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Dynamics of boolean networks
1.  Department of Mathematical Sciences, University of WisconsinMilwaukee, Milwaukee, United States 
References:
[1] 
R. Albert and H. Othmer, The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster, J. Theor. Biol., 223 (2003), 118. doi: 10.1016/S00225193(03)000353. 
[2] 
C. L. Barrett, W. Y. C. Chen and M. J. Zheng, Discrete dynamical systems on graphs and boolean functions, Math. Comput. Simul., 66 (2004), 487497. doi: 10.1016/j.matcom.2004.03.003. 
[3] 
D. Bollman, O. ColóReyes and E. Orozco, Fixed points in discrete models for regulatory genetic networks, EURASIP Journal on Bioinformatics and System Biology, (2007), Online ID97356. 
[4] 
G. Boole, "The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning," Macmillan, Barclay and Macmillan, Cambridge; George Bell, London, 1847. Reprints (1948, 1951), Basil Blackwell, Oxford. 
[5] 
G. Boole, "An Investigation of the Laws of Thought, on Which are Founded the Mathematical Theories of Logic and Probabilities," Macmillan, Barclay and Macmillan, Cambridge; Walton and Maberly, London, 1854. Reprint (1960), Dover, New York. 
[6] 
M. Brickenstein and A. Dreyer, PolyBoRi: A framework for Gröbnerbasis computations with Boolean polynomials, J. Symbolic Comput., 44 (2009), 13261345 doi: 10.1016/j.jsc.2008.02.017. 
[7] 
M. Brickenstein, A. Dreyer, GM. Greuel, M. Wedler and O. Wienand, New developments in the theory of Gröbner bases and applications to formal verification, J. Pure Appl. Algebra, 213 (2009), 16121635. arXiv:0801.1177. 
[8] 
O. ColónReyes, R. Laubenbacher and B. Pareigis, Boolean monomial dynamical systems, Annals of Combinatorics, 8 (2004), 425439. arXiv:math/0403166v1. 
[9] 
M. I. Davidich and S. Bornholdt, Boolean network model predicts cell cycle sequence of fission yeast, PLoS ONE, 3 (2008), e1672. 
[10] 
E. Dubrova, M. Teslenko and A. Martinelli, Kauffman networks: Analysis and applications, ComputerAided Design, IEEE/ACM International Conference (2005), 479484, doi: 10.1109/ICCAD.2005.1560115. 
[11] 
E. D. Fabricius, "Modern Digital Design and Switching Theory," CRC Press 1992. 
[12] 
J. F. Groote and M. Keinänen, A subquadratic algorithm for conjunctive and disjunctive BESs, Theoretical aspects of computingICTAC 2005, 532545, Lecture Notes in Comput. Sci., 3722, Springer, Berlin 2005. 
[13] 
R. A. HernádezToledo, Linear finite dynamical systems, Communications in Algebra, 33 (2005), 29772989. doi: 10.1081/AGB200066211. 
[14] 
A. Ilichinsky, "Cellular Automata: A Discrete Universe," World Scientific Publishing Company, 2001. 
[15] 
A. Jarrah, R. Laubenbacher, B. Stigler and M. Stillman, Reverseengineering of polynomial dynamical systems, Adv. in Appl. Math., 39 (2007), 477489. arXiv:qbio/0605032v1. 
[16] 
A. Jarrah, B. Raposa and R. Laubenbacher, Nested canalyzing, unate cascade, and polynomial functions, Physica D, 233 (2007), 167174. arXiv:qbio/0606013v3. 
[17] 
A. Jarrah, R. Laubenbacher and A. VelizCuba, The dynamics of conjunctive and disjunctive Boolean networks,, preprint available at: \arXiv{0805.0275v1}., (). 
[18] 
W. Just, The steady state system problem is NPhard even for monotone quadratic Boolean dynamical systems,, preprint available at: \url{http://www.math.ohiou.edu/ just/publ.html}., (). 
[19] 
S. Kauffman, C. Peterson, B. Samuelsson and C. Troein, Genetic networks with canalyzing Boolean rules are always stable, PNAS, 101 (2004), 1710217107. doi: 10.1073/pnas.0407783101. 
[20] 
R. Laubenbacher and B. Stigler, A computational algebra approach to the reverse engineering of gene regulatory networks, J. Theor. Biol., 229 (2004), 523537. arXiv:qbio/0312026. 
[21] 
T. E. Malloy, J. Butner and G. C. Jensen, The emergence of dynamic form through phase relations in dynamic systems, Nonlinear Dynamics, Psychology, and Life Sciences, 12 (2008), 371395. 
[22] 
R. Pal, I. Ivanov, A. Datta, M. L. Bittner and E. R. Dougherty, Generating Boolean networks with a prescribed attractor structure, Bioinformatics, 21 (2005), 40214025. doi: 10.1093/bioinformatics/bti664. 
[23] 
J. Reger and K. Schmidt, Modeling and analyzing finite state automata in the finite field $F_2$, Mathematics and Computers in Simulation, 66 (2004), 193206. doi: 10.1016/j.matcom.2003.11.005. 
[24] 
N. A. W. Riel, Dynamic modelling and analysis of biochemical networks: mechanismbased models and modelbased experiments, Briefings in Bioinformatics, 7 (2006), 364374. doi: 10.1093/bib/bbl040. 
[25] 
S. Rudeanu, "Boolean Functions and Equations," NorthHolland, Amsterdam, 1974. 
[26] 
M. H. Stone, The theory of representation for Boolean algebras, Transactions of American Mathematical Society, 40 (1936), 37111. 
[27] 
T. Tamura and T. Akutsu, Algorithms for singleton attractor detection in planar and nonplanar AND/OR Boolean networks, Math. Comput. Sci., 2 (2009), 401420. doi: 10.1007/s1178600800635. 
[28] 
C. J. Tomlin and J. D. Aelrod, Biology by numbers: Mathematical modelling in developmental biology, Nature Reviews Genetics, 8 (2007), 331340. doi: 10.1038/nrg2098. 
[29] 
SQ. Zhang, M. Hayashida, T. Akutsu, WK. Ching and M. K. Ng, Algorithms for finding small attractors in Boolean networks, EURASIP Journal on Bioinformatics and Systems Biology (2007). doi: 10.1155/2007/20180. 
show all references
References:
[1] 
R. Albert and H. Othmer, The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster, J. Theor. Biol., 223 (2003), 118. doi: 10.1016/S00225193(03)000353. 
[2] 
C. L. Barrett, W. Y. C. Chen and M. J. Zheng, Discrete dynamical systems on graphs and boolean functions, Math. Comput. Simul., 66 (2004), 487497. doi: 10.1016/j.matcom.2004.03.003. 
[3] 
D. Bollman, O. ColóReyes and E. Orozco, Fixed points in discrete models for regulatory genetic networks, EURASIP Journal on Bioinformatics and System Biology, (2007), Online ID97356. 
[4] 
G. Boole, "The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning," Macmillan, Barclay and Macmillan, Cambridge; George Bell, London, 1847. Reprints (1948, 1951), Basil Blackwell, Oxford. 
[5] 
G. Boole, "An Investigation of the Laws of Thought, on Which are Founded the Mathematical Theories of Logic and Probabilities," Macmillan, Barclay and Macmillan, Cambridge; Walton and Maberly, London, 1854. Reprint (1960), Dover, New York. 
[6] 
M. Brickenstein and A. Dreyer, PolyBoRi: A framework for Gröbnerbasis computations with Boolean polynomials, J. Symbolic Comput., 44 (2009), 13261345 doi: 10.1016/j.jsc.2008.02.017. 
[7] 
M. Brickenstein, A. Dreyer, GM. Greuel, M. Wedler and O. Wienand, New developments in the theory of Gröbner bases and applications to formal verification, J. Pure Appl. Algebra, 213 (2009), 16121635. arXiv:0801.1177. 
[8] 
O. ColónReyes, R. Laubenbacher and B. Pareigis, Boolean monomial dynamical systems, Annals of Combinatorics, 8 (2004), 425439. arXiv:math/0403166v1. 
[9] 
M. I. Davidich and S. Bornholdt, Boolean network model predicts cell cycle sequence of fission yeast, PLoS ONE, 3 (2008), e1672. 
[10] 
E. Dubrova, M. Teslenko and A. Martinelli, Kauffman networks: Analysis and applications, ComputerAided Design, IEEE/ACM International Conference (2005), 479484, doi: 10.1109/ICCAD.2005.1560115. 
[11] 
E. D. Fabricius, "Modern Digital Design and Switching Theory," CRC Press 1992. 
[12] 
J. F. Groote and M. Keinänen, A subquadratic algorithm for conjunctive and disjunctive BESs, Theoretical aspects of computingICTAC 2005, 532545, Lecture Notes in Comput. Sci., 3722, Springer, Berlin 2005. 
[13] 
R. A. HernádezToledo, Linear finite dynamical systems, Communications in Algebra, 33 (2005), 29772989. doi: 10.1081/AGB200066211. 
[14] 
A. Ilichinsky, "Cellular Automata: A Discrete Universe," World Scientific Publishing Company, 2001. 
[15] 
A. Jarrah, R. Laubenbacher, B. Stigler and M. Stillman, Reverseengineering of polynomial dynamical systems, Adv. in Appl. Math., 39 (2007), 477489. arXiv:qbio/0605032v1. 
[16] 
A. Jarrah, B. Raposa and R. Laubenbacher, Nested canalyzing, unate cascade, and polynomial functions, Physica D, 233 (2007), 167174. arXiv:qbio/0606013v3. 
[17] 
A. Jarrah, R. Laubenbacher and A. VelizCuba, The dynamics of conjunctive and disjunctive Boolean networks,, preprint available at: \arXiv{0805.0275v1}., (). 
[18] 
W. Just, The steady state system problem is NPhard even for monotone quadratic Boolean dynamical systems,, preprint available at: \url{http://www.math.ohiou.edu/ just/publ.html}., (). 
[19] 
S. Kauffman, C. Peterson, B. Samuelsson and C. Troein, Genetic networks with canalyzing Boolean rules are always stable, PNAS, 101 (2004), 1710217107. doi: 10.1073/pnas.0407783101. 
[20] 
R. Laubenbacher and B. Stigler, A computational algebra approach to the reverse engineering of gene regulatory networks, J. Theor. Biol., 229 (2004), 523537. arXiv:qbio/0312026. 
[21] 
T. E. Malloy, J. Butner and G. C. Jensen, The emergence of dynamic form through phase relations in dynamic systems, Nonlinear Dynamics, Psychology, and Life Sciences, 12 (2008), 371395. 
[22] 
R. Pal, I. Ivanov, A. Datta, M. L. Bittner and E. R. Dougherty, Generating Boolean networks with a prescribed attractor structure, Bioinformatics, 21 (2005), 40214025. doi: 10.1093/bioinformatics/bti664. 
[23] 
J. Reger and K. Schmidt, Modeling and analyzing finite state automata in the finite field $F_2$, Mathematics and Computers in Simulation, 66 (2004), 193206. doi: 10.1016/j.matcom.2003.11.005. 
[24] 
N. A. W. Riel, Dynamic modelling and analysis of biochemical networks: mechanismbased models and modelbased experiments, Briefings in Bioinformatics, 7 (2006), 364374. doi: 10.1093/bib/bbl040. 
[25] 
S. Rudeanu, "Boolean Functions and Equations," NorthHolland, Amsterdam, 1974. 
[26] 
M. H. Stone, The theory of representation for Boolean algebras, Transactions of American Mathematical Society, 40 (1936), 37111. 
[27] 
T. Tamura and T. Akutsu, Algorithms for singleton attractor detection in planar and nonplanar AND/OR Boolean networks, Math. Comput. Sci., 2 (2009), 401420. doi: 10.1007/s1178600800635. 
[28] 
C. J. Tomlin and J. D. Aelrod, Biology by numbers: Mathematical modelling in developmental biology, Nature Reviews Genetics, 8 (2007), 331340. doi: 10.1038/nrg2098. 
[29] 
SQ. Zhang, M. Hayashida, T. Akutsu, WK. Ching and M. K. Ng, Algorithms for finding small attractors in Boolean networks, EURASIP Journal on Bioinformatics and Systems Biology (2007). doi: 10.1155/2007/20180. 
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