text:
# American Institute of Mathematical Sciences

December  2021, 29(6): 4297-4314. doi: 10.3934/era.2021086

## Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop

 School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

* Corresponding author: Liangchen Wang

Received  August 2021 Revised  September 2021 Published  December 2021 Early access  October 2021

This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop
 $\begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \Delta u-\chi_1\nabla\cdot(u\nabla v)+\mu_1 u(1-u-a_1w),\, x\in \Omega,\, t>0,\\ 0 = \Delta v-v+w,\,x\in\Omega,\, t>0,\\ w_t = \Delta w-\chi_2\nabla\cdot(w\nabla z)-\chi_3\nabla\cdot(w\nabla v)+\mu_2 w(1-w-a_2u), \,x\in \Omega,\,t>0,\\ 0 = \Delta z-z+u, \,x\in\Omega,\, t>0, \end{array} \right. \end{eqnarray*}$
under homogeneous Neumann boundary conditions in a bounded domain
 $\Omega\subset \mathbb{R}^n$
with
 $n\geq1$
, where the parameters
 $a_1,a_2$
,
 $\chi_1, \chi_2, \chi_3$
,
 $\mu_1, \mu_2$
are positive constants. We first showed some conditions between
 $\frac{\chi_1}{\mu_1}$
,
 $\frac{\chi_2}{\mu_2}$
,
 $\frac{\chi_3}{\mu_2}$
and other ingredients to guarantee boundedness. Moreover, the large time behavior and rates of convergence have also been investigated under some explicit conditions.
Citation: Rong Zhang, Liangchen Wang. Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop. Electronic Research Archive, 2021, 29 (6) : 4297-4314. doi: 10.3934/era.2021086
##### References:
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Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R.I. 1968.  Google Scholar [14] K. Lin and C. Mu, Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2233-2260.  doi: 10.3934/dcdsb.2017094.  Google Scholar [15] K. Lin, C. Mu and L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.  doi: 10.1002/mma.3429.  Google Scholar [16] K. Lin and T. Xiang, On global solutions and blow-up for a short-ranged chemical signaling loop, J. Nonlinear Sci., 29 (2019), 551-591.  doi: 10.1007/s00332-018-9494-6.  Google Scholar [17] K. Lin and T. Xiang, On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop, Calc. Var. Partial Differential Equations, 59 (2020), 1-35.  doi: 10.1007/s00526-020-01777-7.  Google Scholar [18] A. Liu and B. 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Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.  Google Scholar [23] M. Negreanu and J. I. Tello, Global existence and asymptotic behavior of solutions to a predator-prey chemotaxis system with two chemicals, J. Math. Anal. Appl., 474 (2019), 1116-1131.  doi: 10.1016/j.jmaa.2019.02.007.  Google Scholar [24] K. J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bull. Math. Biol., 71 (2009), 1117-1147.  doi: 10.1007/s11538-009-9396-8.  Google Scholar [25] X. Pan, L. Wang and J. Zhang, Boundedness in a three-dimensional two-species and two-stimuli chemotaxis system with chemical signalling loop, Math. Methods Appl. Sci., 43 (2020), 9529-9542.  doi: 10.1002/mma.6621.  Google Scholar [26] X. Pan, L. Wang, J. Zhang and J. Wang, Boundedness in a three-dimensional two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 71 (2020). doi: 10.1007/s00033-020-1248-2.  Google Scholar [27] H. Qiu and S. Guo, Global existence and stability in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1569-1587.  doi: 10.3934/dcdsb.2018220.  Google Scholar [28] P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2007.  Google Scholar [29] G. Ren and B. Liu, Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Anal. Real World Appl., 48 (2019), 288-325.  doi: 10.1016/j.nonrwa.2019.01.017.  Google Scholar [30] C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.  doi: 10.1007/s00285-013-0681-7.  Google Scholar [31] Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar [32] Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst.-Ser. B, 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.  Google Scholar [33] J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.  doi: 10.1088/0951-7715/25/5/1413.  Google Scholar [34] X. Tu, C. Mu and S. Qiu, Boundedness and convergence of constant equilibria in a two-species chemotaxis-competition system with loop, Nonlinear Anal., 198 (2020), 111923.  doi: 10.1016/j.na.2020.111923.  Google Scholar [35] X. Tu, C. Mu and S. Qiu, Global asymptotic stability in a parabolic-elliptic chemotaxis system with competitive kinetics and loop, Appl. Anal., 2020. doi: 10.1080/00036811.2020.1783536.  Google Scholar [36] X. Tu, C. Mu, P. Zheng and K. Lin, Global dynamics in a two-species chemotaxis-competition system with two signals, Discrete Contin. Dyn. Syst., 38 (2018), 3617-3636.  doi: 10.3934/dcds.2018156.  Google Scholar [37] L. Wang, Improvement of conditions for boundedness in a two-species chemotaxis competition system of parabolic-parabolic-elliptic type, J. Math. Anal. Appl., 484 (2020), 123705.  doi: 10.1016/j.jmaa.2019.123705.  Google Scholar [38] L. Wang and C. Mu, A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 4585-4601.  doi: 10.3934/dcdsb.2020114.  Google Scholar [39] L. Wang, C. Mu, X. Hu and P. Zheng, Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant, J. Differential Equations, 264 (2018), 3369-3401.  doi: 10.1016/j.jde.2017.11.019.  Google Scholar [40] L. Wang, J. Zhang, C. Mu and X. Hu, Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 191-221.  doi: 10.3934/dcdsb.2019178.  Google Scholar [41] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar [42] G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661.  doi: 10.1017/S0956792501004843.  Google Scholar [43] L. Xie and Y. Wang, Boundedness in a two-species chemotaxis parabolic system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2717-2729.  doi: 10.3934/dcdsb.2017132.  Google Scholar [44] L. Xie and Y Wang, On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 471 (2019), 584-598.  doi: 10.1016/j.jmaa.2018.10.093.  Google Scholar [45] H. Yu, W. Wang and S. Zheng, Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity, 31 (2018), 502-514.  doi: 10.1088/1361-6544/aa96c9.  Google Scholar [46] Q. Zhang, Competitive exclusion for a two-species chemotaxis system with two chemicals, Appl. Math. Lett., 83 (2018), 27-32.  doi: 10.1016/j.aml.2018.03.012.  Google Scholar [47] Q. Zhang, X. Liu and X. Yang, Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals, J. Math. Phys., 58 (2017), 111504.  doi: 10.1063/1.5011725.  Google Scholar

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##### References:
 [1] X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.  Google Scholar [2] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar [3] T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1253-1272.  doi: 10.3934/dcdsb.2017061.  Google Scholar [4] T. Black, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.  doi: 10.1093/imamat/hxw036.  Google Scholar [5] E. Cruz, M. Negreanu and J. I. Tello, Asymptotic behavior and global existence of solutions to a two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 69 (2018), 20pp. doi: 10.1007/s00033-018-1002-1.  Google Scholar [6] T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar [7] M. Hirata, S. Kurima, M. Mizukami and T. Yokota, Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations, 263 (2017), 470-490.  doi: 10.1016/j.jde.2017.02.045.  Google Scholar [8] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.   Google Scholar [9] D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270.  doi: 10.1007/s00332-010-9082-x.  Google Scholar [10] C. Huang, Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop, Elec. Res. Arch. doi: 10.3934/era.2021037.  Google Scholar [11] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar [12] H. Knútsdóttir, E. Pálsson and L. Edelstein-Keshet, Mathematical model of macrophage-facilitated breast cancer cells invasion, J. Theoret. Biol., 357 (2014), 184-199.  doi: 10.1016/j.jtbi.2014.04.031.  Google Scholar [13] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R.I. 1968.  Google Scholar [14] K. Lin and C. Mu, Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2233-2260.  doi: 10.3934/dcdsb.2017094.  Google Scholar [15] K. Lin, C. Mu and L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.  doi: 10.1002/mma.3429.  Google Scholar [16] K. Lin and T. Xiang, On global solutions and blow-up for a short-ranged chemical signaling loop, J. Nonlinear Sci., 29 (2019), 551-591.  doi: 10.1007/s00332-018-9494-6.  Google Scholar [17] K. Lin and T. Xiang, On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop, Calc. Var. Partial Differential Equations, 59 (2020), 1-35.  doi: 10.1007/s00526-020-01777-7.  Google Scholar [18] A. Liu and B. Dai, Boundedness and stabilization in a two-species chemotaxis system with two chemicals, J. Math. Anal. Appl., 506 (2022), 125609.  doi: 10.1016/j.jmaa.2021.125609.  Google Scholar [19] M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.  doi: 10.3934/dcdsb.2017097.  Google Scholar [20] M. Mizukami, Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type, Math. Methods Appl. Sci., 41 (2018), 234-249.  doi: 10.1002/mma.4607.  Google Scholar [21] M. Mizukami, Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 269-278.  doi: 10.3934/dcdss.2020015.  Google Scholar [22] J. D. Murray, Mathematical Biology, 2$^nd$ edition, Biomathematics, 19. Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.  Google Scholar [23] M. Negreanu and J. I. Tello, Global existence and asymptotic behavior of solutions to a predator-prey chemotaxis system with two chemicals, J. Math. Anal. Appl., 474 (2019), 1116-1131.  doi: 10.1016/j.jmaa.2019.02.007.  Google Scholar [24] K. J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bull. Math. Biol., 71 (2009), 1117-1147.  doi: 10.1007/s11538-009-9396-8.  Google Scholar [25] X. Pan, L. Wang and J. Zhang, Boundedness in a three-dimensional two-species and two-stimuli chemotaxis system with chemical signalling loop, Math. Methods Appl. Sci., 43 (2020), 9529-9542.  doi: 10.1002/mma.6621.  Google Scholar [26] X. Pan, L. Wang, J. Zhang and J. Wang, Boundedness in a three-dimensional two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 71 (2020). doi: 10.1007/s00033-020-1248-2.  Google Scholar [27] H. Qiu and S. Guo, Global existence and stability in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1569-1587.  doi: 10.3934/dcdsb.2018220.  Google Scholar [28] P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2007.  Google Scholar [29] G. Ren and B. Liu, Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Anal. Real World Appl., 48 (2019), 288-325.  doi: 10.1016/j.nonrwa.2019.01.017.  Google Scholar [30] C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.  doi: 10.1007/s00285-013-0681-7.  Google Scholar [31] Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar [32] Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst.-Ser. B, 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.  Google Scholar [33] J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.  doi: 10.1088/0951-7715/25/5/1413.  Google Scholar [34] X. Tu, C. Mu and S. Qiu, Boundedness and convergence of constant equilibria in a two-species chemotaxis-competition system with loop, Nonlinear Anal., 198 (2020), 111923.  doi: 10.1016/j.na.2020.111923.  Google Scholar [35] X. Tu, C. Mu and S. Qiu, Global asymptotic stability in a parabolic-elliptic chemotaxis system with competitive kinetics and loop, Appl. Anal., 2020. doi: 10.1080/00036811.2020.1783536.  Google Scholar [36] X. Tu, C. Mu, P. Zheng and K. Lin, Global dynamics in a two-species chemotaxis-competition system with two signals, Discrete Contin. Dyn. Syst., 38 (2018), 3617-3636.  doi: 10.3934/dcds.2018156.  Google Scholar [37] L. Wang, Improvement of conditions for boundedness in a two-species chemotaxis competition system of parabolic-parabolic-elliptic type, J. Math. Anal. Appl., 484 (2020), 123705.  doi: 10.1016/j.jmaa.2019.123705.  Google Scholar [38] L. Wang and C. Mu, A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 4585-4601.  doi: 10.3934/dcdsb.2020114.  Google Scholar [39] L. Wang, C. Mu, X. Hu and P. Zheng, Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant, J. Differential Equations, 264 (2018), 3369-3401.  doi: 10.1016/j.jde.2017.11.019.  Google Scholar [40] L. Wang, J. Zhang, C. Mu and X. Hu, Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 191-221.  doi: 10.3934/dcdsb.2019178.  Google Scholar [41] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar [42] G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661.  doi: 10.1017/S0956792501004843.  Google Scholar [43] L. Xie and Y. Wang, Boundedness in a two-species chemotaxis parabolic system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2717-2729.  doi: 10.3934/dcdsb.2017132.  Google Scholar [44] L. Xie and Y Wang, On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 471 (2019), 584-598.  doi: 10.1016/j.jmaa.2018.10.093.  Google Scholar [45] H. Yu, W. Wang and S. Zheng, Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity, 31 (2018), 502-514.  doi: 10.1088/1361-6544/aa96c9.  Google Scholar [46] Q. Zhang, Competitive exclusion for a two-species chemotaxis system with two chemicals, Appl. Math. Lett., 83 (2018), 27-32.  doi: 10.1016/j.aml.2018.03.012.  Google Scholar [47] Q. Zhang, X. Liu and X. Yang, Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals, J. Math. Phys., 58 (2017), 111504.  doi: 10.1063/1.5011725.  Google Scholar
 [1] Xie Li, Yilong Wang. Boundedness in a two-species chemotaxis parabolic system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2717-2729. doi: 10.3934/dcdsb.2017132 [2] Liangchen Wang, Jing Zhang, Chunlai Mu, Xuegang Hu. Boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 191-221. doi: 10.3934/dcdsb.2019178 [3] Liangchen Wang, Chunlai Mu. A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4585-4601. doi: 10.3934/dcdsb.2020114 [4] Youshan Tao, Michael Winkler. Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3165-3183. doi: 10.3934/dcdsb.2015.20.3165 [5] Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097 [6] Xu Pan, Liangchen Wang. Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2211-2236. doi: 10.3934/cpaa.2021064 [7] Yu Ma, Chunlai Mu, Shuyan Qiu. Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021218 [8] Alexander Kurganov, Mária Lukáčová-Medvidová. Numerical study of two-species chemotaxis models. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 131-152. doi: 10.3934/dcdsb.2014.19.131 [9] Hai-Yang Jin, Tian Xiang. Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes sytstem with competitive kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1919-1942. doi: 10.3934/dcdsb.2018249 [10] Ke Lin, Chunlai Mu. Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2233-2260. doi: 10.3934/dcdsb.2017094 [11] Xu Pan, Liangchen Wang. On a quasilinear fully parabolic two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021047 [12] Casimir Emako, Luís Neves de Almeida, Nicolas Vauchelet. Existence and diffusive limit of a two-species kinetic model of chemotaxis. Kinetic & Related Models, 2015, 8 (2) : 359-380. doi: 10.3934/krm.2015.8.359 [13] Tai-Chia Lin, Zhi-An Wang. Development of traveling waves in an interacting two-species chemotaxis model. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2907-2927. doi: 10.3934/dcds.2014.34.2907 [14] Huanhuan Qiu, Shangjiang Guo. Global existence and stability in a two-species chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1569-1587. doi: 10.3934/dcdsb.2018220 [15] Wenji Zhang, Pengcheng Niu. Asymptotics in a two-species chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4281-4298. doi: 10.3934/dcdsb.2020288 [16] Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061 [17] Xinyu Tu, Chunlai Mu, Pan Zheng, Ke Lin. Global dynamics in a two-species chemotaxis-competition system with two signals. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3617-3636. doi: 10.3934/dcds.2018156 [18] Masaaki Mizukami. Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 269-278. doi: 10.3934/dcdss.2020015 [19] Yan Li. Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5461-5480. doi: 10.3934/dcdsb.2019066 [20] Tahir Bachar Issa, Rachidi Bolaji Salako. Asymptotic dynamics in a two-species chemotaxis model with non-local terms. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3839-3874. doi: 10.3934/dcdsb.2017193

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# Law of Sines (Elliptic, Hyperbolic, Euclidean)

1. Jul 8, 2011

### DarthPickley

Well, I created this thread (under Geometry/Topology) about the Law of Sines, specifically for the three kinds of geometries.

http://en.wikipedia.org/wiki/Law_of_sines
http://mathworld.wolfram.com/LawofSines.html

The Law of Sines states that, for a triangle ABC with angles A, B, C, and side lengths a = BC, b = AC, & c = AB, which is in:
The Euclidean Plane:
a/Sin(A) = b/Sin(B) = c/Sin(C)
The Sphere:
Sin(a)/Sin(A) = Sin(b)/Sin(B) = Sin(c)/Sin(C)
The Hyperbolic Plane:
Sinh(a)/Sin(A) = Sinh(b)/Sin(B) = Sinh(c)/Sin(C)

I also know that for Euclidean geometry, a/Sin(A) is the radius of the circumscribed circle.

Here is a proof for Euclidean geometry:
Given Triangle ABC:
Construct the altitude from C. Let h be the length of this altitude (the height, where AB is the base)
By a definition of Sine, Sin(A) = h/b and similarly, Sin(B) = h/a.
h = b*Sin(A) and h = a*Sin(B)
b*Sin(A) = h = a*Sin(B) ==> a*Sin(B) = b*Sin(A)
a/Sin(A) = b/Sin(B)
since A and B can be chosen to be any two vertices of the same triangle, it is also true that a/Sin(A) = c/Sin(C) = b/Sin(B) : which is that which was to be demonstrated.

Now, I have been able to prove the Spherical law of cosines using the Sphere and some basic Linear Algebra like dot product = cosine of angle. I don't really understand the hyperbolic plane well, but the formula is similar for hyperbolic law of cosines.

I have been less successful proving the Spherical law of sines, not to mention Hyperbolic law of sines.
This is what I am asking for help with. It may require some stuff with vectors I don't understand right now, but if someone can explain it in a way that makes more sense.

Is there a way to say the theorems I used for the Euclidean proof that is for the other geometries? Of course, you can derive from the law of sines itself that if C is a right angle, then Sin(A) = Sinh(a)/Sinh(c). This is sufficient to prove the entire Law of Sines. But is there a way to prove these Right-Triangle Trigonometric Definitions for Sine / Sinh for Spherical and Hyperbolic Geometry? Is this easier or simpler than the whole thing?

http://mathworld.wolfram.com/SphericalTrigonometry.html

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# Determine the constant term in.

• October 28th 2012, 04:45 AM
Petrus
Determine the constant term in.
Determine the constant term in.https://webwork.math.su.se/webwork2_...1d3f2168b1.png
idk how to solve it :(
• October 28th 2012, 06:37 AM
skeeter
Re: Determine the constant term in.
each term in a binomial expansion is of the form ...

$\binom{n}{k} a^{n-k} b^k$

In this case, $n = 21$ ... the value of $k$ will be such that the variable factors cancel.

$\binom{21}{18} (3x^6)^3\left(\frac{5}{x}\right)^{18}$
• October 28th 2012, 07:05 AM
Petrus
Re: Determine the constant term in.
i did not get it, i dont think i really understand what they want me to answer, with other words i maybe dont understand problem... im confused
• October 28th 2012, 07:59 AM
skeeter
Re: Determine the constant term in.
Quote:

Originally Posted by Petrus
i did not get it, i dont think i really understand what they want me to answer, with other words i maybe dont understand problem... im confused

1. have you learned the binomial theorem?

2. do you understand that a constant term has no variable factors?
• October 28th 2012, 08:05 AM
Petrus
Re: Determine the constant term in.
Quote:

Originally Posted by skeeter
1. have you learned the binomial theorem?

2. do you understand that a constant term has no variable factors?

hi!:)
We have not come here yet, but I want to study in advance. I did read but did not understand clearly i dont like my book :(
• October 28th 2012, 08:18 AM
skeeter
Re: Determine the constant term in.
Quote:

Originally Posted by Petrus
We have not come here yet, but I want to study in advance. I did read but did not understand clearly i dont like my book

your book is not the sole source of knowledge. seek and ye shall find ...

Binomial Theorem (part 1) | Precalculus | Khan Academy
• October 28th 2012, 08:57 AM
Petrus
Re: Determine the constant term in.
ok i did watch and i kinda understand but idk how in my problem to get out the constant term :S
• October 28th 2012, 01:04 PM
skeeter
Re: Determine the constant term in.
I showed you how in post #2.
• October 30th 2012, 10:17 PM
Salahuddin559
Re: Determine the constant term in.
A constant terms comes out when a^n-k * b^k is a constant, and does not involve any variables. In your case, the binomial expansion has only one variable, x. That means, in (3x^6)^n - k * (5/x)^k does not contain any x. For it to happen, when n = 21, you need to equate the power of x to zero, since x^0 = 1, is a constant, and other powers yield terms with x, and so, not a constant term.

(x^6)^21 - k * (1/x)^k = x^0
(x^6)^21 - k * (x^-1)^k = x^0
(x^6)^21 - k * x^-k = x^0
x^(6(21-k) - k) = x^0

Equating coefficients, you get k = 18, which is what he used to get the constant term above.

Salahuddin
Maths online
• October 31st 2012, 04:14 AM
Petrus
Re: Determine the constant term in.
I solved this problem :) forgot to say it ty anyway :)


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Standard errors for estimates of smoothing parameters

My question is based on the "forecast" package for R used in Forecasting with Exponential Smoothing. The State Space Approach - Hyndman et al. 2008. I am using the ets function to estimate the parameters of a model.

Is there a way to obtain standard errors for the estimates of the smoothing parameters?

-

Not all methods lead to analytic expressions (preferably based on proper asymptotic results) that provides this.

But the bootstrap allows you to approximate this via simulation. In essence, you generate (lots of) surrogate 'fake' data sets, employ your estimator on each of these and then use the population of your estimates to make inferences. However, doing bootstrapping in a time series context has its own challenges...

-
+1 it is always nice to bootstrap something. On the other hand every challenge is easy with R: stat.ethz.ch/R-manual/R-devel/library/boot/html/tsboot.html –  mbq Aug 31 '10 at 19:47
Thanks Dirk and mbq. I have one time series I collected. I am not using a data generating process to produce different series. So in this case I understand that I cannot get std deviations? Please correct me if I'm wrong. Thanks again! –  noworries Aug 31 '10 at 23:51
You could try 'chunked' resampling. Google for block bootstrap. –  Dirk Eddelbuettel Sep 1 '10 at 1:14


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# 3 Couples in a Row

Discrete Mathematics Level 5

3 Couples are seated at random in a row on 6 chairs. What is the probability that none of the spouses are seated together?

×


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# Proof Ideas - Strong Induction, Pascal's Triangle and Fibonacci Numbers

I'm looking for attributes/characteristics/properties to prove about Pascal's Triangle or the Fibonacci numbers. Preferably something that requires a strong induction proof that is on the same level as proving things such as the sum of the elements in the nth row of Pascal's triangle is $$2^n$$. Simple induction proof ideas are also fine.

• How about Pascal's triangle and the Fibonacci numbers? Try proving that $F_{n+1} = \sum_{k \ge 0} {n-k \choose k}$. – Qiaochu Yuan Dec 5 '18 at 23:45
• The known properties of Pascal's triangle, and of the Fibonacci sequence, that can be proved by induction, can each fill a library. – DanielWainfleet Dec 6 '18 at 9:11


text:
Search
Question: DiffBind DESeq2 MA plot skewed
0
11 months ago by
rbronste60
rbronste60 wrote:

Wondering what can cause skewing in an MAplot as seen below?

https://www.dropbox.com/s/wpittr9xsw4mjn7/MAplot1.png?dl=0

modified 11 months ago by Rory Stark2.5k • written 11 months ago by rbronste60
1
11 months ago by
Michael Love19k
United States
Michael Love19k wrote:

The MA plot for ChIP-seq is always going to look different than RNA-seq, for various reasons. Here on the horizontal you have comparable peaks, and the diagonal pointing up and right shows those peaks which are only present in whatever group is in the numerator of the LFC, with little to no signal in the group that's in the denominator. (For normalization for ChIP-seq, I could see making small alterations to the default RNA-seq approach to estimating size factors. But I don't have any code for this.)

0
11 months ago by
Rory Stark2.5k
CRUK, Cambridge, UK
Rory Stark2.5k wrote:

As Michael Love says, the diagonals represent sites that are at or near zero on one condition but have higher binding in the other. What is unusual is that the most dense part of the plot lies on one of these diagonals (sites with binding in group1 but no binding in group2). The usual dense area in the middle of the plot, while less prominent here, is actually skewed a bit towards greater binding in group2. To see these two observations together is unusual.

It would be interesting to compare this to a non-normalized plot (set bNormalized=FALSE in the call to dba.plotMA). This would be to make sure that the normalization is not altering the binding characteristics too much. In RNA-seq, it is rare to see a large population of genes that are completely unexpressed in one group but expressed in the other, so normalization methods derived for RNA-seq can sometimes distort a ChIP-seq signal in ways that lead to misleading conclusions.

-R


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# nLab equifibered natural transformation

Equifibered natural transformation

category theory

## Applications

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Equifibered natural transformation

## Definition

Let $F,G:C\to D$ be functors. A natural transformation $\alpha:F\to G$ is equifibered (also called cartesian) if for any morphism $f:x\to y$ in $C$, the naturality square

$\array{ F x & \overset{F f}{\to} & F y\\ ^{\alpha_x}\downarrow & & \downarrow^{\alpha_y} \\ G x & \underset{G f}{\to} & G y}$

is a pullback.

The name “equifibered” comes from the fact that since $\alpha_x$ is a pullback of $\alpha_y$, they must have isomorphic fibers. (Of course, if $C$ is not connected, then being equifibered does not imply that all components of $\alpha$ have isomorphic fibers.)

There is an evident generalization to natural transformations between higher categories.

## Properties

Given a functor $G:C\to D$, if $C$ has a terminal object $1$, then to give a functor $F$ and an equifibered transformation $F\to G$ is equivalent to giving a single object $F1$ and a morphism $F1 \to G1$. The rest of $F$ can then be constructed uniquely by taking pullbacks. This construction is important in the theory of clubs.

## References

In the context of category theory the concept is discussed in

• Aurelio Carboni and Peter Johnstone, Connected limits, familial representability and Artin glueing, Mathematical Structures in Computer Science, Vol. 5 Iss. 4, Cambridge U. Press (December 1995), 441-459.

doi: https://doi.org/10.1017/S0960129500001183. (web)

• Tom Leinster, Higher Operads, Higher Categories, Cambridge University Press 2003. (arXiv link)

In the context of (infinity,1)-categories (with an eye towards (infinity,1)-toposes) the concept is considered in

• Charles Rezk, p. 9 of Toposes and homotopy toposes (2010) (pdf)

• Charles Rezk, p. 2 of When are homotopy colimits compatible with homotopy base change? (2014) (pdf)

Last revised on February 5, 2017 at 15:30:20. See the history of this page for a list of all contributions to it.


text: Generalized Weighted Morrey Estimates for Marcinkiewicz Integrals with Rough Kernel Associated with Schrödinger Operator and Their Commutators Citation： Ferit G"{U}RB"{U}Z.Generalized Weighted Morrey Estimates for Marcinkiewicz Integrals with Rough Kernel Associated with Schrödinger Operator and Their Commutators[J].Chinese Annals of Mathematics B,2020,41(1):77~98 Page view： 33        Net amount： 40 Authors： Ferit G"{U}RB"{U}Z; Abstract： Let $L=-\Delta+V(x)$ be a Schr\"{o}dinger operator, where $\Delta$ is the Laplacian on ${\mathbb{R}^{n}}$, while nonnegative potential $V(x)$ belonging to the reverse H\"{o}lder class. The aim of this paper is to give generalized weighted Morrey estimates for the boundedness of Marcinkiewicz integrals with rough kernel associated with Schr\"{o}dinger operator and their commutators. Moreover, the boundedness of the commutator operators formed by {\rm BMO} functions and Marcinkiewicz integrals with rough kernel associated with Schr\"{o}dinger operators is discussed on the generalized weighted Morrey spaces. As its special cases, the corresponding results of Marcinkiewicz integrals with rough kernel associated with Schr\"{o}dinger operator and their commutators have been deduced, respectively. Also, Marcinkiewicz integral operators, rough Hardy-Littlewood (H-L for short) maximal operators, Bochner-Riesz means and parametric {Marcinkiewicz integral} operators which satisfy the conditions of our main results can be considered as some examples. Keywords： Marcinkiewicz operator, Rough kernel Schr"{o}dinger operatorgeneralized weighted Morrey space, Commutator, {rm BMO} Classification： 42B20, 42B35 Download PDF Full-Text


text:
Glue is a simple command line tool to generate sprites.

## Project description

Glue is a simple command line tool to generate sprites using any kind of source images like PNG, JPEG or GIF. Glue will generate a unique PNG file containing every source image and a map file including the necessary information to use it.

## Project details

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## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$x=-126$
Raising both sides to the third power and isolating the variable, the given equation is equivalent to \begin{array}{l}\require{cancel} (x+1)^{1/3}=-5 \\\\ \left[(x+1)^{1/3}\right]^3=(-5)^3 \\\\ x+1=-125 \\\\ x=-125-1 \\\\ x=-126 .\end{array} Upon checking, $x=-126$ satisfies the original equation.


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## G = C2×He3⋊5D4order 432 = 24·33

### Direct product of C2 and He3⋊5D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — C2×He3⋊5D4
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×He3⋊C2 — C22×He3⋊C2 — C2×He3⋊5D4
 Lower central He3 — C2×He3 — C2×He3⋊5D4
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C2×He35D4
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, be=eb, fbf=b-1, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 1313 in 297 conjugacy classes, 67 normal (13 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C12, C12, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3×C6, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, He3, C3×C12, S3×C6, C62, C2×D12, C6×D4, He3⋊C2, C2×He3, C2×He3, C3×D12, C6×C12, S3×C2×C6, C4×He3, C2×He3⋊C2, C2×He3⋊C2, C22×He3, C6×D12, He35D4, C2×C4×He3, C22×He3⋊C2, C2×He35D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, D12, C22×S3, C2×C3⋊S3, C2×D12, He3⋊C2, C12⋊S3, C22×C3⋊S3, C2×He3⋊C2, C2×C12⋊S3, He35D4, C22×He3⋊C2, C2×He35D4

Smallest permutation representation of C2×He35D4
On 72 points
Generators in S72
(1 42)(2 43)(3 44)(4 41)(5 32)(6 29)(7 30)(8 31)(9 28)(10 25)(11 26)(12 27)(13 38)(14 39)(15 40)(16 37)(17 47)(18 48)(19 45)(20 46)(21 68)(22 65)(23 66)(24 67)(33 71)(34 72)(35 69)(36 70)(49 61)(50 62)(51 63)(52 64)(53 58)(54 59)(55 60)(56 57)
(1 23 32)(2 24 29)(3 21 30)(4 22 31)(5 42 66)(6 43 67)(7 44 68)(8 41 65)(9 19 54)(10 20 55)(11 17 56)(12 18 53)(13 52 36)(14 49 33)(15 50 34)(16 51 35)(25 46 60)(26 47 57)(27 48 58)(28 45 59)(37 63 69)(38 64 70)(39 61 71)(40 62 72)
(1 37 19)(2 38 20)(3 39 17)(4 40 18)(5 35 28)(6 36 25)(7 33 26)(8 34 27)(9 32 69)(10 29 70)(11 30 71)(12 31 72)(13 46 43)(14 47 44)(15 48 41)(16 45 42)(21 61 56)(22 62 53)(23 63 54)(24 64 55)(49 57 68)(50 58 65)(51 59 66)(52 60 67)
(1 69 63)(2 70 64)(3 71 61)(4 72 62)(5 66 45)(6 67 46)(7 68 47)(8 65 48)(9 54 37)(10 55 38)(11 56 39)(12 53 40)(13 25 60)(14 26 57)(15 27 58)(16 28 59)(17 30 21)(18 31 22)(19 32 23)(20 29 24)(33 49 44)(34 50 41)(35 51 42)(36 52 43)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)(65 66 67 68)(69 70 71 72)
(1 3)(5 68)(6 67)(7 66)(8 65)(9 56)(10 55)(11 54)(12 53)(14 16)(17 19)(21 32)(22 31)(23 30)(24 29)(25 60)(26 59)(27 58)(28 57)(33 51)(34 50)(35 49)(36 52)(37 39)(42 44)(45 47)(61 69)(62 72)(63 71)(64 70)

G:=sub<Sym(72)| (1,42)(2,43)(3,44)(4,41)(5,32)(6,29)(7,30)(8,31)(9,28)(10,25)(11,26)(12,27)(13,38)(14,39)(15,40)(16,37)(17,47)(18,48)(19,45)(20,46)(21,68)(22,65)(23,66)(24,67)(33,71)(34,72)(35,69)(36,70)(49,61)(50,62)(51,63)(52,64)(53,58)(54,59)(55,60)(56,57), (1,23,32)(2,24,29)(3,21,30)(4,22,31)(5,42,66)(6,43,67)(7,44,68)(8,41,65)(9,19,54)(10,20,55)(11,17,56)(12,18,53)(13,52,36)(14,49,33)(15,50,34)(16,51,35)(25,46,60)(26,47,57)(27,48,58)(28,45,59)(37,63,69)(38,64,70)(39,61,71)(40,62,72), (1,37,19)(2,38,20)(3,39,17)(4,40,18)(5,35,28)(6,36,25)(7,33,26)(8,34,27)(9,32,69)(10,29,70)(11,30,71)(12,31,72)(13,46,43)(14,47,44)(15,48,41)(16,45,42)(21,61,56)(22,62,53)(23,63,54)(24,64,55)(49,57,68)(50,58,65)(51,59,66)(52,60,67), (1,69,63)(2,70,64)(3,71,61)(4,72,62)(5,66,45)(6,67,46)(7,68,47)(8,65,48)(9,54,37)(10,55,38)(11,56,39)(12,53,40)(13,25,60)(14,26,57)(15,27,58)(16,28,59)(17,30,21)(18,31,22)(19,32,23)(20,29,24)(33,49,44)(34,50,41)(35,51,42)(36,52,43), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72), (1,3)(5,68)(6,67)(7,66)(8,65)(9,56)(10,55)(11,54)(12,53)(14,16)(17,19)(21,32)(22,31)(23,30)(24,29)(25,60)(26,59)(27,58)(28,57)(33,51)(34,50)(35,49)(36,52)(37,39)(42,44)(45,47)(61,69)(62,72)(63,71)(64,70)>;

G:=Group( (1,42)(2,43)(3,44)(4,41)(5,32)(6,29)(7,30)(8,31)(9,28)(10,25)(11,26)(12,27)(13,38)(14,39)(15,40)(16,37)(17,47)(18,48)(19,45)(20,46)(21,68)(22,65)(23,66)(24,67)(33,71)(34,72)(35,69)(36,70)(49,61)(50,62)(51,63)(52,64)(53,58)(54,59)(55,60)(56,57), (1,23,32)(2,24,29)(3,21,30)(4,22,31)(5,42,66)(6,43,67)(7,44,68)(8,41,65)(9,19,54)(10,20,55)(11,17,56)(12,18,53)(13,52,36)(14,49,33)(15,50,34)(16,51,35)(25,46,60)(26,47,57)(27,48,58)(28,45,59)(37,63,69)(38,64,70)(39,61,71)(40,62,72), (1,37,19)(2,38,20)(3,39,17)(4,40,18)(5,35,28)(6,36,25)(7,33,26)(8,34,27)(9,32,69)(10,29,70)(11,30,71)(12,31,72)(13,46,43)(14,47,44)(15,48,41)(16,45,42)(21,61,56)(22,62,53)(23,63,54)(24,64,55)(49,57,68)(50,58,65)(51,59,66)(52,60,67), (1,69,63)(2,70,64)(3,71,61)(4,72,62)(5,66,45)(6,67,46)(7,68,47)(8,65,48)(9,54,37)(10,55,38)(11,56,39)(12,53,40)(13,25,60)(14,26,57)(15,27,58)(16,28,59)(17,30,21)(18,31,22)(19,32,23)(20,29,24)(33,49,44)(34,50,41)(35,51,42)(36,52,43), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72), (1,3)(5,68)(6,67)(7,66)(8,65)(9,56)(10,55)(11,54)(12,53)(14,16)(17,19)(21,32)(22,31)(23,30)(24,29)(25,60)(26,59)(27,58)(28,57)(33,51)(34,50)(35,49)(36,52)(37,39)(42,44)(45,47)(61,69)(62,72)(63,71)(64,70) );

G=PermutationGroup([[(1,42),(2,43),(3,44),(4,41),(5,32),(6,29),(7,30),(8,31),(9,28),(10,25),(11,26),(12,27),(13,38),(14,39),(15,40),(16,37),(17,47),(18,48),(19,45),(20,46),(21,68),(22,65),(23,66),(24,67),(33,71),(34,72),(35,69),(36,70),(49,61),(50,62),(51,63),(52,64),(53,58),(54,59),(55,60),(56,57)], [(1,23,32),(2,24,29),(3,21,30),(4,22,31),(5,42,66),(6,43,67),(7,44,68),(8,41,65),(9,19,54),(10,20,55),(11,17,56),(12,18,53),(13,52,36),(14,49,33),(15,50,34),(16,51,35),(25,46,60),(26,47,57),(27,48,58),(28,45,59),(37,63,69),(38,64,70),(39,61,71),(40,62,72)], [(1,37,19),(2,38,20),(3,39,17),(4,40,18),(5,35,28),(6,36,25),(7,33,26),(8,34,27),(9,32,69),(10,29,70),(11,30,71),(12,31,72),(13,46,43),(14,47,44),(15,48,41),(16,45,42),(21,61,56),(22,62,53),(23,63,54),(24,64,55),(49,57,68),(50,58,65),(51,59,66),(52,60,67)], [(1,69,63),(2,70,64),(3,71,61),(4,72,62),(5,66,45),(6,67,46),(7,68,47),(8,65,48),(9,54,37),(10,55,38),(11,56,39),(12,53,40),(13,25,60),(14,26,57),(15,27,58),(16,28,59),(17,30,21),(18,31,22),(19,32,23),(20,29,24),(33,49,44),(34,50,41),(35,51,42),(36,52,43)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64),(65,66,67,68),(69,70,71,72)], [(1,3),(5,68),(6,67),(7,66),(8,65),(9,56),(10,55),(11,54),(12,53),(14,16),(17,19),(21,32),(22,31),(23,30),(24,29),(25,60),(26,59),(27,58),(28,57),(33,51),(34,50),(35,49),(36,52),(37,39),(42,44),(45,47),(61,69),(62,72),(63,71),(64,70)]])

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 4A 4B 6A ··· 6F 6G ··· 6R 6S ··· 6Z 12A 12B 12C 12D 12E ··· 12T order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 1 1 18 18 18 18 1 1 6 6 6 6 2 2 1 ··· 1 6 ··· 6 18 ··· 18 2 2 2 2 6 ··· 6

62 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 3 3 3 6 type + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 D6 D12 He3⋊C2 C2×He3⋊C2 C2×He3⋊C2 He3⋊5D4 kernel C2×He3⋊5D4 He3⋊5D4 C2×C4×He3 C22×He3⋊C2 C6×C12 C2×He3 C3×C12 C62 C3×C6 C2×C4 C4 C22 C2 # reps 1 4 1 2 4 2 8 4 16 4 8 4 4

Matrix representation of C2×He35D4 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 12 0 0 0 0 0 12 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 12 12 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 4 0 0 3 0 0
,
 10 7 0 0 0 6 3 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 12

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[12,1,0,0,0,12,0,0,0,0,0,0,0,0,3,0,0,12,0,0,0,0,0,4,0],[10,6,0,0,0,7,3,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,12,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12] >;

C2×He35D4 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes_5D_4
% in TeX

G:=Group("C2xHe3:5D4");
// GroupNames label

G:=SmallGroup(432,386);
// by ID

G=gap.SmallGroup(432,386);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,58,1124,4037,537]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,d*b*d^-1=b*c^-1,b*e=e*b,f*b*f=b^-1,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations

׿
×
𝔽


text:
## Calculus (3rd Edition)

$3920 \ J$
The force of one layer is equal to: $\ Force= \ Mass \times \ gravity = 78.4 \Delta y \ N$ The work done on the $i$th segment is the sum of the $W_i$; that is, $W=\Sigma_{j=1}^N W_i \approx \Sigma_{j=1}^N 78.4 y_j \Delta y$ Therefore, the work done when $N \to \infty$ can be computed as: $W=\int_{0}^{10} 78.4 \ y \ dy \\=39.2 [y^2]_{0}^{10} \\ =3920 \ J$


text:
We think you are located in South Africa. Is this correct?

# Analysing Investment And Loan Options

## 3.5 Analysing investment and loan options (EMCG6)

In the next worked example we consider the effects of the duration of the repayment period on the total amount repaid (the amount borrowed plus the accrued interest) for a loan.

## Worked example 11: Repayment periods

David and Julie take out a home loan of R $$\text{2,6}$$ million with an interest rate of $$\text{10}\%$$ per annum compounded monthly.

1. Calculate the monthly repayments for a repayment period of $$\text{30}$$ years.
2. Calculate the interest paid on the loan at the end of the $$\text{30}$$ year period.
3. Determine the monthly repayments for a repayment period of $$\text{20}$$ years.
4. Determine the interest paid on the loan at the end of the $$\text{20}$$ year period.
5. What is the difference in the monthly repayment amounts?
6. Comment on the difference in the interest paid for the two different time periods.

### Consider a $$\text{30}$$ year repayment period on the loan

$x = \frac{P \times i}{\left[1 - (1 + i)^{-n}\right]}$ \begin{align*} P &= \text{R}\,\text{2 600 000} \\ i &= \frac{\text{0,1}}{12} \\ n &= 30 \times 12 = \text{360} \end{align*}\begin{align*} x &= \dfrac{\text{2 600 000} \times \frac{\text{0,1}}{12}}{\left[1 - \left(1 + \frac{\text{0,1}}{12}\right)^{-360}\right]} \\ &= \text{R}\,\text{22 816,86} \end{align*}

Therefore, the monthly repayment is $$\text{R}\,\text{22 816,86}$$ for a $$\text{30}$$ year period.

At the end of the $$\text{30}$$ years, David and Julie will have paid a total amount of:

\begin{align*} &= 30 \times 12 \times \text{R}\,\text{22 816,86} \\ &= \text{R}\,\text{8 214 069,60} \end{align*}

The total amount of interest on the loan:

\begin{align*} \text{Interest} &= \text{total amount paid} - \text{loan amount} \\ &= \text{R}\,\text{8 214 069,60} - \text{R}\,\text{2 600 000} \\ &= \text{R}\,\text{5 614 069,60} \end{align*}

We notice that the interest on the loan is more than double the amount borrowed.

### Consider a $$\text{20}$$ year repayment period on the loan

$x = \frac{P \times i}{\left[1 - (1 + i)^{-n}\right]}$ \begin{align*} P &= \text{R}\,\text{2 600 000} \\ i &= \frac{\text{0,1}}{12} \\ n &= 20 \times 12 = \text{240} \end{align*}\begin{align*} x &= \dfrac{\text{2 600 000} \times \frac{\text{0,1}}{12}}{\left[1 - \left(1 + \frac{\text{0,1}}{12}\right)^{-240}\right]} \\ &= \text{R}\,\text{25 090,56} \end{align*}

Therefore, the monthly repayment is $$\text{R}\,\text{25 090,56}$$ for a $$\text{20}$$ year period.

At the end of the $$\text{20}$$ years, David and Julie will have paid a total amount of:

\begin{align*} &= 20 \times 12 \times \text{R}\,\text{25 090,56} \\ &= \text{R}\,\text{6 021 734,40} \end{align*}

The total amount of interest on the loan:

\begin{align*} \text{Interest} &= \text{total amount paid} - \text{loan amount} \\ &= \text{R}\,\text{6 021 734,40} - \text{R}\,\text{2 600 000} \\ &= \text{R}\,\text{3 421 734,40} \end{align*}

We notice that the interest on the loan is about $$\text{1,3}$$ times the borrowed amount.

### Consider the difference in the repayment and interest amounts

\begin{align*} \text{Difference in repayments } &= \text{R}\,\text{25 090,56} - \text{R}\,\text{22 816,86} \\ &= \text{R}\,\text{2 273,70} \end{align*}

It is also very interesting to look at the difference in the total interest paid:

\begin{align*} \text{Difference in interest } &= \text{R}\,\text{5 614 069,60} - \text{R}\,\text{3 421 734,40} \\ &= \text{R}\,\text{2 192 335,20} \end{align*}

Therefore, by paying an extra $$\text{R}\,\text{2 273,70}$$ each month over a shorter repayment period, David and Julie could save more than $$\text{R}\,\text{2}$$ million on the repayment of their home loan.

When considering taking out a loan, it is advisable to investigate and compare a few options offered by financial institutions. It is very important to make informed decisions regarding personal finances and to make sure that the monthly repayment amount is serviceable (payable). A credit rating is an estimate of a person's ability to fulfill financial commitments based on their previous payment history. Defaulting on a loan can affect a person's credit rating and their chances of taking out another loan in the future.

## Worked example 12: Analysing investment opportunities

Marlene wants to start saving for a deposit on a house. She can afford to invest between $$\text{R}\,\text{400}$$ and $$\text{R}\,\text{600}$$ each month and gets information from four different investment firms. Each firm quotes a different interest rate and a prescribed monthly installment amount. She plans to buy a house in $$\text{7}$$ years' time. Calculate which company offers the best investment opportunity for Marlene.

 Interest rate (compounded monthly) Monthly payment TBS Investments $$\text{13,5}\%$$ p.a. $$\text{R}\,\text{450}$$ Taylor Anderson $$\text{13}\%$$ p.a. $$\text{R}\,\text{555}$$ PHK $$\text{12,5}\%$$ p.a. $$\text{R}\,\text{575}$$ Simfords Consulting $$\text{11}\%$$ p.a. $$\text{R}\,\text{600}$$

### Consider the different investment options

To compare the different investment options, we need to calculate the following for each option at the end of the seven year period:

• The future value of the monthly payments.
• The total amount paid into the investment fund.
• The total interest earned.
$F = \frac{x\left[(1 + i)^{n}-1\right]}{i}$

TBS Investments:

\begin{align*} F &= \frac{\text{450}\left[(1 + \frac{\text{0,135}}{12})^{84}-1\right]}{\frac{\text{0,135}}{12}} \\ &= \text{R}\,\text{62 370,99} \\ \text{Total amount } (T): \enspace &= 7 \times 12 \times \text{R}\,\text{450} \\ &= \text{R}\,\text{37 800} \\ \text{Total interest } (I): \enspace &= \text{R}\,\text{62 370,99} - \text{R}\,\text{37 800} \\ &= \text{R}\,\text{24 570,99} \end{align*}

Taylor Anderson:

\begin{align*} F &= \frac{\text{555}\left[(1 + \frac{\text{0,13}}{12})^{84}-1\right]}{\frac{\text{0,13}}{12}} \\ &= \text{R}\,\text{75 421,65} \\ \text{Total amount } (T): \enspace &= 7 \times 12 \times \text{R}\,\text{555} \\ &= \text{R}\,\text{46 620} \\ \text{Total interest } (I): \enspace &= \text{R}\,\text{75 421,65} - \text{R}\,\text{46 620} \\ &= \text{R}\,\text{28 801,65} \end{align*}

PHK:

\begin{align*} F &= \frac{\text{575}\left[(1 + \frac{\text{0,125}}{12})^{84}-1\right]}{\frac{\text{0,125}}{12}} \\ &= \text{R}\,\text{76 619,96} \\ \text{Total amount } (T): \enspace &= 7 \times 12 \times \text{R}\,\text{575} \\ &= \text{R}\,\text{48 300} \\ \text{Total interest } (I): \enspace &= \text{R}\,\text{76 619,96} - \text{R}\,\text{48 300} \\ &= \text{R}\,\text{28 319,96} \end{align*}

Simfords Consulting:

\begin{align*} F &= \frac{\text{600}\left[(1 + \frac{\text{0,11}}{12})^{84}-1\right]}{\frac{\text{0,11}}{12}} \\ &= \text{R}\,\text{75 416,96} \\ \text{Total amount } (T): \enspace &= 7 \times 12 \times \text{R}\,\text{600} \\ &= \text{R}\,\text{50 400} \\ \text{Total interest } (I): \enspace &= \text{R}\,\text{75 416,96} - \text{R}\,\text{50 400} \\ &= \text{R}\,\text{25 016,96} \end{align*}

### Draw a table of the results to compare the answers

 $$F$$ $$T$$ $$I$$ TBS Investments $$\text{R}\,\text{62 370,99}$$ $$\text{R}\,\text{37 800}$$ $$\text{R}\,\text{24 570,99}$$ Taylor Anderson $$\text{R}\,\text{75 421,65}$$ $$\text{R}\,\text{46 620}$$ $$\text{R}\,\text{28 801,65}$$ PHK $$\text{R}\,\text{76 619,96}$$ $$\text{R}\,\text{48 300}$$ $$\text{R}\,\text{28 319,96}$$ Simfords Consulting $$\text{R}\,\text{75 416,96}$$ $$\text{R}\,\text{50 400}$$ $$\text{R}\,\text{25 016,96}$$

### Make a conclusion

An investment with PHK would provide Marlene with the highest deposit ($$\text{R}\,\text{76 619,96}$$) for her house at the end of the $$\text{7}$$ year period. However, we notice that an investment with Taylor Anderson would earn the highest amount of interest ($$\text{R}\,\text{28 801,65}$$) and is therefore the better investment option.

## Worked example 13: Analysing loan options

William wants to take out a loan of $$\text{R}\,\text{750 000}$$, so he approaches three different banks. He plans to start repaying the loan immediately and he calculates that he can afford a monthly repayment amount between $$\text{R}\,\text{5 500}$$ and $$\text{R}\,\text{7 000}$$.

Calculate which of the three options would be best for William.

• West Bank offers a repayment period of $$\text{30}$$ years and an interest rate of prime compounded monthly.
• AcuBank offers a repayment period of $$\text{20}$$ years and an interest rate of prime $$+ \text{0,5}\%$$ compounded monthly.
• FinTrust Bank offers a repayment period of $$\text{15}$$ years and an interest rate of prime $$+ \text{2}\%$$ compounded monthly.

### Consider the different loan options

To compare the different loan options, we need to calculate the following for each option:

• The monthly payment amount.
• The total amount paid to repay the loan.
• The amount of interest on the loan.
$x = \frac{P \times i}{\left[1 - (1 + i)^{-n}\right]}$

West Bank:

\begin{align*} x & = \frac{ \text{750 000} \times \frac{\text{0,085}}{12}}{\left[1 - (1 + \frac{\text{0,085}}{12})^{-\text{360}}\right]} \\ &= \text{R}\,\text{5 766,85} \\ \text{Total amount } (T): \enspace &= 30 \times 12 \times \text{R}\,\text{5 766,85} \\ &= \text{R}\,\text{2 076 066} \\ \text{Total interest } (I): \enspace &= \text{R}\,\text{2 076 066} - \text{R}\,\text{750 000} \\ &= \text{R}\,\text{1 326 066} \end{align*}

AcuBank:

\begin{align*} x &= \frac{ \text{750 000} \times \frac{\text{0,09}}{12}}{\left[1 - (1 + \frac{\text{0,09}}{12})^{-\text{240}}\right]} \\ &= \text{R}\,\text{6 747,94} \\ \text{Total amount } (T): \enspace &= 20 \times 12 \times \text{R}\,\text{6 747,94} \\ &= \text{R}\,\text{1 619 505,60} \\ \text{Total interest } (I): \enspace &= \text{R}\,\text{1 619 505,60} - \text{R}\,\text{750 000} \\ &= \text{R}\,\text{869 505,60} \end{align*}

FinTrust Bank:

\begin{align*} x &= \frac{ \text{750 000} \times \frac{\text{0,105}}{12}}{\left[1 - (1 + \frac{\text{0,105}}{12})^{-\text{180}}\right]} \\ &= \text{R}\,\text{8 290,49} \\ \text{Total amount } (T): \enspace &= 15 \times 12 \times \text{R}\,\text{8 290,49} \\ &= \text{R}\,\text{1 492 288,20} \\ \text{Total interest } (I): \enspace &= \text{R}\,\text{1 492 288,20} - \text{R}\,\text{750 000} \\ &= \text{R}\,\text{742 288,20} \end{align*}

### Draw a table of the results to compare the answers

 $$x$$ $$T$$ $$I$$ West Bank $$\text{R}\,\text{5 766,85}$$ $$\text{R}\,\text{2 076 066,00}$$ $$\text{R}\,\text{1 326 066,00}$$ AcuBank $$\text{R}\,\text{6 747,94}$$ $$\text{R}\,\text{1 619 505,60}$$ $$\text{R}\,\text{869 505,60}$$ FinTrust Bank $$\text{R}\,\text{8 290,49}$$ $$\text{R}\,\text{1 492 288,20}$$ $$\text{R}\,\text{742 288,20}$$

### Make a conclusion

A loan from FinTrust Bank would accumulate the lowest amount of interest but the monthly repayment amounts are not within William's budget. Although West Bank offers the lowest interest rate and monthly repayment amount, the interest earned on the loan is very high as a result of the longer repayment period. If we assume that William must repay the loan over the given time periods, then AcuBank offers the best option.

However, we know that William can afford to pay more than $$\text{R}\,\text{5 766,85}$$ per month, and if the bank allows him to pay back the loan earlier, he should consider taking out a loan with West Bank and take advantage of the lower interest rate.

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## Analysing investment and loan options

Exercise 3.5

Cokisa is $$\text{31}$$ years old and starting to plan for her future. She has been thinking about her retirement and wants to open an annuity so that she will have money when she retires. Her intention is to retire when she is 65 years old. Cokisa visits the Trader's Bank of Tembisa and learns that there are two investment options from which she can choose:

• Option A: $$\text{7,76}\%$$ p.a. compounded once every four months
• Option B: $$\text{7,78}\%$$ p.a. compounded half-yearly

Which is the better investment option for Cokisa if the amount she will deposit will always be the same?

There are two separate account options for Cokisa to consider; to determine which is the better option we need to compare the effective interest of each option. The higher the effective interest, the quicker the account will grow. The effective interest formula is:

\begin{align*} i + 1 & = \left(1 + \frac{i^{m}}{m} \right)^{m} \\ \text{Where: } \quad & \\ i & = \text{the effective interest rate} \\ i^{m} & = \text{the nominal interest rate} \\ m & = \text{the number of compounding periods each year} \end{align*}

Work out the effective interest for option A:

\begin{align*} i & = \left(1 + \frac{\text{0,0776}}{\text{3}} \right)^{\text{3}} - 1 \\ & = \text{0,07962} \ldots \end{align*}

This calculation shows that option A an effective interest rate of about $$\text{7,9625}\%$$.

Now work out the effective interest for option B:

\begin{align*} i & = \left(1 + \frac{\text{0,0778}}{\text{2}} \right)^{\text{2}} - 1 \\ & = \text{0,07931} \ldots \end{align*}

For option B, the effective interest rate will be approximately $$\text{7,9313}\%$$.

By comparing the two calculations above, we see that the option A is the better one.

NOTE: It may seem that we could answer this question by picking an amount for the regular payment and then using the future value formula for each of the two options to see which one produces more money. However, this will not work because the compounding periods are different. If we want to work it out this way, we MUST adjust the payment value we use according to the compounding period, for example if we choose a regular payment of $$\text{R}\,\text{100}$$ once every four months for option A, then we will have to use $$\text{R}\,\text{150}$$ semi-annually for option B (because $$\text{R}\,\text{100} \times \text{3} = \text{300}$$ is the same total amount per year as $$\text{R}\,\text{150} \times \text{2} = \text{300}$$).

Cokisa opens an account and starts saving $$\text{R}\,\text{4 000}$$ every four months. How much money (to the nearest rand) will she have saved when she reaches her planned retirement?

\begin{align*} F & = \frac{\text{4 000} \left[ \left(1 + \frac{\text{0,0776}}{\text{3}}\right) ^{(\text{34} \times \text{3})} - 1 \right]} {\left(\frac{\text{0,0776}}{\text{3}} \right)} \\ & = \text{R}\,\text{1 937 512,76} \end{align*}

Cokisa will have $$\text{R}\,\text{1 937 512,76}$$ for her retirement.

Phoebe wants to take out a home loan of R $$\text{1,6}$$ million. She approaches three different banks for their loan options:

• Bank A offers a repayment period of $$\text{30}$$ years and an interest rate of $$\text{12}\%$$ per annum compounded monthly.
• Bank B offers a repayment period of $$\text{20}$$ years and an interest rate of $$\text{14}\%$$ per annum compounded monthly.
• Bank C offers a repayment period of $$\text{30}$$ years and an interest rate of $$\text{14}\%$$ per annum compounded monthly.

If Phoebe intends to start her monthly repayments immediately, calculate which of the three options would be best for her.

$x = \frac{P \times i}{\left[1 - (1 + i)^{-n}\right]}$

Bank A:

\begin{align*} x & = \frac{ \text{1 600 000} \times \frac{0.12}{12}}{\left[1 - (1 + \frac{0.12}{12})^{-\text{360}}\right]} \\ &= \text{R}\,\text{16 457,80} \\ \text{Total amount } (T): \enspace &= 30 \times 12 \times \text{R}\,\text{16 457,80} \\ &= \text{R}\,\text{5 924 808} \\ \text{Total interest } (I): \enspace &= \text{R}\,\text{5 924 808} - \text{R}\,\text{1 600 000} \\ &= \text{R}\,\text{4 324 808} \end{align*}

Bank B:

\begin{align*} x &= \frac{ \text{1 600 000} \times \frac{0.14}{12}}{\left[1 - (1 + \frac{0.14}{12})^{-\text{240}}\right]} \\ &= \text{R}\,\text{19 896,33} \\ \text{Total amount } (T): \enspace &= 20 \times 12 \times \text{R}\,\text{19 896,33} \\ &= \text{R}\,\text{4 775 119,20} \\ \text{Total interest } (I): \enspace &= \text{R}\,\text{4 775 119,20} - \text{R}\,\text{1 600 000} \\ &= \text{R}\,\text{3 175 119,20} \end{align*}

Bank C:

\begin{align*} x &= \frac{ \text{1 600 000} \times \frac{0.14}{12}}{\left[1 - (1 + \frac{0.14}{12})^{-\text{360}}\right]} \\ &= \text{R}\,\text{18 957,95} \\ \text{Total amount } (T): \enspace &= 30 \times 12 \times \text{R}\,\text{18 957,95} \\ &= \text{R}\,\text{6 824 862} \\ \text{Total interest } (I): \enspace &= \text{R}\,\text{6 824 862} - \text{R}\,\text{1 600 000} \\ &= \text{R}\,\text{5 224 862} \end{align*}
 $$x$$ $$T$$ $$I$$ Bank A $$\text{R}\,\text{16 457,80}$$ $$\text{R}\,\text{5 924 808,00}$$ $$\text{R}\,\text{4 324 808,00}$$ Bank B $$\text{R}\,\text{19 896,33}$$ $$\text{R}\,\text{4 775 119,20}$$ $$\text{R}\,\text{3 175 119,20}$$ Bank C $$\text{R}\,\text{18 957,95}$$ $$\text{R}\,\text{6 824 862,00}$$ $$\text{R}\,\text{5 224 862,00}$$

A loan from Bank A would have the lowest monthly repayments, however the interest paid is high as a result of the longer repayment period. Therefore, Phoebe should consider taking out a loan with Bank B, as this has the lowest total repayment amount.

### Pyramid schemes (EMCG7)

A pyramid scheme is a moneymaking scheme that promises investors unusually high returns on their investment. The concept of a pyramid scheme is quite simple and should be easy to identify, however, it is often cleverly disguised as a legitimate business. In some cases a product is offered and in others the scheme is marketed as a highly profitable investment opportunity. Unfortunately, many of these schemes have cost millions of people their savings. Pyramid schemes are illegal in South Africa.

• "Ponzi schemes" are named after Charles Ponzi, a Italian businessman who lived in the U.S. and Canada. One of his investment schemes involved buying postal reply coupons in other countries and redeeming them for a higher value in the United States. He promised investors $$\text{100}\%$$ profit within $$\text{90}$$ days of their investment. As the scheme grew, Ponzi paid off early investors with money from investors who joined the scheme at a later stage. Ponzi's fraudulent scheme was exposed and investors lost millions of dollars. He was sent to prison for a number of years.
• South African Adriaan Nieuwoudt started a pyramid scheme commonly referred to as the "Kubus" scheme. Participants bought a biological substance called an activator that was supposed to be used in beauty products. The activator was used to grow cultures in milk that were then dried and ground up and resold to new participants. The scheme had thousands of investors and took in approximately $$\text{R}\,\text{140}$$ million before it was declared an illegal lottery.

A pyramid scheme starts with one person, who finds other people to invest their money in the scheme. These people then enrol more people into the scheme and the base of the pyramid grows. The money invested by new investors goes to participants closer to the top of the pyramid. This is unsustainable because it requires that more and more people join the scheme and the growth must end at some point because there are a finite number of people. Therefore, most of the investors lose their money when the scheme collapses.

The South African Reserve Bank has launched a public awareness campaign, “Beware of oMashayana (crooks)”, to help educate people about pyramid schemes and how to identify them.

Beware of oMashayana

So, you think you’ve found the perfect investment? Regular high returns with no risk? Be careful. Don’t lose all your money. If it sounds too good to be true, it’s probably a Pyramid or Ponzi scheme.

What is a Ponzi scheme?

A Ponzi con-artist will only ask you to give them money that they say will be invested in a scheme or project, such as supposed property developments; bridging finance; foreign-exchange transactions; venture capital to other companies; or share units. The scheme operator promises to give you back much more money than you have given them initially, in a very short space of time.

What is a pyramid scheme?

A pyramid con-artist will offer you the chance to make quick money for yourself, often by selling something. You pay a joining fee, buy the product and then sell it. They tell you that the more people you get to sell for you, the more money you will make. It’s easy for the first people to introduce new members, but soon everyone is part of the scheme and it gets harder to find new members to join. For example, the con-artist recruits $$\text{6}$$ people who each pay a $$\text{R}\,\text{100}$$ fee. Each of those people has to recruit $$\text{6}$$ people. There are now up to $$\text{36}$$ people. Now, those $$\text{36}$$ people each have to sign up $$\text{6}$$ people – this equals $$\text{216}$$, and by the time you have to get to level $$\text{10}$$, you have to get $$\text{60}$$ million people to join to keep the scheme going. This can never work.

What is the difference between a pyramid and a Ponzi scheme?

The main difference is that with a pyramid scheme you have to work or sell to recruit investors, while with a Ponzi scheme the con-artist will only ask you to invest in something (for example, property development). Both schemes are illegal.

What can I do to protect myself?

You are your own best protection. It is your responsibility to make sure that you never give your money to any company or person that is not registered as a deposit-taking institution in terms of the Banks Act.

Why is handing my money over so risky?

When you hand over your money (notes and coins) to another person who then loses the money, steals it or goes bankrupt, you only have an unsecured claim against that person or their estate and you might not get all your money back.

Why is it safer to hand my money to a bank?

Banks and investment companies have to be registered so that they can be regulated and supervised, to make sure that your money is safe. Unregulated and unsupervised persons and groups don’t follow these rules and your money is at great risk with them.


text:
# using \newcommand with \immediate\write inside beamer

I'm trying to write some strings to an external file using the \immediate\write as in the MWE below

\documentclass[a4paper,9pt]{beamer}
\begin{document}

\begin{frame}[plain] %< comment out for article

\newcommand{\createtexfile}[1]{%
\newwrite\file
\immediate\openout\file=output.tex
\immediate\write\file{\string\textbf{foo bar} }
\immediate\write\file{\string #1}
\immediate\closeout\file
}

\createtexfile{some text}

\end{frame} %< comment out for article
\end{document}

which should output:

> cat output.tex

\textbf{foo bar}
some text

If I run this using 'article' as a document class (frame should be commented out) it compiles without errors. When using the 'beamer' documentclass I get the following error:

! Illegal parameter number in definition of \iterate. 1 l.17 \end{frame} %< comment out for article You meant to type ## instead of #, right? Or maybe a } was forgotten somewhere earlier, and things are all screwed up? I'm going to assume that you meant ##.

Does anybody else get this error as well? and if so, any ideas how to get around it? I would appreciate you inputs.

• don't define the command inside the frame and always never have the \newwrite inside a macro you just want to do that once, in the preamble. You are also missing % from ends of lines. Aug 1, 2018 at 18:19

Don't define the command inside the frame, define it in the preamble.

Also, always (not just in beamer) never have the \newwrite inside a macro you just want to do that once, in the preamble.

You are also missing % from ends of lines.

Just for completion, taking the answer from David Carlisle, this is how should be implemented.

\documentclass[a4paper,9pt]{beamer}

\newwrite\file %<-

\newcommand{\createtexfile}[1]{%
\immediate\openout\file=output.tex
\immediate\write\file{\string\textbf{foo bar} }%<-
\immediate\write\file{\string #1}%<-
\immediate\closeout\file%<-
}

\begin{document}

\begin{frame}[plain] %< comment out for article

\createtexfile{some text}

\end{frame} %< comment out for article
\end{document}


text:
# Chevalley Groups over an arbitrary ring.

My question is simply about the Chevalley groups over rings. In many books, including Carter's book on "Simple groups of Lie types", the groups are considered over fields. I have checked the computations and I noticed that the computations works over any commutative $\mathbb{Z}$-algebra. Why these groups are not introduced in the general format? Of course if these groups are defined over rings then they are not necessarily simple. Is it the reason why most people want to define them over fields?

Just to give a general definition of Chevalley groups over rings let me review the construction.

Let $L$ be a finite dimension complex simple Lie algebra with a Chevalley basis $$\{ e_r: r\in\Pi; e_r: r\in\Phi \},$$ where $\Pi$ is basis for the root system $\Phi$. Pick $\zeta\in\mathbb{C}$. Therefore $$\exp(\zeta ad_{e_r}),$$ is an Lie-algebra automorphism of $L$. One can observe that the entries of the matrix associated to $\exp(\zeta ad_{e_r})$, denoted by $A_r(\zeta)$, with respect to a Chevalley basis, are of the form $a\zeta^l$ where $a\in\mathbb{Z}$ and $l\in\mathbb{Z}_{\geq 0}$. Now let $B$ be a commutative $Z$ algebra with the structure map $\rho: \mathbb{Z}\to B$. Pick $b\in B$, then we consider the matrix $\overline{A_r(b)}$ which is obtained by transforming the entry of $a\zeta^l$ of the matrix $A_r(\zeta)$ into $\rho(a)b^l$. Then One can consider the linear transformation, denoted by $\overline{x_r(b)}$, obtained by $\overline{A_r(b)}$ on $L(B):=L(\mathbb{Z})\otimes_{\mathbb{Z}}B$. One can show that $\overline{x_r(b)}$ is indeed a Lie-algebra automorphism of $L(B)$. Then the Chevalley group $G_{ad}(B,\Phi)$, is defined by the subgroup of $GL(L(B))$ generated by $\overline{x_r(b)}$.

Is there a problem in this construction that I am not considering?

Many thanks for your answers.

• Volumes II and III of SGA3 are all about this story (giving an intrinsic definition of "Chevalley groups" over $\mathbf{Z}$ was Grothendieck's motivation for Demazure in his thesis which entailed much of SGA3). The reason "most people" define them only over fields is because the community of specialists in the representation and structure theory of algebraic (and arithmetic) groups and the community of experts in the level of scheme theory which pervades SGA3 has small overlap. Your construction is wrong even for ${\rm{PGL}}_2$ over $\mathbf{Q}$ (if meant at the level of $B$-valued points). Nov 14, 2013 at 0:35
• Marguax: I don't know much about the Scheme theoretical part of algebraic groups. You mentioned about the $PGL_2$ over $Q$. I don't see how to construct this by simple Lie algebra. Indeed the simple Lie algebra with the root system $A_1$ gives $PSL_2$. So I don't understand the mistake in my construction. Would you please explain more?
– M.B
Nov 14, 2013 at 0:51
• Dear M.B.: Sorry, I wasn't aware of your background, so my answer below will likely be useless to you. But in my answer I do address the distinction among ${\rm{SL}}_n(k)$, ${\rm{PGL}}_n(k)$, and ${\rm{SL}}_n(k)/\mu_n(k)$, and put your suggested construction into a broader framework. Nov 14, 2013 at 0:53
• M.B.: To construct the non-perfect group ${\rm{PGL}}_2(\mathbf{Q})$ as an abstract group, you have to manually fatten the diagonal by tossing in more generators on the diagonal. Or, at the cost of explicitness, you could take the algebraic $\mathbf{Q}$-group generated (in the sense of algebraic groups!) by the root groups in the adjoint representation of the Lie algebra; that is ${\rm{PGL}}_2$ as a group variety. Then form its group of rational points. Nov 14, 2013 at 4:59

## 2 Answers

The clean definition of adjoint Chevalley groups can be given in the spirit of what you are trying to do, but there is a hidden subtlety because it is only the torus in the simply connected case that is literally generated by the coroot groups (i.e., the simple positive coroots are a basis of its cocharacter group) whereas in the adjoint case it is the character group having a basis related to those simple positive coroots.

Since the root groups "come from" the simply connected central cover via isomorphisms, all you can hope to get using the root groups is at best the image of the points from the simply connected central cover in the adjoint form. And even that would only tend to work over fields, where we have Bruhat decomposition on rational points; already over a dvr there isn't quite a Bruhat decomposition, so it becomes rather subtle to try to get by with the group generated by points of root groups.

For example, in the case of ${\rm{PGL}}_n$ over a field $k$, what you get from the root groups is the image of ${\rm{SL}}_n(k)$ in ${\rm{PGL}}_n(k)$ (recall that ${\rm{SL}}_2(k)$ is generated by the standard unipotent points, so applied with the ${\rm{SL}}_2$ associated to each simple positive root this is "enough" to capture the $k$-points of the diagonal torus in ${\rm{SL}}_n$). Of course, the map ${\rm{SL}}_n(k) \rightarrow {\rm{PGL}}_n(k)$ has normal image with cokernel $k^{\times}/(k^{\times})^n$ that is typically huge (e.g., $k = \mathbf{Q}$).

It is true that with the correct definition of $G^{\rm{ad}}(\Phi)$, the natural action on its Lie algebra $L$ over $\mathbf{Z}$ (the latter having a definition as you indicate, also in Carter's book) does make this a closed subgroup scheme of the automorphism scheme of $L$. So it does occur inside ${\rm{GL}}(L)$ "defined by some equations". But this is not a good way to understand its structure and properties over rings.

But let's come back to Carter's book. The point is that if $G$ is a split simply connected semisimple group over a finite field $k$ with $G^{\rm{ad}} = G/Z$ its adjoint central quotient (with $Z$ the schematic center of $G$), the isogeny $G \rightarrow G^{\rm{ad}}$ induces an injection of groups $G(k)/Z(k) \rightarrow G^{\rm{ad}}(k) \subset {\rm{GL}}(L_k)$. So voila, the group $G(k)/Z(k)$ is found inside ${\rm{GL}}(L_k)$, and as such it is literally generated by the $k$-points of the root groups. And away from some very low-rank cases over fields of size 2 or 3, this group is simple. But it is neither $G(k)$ nor $G^{\rm{ad}}(k)$ in general! That is, $G(k)/Z(k)$ is not "an algebraic group" in a reasonable sense in general.

So if those "simple finite groups of Lie type" $G(k)/Z(k)$ may be called "Chevalley groups", that is a misnomer in the sense that "Chevalley group" should be either an algebro-geometric structure or at least its group of points over some ring of interest, and $G(k)/Z(k)$ is typically neither of those. For example, if $G = {\rm{SL}}_n$ then $G(k)/Z(k)$ is the commutator subgroup of ${\rm{PGL}}_n(k)$ (sometimes called ${\rm{PSL}}_n(k)$, which is dangerous notation) but it is not an "algebraic group" in any reasonable sense in general. (Nonetheless, the theory of algebraic groups is very powerful to tell us things about $G(k)/Z(k)$!)

EDIT: In view of the OP's comments, probably I should explain why I say that the commonly seen notation "${\rm{PSL}}_n(k)$" for the group ${\rm{SL}}_n(k)/\mu_n(k)$ is quite dangerous. The notation suggests that if $K/k$ is a Galois extension of fields then the injection of groups ${\rm{SL}}_n(k)/\mu_n(k) \rightarrow {\rm{SL}}_n(K)/\mu_n(K)$ has image that is the ${\rm{Gal}}(K/k)$-invariants of the target (as is the case for the injection $X(k) \rightarrow X(K)$ of rational points of a variety $X$ over $k$). But this is totally false: there is an obstruction coming from the part of the Brauer group of $k$ killed by $K$.

So truly the construction $k \rightsquigarrow {\rm{SL}}_n(k)/\mu_n(k)$ is not at all like forming rational points of a variety (over a ground field that may be increased tomorrow).

In fact, the same phenomenon is even seen for ${\rm{PGL}}_n$ over rings more general than fields (or rather, more general than local rings and PID's). To be precise, if we consider the construction $R \rightsquigarrow {\rm{GL}}_n(R)/R^{\times}$ on rings (not just fields) then it is equally bad as above; e.g., one can make examples of number fields $K$ with nontrivial 2-torsion in the class group and finite Galois extensions $K'/K$ such that ${\rm{GL}}_2(O_K)/O_K^{\times}$ is strictly smaller than the subgroup of ${\rm{Gal}}(K'/K)$-invariants in ${\rm{GL}}_2(O_{K'})/O_{K'}^{\times}$. This may feel "weird" if you haven't seen it before, but it is entirely intuitive from the scheme-theoretic perspective. All of the same issues also arise in the more general Chevalley group situation (for $G(k)/Z(k)$ as above).

The only truly satisfactory way I have ever seen to come to grips with this kind of behavior for quotient constructions (over rings, and even over fields in tricky situations) in a systematic manner that "always works" (with good technique) is via group schemes. There are ways to grapple with it over fields without using schemes, but it tends to cause arguments to deviate a bit from group-theoretic intuition. And this includes even the case of Chevalley groups, over rings or fields. But it could just be a matter of taste; plenty of people have done very deep work in these matters without mastering the fancy algebraic geometry (e.g., Borel, Bruhat-Tits, et al.).

• Dear Marguax. I am so thankful for answering to my question But unfortunately since I don't know much about the theory of algebraic groups I can not understand many points that you have made in your answer. It would be very good if you would please give me a reference with more details for these.
– M.B
Nov 14, 2013 at 3:05
• Moreover I have seen the following paper by Borel on "Properties and linear representations of Chevalley groups". In this paper he also has defined the Chevalley groups over field with the way that I presented in my question. So I just want to know if there is any mistake if one define it over rings. Perhaps what you are trying to mention is that this construction is not general enough. Do I see it correctly?
– M.B
Nov 14, 2013 at 3:06
• @M.B.: There is a typo in 3.3(1): $G_{\pi,k}$ at the start should be $G_{\pi,K}$, with $K$ an algebraically closed field containing a given field $k$. In 3.3(5) he says (as I did) that for the "Chevalley group" $G_{\pi}$ as a $\mathbf{Z}$-group, the "group generated by $x_a(t)$'s" is the image in $G_{\pi}(k)$ of the $k$-points of the simply connected central cover of $G_{\pi}$. In section 4 he abandons pure group theory to make $G_{\pi}$ as a $\mathbf{Z}$-group scheme. Upshot: his abstract group $G_{\pi,k}$ is not $G_{\pi}(k)$. He calls $G_{\pi,k}$ a "Chevalley group", but this is dangerous! Nov 14, 2013 at 4:24
• In the end, the main issue is probably want you want to do with these things. There may be entrenched incompatible traditions in the group theory and group scheme communities for the meaning of "Chevalley group", and the arbiter of one's preference is what one wants to do with it. The algebraic geometer's viewpoint is very robust for varying rings and fields, and recovers the other viewpoint in a clean way. I am skeptical that "group generated by root groups" is a useful notion except over fields (where Bruhat decomposition is available), but I am open to hearing to the contrary. Nov 14, 2013 at 4:36
• The 1996 paper of Vavilov and Plotkin mentioned in my comment to Jim's answer addresses the role of "group generated by root groups" beyond the setting of fields. (It seems to be of more interest than I had expected, but nonetheless they also discuss the fact that with general rings these are smaller than the groups of ring-valued points of the Chevalley group schemes, and it is the latter groups of points which they call "Chevalley groups".) Nov 14, 2013 at 15:17

A couple of further clarifications, to supplement the extensive answer by marguax and the many comments:

1. Chevalley's influential 1955 paper was mainly concerned with finding a uniform approach to most of the known simple finite groups of Lie type (supplemented soon afterward by the introduction of twisted groups as well as the groups of Suszuki and Ree). The original "Chevalley group" was defined as a group of automorphisms of a certain Lie algebra, generated by unipotent elements; this group is simple over almost all fields (except a few very small ones). The Lie algebra here is obtained by first reducing mod $$p$$ a Chevalley $$\mathbb{Z}$$-basis of a simple Lie algebra over $$\mathbb{C}$$, then tensoring with a field of prime characteristic $$p$$.

A subtle point here is that the "Chevalley Lie algebra" obtained over an algebraically closed field is actually the Lie algebra of the corresponding simply connected algebraic group, whereas the Chevalley groups themselves are closer to the adjoint groups.

Anyway, Steinberg in his 1967-68 Yale lectures broadened the notion of "Chevalley group" by using other faithful Lie algebra representations, to include special linear groups and the like.

1. In fact, there is a huge amount of literature about Chevalley groups over (commutative) rings. Here the ideal structure of the ring contributes to normal subgroup structure in the group, so the groups are typically non-simple. But they do come up naturally in algebraic K-theory, including the study of the congruence subgroup problem. N.A. Vavilov and many others have written extensively about such groups. There is less textbook literature along these lines, except for a few books on algebraic K-theory including one by Hahn and O'Meara. And the entire subject becomes quite technical.

ADDED: Concerning the more sophisticated viewpoint of Chevalley-Demazure group schemes, there is an important recent paper by Lusztig (not yet freely available online) Study of a $$\mathbf{Z}$$-form of the coordinate ring of a reductive group. The arXiv preprint is here.

• Steinberg's Yale notes build only simply connected forms, yes? (Section 4 of Borel's article which the OP looked at goes further.) For your #2, the "Chevalley groups" in Vavilov et al. are the ring-valued points of Chevalley-Deamzure group schemes, whereas what the OP and Borel call a "Chevalley group" is called there an "elementary subgroup". The link between them and "rare" cases of equality (and role of Bruhat decomposition in that) are addressed in the first 2 pages of the 1996 Acta Applicandae paper of Vavilov and Plotkin; also see the last paragraph of section 1. Nov 14, 2013 at 15:11
• Dear Jim: Many thanks for the answer. You have mentioned that the Chevalley group, constructed by the Chevalley basis, has been considered for commutative rings. So I would like to know if the construction is the same as I presented in my question. This is the reason I asked the question.
– M.B
Nov 14, 2013 at 16:02
• @marguax: No, Steinberg works with an arbitrary lattice between the root lattice and the weight lattice. His "Chevalley groups" are generated by elementary (unipotent) matrices, but in his lectures and related papers he deals with related covering groups as well. The notes are online at www.math.ucla.edu/~rst/ Your further comments are helpful. Nov 14, 2013 at 17:27
• @M.B.: The definition of "Chevalley group" varies in different sources, so it's always important to pin it down. In his many papers, Vavilov works with the points of a Chevalley-Demazure group scheme but also with the usually smaller "elementary Chevalley group"; the difference is an important part of studying the groups over rings. See his long survey: Structure of Chevalley groups over commutative rings.* Nonassociative algebras and related topics (Hiroshima, 1990), 219–335, World Sci. Publ., River Edge, NJ, 1991. Your construction is much narrower, similar to Chevalley's. Nov 14, 2013 at 17:35


text:
# Prove that an equation has a unique (convergent?) solution

I'm stuck on this problem. Any advice is appreciated:

i. Prove that $\forall n \in \mathbb{N}$ the equation $x^n +x - 1 = 0$ has a unique solution $r_n$.

ii. Find the limit of the sequence $(r_n)_{n\geq1}$.

Here's what I have so far (not much, sadly)

I rewrote the equation in i. to get 1 to the other side. I also believe that the solution should be positive for the few numbers that I have tried. I'm thinking about using Newton's Method but I'm not sure how effective it would be. Any advice?

• (i) isn't even true. For $n = 2$, and it appears for all even $n$, there are two distinct solutions. – Mike Pierce Mar 15 '16 at 2:44
• And indeed, when $n$ is even there are always two solutions. – Robert Israel Mar 15 '16 at 2:48
• Wait, I'm sorry. I missed something. It should say "unique positive." my apologies. – Rethink123 Mar 15 '16 at 2:49

Hint for (2): Is $r_n > 1$ or $< 1$? For fixed $x$ in the appropriate interval, what happens to $x^n + x - 1$ as $n \to \infty$ what does that tell you about how $r_n$ relates to $x$?