doi: 10.3934/jimo.2018169

Evaluation strategy and mass balance for making decision about the amount of aluminum fluoride addition based on superheat degree

School of Information Science and Engineering, Central South University, Changsha 410083, China

*Corresponding author: Xiaofang Chen (e-mail: xiaofangchen@csu.edu.cn)

Received  June 2018 Revised  August 2018 Published  October 2018

Fund Project: This project was supported by the National Natural Science Foundation of China (61773405, 61533020, 61621062 and 61725306); and the innovation project of Central South University (502390003)

The purpose of aluminum fluoride (AlF3) addition is to adjust the superheat degree (SD) in the aluminum reduction process. Determining the appropriate amount of AlF3 to add has long been a challenging industrial issue as a result of its inherent complexity. Because of the decreasing number of experienced technicians, the manual addition of AlF3 is usually inexact, which easily leads to an unstable cell condition. In this paper, an evaluation strategy based on the SD for AlF3 addition is proposed. An extended naïve Bayesian classifier (ENBC) is designed to estimate the states of SD and its trends that represent the current and potential cell condition respectively, and then the process is graded by evaluating the estimated results based on fuzzy theory. The reduction process is divided into a few situations based on the evaluation grades, and mass balance is introduced to determine the amount of AlF3 addition in each situation. The results of experiments show that the proposed strategy is feasible, and the effectiveness of AlF3 addition is improved compared to the existing method. Moreover, automatic AlF3 addition is promising based on the proposed strategy.

Citation: Weichao Yue, Weihua Gui, Xiaofang Chen, Zhaohui Zeng, Yongfang Xie. Evaluation strategy and mass balance for making decision about the amount of aluminum fluoride addition based on superheat degree. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018169
References:
[1]

I. Barbeito and R. Cao, Smoothed stationary bootstrap bandwidth selection for density estimation with dependent data, Computational Statistics & Data Analysis, 104 (2016), 130-147. doi: 10.1016/j.csda.2016.06.015.

[2]

K. BarbéL. G. FuentesL. Barford and L. Lauwers, A guaranteed blind and automatic probability density estimation of raw measurements, IEEE Transactions on Instrumentation & Measurement, 63 (2014), 2120-2128.

[3]

Z. G. ChenY. G. LiX. F. Chen and W. H. Gui, Semantic Network Based on Intuitionistic Fuzzy Directed Hyper-Graphs and Application to Aluminum Electrolysis Cell Condition Identification, IEEE Access, 5 (2017), 20145-20156. doi: 10.1109/ACCESS.2017.2752200.

[4]

Y. Chien, Pattern classification and scene analysis, IEEE Transactions Automatic Control, 19 (1974), 462-463. doi: 10.1109/TAC.1974.1100577.

[5]

K. S. ChuangH. L. Tzeng and S. Chen, Fuzzy c-means clustering with spatial information for image segmentation, Computerized Medical Imaging and Graphics, 30 (2006), 9-15. doi: 10.1016/j.compmedimag.2005.10.001.

[6]

P. Desclaux, AlF3 additions based on bath temperature measurements, Light Metals, (1987), 309-313.

[7]

T. DrengstigD. Ljungquist and B. A. Foss, On the AlF3 and temperature control of an aluminum electrolysis cell, IEEE Transactions on Control Systems Technology, 6 (1998), 157-171.

[8]

M. Dupuis and I. GéniSim, Excess AlF3 concentration in bath control logic, National Conference on Advancements in Aluminium Electrolysis, Indian Institute of Metals, Angul, (2006), 309-313.

[9]

P. M. Entner and G. A. Gudmundsson, Further development of the temperature model, Light Metals, (1996), 445-449.

[10]

W. Haupin and H. Kvande, Mathematical model of fluoride evolution from Hall-Héroult Cells, Essential Readings in Light Metals, (2016), 903-909.

[11]

Y. J. HeY. MaoW. L. Chen and Y. X. Chen, Nonlinear metric learning with kernel density estimation, IEEE Transactions Knowledge and Data Engineering, 27 (2015), 1602-1614. doi: 10.1109/TKDE.2014.2384522.

[12]

Y. L. HeR. WangS. Kwong and X. Z. Wang, Bayesian classifiers based on probability density estimation and their applications to simultaneous fault diagnosis, Information Sciences, 259 (2014), 252-268. doi: 10.1016/j.ins.2013.09.003.

[13]

Y. B. HuangX. D. Qu and J. M. Zhou, Coupled heat/mass balance model for analyzing correlation between excess AlF3 concentration and aluminum electrolyte temperature, Transactions of Nonferrous Metals Society of China, 19 (2009), 724-729.

[14]

M. M. HylandE. C. PattersonF. S. Mcfadden and B. J. Welch, Aluminium fluoride consumption and control in smelting cells, Scand. J. Metall., 30 (2001), 404-414.

[15]

H. J. Kim, S. N. MacEachern and Y. Jung, Bandwidth selection for kernel density estimation with a markov chain monte carlo sample, arXiv. Preprint. arXiv., 1607.08274 (2016), 1-16.

[16]

K. R. Kloetstra, S. Benninghoff, M. A. Stam and B. W. Toebes, Optimisation of Aluminium Fluoride Control at Aluminium Delfzijl, Proceedings of the 7th Australasian Aluminium Smelting Workshop, (2001), 506-514.

[17]

M. KöhlerA. Schindler and S. Sperlich, A review and comparison of bandwidth selection methods for kernel regression, International Statistical Review, 82 (2014), 243-274. doi: 10.1111/insr.12039.

[18]

S. Kolås, Defining and verifying the 'correlation line' in aluminum electrolysis, JOM, 59 (2007), 55-60.

[19]

S Kolås and T. Støre, Bath temperature and AlF3 control of an aluminium electrolysis cell, Control and Engineering Practice, 17 (2009), 1035-1043.

[20]

M. E. Maron and J. L. Kuhns, On relevance, probabilistic indexing and information retrieval, Journal of the ACM, 7 (1960), 216-244. doi: 10.1145/321033.321035.

[21]

A. Meghlaoui, Y. A. A. Farsi and N. H. Aljabri, Analytical and experimental study of fluoride evolution, Light Metals-Warrendale-Proceedings. TMS, 2001, 283-288.

[22]

A. R. Mugdadi and I. A. Ahmad, A bandwidth selection for kernel density estimation of functions of random variables, Computational Statistics & Data Analysis, 47 (2004), 49-62. doi: 10.1016/j.csda.2003.10.013.

[23]

R. J. Pak, The influence function of the optimal bandwidth for kernel density estimation, Communications in Statistics, 46 (2017), 602-608. doi: 10.1080/03610926.2014.1000501.

[24]

D. J. Salt, Bath chemistry control system, Essential Readings in Light Metals, (2016), 798-803.

[25]

A. T. TabereauxT. R. Alcorn and L Trembley, Lithium-Modified Low Ratio Electrolyte Chemistry for Improved Performance in Modern Reduction Cells, Essential Readings in Light Metals, (2016), 83-88.

[26]

J. Thonstand and S. Roselth, Equilibrium between bath and side ledge in aluminum cells-Basic principles, Light Metals, (1983), 414-424.

[27]

L. Y. WangW. H. GuiK. L. TeoR. C. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications, Journal of Industrial & Management Optimization, 5 (2009), 705-718. doi: 10.3934/jimo.2009.5.705.

[28]

X. Z. WangY. L. He and D. D. Wang, Non-naive bayesian classifiers for classification problems with continuous attributes, IEEE Transactions on Cybernetics, 44 (2013), 21-39. doi: 10.1109/TCYB.2013.2245891.

[29]

M. J. Wilson, Practical considerations used in the development of a method for calculating aluminum fluoride additions based on cell temperatures, Light Metals (1992), 37-5-378.

[30]

J. YeH. XuE. Feng and Z. Xiu, Optimization of a fed-batch bioreactor for 1, 3-propanediol production using hybrid nonlinear optimal control, Journal of Process Control, 24 (2014), 1556-1569.

[31]

J. YiD. HuangS. Fu and T. Li, Optimized relative transformation matrix using bacterial foraging algorithm for process fault detection, IEEE Transactions on Industrial Electronics, 63 (2016), 2595-2605. doi: 10.1109/TIE.2016.2515057.

[32]

W. C. YueX. F. ChenW. H. Gui and H. L. Zhang, A knowledge reasoning Fuzzy-Bayesian network for root cause analysis of abnormal aluminum electrolysis cell condition, Fronters of Chemical Science and Engineering, 11 (2017), 414-428. doi: 10.1007/s11705-017-1663-x.

[33]

S. P. Zeng and F. W. Cui, Dynamic decision model for amount of AlF3 addition in industrial aluminum electrolysis, International Conference on Mechatronics, Robotics and Automation, (2015), 307-318.

[34]

S. P. ZengS. S. Wang and Y. X. Qu, Control of temperature and aluminum fluoride concentration based on model prediction in aluminum electrolysis, Advances in Materials Science & Engineering, (2014), 1-5.

[35]

E. ZentenoZ. A. KhanM. Isaksson and P. Handel, Finding structural information about RF power amplifiers using an orthogonal nonparametric kernel smoothing estimator, IEEE Transactions on Vehicular Technology, 65 (2016), 2883-2889. doi: 10.1109/TVT.2015.2434497.

[36]

S. Q. ZhanM. LiJ. M. Zhou and Y. W. Zhou, CFD simulation of dissolution process of alumina in an aluminum reduction cell with two-particle phase population balance model, Applied Thermal Engineering, 73 (2014), 805-818. doi: 10.1016/j.applthermaleng.2014.08.040.

show all references

References:
[1]

I. Barbeito and R. Cao, Smoothed stationary bootstrap bandwidth selection for density estimation with dependent data, Computational Statistics & Data Analysis, 104 (2016), 130-147. doi: 10.1016/j.csda.2016.06.015.

[2]

K. BarbéL. G. FuentesL. Barford and L. Lauwers, A guaranteed blind and automatic probability density estimation of raw measurements, IEEE Transactions on Instrumentation & Measurement, 63 (2014), 2120-2128.

[3]

Z. G. ChenY. G. LiX. F. Chen and W. H. Gui, Semantic Network Based on Intuitionistic Fuzzy Directed Hyper-Graphs and Application to Aluminum Electrolysis Cell Condition Identification, IEEE Access, 5 (2017), 20145-20156. doi: 10.1109/ACCESS.2017.2752200.

[4]

Y. Chien, Pattern classification and scene analysis, IEEE Transactions Automatic Control, 19 (1974), 462-463. doi: 10.1109/TAC.1974.1100577.

[5]

K. S. ChuangH. L. Tzeng and S. Chen, Fuzzy c-means clustering with spatial information for image segmentation, Computerized Medical Imaging and Graphics, 30 (2006), 9-15. doi: 10.1016/j.compmedimag.2005.10.001.

[6]

P. Desclaux, AlF3 additions based on bath temperature measurements, Light Metals, (1987), 309-313.

[7]

T. DrengstigD. Ljungquist and B. A. Foss, On the AlF3 and temperature control of an aluminum electrolysis cell, IEEE Transactions on Control Systems Technology, 6 (1998), 157-171.

[8]

M. Dupuis and I. GéniSim, Excess AlF3 concentration in bath control logic, National Conference on Advancements in Aluminium Electrolysis, Indian Institute of Metals, Angul, (2006), 309-313.

[9]

P. M. Entner and G. A. Gudmundsson, Further development of the temperature model, Light Metals, (1996), 445-449.

[10]

W. Haupin and H. Kvande, Mathematical model of fluoride evolution from Hall-Héroult Cells, Essential Readings in Light Metals, (2016), 903-909.

[11]

Y. J. HeY. MaoW. L. Chen and Y. X. Chen, Nonlinear metric learning with kernel density estimation, IEEE Transactions Knowledge and Data Engineering, 27 (2015), 1602-1614. doi: 10.1109/TKDE.2014.2384522.

[12]

Y. L. HeR. WangS. Kwong and X. Z. Wang, Bayesian classifiers based on probability density estimation and their applications to simultaneous fault diagnosis, Information Sciences, 259 (2014), 252-268. doi: 10.1016/j.ins.2013.09.003.

[13]

Y. B. HuangX. D. Qu and J. M. Zhou, Coupled heat/mass balance model for analyzing correlation between excess AlF3 concentration and aluminum electrolyte temperature, Transactions of Nonferrous Metals Society of China, 19 (2009), 724-729.

[14]

M. M. HylandE. C. PattersonF. S. Mcfadden and B. J. Welch, Aluminium fluoride consumption and control in smelting cells, Scand. J. Metall., 30 (2001), 404-414.

[15]

H. J. Kim, S. N. MacEachern and Y. Jung, Bandwidth selection for kernel density estimation with a markov chain monte carlo sample, arXiv. Preprint. arXiv., 1607.08274 (2016), 1-16.

[16]

K. R. Kloetstra, S. Benninghoff, M. A. Stam and B. W. Toebes, Optimisation of Aluminium Fluoride Control at Aluminium Delfzijl, Proceedings of the 7th Australasian Aluminium Smelting Workshop, (2001), 506-514.

[17]

M. KöhlerA. Schindler and S. Sperlich, A review and comparison of bandwidth selection methods for kernel regression, International Statistical Review, 82 (2014), 243-274. doi: 10.1111/insr.12039.

[18]

S. Kolås, Defining and verifying the 'correlation line' in aluminum electrolysis, JOM, 59 (2007), 55-60.

[19]

S Kolås and T. Støre, Bath temperature and AlF3 control of an aluminium electrolysis cell, Control and Engineering Practice, 17 (2009), 1035-1043.

[20]

M. E. Maron and J. L. Kuhns, On relevance, probabilistic indexing and information retrieval, Journal of the ACM, 7 (1960), 216-244. doi: 10.1145/321033.321035.

[21]

A. Meghlaoui, Y. A. A. Farsi and N. H. Aljabri, Analytical and experimental study of fluoride evolution, Light Metals-Warrendale-Proceedings. TMS, 2001, 283-288.

[22]

A. R. Mugdadi and I. A. Ahmad, A bandwidth selection for kernel density estimation of functions of random variables, Computational Statistics & Data Analysis, 47 (2004), 49-62. doi: 10.1016/j.csda.2003.10.013.

[23]

R. J. Pak, The influence function of the optimal bandwidth for kernel density estimation, Communications in Statistics, 46 (2017), 602-608. doi: 10.1080/03610926.2014.1000501.

[24]

D. J. Salt, Bath chemistry control system, Essential Readings in Light Metals, (2016), 798-803.

[25]

A. T. TabereauxT. R. Alcorn and L Trembley, Lithium-Modified Low Ratio Electrolyte Chemistry for Improved Performance in Modern Reduction Cells, Essential Readings in Light Metals, (2016), 83-88.

[26]

J. Thonstand and S. Roselth, Equilibrium between bath and side ledge in aluminum cells-Basic principles, Light Metals, (1983), 414-424.

[27]

L. Y. WangW. H. GuiK. L. TeoR. C. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications, Journal of Industrial & Management Optimization, 5 (2009), 705-718. doi: 10.3934/jimo.2009.5.705.

[28]

X. Z. WangY. L. He and D. D. Wang, Non-naive bayesian classifiers for classification problems with continuous attributes, IEEE Transactions on Cybernetics, 44 (2013), 21-39. doi: 10.1109/TCYB.2013.2245891.

[29]

M. J. Wilson, Practical considerations used in the development of a method for calculating aluminum fluoride additions based on cell temperatures, Light Metals (1992), 37-5-378.

[30]

J. YeH. XuE. Feng and Z. Xiu, Optimization of a fed-batch bioreactor for 1, 3-propanediol production using hybrid nonlinear optimal control, Journal of Process Control, 24 (2014), 1556-1569.

[31]

J. YiD. HuangS. Fu and T. Li, Optimized relative transformation matrix using bacterial foraging algorithm for process fault detection, IEEE Transactions on Industrial Electronics, 63 (2016), 2595-2605. doi: 10.1109/TIE.2016.2515057.

[32]

W. C. YueX. F. ChenW. H. Gui and H. L. Zhang, A knowledge reasoning Fuzzy-Bayesian network for root cause analysis of abnormal aluminum electrolysis cell condition, Fronters of Chemical Science and Engineering, 11 (2017), 414-428. doi: 10.1007/s11705-017-1663-x.

[33]

S. P. Zeng and F. W. Cui, Dynamic decision model for amount of AlF3 addition in industrial aluminum electrolysis, International Conference on Mechatronics, Robotics and Automation, (2015), 307-318.

[34]

S. P. ZengS. S. Wang and Y. X. Qu, Control of temperature and aluminum fluoride concentration based on model prediction in aluminum electrolysis, Advances in Materials Science & Engineering, (2014), 1-5.

[35]

E. ZentenoZ. A. KhanM. Isaksson and P. Handel, Finding structural information about RF power amplifiers using an orthogonal nonparametric kernel smoothing estimator, IEEE Transactions on Vehicular Technology, 65 (2016), 2883-2889. doi: 10.1109/TVT.2015.2434497.

[36]

S. Q. ZhanM. LiJ. M. Zhou and Y. W. Zhou, CFD simulation of dissolution process of alumina in an aluminum reduction cell with two-particle phase population balance model, Applied Thermal Engineering, 73 (2014), 805-818. doi: 10.1016/j.applthermaleng.2014.08.040.

Figure 1.  Sketch of aluminum reduction cell
Figure 2.  Sketch of binary phase diagram of NaF-AlF3
Figure 3.  Internal and external environments of aluminum reduction process
Figure 4.  Response of AlF3 addition with respect to relationship between electrolyte temperature and SD
Figure 5.  Solution for amount decision concerning AlF3 addition
Figure 6.  Naïve Bayes for SD state evaluation
Figure 7.  Naïve Bayes classifier for dSD state estimation
Figure 8.  (a) Fuzzy inference rules for evaluation and (b) evaluation grade based on SD and dSD
Figure 9.  Changes with balance point variation
Figure 10.  Classification results for NBC and ENBC with seeds data set
Figure 11.  Classification results for NBC and ENBC with banknote data set
Figure 12.  Classification result-based NBC and ENBC with data set of cell
Figure 13.  The evaluation grades with the better AlF3 addition
Figure 14.  Values of characteristic parameters over two months
Figure 15.  Evaluation grades for cell condition
Figure 16.  (a) Comparison between actual feeding times and those based on proposed strategy; (b) comparison between actual feeding times and those based on linear programming model; (c) Comparison between actual feeding times and those based on NBC and mass balance
Figure 17.  Error comparison of feeding times based on proposed strategy and existing strategy
Table 1.  Eight characteristic parameters.
ParameterAb.ValueRole analysis
Aluminum levelAL20-23 cmThe height of the molten aluminum. A higher AL leads to greater heat loss, and vice versa. A suitable AL can stabilize the cell voltage.
Molecular ratioMR2.64-3.0This affects the dissolution of the alumina in the electrolyte, with a higher MR leading to a lower SD, and vice versa.
Electrolyte levelEL23-28 cmThis stabilize the thermal balance of the cell. Thus, the thermal balance is robust with a suitable EL.
WavingWA0-20 mvA strong low-frequency noise may be due to insufficient energy intake for the cell.
VibrationVI0-50 mvVI is an indicator of the stability of the cell. A greater VI is more likely for a cold cell.
Under/over number ratioUO0.75-1The UO is the ratio between the under and over feeding times. A smaller UO is more likely for a cold cell, and vice versa.
Tapping amountTA2.9-3.05 tonThe TA has a great influence on the energy balance. A greater TA is more likely for a hot cell, and vice versa.
Electrolyte temperatureET955-965℃This affects the entire operation condition of the cell. A higher temperature is more likely for a hot cell, and vice versa.
ParameterAb.ValueRole analysis
Aluminum levelAL20-23 cmThe height of the molten aluminum. A higher AL leads to greater heat loss, and vice versa. A suitable AL can stabilize the cell voltage.
Molecular ratioMR2.64-3.0This affects the dissolution of the alumina in the electrolyte, with a higher MR leading to a lower SD, and vice versa.
Electrolyte levelEL23-28 cmThis stabilize the thermal balance of the cell. Thus, the thermal balance is robust with a suitable EL.
WavingWA0-20 mvA strong low-frequency noise may be due to insufficient energy intake for the cell.
VibrationVI0-50 mvVI is an indicator of the stability of the cell. A greater VI is more likely for a cold cell.
Under/over number ratioUO0.75-1The UO is the ratio between the under and over feeding times. A smaller UO is more likely for a cold cell, and vice versa.
Tapping amountTA2.9-3.05 tonThe TA has a great influence on the energy balance. A greater TA is more likely for a hot cell, and vice versa.
Electrolyte temperatureET955-965℃This affects the entire operation condition of the cell. A higher temperature is more likely for a hot cell, and vice versa.
Table 2.  Fuzzy numbers definitions for SD and its trends.
Definitions for SDDefinitions for dSD
LabelMeaningMembershipLabelMeaningMembership
VLVery low $\mu \left( VL \right)=P\left( VL\left| {{{\bf{x}}}_{i}} \right. \right)$HNHigh negative $\mu \left( HN \right)=P\left( HN\left| \Delta {{{\bf{x}}}_{i}} \right. \right)$
LLLittle low $\mu \left( LL \right)=P\left( LL\left| {{{\bf{x}}}_{i}} \right. \right)$LNLow negative $\mu \left( LN \right)=P\left( LN\left| \Delta {{{\bf{x}}}_{i}} \right. \right)$
NNormal $\mu \left( N\right)=P\left( N\left| {{{\bf{x}}}_{i}} \right. \right)$Zzero $\mu \left( N \right)=P\left( N\left| \Delta {{{\bf{x}}}_{i}} \right. \right)$
LHLittle high $\mu \left( LP \right)=P\left( LP\left| {{{\bf{x}}}_{i}} \right. \right)$LPLow positive $\mu \left( LH \right)=P\left( LH\left| \Delta {{{\bf{x}}}_{i}} \right. \right)$
VHVery high $\mu \left( LH \right)=P\left( LH\left| {{{\bf{x}}}_{i}} \right. \right)$HPHigh positive $\mu \left( HP \right)=P\left( HP\left| \Delta {{{\bf{x}}}_{i}} \right. \right)$
Definitions for SDDefinitions for dSD
LabelMeaningMembershipLabelMeaningMembership
VLVery low $\mu \left( VL \right)=P\left( VL\left| {{{\bf{x}}}_{i}} \right. \right)$HNHigh negative $\mu \left( HN \right)=P\left( HN\left| \Delta {{{\bf{x}}}_{i}} \right. \right)$
LLLittle low $\mu \left( LL \right)=P\left( LL\left| {{{\bf{x}}}_{i}} \right. \right)$LNLow negative $\mu \left( LN \right)=P\left( LN\left| \Delta {{{\bf{x}}}_{i}} \right. \right)$
NNormal $\mu \left( N\right)=P\left( N\left| {{{\bf{x}}}_{i}} \right. \right)$Zzero $\mu \left( N \right)=P\left( N\left| \Delta {{{\bf{x}}}_{i}} \right. \right)$
LHLittle high $\mu \left( LP \right)=P\left( LP\left| {{{\bf{x}}}_{i}} \right. \right)$LPLow positive $\mu \left( LH \right)=P\left( LH\left| \Delta {{{\bf{x}}}_{i}} \right. \right)$
VHVery high $\mu \left( LH \right)=P\left( LH\left| {{{\bf{x}}}_{i}} \right. \right)$HPHigh positive $\mu \left( HP \right)=P\left( HP\left| \Delta {{{\bf{x}}}_{i}} \right. \right)$
Table 3.  Detailed statistical results.
Data set namesNumber of instancesAccuracy rate
trainingtestattributesENBCNBC
hline seeds1357570.97330.8933
banknote12977550.94670.8267
Data set namesNumber of instancesAccuracy rate
trainingtestattributesENBCNBC
hline seeds1357570.97330.8933
banknote12977550.94670.8267
Table 4.  Labels for each data group.
LabelsIndexParameters
MREL (cm)WA (mv)VI (mv)UOTA kgET (℃)AL (cm)
VL13.05335377990.76302395922.0
LL22.9832891140.52298896023.0
N32.79236100.40281197425.5
LH42.8731261.18278797621.0
VH52.5225350.33289698524.0
LabelsIndexParameters
MREL (cm)WA (mv)VI (mv)UOTA kgET (℃)AL (cm)
VL13.05335377990.76302395922.0
LL22.9832891140.52298896023.0
N32.79236100.40281197425.5
LH42.8731261.18278797621.0
VH52.5225350.33289698524.0
[1]

Joseph Bayara, André Conseibo, Moussa Ouattara, Artibano Micali. Train algebras of degree 2 and exponent 3. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1371-1386. doi: 10.3934/dcdss.2011.4.1371

[2]

Carolin Kreisbeck. A note on $3$d-$1$d dimension reduction with differential constraints. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 55-73. doi: 10.3934/dcdss.2017003

[3]

Francisco Braun, José Ruidival dos Santos Filho. The real jacobian conjecture on $\R^2$ is true when one of the components has degree 3. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 75-87. doi: 10.3934/dcds.2010.26.75

[4]

Cuncai Hua, Guanrong Chen, Qunhong Li, Juhong Ge. Converting a general 3-D autonomous quadratic system to an extended Lorenz-type system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 475-488. doi: 10.3934/dcdsb.2011.16.475

[5]

Carles Simó. Measuring the total amount of chaos in some Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5135-5164. doi: 10.3934/dcds.2014.34.5135

[6]

Alan Beggs. Learning in monotone bayesian games. Journal of Dynamics & Games, 2015, 2 (2) : 117-140. doi: 10.3934/jdg.2015.2.117

[7]

Yinying Duan, Yong Ye, Zhichao Liu. Risk assessment for enterprise merger and acquisition via multiple classifier fusion. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 747-759. doi: 10.3934/dcdss.2019049

[8]

Masoumeh Dashti, Stephen Harris, Andrew Stuart. Besov priors for Bayesian inverse problems. Inverse Problems & Imaging, 2012, 6 (2) : 183-200. doi: 10.3934/ipi.2012.6.183

[9]

Mila Nikolova. Model distortions in Bayesian MAP reconstruction. Inverse Problems & Imaging, 2007, 1 (2) : 399-422. doi: 10.3934/ipi.2007.1.399

[10]

Matthew M. Dunlop, Andrew M. Stuart. The Bayesian formulation of EIT: Analysis and algorithms. Inverse Problems & Imaging, 2016, 10 (4) : 1007-1036. doi: 10.3934/ipi.2016030

[11]

M.T. Boudjelkha. Extended Riemann Bessel functions. Conference Publications, 2005, 2005 (Special) : 121-130. doi: 10.3934/proc.2005.2005.121

[12]

Carsten Burstedde. On the numerical evaluation of fractional Sobolev norms. Communications on Pure & Applied Analysis, 2007, 6 (3) : 587-605. doi: 10.3934/cpaa.2007.6.587

[13]

Guillaume Bal, Ian Langmore, Youssef Marzouk. Bayesian inverse problems with Monte Carlo forward models. Inverse Problems & Imaging, 2013, 7 (1) : 81-105. doi: 10.3934/ipi.2013.7.81

[14]

Nicolas Lermé, François Malgouyres, Dominique Hamoir, Emmanuelle Thouin. Bayesian image restoration for mosaic active imaging. Inverse Problems & Imaging, 2014, 8 (3) : 733-760. doi: 10.3934/ipi.2014.8.733

[15]

Liming Yang, Yannan Chao. A new semi-supervised classifier based on maximum vector-angular margin. Journal of Industrial & Management Optimization, 2017, 13 (2) : 609-622. doi: 10.3934/jimo.2016035

[16]

Zuguo Chen, Yonggang Li, Xiaofang Chen, Chunhua Yang, Weihua Gui. Anode effect prediction based on collaborative two-dimensional forecast model in aluminum electrolysis production. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-24. doi: 10.3934/jimo.2018060

[17]

Zalman Balanov, Wieslaw Krawcewicz, Haibo Ruan. Applied equivariant degree, part I: An axiomatic approach to primary degree. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 983-1016. doi: 10.3934/dcds.2006.15.983

[18]

Anna M. Barry, Esther WIdiasih, Richard Mcgehee. Nonsmooth frameworks for an extended Budyko model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2447-2463. doi: 10.3934/dcdsb.2017125

[19]

Zari Dzalilov, Iradj Ouveysi, Alexander Rubinov. An extended lifetime measure for telecommunication network. Journal of Industrial & Management Optimization, 2008, 4 (2) : 329-337. doi: 10.3934/jimo.2008.4.329

[20]

Michael Baake, John A. G. Roberts, Reem Yassawi. Reversing and extended symmetries of shift spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 835-866. doi: 10.3934/dcds.2018036

2017 Impact Factor: 0.994

Metrics

  • PDF downloads (8)
  • HTML views (62)
  • Cited by (0)

[Back to Top]