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Chen, Lansun

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 Author ID: chen.lansun Published as: Chen Lan-sun; Chen, L.; Chen, L. S.; Chen, L.-S.; Chen, Lan Sun; Chen, Lan-Sun; Chen, Lan-sun; Chen, Lansun External Links: MGP · dblp
 Documents Indexed: 527 Publications since 1970, including 2 Books Reviewing Activity: 130 Reviews
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Co-Authors

 4 single-authored 50 Jiao, Jianjun 32 Song, Xinyu 26 Meng, Xinzhu 21 Zhang, Shuwen 19 Guo, Hongjian 19 Pang, Guoping 17 Cai, Shaohong 17 Wei, Chunjin 17 Zhang, Yujuan 16 Gao, Shujing 16 Liu, Bing 16 Teng, Zhi-dong 16 Xiao, Yanni 16 Xu, Rui 16 Zhang, Hong 16 Zhao, Zhong 15 Lu, Zhonghua 14 Pei, Yongzhen 14 Sun, Lihua 13 Tang, Sanyi 13 Wang, Fengyan 11 Fu, Jinbo 11 Li, Changguo 10 Cui, Jingan 10 Hui, Jing 10 Liu, Zhijun 9 Dong, Lingzhen 9 Liu, Shengqiang 9 Wang, Tieying 8 Li, Zuxiong 8 Shi, Ruiqing 8 Zhang, Xinan 7 Liu, Kaiyuan 7 Sun, Shulin 7 Xu, Weijian 6 Chi, Xuebin 6 Liang, Zhaojun 6 Liu, Xianning 6 Tan, Dejun 6 Tan, Yuanshun 6 Tian, Yuan 6 Zeng, Guangzhao 5 Chaplain, Mark A. J. 5 Lu, Zhengyi 5 Luo, Guilie 5 Sun, Kaibiao 5 Tao, Fengmei 5 Wang, Limin 5 Zhao, Min 4 Dai, Chuanjun 4 Georgescu, Paul 4 Hao, Feilong 4 Li, Hua 4 Nieto Roig, Juan Jose 4 Wang, Chunhua 4 Wang, Wendi 3 Chen, Jufang 3 Chen, Shidong 3 Cheng, Rongfu 3 Gui, Zhanji 3 Hao, Chunping 3 Li, Limei 3 Liang, Zhiqing 3 Liu, Shaoying 3 Sun, Mingjing 3 Xiu, Zhilong 3 Zhang, Jingru 3 Zhang, Meng 3 Zhu, Rongping 2 Agarwal, Ravi P. 2 Cao, Xiantong 2 Cheng, Huidong 2 Dou, Jiawei 2 Fan, Meng 2 Huang, Mingzhan 2 Ji, Xuehui 2 Jiang, Xiaowu 2 Kasperski, Andrzej 2 Li, Zeyu 2 Liu, Pingzhou 2 Long, Wen 2 Peng, Qiuliang 2 Song, Zhitao 2 Tan, Ronghua 2 Wang, Dongda 2 Wang, Mingshu 2 Wang, Xuelei 2 Xie, Dehui 2 Yang, Li 2 Yang, Xia 2 Yang, Xiaosong 2 Ye, Yanqian 2 Zhang, Qingrui 2 Zhang, Tongqian 2 Zhu, Guanghu 1 Angela, Torres 1 Bao, Lei 1 Cai, Suilin 1 Cao, Feng 1 Cheke, Robert A. ...and 83 more Co-Authors
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Serials

 40 Chaos, Solitons and Fractals 36 Applied Mathematics and Computation 18 Nonlinear Dynamics 16 Nonlinear Analysis. Real World Applications 15 Journal of Mathematical Analysis and Applications 14 Communications in Nonlinear Science and Numerical Simulation 13 International Journal of Biomathematics 12 Computers & Mathematics with Applications 12 Journal of Dalian University of Technology 12 Journal of Mathematical Chemistry 12 Journal of Biological Systems 10 Journal of Biomathematics 10 International Journal of Pure and Applied Mathematics 10 Journal of Applied Mathematics and Computing 9 Advances in Complex Systems 8 Journal of Computational and Applied Mathematics 8 Mathematica Applicata 8 Applied Mathematical Modelling 8 Acta Mathematica Scientia. Series A. (Chinese Edition) 7 Acta Mathematica Sinica 7 Mathematics and Computers in Simulation 7 Journal of Systems Science and Mathematical Sciences 7 Acta Mathematicae Applicatae Sinica. English Series 7 Discrete Dynamics in Nature and Society 7 Journal of Systems Science and Complexity 6 Applicable Analysis 6 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 6 Acta Mathematicae Applicatae Sinica 6 Applied Mathematics and Mechanics. (English Edition) 6 Mathematical and Computer Modelling 6 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 6 Differential Equations and Dynamical Systems 6 Discrete and Continuous Dynamical Systems. Series B 6 Journal of Theoretical Biology 5 Journal of Mathematics. Wuhan University 5 Systems Science and Mathematical Sciences 5 Acta Mathematica Sinica. English Series 5 Journal of Applied Mathematics 5 Advances in Difference Equations 4 Mathematical Biosciences 4 Mathematical Methods in the Applied Sciences 4 Mathematics in Practice and Theory 4 Annals of Differential Equations 4 Acta Mathematica Scientia. Series B. (English Edition) 3 Rocky Mountain Journal of Mathematics 3 Bulletin of Mathematical Biology 3 Journal of Nanjing Normal University. Natural Science Edition 3 Journal of Jilin University. Science Edition 3 Mathematical Medicine and Biology 2 Journal of Mathematical Biology 2 Journal of Mathematical Research & Exposition 2 Applied Mathematics Letters 2 Acta Mathematica Scientia 2 The Canadian Applied Mathematics Quarterly 2 Applied Mathematics. Series A (Chinese Edition) 2 Applied Mathematics. Series B (English Edition) 2 Pure and Applied Mathematics 2 Acta Mathematica Scientia. Series A. (Chinese Edition) 2 Bulletin of Biomathematics 2 M2AN. Mathematical Modelling and Numerical Analysis. ESAIM, European Series in Applied and Industrial Mathematics 2 International Journal of Information & Systems Sciences 1 IMA Journal of Applied Mathematics 1 Indian Journal of Pure & Applied Mathematics 1 Theoretical Population Biology 1 Journal of Lanzhou University. Natural Sciences 1 Acta Scientiarum Naturalium Universitatis Sunyatseni 1 Chinese Annals of Mathematics. Series B 1 Journal of Wuhan University. Natural Science Edition 1 Journal of Engineering Mathematics (Xi’an) 1 Northeastern Mathematical Journal 1 Journal of Central China Normal University. Natural Sciences 1 Computational Mathematics and Modeling 1 Dynamic Systems and Applications 1 Neural, Parallel & Scientific Computations 1 Complexity 1 Communications on Applied Nonlinear Analysis 1 Abstract and Applied Analysis 1 Taiwanese Journal of Mathematics 1 Chinese Quarterly Journal of Mathematics 1 Journal of Mathematical Study 1 Acta Analysis Functionalis Applicata. AAFA 1 International Journal of Nonlinear Sciences and Numerical Simulation 1 International Journal of Differential Equations and Applications 1 The ANZIAM Journal 1 Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis 1 Stochastics and Dynamics 1 Chinese Journal of Contemporary Mathematics 1 Journal of University of Science and Technology of China 1 Mathematical Biosciences and Engineering 1 Journal of Guangxi Normal University. Natural Science Edition 1 Translations of Mathematical Monographs 1 Computational & Mathematical Methods in Medicine 1 Nonlinear Analysis. Hybrid Systems 1 Journal of Mathematical Control Science and Applications 1 Chinese Journal of Engineering Mathematics 1 Advances and Applications in Mathematical Sciences 1 Journal of Beihua University. Natural Science 1 Nonlinear Analysis. Theory, Methods & Applications
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Fields

 448 Ordinary differential equations (34-XX) 442 Biology and other natural sciences (92-XX) 96 Dynamical systems and ergodic theory (37-XX) 51 Systems theory; control (93-XX) 20 Difference and functional equations (39-XX) 16 Numerical analysis (65-XX) 13 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 8 Operator theory (47-XX) 6 Calculus of variations and optimal control; optimization (49-XX) 4 Partial differential equations (35-XX) 3 Integral equations (45-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Operations research, mathematical programming (90-XX) 1 General and overarching topics; collections (00-XX) 1 Probability theory and stochastic processes (60-XX) 1 Statistics (62-XX) 1 Computer science (68-XX) 1 Mechanics of particles and systems (70-XX)

Citations contained in zbMATH

397 Publications have been cited 6,610 times in 2,724 Documents Cited by Year
Complex dynamics of Holling type II Lotka–Volterra predator–prey system with impulsive perturbations on the predator. Zbl 1085.34529
Liu, Xianning; Chen, Lansun
2003
Modeling and analysis of a predator-prey model with disease in the prey. Zbl 0978.92031
Xiao, Yanni; Chen, Lansun
2001
Density-dependent birth rate, birth pulses and their population dynamic consequences. Zbl 0990.92033
Tang, Sanyi; Chen, Lansun
2002
A delayed epidemic model with stage-structure and pulses for pest management strategy. Zbl 1154.34394
Zhang, Hong; Chen, Lansun; Nieto, Juan J.
2008
Optimal harvesting and stability for two-species competitive system with stage structure. Zbl 1028.34049
Song, Xinyu; Chen, Lansun
2001
Integrated pest management models and their dynamical behaviour. Zbl 1334.91058
Tang, Sanyi; Xiao, Yanni; Chen, Lansun; Cheke, Robert A.
2005
The stage-structured predator-prey model and optimal harvesting policy. Zbl 0961.92037
Zhang, Xinan; Chen, Lansun; Neumann, Avidan U.
2000
The dynamical behavior of a Lotka-Volterra predator-prey model concerning integrated pest management. Zbl 1082.34039
Liu, Bing; Zhang, Yujuan; Chen, Lansun
2005
Permanence and positive periodic solution for a single-species nonautonomous delay diffusive models. Zbl 0873.34061
Yang, Xia; Chen, Lansun; Chen, Jufang
1996
The dynamics of an epidemic model for pest control with impulsive effect. Zbl 1188.93038
Wang, Limin; Chen, Lansun; Nieto, Juan J.
2010
The effects of impulsive toxicant input on a population in a polluted environment. Zbl 1041.92044
Liu, Bing; Chen, Lansun; Zhang, Yujuang
2003
The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission. Zbl 1023.92026
Lu, Zhonghua; Chi, Xuebin; Chen, Lansun
2002
Analysis of a predator-prey model with Holling II functional response concerning impulsive control strategy. Zbl 1089.92060
Liu, Bing; Teng, Zhidong; Chen, Lansun
2006
The study of predator–prey system with defensive ability of prey and impulsive perturbations on the predator. Zbl 1081.34041
Zhang, Shuwen; Dong, Lingzhen; Chen, Lansun
2005
Global dynamics of the periodic logistic system with periodic impulsive perturbations. Zbl 1054.34015
Liu, Xianning; Chen, Lansun
2004
Dynamic complexities of a Holling I predator-prey model concerning periodic biological and chemical control. Zbl 1058.92047
Liu, Bing; Zhang, Yujuan; Chen, Lansun
2004
Positive periodic solutions to periodic Kolmogorov-type systems with delays. Zbl 0976.34063
Teng, Zhidong; Chen, Lansun
1999
A ratio-dependent predator-prey model with disease in the prey. Zbl 1024.92017
Xiao, Yanni; Chen, Lansun
2002
Modelling and analysis of integrated pest management strategy. Zbl 1114.92074
Tang, Sanyi; Chen, Lansun
2004
Existence of periodic solution of order one of planar impulsive autonomous system. Zbl 1088.34040
Zeng, Guangzhao; Chen, Lansun; Sun, Lihua
2006
Asymptotic behaviors of competitive Lotka-Volterra system with stage structure. Zbl 1022.34039
Liu, Shengqiang; Chen, Lansun; Luo, Guilie; Jiang, Youlin
2002
Global attractivity of a stage-structure variable coefficients predator-prey system with time delay and impulsive perturbations on predators. Zbl 1155.92355
Jiao, Jianjun; Chen, Lansun
2008
Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination. Zbl 1111.92049
Meng, Xinzhu; Chen, Lansun; Cheng, Huidong
2007
Multiple attractors in stage-structured population models with birth pulses. Zbl 1334.92371
Tang, Sanyi; Chen, Lansun
2003
Permanence and extinction of periodic predator–prey systems in a patchy environment with delay. Zbl 1018.92033
Teng, Zhidong; Chen, Lansun
2003
Recent progress on stage-structured population dynamics. Zbl 1077.92516
Liu, Shengqiang; Chen, Lansun; Agarwal, R.
2002
The effect of dispersal on population growth with stage-structure. Zbl 0968.92018
Cui, Jingan; Chen, Lansun; Wang, Wendi
2000
A predator-prey model with disease in the prey and two impulses for integrated pest management. Zbl 1185.34015
Shi, Ruiqing; Jiang, Xiaowu; Chen, Lansun
2009
The dynamics of an age structured predator-prey model with disturbing pulse and time delays. Zbl 1142.34054
Meng, Xinzhu; Jiao, Jianjun; Chen, Lansun
2008
The dynamics of a new SIR epidemic model concerning pulse vaccination strategy. Zbl 1131.92056
Meng, Xinzhu; Chen, Lansun
2008
Extinction and permanence in nonautonomous competitive systems with stage structure. Zbl 1039.34068
Liu, Shengqiang; Chen, Lansun; Liu, Zhuojun
2002
Persistence and stability for a two-species ratio-dependent predator-prey system with time delay in a two-patch environment. Zbl 0949.92028
Xu, Rui; Chen, Lansun
2000
The relative position and number of limit cycles of the quadratic differential system. Zbl 0433.34022
Chen, Lansun; Wang, Mingshu
1979
Impulsive vaccination of SIR epidemic models with nonlinear incidence rates. Zbl 1100.92040
Hui, Jing; Chen, Lansun
2004
Permanence and global stability in an impulsive Lotka-Volterra $$N$$-species competitive system with both discrete delays and continuous delays. Zbl 1155.92356
Meng, Xinzhu; Chen, Lansun
2008
A delay SIR epidemic model with pulse vaccination and incubation times. Zbl 1184.92044
Meng, Xinzhu; Chen, Lansun; Wu, Bo
2010
The effect of seasonal harvesting on stage-structured population models. Zbl 1058.92051
Tang, Sanyi; Chen, Lansun
2004
Chaos in functional response host - parasitoid ecosystem models. Zbl 1022.92042
Tang, Sanyi; Chen, Lansun
2002
The dynamics of a prey-dependent consumption model concerning impulsive control strategy. Zbl 1074.92042
Liu, Bing; Chen, Lansun; Zhang, Yujuan
2005
Impulsive control strategies in biological control of pesticide. Zbl 1100.92071
Lu, Zhonghua; Chi, Xuebin; Chen, Lansun
2003
Periodic solution of a two-species competitive system with toxicant and birth pulse. Zbl 1137.34017
Liu, Zhijun; Chen, Lansun
2007
Permanence and extinction in logistic and Lotka-Volterra systems with diffusion. Zbl 0985.34061
Cui, Jingan; Chen, Lansun
2001
The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses. Zbl 1061.92059
Gao, Shujing; Chen, Lansun
2005
Global attractivity in a ”food-limited” population model with impulsive effects. Zbl 1062.34055
Tang, Sanyi; Chen, Lansun
2004
Hopf bifurcation and global stability for a delayed predator-prey system with stage structure for predator. Zbl 1151.34067
Gao, Shujing; Chen, Lansun; Teng, Zhidong
2008
Bifurcation of nontrivial periodic solutions for an impulsively controlled pest management model. Zbl 1151.34037
Georgescu, Paul; Zhang, Hong; Chen, Lansun
2008
A delayed SIRS epidemic model with pulse vaccination. Zbl 1152.34379
Pang, Guoping; Chen, Lansun
2007
Impulsive vaccination of an SEIRS model with time delay and varying total population size. Zbl 1139.92314
Gao, Shujing; Chen, Lansun; Teng, Zhidong
2007
Analysis of a three species eco-epidemiological model. Zbl 0967.92017
Xiao, Yanni; Chen, Lansun
2001
The effect of diffusion on the time varying logistic population growth. Zbl 0934.92025
Cui, Jingan; Chen, Lansun
1998
An impulsive predator-prey system with Beddington-DeAngelis functional response and time delay. Zbl 1155.92045
Zhang, Hong; Georgescu, Paul; Chen, Lansun
2008
A Holling II functional response food chain model with impulsive perturbations. Zbl 1086.34043
Zhang, Shuwen; Chen, Lansun
2005
The periodic competing Lotka-Volterra model with impulsive effect. Zbl 1055.92056
Liu, Bing; Chen, Lansun
2004
Analysis of a stage-structured predator-prey system with birth pulse and impulsive harvesting at different moments. Zbl 1220.34067
Jiao, Jianjun; Cai, Shaohong; Chen, Lansun
2011
Optimal harvesting and stability for a predator-prey system with stage structure. Zbl 1054.34125
Song, Xin-yu; Chen, Lan-sun
2002
Persistence and global stability for nonautonomous predator-prey system with diffusion and time delay. Zbl 0903.92029
Song, Xinyu; Chen, Lansun
1998
Dynamic complexity of a two-prey one-predator system with impulsive effect. Zbl 1076.34055
Zhang, Yujuan; Xiu, Zhilong; Chen, Lansun
2005
Complexity of an SIR epidemic dynamics model with impulsive vaccination control. Zbl 1065.92050
Zeng, Guang Zhao; Chen, Lan Sun; Sun, Li Hua
2005
The geometrical analysis of a predator-prey model with two state impulses. Zbl 1250.92047
Zhao, Lichun; Chen, Lansun; Zhang, Qingling
2012
A delayed stage-structured predator-prey model with impulsive stocking on prey and continuous harvesting on predator. Zbl 1126.92052
Jiao, Jianjun; Pang, Guoping; Chen, Lansun; Luo, Guilie
2008
A stage-structured Holling mass defence predator-prey model with impulsive perturbations on predators. Zbl 1117.92053
Jiao, Jianjun; Meng, Xinzhu; Chen, Lansun
2007
Chaos in three species food chain system with impulsive perturbations. Zbl 1066.92060
Zhang, Shuwen; Chen, Lansun
2005
Modelling and analysis of a single-species system with stage structure and harvesting. Zbl 1024.92015
Song, Xinyu; Chen, Lansun
2002
Persistence and global stability for two-species nonautonomous competition Lotka-Volterra patch-system with time delay. Zbl 0949.34060
Zhang, Jingru; Chen, Lansun; Chen, Xiu Dong
1999
Global stability of a population dispersal in a two-patch environment. Zbl 0892.92026
Wang, Wendi; Chen, Lansun; Lu, Zhengyi
1997
Extinction and permanence in competitive stage structured system with time-delays. Zbl 1021.34065
Liu, Shengqiang; Chen, Lansun; Luo, Guilie
2002
Impulsive diffusion in single species model. Zbl 1131.92071
Wang, Limin; Liu, Zhijun; Jinghui; Chen, Lansun
2007
Extinction and permanence of chemostat model with pulsed input in a polluted environment. Zbl 1221.37209
Zhao, Zhong; Chen, Lansun; Song, Xinyu
2009
Impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate. Zbl 1151.92030
Zhao, Zhong; Chen, Lansun; Song, Xinyu
2008
Permanence and global attractivity of stage-structured predator-prey model with continuous harvesting on predator and impulsive stocking on prey. Zbl 1231.34021
Jiao, Jian-Jun; Chen, Lan-Sun; Nieto, Juan J.; Angela, Torres
2008
Pulse vaccination of an SEIR epidemic model with time delay. Zbl 1144.34390
Gao, Shujing; Chen, Lansun; Teng, Zhidong
2008
An appropriate pest management $$SI$$ model with biological and chemical control concern. Zbl 1130.92046
Jiao, Jianjun; Chen, Lansun; Luo, Guilie
2008
A study of predator–prey models with the Beddington–DeAnglis functional response and impulsive effect. Zbl 1102.34032
Zhang, Shuwen; Chen, Lansun
2006
Periodic solution of a chemostat model with variable yield and impulsive state feedback control. Zbl 1243.34017
Li, Zuxiong; Chen, Lansun; Liu, Zhijun
2012
Periodic solution and almost periodic solution for a nonautonomous Lotka-Volterra dispersal system with infinite delay. Zbl 1141.34043
Meng, Xinzhu; Chen, Lansun
2008
Dynamical behavior for a stage-structured SIR infectious disease model. Zbl 1007.92032
Xiao, Yanni; Chen, Lansun; van den Bosch, Frank
2002
The periodic solution of a class of epidemic models. Zbl 0939.92031
Zhang, Xin-An; Chen, Lansun
1999
A delayed epidemic model with pulse vaccination. Zbl 1149.92329
Wei, Chunjin; Chen, Lansun
2008
A new stage structured predator-prey Gompertz model with time delay and impulsive perturbations on the prey. Zbl 1131.92064
Liu, Kaiyuan; Meng, Xinzhu; Chen, Lansun
2008
Dynamic behaviors of Monod type chemostat model with impulsive perturbation on the nutrient concentration. Zbl 1217.34125
Sun, Shulin; Chen, Lansun
2007
Dynamic complexitities in a Lotka-Volterra predator-prey model concerning impulsive control strategy. Zbl 1080.34026
Liu, Bing; Zhang, Yujuan; Chen, Lansun
2005
Global stability of a predator-prey system with stage structure for the predator. Zbl 1062.34056
Xiao, Yan Ni; Chen, Lansun
2004
Extinction and permanence of a two-prey one-predator system with impulsive effect. Zbl 1046.92051
Zhang, Yujuan; Liu, Bing; Chen, Lansun
2003
Chaos in periodically forced Holling type II predator-prey system with impulsive perturbations. Zbl 1083.37537
Zhang, Shuwen; Tan, Dejun; Chen, Lansun
2006
A food chain model with impulsive perturbations and Holling IV functional response. Zbl 1066.92061
Zhang, Shuwen; Wang, Fengyan; Chen, Lansun
2005
Persistence and periodic orbits for two-species predator-prey system with diffusion. Zbl 0941.92032
Song, Xinyu; Chen, Lansun
1998
Dynamics of a stage-structured predator-prey model with prey impulsively diffusing between two patches. Zbl 1204.34108
Jiao, Jianjun; Chen, Lansun; Cai, Shaohong; Wang, Limin
2010
The effect of impulsive vaccination on an SIR epidemic model. Zbl 1186.92040
Shi, Ruiqing; Jiang, Xiaowu; Chen, Lansun
2009
The dynamics of an impulsive delay predator-prey model with variable coefficients. Zbl 1133.92029
Meng, Xin-Zhu; Chen, Lan-Sun; Li, Qing-Xue
2008
Optimal harvesting policies for periodic Gompertz systems. Zbl 1152.34333
Dong, Lingzhen; Chen, Lansun; Sun, Lihua
2007
Periodic solutions of a discrete time Lotka–Volterra type food-chain model with delays. Zbl 1081.92043
Xu, Rui; Chen, Lansun; Hao, Feilong
2005
Persistence and periodic orbits for two-species nonautonomous diffusion Lotka- Volterra models. Zbl 0827.34040
Zeng, Guangzhao; Chen, Lansun; Chen, Jufang
1994
Periodic solution and heteroclinic bifurcation in a predator-prey system with Allee effect and impulsive harvesting. Zbl 1306.92052
Wei, Chunjin; Chen, Lansun
2014
The study of a ratio-dependent predator-prey model with stage structure in the prey. Zbl 1183.92083
Shi, Ruiqing; Chen, Lansun
2009
A single stage-structured population model with mature individuals in a polluted environment and pulse input of environmental toxin. Zbl 1162.92330
Jiao, Jianjun; Long, Wen; Chen, Lansun
2009
Bifurcation and complexity of Monod type predator–prey system in a pulsed chemostat. Zbl 1096.34029
Wang, Fengyan; Zhang, Shuwen; Chen, Lansun; Sun, Lihua
2006
A single species model with impulsive diffusion. Zbl 1180.92072
Hui, Jing; Chen, Lansun
2005
An SIS epidemic model with stage structure and a delay. Zbl 1035.34054
Xiao, Yanni; Chen, Lansun
2002
Global stability of a competition model with periodic coefficients and time delays. Zbl 0845.92020
Wang, Wendi; Chen, Lansun; Lu, Zhengyi
1995
The effect of delays on the permanence for Lotka-Volterra systems. Zbl 0833.34071
Chen, Lansun; Lu, Zhengyi; Wang, Wendi
1995
A solution operator for $$\bar\partial$$ on the Hartogs triangle and $$L^p$$ estimates. Zbl 1439.32098
Chen, L.; McNeal, J. D.
2020
Product domains, multi-Cauchy transforms, and the $$\bar{\partial}$$ equation. Zbl 1439.32097
Chen, L.; McNeal, J. D.
2020
A new vector valued similarity measure for intuitionistic fuzzy sets based on OWA operators. Zbl 1429.03167
Fei, L.; Wang, H.; Chen, L.; Deng, Y.
2019
State-dependent pulse vaccination and therapeutic strategy in an SI epidemic model with nonlinear incidence rate. Zbl 1423.92198
Liu, Kaiyuan; Zhang, Tongqian; Chen, Lansun
2019
On the arrangement of tidal turbines in rough and oscillatory channel flow. Zbl 1415.86003
Bonar, P. A. J.; Chen, L.; Schnabl, A. M.; Venugopal, V.; Borthwick, A. G. L.; Adcock, T. A. A.
2019
The complex variable reproducing kernel particle method for bending problems of thin plates on elastic foundations. Zbl 06965870
Chen, L.; Cheng, Y. M.
2018
Estimating the index of increase via balancing deterministic and random data. Zbl 1401.62043
Chen, L.; Davydov, Y.; Gribkova, N.; Zitikis, R.
2018
The optimal kinematic dynamo driven by steady flows in a sphere. Zbl 1419.76704
Chen, L.; Herreman, W.; Li, K.; Livermore, P. W.; Luo, J. W.; Jackson, A.
2018
State feedback impulsive control of computer worm and virus with saturated incidence. Zbl 1414.68014
Zhang, Meng; Liu, Kaiyuan; Chen, Lansun; Li, Zeyu
2018
Commuting mappings on the Hochschild extension of an algebra. Zbl 1410.16040
Chen, L.; Cai, J.
2018
Dynamical properties of a kind of SIR model with constant vaccination rate and impulsive state feedback control. Zbl 1376.34047
Guo, Hongjian; Chen, Lansun; Song, Xinyu
2017
Periodicity induced by state feedback controls and driven by disparate dynamics of a herbivore-plankton model with cannibalism. Zbl 1393.92044
Fang, Dandan; Pei, Yongzhen; Lv, Yunfei; Chen, Lansun
2017
Homoclinic bifurcation of a ratio-dependent predator-prey system with impulsive harvesting. Zbl 1375.92057
Wei, Chunjin; Liu, Junnan; Chen, Lansun
2017
Stability of a convex order one periodic solution of unilateral asymptotic type. Zbl 1390.34041
Huang, Mingzhan; Chen, Lansun; Song, Xinyu
2017
Increasing eigenstructure assignment design degree of freedom using lifting. Zbl 1380.93125
Chen, L.; Pomfret, A. J.; Clarke, T.
2017
A kind of non-traditional biomanipulation model with constant releasing fish. Zbl 1371.34068
Guo, Hongjian; Chen, Lansun; Song, Xinyu
2017
Dynamics of a plankton-nutrient chemostat model with hibernation and it described by impulsive switched systems. Zbl 1360.34108
Jiao, Jianjun; Cai, Shaohong; Chen, Lansun
2017
A state feedback impulse model for computer worm control. Zbl 1349.93172
Zhang, Meng; Song, Guohua; Chen, Lansun
2016
A NURBS based Galerkin approach for the analysis of solids in boundary representation. Zbl 1425.74456
Chen, L.; Simeon, B.; Klinkel, S.
2016
A novel method for analyzing the stability of periodic solution of impulsive state feedback model. Zbl 1410.34141
Sun, Mingjing; Liu, Yinli; Liu, Sujuan; Hu, Zuoliang; Chen, Lansun
2016
A pest management model with state feedback control. Zbl 1418.92176
Liu, Qiong; Huang, Lizhuang; Chen, Lansun
2016
A NURBS based hybrid collocation-Galerkin method for the analysis of boundary represented solids. Zbl 1425.65166
Klinkel, S.; Chen, L.; Dornisch, W.
2015
Geometric properties of solution of a cylindrical dynamic system with impulsive state feedback control. Zbl 1309.34012
Guo, Hongjian; Chen, Lansun; Song, Xinyu
2015
An integrated fast Fourier transform-based phase-field and crystal plasticity approach to model recrystallization of three dimensional polycrystals. Zbl 1423.74712
Chen, L.; Chen, J.; Lebensohn, R. A.; Ji, Y. Z.; Heo, T. W.; Bhattacharyya, S.; Chang, K.; Mathaudhu, S.; Liu, Z. K.; Chen, L.-Q.
2015
A stage structure pest management model with impulsive state feedback control. Zbl 1353.91032
Pang, Guoping; Chen, Lansun; Xu, Weijian; Fu, Gang
2015
Hybrid collocation-Galerkin approach for the analysis of surface represented 3D-solids employing SB-FEM. Zbl 1423.74869
Chen, L.; Dornisch, W.; Klinkel, S.
2015
Qualitative analysis of impulsive state feedback control to an algae-fish system with bistable property. Zbl 1410.93053
Guo, Hongjian; Chen, Lansun; Song, Xinyu
2015
A pest control model with state-dependent impulses. Zbl 1323.92172
Ji, Xuehui; Yuan, Sanling; Chen, Lansun
2015
Dynamics of a periodic impulsive switched predator-prey system with hibernation and birth pulse. Zbl 1422.92116
Jiao, Jianjun; Cai, Shaohong; Li, Limei; Chen, Lansun
2015
Lattice Boltzmann simulation of gas-solid adsorption processes at pore scale level. Zbl 1349.76757
Zhou, L.; Qu, Z. G.; Chen, L.; Tao, W. Q.
2015
Analysis of household data on influenza epidemic with Bayesian hierarchical model. Zbl 1315.92080
Hsu, C. Y.; Yen, A. M. F.; Chen, L. S.; Chen, H. H.
2015
Periodic solution and heteroclinic bifurcation in a predator-prey system with Allee effect and impulsive harvesting. Zbl 1306.92052
Wei, Chunjin; Chen, Lansun
2014
Periodic solution of the system with impulsive state feedback control. Zbl 1314.93024
Pang, Guoping; Chen, Lansun
2014
Homoclinic bifurcation of prey-predator model with impulsive state feedback control. Zbl 1334.92378
Wei, Chunjin; Chen, Lansun
2014
Geometric approach to the stability analysis of the periodic solution in a semi-continuous dynamic system. Zbl 1395.34051
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2014
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Guo, Hongjian; Song, Xinyu; Chen, Lansun
2014
Periodic solutions and homoclinic bifurcation of a predator-prey system with two types of harvesting. Zbl 1281.92069
Huang, Mingzhan; Liu, Shouzong; Song, Xinyu; Chen, Lansun
2013
Heteroclinic bifurcations of a prey-predator fishery model with impulsive harvesting. Zbl 1417.37288
Wei, Chunjin; Chen, Lansun
2013
An epidemic model with pulses for pest management. Zbl 1402.92389
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2013
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Wei, Chunjin; Zhang, Shuwen; Chen, Lansun
2013
Bifurcations and periodic solutions for an algae-fish semicontinuous system. Zbl 07095175
Dai, Chuanjun; Zhao, Min; Chen, Lansun
2013
Periodic solutions of a microbial pesticide model with the Monod growth rate and impulsive state feedback control. Zbl 1313.34144
Wang, Tieying; Chen, Changhai; Chen, Lansun
2013
The geometrical analysis of a predator-prey model with two state impulses. Zbl 1250.92047
Zhao, Lichun; Chen, Lansun; Zhang, Qingling
2012
Periodic solution of a chemostat model with variable yield and impulsive state feedback control. Zbl 1243.34017
Li, Zuxiong; Chen, Lansun; Liu, Zhijun
2012
Homoclinic bifurcation in semi-continuous dynamic systems. Zbl 1305.34065
Dai, Chuanjun; Zhao, Min; Chen, Lansun
2012
Periodic solution of prey-predator model with Beddington-DeAngelis functional response and impulsive state feedback control. Zbl 1263.92053
Wei, Chunjin; Chen, Lansun
2012
Complex dynamic behavior of three-species ecological model with impulse perturbations and seasonal disturbances. Zbl 1257.92040
Dai, Chuanjun; Zhao, Min; Chen, Lansun
2012
Dynamic complexity of an Ivlev-type prey-predator system with impulsive state feedback control. Zbl 1256.34032
Dai, Chuanjun; Zhao, Min; Chen, Lansun
2012
A Leslie-Gower pest management model with impulsive state feedback control. Zbl 1289.92087
Wei, Chunjin; Chen, Lansun
2012
Dynamic analysis of a predator-prey (pest) model with disease in prey and involving an impulsive control strategy. Zbl 1251.93092
Zhao, Min; Wang, Yanzhen; Chen, Lansun
2012
On the modeling of surface tension and its applications by the generalized interpolation material point method. Zbl 1356.76226
Chen, L.; Lee, J. H.; Chen, C.-F.
2012
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Jiao, Jianjun; Cai, Shaohong; Chen, Lansun
2012
Dynamical analysis of a biological resource management model with impulsive releasing and harvesting. Zbl 1278.92040
Jiao, Jianjun; Chen, Lansun; Cai, Shaohong
2012
The genic mutation on dynamics of a predator-prey system with impulsive effect. Zbl 1267.34098
Jiao, Jianjun; Chen, Lansun
2012
Dynamical analysis on a single population model with state-dependent impulsively unilateral diffusion between two patches. Zbl 1377.92070
Jiao, Jianjun; Chen, Lansun; Cai, Shaohong
2012
The periodic Volterra model with mutual interference and impulsive effect. Zbl 1280.34079
Zhang, Yujuan; Chen, Lansun
2012
Periodic solutions of a class of semi-continuous dynamical systems with center equilibrium. Zbl 1265.34146
Guo, Hongjian; Chen, Lansun; Song, Xinyu
2012
Analysis of a stage-structured predator-prey system with birth pulse and impulsive harvesting at different moments. Zbl 1220.34067
Jiao, Jianjun; Cai, Shaohong; Chen, Lansun
2011
Modelling and qualitative analysis of a predator-prey system with state-dependent impulsive effects. Zbl 1236.92072
Tian, Yuan; Sun, Kaibiao; Chen, Lansun
2011
Dynamical analysis of a five-dimensioned chemostat model with impulsive diffusion and pulse input environmental toxicant. Zbl 1216.92062
Jiao, Jianjun; Ye, Kaili; Chen, Lansun
2011
Nonlinear analysis of a microbial pesticide model with impulsive state feedback control. Zbl 1235.93108
Wang, Tieying; Chen, Lansun
2011
Dynamical analysis of a three-dimensional predator-prey model with impulsive harvesting and diffusion. Zbl 1210.34114
Jiao, Jianjun; Cai, Shaohong; Chen, Lansun
2011
Impulsive perturbations of a predator-prey system with modified Leslie-Gower and Holling type II schemes. Zbl 1222.34057
Zhao, Zhong; Yang, Li; Chen, Lansun
2011
Dynamical behaviors of a delayed chemostat model with impulsive diffusion on nutrients. Zbl 1216.34083
Jiao, Jianjun; Cai, Shaohong; Chen, Lansun
2011
Bifurcation of a three molecular saturated reaction with impulsive input. Zbl 1223.34072
Li, Zuxiong; Zhao, Zhong; Chen, Lansun
2011
The global dynamics of a class of vector fields in $$\mathbb R^3$$. Zbl 1275.34063
Zhang, Xin An; Liang, Zhao Jun; Chen, Lan Sun
2011
Nonlinear modelling of chemostat model with time delay and impulsive effect. Zbl 1215.34081
Zhao, Zhong; Zhang, Xiuquan; Chen, Lansun
2011
Dynamics on a stage-structured single population model with birth pulse and impulsive harvesting. Zbl 1247.92027
Jiao, Jianjun; Bao, Lei; Chen, Lansun
2011
Dynamical behaviors of a biological management model with impulsive stocking juvenile predators and continuous harvesting adult predators. Zbl 1216.34041
Jiao, Jianjun; Cai, Shaohong; Chen, Lansun
2011
Global analysis of a three-dimensional delayed Michaelis-Menten chemostat-type models with pulsed input. Zbl 1216.34085
Wang, Tieying; Chen, Lansun
2011
The dynamics of an epidemic model for pest control with impulsive effect. Zbl 1188.93038
Wang, Limin; Chen, Lansun; Nieto, Juan J.
2010
A delay SIR epidemic model with pulse vaccination and incubation times. Zbl 1184.92044
Meng, Xinzhu; Chen, Lansun; Wu, Bo
2010
Dynamics of a stage-structured predator-prey model with prey impulsively diffusing between two patches. Zbl 1204.34108
Jiao, Jianjun; Chen, Lansun; Cai, Shaohong; Wang, Limin
2010
The dynamics of a mutual interference age structured predator-prey model with time delay and impulsive perturbations on predators. Zbl 1184.92059
Zhu, Guanghu; Meng, Xinzhu; Chen, Lansun
2010
An impulsive predator-prey model with disease in the prey for integrated pest management. Zbl 1221.34037
Shi, Ruiqing; Chen, Lansun
2010
Dynamic analysis of a turbidostat model with the feedback control. Zbl 1221.93052
Zhao, Zhong; Wang, Tieying; Chen, Lansun
2010
Nonlinear modelling of a synchronized chemostat with impulsive state feedback control. Zbl 1201.92065
Sun, Kaibiao; Tian, Yuan; Chen, Lansun; Kasperski, Andrzej
2010
Impulsive state feedback control of the microorganism culture in a turbidostat. Zbl 1190.92047
Zhao, Zhong; Yang, Li; Chen, Lansun
2010
Comment on “Existence and stability of periodic solution of a Lotka-Volterra predator-prey model with state dependent impulsive effects” [J. Comput. Appl. Math. 224, 544–555 (2009)]. Zbl 1203.34075
Tian, Yuan; Sun, Kaibiao; Chen, Lansun
2010
Qualitative analysis of a variable yield turbidostat model with impulsive state feedback control. Zbl 1214.34038
Guo, Hongjian; Chen, Lansun
2010
Dynamical behaviors of a trimolecular response model with impulsive input. Zbl 1209.34057
Li, Zuxiong; Chen, Lansun
2010
Mathematical models of restoration and control of a single species with Allee effect. Zbl 1201.34016
Guo, Hongjian; Chen, Lansun; Song, Xinyu
2010
A study on time-limited control of single-pest with stage-structure. Zbl 1198.92039
Guo, Hongjian; Chen, Lansun
2010
Nonlinear modelling and qualitative analysis of a real chemostat with pulse feeding. Zbl 1195.92031
Tian, Yuan; Sun, Kaibiao; Kasperski, Andrzej; Chen, Lansun
2010
Dynamical analysis of a delayed predator-prey system with birth pulse and impulsive harvesting at different moments. Zbl 1229.34123
Jiao, Jianjun; Chen, Lansun
2010
Extinction and permanence of two-nutrient and two-microorganism chemostat model with pulsed input. Zbl 1222.37104
Wang, Tieying; Chen, Lansun; Zhang, Ping
2010
Existence and global stability of periodic solution for impulsive predator-prey model with diffusion and distributed delay. Zbl 1219.34106
Zhao, Zhong; Li, Zuxiong; Chen, Lansun
2010
Dynamic analysis of lactic acid fermentation with impulsive input. Zbl 1190.92010
Zhao, Zhong; Chen, Lansun
2010
Feasibility of time-limited control of a competition system with impulsive harvest. Zbl 1190.34056
Guo, Hongjian; Chen, Lansun; Song, Xinyu
2010
Analysis of a delayed chemostat model with impulsive perturbation on the polluted nutrient. Zbl 1240.34408
Jiao, Jianjun; Chen, Lansun
2010
Global attractivity of the positive periodic solution of a delay logistic population model with impulses. Zbl 1240.34417
Pei, Yongzhen; Li, Changguo; Chen, Lansun
2010
A predator-prey model with disease in the prey and two impulses for integrated pest management. Zbl 1185.34015
Shi, Ruiqing; Jiang, Xiaowu; Chen, Lansun
2009
Extinction and permanence of chemostat model with pulsed input in a polluted environment. Zbl 1221.37209
Zhao, Zhong; Chen, Lansun; Song, Xinyu
2009
The effect of impulsive vaccination on an SIR epidemic model. Zbl 1186.92040
Shi, Ruiqing; Jiang, Xiaowu; Chen, Lansun
2009
The study of a ratio-dependent predator-prey model with stage structure in the prey. Zbl 1183.92083
Shi, Ruiqing; Chen, Lansun
2009
A single stage-structured population model with mature individuals in a polluted environment and pulse input of environmental toxin. Zbl 1162.92330
Jiao, Jianjun; Long, Wen; Chen, Lansun
2009
Periodic solution of a chemostat model with monod growth rate and impulsive state feedback control. Zbl 1402.92422
Guo, Hongjian; Chen, Lansun
2009
Periodic solution of a turbidostat model with impulsive state feedback control. Zbl 1183.92003
Li, Zuxiong; Chen, Lansun
2009
Dynamical analysis of a delayed predator-prey model with impulsive diffusion between two patches. Zbl 1190.34107
Jiao, Jianjun; Yang, Xiaosong; Cai, Shaohong; Chen, Lansun
2009
The effects of impulsive harvest on a predator-prey system with distributed time delay. Zbl 1221.34218
Guo, Hongjian; Chen, Lansun
2009
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text:
# How do you check a linear equation and find if 7 is a solution to 5x-5=30?

In general, to evaluate any function $f \left(x\right)$ in a specific point, you simply need to substitute every $x$ with that specific value.
In your case, rather than evaluating a function, you want to check an identity. The spirit is the same, you only need to substitue $x \setminus \rightarrow 7$, and see if you obtain a true statement, or a false one.
$5 x - 5 = 30 \setminus \rightarrow 5 \setminus \cdot 7 - 5 = 30 \setminus \iff 35 - 5 = 30 \setminus \iff 30 = 30$

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The Achievements of Students at the Stages of Education from the Second to Fourth Using Functional Principal Component Analysis

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#### Statistics in Transition New Series

Polish Statistical Association

Central Statistical Office of Poland

Subject: Economics, Statistics & Probability

ISSN: 1234-7655
eISSN: 2450-0291

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VOLUME 18 , ISSUE 1 (March 2017) > List of articles

### The Achievements of Students at the Stages of Education from the Second to Fourth Using Functional Principal Component Analysis

Citation Information : Statistics in Transition New Series. Volume 18, Issue 1, Pages 139-150, DOI: https://doi.org/10.21307/stattrans-2016-062

Published Online: 17-July-2017

### ARTICLE

#### ABSTRACT

Significant demographic phenomena can be observed in Poland – the number of school age population is decreasing. It affects higher education since the immediate effect of demographic changes is the drop in the number of students. The analysis of the level of future students’ knowledge also remains an important aspect of the problem.

The purpose of the article is to compare the level of knowledge presented by students at the subsequent stages of education in the period 2009-2015. The research covers the average exam results received on graduation from the second, third and fourth stage of education. Functional principal component analysis, which is based on functional data, will be applied in the study. This method allows an analysis of dynamic data.

## 1. Introduction

Demographic conditions exert a direct impact on higher education. Since 1990 Polish higher education has been going through a period of continuous and dynamic growth resulting from the population boom, lasting for almost 15 years. Starting from 2006, the first symptoms of this trend collapse became noticeable as the number of students was gradually decreasing. A drop in the number of 10-year-olds was recorded in each consecutive year and thus the reversal of the trend favourable for Polish higher education. This process has been continuing until today causing the ongoing decrease in the number of the traditional college-age population.

Figure 1 presents the number of students graduating from the subsequent stages of education. The students who graduated from primary school (PS) in 2003 are marked with the letter A on x-axis, in 2006 - middle school and 2009 - secondary school, i.e. theoretically those born in the same year. It can be observed that slight local extremes of the PS students correspond to larger fluctuations in the number of secondary school students. The number of PS graduates reached the minimum (M) level in 2006 and of secondary school graduates in 2012, whereas a year later – the local maximum was achieved. The number of PS graduates has been stabilizing since 2014, i.e. the students who will graduate from the middle school (MS) in 2017 and the secondary school in 2020.

##### Figure 1.

The number of students graduating from the subsequent stages of education

Source: author’s compilation based on CKE [Central Examination Commission] data.

The level of knowledge presented by future students is another important aspect affecting higher education. For this reason, the level of knowledge presented by students at the consecutive stages of education in the period 2009-2015 was compared. The research covers average exam results received on graduating from the second, third and fourth stages of education. Functional principal component analysis was applied in the study.

## 2. Methodology

Principal component analysis (PCA) is based on the transformation of original variables into the set of new and mutually orthogonal variables referred to as principal components [Harman 1975]. Functional principal component analysis (FPCA) is characterized by the advantages of a classical principal component analysis and, moreover, allows for the analysis of dynamic data. The type of data is the basic difference between these two methods: PCA is based on multivariate data, whereas FPCA is based on functional data. Functional data take the form of curves and trajectories, i.e. the sequence of individual observations rather than just a single observation [Hall and Hosseini-Nasab 2006, Krzyśko et al. 2012].

In the case of the functional principal component analysis (FPCA) each principal component is presented as the principal component weight function, also referred to as the time dependent eigenfunction ξj(t) [Daniele 2006, Ramsay and Silverman 2005]. Eigenfunction maximizes the principal component function variance:

##### (1)
$v(t,s)=def1n-1∑i=1n[xi(t)-x¯(t)][xi(s)-x¯(s)]$

Similarly to the classical PCA, in the case of the functional one the problem is the function variance distribution:

##### (2)
$v(t,s)=∑jλjξj(t)ξj(s)$
where λj, ξj(t) satisfy eigenequation:
##### (3)
$〈v(u,),ξj〉=λjξj(s).$
and eigenvalues are positive and non-decreasing:
##### (4)
$λj=def∫Tξj(t)v(t,s)ξj(s)dtds.$
Eigenfunctions satisfy the condition:
##### (5)
$∫Tξj2(t)dt=1and∫Tξj(t)ξi(t)dt=0(i

Eigenfunctions define the principal components of variation between the sampling functions xi [Ingrassia and Costanzo 2005, Hall et al. 2006].

## 3. The achievements of students

### 3.1. The second stage of education

Since April 2002 the exam taken by students graduating from the sixth grade of primary school has been held. The standards of examination requirements constitute the basis for carrying out the final test. Until 2014 the standards were grouped into five trans-subject categories:

• writing,

• reasoning,

• use of information,

• practical application of knowledge.

In 2015 the test consisted of two parts:

• the first part – covers tasks in Polish and maths,

• the second part – covers tasks in a modern foreign language.

A sixth grader takes the test in one of the following foreign languages: English, French, Spanish, German, Russian and Italian. A student can choose only the foreign language learned at school as an obligatory subject.

Figure 2 illustrates the average percentage results in particular subjects received by the sixth graders from their graduation test in the period 2002-2014.

##### Figure 2.

Average test results achieved on graduating from the sixth grade of PS in particular subjects [%]

Source: author’s compilation based on CKE [Central Examination Commission] data.

The average result in all subjects is denoted by a solid line and presents the decreasing tendency in the period 2002-2014.

The functional principal component analysis allowed for distinguishing two component functions. The practical explanation of the functional principal components is supported by the graphs showing each component deviation from the average in all subjects (Fig. 3).

##### Figure 3.

Weight functions

Source: author’s compilation using R program.

The first functional principal component explains 72% of the joint variation, whereas the second one 18%. The first component reflects the overall tendency. The plus sign on this component means that the curve describing the result in a particular subject remains above the average. The second component shows the tendency in both outlier and mid years against the average. The plus sign on the second component means that the test result in a given subject before 2005 and after 2011 was below average, whereas in the years 2005-2011 the result was above average.

Based on the results of the functional factor analysis, data visualization and a comparison of analysed objects can be performed. Figure 4 presents data projection on the plane defined by two functional principal components.

##### Figure 4.

Objects in the components’ space

Source: author’s compilation using R program.

Students received the best results in reading – above average. Writing and using information remained on an average level, the situation was better before 2005 and after 2011. Both practical application of knowledge and reasoning were below average.

### 3.2. The third stage of education

The exam for the third graders of middle school covers the acquired knowledge and skills specified in the core curriculum of general education for the selected subjects taught at the third and earlier stages of education.

Before 2009 middle school students took the exam in humanities as well as maths and natural science only. In the period 2009-2011 middle school graduation exam consisted of three parts:

• humanities,

• maths and natural science,

• modern foreign language.

In the years 2012-2015 the following scopes were identified within the framework of each exam part:

• humanities covering history and social studies and also Polish,

• maths and natural science covering subjects teaching natural sciences and maths,

• modern foreign language at either basic or extended level.

A middle school student takes an exam in one of the following foreign languages: English, French, Spanish, German, Russian, Ukrainian and Italian. A student can choose only the foreign language learned at school as an obligatory subject.

Every middle school student is obliged to take an exam in a modern foreign language at the basic level. An extended level exam is obligatory only for those students who choose to take an exam in the language they used to learn in a primary school. The other middle school students can also take it if they wish to check the level of their language skills.

The exam has a written form. Taking it is the condition to graduate from the middle school, however, the minimum result to be achieved by a student is not defined and, therefore, it is not possible to fail the exam.

Figure 5 presents the average percentage test results in particular subjects taken after the third grade of middle school in the years 2006-2015. The average result in all subjects is denoted by a solid line and shows the decreasing tendency in the period 2006-2015.

##### Figure 5.

Average exam results achieved on graduating from the third class of middle school in particular subjects [%]

Source: author’s compilation based on CKE [Central Examination Commission] data.

Two component functions were distinguished by means of the functional principal component analysis. Fig. 6 presents the graphs of each component deviation from the average in all subjects.

##### Figure 6.

FPCA weight functions

Source: author’s compilation using R program.

The first functional principal component explains 74% of the joint variation, whereas the second one 24%. The first component is responsible for the overall tendency. The plus sign of this component means that the curve describing the result in a particular subject remains above the average. The second component shows the tendency in the initial and final years against the average (“the beginning vs. the end”) and compares the period until 2011 and after 2012 against the average result. The plus sign on the second component means that the test result in a given subject at the beginning of the analysed period was higher than the average, whereas in the years 2012-2014 the result was below the average.

The visualisation of objects in the space of the component function (fig. 7) allows for drawing the following conclusions:

• humanities: the test result below the average, the situation was worse at the beginning of the analyzed period than in the years 2012-2014,

• English and German: average test result, higher than average at the beginning of the studied period, after 2012 – lower than average,

• maths and natural science: test result lower than average, at the beginning of the analysed period the situation was worse than after 2012.

##### Figure 7.

Objects in the components’ space

Source: author’s compilation using R program.

### 3.3. The fourth stage of education

A graduate taking “the old type of graduation exam” (before 2015) is obligated to take: two exams in the oral part and three exams in the written part. The obligatory exams in the oral part are as follows:

• an exam in Polish (without defining the level),

• an exam in a modern foreign language (without defining the level).

The obligatory exams in the written part are as follows:

• an exam in Polish (basic level),

• an exam in maths (basic level),

• an exam in a modern foreign language (basic level).

A graduate taking “the new type of graduation exam” (from 2015) in the written part is obliged to take an exam in the chosen additional subject (extended level).

In order to receive the graduation diploma a student has to get at least 30% points at the exam in each obligatory subject in the oral part and receive at least 30% points at the exam in each obligatory subject in the written part.

Figure 8 presents the number of students who passed / failed the secondary school graduation exam in the period 2009-2015. It is noticeable that the number of students taking the graduation exam is decreasing each consecutive year and, moreover, the number of those who failed this exam remains disturbingly high in the recent two years.

##### Figure 8.

The number of students taking graduation exam

Source: author’s compilation based on CKE [Central Examination Commission] data.

Figure 9 presents the average percentage results obtained at the secondary school graduation exam in particular subjects in the period 2008-2015. The bold line shows an average result in all subjects – a decreasing tendency is noticeable.

##### Figure 9.

Average results of the secondary school graduation exam

Source: author’s compilation based on CKE [Central Examination Commission] data.

The functional principal component analysis was applied to distinguish two component functions. The graphs of each component deviation from the average in all subjects are presented in Figure 10.

##### Figure 10.

FPCA components

Source: author’s compilation using R program.

The first functional principal component explains 80% of the joint variation, whereas the second one 15%. The first component is responsible for the overall tendency. The plus sign of this component means that the curve describing the result in a particular subject remains above the average. The second component shows the tendency in the initial and final years against the average (“the beginning vs. the end”) and compares the period until 2010 and after 2010 against the average result. The plus sign on the second component means that the secondary school graduation exam result in a given subject was higher than the average at the beginning of the analysed period, whereas at the end the result was worse than the average.

Next the data were projected on the plane determined by two functional principal components (Fig. 11).

##### Figure 11.

Objects in the component’s space

Source: author’s compilation using R program.

The best exam results were achieved by the secondary school graduates in German and English at the basic level (plus sign on the first and minus sign on the second component). The worst results were achieved in basic physics, basic chemistry and extended maths (minus sign on the first and plus sign on the second component).

## 4. Final remarks

The number of sixth graders in primary schools has stabilized since 2014, the number of final year students in middle schools and secondary schools has been continuously decreasing.

The decreasing tendency in average exam results is noticeable at all levels of education. Starting from primary school, the students receive better results in humanities, whereas science causes more problems. Primary school leavers achieved an above average result in reading, but their practical application of knowledge was below average in all subjects. An above average result was obtained in humanities part at the middle school graduation exam, whereas maths and natural science part showed below average results in all subjects. Secondary school graduates have biggest problems in science, especially in physics (basic and extended level), in basic chemistry and extended maths. The best results were recorded in German and English at the basic level.

Due to the demographic situation and the decrease in the population representing basic educational age groups the number of students has been decreasing since 2006. This situation will persist until 2020, when the students who graduated from primary schools in 2014 will start their university education.

Similarly to the classic analysis of principal components, FPCA makes this visualisation possible. It makes the analysis easier and shows nontrivial correlations, which are difficult to find in a different way. A vital property of both methods is data reduction with maximum information being kept. Thanks to the functional analysis of principal components, which enriches possibilities of the classic analysis of principal components, it is possible to analyse dynamic data, showing both the tendency as well as the pace of changes in time.

## References

1. Daniele, M., (2006). Functional principal components analysis to study environmental data, the article available on the website http://www.old.sis-statistica.org/files/pdf/atti/Spontanee%202006_677-680.pdf
[URL]
2. Hall, P., Hosseini-Nasab, M., (2006). On properties of functional principal components analysis, Journal of the Royal Statistical Society, Series B (Statistical Methodology), Vol. 68, No.1, pp. 109–126.
[CROSSREF]
3. Hall, P., Müller, H. G., Wang, J. L., (2006). Properties of Principal Component Methods for Functional and Longitudinal Data Analysis, The Annals of Statistics, Vol. 34, No. 3., pp. 1493–1517.
[CROSSREF]
4. Harman, H., (1975). Modern Factor Analysis, The University of Chicago Press.
5. Ingrassia, S., Costanzo, G. D., (2005). Functional principal component analysis of financial time series, in: Vichi M., Monari P., Mignani S., Montanari A. (ed.), New Developments in Classification and Data Analysis, Springer-Verlag, Berlin, pp. 351–358.
6. Krzyśko, M., Górecki, T., Deręgowski, K., (2012). Jądrowa i Funkcjonalna Analiza Składowych Głównych [Nuclear and Functional Analysis of Principal Components], the meeting of Polish Statistical Association Division in Poznań, presentation available on http://stat.gov.pl/cps/rde/xbcr/pts/Krzysko_wyklad_7_11_12.pdf (access date 1st March 2015).
[URL]
7. Ramsay, J. O., Silverman, B. W., (2005). Functional Data Analysis, Springer.
[CROSSREF]
8. Ramsay, J. O., Hooker, G., Graves, S., (2009). Functional Data Analysis with R and MATLAB, Springer.
[CROSSREF]

### FIGURES & TABLES

Figure 1.

The number of students graduating from the subsequent stages of education

Source: author’s compilation based on CKE [Central Examination Commission] data.

Figure 2.

Average test results achieved on graduating from the sixth grade of PS in particular subjects [%]

Source: author’s compilation based on CKE [Central Examination Commission] data.

Figure 3.

Weight functions

Source: author’s compilation using R program.

Figure 4.

Objects in the components’ space

Source: author’s compilation using R program.

Figure 5.

Average exam results achieved on graduating from the third class of middle school in particular subjects [%]

Source: author’s compilation based on CKE [Central Examination Commission] data.

Figure 6.

FPCA weight functions

Source: author’s compilation using R program.

Figure 7.

Objects in the components’ space

Source: author’s compilation using R program.

Figure 8.

The number of students taking graduation exam

Source: author’s compilation based on CKE [Central Examination Commission] data.

Figure 9.

Average results of the secondary school graduation exam

Source: author’s compilation based on CKE [Central Examination Commission] data.

Figure 10.

FPCA components

Source: author’s compilation using R program.

Figure 11.

Objects in the component’s space

Source: author’s compilation using R program.

### REFERENCES

1. Daniele, M., (2006). Functional principal components analysis to study environmental data, the article available on the website http://www.old.sis-statistica.org/files/pdf/atti/Spontanee%202006_677-680.pdf
[URL]
2. Hall, P., Hosseini-Nasab, M., (2006). On properties of functional principal components analysis, Journal of the Royal Statistical Society, Series B (Statistical Methodology), Vol. 68, No.1, pp. 109–126.
[CROSSREF]
3. Hall, P., Müller, H. G., Wang, J. L., (2006). Properties of Principal Component Methods for Functional and Longitudinal Data Analysis, The Annals of Statistics, Vol. 34, No. 3., pp. 1493–1517.
[CROSSREF]
4. Harman, H., (1975). Modern Factor Analysis, The University of Chicago Press.
5. Ingrassia, S., Costanzo, G. D., (2005). Functional principal component analysis of financial time series, in: Vichi M., Monari P., Mignani S., Montanari A. (ed.), New Developments in Classification and Data Analysis, Springer-Verlag, Berlin, pp. 351–358.
6. Krzyśko, M., Górecki, T., Deręgowski, K., (2012). Jądrowa i Funkcjonalna Analiza Składowych Głównych [Nuclear and Functional Analysis of Principal Components], the meeting of Polish Statistical Association Division in Poznań, presentation available on http://stat.gov.pl/cps/rde/xbcr/pts/Krzysko_wyklad_7_11_12.pdf (access date 1st March 2015).
[URL]
7. Ramsay, J. O., Silverman, B. W., (2005). Functional Data Analysis, Springer.
[CROSSREF]
8. Ramsay, J. O., Hooker, G., Graves, S., (2009). Functional Data Analysis with R and MATLAB, Springer.
[CROSSREF]


text:
Proof for Sturm Liouville eigenfunction expantion pointwise convergence theorem

In "Elementary Partial Differential Equation" by Berg and McGregor, the following theorem is given without proof:

Let $f(x)$ be piecewise smooth on the interval $[a,b]$ and let $\{\varphi_n(x)\}$ be the eigenfunctions of a self-adjoint regular Sturm-Liouville problem. Then for each value of $x$ in $[a,b]$, the Fourier series of $f(x)$ relative to $\{\varphi_n(x)\}$ converges, and $$\sum_{k=0}^\infty a_k\varphi_k(x)=\frac{f(x^+)+f(x^-)}{2}\qquad a<x<b$$

I am wondering about two things:

1. Where can I find the proof for this theorem?
2. Can the condition on $f$ can be weakened to piecewise continuity and the existence of one-sided derivatives on each point, as in Dirichlet theorem for periodical Fourier series (where the Sturm-Liouville problem is with periodical BC)?

The Sturm-Liouville eigenvalue problem is $$-(py')'+qy = \lambda wy,\;\;\; a \le x \le b,$$ subject to separated endpoint conditions at $x=a,b$.
Suppose that $p$, $w$ are positive and twice continuously differentiable on $[a,b]$, and further suppose that $q$ is continuous on $[a,b]$. Then, after a change of independent and dependent variables, the equation can be put into potential form on a new interval: $$-f''+Qf = \lambda f,\;\;\; c \le x \le d.$$ The function $Q$ will be continuous under the stated assumptions. The regular endpoint conditions for $y$ are transformed to regular endpoint conditions for $f$. If $|Q|\le M$ on $[c,d]$, then, for eigenvalues $\lambda >> M$, it follows that the solutions $f$ are asymptotically close to solutions of $-f''=\lambda f$ with the transformed endpoint conditions. In this way, the Fourier series of eigenfunctions for the regular problem is shown to be converge in the same was as the Fourier series of trigonometric functions associated with the second problem in $f$.
So convergence is reduced to the study of the Fourier expansions associated with the problem $$-f'' = \lambda f, \;\;\; c \le x \le d,$$ with regular endpoint conditions at $c$, $d$. The trigonometric expansions must be handled in order to complete the proof of a general problem, but that's the general idea. The end result is this: The Fourier series for a function $g$ on $[c,d]$ in the trigonometric series associated with the reduced problem converges to a limit $L$ iff the Sturm-Liouville eigenfunction expansion in the eigenfunctions of $-f''+Qf=\lambda f$ converges to $L$.
• @MOMO : The result for the potential form $-f''+Qf=\lambda f$ with separated endpoint conditions is handled in the book by E. C. Titchmarsh I referenced, in the first chapter. I don't know if the substitution is recorded in Titchmarsh, but that should be easy to find otherwise. – DisintegratingByParts Mar 15 '17 at 21:10


text:
×

# ← Which has a greater mass? · Level 1 (20 points)

this question is false mass and weight are two different things there wasnt enough information provided if i have 30N of mercury and 30N of liquid nitrogen which has a greater mass... which takes up more space.

Note by Keith Schroader
3 years, 1 month ago

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Depends on the structural integrity of the flux related to differing states of matter.

- 2 years, 9 months ago


text:
## 10.3 – Applications of Quadratic Functions

### Learning Objectives

• (10.3.1) – Solve application problems involving quadratic functions
• Objects in free fall
• Determining the width of a border
• Finding the maximum and minimum values of a quadratic function

# (10.3.1) – Solve application problems involving quadratic functions

Quadratic equations are widely used in science, business, and engineering. Quadratic equations are commonly used in situations where two things are multiplied together and they both depend on the same variable. For example, when working with area, if both dimensions are written in terms of the same variable, you use a quadratic equation. Because the quantity of a product sold often depends on the price, you sometimes use a quadratic equation to represent revenue as a product of the price and the quantity sold. Quadratic equations are also used when gravity is involved, such as the path of a ball or the shape of cables in a suspension bridge.

### Objects in free fall

A very common and easy-to-understand application is the height of a ball thrown at the ground off a building. Because gravity will make the ball speed up as it falls, a quadratic equation can be used to estimate its height any time before it hits the ground. Note: The equation isn’t completely accurate, because friction from the air will slow the ball down a little. For our purposes, this is close enough.

### Example

A ball is thrown off a building from 200 feet above the ground. Its starting velocity (also called initial velocity) is $−10$ feet per second. (The negative value means it’s heading toward the ground.)

The equation $h=-16t^{2}-10t+200$ can be used to model the height of the ball after $t$ seconds. About how long does it take for the ball to hit the ground?

In the next video we show another example of how the quadratic equation can be used to find the time it takes for an object in free fall to hit the ground.

Here are some more similar objects in free fall examples.

### Example: Applying the Vertex and $x$-Intercepts of a Parabola

A ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. The ball’s height above ground can be modeled by the equation $H\left(t\right)=-16{t}^{2}+80t+40$.

a. When does the ball reach the maximum height?

b. What is the maximum height of the ball?

c. When does the ball hit the ground?

### EXAMPLE

A rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of 96 feet per second. The rock’s height above ocean can be modeled by the equation $H\left(t\right)=-16{t}^{2}+96t+112$.

a. When does the rock reach the maximum height?

b. What is the maximum height of the rock?

c. When does the rock hit the ocean?

### Applications of quadratic functions: determining the width of a border

The area problem below does not look like it includes a Quadratic Formula of any type, and the problem seems to be something you have solved many times before by simply multiplying. But in order to solve it, you will need to use a quadratic equation.

### Example

Bob made a quilt that is 4 ft $\times$ 5 ft. He has 10 sq. ft. of fabric he can use to add a border around the quilt. How wide should he make the border to use all the fabric? (The border must be the same width on all four sides.)

Here is a video which gives another example of using the quadratic formula for a geometry problem involving the border around a quilt.

### Finding the maximum and minimum values of a quadratic function

There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue.

### ExAMPLE

Find two numbers $x$ and $y$ whose difference is 100 and whose product is a minimum.

### Example: Finding the Maximum Value of a Quadratic Function

A backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has purchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side.

1. Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length $L$.
2. What dimensions should she make her garden to maximize the enclosed area?

### How To: Given an application involving revenue, use a quadratic equation to find the maximum.

1. Write a quadratic equation for revenue.
2. Find the vertex of the quadratic equation.
3. Determine the $y$-value of the vertex.

### Example: Finding Maximum Revenue

The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a local newspaper currently has 84,000 subscribers at a quarterly charge of $30. Market research has suggested that if the owners raise the price to$32, they would lose 5,000 subscribers. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue?

### Try It

A coordinate grid has been superimposed over the quadratic path of a basketball in the picture below. Find an equation for the path of the ball. Does the shooter make the basket?

(credit: modification of work by Dan Meyer)


text:
# Heine-Borel - Finite Intersection equivalence in subspaces

I'm doing a course in Calculus/Real Analysis, and we're covering really basic topological ideas (metric spaces, open/closed sets etc.). From my notes it looks like we proved the following in class (Heine-Borel):

(1) A set K is compact iff (2) for every open cover of K there is a finite subcover.

And then we claimed without proof that the right hand side is clearly equivalent to the following property: (3) for every set of closed subsets from K that have the finite intersection property, the intersection of said sets is non-empty.

It looks pretty easy to prove this equivalence when K is an entire space. Like this proof here. But from my notes it appears to be a theorem that holds true with any subset of the metric space.

My question is how that proof holds up when K (as mentioned in class) is not the whole space but a subset of the space. The proof relies on the complement of the sets, which means that given an open cover, the complement of the sets are closed but are still in the space (obviously, since the complement is with respect to the space). However when we take the complement w.r.t. the space and K is no longer the whole space, the sets we have are no longer from K necessarily. It also appears of no help to intersect them with K, since we cannot assume K is closed (that's part of what we're trying to prove in the sense that compactness is closed+bounded).

The difference seems petty but it's bugging me to no end.

Apologies about the wordiness, my latex skills are about as poor as my topology skills. Thanks.

-
Note that (2) is the usual definition of compactness. I don’t know what definition your course is using, but it may be something that’s actually not equivalent to compactness in all topological spaces. –  Brian M. Scott Sep 22 '11 at 18:44
@BrianM.Scott, we defined compactness of K as: every sequence in K has a subsequence that converges to an element in K, which I think is pretty general. –  davin Sep 23 '11 at 0:27
@davin: It is actually a property called sequential compactness, which in metric spaces is equivalent to compactness. –  Asaf Karagila Sep 23 '11 at 6:38

In particular, if $X$ is any space (metric, in this case) and $K\subseteq X$ is a compact subset of $X$ then $K$ is a space equipped with the relative topology.
$U\subseteq K$ is open if and only if there exists some $V\subseteq X$ which is open in the original topology and $K\cap V=U$.
Now if $K$ is a compact subset of $X$ then $K$ forms a compact space with the relative topology. This is since every cover of $K$ by open sets can be extended to an open cover in $X$, where we know $K$ is compact.
From here we can just assume that $K$ is now the entire space, and prove as in the link.


text:
# Displaying Numbers in MATLAB

The way MATLAB displays numbers sometimes confuses users. Occasionally someone posts a concern to the MATLAB newsgroup that the calculation just performed was done to only 4 digits but the user expected more decimals. Assuming the code is "typical" in some vague sense, the user's numeric information at this point is almost always sitting in double precision MATLAB variables.

### Numeric Formats in MATLAB

Helpful posts typically follow up with the explanation that the display of output is controlled by the format setting and that the default numeric variables in MATLAB are indeed double precision, holding about 15 significant decimal digits. By default, MATLAB displays floating point values using what's called a scaled fixed point format, resulting in 4 digits displayed after the decimal point.

Here's the value of pi displayed with the default format in MATLAB.

format
mypi = pi
mypi =

3.1416

and we can see that the variable mypi is stored as a double precision value.

class(mypi)
ans =

double

To see more digits, I can now use some of the other format options, for example

format long
mypi
mypi =

3.141592653589793

### Engineering Notation

In Release R14SP2, we enhanced format to include the engineering formats (short and long), resulting in values displayed with powers of 10 appearing in multiples of 3.

format short eng
manypis = mypi*[1 10 100 1000]
manypis =

3.1416e+000    31.4159e+000   314.1593e+000     3.1416e+003

The new formats allow you some extra flexibility on how to view data before resorting to writing your own specialized formatting routines, for example using fprintf.

### Interesting Diversion

While thinking about this post, I wondered about the history of scientific and engineering notation. Using some of my favorite web tools, I found out that scientific notation was first mentioned historically by Florian Cajori who wrote A History of Mathematical Notations (1928-29, 2 vols.).

I next wandered down the hall to talk to Steve of Image Processing blog fame. Wouldn't you know! He sent a letter to his son's teacher last year when the class was asking about what engineering notation was useful for. Steve found some interesting information engineering format for calculators. How cool is it that I can walk into his office and talk about something tangential like this bit of history and find out that not only does the topic interest him too, but he had his own reason at looking into it from a slightly different perspective. It made my day!

### References

Here are some other references that might be helpful for those sorting out questions of display and precision.

### Year End

It's nearly the end of the year 2006 as well as being approximately the first year birthday for this blog. It's been wonderful meeting you in this forum (and occasionally other ways as well!). I'd encourage you to continue sharing your thoughts with my during the next year as well. Happy New Year to all!

Published with MATLAB® 7.3

|


text:
## The dissection of five-dimensional simplices into orthoschemes.(English)Zbl 0806.52012

Summary: In this paper the dissection of five-dimensional simplices into orthoschemes is shown. Firstly, some general methods for dissecting $$n$$- dimensional Euclidean simplices are described. For this, a description of simplices by graphs is given. All methods for cutting a simplex are investigated with the help of these graphs. The dissection of the five- dimensional Euclidean simplices is thoroughly investigated and it is shown that each of them is decomposable into orthoschemes. Furthermore, the validity of the propositions in spaces of constant curvature is checked and it is proved that every five-dimensional spherical or hyperbolic simplex can be dissected into orthoschemes.

### MSC:

 52B45 Dissections and valuations (Hilbert’s third problem, etc.) 52A55 Spherical and hyperbolic convexity
Full Text:


text:
# How to typeset a proof via structural induction

Structural induction differs from "mathmatical" induction in the number of cases: While mathmatical induction requires exactly two cases, structural induction may require many more.

A proof via structural induction thus requires:

1. An environment for Cases (preferably labeled and numbered, preferably without further indentation)
2. A sub-proof environment that contains the proof for the specific case (preferably with a distinctive q.e.d. sign)

Is there any support inside a \begin{proof}\end{\proof} environment that covers this?

• This definition of 'mathematical induction' is new to me. Maybe I'm just out of date but, when I learnt it, what you are calling 'structural induction' would have (also) been mathematical induction. – cfr Oct 20 '15 at 12:34
• @cfr I think what the OP is calling "mathematical induction" is induction over the naturals, while structural induction works on any algebra (as in universal algebra). The induction principle on naturals is just a special case – Bordaigorl Oct 20 '15 at 12:47

Is this what you are looking for? If the way Cases work here does not please you, you should clarify the question because it's not clear what do you want exactly.

\documentclass{scrartcl}
\usepackage{mathtools,amssymb,amsthm,enumitem}

\SetEnumitemKey{ncases}{itemindent=!,before=\let\makelabel\ncasesmakelabel}
\newcommand*\ncasesmakelabel[1]{Case #1}

\newenvironment{subproof}
{\def\proofname{Subproof}%
\def\qedsymbol{$\triangleleft$}%
\proof}
{\endproof}

\begin{document}
\begin{proof}
Some text or \texttt{\string\leavevmode} (so the enumerate starts in another line)
\begin{enumerate}[ncases]
\item Foo.
\begin{subproof}
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
\end{subproof}
\item Bar.
\begin{subproof}
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
\end{subproof}
\end{enumerate}
And more text.
\end{proof}

\end{document}

• Thanks! that looks great. \edef extends a definition? – choeger Oct 20 '15 at 12:30
• No, in this case it was wrong what I used (not entirely wrong, but didn't achive what I wanted), I updated the code. – Manuel Oct 20 '15 at 12:32
• \MakeLowercase, not \lowercase: the latter works for English, not necessarily for other languages. – egreg Oct 20 '15 at 12:33
• You can put any name you want inside the definition of \proofname I just used a trick that adds Sub before the original name of the proof (after being converted to lowercase). In English that definition is the same as \def\proofname{Subproof}. @egreg True, corrected. – Manuel Oct 20 '15 at 12:33
• I suggest forgetting such clever tricks and defining a \subproofname command. – egreg Oct 20 '15 at 12:49


text:
# Re: Between Regular and Context Free Languages

## torbenm@diku.dk (Torben AEgidius Mogensen)25 Jan 1997 21:46:46 -0500

From comp.compilers

Related articles
Between Regular and Context Free Languages loewe@ipd.info.uni-karlsruhe.de (Welf Loewe) (1997-01-22)
Re: Between Regular and Context Free Languages torbenm@diku.dk (1997-01-25)
Re: Between Regular and Context Free Languages nixon@softlab.se (Leif Nixon) (1997-01-25)
| List of all articles for this month |

 From: torbenm@diku.dk (Torben AEgidius Mogensen) Newsgroups: comp.compilers Date: 25 Jan 1997 21:46:46 -0500 Organization: Department of Computer Science, U of Copenhagen References: 97-01-176 Keywords: syntax, theory

Welf Loewe <loewe@ipd.info.uni-karlsruhe.de> writes:

>Are there classes of languages R* with

>Regular Languages \subset R* \subset context free languages

>such that

>L, L' \in R* ==> L \subset L' is decidable?
>L, L' \in R* ==> L \subset L' is efficiently computable?

The question is undecidable for the most commonly quoted language
class that is strictly between regular and CFLs: the deterministic
context-free languages. Deterministic CFLs are those that can be
recognized by deterministic one-way pushdown automata (DPDA). So your
class can not contain the DCFLs.

You can, of course, add a finite number of non-regular languages to
the class of regular languages and get a class where subset is
decidable. This is, however, not very interesting; you would really
like R*\R to be infinite.

I can construct such a class. I'm not sure it is very interesting,
though. The class is those languages that can be recognized by
deterministic finite automata with the following extension: Each
transition is labelled by 1 or 2 in addition to the alphabet symbol. A
transition labelled 1 is valid in the first half of the input string
and a transition labelled 2 is valid in the second half of the input
string. A state can have two transitions on the same input symbol if
they are labelled by different numbers. This class is obviosuly closed
under intersection, union and complement, and it is decidable whether
the language is empty. You basically use the same methods that are
used for deterministic automata with trivial modifications. Hence,
subset is decidable.

The class is also a subset of the CFLs, as such an automaton can be
containing the symbol $. You can take a transition labelled 1 if there is a 1 or$ on the stack top. If you do so, push a 1 on the stack. You
can take a transition labelled 2 if there is a 1 or 2 on the stack
top. If there is a 1, pop it and push a 2. If there is a 2, pop that
and fail if the stack symbol under that is a $. Otherwise, pop the 1 (as it must be) and push a 2. If the stack top is$ at then end of the
input, you accept.

Hence, if you have a 1 on the stack top, you can nondeterministically
take either a 1-labelled or a 2-labelled transition. But if you are
not at the halfway point when you chose the 2-labelled transition, you
will fail later on.

The class also contains non-regular languages. It can e.g. recognize
the language {a^n b^n | n \in N}. More precisely, it contains all
languages of the form {vw | v \in L1, w \in L2, |v|=|w|} where L1 and
L2 are regular languages. To make it a strict superset of the regular
languages, odd-length strings must be handled as well. This is a
trivial modification.

Torben Mogensen (torbenm@diku.dk)
--

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text:
## Alex's Predictive Knowledge Primer

This is a really tiny bare-bones explanation of predictive knowledge I wrote a while ago to share with someone. WIP.

#### What is Predictive Knowledge?

A key challenge for machine intelligence is that of representation. The performance of a system is tied to its ability to represent and perceive its environment. Predictive knowledge is a proposal for constructing representations of the environment by making large collections of predictions. An agent is able to continually anticipate its sensation from its environment by making many predictions about the dynamics of its environment with respect to its behaviour (Modayil, 2014). These predictions about expected sensation can then be used to inform an agent's internal representation of its environment (Littman and Sutton, 2002). Other proposals describe inter-relations of predictions, similar to Temporal Difference Networks (Tanner and Sutton 2005; Makino, and Takagi, 2008) to enable abstract, conceptual representations by making predictions of predictions (Schapire and Rivest, 1988).

Predictive knowledge systems have been shown to be a scalable way to update and verify an agent's representation of the world, with examples of real-world robotic prediction tasks making thousands or tens of thousands of predictions in real-time on consumer-grade devices (Sutton, 2011).

We specifically consider predictive knowledge methods that 1) are able to expand their representations by proposing new predictions---possibly increasing required resources, 2) are able to self-verify their predictions through interaction with their environment, and 3) are able continually learn their predictions on-line.

#### General Value Functions

An integral part of predictive knowledge is the formulation of the predictor. A method of making predictions are General Value Functions (GVFs) (White, 2015). GVFs estimate the expected discounted return of a signal $C$ defined as . Value is estimated with respect to a specific policy $\pi$, discount function $\gamma$, and cumulant $c$: or, $v(s;\pi, \gamma, c) =\mathop{\mathbb{E}}_\pi[G_t|S_t = s]$ where $s$ is the state of the environment.

A policy $\pi$ specifies the behaviour over which the prediction is being made as the probability of taking an action $a_t$ in state $s_t$: $\pi(a_t|s_t)$. The cumulant $c$ describes the signal of interest---the value which the GVF is accumulating over time. The cumulant $c$ may be a function of observations an agent makes, or internal signals produced by the system (i.e. predictions made by the system, or other learned values).The discounting function $\gamma(s_t,a_t,s_{t+1})$ describes how future rewards are discounted.

The simplest discounting functions use a constant value where $0 \leq \gamma \leq 1$. GVFs where $\gamma = 0$ are myopic: the return at any given state $t$ will be the next observed $c_{t+1}$. Where $\gamma = 1$, the return is undiscounted: all signals equally contribute to the return $G_t$. When varying the discount between $0$ and $1$ we are describing how much future $c$ values should impact value of the current the return.

Another variant of discount functions are termination functions: functions where $0 < \gamma \leq 1$ until some event, where $\gamma = 0$. For example:

In this example, $\gamma = 1$ until a robot runs into something which causes its bump sensor to depress. At this point, $\gamma = 0$. This has the effect of acting as a timer. We accumulate the cumulant $c$ until some event occurs, at which point the accumulation is terminated.

The parameters $c$, $\pi$, and $\gamma$ are the question parameters which specify what a GVF is about. In addition, any learning algorithm specific parameters are considered answer parameters describe how a learning method learns the GVF. For example, in Temporal Difference (TD) learning, the step size $\alpha$ and eligibility decay $\lambda$ for TD learning are the answer parameters.

#### Learning General Value Functions: Temporal-difference Learning

GVFs can be learnt online, incrementally through methods such as Temporal-difference (TD) learning (Sutton 1988). TD methods are of interest, as they are able to update their estimates based on previous estimates: they do not need to wait to observe a final outcome to update their estimates. For instance, if we were to formulate a GVF where we considered what the expectation of the position of the gripper on a robot arm over 10 seconds was, using TD learning we would not need to wait 10 seconds and observe the true return $G$ to update our estimate.

Using TD learning we update a value function $v$ which estimates the true value of a given state $v_*(s) := \mathop{\mathbb{E}} \lbrace G | S_t = s \rbrace$. When considering traditional Reinforcement Learning problems, value could be the estimated accumulation of some reward $r$ associated with a goal-oriented behaviour; however, value is not limited to reward: it can be other signals we wish to estimate, including cumulants of GVFs.

In this primer we describe TD learning with linear function approximation where $\phi(s)$ is a binary feature vector which describes the state $s$. The estimate of a state is then the linear combination of the feature vector $\phi$ and a set of learned weights $w$: $V(s) = w^\top \phi(s)$.

The TD error is:

We do not use the value $v_*$ in our formulation of the target value of our TD error. Instead, the target value is $R_{t+1} + \gamma_t w^\top \phi(s_{t+1})$---a biased estimate of the true return using our current learned estimate $V$. We bootstrap our update by using our learned estimate of the value of the following state $V(s_{t+1})$ to construct the target value. As a result, when performing TD learning, the target value is updated through learning.

Using the TD error $\delta$, we update the weights by taking a step of size $0 < \alpha$ to reduce the error for the presently active features $\phi(s)$.

We can assign credit to previously visited states for currently observed rewards by using elegibility traces. Eligibility traces $z$ maintain a decaying trace of previously active features. At each time-step, the elegibility traces are decayed by $0 \leq \lambda \leq 1$ and the discounting function $\gamma$ and incremented by the currently visited state $\phi(s)$. Where $\lambda = 0$, the weight update is equivalent to $\alpha \delta \phi(s)$, and when $\lambda = 1$ the TD update is a Monte Carlo update.

With eligibility traces the weights are updated by $w \gets w + \alpha \delta z$.

TD methods can be applied online, in real-time with little computational cost. TD's bootstrapped update makes it effective in lifelong continual learning systems where the true return $G_t$ may never be observed, or where the dynamics of the environment are too complex to model.


text:
Question

# Calculate the number of atoms of each element present in $122.5g$of $KCI{O_3}$.

Hint: $1$ mole of a substance contains $6.022\, \times \,{10^{23}}$amount of that substance, which is Avogadro's number. A mole is the amount of substance which has mass equal to the gram atomic mass or gram molecular mass.
Molecular mass$=$sum of atomic masses of all atoms present in that molecule.
Number of moles in a substance $= $\dfrac{{Given\,\,mass\,of\,the\,subs\tan ce}}{{Molecular\,mass}} 1 mole of an atom = 6.022\, \times \,{10^{23}}atoms Complete step by step answer: 1 mole of a molecule is the amount of substance which has mass equal to the gram molecular mass. A gram molecular mass of a molecule is the sum of the atomic masses of all the atoms present in that molecule. 1 mole of each atom is present in 1 mole of a molecule. So, 1 mole of a KCl{O_3} molecule contains 1 mole of K atom, 1 mole of Cl atom, and 3 moles of O atom. Thus, the molecular mass of KCl{O_3}$= \,1 \times 39 + 1 \times 35.5 + 3 \times 16 = 122.5\,g/mol$ By the mole concept, Number of moles in a substance =$\dfrac{{Given\,\,mass\,of\,the\,subs\tan ce}}{{Molecular\,mass}}$
Here, the given mass of $KCI{O_3}$=$122.5g$
Thus, the number of moles in $122.5g $KCI{O_3}$ = $\frac{{122.5g}}{{122.5g/mol}} = 1mole Therefore, 1 mole of KCI{O_3} contains 1 mole of K atom = 6.022\, \times \,{10^{23}}$K$atoms
$1$ mole of $KCI{O_3}$ contains $1$ mole of $Cl$ atom = $6.022\, \times \,{10^{23}}$Clatoms 1 mole of KCI{O_3} contains 3 moles of O atom = 3 \times 6.022\, \times \,{10^{23}}$O$atoms$= 18.066 \times {10^{23}}$$O$atoms
To calculate the molecular mass it is necessary to know the atomic masses of elements. There is 1 mole of each atom present in 1 mole of a molecule. Since, the given mass is equal to the molecular mass of $KCI{O_3}$, it can be easily interpreted that $6.022\, \times \,{10^{23}}$number of each atoms are present in $KCI{O_3}$. But, if the given mass is not equal to the molecular mass, the number of moles of the molecule present in the given mass should be calculated using the above formula, and for such a situation the number of atoms of each element will change.