2019, 15: 427-435. doi: 10.3934/jmd.2019026

The work of Mike Hochman on multidimensional symbolic dynamics and Borel dynamics

Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA

Received  October 24, 2018 Published  December 2019

Fund Project: This article is based on the talk I gave on the occasion of Mike Hochman's being awarded the Brin Prize. I am grateful to Nishant Chandgotia, Emmanuel Jeandel, Doug Lind, and Anush Tserunyan for very helpful feedback.

We review the impact of Mike Hochman's work on mutlidimensional symbolic dynamics and Borel dynamics.

Citation: Mike Boyle. The work of Mike Hochman on multidimensional symbolic dynamics and Borel dynamics. Journal of Modern Dynamics, 2019, 15: 427-435. doi: 10.3934/jmd.2019026
References:
[1]

R. L. Adler and B. Marcus, Topological entropy and equivalence of dynamical systems, Mem. Amer. Math. Soc., 20 (1979). doi: 10.1090/memo/0219.  Google Scholar

[2]

N. AubrunS. Barbieri and M. Sablik, A notion of effectiveness for subshifts on finitely generated groups, Theoret. Comput. Sci., 661 (2017), 35-55.  doi: 10.1016/j.tcs.2016.11.033.  Google Scholar

[3]

N. Aubrun and M. Sablik, Simulation of effective subshifts by two-dimensional subshifts of finite type, Acta Appl. Math., 126 (2013), 35-63.  doi: 10.1007/s10440-013-9808-5.  Google Scholar

[4]

R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1989.  Google Scholar

[5]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar

[6]

M. Boyle and J. Buzzi, The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors, J. Eur. Math. Soc. (JEMS), 19 (2017), 2739-2782.  doi: 10.4171/JEMS/727.  Google Scholar

[7]

M. BoyleJ. Buzzi and R. Gómez, Almost isomorphism for countable state Markov shifts, J. Reine Angew. Math., 592 (2006), 23-47.  doi: 10.1515/CRELLE.2006.021.  Google Scholar

[8]

D. Burguet, Topological and almost Borel universality for systems with the weak specification property, Ergodic Theory Dynam. Systems, (first published online February 2019). doi: 10.1017/etds.2019.7.  Google Scholar

[9]

D. Burguet, Topological and almost Borel universality for systems with the weak specification property, preprint, arXiv: 1901.00666. Google Scholar

[10]

J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math., 100 (1997), 125-161.  doi: 10.1007/BF02773637.  Google Scholar

[11]

N. Chandgotia and T. Meyerovitch, Borel subsystems and ergodic universality for compact $\mathbb Z^d$-systems via specification and beyond, preprint, arXiv: 1903.05716. Google Scholar

[12]

C. T. ConleyA. S. Kechris and B. D. Miller, Stationary probability measures and topological realizations, Israel J. Math., 198 (2013), 333-345.  doi: 10.1007/s11856-013-0025-8.  Google Scholar

[13]

M. Delacourt and B. Hellouin de Menibus, Characterisation of limit measures of higher-dimensional cellular automata, Theory Comput. Syst., 61 (2017), 1178-1213.  doi: 10.1007/s00224-017-9753-1.  Google Scholar

[14]

R. DoughertyS. Jackson and A. S. Kechris, The structure of hyperfinite Borel equivalence relations, Trans. Amer. Math. Soc., 341 (1994), 193-225.  doi: 10.1090/S0002-9947-1994-1149121-0.  Google Scholar

[15]

B. Durand and A. Romashchenko, On the expressive power of quasiperiodic SFT, 42nd International Symposium on Mathematical Foundations of Computer Science, 2017.  Google Scholar

[16]

B. Durand, A. Romashchenko and A. Shen, Effective closed subshifts in 1D can be implemented in 2D, in Fields of Logic and Computation, Lecture Notes in Computer Science, 6300, Springer, Berlin, 2010,208–226. doi: 10.1007/978-3-642-15025-8_12.  Google Scholar

[17]

P. Guillon and C. Zinoviadis, Densities and entropies in cellular automata, in How the World Computes, Lecture Notes in Computer Science, 7318, Springer, Heidelberg, 2012,253–263. doi: 10.1007/978-3-642-30870-3_26.  Google Scholar

[18]

B. Hellouin de Menibus and M. Sablik, Characterization of sets of limit measures of a cellular automaton iterated on a random configuration, Ergodic Theory Dynam. Systems, 38 (2018), 601-650.  doi: 10.1017/etds.2016.46.  Google Scholar

[19]

M. Hochman, On the dynamics and recursive properties of multidimensional symbolic systems, Invent. Math., 176 (2009), 131-167.  doi: 10.1007/s00222-008-0161-7.  Google Scholar

[20]

M. Hochman, Isomorphism and embedding of Borel systems on full sets, Acta Appl. Math., 126 (2013), 187-201.  doi: 10.1007/s10440-013-9813-8.  Google Scholar

[21]

M. Hochman, Every Borel automorphism without finite invariant measures admits a two-set generator, J. Eur. Math. Soc. (JEMS), 21 (2019), 271-317.  doi: 10.4171/JEMS/836.  Google Scholar

[22]

M. Hochman, Multidimensional shifts of finite type and sofic shifts, in Combinatorics, Words and Symbolic Dynamics, Encyclopedia Math. Appl., 159, Cambridge Univ. Press, Cambridge, 2016, 296–358.  Google Scholar

[23]

M. Hochman and T. Meyerovitch, A characterization of the entropies of multidimensional shifts of finite type, Ann. of Math. (2), 171 (2010), 2011–2038. doi: 10.4007/annals.2010.171.2011.  Google Scholar

[24]

S. JacksonA. S. Kechris and A. Louveau, Countable Borel equivalence relations, J. Math. Log., 2 (2002), 1-80.  doi: 10.1142/S0219061302000138.  Google Scholar

[25]

E. Jeandel, Computability in symbolic dynamics, in Pursuit of the Universal, Lecture Notes in Computer Science, 9709, Springer, 2016,124–131. doi: 10.1007/978-3-319-40189-8_13.  Google Scholar

[26]

E. Jeandel and P. Vanier, Characterizations of periods of multi-dimensional shifts, Ergodic Theory Dynam. Systems, 35 (2015), 431-460.  doi: 10.1017/etds.2013.60.  Google Scholar

[27]

J. Kari, Rice's theorem for the limit sets of cellular automata, Theoret. Comput. Sci., 127 (1994), 229-254.  doi: 10.1016/0304-3975(94)90041-8.  Google Scholar

[28]

J. Kari, Cellular automata, tilings and (un)computability, in Combinatorics, Words and Symbolic Dynamics, Encyclopedia Math. Appl., 159, Cambridge Univ. Press, Cambridge, 2016, 241–295.  Google Scholar

[29]

P. W. Kasteleyn, The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice, Physica, 27 (1961), 1209-1225.  doi: 10.1016/0031-8914(61)90063-5.  Google Scholar

[30]

W. Krieger, On entropy and generators of measure-preserving transformations, Trans. Amer. Math. Soc., 149 (1970), 453-464.  doi: 10.1090/S0002-9947-1970-0259068-3.  Google Scholar

[31]

E. H. Lieb, Residual entropy of square ice, Phys. Rev., 162, 1967. doi: 10.1103/PhysRev.162.162.  Google Scholar

[32]

D. A. Lind, The entropies of topological Markov shifts and a related class of algebraic integers, Ergodic Theory Dynam. Systems, 4 (1984), 283-300.  doi: 10.1017/S0143385700002443.  Google Scholar

[33]

D. Lind, Multi-dimensional symbolic dynamics, in Symbolic Dynamics and Its Applications, Proc. Sympos. Appl. Math., 60, Amer. Math. Soc., Providence, RI, 2004, 61–79. doi: 10.1090/psapm/060/2078846.  Google Scholar

[34]

K. McGoff and R. Pavlov, Random $\Bbb{Z}^d$-shifts of finite type, J. Mod. Dyn., 10 (2016), 287-330.  doi: 10.3934/jmd.2016.10.287.  Google Scholar

[35]

T. Meyerovitch, Growth-type invariants for $\Bbb Z^d$ subshifts of finite type and arithmetical classes of real numbers, Invent. Math., 184 (2011), 567-589.  doi: 10.1007/s00222-010-0296-1.  Google Scholar

[36]

S. Mozes, Tilings, substitution systems and dynamical systems generated by them, J. Analyse Math., 53 (1989), 139-186.  doi: 10.1007/BF02793412.  Google Scholar

[37]

R. Pavlov and M. Schraudner, Entropies realizable by block gluing $\Bbb{Z}^d$ shifts of finite type, J. Anal. Math., 126 (2015), 113-174.  doi: 10.1007/s11854-015-0014-4.  Google Scholar

[38]

A. Quas and T. Soo, Ergodic universality of some topological dynamical systems, Trans. Amer. Math. Soc., 368 (2016), 4137–4170. doi: 10.1090/tran/6489.  Google Scholar

[39]

R. M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math., 12 (1971), 177-209.  doi: 10.1007/BF01418780.  Google Scholar

[40]

S. Shelah and B. Weiss, Measurable recurrence and quasi-invariant measures, Israel J. Math., 43 (1982), 154-160.  doi: 10.1007/BF02761726.  Google Scholar

[41]

S. G. Simpson, Medvedev degrees of two-dimensional subshifts of finite type, Ergodic Theory Dynam. Systems, 34 (2014), 679-688.  doi: 10.1017/etds.2012.152.  Google Scholar

[42]

A. Tserunyan, Finite generators for countable group actions in the Borel and Baire category settings, Adv. Math., 269 (2015), 585-646.  doi: 10.1016/j.aim.2014.10.013.  Google Scholar

[43]

B. Weiss, Measurable dynamics, in Conference in Modern Analysis and Probability, Contemp. Math., 26, Amer. Math. Soc., Providence, RI, 1984,395–421. doi: 10.1090/conm/026/737417.  Google Scholar

[44]

B. Weiss, Countable generators in dynamics–-universal minimal models, in Measure and Measurable Dynamics, Contemp. Math., 94, Amer. Math. Soc., Providence, RI, 1989,321–326. doi: 10.1090/conm/094/1013000.  Google Scholar

[45]

C. Zinoviadis, Hierarchy and expansiveness in 2D subshifts of finite type, in Language and Automata Theory and Applications, Lecture Notes in Computer Science, 8977, Springer, Cham, 2015,365–377. doi: 10.1007/978-3-319-15579-1_28.  Google Scholar

show all references

References:
[1]

R. L. Adler and B. Marcus, Topological entropy and equivalence of dynamical systems, Mem. Amer. Math. Soc., 20 (1979). doi: 10.1090/memo/0219.  Google Scholar

[2]

N. AubrunS. Barbieri and M. Sablik, A notion of effectiveness for subshifts on finitely generated groups, Theoret. Comput. Sci., 661 (2017), 35-55.  doi: 10.1016/j.tcs.2016.11.033.  Google Scholar

[3]

N. Aubrun and M. Sablik, Simulation of effective subshifts by two-dimensional subshifts of finite type, Acta Appl. Math., 126 (2013), 35-63.  doi: 10.1007/s10440-013-9808-5.  Google Scholar

[4]

R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1989.  Google Scholar

[5]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar

[6]

M. Boyle and J. Buzzi, The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors, J. Eur. Math. Soc. (JEMS), 19 (2017), 2739-2782.  doi: 10.4171/JEMS/727.  Google Scholar

[7]

M. BoyleJ. Buzzi and R. Gómez, Almost isomorphism for countable state Markov shifts, J. Reine Angew. Math., 592 (2006), 23-47.  doi: 10.1515/CRELLE.2006.021.  Google Scholar

[8]

D. Burguet, Topological and almost Borel universality for systems with the weak specification property, Ergodic Theory Dynam. Systems, (first published online February 2019). doi: 10.1017/etds.2019.7.  Google Scholar

[9]

D. Burguet, Topological and almost Borel universality for systems with the weak specification property, preprint, arXiv: 1901.00666. Google Scholar

[10]

J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math., 100 (1997), 125-161.  doi: 10.1007/BF02773637.  Google Scholar

[11]

N. Chandgotia and T. Meyerovitch, Borel subsystems and ergodic universality for compact $\mathbb Z^d$-systems via specification and beyond, preprint, arXiv: 1903.05716. Google Scholar

[12]

C. T. ConleyA. S. Kechris and B. D. Miller, Stationary probability measures and topological realizations, Israel J. Math., 198 (2013), 333-345.  doi: 10.1007/s11856-013-0025-8.  Google Scholar

[13]

M. Delacourt and B. Hellouin de Menibus, Characterisation of limit measures of higher-dimensional cellular automata, Theory Comput. Syst., 61 (2017), 1178-1213.  doi: 10.1007/s00224-017-9753-1.  Google Scholar

[14]

R. DoughertyS. Jackson and A. S. Kechris, The structure of hyperfinite Borel equivalence relations, Trans. Amer. Math. Soc., 341 (1994), 193-225.  doi: 10.1090/S0002-9947-1994-1149121-0.  Google Scholar

[15]

B. Durand and A. Romashchenko, On the expressive power of quasiperiodic SFT, 42nd International Symposium on Mathematical Foundations of Computer Science, 2017.  Google Scholar

[16]

B. Durand, A. Romashchenko and A. Shen, Effective closed subshifts in 1D can be implemented in 2D, in Fields of Logic and Computation, Lecture Notes in Computer Science, 6300, Springer, Berlin, 2010,208–226. doi: 10.1007/978-3-642-15025-8_12.  Google Scholar

[17]

P. Guillon and C. Zinoviadis, Densities and entropies in cellular automata, in How the World Computes, Lecture Notes in Computer Science, 7318, Springer, Heidelberg, 2012,253–263. doi: 10.1007/978-3-642-30870-3_26.  Google Scholar

[18]

B. Hellouin de Menibus and M. Sablik, Characterization of sets of limit measures of a cellular automaton iterated on a random configuration, Ergodic Theory Dynam. Systems, 38 (2018), 601-650.  doi: 10.1017/etds.2016.46.  Google Scholar

[19]

M. Hochman, On the dynamics and recursive properties of multidimensional symbolic systems, Invent. Math., 176 (2009), 131-167.  doi: 10.1007/s00222-008-0161-7.  Google Scholar

[20]

M. Hochman, Isomorphism and embedding of Borel systems on full sets, Acta Appl. Math., 126 (2013), 187-201.  doi: 10.1007/s10440-013-9813-8.  Google Scholar

[21]

M. Hochman, Every Borel automorphism without finite invariant measures admits a two-set generator, J. Eur. Math. Soc. (JEMS), 21 (2019), 271-317.  doi: 10.4171/JEMS/836.  Google Scholar

[22]

M. Hochman, Multidimensional shifts of finite type and sofic shifts, in Combinatorics, Words and Symbolic Dynamics, Encyclopedia Math. Appl., 159, Cambridge Univ. Press, Cambridge, 2016, 296–358.  Google Scholar

[23]

M. Hochman and T. Meyerovitch, A characterization of the entropies of multidimensional shifts of finite type, Ann. of Math. (2), 171 (2010), 2011–2038. doi: 10.4007/annals.2010.171.2011.  Google Scholar

[24]

S. JacksonA. S. Kechris and A. Louveau, Countable Borel equivalence relations, J. Math. Log., 2 (2002), 1-80.  doi: 10.1142/S0219061302000138.  Google Scholar

[25]

E. Jeandel, Computability in symbolic dynamics, in Pursuit of the Universal, Lecture Notes in Computer Science, 9709, Springer, 2016,124–131. doi: 10.1007/978-3-319-40189-8_13.  Google Scholar

[26]

E. Jeandel and P. Vanier, Characterizations of periods of multi-dimensional shifts, Ergodic Theory Dynam. Systems, 35 (2015), 431-460.  doi: 10.1017/etds.2013.60.  Google Scholar

[27]

J. Kari, Rice's theorem for the limit sets of cellular automata, Theoret. Comput. Sci., 127 (1994), 229-254.  doi: 10.1016/0304-3975(94)90041-8.  Google Scholar

[28]

J. Kari, Cellular automata, tilings and (un)computability, in Combinatorics, Words and Symbolic Dynamics, Encyclopedia Math. Appl., 159, Cambridge Univ. Press, Cambridge, 2016, 241–295.  Google Scholar

[29]

P. W. Kasteleyn, The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice, Physica, 27 (1961), 1209-1225.  doi: 10.1016/0031-8914(61)90063-5.  Google Scholar

[30]

W. Krieger, On entropy and generators of measure-preserving transformations, Trans. Amer. Math. Soc., 149 (1970), 453-464.  doi: 10.1090/S0002-9947-1970-0259068-3.  Google Scholar

[31]

E. H. Lieb, Residual entropy of square ice, Phys. Rev., 162, 1967. doi: 10.1103/PhysRev.162.162.  Google Scholar

[32]

D. A. Lind, The entropies of topological Markov shifts and a related class of algebraic integers, Ergodic Theory Dynam. Systems, 4 (1984), 283-300.  doi: 10.1017/S0143385700002443.  Google Scholar

[33]

D. Lind, Multi-dimensional symbolic dynamics, in Symbolic Dynamics and Its Applications, Proc. Sympos. Appl. Math., 60, Amer. Math. Soc., Providence, RI, 2004, 61–79. doi: 10.1090/psapm/060/2078846.  Google Scholar

[34]

K. McGoff and R. Pavlov, Random $\Bbb{Z}^d$-shifts of finite type, J. Mod. Dyn., 10 (2016), 287-330.  doi: 10.3934/jmd.2016.10.287.  Google Scholar

[35]

T. Meyerovitch, Growth-type invariants for $\Bbb Z^d$ subshifts of finite type and arithmetical classes of real numbers, Invent. Math., 184 (2011), 567-589.  doi: 10.1007/s00222-010-0296-1.  Google Scholar

[36]

S. Mozes, Tilings, substitution systems and dynamical systems generated by them, J. Analyse Math., 53 (1989), 139-186.  doi: 10.1007/BF02793412.  Google Scholar

[37]

R. Pavlov and M. Schraudner, Entropies realizable by block gluing $\Bbb{Z}^d$ shifts of finite type, J. Anal. Math., 126 (2015), 113-174.  doi: 10.1007/s11854-015-0014-4.  Google Scholar

[38]

A. Quas and T. Soo, Ergodic universality of some topological dynamical systems, Trans. Amer. Math. Soc., 368 (2016), 4137–4170. doi: 10.1090/tran/6489.  Google Scholar

[39]

R. M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math., 12 (1971), 177-209.  doi: 10.1007/BF01418780.  Google Scholar

[40]

S. Shelah and B. Weiss, Measurable recurrence and quasi-invariant measures, Israel J. Math., 43 (1982), 154-160.  doi: 10.1007/BF02761726.  Google Scholar

[41]

S. G. Simpson, Medvedev degrees of two-dimensional subshifts of finite type, Ergodic Theory Dynam. Systems, 34 (2014), 679-688.  doi: 10.1017/etds.2012.152.  Google Scholar

[42]

A. Tserunyan, Finite generators for countable group actions in the Borel and Baire category settings, Adv. Math., 269 (2015), 585-646.  doi: 10.1016/j.aim.2014.10.013.  Google Scholar

[43]

B. Weiss, Measurable dynamics, in Conference in Modern Analysis and Probability, Contemp. Math., 26, Amer. Math. Soc., Providence, RI, 1984,395–421. doi: 10.1090/conm/026/737417.  Google Scholar

[44]

B. Weiss, Countable generators in dynamics–-universal minimal models, in Measure and Measurable Dynamics, Contemp. Math., 94, Amer. Math. Soc., Providence, RI, 1989,321–326. doi: 10.1090/conm/094/1013000.  Google Scholar

[45]

C. Zinoviadis, Hierarchy and expansiveness in 2D subshifts of finite type, in Language and Automata Theory and Applications, Lecture Notes in Computer Science, 8977, Springer, Cham, 2015,365–377. doi: 10.1007/978-3-319-15579-1_28.  Google Scholar

[1]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270

[2]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[3]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[4]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[5]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116

[6]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426

[7]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[8]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

[9]

Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy, Miles H. Wheeler. Stuart-type polar vortices on a rotating sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 201-215. doi: 10.3934/dcds.2020263

[10]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[11]

Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117

[12]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[13]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120

[14]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

[15]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[16]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351

[17]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

[18]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

2019 Impact Factor: 0.465

Metrics

  • PDF downloads (101)
  • HTML views (192)
  • Cited by (0)

Other articles
by authors

[Back to Top]