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.

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.

Inter-Fraction Analysis of One Week Hypo-Fractionation of Deep Inspiration Breath Hold (DIBH) Technique for Left Sided Breast Cancer Radiation Treatment

The aim of this study was to investigate the inter-fraction variations, patient comfort and knowledge at Charlotte Maxeke Johannesburg Academic Hospital (CMJAH). The differences in set-up that occurred between treatment sessions for the left sided breast patients were observed and recorded. Measurements of routine set-up variation for 24 patients were performed by matching the cone beam computed tomography (CBCT) and the planning computed tomography (CT). Scans of all five fractions per patient were used to quantify the setup variations with standard deviation (SD) in all the three directions (anterior posterior, left right, and superior inferior). The patients DIBH comfort and knowledge was also evaluated. The average translational errors for the anterior posterior (AP, z), left-right (LR, x), and Superior-inferior (SI, y) directions were 0.40 cm, 0.40 cm, and 0.40 cm, respectively. The translation variation of the three directions showed statistical significance (P < 0.05). On comfort and knowledge investigation, among all participants, 80% moderately agreed that the therapist’s instructions for operating the deep inspiration breath hold (DIBH) technique were easy to understand, and 63.33% indicated that their comfort with the DIBH technique was neutral or average. The inter-fraction variations in patients with left-sided breast cancer were qualitatively analyzed. Significant shifts between CBCT and planning CT images were observed. The daily treatment verification could assist accurate dose delivery.

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.

Efficacy of stromal vascular fraction for knee osteoarthritis:A prospective,single-centre,non-randomized study with 2 years follow-up

BACKGROUND Current osteoarthritis(OA)treatments focus on symptom relief without addressing the underlying disease process.In regenerative medicine,current treatments have limitations.In regenerative medicine,more research is needed for intra-articular stromal vascular fraction(SVF)injections in OA,including dosage optimization,long-term efficacy,safety,comparisons with other treatments,and mechanism exploration.AIM To compare the efficacy of intra-articular SVF with corticosteroid(ICS)injections in patients with primary knee OA.METHODS The study included 50 patients with Kellgren-Lawrence grades II and III OA.Patients were randomly assigned(1:1)to receive either a single intra-articular SVF injection(group A)or a single intra-articular ICS(triamcinolone)(group B)injection.Patients were followed up at 1,3,6,12,and 24 months.Visual analog score(VAS)and International Knee Documentation Committee(IKDC)scores were administered before the procedure and at all followups.The safety of SVF in terms of adverse and severe adverse events was recorded.Statistical analysis was performed with SPSS Version 26.0,IBM Corp,Chicago,IL,United States.RESULTS Both groups had similar demographics and baseline clinical characteristics.Follow-up showed minor patient loss,resulting in 23 and 24 in groups A and B respectively.Group A experienced a notable reduction in pain,with VAS scores decreasing from 7.7 to 2.4 over 24 months,compared to a minor reduction from 7.8 to 6.2 in Group B.This difference in pain reduction in group A was statistically significant from the third month onwards.Additionally,Group A showed significant improvements in knee functionality,with IKDC scores rising from 33.4 to 83.10,whereas Group B saw a modest increase from 36.7 to 45.16.The improvement in Group A was statistically significant from 6 months and maintained through 24 months.CONCLUSION Our study demonstrated that intra-articular administration of SVF can lead to reduced pain and improved knee function in patients with primary knee OA.More adequately powered,multi-center,double-blinded,randomised clinical trials with longer follow-ups are needed to further establish safety and justify its clinical use.

Ethyl acetate fraction of Sargassum pallidum extract attenuates particulate matter-induced oxidative stress and inflammation in keratinocytes and zebrafish

Objective:To evaluate the effect of the ethyl acetate fraction derived from Sargassum pallidum extract against particulate matter(PM)-induced oxidative stress and inflammation in HaCaT cells and zebrafish.Methods:HaCaT cells and zebrafish were used to evaluate the protective effects of the ethyl acetate fraction of Sargassum pallidum extract against PM-induced oxidative stress and inflammation.The production of nitric oxide(NO),intracellular ROS,prostaglandin E_(2)(PGE_(2)),and pro-inflammatory cytokines,and the expression levels of COX-2,iNOS,and NF-κB were evaluated in PM-induced HaCaT cells.Furthermore,the levels of ROS,NO,and lipid peroxidation were assessed in the PM-exposed zebrafish model.Results:The ethyl acetate fraction of Sargassum pallidum extract significantly decreased the production of NO,intracellular ROS,and PGE_(2) in PM-induced HaCaT cells.In addition,the fraction markedly suppressed the levels of pro-inflammatory cytokines and inhibited the expression levels of COX-2,iNOS,and NF-κB.Furthermore,it displayed remarkable protective effects against PM-induced inflammatory response and oxidative stress,represented by the reduction of NO,ROS,and lipid peroxidation in zebrafish.Conclusions:The ethyl acetate fraction of Sargassum pallidum extract exhibits a protective effect against PM-induced oxidative stress and inflammation both in vitro and in vivo and has the potential as a candidate for the development of pharmaceutical and cosmeceutical products.

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.

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.

How to estimate isotope fractionations of a Rayleigh-like but diffusion-limited disequilibrium process?

The Rayleigh distillation isotope fractionation(RDIF) model is one of the most popular methods used in isotope geochemistry. Numerous isotope signals observed in geologic processes have been interpreted with this model. The RDIF model provides a simple mathematic solution for the reservoir-limited equilibrium isotope fractionation effect. Due to the reservoir effect, tremendously large isotope fractionations will always be produced if the reservoir is close to being depleted. However, in real situations, many prerequisites assumed in the RDIF model are often difficult to meet. For instance, it requires the relocated materials, which are removed step by step from one reservoir to another with different isotope compositions(i.e., with isotope fractionation), to be isotopically equilibrated with materials in the first reservoir simultaneously. This ‘‘quick equilibrium requirement’’ is indeed hard to meet if the first reservoir is sufficiently large or the removal step is fast. The whole first reservoir will often fail to re-attain equilibrium in time before the next removal starts.This problem led the RDIF model to fail to interpret isotope signals of many real situations. Here a diffusion-coupled and Rayleigh-like(i.e., reservoir-effect included) separation process is chosen to investigate this problem. We find that the final isotope fractionations are controlled by both the diffusion process and the reservoir effects via the disequilibrium separation process. Due to its complexity, we choose to use a numerical simulation method to solve this problem by developing specific computing codes for the working model.According to our simulation results, the classical RDIF model only governs isotope fractionations correctly at the final stages of separation when the reservoir scale(or thickness of the system) is reduced to the order of magnitude of the quotient of the diffusivity and the separation rate. The RDIF model fails in other situations and the isotope fractionations will be diffusion-limited when the reservoir is relatively large, or the separation rate is fast. We find that the effect of internal isotope distribution inhomogeneity caused by diffusion on the Rayleigh-like separation process is significant and cannot be ignored. This method can be applied to study numerous geologic and planetary processes involving diffusion-limited disequilibrium separation processes including partial melting,evaporation, mineral precipitation, core segregation, etc.Importantly, we find that far more information can be extracted through analyzing isotopic signals of such ‘‘disequilibrium’’processes than those of fully equilibrated ones, e.g., reservoir size and the separation rate. Such information may provide a key to correctly interpreting many isotope signals observed from geochemical and cosmochemical processes.

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.

A unified fractional flow framework for predicting the liquid holdup in two-phase pipe flows

Two-phase pipe flow occurs frequently in oil&gas industry,nuclear power plants,and CCUS.Reliable calculations of gas void fraction(or liquid holdup)play a central role in two-phase pipe flow models.In this paper we apply the fractional flow theory to multiphase flow in pipes and present a unified modeling framework for predicting the fluid phase volume fractions over a broad range of pipe flow conditions.Compared to existing methods and correlations,this new framework provides a simple,approximate,and efficient way to estimate the phase volume fraction in two-phase pipe flow without invoking flow patterns.Notably,existing correlations for estimating phase volume fraction can be transformed and expressed under this modeling framework.Different fractional flow models are applicable to different flow conditions,and they demonstrate good agreement against experimental data within 5%errors when compared with an experimental database comprising of 2754 data groups from 14literature sources,covering various pipe geometries,flow patterns,fluid properties and flow inclinations.The gas void fraction predicted by the framework developed in this work can be used as inputs to reliably model the hydraulic and thermal behaviors of two-phase pipe flows.

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.

Understanding and controlling the variables for stromal vascular fraction therapy

In regenerative medicine,the isolation of mesenchymal stromal cells(MSCs)from the adipose tissue’s stromal vascular fraction(SVF)is a critical area of study.Our review meticulously examines the isolation process of MSCs,starting with the extraction of adipose tissue.The choice of liposuction technique,anatomical site,and immediate processing are essential to maintain cell functionality.We delve into the intricacies of enzymatic digestion,emphasizing the fine-tuning of enzyme concentrations to maximize cell yield while preventing harm.The review then outlines the filtration and centrifugation techniques necessary for isolating a purified SVF,alongside cell viability assessments like flow cytometry,which are vital for confirming the efficacy of the isolated MSCs.We discuss the advantages and drawbacks of using autologous vs allogeneic SVF sources,touching upon immunocompatibility and logistical considerations,as well as the variability inherent in donor-derived cells.Anesthesia choices,the selection between hypo-dermic needles vs liposuction cannulas,and the role of adipose tissue lysers in achieving cellular dissociation are evaluated for their impact on SVF isolation.Centrifugation protocols are also analyzed for their part in ensuring the integrity of the SVF.The necessity for standardized MSC isolation protocols is highlighted,promoting reproducibility and successful clinical application.We encourage ongoing research to deepen the understanding of MSC biology and therapeutic action,aiming to further the field of regenerative medicine.The review concludes with a call for rigorous research,interdisciplinary collaboration,and strict adherence to ethical and regulatory standards to safeguard patient safety and optimize treatment outcomes with MSCs.

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.

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.

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.

THE LONG TIME BEHAVIOR OF THE FRACTIONAL ORNSTEIN-UHLENBECK PROCESS WITH LINEAR SELF-REPELLING DRIFT

Let B^(H) be a fractional Brownian motion with Hurst index 1/2≤H<1.In this paper,we consider the equation(called the Ornstein-Uhlenbeck process with a linear self-repelling drift)dX_(t)^(H)=dB_(t)^(H)+σ X_(t)^(H)dt+vdt-θ(∫_(0)^(t)(X_(t)^(H)-X_(s)^(H))ds)dt,whereθ<0,σ,v∈ℝ.The process is an analogue of self-attracting diffusion(Cranston,Le Jan.Math Ann,1995,303:87–93).Our main aim is to study the large time behaviors of the process.We show that the solution X^(H)diverges to infinity as t tends to infinity,and obtain the speed at which the process X^(H)diverges to infinity.