2012, 9(1): 123-145. doi: 10.3934/mbe.2012.9.123

A statistical approach to the use of control entropy identifies differences in constraints of gait in highly trained versus untrained runners

1. 

Department of Mathematics & Computer Science, Clarkson University, Potsdam, NY 13676, United States, United States, United States

2. 

Applied Physiology Laboratory, Eastern Michigan University, Ypsilanti, MI 48197, United States, United States

Received  February 2011 Revised  July 2011 Published  December 2011

Control entropy (CE) is a complexity analysis suitable for dynamic, non-stationary conditions which allows the inference of the control effort of a dynamical system generating the signal [4]. These characteristics make CE a highly relevant time varying quantity relevant to the dynamic physiological responses associated with running. Using High Resolution Accelerometry (HRA) signals we evaluate here constraints of running gait, from two different groups of runners, highly trained collegiate and untrained runners. To this end, we further develop the control entropy (CE) statistic to allow for group analysis to examine the non-linear characteristics of movement patterns in highly trained runners with those of untrained runners, to gain insight regarding gaits that are optimal for running. Specifically, CE develops response time series of individuals descriptive of the control effort; a group analysis of these shapes developed here uses Karhunen Loeve Analysis (KL) modes of these time series which are compared between groups by application of a Hotelling $T^{2}$ test to these group response shapes. We find that differences in the shape of the CE response exist within groups, between axes for untrained runners (vertical vs anterior-posterior and mediolateral vs anterior-posterior) and trained runners (mediolateral vs anterior-posterior). Also shape differences exist between groups by axes (vertical vs mediolateral). Further, the CE, as a whole, was higher in each axis in trained vs untrained runners. These results indicate that the approach can provide unique insight regarding the differing constraints on running gait in highly trained and untrained runners when running under dynamic conditions. Further, the final point indicates trained runners are less constrained than untrained runners across all running speeds.
Citation: Rana D. Parshad, Stephen J. McGregor, Michael A. Busa, Joseph D. Skufca, Erik Bollt. A statistical approach to the use of control entropy identifies differences in constraints of gait in highly trained versus untrained runners. Mathematical Biosciences & Engineering, 2012, 9 (1) : 123-145. doi: 10.3934/mbe.2012.9.123
References:
[1]

W. Aziz and M. Arif, Complexity analysis of stride interval time series by threshold dependent symbolic entropy, Eur. J. Appl. Physiol., 98 (2006), 30-40. doi: 10.1007/s00421-006-0226-5.

[2]

O. Beauchet, V. Dubost, F. R. Herrmann and R. W. Kressig, Stride-to-stride variability while backward counting among healthy young adults, J. Neuroeng. Rehabil., 2 (2005), 26. doi: 10.1186/1743-0003-2-26.

[3]

B. R. Bloem, V. V. Valkenburg, M. Slabbekoorn and M. D. Willemsen, The multiple tasks test: Development and normal strategies, Gait Posture, 14 (2001), 191-202.

[4]

E. M. Bollt, J. D. Skufca and S. J. McGregor, Control entropy: A complexity measure for nonstationary signals, Mathematical Biosciences and Engineering, 6 (2009), 1-25. doi: 10.3934/mbe.2009.6.1.

[5]

U. H. Buzzi and B. D. Ulrich, Dynamic stability of gait cycles as a function of speed and system constraints, Motor Control, 8 (2004), 241-254.

[6]

P. Cavanagh, The mechanics of distance running: A historical perspective, in "Biomechanics of Distance Running" (ed. P. Cavanagh), Human Kinetics, 1990.

[7]

T. M. Cover and J. A. Thomas, "Elements of Information Theory," Wiley Series in Telecommunications, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1991. doi: 10.1002/0471200611.

[8]

K. Davids, S. Bennett and K. M. Newell, "Movement System Variability," Human Kinetics, Champaign, IL, 2006.

[9]

D. P. Ferris, G. S. Sawicki and M. A. Daley, A physiologist's perspective on robotic exoskeletons for human locomotion, Int. J. HR, 4 (2007), 507-528. doi: 10.1142/S0219843607001138.

[10]

A. D. Georgoulis, C. Moraiti, S. Ristanis and N. Stergiou, A novel approach to measure variability in the anterior cruciate ligament deficient knee during walking: The use of the approximate entropy in orthopaedics, J. Clin. Monit. Comput., 20 (2006), 11-18. doi: 10.1007/s10877-006-1032-7.

[11]

P. S. Glazier and K. Davids, Constraints on the complete optimization of human motion, Sports Med., 39 (2009), 15-28. doi: 10.2165/00007256-200939010-00002.

[12]

G. H. Golub and C. F. Van Loan, "Matrix Computations," The Johns Hopkins University Press, 1996.

[13]

P. Grassberger and I. Procaccia, Estimation of the Kolmogorov entropy from a chaotic signal, Physical Review A, 28 (1983), 2591-2593. doi: 10.1103/PhysRevA.28.2591.

[14]

J. M. Hausdorff, Gait dynamics, fractals and falls: Finding meaning in the stride-to-stride fluctuations of human walking, Hum. Mov. Sci., 26 (2007), 555-589.

[15]

H. Kantz and T. Schreiber, "Nonlinear Time Series Analysis," Second edition, Cambridge University Press, Cambridge, 2004.

[16]

C. K. Karmakar, A. H. Khandoker, R. K. Begg, M. Palaniswami and S. Taylor, Understanding ageing effects by approximate entropy analysis of gait variability, Conf. Proc. IEEE Eng. Med. Biol. Soc., (2007), 1965-1968. doi: 10.1109/IEMBS.2007.4352703.

[17]

A. H. Khandoker, M. Palaniswami and R. K. Begg, A comparative study on approximate entropy measure and poincaire plot indexes of minimum foot clearance variability in the elderly during walking, J. Neuroeng. Rehabil., 5 (2008), 4. doi: 10.1186/1743-0003-5-4.

[18]

M. J. Kurz and N. Stergiou, The aging human neuromuscular system expresses less certainty for selecting joint kinematics during gait, Neurosci. Lett., 348 (2003), 155-158. doi: 10.1016/S0304-3940(03)00736-5.

[19]

F. Liao, J. Wang and P. He, Multi-resolution entropy analysis of gait symmetry in neurological degenerative diseases and amyotrophic lateral sclerosis, Med. Eng. Phys., 30 (2008), 299-310. doi: 10.1016/j.medengphy.2007.04.014.

[20]

K. V. Mardia, J. T. Kent and J. M. Bibby, "Multivariate Analysis," Probability and Mathematical Statistics: A Series of Monographs and Textbooks, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York-Toronto, Ont., 1979.

[21]

S. J. McGregor, M. A. Busa, J. D. Skufca, J. A. Yaggie and E. M. Bollt, Control entropy identifies differential changes in complexity of walking and running gait patterns with increasing speed in highly trained runners, Chaos, 19 (2009), 026109, 13 pp.

[22]

S. J. McGregor, M. A. Busa, J. A. Yaggie and E. M. Bollt, High resolution MEMS accelerometers to estimate VO2 and compare running mechanics between highly trained inter-collegiate and untrained runners, PLoS One, 4 (2009), e7355. doi: 10.1371/journal.pone.0007355.

[23]

S. P. Messier, C. Legault, C. R. Schoenlank, J. J. Newman, D. F. Martin and P. Devita, Risk factors and mechanisms of knee injury in runners, Med. Sci. Sports Exerc., 40 (2008), 1873-1879. doi: 10.1249/MSS.0b013e31817ed272.

[24]

D. J. Miller, N. Stergiou and M. J. Kurz, An improved surrogate method for detecting the presence of chaos in gait, J. Biomech., 39 (2006), 2873-2876. doi: 10.1016/j.jbiomech.2005.10.019.

[25]

R. Moe-Nilssen, A new method for evaluating motor control in gait under real-life environmental conditions. Part 2: Gait analysis, Clin. Biomech. (Bristol, Avon), 13 (1998), 328-335. doi: 10.1016/S0268-0033(98)00090-4.

[26]

K. M. Newell, Constraints on the development of coordination, in "Motor Development in Children: Aspects of Coordination and Control" (eds. M. G. Wade and W. H. Dordect), Nihjoff, (1986), 341-360.

[27]

K. M. Newell and D. E. Vaillancourt, Dimensional change in motor learning, Hum. Mov. Sci., 20 (2001), 695-715. doi: 10.1016/S0167-9457(01)00073-2.

[28]

S. M. Pincus, Approximate entropy as a measure of system complexity, Proceedings of the National Academy of Sciences of the United States of America, 88 (1991), 2297-2301.

[29]

S. M. Pincus, Assessing serial irregularity and its implications for health, Annals of the New York Academy of Sciences, 954 (2001), 245. doi: 10.1111/j.1749-6632.2001.tb02755.x.

[30]

A. Renyi, On measures of entropy and information, Proceedings of the 4th Berkeley Sympo- sium on Mathematical Statistics and Probability, 1 (1961), 547-561.

[31]

J. S. Richman and J. R. Moorman, Physiological time-series analysis using approximate en- tropy and sample entropy, American Journal of Physiology- Heart and Circulatory Physiology, 278 (2000), 2039-2049.

[32]

C. Robinson, "Infinite Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDE and the Theory of Global Attractors," 2nd edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001.

[33]

L. A. Schrodt, V. S. Mercer, C. A. Giuliani and M. Hartman, Characteristics of stepping over an obstacle in community dwelling older adults under dual-task conditions, Gait Posture, 19 (2004), 279-287. doi: 10.1016/S0966-6362(03)00067-5.

[34]

C. E. Shannon and W. Weaver, "The Mathematical Theory of Information," Uni- versity of Illinois Press, 97, Urbana, 1949.

[35]

J. S. Slawinski and V. L. Billat, Difference in mechanical and energy cost between highly, well, and nontrained runners, Med. Sci. Sports Exerc., 36 (2004), 1440-1446. doi: 10.1249/01.MSS.0000135785.68760.96.

[36]

G. Yogev-Seligmann, J. M. Hausdorff and N. Giladi, The role of executive function and attention in gait, Mov. Disord., 23 (2008), 329-342. doi: 10.1002/mds.21720.

[37]

M. Joyner and E. Coyle, Endurance exercise performance: The physiology of champions, The Journal of Physiology, 586 (2008), 35-44.

[38]

J. Lin, E. Keogh, S. Lonardi and B. Chiu, A symbolic representation of time series, with implications for streaming algorithms, Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery, (2003), 2-11.

[39]

Y. Nakayama, K. Kudo and T. Ohtsuki, Variability and fluctuation in running gait cycle of trained runners and non-runners, Gait Posture, 31 (2009), 331-333. doi: 10.1016/j.gaitpost.2009.12.003.

[40]

K. Jordan and K. M. Newell, The structure of variability in human walking and running is speed-dependent, Exerc. Sport Sci. Rev., 36 (2008), 200-204. doi: 10.1097/JES.0b013e3181877d71.

show all references

References:
[1]

W. Aziz and M. Arif, Complexity analysis of stride interval time series by threshold dependent symbolic entropy, Eur. J. Appl. Physiol., 98 (2006), 30-40. doi: 10.1007/s00421-006-0226-5.

[2]

O. Beauchet, V. Dubost, F. R. Herrmann and R. W. Kressig, Stride-to-stride variability while backward counting among healthy young adults, J. Neuroeng. Rehabil., 2 (2005), 26. doi: 10.1186/1743-0003-2-26.

[3]

B. R. Bloem, V. V. Valkenburg, M. Slabbekoorn and M. D. Willemsen, The multiple tasks test: Development and normal strategies, Gait Posture, 14 (2001), 191-202.

[4]

E. M. Bollt, J. D. Skufca and S. J. McGregor, Control entropy: A complexity measure for nonstationary signals, Mathematical Biosciences and Engineering, 6 (2009), 1-25. doi: 10.3934/mbe.2009.6.1.

[5]

U. H. Buzzi and B. D. Ulrich, Dynamic stability of gait cycles as a function of speed and system constraints, Motor Control, 8 (2004), 241-254.

[6]

P. Cavanagh, The mechanics of distance running: A historical perspective, in "Biomechanics of Distance Running" (ed. P. Cavanagh), Human Kinetics, 1990.

[7]

T. M. Cover and J. A. Thomas, "Elements of Information Theory," Wiley Series in Telecommunications, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1991. doi: 10.1002/0471200611.

[8]

K. Davids, S. Bennett and K. M. Newell, "Movement System Variability," Human Kinetics, Champaign, IL, 2006.

[9]

D. P. Ferris, G. S. Sawicki and M. A. Daley, A physiologist's perspective on robotic exoskeletons for human locomotion, Int. J. HR, 4 (2007), 507-528. doi: 10.1142/S0219843607001138.

[10]

A. D. Georgoulis, C. Moraiti, S. Ristanis and N. Stergiou, A novel approach to measure variability in the anterior cruciate ligament deficient knee during walking: The use of the approximate entropy in orthopaedics, J. Clin. Monit. Comput., 20 (2006), 11-18. doi: 10.1007/s10877-006-1032-7.

[11]

P. S. Glazier and K. Davids, Constraints on the complete optimization of human motion, Sports Med., 39 (2009), 15-28. doi: 10.2165/00007256-200939010-00002.

[12]

G. H. Golub and C. F. Van Loan, "Matrix Computations," The Johns Hopkins University Press, 1996.

[13]

P. Grassberger and I. Procaccia, Estimation of the Kolmogorov entropy from a chaotic signal, Physical Review A, 28 (1983), 2591-2593. doi: 10.1103/PhysRevA.28.2591.

[14]

J. M. Hausdorff, Gait dynamics, fractals and falls: Finding meaning in the stride-to-stride fluctuations of human walking, Hum. Mov. Sci., 26 (2007), 555-589.

[15]

H. Kantz and T. Schreiber, "Nonlinear Time Series Analysis," Second edition, Cambridge University Press, Cambridge, 2004.

[16]

C. K. Karmakar, A. H. Khandoker, R. K. Begg, M. Palaniswami and S. Taylor, Understanding ageing effects by approximate entropy analysis of gait variability, Conf. Proc. IEEE Eng. Med. Biol. Soc., (2007), 1965-1968. doi: 10.1109/IEMBS.2007.4352703.

[17]

A. H. Khandoker, M. Palaniswami and R. K. Begg, A comparative study on approximate entropy measure and poincaire plot indexes of minimum foot clearance variability in the elderly during walking, J. Neuroeng. Rehabil., 5 (2008), 4. doi: 10.1186/1743-0003-5-4.

[18]

M. J. Kurz and N. Stergiou, The aging human neuromuscular system expresses less certainty for selecting joint kinematics during gait, Neurosci. Lett., 348 (2003), 155-158. doi: 10.1016/S0304-3940(03)00736-5.

[19]

F. Liao, J. Wang and P. He, Multi-resolution entropy analysis of gait symmetry in neurological degenerative diseases and amyotrophic lateral sclerosis, Med. Eng. Phys., 30 (2008), 299-310. doi: 10.1016/j.medengphy.2007.04.014.

[20]

K. V. Mardia, J. T. Kent and J. M. Bibby, "Multivariate Analysis," Probability and Mathematical Statistics: A Series of Monographs and Textbooks, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York-Toronto, Ont., 1979.

[21]

S. J. McGregor, M. A. Busa, J. D. Skufca, J. A. Yaggie and E. M. Bollt, Control entropy identifies differential changes in complexity of walking and running gait patterns with increasing speed in highly trained runners, Chaos, 19 (2009), 026109, 13 pp.

[22]

S. J. McGregor, M. A. Busa, J. A. Yaggie and E. M. Bollt, High resolution MEMS accelerometers to estimate VO2 and compare running mechanics between highly trained inter-collegiate and untrained runners, PLoS One, 4 (2009), e7355. doi: 10.1371/journal.pone.0007355.

[23]

S. P. Messier, C. Legault, C. R. Schoenlank, J. J. Newman, D. F. Martin and P. Devita, Risk factors and mechanisms of knee injury in runners, Med. Sci. Sports Exerc., 40 (2008), 1873-1879. doi: 10.1249/MSS.0b013e31817ed272.

[24]

D. J. Miller, N. Stergiou and M. J. Kurz, An improved surrogate method for detecting the presence of chaos in gait, J. Biomech., 39 (2006), 2873-2876. doi: 10.1016/j.jbiomech.2005.10.019.

[25]

R. Moe-Nilssen, A new method for evaluating motor control in gait under real-life environmental conditions. Part 2: Gait analysis, Clin. Biomech. (Bristol, Avon), 13 (1998), 328-335. doi: 10.1016/S0268-0033(98)00090-4.

[26]

K. M. Newell, Constraints on the development of coordination, in "Motor Development in Children: Aspects of Coordination and Control" (eds. M. G. Wade and W. H. Dordect), Nihjoff, (1986), 341-360.

[27]

K. M. Newell and D. E. Vaillancourt, Dimensional change in motor learning, Hum. Mov. Sci., 20 (2001), 695-715. doi: 10.1016/S0167-9457(01)00073-2.

[28]

S. M. Pincus, Approximate entropy as a measure of system complexity, Proceedings of the National Academy of Sciences of the United States of America, 88 (1991), 2297-2301.

[29]

S. M. Pincus, Assessing serial irregularity and its implications for health, Annals of the New York Academy of Sciences, 954 (2001), 245. doi: 10.1111/j.1749-6632.2001.tb02755.x.

[30]

A. Renyi, On measures of entropy and information, Proceedings of the 4th Berkeley Sympo- sium on Mathematical Statistics and Probability, 1 (1961), 547-561.

[31]

J. S. Richman and J. R. Moorman, Physiological time-series analysis using approximate en- tropy and sample entropy, American Journal of Physiology- Heart and Circulatory Physiology, 278 (2000), 2039-2049.

[32]

C. Robinson, "Infinite Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDE and the Theory of Global Attractors," 2nd edition, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001.

[33]

L. A. Schrodt, V. S. Mercer, C. A. Giuliani and M. Hartman, Characteristics of stepping over an obstacle in community dwelling older adults under dual-task conditions, Gait Posture, 19 (2004), 279-287. doi: 10.1016/S0966-6362(03)00067-5.

[34]

C. E. Shannon and W. Weaver, "The Mathematical Theory of Information," Uni- versity of Illinois Press, 97, Urbana, 1949.

[35]

J. S. Slawinski and V. L. Billat, Difference in mechanical and energy cost between highly, well, and nontrained runners, Med. Sci. Sports Exerc., 36 (2004), 1440-1446. doi: 10.1249/01.MSS.0000135785.68760.96.

[36]

G. Yogev-Seligmann, J. M. Hausdorff and N. Giladi, The role of executive function and attention in gait, Mov. Disord., 23 (2008), 329-342. doi: 10.1002/mds.21720.

[37]

M. Joyner and E. Coyle, Endurance exercise performance: The physiology of champions, The Journal of Physiology, 586 (2008), 35-44.

[38]

J. Lin, E. Keogh, S. Lonardi and B. Chiu, A symbolic representation of time series, with implications for streaming algorithms, Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery, (2003), 2-11.

[39]

Y. Nakayama, K. Kudo and T. Ohtsuki, Variability and fluctuation in running gait cycle of trained runners and non-runners, Gait Posture, 31 (2009), 331-333. doi: 10.1016/j.gaitpost.2009.12.003.

[40]

K. Jordan and K. M. Newell, The structure of variability in human walking and running is speed-dependent, Exerc. Sport Sci. Rev., 36 (2008), 200-204. doi: 10.1097/JES.0b013e3181877d71.

[1]

Luis A. Fernández, Cecilia Pola. Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2577-2612. doi: 10.3934/dcdsb.2018266

[2]

Vladimir Turetsky, Valery Y. Glizer. Optimal decision in a Statistical Process Control with cubic loss. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2903-2926. doi: 10.3934/jimo.2021096

[3]

Adriano Da Silva, Christoph Kawan. Invariance entropy of hyperbolic control sets. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 97-136. doi: 10.3934/dcds.2016.36.97

[4]

Leonor Cruzeiro. The VES hypothesis and protein misfolding. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1033-1046. doi: 10.3934/dcdss.2011.4.1033

[5]

Wenming Li, Yongge Tian, Ruixia Yuan. Statistical analysis of a linear regression model with restrictions and superfluous variables. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022079

[6]

Valery Y. Glizer, Vladimir Turetsky, Emil Bashkansky. Statistical process control optimization with variable sampling interval and nonlinear expected loss. Journal of Industrial and Management Optimization, 2015, 11 (1) : 105-133. doi: 10.3934/jimo.2015.11.105

[7]

Zehan Lin, Chongbin Xu, Caidi Zhao, Chujin Li. Statistical solution and Kolmogorov entropy for the impulsive discrete Klein-Gordon-Schrödinger-type equations. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022065

[8]

Xufeng Guo, Gang Liao, Wenxiang Sun, Dawei Yang. On the hybrid control of metric entropy for dominated splittings. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5011-5019. doi: 10.3934/dcds.2018219

[9]

Erik M. Bollt, Joseph D. Skufca, Stephen J . McGregor. Control entropy: A complexity measure for nonstationary signals. Mathematical Biosciences & Engineering, 2009, 6 (1) : 1-25. doi: 10.3934/mbe.2009.6.1

[10]

C. Xiong, J.P. Miller, F. Gao, Y. Yan, J.C. Morris. Testing increasing hazard rate for the progression time of dementia. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 813-821. doi: 10.3934/dcdsb.2004.4.813

[11]

Jean-François Biasse, Michael J. Jacobson, Jr.. Smoothness testing of polynomials over finite fields. Advances in Mathematics of Communications, 2014, 8 (4) : 459-477. doi: 10.3934/amc.2014.8.459

[12]

Fumio Ishizaki. Analysis of the statistical time-access fairness index of one-bit feedback fair scheduler. Numerical Algebra, Control and Optimization, 2011, 1 (4) : 675-689. doi: 10.3934/naco.2011.1.675

[13]

Ikuo Arizono, Yasuhiko Takemoto. Statistical mechanics approach for steady-state analysis in M/M/s queueing system with balking. Journal of Industrial and Management Optimization, 2022, 18 (1) : 25-44. doi: 10.3934/jimo.2020141

[14]

Fabio Ancona, Laura Caravenna, Annalisa Cesaroni, Giuseppe M. Coclite, Claudio Marchi, Andrea Marson. Analysis and control on networks: Trends and perspectives. Networks and Heterogeneous Media, 2017, 12 (3) : i-ii. doi: 10.3934/nhm.201703i

[15]

Fabio Ancona, Laura Caravenna, Annalisa Cesaroni, Giuseppe M. Coclite, Claudio Marchi, Andrea Marson. Analysis and control on networks: Trends and perspectives. Networks and Heterogeneous Media, 2017, 12 (2) : i-ii. doi: 10.3934/nhm.201702i

[16]

Fritz Colonius. Invariance entropy, quasi-stationary measures and control sets. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2093-2123. doi: 10.3934/dcds.2018086

[17]

Tomáš Smejkal, Jiří Mikyška, Jaromír Kukal. Comparison of modern heuristics on solving the phase stability testing problem. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1161-1180. doi: 10.3934/dcdss.2020227

[18]

Philippe Destuynder, Caroline Fabre. Few remarks on the use of Love waves in non destructive testing. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 427-444. doi: 10.3934/dcdss.2016005

[19]

Amina Eladdadi, Noura Yousfi, Abdessamad Tridane. Preface: Special issue on cancer modeling, analysis and control. Discrete and Continuous Dynamical Systems - B, 2013, 18 (4) : i-iii. doi: 10.3934/dcdsb.2013.18.4i

[20]

Yanming Ge. Analysis of airline seat control with region factor. Journal of Industrial and Management Optimization, 2012, 8 (2) : 363-378. doi: 10.3934/jimo.2012.8.363

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (2)

[Back to Top]