doi: 10.3934/amc.2020086

Quasi-symmetric designs on $ 56 $ points

Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, HR-10000 Zagreb, Croatia

* Corresponding author: V. Krčadinac

Received  November 2019 Revised  February 2020 Published  June 2020

Fund Project: This work has been fully supported by the Croatian Science Foundation under the project 6732

Computational techniques for the construction of quasi-symmetric block designs are explored and applied to the case with $ 56 $ points. One new $ (56,16,18) $ and many new $ (56,16,6) $ designs are discovered, and non-existence of $ (56,12,9) $ and $ (56,20,19) $ designs with certain automorphism groups is proved. The number of known symmetric $ (78,22,6) $ designs is also significantly increased.

Citation: Vedran Krčadinac, Renata Vlahović Kruc. Quasi-symmetric designs on $ 56 $ points. Advances in Mathematics of Communications, doi: 10.3934/amc.2020086
References:
[1]

T. Beth, D. Jungnickel and H. Lenz, Hanfried Design Theory. Vol. II, 2nd edition, Cambridge University Press, Cambridge, 1999.  Google Scholar

[2]

W. BosmaJ. Cannon and C. Playoust, The Magma algebra system Ⅰ: The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[3]

A. E. Brouwer, The uniqueness of the strongly regular graph on 77 points, J. Graph Theory, 7 (1983), 455-461.  doi: 10.1002/jgt.3190070411.  Google Scholar

[4]

A. E. Brouwer, Uniqueness and nonexistence of some graphs related to $M_22$, Graphs Combin., 2 (1986), 21-29.  doi: 10.1007/BF01788073.  Google Scholar

[5]

A. R. Calderbank, Geometric invariants for quasisymmetric designs, J. Combin. Theory Ser. A, 47 (1988), 101-110.  doi: 10.1016/0097-3165(88)90044-1.  Google Scholar

[6]

D. CrnkovićD. Dumičić Danilović and S. Rukavina, On symmetric (78, 22, 6) designs and related self-orthogonal codes, Util. Math., 109 (2018), 227-253.   Google Scholar

[7]

D. CrnkovićB. G. RodriguesS. Rukavina and V. D. Tonchev, Quasi-symmetric $2$-$(64, 24, 46)$ designs derived from $AG(3, 4)$, Discrete Math., 340 (2017), 2472-2478.  doi: 10.1016/j.disc.2017.06.008.  Google Scholar

[8]

Y. DingS. HoughtenC. LamS. SmithL. Thiel and V. D. Tonchev, Quasi-symmetric $2$-$(28, 12, 11)$ designs with an automorphism of order $7$, J. Combin. Des., 6 (1998), 213-223.   Google Scholar

[9]

I. A. Faradžev, Constructive enumeration of combinatorial objects, in Problemes combinatoires et théorie des graphes, Colloq. Internat. CNRS, Paris, 1978,131–135. Google Scholar

[10]

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.10, 2018., http://www.gap-system.org Google Scholar

[11]

Z. Janko and T. Van Trung, Construction of a new symmetric block design for $(78, 22, 6)$ with the help of tactical decompositions, J. Combin. Theory Ser. A, 40 (1985), 451-455.  doi: 10.1016/0097-3165(85)90107-4.  Google Scholar

[12]

V. Krčadinac, Steiner $2$-designs $S(2, 4, 28)$ with nontrivial automorphisms, Glas. Mat. Ser. III, 37(57) (2002), 259-268.   Google Scholar

[13]

V. Krčadinac and R. Vlahović, New quasi-symmetric designs by the Kramer-Mesner method, Discrete Math., 339 (2016), 2884-2890.  doi: 10.1016/j.disc.2016.05.030.  Google Scholar

[14]

B. D. McKay, Isomorph-free exhaustive generation, J. Algorithms, 26 (1998), 306-324.  doi: 10.1006/jagm.1997.0898.  Google Scholar

[15]

B. D. McKay and A. Piperno, Practical graph isomorphism, Ⅱ, J. Symbolic Comput., 60 (2014), 94-112.  doi: 10.1016/j.jsc.2013.09.003.  Google Scholar

[16]

A. Munemasa and V. D. Tonchev, A new quasi-symmetric $2$-$(56, 16, 6)$ design obtained from codes, Discrete Math., 284 (2004), 231-234.  doi: 10.1016/j.disc.2003.11.036.  Google Scholar

[17]

A. Neumaier, Regular sets and quasisymmetric 2-designs, in Combinatorial Theory (Schloss Rauischholzhausen, 1982), Lecture Notes in Math., Vol. 969, Springer, Berlin-New York, 1982,258–275.  Google Scholar

[18]

S. Niskanen and P. R. J. Östergård, Cliquer User's Guide, Version 1.0, Communications Laboratory, Helsinki University of Technology, Espoo, Finland, Tech. Rep. T48, 2003. Google Scholar

[19]

P. R. J. Östergård, A fast algorithm for the maximum clique problem, Discrete Appl. Math., 120 (2002), 197-207.  doi: 10.1016/S0166-218X(01)00290-6.  Google Scholar

[20]

R. M. Pawale, M. S. Shrikhande and S. M. Nyayate, Conditions for the parameters of the block graph of quasi-symmetric designs, Electron. J. Combin., 22 (2015), Paper 1.36, 30 pp. doi: 10.37236/3954.  Google Scholar

[21]

R. C. Read, Every one a winner or how to avoid isomorphism search when cataloguing combinatorial configurations, in Ann. Discrete Math., 2, 1978,107–120. doi: 10.1016/S0167-5060(08)70325-X.  Google Scholar

[22]

M. S. Shrikhande, Quasi-symmetric designs, in The CNC Handbook of Combinatorial Designs, Second Edition, CRC Press, Boca Raton, FL, 2007,578–582. Google Scholar

[23] M. S. Shrikhande and S. S. Sane, Quasi-symmetric Designs, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511665615.  Google Scholar
[24]

V. D. Tonchev, Embedding of the Witt-Mathieu system $S(3, 6, 22)$ in a symmetric $2$-$(78, 22, 6)$ design, Geom. Dedicata, 22 (1987), 49-75.  doi: 10.1007/BF00183053.  Google Scholar

show all references

References:
[1]

T. Beth, D. Jungnickel and H. Lenz, Hanfried Design Theory. Vol. II, 2nd edition, Cambridge University Press, Cambridge, 1999.  Google Scholar

[2]

W. BosmaJ. Cannon and C. Playoust, The Magma algebra system Ⅰ: The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[3]

A. E. Brouwer, The uniqueness of the strongly regular graph on 77 points, J. Graph Theory, 7 (1983), 455-461.  doi: 10.1002/jgt.3190070411.  Google Scholar

[4]

A. E. Brouwer, Uniqueness and nonexistence of some graphs related to $M_22$, Graphs Combin., 2 (1986), 21-29.  doi: 10.1007/BF01788073.  Google Scholar

[5]

A. R. Calderbank, Geometric invariants for quasisymmetric designs, J. Combin. Theory Ser. A, 47 (1988), 101-110.  doi: 10.1016/0097-3165(88)90044-1.  Google Scholar

[6]

D. CrnkovićD. Dumičić Danilović and S. Rukavina, On symmetric (78, 22, 6) designs and related self-orthogonal codes, Util. Math., 109 (2018), 227-253.   Google Scholar

[7]

D. CrnkovićB. G. RodriguesS. Rukavina and V. D. Tonchev, Quasi-symmetric $2$-$(64, 24, 46)$ designs derived from $AG(3, 4)$, Discrete Math., 340 (2017), 2472-2478.  doi: 10.1016/j.disc.2017.06.008.  Google Scholar

[8]

Y. DingS. HoughtenC. LamS. SmithL. Thiel and V. D. Tonchev, Quasi-symmetric $2$-$(28, 12, 11)$ designs with an automorphism of order $7$, J. Combin. Des., 6 (1998), 213-223.   Google Scholar

[9]

I. A. Faradžev, Constructive enumeration of combinatorial objects, in Problemes combinatoires et théorie des graphes, Colloq. Internat. CNRS, Paris, 1978,131–135. Google Scholar

[10]

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.10, 2018., http://www.gap-system.org Google Scholar

[11]

Z. Janko and T. Van Trung, Construction of a new symmetric block design for $(78, 22, 6)$ with the help of tactical decompositions, J. Combin. Theory Ser. A, 40 (1985), 451-455.  doi: 10.1016/0097-3165(85)90107-4.  Google Scholar

[12]

V. Krčadinac, Steiner $2$-designs $S(2, 4, 28)$ with nontrivial automorphisms, Glas. Mat. Ser. III, 37(57) (2002), 259-268.   Google Scholar

[13]

V. Krčadinac and R. Vlahović, New quasi-symmetric designs by the Kramer-Mesner method, Discrete Math., 339 (2016), 2884-2890.  doi: 10.1016/j.disc.2016.05.030.  Google Scholar

[14]

B. D. McKay, Isomorph-free exhaustive generation, J. Algorithms, 26 (1998), 306-324.  doi: 10.1006/jagm.1997.0898.  Google Scholar

[15]

B. D. McKay and A. Piperno, Practical graph isomorphism, Ⅱ, J. Symbolic Comput., 60 (2014), 94-112.  doi: 10.1016/j.jsc.2013.09.003.  Google Scholar

[16]

A. Munemasa and V. D. Tonchev, A new quasi-symmetric $2$-$(56, 16, 6)$ design obtained from codes, Discrete Math., 284 (2004), 231-234.  doi: 10.1016/j.disc.2003.11.036.  Google Scholar

[17]

A. Neumaier, Regular sets and quasisymmetric 2-designs, in Combinatorial Theory (Schloss Rauischholzhausen, 1982), Lecture Notes in Math., Vol. 969, Springer, Berlin-New York, 1982,258–275.  Google Scholar

[18]

S. Niskanen and P. R. J. Östergård, Cliquer User's Guide, Version 1.0, Communications Laboratory, Helsinki University of Technology, Espoo, Finland, Tech. Rep. T48, 2003. Google Scholar

[19]

P. R. J. Östergård, A fast algorithm for the maximum clique problem, Discrete Appl. Math., 120 (2002), 197-207.  doi: 10.1016/S0166-218X(01)00290-6.  Google Scholar

[20]

R. M. Pawale, M. S. Shrikhande and S. M. Nyayate, Conditions for the parameters of the block graph of quasi-symmetric designs, Electron. J. Combin., 22 (2015), Paper 1.36, 30 pp. doi: 10.37236/3954.  Google Scholar

[21]

R. C. Read, Every one a winner or how to avoid isomorphism search when cataloguing combinatorial configurations, in Ann. Discrete Math., 2, 1978,107–120. doi: 10.1016/S0167-5060(08)70325-X.  Google Scholar

[22]

M. S. Shrikhande, Quasi-symmetric designs, in The CNC Handbook of Combinatorial Designs, Second Edition, CRC Press, Boca Raton, FL, 2007,578–582. Google Scholar

[23] M. S. Shrikhande and S. S. Sane, Quasi-symmetric Designs, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511665615.  Google Scholar
[24]

V. D. Tonchev, Embedding of the Witt-Mathieu system $S(3, 6, 22)$ in a symmetric $2$-$(78, 22, 6)$ design, Geom. Dedicata, 22 (1987), 49-75.  doi: 10.1007/BF00183053.  Google Scholar

Table 1.  The known QSDs with $ v = 56 $
No. $ v $ $ k $ $ \lambda $ $ r $ $ b $ $ x $ $ y $ NQSD Ref.
47 56 16 18 66 231 4 8 $ \ge 3 $ [13]
48 56 15 42 165 616 3 6 0 [5]
49 56 12 9 45 210 0 3 ?
50 56 21 24 66 176 6 9 0 [5]
51 56 20 19 55 154 5 8 ?
52 56 16 6 22 77 4 6 $ \ge 2 $ [24,16]
No. $ v $ $ k $ $ \lambda $ $ r $ $ b $ $ x $ $ y $ NQSD Ref.
47 56 16 18 66 231 4 8 $ \ge 3 $ [13]
48 56 15 42 165 616 3 6 0 [5]
49 56 12 9 45 210 0 3 ?
50 56 21 24 66 176 6 9 0 [5]
51 56 20 19 55 154 5 8 ?
52 56 16 6 22 77 4 6 $ \ge 2 $ [24,16]
Table 3.  Dimensions and weight distributions of self-orthogonal binary codes spanned by $ (56,16,6) $ QSDs
$ \dim $ $ a_0 $ $ a_8 $ $ a_{12} $ $ a_{16} $ $ a_{20} $ $ a_{24} $ $ a_{28} $
$ C_1 $ 26 1 91 2 016 152 425 2 939 776 16 194 619 28 531 008
$ C_2 $ 26 1 7 $ 2\,016 $ $ 155\,365 $ $ 2\,926\,336 $ $ 16\,224\,019 $ $ 28\,493\,376 $
$ C_3 $ 24 1 75 0 40 089 730 368 4 055 835 7 124 480
$ C_{4\hbox{-}6,9,10} $ 22 1 15 0 9 933 183 168 1 012 515 1 783 040
$ C_{7,11\hbox{-}13} $ 25 1 75 672 77 721 1 465 984 8 103 963 14 257 600
$ C_8 $ 25 1 75 960 75 417 1 474 048 8 087 835 14 277 760
$ C_{14} $ 22 1 15 0 10 701 178 560 1 024 035 1 767 680
$ C_{15} $ 23 1 15 288 19 917 361 216 2 040 867 3 544 000
$ C_{16} $ 23 1 15 96 19 917 365 056 2 028 579 3 561 280
$ C_{17} $ 24 1 75 160 39 833 728 704 4 062 235 7 115 200
$ C_{18} $ 22 1 15 64 9 677 183 424 1 012 771 1 782 400
$ C_{19,21,24} $ 22 1 15 16 10 061 182 080 1 015 459 1 779 040
$ C_{20,22} $ 22 1 15 64 10 445 178 816 1 024 291 1 767 040
$ C_{23} $ 25 1 75 1 280 74 905 1 470 720 8 100 635 14 259 200
$ C_{25} $ 25 1 75 992 77 209 1 462 656 8 116 763 14 239 040
$ C_{26} $ 27 1 139 4 992 307 161 5 848 832 32 477 083 56 941 312
$ C_{27} $ 27 1 99 4 304 305 873 5 872 320 32 406 731 57 039 072
$ C_{28,29} $ 27 1 99 4 112 307 409 5 866 944 32 417 483 57 025 632
$ C_{30} $ 26 1 147 1 008 158 529 2 920 512 16 231 467 28 485 536
$ C_{31,32,34,35} $ 27 1 147 3 696 309 057 5 862 976 32 423 979 57 018 016
$ C_{33,39} $ 27 1 147 4 976 307 009 5 849 664 32 475 179 56 943 776
$ C_{36} $ 26 1 75 2 240 153 241 2 931 200 16 218 395 28 498 560
$ C_{37,38} $ 27 1 75 4 416 305 817 5 871 616 32 408 859 57 036 160
$ \dim $ $ a_0 $ $ a_8 $ $ a_{12} $ $ a_{16} $ $ a_{20} $ $ a_{24} $ $ a_{28} $
$ C_1 $ 26 1 91 2 016 152 425 2 939 776 16 194 619 28 531 008
$ C_2 $ 26 1 7 $ 2\,016 $ $ 155\,365 $ $ 2\,926\,336 $ $ 16\,224\,019 $ $ 28\,493\,376 $
$ C_3 $ 24 1 75 0 40 089 730 368 4 055 835 7 124 480
$ C_{4\hbox{-}6,9,10} $ 22 1 15 0 9 933 183 168 1 012 515 1 783 040
$ C_{7,11\hbox{-}13} $ 25 1 75 672 77 721 1 465 984 8 103 963 14 257 600
$ C_8 $ 25 1 75 960 75 417 1 474 048 8 087 835 14 277 760
$ C_{14} $ 22 1 15 0 10 701 178 560 1 024 035 1 767 680
$ C_{15} $ 23 1 15 288 19 917 361 216 2 040 867 3 544 000
$ C_{16} $ 23 1 15 96 19 917 365 056 2 028 579 3 561 280
$ C_{17} $ 24 1 75 160 39 833 728 704 4 062 235 7 115 200
$ C_{18} $ 22 1 15 64 9 677 183 424 1 012 771 1 782 400
$ C_{19,21,24} $ 22 1 15 16 10 061 182 080 1 015 459 1 779 040
$ C_{20,22} $ 22 1 15 64 10 445 178 816 1 024 291 1 767 040
$ C_{23} $ 25 1 75 1 280 74 905 1 470 720 8 100 635 14 259 200
$ C_{25} $ 25 1 75 992 77 209 1 462 656 8 116 763 14 239 040
$ C_{26} $ 27 1 139 4 992 307 161 5 848 832 32 477 083 56 941 312
$ C_{27} $ 27 1 99 4 304 305 873 5 872 320 32 406 731 57 039 072
$ C_{28,29} $ 27 1 99 4 112 307 409 5 866 944 32 417 483 57 025 632
$ C_{30} $ 26 1 147 1 008 158 529 2 920 512 16 231 467 28 485 536
$ C_{31,32,34,35} $ 27 1 147 3 696 309 057 5 862 976 32 423 979 57 018 016
$ C_{33,39} $ 27 1 147 4 976 307 009 5 849 664 32 475 179 56 943 776
$ C_{36} $ 26 1 75 2 240 153 241 2 931 200 16 218 395 28 498 560
$ C_{37,38} $ 27 1 75 4 416 305 817 5 871 616 32 408 859 57 036 160
Table 2.  Dimensions and weight distributions of self-orthogonal binary codes spanned by $ (56,16,18) $ QSDs
$ \dim $ $ a_0 $ $ a_8 $ $ a_{12} $ $ a_{16} $ $ a_{20} $ $ a_{24} $ $ a_{28} $
$ C_{1,2} $ 23 1 75 0 21 657 353 536 2 059 035 3 520 000
$ C_3 $ 19 1 0 0 1 722 19 936 134 085 212 800
$ C_4 $ 23 1 15 216 20 493 359 200 2 044 899 3 538 960
$ \dim $ $ a_0 $ $ a_8 $ $ a_{12} $ $ a_{16} $ $ a_{20} $ $ a_{24} $ $ a_{28} $
$ C_{1,2} $ 23 1 75 0 21 657 353 536 2 059 035 3 520 000
$ C_3 $ 19 1 0 0 1 722 19 936 134 085 212 800
$ C_4 $ 23 1 15 216 20 493 359 200 2 044 899 3 538 960
Table 4.  Distribution of the known $ (56,16,6) $ QSDs and symmetric $ (78,22,6) $ designs by full automorphism group order
$ | \mathrm{Aut}| $ #$ (56,16,6) $ #$ (78,22,6) $
$ 168 $ $ 1 $ $ 2 $
$ 78 $ $ 0 $ $ 1 $
$ 48 $ $ 876 $ $ 1664 $
$ 24 $ $ 1 $ $ 378 $
$ 21 $ $ 1 $ $ 2 $
$ 16 $ $ 228 $ $ 456 $
$ 12 $ $ 303 $ $ 606 $
$ 6 $ $ 0 $ $ 32 $
$ | \mathrm{Aut}| $ #$ (56,16,6) $ #$ (78,22,6) $
$ 168 $ $ 1 $ $ 2 $
$ 78 $ $ 0 $ $ 1 $
$ 48 $ $ 876 $ $ 1664 $
$ 24 $ $ 1 $ $ 378 $
$ 21 $ $ 1 $ $ 2 $
$ 16 $ $ 228 $ $ 456 $
$ 12 $ $ 303 $ $ 606 $
$ 6 $ $ 0 $ $ 32 $
Table 5.  An updated table of QSDs with $ v = 56 $
No. $ v $ $ k $ $ \lambda $ $ r $ $ b $ $ x $ $ y $ NQSD
47 56 16 18 66 231 4 8 $ \ge 4 $
48 56 15 42 165 616 3 6 0
49 56 12 9 45 210 0 3 ?
50 56 21 24 66 176 6 9 0
51 56 20 19 55 154 5 8 ?
52 56 16 6 22 77 4 6 $ \ge 1410 $
No. $ v $ $ k $ $ \lambda $ $ r $ $ b $ $ x $ $ y $ NQSD
47 56 16 18 66 231 4 8 $ \ge 4 $
48 56 15 42 165 616 3 6 0
49 56 12 9 45 210 0 3 ?
50 56 21 24 66 176 6 9 0
51 56 20 19 55 154 5 8 ?
52 56 16 6 22 77 4 6 $ \ge 1410 $
[1]

Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028

[2]

Sabyasachi Dey, Tapabrata Roy, Santanu Sarkar. Revisiting design principles of Salsa and ChaCha. Advances in Mathematics of Communications, 2019, 13 (4) : 689-704. doi: 10.3934/amc.2019041

[3]

Ziteng Wang, Shu-Cherng Fang, Wenxun Xing. On constraint qualifications: Motivation, design and inter-relations. Journal of Industrial & Management Optimization, 2013, 9 (4) : 983-1001. doi: 10.3934/jimo.2013.9.983

[4]

Mao Okada. Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces. Journal of Modern Dynamics, 2021, 17: 111-143. doi: 10.3934/jmd.2021004

[5]

Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023

[6]

Rabiaa Ouahabi, Nasr-Eddine Hamri. Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2361-2370. doi: 10.3934/dcdsb.2020182

[7]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

[8]

Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149

[9]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[10]

Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637

[11]

Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181

[12]

Zhimin Chen, Kaihui Liu, Xiuxiang Liu. Evaluating vaccination effectiveness of group-specific fractional-dose strategies. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021062

[13]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

[14]

Lars Grüne, Luca Mechelli, Simon Pirkelmann, Stefan Volkwein. Performance estimates for economic model predictive control and their application in proper orthogonal decomposition-based implementations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021013

[15]

Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021025

[16]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[17]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[18]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81

[19]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[20]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (87)
  • HTML views (279)
  • Cited by (1)

Other articles
by authors

[Back to Top]