more pronounced in soils under the influence of deadwood located in lower locations of warmer exposure.Nutrients released from decaying wood stimulate the enzymatic activity of soils that are within the range of deadwood influence.

Effects of phosphorus fertilizer application rate on transformation processes of phosphorus fractions in the purple alluvial soil of a riparian zone

Effects of phosphorus(P)fertilizer application rate on soil transformation processes of P fraction are still unclear in the riparian zone.Purple alluvial soils in the riparian zone of the Three Gorges Reservoir were collected to conduct a 21-day incubation executed by two hydrological environments(drying and flooding)and ten application rates of P fertilizer.Transformation percentages of P fertilizer(TPPF)were calculated as content differences of soil P fractions between fertilizer addition and none fertilizer addition divided by soil total P increases caused by fertilizer addition.TPPF to inorganic P extracted by sodium hydroxide(NaOH-Pi)and hydrochloric acid(HCl-Pi)increase by 20.91%(9.71%)and 24.26%(40.72%)under the drying(flooding)environment.Instead,TPPF to the other fractions decrease.Phosphorus fertilizer input mainly has indirect positive and negative effects on organic P via precipitated P under the drying and flooding environments and finally has indirect positive effects on labile P(p<0.001).Percentage changes of water-soluble inorganic P(H2O-Pi)and HCl-Pi under the flooding environment are higher than that under the drying environment,and percentage changes of organic P extracted by sodium hydrogen carbonate(Na HCO3-Po)and NaOH-Pi show an opposite trend(p<0.01).?(differences in soil P fraction content between flooding and drying incubations)H2O-Pi is negatively correlated with?NaHCO3-Po,and?NaHCO3-Po is positively correlated with?NaOH-Pi(p<0.001).In conclusion,P fertilizer is transformed more into precipitated P than into other P fractions with an application rate increase.Phosphorus fertilizer input mainly increases organic P via precipitated P under the drying environment and decreases organic P via precipitated P under the flooding environment,and organic P is further transformed into labile P.With P fertilizer input,P release caused by flooding is derived from NaHCO3-Po release triggered by NaOH-Pi release.The results can be helpful for the understanding of P fertilizer migration processes from the riparian zone soil to the Three Gorges Reservoir under rain leaching and flooding.

Interaction between systemic iron parameters and left ventricular structure and function in the preserved ejection fraction population:a two-sample bidirectional Mendelian randomization study

BACKGROUND Left ventricular(LV)remodeling and diastolic function in people with heart failure(HF)are correlated with iron status;however,the causality is uncertain.This Mendelian randomization(MR)study investigated the bidirectional causal relationship between systemic iron parameters and LV structure and function in a preserved ejection fraction population.METHODS Transferrin saturation(TSAT),total iron binding capacity(TIBC),and serum iron and ferritin levels were extracted as instrumental variables for iron parameters from meta-analyses of public genome-wide association studies.Individuals without myocardial infarction history,HF,or LV ejection fraction(LVEF)<50%(n=16,923)in the UK Biobank Cardiovascular Magnetic Resonance Imaging Study constituted the outcome dataset.The dataset included LV end-diastolic volume,LV endsystolic volume,LV mass(LVM),and LVM-to-end-diastolic volume ratio(LVMVR).We used a two-sample bidirectional MR study with inverse variance weighting(IVW)as the primary analysis method and estimation methods using different algorithms to improve the robustness of the results.RESULTS In the IVW analysis,one standard deviation(SD)increased in TSAT significantly correlated with decreased LVMVR(β=-0.1365;95%confidence interval[CI]:-0.2092 to-0.0638;P=0.0002)after Bonferroni adjustment.Conversely,no significant relationships were observed between other iron and LV parameters.After Bonferroni correction,reverse MR analysis showed that one SD increase in LVEF significantly correlated with decreased TSAT(β=-0.0699;95%CI:-0.1087 to-0.0311;P=0.0004).No heterogeneity or pleiotropic effects evidence was observed in the analysis.CONCLUSIONS We demonstrated a causal relationship between TSAT and LV remodeling and function in a preserved ejection fraction population.

Comparative pyrolysis kinetics of heavy oil and its SARA fractions using distributed activation energy model

The pyrolysis behaviors of Qingdao vacuum residue(QD-VR)and its SARA(saturates,aromatics,reins,and asphaltenes)fractions were evaluated by thermo-gravimetric with mass spectrometer(TG-MS).The pyrolysis kinetics were determined by Friedman,one-parallel and four-parallel distributed activation energy model(DAEM),respectively.The results indicated that the pyrolysis behavior of QD-VR was similar to that of aromatics.For saturates,the release of H_(2),CH_(4),CO,and CO_(2) occurred in 80-400℃,while the temperature range for QD-VR and other fractions is 200-800℃.The average activation energy(Ea)via Friedman method was 179.72 kJ/mol and increased with the conversion ratio.One-parallel Gaussian DAEM was more suitable to describe the pyrolysis process of the single SARA fractions,while four-parallel Gaussian DAEM was more suitable to describe the pyrolysis process of heavy oil.Furthermore,comparing the weighted E_(a) from one-parallel(227.64 kJ/mol)and four-parallel Gaussian DAEM(204.63 kJ/mol),the results suggested that during pyrolysis process of heavy oil,there was an interaction between the SARA fractions,which could reduce the E_(a) of heavy oil pyrolysis.Specifically,during heavy oil pyrolysis,resins and asphaltenes could increase the E_(a) of saturates and aromatics,while saturates and aromatics could decrease the E_(a) of resins and asphaltenes.

Model reduction of fractional impedance spectra for time–frequency analysis of batteries, fuel cells, and supercapacitors

Joint time–frequency analysis is an emerging method for interpreting the underlying physics in fuel cells,batteries,and supercapacitors.To increase the reliability of time–frequency analysis,a theoretical correlation between frequency-domain stationary analysis and time-domain transient analysis is urgently required.The present work formularizes a thorough model reduction of fractional impedance spectra for electrochemical energy devices involving not only the model reduction from fractional-order models to integer-order models and from high-to low-order RC circuits but also insight into the evolution of the characteristic time constants during the whole reduction process.The following work has been carried out:(i)the model-reduction theory is addressed for typical Warburg elements and RC circuits based on the continued fraction expansion theory and the response error minimization technique,respectively;(ii)the order effect on the model reduction of typical Warburg elements is quantitatively evaluated by time–frequency analysis;(iii)the results of time–frequency analysis are confirmed to be useful to determine the reduction order in terms of the kinetic information needed to be captured;and(iv)the results of time–frequency analysis are validated for the model reduction of fractional impedance spectra for lithium-ion batteries,supercapacitors,and solid oxide fuel cells.In turn,the numerical validation has demonstrated the powerful function of the joint time–frequency analysis.The thorough model reduction of fractional impedance spectra addressed in the present work not only clarifies the relationship between time-domain transient analysis and frequency-domain stationary analysis but also enhances the reliability of the joint time–frequency analysis for electrochemical energy devices.

Pectin fractions extracted sequentially from Cerasus humilis:Their compositions,structures,functional properties and antioxidant activities

Three pectin fractions(water-soluble fraction(WSF),chelator-soluble fraction(CSF),and sodium carbonatesoluble fraction(NSF))were obtained from Chinese dwarf cherry(Cerasus humilis)fruits.All of them were branched low methoxylated pectins with an amorphous or partially nanocrystalline nature and eight neutral monosaccharides(arabinose and galactose were most abundant).WSF,CSF and NSF had a degree of methylation(DM)of 35.82%,14.85%and 7.13%,uronic acid(UA)content of 76.02%,83.71%and 69.01%,and total protein content of 2.4%,2.1%and 8.8%,respectively.Their molecular weights were 340.31,330.16 and 141.31 kg/mol,respectively(analyzed by gel permeation chromatography(GPC)).WSF,CSF and NSF exhibited good rheological,thermal,emulsifying,emulsion-stabilizing,water-adsorbing,oil-binding,cholesterol-binding and antioxidant properties.NSF had the highest emulsifying,emulsion stabilizing,water-/oil-/cholesterol-binding and antioxidant capacities,followed by CSF.NSF had the highest viscosity(406.77 m Pa·s),flowability,and resistance to heat-induced changes/damage,which may be related to its lowest polydispersity index,DM and UA content and highest protein content.The three pectin fractions with desirable characteristics can be used as food additives/ingredients and dietary supplements.

Determination of Atomic Fractions of Isotopes Carbon-13 and Nitrogen-15 Directly in Glicine, L-Leucine, Isoleucine and Alanine

Using compounds modified by the isotopes carbon-13 and nitrogen-15 helps conduct research in various fields of science, such as medicine, pharmacology, pharmacokinetics, metabolism, agriculture, and others. In the case of the availability of reliable, express, and cheap methods, the area of their use will gradually expand. A determination of the atomic fraction of the isotopes carbon-13 and nitrogen-15 directly in glycine, leucine, isoleucine, and alanine is proposed;the modification concerns all centers or one or more identical carbon and nitrogen centers separately, as well as both isotopes at the same time. There are defined mass lines of the mass spectrum of each amino acid, through which the isotopic content of carbon and nitrogen is calculated. The processes that must be taken into account for the determination of the isotopic content are also established. Isotopic analysis of these compounds until now was carried out by transforming them into carbon oxide, dioxide, and molecular nitrogen, and determination of their content in individual centers was impossible.

Intermittent disturbance mechanical behavior and fractional deterioration mechanical model of rock under complex true triaxial stress paths

Mechanical excavation,blasting,adjacent rockburst and fracture slip that occur during mining excavation impose dynamic loads on the rock mass,leading to further fracture of damaged surrounding rock in three-dimensional high-stress and even causing disasters.Therefore,a novel complex true triaxial static-dynamic combined loading method reflecting underground excavation damage and then frequent intermittent disturbance failure is proposed.True triaxial static compression and intermittent disturbance tests are carried out on monzogabbro.The effects of intermediate principal stress and amplitude on the strength characteristics,deformation characteristics,failure characteristics,and precursors of monzogabbro are analyzed,intermediate principal stress and amplitude increase monzogabbro strength and tensile fracture mechanism.Rapid increases in microseismic parameters during rock loading can be precursors for intermittent rock disturbance.Based on the experimental result,the new damage fractional elements and method with considering crack initiation stress and crack unstable stress as initiation and acceleration condition of intermittent disturbance irreversible deformation are proposed.A novel three-dimensional disturbance fractional deterioration model considering the intermediate principal stress effect and intermittent disturbance damage effect is established,and the model predicted results align well with the experimental results.The sensitivity of stress states and model parameters is further explored,and the intermittent disturbance behaviors at different f are predicted.This study provides valuable theoretical bases for the stability analysis of deep mining engineering under dynamic loads.

Analytical solutions fractional order partial differential equations arising in fluid dynamics

This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectional propagation of long-wave in dispersive media and GSEs are used to model the interaction between one-dimensional high,and low-frequency waves.Classes of trigonometric and hyperbolic function solutions in fractional calculus are discussed.Graphical simulations of the numerical solutions are flaunted by MATLAB.

Dynamics of the Fractional-Order Lorenz System Based on Adomian Decomposition Method and Its DSP Implementation

Dear Editor,Dynamics and digital circuit implementation of the fractional-order Lorenz system are investigated by employing Adomian decomposition method(ADM).Dynamics of the fractional-order Lorenz system with derivative order and parameter varying is analyzed by means of Lyapunov exponents(LEs),bifurcation diagram.

Diagnostic performance of intravascular ultrasound-based fractional flow reserve in evaluating of intermediate left main stenosis

BACKGROUND The recently introduced ultrasonic flow ratio(UFR),is a novel fast computational method to derive fractional flow reserve(FFR)from intravascular ultrasound(IVUS)images.In the present study,we evaluate the diagnostic performance of UFR in patients with intermediate left main(LM)stenosis.METHODS This is a prospective,single center study enrolling consecutive patients with presence of intermediated LM lesions(diameter stenosis of 30%-80%by visual estimation)underwent IVUS and FFR measurement.An independent core laboratory assessed offline UFR and IVUS-derived minimal lumen area(MLA)in a blinded fashion.RESULTS Both UFR and FFR were successfully achieved in 41 LM patients(mean age,62.0±9.9 years,46.3%diabetes).An acceptable correlation between UFR and FFR was identified(r=0.688,P<0.0001),with an absolute numerical difference of 0.03(standard difference:0.01).The area under the curve(AUC)in diagnosis of physiologically significant coronary stenosis for UFR was 0.94(95%CI:0.87-1.01),which was significantly higher than angiographic identified stenosis>50%(AUC=0.66,P<0.001)and numerically higher than IVUS-derived MLA(AUC=0.82;P=0.09).Patient level diagnostic accuracy,sensitivity and specificity for UFR to identify FFR≤0.80 was 82.9%(95%CI:70.2-95.7),93.1%(95%CI:82.2-100.0),58.3%(95%CI:26.3-90.4),respectively.CONCLUSION In patients with intermediate LM diseases,UFR was proved to be associated with acceptable correlation and high accuracy with pressure wire-based FFR as standard reference.The present study supports the use of UFR for functional evaluation of intermediate LM stenosis.

A novel variable-order fractional chaotic map and its dynamics

In recent years,fractional-order chaotic maps have been paid more attention in publications because of the memory effect.This paper presents a novel variable-order fractional sine map(VFSM)based on the discrete fractional calculus.Specially,the order is defined as an iterative function that incorporates the current state of the system.By analyzing phase diagrams,time sequences,bifurcations,Lyapunov exponents and fuzzy entropy complexity,the dynamics of the proposed map are investigated comparing with the constant-order fractional sine map.The results reveal that the variable order has a good effect on improving the chaotic performance,and it enlarges the range of available parameter values as well as reduces non-chaotic windows.Multiple coexisting attractors also enrich the dynamics of VFSM and prove its sensitivity to initial values.Moreover,the sequence generated by the proposed map passes the statistical test for pseudorandom number and shows strong robustness to parameter estimation,which proves the potential applications in the field of information security.

Chromatin condensation but not DNA integrity of pig sperm is greater in the sperm-rich fraction

Background Protamination and condensation of sperm chromatin as well as DNA integrity play an essential role during fertilization and embryo development.In some mammals,like pigs,ejaculates are emitted in three separate fractions:pre-sperm,sperm-rich(SRF)and post sperm-rich(PSRF).These fractions are known to vary in volume,sperm concentration and quality,as well as in the origin and composition of seminal plasma(SP),with differences being also observed within the SRF one.Yet,whether disparities in the DNA integrity and chromatin condensation and pro-tamination of their sperm exist has not been interrogated.Results This study determined chromatin protamination(Chromomycin A3 test,CMA_(3)),condensation(Dibromobi-mane test,DBB),and DNA integrity(Comet assay)in the pig sperm contained in the first 10 m L of the SRF(SRF-P1),the remaining portion of the sperm-rich fraction(SRF-P2),and the post sperm-rich fraction(PSRF).While chromatin protamination was found to be similar between the different ejaculate fractions(P>0.05),chromatin condensation was seen to be greater in SRF-P1 and SRF-P2 than in the PSRF(P=0.018 and P=0.004,respectively).Regarding DNA integrity,no differences between fractions were observed(P>0.05).As the SRF-P1 has the highest sperm concentra-tion and ejaculate fractions are known to differ in antioxidant composition,the oxidative stress index(OSi)in SP,calcu-lated as total oxidant activity divided by total antioxidant capacity,was tested and confirmed to be higher in the SRF-P1 than in SRF-P2 and PSRF(0.42±0.06 vs.0.23±0.09 and 0.08±0.00,respectively;P<0.01);this index,in addition,was observed to be correlated to the sperm concentration of each fraction(Rs=0.973;P<0.001).Conclusion While sperm DNA integrity was not found to differ between ejaculate fractions,SRF-P1 and SRF-P2 were observed to exhibit greater chromatin condensation than the PSRF.This could be related to the OSi of each fraction.

Fractional-order heterogeneous memristive Rulkov neuronal network and its medical image watermarking application

This article proposes a novel fractional heterogeneous neural network by coupling a Rulkov neuron with a Hopfield neural network(FRHNN),utilizing memristors for emulating neural synapses.The study firstly demonstrates the coexistence of multiple firing patterns through phase diagrams,Lyapunov exponents(LEs),and bifurcation diagrams.Secondly,the parameter related firing behaviors are described through two-parameter bifurcation diagrams.Subsequently,local attraction basins reveal multi-stability phenomena related to initial values.Moreover,the proposed model is implemented on a microcomputer-based ARM platform,and the experimental results correspond to the numerical simulations.Finally,the article explores the application of digital watermarking for medical images,illustrating its features of excellent imperceptibility,extensive key space,and robustness against attacks including noise and cropping.

Rebuilding the theory of isotope fractionation for evaporation of silicate melts under vacuum condition

Isotope eff ects are pivotal in understanding silicate melt evaporation and planetary accretion processes.Based on the Hertz-Knudsen equation,the current theory often fails to predict observed isotope fractionations of laboratory experiments due to its oversimplified assumptions.Here,we point out that the Hertz-Knudsen-equation-based theory is incomplete for silicate melt evaporation cases and can only be used for situations where the vaporized species is identical to the one in the melt.We propose a new model designed for silicate melt evaporation under vacuum.Our model considers multiple steps including mass transfer,chemical reaction,and nucleation.Our derivations reveal a kinetic isotopic fractionation factor(KIFF orα)αour model=[m(^(1)species)/m(^(2)species)]^(0.5),where m(species)is the mass of the reactant of reaction/nucleation-limiting step or species of diffusion-limiting step and superscript 1 and 2 represent light and heavy isotopes,respectively.This model can eff ectively reproduce most reported KIFFs of laboratory experiments for various elements,i.e.,Mg,Si,K,Rb,Fe,Ca,and Ti.And,the KIFF-mixing model referring that an overall rate of evaporation can be determined by two steps jointly can account for the eff ects of low P_(H2)pressure,composition,and temperature.In addition,we find that chemical reactions,diffusion,and nucleation can control the overall rate of evaporation of silicate melts by using the fitting slope in ln(−ln f)versus ln(t).Notably,our model allows for the theoretical calculations of parameters like activation energy(E_(a)),providing a novel approach to studying compositional and environmental eff ects on evaporation processes,and shedding light on the formation and evolution of the proto-solar and Earth-Moon systems.

Memory effect in time fractional Schrödinger equation

A significant obstacle impeding the advancement of the time fractional Schrodinger equation lies in the challenge of determining its precise mathematical formulation.In order to address this,we undertake an exploration of the time fractional Schrodinger equation within the context of a non-Markovian environment.By leveraging a two-level atom as an illustrative case,we find that the choice to raise i to the order of the time derivative is inappropriate.In contrast to the conventional approach used to depict the dynamic evolution of quantum states in a non-Markovian environment,the time fractional Schrodinger equation,when devoid of fractional-order operations on the imaginary unit i,emerges as a more intuitively comprehensible framework in physics and offers greater simplicity in computational aspects.Meanwhile,we also prove that it is meaningless to study the memory of time fractional Schrodinger equation with time derivative 1<α≤2.It should be noted that we have not yet constructed an open system that can be fully described by the time fractional Schrodinger equation.This will be the focus of future research.Our study might provide a new perspective on the role of time fractional Schrodinger equation.

Nuclear volume effects in kinetic isotope fractionation:A case study of mercury oxidation by chlorine species

It is well-known that the equilibrium isotope fractionation of mercury(Hg)includes classical massdependent fractionations(MDFs)and nuclear volume effect(NVE)induced mass-independent fractionations(MIFs).However,the effect of the NVE on these kinetic processes is not known.The total fractionations(MDFs+NVEinduced MIFs)of several representative Hg-incorporated substances were selected and calculated with ab initio calculations in this work for both equilibrium and kinetic processes.NVE-induced MIFs were calculated with scaled contact electron densities at the nucleus through systematic evaluations of their accuracy and errors using the Gaussian09 and DIRAC19 packages(named the electron density scaling method).Additionally,the NVE-induced kinetic isotope effect(KIE)of Hg isotopes are also calculated with this method for several representative Hg oxidation reactions by chlorine species.Total KIEs for 202 Hg/^(198)Hg ranging from−2.27‰to 0.96‰are obtained.Three anomalous^(202)Hg-enriched KIEs(δ^(202)Hg/^(198)Hg=0.83‰,0.94‰,and 0.96‰,)caused by the NVE are observed,which are quite different from the classical view(i.e.,light isotopes react faster than the heavy ones).The electron density scaling method we developed in this study can provide an easier way to calculate the NVE-induced KIEs for heavy isotopes and serve to better understand the fractionation mechanisms of mercury isotope systems.