-
Previous Article
The boundedness and upper semicontinuity of the pullback attractors for a 2D micropolar fluid flows with delay
- ERA Home
- This Issue
-
Next Article
Recent progress on the mathematical study of anomalous localized resonance in elasticity
New series for powers of $ \pi $ and related congruences
Department of Mathematics, Nanjing University, Nanjing 210093, China |
$ \pi $ |
$ \sum\limits_{k = 0}^\infty\frac{P(k)\binom{2k}k\binom{3k}k\binom{6k}{3k}}{(k+1)(2k-1)(6k-1)(-640320)^{3k}} = \frac{18\times557403^3\sqrt{10005}}{5\pi} $ |
$ \begin{align*} P(k) = &637379600041024803108 k^2 + 657229991696087780968 k \\&+ 19850391655004126179, \end{align*} $ |
$ \sum\limits_{k = 1}^\infty \frac{(3k+1)16^k}{(2k+1)^2k^3 \binom{2k}k^3} = \frac{\pi^2-8}2, $ |
$ \sum\limits_{n = 0}^\infty\frac{3n+1}{(-100)^n}\sum\limits_{k = 0}^n{n\choose k}^2T_k(1,25)T_{n-k}(1,25) = \frac{25}{8\pi}, $ |
$ T_k(b,c) $ |
$ x^k $ |
$ (x^2+bx+c)^k $ |
$ 1/\pi $ |
$ \pi $ |
$ \sum\limits_{k = 0}^\infty\frac{39480k+7321}{(-29700)^k}T_k(14,1)T_k(11,-11)^2 = \frac{6795\sqrt5}{\pi}. $ |
$ 8 $ |
References:
[1] |
N. D. Baruah and B. C. Berndt,
Eisenstein series and Ramanujan-type series for $1/\pi$, Ramanujan J., 23 (2010), 17-44.
doi: 10.1007/s11139-008-9155-8. |
[2] |
B. C. Berndt, Ramanujan's Notebooks. Part IV, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-0879-2. |
[3] |
H. H. Chan, S. H. Chan and Z. Liu,
Domb's numbers and Ramanujan-Sato type series for $1/\pi$, Adv. Math., 186 (2004), 396-410.
doi: 10.1016/j.aim.2003.07.012. |
[4] |
H. H. Chan and S. Cooper,
Rational analogues of Ramanujan's series for $1/\pi$, Math. Proc. Cambridge Philos. Soc., 153 (2012), 361-383.
doi: 10.1017/S0305004112000254. |
[5] |
H. H. Chan, Y. Tanigawa, Y. Yang and W. Zudilin,
New analogues of Clausen's identities arising from the theory of modular forms, Adv. Math., 228 (2011), 1294-1314.
doi: 10.1016/j.aim.2011.06.011. |
[6] |
H. H. Chan, J. Wan and W. Zudilin,
Legendre polynomials and Ramanujan-type series for $1/\pi$, Israel J. Math., 194 (2013), 183-207.
doi: 10.1007/s11856-012-0081-5. |
[7] |
W. Y. C. Chen, Q.-H. Hou and Y.-P. Mu,
A telescoping method for double summations, J. Comput. Appl. Math., 196 (2006), 553-566.
doi: 10.1016/j.cam.2005.10.010. |
[8] |
D. V. Chudnovsky and G. V. Chudnovsky, Approximations and complex multiplication according to Ramanujan in Ramanujan Revisited, Academic Press, Boston, MA, 1988.
![]() |
[9] |
S. Cooper,
Sporadic sequences, modular forms and new series for $1/\pi$, Ramanujan J., 29 (2012), 163-183.
doi: 10.1007/s11139-011-9357-3. |
[10] |
S. Cooper, Ramanujan's Theta Functions, Springer, Cham, 2017.
doi: 10.1007/978-3-319-56172-1. |
[11] |
S. Cooper, J. G. Wan and W. Zudilin, Holonomic alchemy and series for $1/\pi$, in Analytic Number Theory, Modular Forms and $q$-Hypergeometric Series, Springer Proc. Math. Stat., 221, Springer, Cham, 2017,179–205.
doi: 10.1007/978-3-319-68376-8_12. |
[12] |
D. A. Cox, Primes of the Form $x^2+ny^2$. Fermat, Class Field Theory and Complex Multiplication, John Wiley & Sons, Inc., New York, 1989. |
[13] |
H. R. P. Ferguson, D. H. Bailey and S. Arno,
Analysis of PSLQ, an integer relation finding algorithm, Math. Comp., 68 (1999), 351-369.
doi: 10.1090/S0025-5718-99-00995-3. |
[14] |
J. Franel, On a question of Laisant, L'Intermédiaire des Mathématiciens, 1 (1894), 45-47. Google Scholar |
[15] |
J. W. L. Glaisher, On series for $1/\pi$ and $1/\pi^2$, Quart. J. Pure Appl. Math., 37 (1905), 173-198. Google Scholar |
[16] |
J. Guillera, Tables of Ramanujan series with rational values of $z$., Available from: http://personal.auna.com/jguillera/ramatables.pdf. Google Scholar |
[17] |
J. Guillera,
Hypergeometric identities for 10 extended Ramanujan-type series, Ramanujan J., 15 (2008), 219-234.
doi: 10.1007/s11139-007-9074-0. |
[18] |
J. Guillera and M. Rogers,
Ramanujan series upside-down, J. Aust. Math. Soc., 97 (2014), 78-106.
doi: 10.1017/S1446788714000147. |
[19] |
V. J. W. Guo and J.-C. Liu,
Some congruences related to a congruence of Van Hamme, Integral Transforms Spec. Funct., 31 (2020), 221-231.
doi: 10.1080/10652469.2019.1685991. |
[20] |
V. J. W. Guo, G.-S. Mao and H. Pan,
Proof of a conjecture involving Sun polynomials, J. Difference Equ. Appl., 22 (2016), 1184-1197.
doi: 10.1080/10236198.2016.1188088. |
[21] |
V. J. W. Guo and M. J. Schlosser, Some $q$-supercongruences from transfromation formulas for basic hypergeometric series, Constr. Approx., to appear. Google Scholar |
[22] |
K. Hessami Pilehrood and T. Hessami Pilehrood, Bivariate identities for values of the Hurwitz zeta function and supercongruences, Electron. J. Combin., 18 (2011), Research paper 35, 30pp.
doi: 10.37236/2049. |
[23] |
K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, 84, Springer-Verlag, New York, 1990.
doi: 10.1007/978-1-4757-2103-4. |
[24] |
F. Jarvis and H. A. Verrill,
Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J., 22 (2010), 171-186.
doi: 10.1007/s11139-009-9218-5. |
[25] |
E. Mortenson,
Supercongruences between truncated ${}_2F_1$ hypergeometric functions and their Gaussian analogs, Trans. Amer. Math. Soc., 355 (2003), 987-1007.
doi: 10.1090/S0002-9947-02-03172-0. |
[26] |
Y.-P. Mu and Z.-W. Sun,
Telescoping method and congruences for double sums, Int. J. Number Theory, 14 (2018), 143-165.
doi: 10.1142/S1793042118500100. |
[27] |
S. Ramanujan, Modular equations and approximations to $\pi$, in Collected Papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI, 2000, 23–39. |
[28] |
F. Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, in Calabi-Yau Varieties and Mirror Symmetry, Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003,223–231. |
[29] |
M. D. Rogers,
New ${}_5F_4$ hypergeometric transformations, three-variable Mahler measures, and formulas for $1/\pi$, Ramanujan J., 18 (2009), 327-340.
doi: 10.1007/s11139-007-9040-x. |
[30] |
M. Rogers and A. Straub,
A solution of Sun's fanxiexian_myfh520 challenge concerning $520/\pi$, Int. J. Number Theory, 9 (2013), 1273-1288.
doi: 10.1142/S1793042113500267. |
[31] |
Z.-H. Sun,
Congruences involving binomial coefficients and Apéry-like numbers, Publ. Math. Debrecen, 96 (2020), 315-346.
doi: 10.5486/PMD.2020.8577. |
[32] |
Z.-W. Sun,
On congruences related to central binomial coefficients, J. Number Theory, 131 (2011), 2219-2238.
doi: 10.1016/j.jnt.2011.04.004. |
[33] |
Z.-W. Sun,
Super congruences and Euler numbers, Sci. China Math., 54 (2011), 2509-2535.
doi: 10.1007/s11425-011-4302-x. |
[34] |
Z.-W. Sun, List of conjectural series for powers of $\pi$ and other constants, preprint, arXiv: 1102.5649. Google Scholar |
[35] |
Z.-W. Sun, Conjectures and results on x2 mod p2 with 4p = x2 + dy2, in Number Theory and Related Area, Adv. Lect. Math., 27, Int. Press, Somerville, MA, 2013, 149-197. |
[36] |
Z.-W. Sun,
Congruences for Franel numbers, Adv. in Appl. Math., 51 (2013), 524-535.
doi: 10.1016/j.aam.2013.06.004. |
[37] |
Z.-W. Sun,
Connections between $p = x^2+3y^2$ and Franel numbers, J. Number Theory, 133 (2013), 2914-2928.
doi: 10.1016/j.jnt.2013.02.014. |
[38] |
Z.-W. Sun,
$p$-adic congruences motivated by series, J. Number Theory, 134 (2014), 181-196.
doi: 10.1016/j.jnt.2013.07.011. |
[39] |
Z.-W. Sun,
Congruences involving generalized central trinomial coefficients, Sci. China Math., 57 (2014), 1375-1400.
doi: 10.1007/s11425-014-4809-z. |
[40] |
Z.-W. Sun, On sums related to central binomial and trinomial coefficients, in Combinatorial and Additive Number Theory: CANT 2011 and 2012, Springer Proc. Math. Stat., 101, Springer, New York, 2014,257–312.
doi: 10.1007/978-1-4939-1601-6_18. |
[41] |
Z.-W. Sun,
Some new series for $1/\pi$ and related congruences, Nanjing Daxue Xuebao Shuxue Bannian Kan, 31 (2014), 150-164.
|
[42] |
Z.-W. Sun,
New series for some special values of $L$-functions, Nanjing Daxue Xuebao Shuxue Bannian Kan, 32 (2015), 189-218.
|
[43] |
Z.-W. Sun,
Congruences involving $g_n(x) = \sum_{k = 0}^n \binom nk^2 \binom2kkx^k$, Ramanujan J., 40 (2016), 511-533.
doi: 10.1007/s11139-015-9727-3. |
[44] |
Z.-W. Sun, Supercongruences involving Lucas sequences, preprint, arXiv: 1610.03384. Google Scholar |
[45] |
Z.-W. Sun,
Open conjectures on congruences, Nanjing Daxue Xuebao Shuxue Bannian Kan, 36 (2019), 1-99.
doi: 10.3969/j.issn.0469-5097.2019.01.01. |
[46] |
Z.-W. Sun and R. Tauraso,
On some new congruences for binomial coefficients, Int. J. Number Theory, 7 (2011), 645-662.
doi: 10.1142/S1793042111004393. |
[47] |
L. Van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, in $p$-adic Functional Analysis, Lecture Notes in Pure and Appl. Math., 192, Dekker, New York, 1997,223–236. |
[48] |
S. Wagner, Asymptotics of generalised trinomial coefficients, preprint, arXiv: 1205.5402. Google Scholar |
[49] |
J. Wan and W. Zudilin,
Generating functions of Legendre polynomials: A tribute to Fred Brafman, J. Approx. Theory, 164 (2012), 488-503.
doi: 10.1016/j.jat.2011.12.001. |
[50] |
C. Wang, Symbolic summation methods and hypergeometric supercongruences, J. Math. Anal. Appl., 488 (2020), Article ID 124068, 11pp.
doi: 10.1016/j.jmaa.2020.124068. |
[51] |
D. Zagier, Integral solutions of Apéry-like recurrence equations, in Groups and Symmetries, CRM Proc. Lecture Notes, 47, Amer. Math. Soc., Providence, RI, 2009,349–366. |
[52] |
D. Zeilberger, Closed form (pun intended!), in A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemp. Math., 143, Amer. Math. Soc., Providence, RI, 1993,579–607. |
[53] |
W. Zudilin,
Ramanujan-type supercongruences, J. Number Theory, 129 (2009), 1848-1857.
doi: 10.1016/j.jnt.2009.01.013. |
[54] |
W. Zudilin,
A generating function of the squares of Legendre polynomials, Bull. Aust. Math. Soc., 89 (2014), 125-131.
doi: 10.1017/S0004972713000233. |
show all references
References:
[1] |
N. D. Baruah and B. C. Berndt,
Eisenstein series and Ramanujan-type series for $1/\pi$, Ramanujan J., 23 (2010), 17-44.
doi: 10.1007/s11139-008-9155-8. |
[2] |
B. C. Berndt, Ramanujan's Notebooks. Part IV, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-0879-2. |
[3] |
H. H. Chan, S. H. Chan and Z. Liu,
Domb's numbers and Ramanujan-Sato type series for $1/\pi$, Adv. Math., 186 (2004), 396-410.
doi: 10.1016/j.aim.2003.07.012. |
[4] |
H. H. Chan and S. Cooper,
Rational analogues of Ramanujan's series for $1/\pi$, Math. Proc. Cambridge Philos. Soc., 153 (2012), 361-383.
doi: 10.1017/S0305004112000254. |
[5] |
H. H. Chan, Y. Tanigawa, Y. Yang and W. Zudilin,
New analogues of Clausen's identities arising from the theory of modular forms, Adv. Math., 228 (2011), 1294-1314.
doi: 10.1016/j.aim.2011.06.011. |
[6] |
H. H. Chan, J. Wan and W. Zudilin,
Legendre polynomials and Ramanujan-type series for $1/\pi$, Israel J. Math., 194 (2013), 183-207.
doi: 10.1007/s11856-012-0081-5. |
[7] |
W. Y. C. Chen, Q.-H. Hou and Y.-P. Mu,
A telescoping method for double summations, J. Comput. Appl. Math., 196 (2006), 553-566.
doi: 10.1016/j.cam.2005.10.010. |
[8] |
D. V. Chudnovsky and G. V. Chudnovsky, Approximations and complex multiplication according to Ramanujan in Ramanujan Revisited, Academic Press, Boston, MA, 1988.
![]() |
[9] |
S. Cooper,
Sporadic sequences, modular forms and new series for $1/\pi$, Ramanujan J., 29 (2012), 163-183.
doi: 10.1007/s11139-011-9357-3. |
[10] |
S. Cooper, Ramanujan's Theta Functions, Springer, Cham, 2017.
doi: 10.1007/978-3-319-56172-1. |
[11] |
S. Cooper, J. G. Wan and W. Zudilin, Holonomic alchemy and series for $1/\pi$, in Analytic Number Theory, Modular Forms and $q$-Hypergeometric Series, Springer Proc. Math. Stat., 221, Springer, Cham, 2017,179–205.
doi: 10.1007/978-3-319-68376-8_12. |
[12] |
D. A. Cox, Primes of the Form $x^2+ny^2$. Fermat, Class Field Theory and Complex Multiplication, John Wiley & Sons, Inc., New York, 1989. |
[13] |
H. R. P. Ferguson, D. H. Bailey and S. Arno,
Analysis of PSLQ, an integer relation finding algorithm, Math. Comp., 68 (1999), 351-369.
doi: 10.1090/S0025-5718-99-00995-3. |
[14] |
J. Franel, On a question of Laisant, L'Intermédiaire des Mathématiciens, 1 (1894), 45-47. Google Scholar |
[15] |
J. W. L. Glaisher, On series for $1/\pi$ and $1/\pi^2$, Quart. J. Pure Appl. Math., 37 (1905), 173-198. Google Scholar |
[16] |
J. Guillera, Tables of Ramanujan series with rational values of $z$., Available from: http://personal.auna.com/jguillera/ramatables.pdf. Google Scholar |
[17] |
J. Guillera,
Hypergeometric identities for 10 extended Ramanujan-type series, Ramanujan J., 15 (2008), 219-234.
doi: 10.1007/s11139-007-9074-0. |
[18] |
J. Guillera and M. Rogers,
Ramanujan series upside-down, J. Aust. Math. Soc., 97 (2014), 78-106.
doi: 10.1017/S1446788714000147. |
[19] |
V. J. W. Guo and J.-C. Liu,
Some congruences related to a congruence of Van Hamme, Integral Transforms Spec. Funct., 31 (2020), 221-231.
doi: 10.1080/10652469.2019.1685991. |
[20] |
V. J. W. Guo, G.-S. Mao and H. Pan,
Proof of a conjecture involving Sun polynomials, J. Difference Equ. Appl., 22 (2016), 1184-1197.
doi: 10.1080/10236198.2016.1188088. |
[21] |
V. J. W. Guo and M. J. Schlosser, Some $q$-supercongruences from transfromation formulas for basic hypergeometric series, Constr. Approx., to appear. Google Scholar |
[22] |
K. Hessami Pilehrood and T. Hessami Pilehrood, Bivariate identities for values of the Hurwitz zeta function and supercongruences, Electron. J. Combin., 18 (2011), Research paper 35, 30pp.
doi: 10.37236/2049. |
[23] |
K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, 84, Springer-Verlag, New York, 1990.
doi: 10.1007/978-1-4757-2103-4. |
[24] |
F. Jarvis and H. A. Verrill,
Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J., 22 (2010), 171-186.
doi: 10.1007/s11139-009-9218-5. |
[25] |
E. Mortenson,
Supercongruences between truncated ${}_2F_1$ hypergeometric functions and their Gaussian analogs, Trans. Amer. Math. Soc., 355 (2003), 987-1007.
doi: 10.1090/S0002-9947-02-03172-0. |
[26] |
Y.-P. Mu and Z.-W. Sun,
Telescoping method and congruences for double sums, Int. J. Number Theory, 14 (2018), 143-165.
doi: 10.1142/S1793042118500100. |
[27] |
S. Ramanujan, Modular equations and approximations to $\pi$, in Collected Papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI, 2000, 23–39. |
[28] |
F. Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, in Calabi-Yau Varieties and Mirror Symmetry, Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003,223–231. |
[29] |
M. D. Rogers,
New ${}_5F_4$ hypergeometric transformations, three-variable Mahler measures, and formulas for $1/\pi$, Ramanujan J., 18 (2009), 327-340.
doi: 10.1007/s11139-007-9040-x. |
[30] |
M. Rogers and A. Straub,
A solution of Sun's fanxiexian_myfh520 challenge concerning $520/\pi$, Int. J. Number Theory, 9 (2013), 1273-1288.
doi: 10.1142/S1793042113500267. |
[31] |
Z.-H. Sun,
Congruences involving binomial coefficients and Apéry-like numbers, Publ. Math. Debrecen, 96 (2020), 315-346.
doi: 10.5486/PMD.2020.8577. |
[32] |
Z.-W. Sun,
On congruences related to central binomial coefficients, J. Number Theory, 131 (2011), 2219-2238.
doi: 10.1016/j.jnt.2011.04.004. |
[33] |
Z.-W. Sun,
Super congruences and Euler numbers, Sci. China Math., 54 (2011), 2509-2535.
doi: 10.1007/s11425-011-4302-x. |
[34] |
Z.-W. Sun, List of conjectural series for powers of $\pi$ and other constants, preprint, arXiv: 1102.5649. Google Scholar |
[35] |
Z.-W. Sun, Conjectures and results on x2 mod p2 with 4p = x2 + dy2, in Number Theory and Related Area, Adv. Lect. Math., 27, Int. Press, Somerville, MA, 2013, 149-197. |
[36] |
Z.-W. Sun,
Congruences for Franel numbers, Adv. in Appl. Math., 51 (2013), 524-535.
doi: 10.1016/j.aam.2013.06.004. |
[37] |
Z.-W. Sun,
Connections between $p = x^2+3y^2$ and Franel numbers, J. Number Theory, 133 (2013), 2914-2928.
doi: 10.1016/j.jnt.2013.02.014. |
[38] |
Z.-W. Sun,
$p$-adic congruences motivated by series, J. Number Theory, 134 (2014), 181-196.
doi: 10.1016/j.jnt.2013.07.011. |
[39] |
Z.-W. Sun,
Congruences involving generalized central trinomial coefficients, Sci. China Math., 57 (2014), 1375-1400.
doi: 10.1007/s11425-014-4809-z. |
[40] |
Z.-W. Sun, On sums related to central binomial and trinomial coefficients, in Combinatorial and Additive Number Theory: CANT 2011 and 2012, Springer Proc. Math. Stat., 101, Springer, New York, 2014,257–312.
doi: 10.1007/978-1-4939-1601-6_18. |
[41] |
Z.-W. Sun,
Some new series for $1/\pi$ and related congruences, Nanjing Daxue Xuebao Shuxue Bannian Kan, 31 (2014), 150-164.
|
[42] |
Z.-W. Sun,
New series for some special values of $L$-functions, Nanjing Daxue Xuebao Shuxue Bannian Kan, 32 (2015), 189-218.
|
[43] |
Z.-W. Sun,
Congruences involving $g_n(x) = \sum_{k = 0}^n \binom nk^2 \binom2kkx^k$, Ramanujan J., 40 (2016), 511-533.
doi: 10.1007/s11139-015-9727-3. |
[44] |
Z.-W. Sun, Supercongruences involving Lucas sequences, preprint, arXiv: 1610.03384. Google Scholar |
[45] |
Z.-W. Sun,
Open conjectures on congruences, Nanjing Daxue Xuebao Shuxue Bannian Kan, 36 (2019), 1-99.
doi: 10.3969/j.issn.0469-5097.2019.01.01. |
[46] |
Z.-W. Sun and R. Tauraso,
On some new congruences for binomial coefficients, Int. J. Number Theory, 7 (2011), 645-662.
doi: 10.1142/S1793042111004393. |
[47] |
L. Van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, in $p$-adic Functional Analysis, Lecture Notes in Pure and Appl. Math., 192, Dekker, New York, 1997,223–236. |
[48] |
S. Wagner, Asymptotics of generalised trinomial coefficients, preprint, arXiv: 1205.5402. Google Scholar |
[49] |
J. Wan and W. Zudilin,
Generating functions of Legendre polynomials: A tribute to Fred Brafman, J. Approx. Theory, 164 (2012), 488-503.
doi: 10.1016/j.jat.2011.12.001. |
[50] |
C. Wang, Symbolic summation methods and hypergeometric supercongruences, J. Math. Anal. Appl., 488 (2020), Article ID 124068, 11pp.
doi: 10.1016/j.jmaa.2020.124068. |
[51] |
D. Zagier, Integral solutions of Apéry-like recurrence equations, in Groups and Symmetries, CRM Proc. Lecture Notes, 47, Amer. Math. Soc., Providence, RI, 2009,349–366. |
[52] |
D. Zeilberger, Closed form (pun intended!), in A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemp. Math., 143, Amer. Math. Soc., Providence, RI, 1993,579–607. |
[53] |
W. Zudilin,
Ramanujan-type supercongruences, J. Number Theory, 129 (2009), 1848-1857.
doi: 10.1016/j.jnt.2009.01.013. |
[54] |
W. Zudilin,
A generating function of the squares of Legendre polynomials, Bull. Aust. Math. Soc., 89 (2014), 125-131.
doi: 10.1017/S0004972713000233. |
[1] |
Cristina M. Ballantine. Ramanujan type graphs and bigraphs. Conference Publications, 2003, 2003 (Special) : 78-82. doi: 10.3934/proc.2003.2003.78 |
[2] |
J.W. Bruce, F. Tari. Generic 1-parameter families of binary differential equations of Morse type. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 79-90. doi: 10.3934/dcds.1997.3.79 |
[3] |
Zhi Liu, Tie Zhang. An improved ARMA(1, 1) type fuzzy time series applied in predicting disordering. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 355-366. doi: 10.3934/naco.2020007 |
[4] |
Mrinal Kanti Roychowdhury. Quantization coefficients for ergodic measures on infinite symbolic space. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2829-2846. doi: 10.3934/dcds.2014.34.2829 |
[5] |
Lorenzo Sella, Pieter Collins. Computation of symbolic dynamics for two-dimensional piecewise-affine maps. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 739-767. doi: 10.3934/dcdsb.2011.15.739 |
[6] |
R.D.S. Oliveira, F. Tari. On pairs of differential $1$-forms in the plane. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 519-536. doi: 10.3934/dcds.2000.6.519 |
[7] |
Shrikrishna G. Dani. Simultaneous diophantine approximation with quadratic and linear forms. Journal of Modern Dynamics, 2008, 2 (1) : 129-138. doi: 10.3934/jmd.2008.2.129 |
[8] |
Anish Ghosh, Dubi Kelmer. A quantitative Oppenheim theorem for generic ternary quadratic forms. Journal of Modern Dynamics, 2018, 12: 1-8. doi: 10.3934/jmd.2018001 |
[9] |
Tomas Johnson, Warwick Tucker. Automated computation of robust normal forms of planar analytic vector fields. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 769-782. doi: 10.3934/dcdsb.2009.12.769 |
[10] |
Richard D. Neidinger. Efficient recurrence relations for univariate and multivariate Taylor series coefficients. Conference Publications, 2013, 2013 (special) : 587-596. doi: 10.3934/proc.2013.2013.587 |
[11] |
V. Mastropietro, Michela Procesi. Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities. Communications on Pure & Applied Analysis, 2006, 5 (1) : 1-28. doi: 10.3934/cpaa.2006.5.1 |
[12] |
Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362 |
[13] |
Xavier Blanc, Claude Le Bris. Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings. Networks & Heterogeneous Media, 2010, 5 (1) : 1-29. doi: 10.3934/nhm.2010.5.1 |
[14] |
Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107 |
[15] |
Constantin N. Beli. Representations of integral quadratic forms over dyadic local fields. Electronic Research Announcements, 2006, 12: 100-112. |
[16] |
Jyrki Lahtonen, Gary McGuire, Harold N. Ward. Gold and Kasami-Welch functions, quadratic forms, and bent functions. Advances in Mathematics of Communications, 2007, 1 (2) : 243-250. doi: 10.3934/amc.2007.1.243 |
[17] |
Jiyoung Han. Quantitative oppenheim conjecture for $ S $-arithmetic quadratic forms of rank $ 3 $ and $ 4 $. Discrete & Continuous Dynamical Systems - A, 2021, 41 (5) : 2205-2225. doi: 10.3934/dcds.2020359 |
[18] |
Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141 |
[19] |
Kai-Uwe Schmidt. The merit factor of binary arrays derived from the quadratic character. Advances in Mathematics of Communications, 2011, 5 (4) : 589-607. doi: 10.3934/amc.2011.5.589 |
[20] |
Arne Winterhof, Zibi Xiao. Binary sequences derived from differences of consecutive quadratic residues. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020100 |
Impact Factor: 0.263
Tools
Metrics
Other articles
by authors
[Back to Top]