Separating He from CH_(4)or N_(2)is crucial for natural gas He extraction,a prevailing industrial approach.Herein,molecular simulation and machine learning(ML)were combined to screen 801 experimentally synthesized COF...Separating He from CH_(4)or N_(2)is crucial for natural gas He extraction,a prevailing industrial approach.Herein,molecular simulation and machine learning(ML)were combined to screen 801 experimentally synthesized COFs for He/CH_(4)and He/N_(2)separation,either by means of adsorption or membrane separation.Top 10 COFs for 4 different gas separation purposes(CH_(4)/He or N_(2)/He separation with either adsorption or membrane)were identified respectively.The highest adsorption performance score(APSmix,defined as the product of working capacity and adsorption selectivity for mixture gas)reached 447.88 mol/kg and 49.45 mol/kg for CH_(4)/He and N_(2)/He,with corresponding adsorption selectivity of 115.56 and 30.33.He permeabilities of 1.5×10^(6)or 1.2×10^(6)Barrer were achieved for equimolar He/CH_(4)or He/N_(2)mixture gas separations,accompanied by permselectivity of 5.47 and 11.80 well surpassing 2008 Robeson's upper bound.Best performing COFs for adsorption separation are 3D COFs with pore diameter below 0.8 nm while those for membrane separation are 2D COFs with large pores.Additionally,ML models were developed to predict separation performance,with key descriptors identified.The mechanism for how COFs'structure affects their separation performance was also revealed.展开更多
The capture of CO_(2)from CO_(2)/H_(2)gas mixtures in syngas is a crucial issue for hydrogen production from steam methane reforming in industry,as the presence of CO_(2)directly affects the purity of H_(2).A combinat...The capture of CO_(2)from CO_(2)/H_(2)gas mixtures in syngas is a crucial issue for hydrogen production from steam methane reforming in industry,as the presence of CO_(2)directly affects the purity of H_(2).A combination of a high-throughput screening method and grand canonical Monte Carlo simulation was utilized to evaluate and screen 1725 metal–organic frameworks(MOFs)in detail as a means of determining their adsorption performance for CO_(2)/H_(2)gas mixtures.The adsorption and separation performance of double-linker MOFs was comprehensively evaluated using eight evaluation indicators,namely,the largest cavity diameter,accessible surface area,pore occupied accessible volume,porosity,adsorption selectivity,working capacity,adsorbent performance score and percent regeneration.Six optimal performance frameworks were screened to further study their single-component adsorption and binary competitive adsorption of CO_(2)/H_(2)respectively.The CO_(2)adsorption selectivity at different CO_(2)/H_(2)feed ratios was also evaluated,which indicated their excellent adsorption and separation performance.The microscopic adsorption mechanisms for CO_(2)and H_(2)at the molecular level were investigated by analyzing the radial distribution function and density distribution.This study may provide directional guidance and reference for subsequent experiments on the adsorption and separation of CO_(2)/H_(2).展开更多
Deuterium(D_(2)) is one of the important fuel sources that power nuclear fusion reactors. The existing D_(2)/H_(2) separation technologies that obtain high-purity D_(2) are cost-intensive. Recent research has shown th...Deuterium(D_(2)) is one of the important fuel sources that power nuclear fusion reactors. The existing D_(2)/H_(2) separation technologies that obtain high-purity D_(2) are cost-intensive. Recent research has shown that metal-organic frameworks(MOFs) are of good potential for D_(2)/H_(2) separation application. In this work, a high-throughput computational screening of 12020 computation-ready experimental MOFs is carried out to determine the best MOFs for hydrogen isotope separation application. Meanwhile, the detailed structure-performance correlation is systematically investigated with the aid of machine learning. The results indicate that the ideal D_(2)/H_(2) adsorption selectivity calculated based on Henry coefficient is strongly correlated with the 1/ΔAD feature descriptor;that is, inverse of the adsorbility difference of the two adsorbates. Meanwhile, the machine learning(ML) results show that the prediction accuracy of all the four ML methods is significantly improved after the addition of this feature descriptor. In addition, the ML results based on extreme gradient boosting model also revealed that the 1/ΔAD descriptor has the highest relative importance compared to other commonly-used descriptors. To further explore the effect of hydrogen isotope separation in binary mixture, 1548 MOFs with ideal adsorption selectivity greater than 1.5 are simulated at equimolar conditions. The structure-performance relationship shows that high adsorption selectivity MOFs generally have smaller pore size(0.3-0.5 nm) and lower surface area. Among the top 200 performers, the materials mainly have the sql, pcu, cds, hxl, and ins topologies.Finally, three MOFs with high D_(2)/H_(2) selectivity and good D_(2) uptake are identified as the best candidates,of all which had one-dimensional channel pore. The findings obtained in this work may be helpful for the identification of potentially promising candidates for hydrogen isotope separation.展开更多
A series of laboratory screening tests have been performed with the objective of evaluating the effect of feed rate, screen length and screen motion on the quality of separation. The separation performance has been ev...A series of laboratory screening tests have been performed with the objective of evaluating the effect of feed rate, screen length and screen motion on the quality of separation. The separation performance has been evaluated by the use of proposed ISO Standard measures which are based on the partition curve.The laboratory rig, the test programme and results derived from it are discussed. The effect of screen length, feed rate and screen motion on the levels of screen performance as indicated by these measures and a normalised separation size parameter are discussed in relation to the assumed screening conditions for the tests reported and conclusions are summarised.展开更多
[Objectives]To study the phosphorus removal performance of phosphate accumulating organisms(PAOs).[Methods]Activated sludge from domestic sewage treatment plant was used as the strain source,and phosphate accumulating...[Objectives]To study the phosphorus removal performance of phosphate accumulating organisms(PAOs).[Methods]Activated sludge from domestic sewage treatment plant was used as the strain source,and phosphate accumulating organisms were screened by plate streaking method and dilution coating plate method.Six kinds of excellent phosphate accumulating organisms were obtained by metachromatic granule staining experiment,total phosphorus experiment and simulated sewage phosphorus removal experiment to assist the observation of bac-terial morphology and experiment of phosphorus removal capacity.In addition,the influencing factors of phosphorus removal capacity(nitrogen source,trace metal ions)were analyzed.[Results]In the case of simulated sewage,the phosphorus removal rate of strain b was the highest,reaching 66.25%,while the phosphorus removal rate of strain e and f was about 10%lower than that of the phosphorus uptake experiment.[Conclusions]This study is expected to provide a theoretical reference for the gradual optimization of the screening method of phosphorus re-moval bacteria in domestic sewage treatment.展开更多
Kinetic-based adsorptive separation is deemed as an energy-efficient approach for gas purification,yet its underlying mechanism is difficult to justify.Herein,we propose an intriguing interaction-screening mechanism w...Kinetic-based adsorptive separation is deemed as an energy-efficient approach for gas purification,yet its underlying mechanism is difficult to justify.Herein,we propose an intriguing interaction-screening mechanism with a cage-interconnected metal-organic framework(JNU-2)as a model via a multi-scale theoretical approach.Grand Canonical Monte Carlo(GCMC)simulations establish gas diffusion channels with the calculated C_(2)H_(4) and C_(2)H_(6) adsorptions comparable to the experimental ones.Molecular dynamic(MD)simulations reveal single-molecule passages along the diffusion channel and that the probability of C_(2)H_(6) diffusing into the passage is nine times higher than that of C_(2)H_(4).Density functional theory(DFT)calculations further confirm an overall preferential interaction with C_(2)H_(6) passing through the singlemolecule passage.This work has successfully demonstrated a theoretical methodology of multi-scale simulations and depicted a rarely observed interaction-screening mechanism in JNU-2 that corroborates its balanced adsorption capacity and C_(2)H_(6)/C_(2)H_(4) adsorption selectivity.Such a methodology should be applicable to other well-defined structures for a better understanding of their gas adsorption/separation behaviours.展开更多
Vibrating flip-flow screens(VFFS)with stretchable polyurethane sieve mats have been widely used in screening fine-grained materials in recent years.In this work,the discrete element method(DEM)is used to study the scr...Vibrating flip-flow screens(VFFS)with stretchable polyurethane sieve mats have been widely used in screening fine-grained materials in recent years.In this work,the discrete element method(DEM)is used to study the screening process in VFFS to explain particle flow and separation behavior at the particle scale.Unlike traditional vibrating screens,for VFFS,the amplitude response of each point on the elastic sieve mat is different everywhere.This study measures the kinematics of the elastic sieve mat under different conditions such as different stretched lengths and material loads.To establish the elastic sieve mat model in a DEM simulation,the continuous elastic sieve mat is discretized into multiple units,and the displacement signal of each unit tested is analyzed by Fourier series.The Fourier series analysis results of each unit are used as the setting parameters for motion.In this way,the movement of the elastic sieve mat is approximately simulated,and a DEM model of VFFS is produced.Through the simulation,the flow and separation of different-sized particles in VFFS are studied,and the reasonability of the simulation is verified by a pilot-scale screening experiment.The present study demonstrates the potential of the DEM method for the analysis of screening processes in VFFS.展开更多
The discrete element method was utilized to construct three-dimensional discrete element models for the rice mixture,and their motions were analyzed numerically on a planar vibration screening device.The results showe...The discrete element method was utilized to construct three-dimensional discrete element models for the rice mixture,and their motions were analyzed numerically on a planar vibration screening device.The results showed that,after falling onto the vibrating screen surface,the mixture undergoes a reciprocating motion within the same cycle.During the screening process,the mixture undergoes segregation,slides along the screen surface,passes through the screen and then falls.In comparing the movement of grains and shriveled grains,it can be seen that the velocity of shriveled grains experiences cyclical changes,which is consistent with the grains’motion cycle.The impact on grains is shown to be greater than that on shriveled grains,and the average speeds of shriveled grains and grains converge.The curve shows that the screening of repetitive movements has a significant effect on the average velocity of shriveled grains,but also the velocity of shriveled grains and the vibration parameters can be well represented by a fitting equation.It is beneficial for the separation of grains from shriveled grains to choose a greater vibration frequency and screen surface inclination in the range of commonly used.展开更多
Protein liquid-liquid phase separation(LLPS),a pivotal phenomenon intricately linked to cellular processes,is regulated by various other proteins.However,there is still a lack of high-throughput methods for screening ...Protein liquid-liquid phase separation(LLPS),a pivotal phenomenon intricately linked to cellular processes,is regulated by various other proteins.However,there is still a lack of high-throughput methods for screening protein regulators of LLPS in target proteins.Here,we developed a CRISPR/Cas9-based screening method to identify protein phase separation regulators by integrating bimolecular fluorescence complementation(BiFC)and fluorescence-activated cell sorting(FACS).Using this newly developed method,we screened the RNA-binding proteins that regulate PABPN1 phase separation and identified the tumor suppressor QKI as a promoter of PABPN1 phase separation.Furthermore,QKI exhibits decreased expression levels and diminished nuclear localization in colorectal cancer cells,resulting in reduced PABPN1 phase separation,which,in turn,promotes alternative polyadenylation(APA),cell proliferation,and migration in colorectal cancer.展开更多
To improve the deteriorated capacity gain and source recovery performance due to channel mismatch problem,this paper reports a research about blind separation method against channel mismatch in multiple-input multiple...To improve the deteriorated capacity gain and source recovery performance due to channel mismatch problem,this paper reports a research about blind separation method against channel mismatch in multiple-input multiple-output(MIMO) systems.The channel mismatch problem can be described as a channel with bounded fluctuant errors due to channel distortion or channel estimation errors.The problem of blind signal separation/extraction with channel mismatch is formulated as a cost function of blind source separation(BSS) subject to the second-order cone constraint,which can be called as second-order cone programing optimization problem.Then the resulting cost function is solved by approximate negentropy maximization using quasi-Newton iterative methods for blind separation/extraction source signals.Theoretical analysis demonstrates that the proposed algorithm has low computational complexity and improved performance advantages.Simulation results verify that the capacity gain and bit error rate(BER) performance of the proposed blind separation method is superior to those of the existing methods in MIMO systems with channel mismatch problem.展开更多
A kind of cone separation theorems is established, by which the extension theorems for cone linear continuous operators are developed. As an application, the extension theorem for positive linear continuous operators ...A kind of cone separation theorems is established, by which the extension theorems for cone linear continuous operators are developed. As an application, the extension theorem for positive linear continuous operators is given.展开更多
基金the support from the Natural Science Foundation of China(U23A20115)the Natural Science Foundation of China(22368027,22078104)+4 种基金Science and Technology Key Project of Guangdong Province(2025B0101060003)the Natural Science Foundation of Guangdong Province(2024A1515012725,2024A1515012724)Guangzhou Municipal Science and Technology Project(2024A04J6251)State Key Laboratory of Pulp and Paper Engineering 2024ZD03Fundamental Research Funds for the Central Universities(2025ZYGXZR023)。
文摘Separating He from CH_(4)or N_(2)is crucial for natural gas He extraction,a prevailing industrial approach.Herein,molecular simulation and machine learning(ML)were combined to screen 801 experimentally synthesized COFs for He/CH_(4)and He/N_(2)separation,either by means of adsorption or membrane separation.Top 10 COFs for 4 different gas separation purposes(CH_(4)/He or N_(2)/He separation with either adsorption or membrane)were identified respectively.The highest adsorption performance score(APSmix,defined as the product of working capacity and adsorption selectivity for mixture gas)reached 447.88 mol/kg and 49.45 mol/kg for CH_(4)/He and N_(2)/He,with corresponding adsorption selectivity of 115.56 and 30.33.He permeabilities of 1.5×10^(6)or 1.2×10^(6)Barrer were achieved for equimolar He/CH_(4)or He/N_(2)mixture gas separations,accompanied by permselectivity of 5.47 and 11.80 well surpassing 2008 Robeson's upper bound.Best performing COFs for adsorption separation are 3D COFs with pore diameter below 0.8 nm while those for membrane separation are 2D COFs with large pores.Additionally,ML models were developed to predict separation performance,with key descriptors identified.The mechanism for how COFs'structure affects their separation performance was also revealed.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11304079,11404094,and 11504088)Science and Technology Research Project of Henan Science and Technology Department(Grant No.182102410076)。
文摘The capture of CO_(2)from CO_(2)/H_(2)gas mixtures in syngas is a crucial issue for hydrogen production from steam methane reforming in industry,as the presence of CO_(2)directly affects the purity of H_(2).A combination of a high-throughput screening method and grand canonical Monte Carlo simulation was utilized to evaluate and screen 1725 metal–organic frameworks(MOFs)in detail as a means of determining their adsorption performance for CO_(2)/H_(2)gas mixtures.The adsorption and separation performance of double-linker MOFs was comprehensively evaluated using eight evaluation indicators,namely,the largest cavity diameter,accessible surface area,pore occupied accessible volume,porosity,adsorption selectivity,working capacity,adsorbent performance score and percent regeneration.Six optimal performance frameworks were screened to further study their single-component adsorption and binary competitive adsorption of CO_(2)/H_(2)respectively.The CO_(2)adsorption selectivity at different CO_(2)/H_(2)feed ratios was also evaluated,which indicated their excellent adsorption and separation performance.The microscopic adsorption mechanisms for CO_(2)and H_(2)at the molecular level were investigated by analyzing the radial distribution function and density distribution.This study may provide directional guidance and reference for subsequent experiments on the adsorption and separation of CO_(2)/H_(2).
基金supported by the National Natural Science Foundation of China (22078004)the Research Development Fund from Xi’an Jiaotong-Liverpool University (RDF-16-02-03 and RDF15-01-23)key program special fund (KSF-E-03)。
文摘Deuterium(D_(2)) is one of the important fuel sources that power nuclear fusion reactors. The existing D_(2)/H_(2) separation technologies that obtain high-purity D_(2) are cost-intensive. Recent research has shown that metal-organic frameworks(MOFs) are of good potential for D_(2)/H_(2) separation application. In this work, a high-throughput computational screening of 12020 computation-ready experimental MOFs is carried out to determine the best MOFs for hydrogen isotope separation application. Meanwhile, the detailed structure-performance correlation is systematically investigated with the aid of machine learning. The results indicate that the ideal D_(2)/H_(2) adsorption selectivity calculated based on Henry coefficient is strongly correlated with the 1/ΔAD feature descriptor;that is, inverse of the adsorbility difference of the two adsorbates. Meanwhile, the machine learning(ML) results show that the prediction accuracy of all the four ML methods is significantly improved after the addition of this feature descriptor. In addition, the ML results based on extreme gradient boosting model also revealed that the 1/ΔAD descriptor has the highest relative importance compared to other commonly-used descriptors. To further explore the effect of hydrogen isotope separation in binary mixture, 1548 MOFs with ideal adsorption selectivity greater than 1.5 are simulated at equimolar conditions. The structure-performance relationship shows that high adsorption selectivity MOFs generally have smaller pore size(0.3-0.5 nm) and lower surface area. Among the top 200 performers, the materials mainly have the sql, pcu, cds, hxl, and ins topologies.Finally, three MOFs with high D_(2)/H_(2) selectivity and good D_(2) uptake are identified as the best candidates,of all which had one-dimensional channel pore. The findings obtained in this work may be helpful for the identification of potentially promising candidates for hydrogen isotope separation.
文摘A series of laboratory screening tests have been performed with the objective of evaluating the effect of feed rate, screen length and screen motion on the quality of separation. The separation performance has been evaluated by the use of proposed ISO Standard measures which are based on the partition curve.The laboratory rig, the test programme and results derived from it are discussed. The effect of screen length, feed rate and screen motion on the levels of screen performance as indicated by these measures and a normalised separation size parameter are discussed in relation to the assumed screening conditions for the tests reported and conclusions are summarised.
基金Supported by Zhaoqing University Innovation and Entrepreneurship Training Program for College Students(202310580018).
文摘[Objectives]To study the phosphorus removal performance of phosphate accumulating organisms(PAOs).[Methods]Activated sludge from domestic sewage treatment plant was used as the strain source,and phosphate accumulating organisms were screened by plate streaking method and dilution coating plate method.Six kinds of excellent phosphate accumulating organisms were obtained by metachromatic granule staining experiment,total phosphorus experiment and simulated sewage phosphorus removal experiment to assist the observation of bac-terial morphology and experiment of phosphorus removal capacity.In addition,the influencing factors of phosphorus removal capacity(nitrogen source,trace metal ions)were analyzed.[Results]In the case of simulated sewage,the phosphorus removal rate of strain b was the highest,reaching 66.25%,while the phosphorus removal rate of strain e and f was about 10%lower than that of the phosphorus uptake experiment.[Conclusions]This study is expected to provide a theoretical reference for the gradual optimization of the screening method of phosphorus re-moval bacteria in domestic sewage treatment.
基金supported by the National Natural Science Foundation of China(No.21731002,21975104,22150004,and 22101099)the Guangdong Major Project of Basic and Applied Research(No.2019B030302009)+2 种基金the Guangdong Basic and Applied Basic Research Foundation(No.2020A1515011005)We thank the high-performance public computing service platform of Jinan University for providing computational resourcesWe also appreciate Xiao-Jing Xie(Jinan University)and Dr.Weijie Zhang(Univ.of North Texas)for their help and discussion。
文摘Kinetic-based adsorptive separation is deemed as an energy-efficient approach for gas purification,yet its underlying mechanism is difficult to justify.Herein,we propose an intriguing interaction-screening mechanism with a cage-interconnected metal-organic framework(JNU-2)as a model via a multi-scale theoretical approach.Grand Canonical Monte Carlo(GCMC)simulations establish gas diffusion channels with the calculated C_(2)H_(4) and C_(2)H_(6) adsorptions comparable to the experimental ones.Molecular dynamic(MD)simulations reveal single-molecule passages along the diffusion channel and that the probability of C_(2)H_(6) diffusing into the passage is nine times higher than that of C_(2)H_(4).Density functional theory(DFT)calculations further confirm an overall preferential interaction with C_(2)H_(6) passing through the singlemolecule passage.This work has successfully demonstrated a theoretical methodology of multi-scale simulations and depicted a rarely observed interaction-screening mechanism in JNU-2 that corroborates its balanced adsorption capacity and C_(2)H_(6)/C_(2)H_(4) adsorption selectivity.Such a methodology should be applicable to other well-defined structures for a better understanding of their gas adsorption/separation behaviours.
基金supported by the Fundamental Research Funds for the Central Universities (grant No.2021YJSHH32)Anhui Province Major Science and Technology Achievements Engineering Research and Development Special Project (grant No.202103c08020007).
文摘Vibrating flip-flow screens(VFFS)with stretchable polyurethane sieve mats have been widely used in screening fine-grained materials in recent years.In this work,the discrete element method(DEM)is used to study the screening process in VFFS to explain particle flow and separation behavior at the particle scale.Unlike traditional vibrating screens,for VFFS,the amplitude response of each point on the elastic sieve mat is different everywhere.This study measures the kinematics of the elastic sieve mat under different conditions such as different stretched lengths and material loads.To establish the elastic sieve mat model in a DEM simulation,the continuous elastic sieve mat is discretized into multiple units,and the displacement signal of each unit tested is analyzed by Fourier series.The Fourier series analysis results of each unit are used as the setting parameters for motion.In this way,the movement of the elastic sieve mat is approximately simulated,and a DEM model of VFFS is produced.Through the simulation,the flow and separation of different-sized particles in VFFS are studied,and the reasonability of the simulation is verified by a pilot-scale screening experiment.The present study demonstrates the potential of the DEM method for the analysis of screening processes in VFFS.
基金This work was supported by the National Natural Science Foundation of China(51305182)the Ministry of Agriculture Key Laboratory of Modern Agricultural Equipment Grant(201303003).
文摘The discrete element method was utilized to construct three-dimensional discrete element models for the rice mixture,and their motions were analyzed numerically on a planar vibration screening device.The results showed that,after falling onto the vibrating screen surface,the mixture undergoes a reciprocating motion within the same cycle.During the screening process,the mixture undergoes segregation,slides along the screen surface,passes through the screen and then falls.In comparing the movement of grains and shriveled grains,it can be seen that the velocity of shriveled grains experiences cyclical changes,which is consistent with the grains’motion cycle.The impact on grains is shown to be greater than that on shriveled grains,and the average speeds of shriveled grains and grains converge.The curve shows that the screening of repetitive movements has a significant effect on the average velocity of shriveled grains,but also the velocity of shriveled grains and the vibration parameters can be well represented by a fitting equation.It is beneficial for the separation of grains from shriveled grains to choose a greater vibration frequency and screen surface inclination in the range of commonly used.
基金supported by the National Natural Science Foundation of China(Nos.32470586 and 31971332 to Y.F.,Nos.91942301 and 81430099 to A.X.)National Key Research and Development Program of China(No.2022YFA1103900 to Y.F.)Guangdong Science and Technology Department(No.2023B1212060028 to A.X.).
文摘Protein liquid-liquid phase separation(LLPS),a pivotal phenomenon intricately linked to cellular processes,is regulated by various other proteins.However,there is still a lack of high-throughput methods for screening protein regulators of LLPS in target proteins.Here,we developed a CRISPR/Cas9-based screening method to identify protein phase separation regulators by integrating bimolecular fluorescence complementation(BiFC)and fluorescence-activated cell sorting(FACS).Using this newly developed method,we screened the RNA-binding proteins that regulate PABPN1 phase separation and identified the tumor suppressor QKI as a promoter of PABPN1 phase separation.Furthermore,QKI exhibits decreased expression levels and diminished nuclear localization in colorectal cancer cells,resulting in reduced PABPN1 phase separation,which,in turn,promotes alternative polyadenylation(APA),cell proliferation,and migration in colorectal cancer.
基金supported by Sichuan Youth Science and Technology Innovation Research Team Project(No.2015TD0022)the Talents Project of Sichuan University of Science and Engineering(No.2017RCL11 and No.2017RCL10)the first batch of science and technology plan key R&D project of Sichuan province(No.2017GZ0068)
文摘To improve the deteriorated capacity gain and source recovery performance due to channel mismatch problem,this paper reports a research about blind separation method against channel mismatch in multiple-input multiple-output(MIMO) systems.The channel mismatch problem can be described as a channel with bounded fluctuant errors due to channel distortion or channel estimation errors.The problem of blind signal separation/extraction with channel mismatch is formulated as a cost function of blind source separation(BSS) subject to the second-order cone constraint,which can be called as second-order cone programing optimization problem.Then the resulting cost function is solved by approximate negentropy maximization using quasi-Newton iterative methods for blind separation/extraction source signals.Theoretical analysis demonstrates that the proposed algorithm has low computational complexity and improved performance advantages.Simulation results verify that the capacity gain and bit error rate(BER) performance of the proposed blind separation method is superior to those of the existing methods in MIMO systems with channel mismatch problem.
文摘A kind of cone separation theorems is established, by which the extension theorems for cone linear continuous operators are developed. As an application, the extension theorem for positive linear continuous operators is given.
