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# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021116
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## Dynamical analysis of a banking duopoly model with capital regulation and asymmetric costs

 1 Polytechnic University of Marche, Department of Management, Piazzale R. Martelli 8, 60121, Ancona (AN), Italy 2 University of Modena and Reggio Emilia, Department of Economics Marco Biagi, Via Jacopo Berengario 51, 41121, Modena, Italy

* Corresponding author: giovanni.campisi@unimore.it

Received  October 2020 Revised  February 2021 Early access April 2021

It is well known that regulation and efficiency are two important issues on banking literature. The goal of the paper is to analyse them through a banking duopoly model with heterogeneous expectations. To this purpose, we consider two scenarios. In the first one, we focus on regulation effects. In particular, empirical literature on Italian banks finds evidence on the asymmetry of the costs of regulation that penalize small banks with respect to the large ones. In this direction, we analyse a duopoly model where small banks and large banks have different forecasting rules and we capture the differences of the regulations' effects assuming asymmetry in the cost functions. We introduce linear cost function for small banks and quadratic cost function for large banks. In the second scenario, we study the relation between regulation and bank efficiency highlighting empirical results showing that large banks register higher level of inefficiency than small banks. Moreover, in order to stress new evidences and to confirm empirical results on banking regulation and efficiency, we conduct an analytical and numerical analysis.

Citation: Serena Brianzoni, Giovanni Campisi. Dynamical analysis of a banking duopoly model with capital regulation and asymmetric costs. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021116
##### References:
 [1] H. N. Agiza and A. A. Elsadany, Nonlinear dynamics in the Cournot duopoly game with heterogeneous players, Physica A: Statistical Mechanics and its Applications, 320 (2003), 512-524.  doi: 10.1016/S0378-4371(02)01648-5.  Google Scholar [2] A. Agliari, C. Chiarella and L. Gardini, A re-evaluation of adaptive expectations in light of global nonlinear dynamic analysis, Journal of Economic Behavior & Organization, 60 (2006), 526-552.  doi: 10.1016/j.jebo.2004.07.006.  Google Scholar [3] F. Aiello and G. Bonanno, Profit and cost efficiency in the italian banking industry (2006-2011), Economics and Business Letters, 2 (2013), 190-205.  doi: 10.17811/ebl.2.4.2013.190-205.  Google Scholar [4] P. Alessandrini, M. Croci and A. Zazzaro, The geography of banking power: The role of functional distance, in The Banks and the Italian Economy, Springer, 2009, 93-123. Google Scholar [5] P. Alessandrini, M. Fratianni, L. Papi and A. 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Michetti, Nonlinear dynamics in a business-cycle model with logistic population growth, Chaos, Solitons & Fractals, 40 (2009), 717-730.  doi: 10.1016/j.chaos.2007.08.041.  Google Scholar [15] S. Brianzoni, C. Mammana and E. Michetti, Local and global dynamics in a discrete time growth model with nonconcave production function, Discrete Dynamics in Nature and Society, 2012 (2012), Art. ID 536570, 1-22. doi: 10.1155/2012/536570.  Google Scholar [16] E. Dia and M. Giuliodori, Portfolio separation and the dynamics of bank interest rates, Scottish Journal of Political Economy, 59 (2012), 28-46.  doi: 10.1111/j.1467-9485.2011.0567.x.  Google Scholar [17] R. Dieci and F. Westerhoff, Heterogeneous speculators, endogenous fluctuations and interacting markets: A model of stock prices and exchange rates, Journal of Economic Dynamics and Control, 34 (2010), 743-764.  doi: 10.1016/j.jedc.2009.11.002.  Google Scholar [18] G. Fagiolo, D. Giachini and A. 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Papi, Technical efficiency and scale efficiency in the Italian banking sector: a non-parametric approach, Applied Economics, 27 (1995), 385-395.  doi: 10.1080/00036849500000123.  Google Scholar [23] L. Gardini, F. Tramontana, V. Avrutin and M. Schanz, Border-collision bifurcations in 1d piecewise-linear maps and Leonov's approach, International Journal of Bifurcation and Chaos, 20 (2010), 3085-3104.  doi: 10.1142/S021812741002757X.  Google Scholar [24] A. Giannola, A. Lopes, C. Ricci and G. Scarfiglieri, Divari territoriali ed efficienza del sistema bancario italiano, Quintieri B. (a cura di), Finanza Istituzioni e Sviluppo Regionale, il Mulino, Bologna, 219-253. Google Scholar [25] L. Giordano and A. Lopes, Preferenza al rischio e qualità degli impieghi come determinanti dell'efficienza del sistema bancario italiano, Giannola, A. (a cura di), Riforme istituzionali e Mutamento Strutturale, Mercati, imprese e istituzioni in un sistema dualisticoi, Carocci, Roma. Google Scholar [26] L. Giordano and A. Lopes, Le banche italiane tra consolidamento e recuperi di efficienza (1999-2008). una promessa mantenuta?, Giannola, A. Lopes, A. Sarno, D. (a cura di), I Problemi Dello Sviluppo Economico e del suo Finanziamento Nelle Aree Deboli, Carocci, Roma, 169-202. Google Scholar [27] C. Girardone, P. Molyneux and E. P. Gardener, Analysing the determinants of bank efficiency: The case of Italian banks, Applied Economics, 36 (2004), 215-227.  doi: 10.1080/0003684042000175334.  Google Scholar [28] M. A. Klein, A theory of the banking firm, Journal of Money, Credit and Banking, 3 (1971), 205-218.  doi: 10.2307/1991279.  Google Scholar [29] F. Lamantia, M. Pezzino and F. Tramontana, Dynamic analysis of discontinuous best response with innovation, Journal of Economic Dynamics and Control, 91 (2018), 120-133.  doi: 10.1016/j.jedc.2018.01.024.  Google Scholar [30] A. Medio and M. Lines, Nonlinear Dynamics: A Primer, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511754050.  Google Scholar [31] M. Monti et al., Deposit, Credit and Interest Rate Determination Under Alternative Bank Objective Function, . Karl Shell, Giorgio P. SzegÃ¶ (Eds. ), Mathematical methods in investment and finance, North-Holland/American Elsevier, 1972, 431-454. Google Scholar [32] A. Resti, Evaluating the cost-efficiency of the Italian banking system: What can be learned from the joint application of parametric and non-parametric techniques, Journal of Banking & Finance, 21 (1997), 221-250.  doi: 10.1016/S0378-4266(96)00036-2.  Google Scholar [33] E. Saltari and G. Travaglini, L'economia Italiana Del Nuovo Millennio, (Vol. 282). Carocci, Roma, 2009. Google Scholar [34] M. L. Stefani, V. Vacca, D. Coin, S. Del Prete, C. Demma, M. Galardo, I. Garri, S. Mocetti and D. Pellegrino, Le Banche Locali e il Finanziamento Dei Territori: Evidenze Per l'Italia (2007-2014), 324, Banca d'Italia, 2016. Google Scholar [35] F. Tramontana, Heterogeneous duopoly with isoelastic demand function, Economic Modelling, 27 (2010), 350-357.  doi: 10.1016/j.econmod.2009.09.014.  Google Scholar [36] F. Tramontana, L. Gardini, V. Avrutin and M. Schanz, Period adding in piecewise linear maps with two discontinuities, International Journal of Bifurcation and Chaos, 22 (2012), 1250068, 1-30. doi: 10.1142/S021812741250068X.  Google Scholar [37] F. Tramontana, L. Gardini, R. Dieci and F. Westerhoff, The emergence of bull and bear dynamics in a nonlinear model of interacting markets, Discrete Dynamics in Nature and Society, 2009 (2009), Art. ID 310471, 1-30. doi: 10.1155/2009/310471.  Google Scholar [38] F. Tramontana, F. Westerhoff and L. Gardini, On the complicated price dynamics of a simple one-dimensional discontinuous financial market model with heterogeneous interacting traders, Journal of Economic Behavior & Organization, 74 (2010), 187-205.  doi: 10.1016/j.jebo.2010.02.008.  Google Scholar [39] F. Tramontana, F. Westerhoff and L. Gardini, A simple financial market model with chartists and fundamentalists: Market entry levels and discontinuities, Mathematics and Computers in Simulation, 108 (2015), 16-40.  doi: 10.1016/j.matcom.2013.06.002.  Google Scholar

show all references

##### References:
 [1] H. N. Agiza and A. A. Elsadany, Nonlinear dynamics in the Cournot duopoly game with heterogeneous players, Physica A: Statistical Mechanics and its Applications, 320 (2003), 512-524.  doi: 10.1016/S0378-4371(02)01648-5.  Google Scholar [2] A. Agliari, C. Chiarella and L. Gardini, A re-evaluation of adaptive expectations in light of global nonlinear dynamic analysis, Journal of Economic Behavior & Organization, 60 (2006), 526-552.  doi: 10.1016/j.jebo.2004.07.006.  Google Scholar [3] F. Aiello and G. Bonanno, Profit and cost efficiency in the italian banking industry (2006-2011), Economics and Business Letters, 2 (2013), 190-205.  doi: 10.17811/ebl.2.4.2013.190-205.  Google Scholar [4] P. Alessandrini, M. Croci and A. Zazzaro, The geography of banking power: The role of functional distance, in The Banks and the Italian Economy, Springer, 2009, 93-123. Google Scholar [5] P. Alessandrini, M. Fratianni, L. Papi and A. Zazzaro, The asymmetric burden of regulation: Will local banks survive, Money and Finance Research group (Mo. Fi. R. ) Working Papers No, 125, 1-19. Google Scholar [6] P. Alessandrini and L. Papi, Banche locali e piccole imprese dopo la crisi tra nuove regole e innovazioni digitali, Money and Finance Research group (Mo. Fi. R. ) Working Papers No, 148, 1-25. Google Scholar [7] P. Alessandrini and L. Papi, L'impatto della "bolla regolamentare" sulle banche: Alcune valutazioni, Money and Finance Research group (Mo. Fi. R. ) Working Papers No, 150, 1-35. Google Scholar [8] A. N. Berger, A. Saunders, J. M. Scalise and G. F. Udell, The effects of bank mergers and acquisitions on small business lending, Journal of Financial Economics, 50 (1998), 187-229.   Google Scholar [9] G. -I. Bischi, C. Chiarella, M. Kopel, F. Szidarovszky et al., Nonlinear Oligopolies, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-02106-0.  Google Scholar [10] G. I. Bischi and F. Lamantia, Coexisting attractors and complex basins in discrete-time economic models, in Nonlinear Dynamical Systems in Economics, Springer, 2005, 187-231. doi: 10.1007/3-211-38043-4_7.  Google Scholar [11] G.-I. Bischi, L. Stefanini and L. Gardini, Synchronization, intermittency and critical curves in a duopoly game, Mathematics and Computers in Simulation, 44 (1998), 559-585.  doi: 10.1016/S0378-4754(97)00100-6.  Google Scholar [12] S. Brianzoni, G. Campisi and A. Russo, Corruption and economic growth with non constant labor force growth, Communications in Nonlinear Science and Numerical Simulation, 58 (2018), 202-219.  doi: 10.1016/j.cnsns.2017.07.007.  Google Scholar [13] S. Brianzoni, L. Gori and E. Michetti, Dynamics of a Bertrand duopoly with differentiated products and nonlinear costs: Analysis, comparisons and new evidences, Chaos, Solitons & Fractals, 79 (2015), 191-203.  doi: 10.1016/j.chaos.2015.05.014.  Google Scholar [14] S. Brianzoni, C. Mammana and E. Michetti, Nonlinear dynamics in a business-cycle model with logistic population growth, Chaos, Solitons & Fractals, 40 (2009), 717-730.  doi: 10.1016/j.chaos.2007.08.041.  Google Scholar [15] S. Brianzoni, C. Mammana and E. Michetti, Local and global dynamics in a discrete time growth model with nonconcave production function, Discrete Dynamics in Nature and Society, 2012 (2012), Art. ID 536570, 1-22. doi: 10.1155/2012/536570.  Google Scholar [16] E. Dia and M. Giuliodori, Portfolio separation and the dynamics of bank interest rates, Scottish Journal of Political Economy, 59 (2012), 28-46.  doi: 10.1111/j.1467-9485.2011.0567.x.  Google Scholar [17] R. Dieci and F. Westerhoff, Heterogeneous speculators, endogenous fluctuations and interacting markets: A model of stock prices and exchange rates, Journal of Economic Dynamics and Control, 34 (2010), 743-764.  doi: 10.1016/j.jedc.2009.11.002.  Google Scholar [18] G. Fagiolo, D. Giachini and A. Roventini, Innovation, finance, and economic growth: An agent-based approach, Journal of Economic Interaction and Coordination, 15 (2020), 703-736.  doi: 10.1007/s11403-019-00258-1.  Google Scholar [19] L. Fanti, The dynamics of a banking duopoly with capital regulations, Economic Modelling, 37 (2014), 340-349.  doi: 10.1016/j.econmod.2013.11.010.  Google Scholar [20] L. Fanti, L. Gori, C. Mammana and E. Michetti, The dynamics of a Bertrand duopoly with differentiated products: Synchronization, intermittency and global dynamics, Chaos, Solitons & Fractals, 52 (2013), 73-86.  doi: 10.1016/j.chaos.2013.04.002.  Google Scholar [21] L. Fanti, L. Gori and M. Sodini, Nonlinear dynamics in a Cournot duopoly with relative profit delegation, Chaos, Solitons & Fractals, 45 (2012), 1469-1478.  doi: 10.1016/j.chaos.2012.08.008.  Google Scholar [22] C. A. Favero and L. Papi, Technical efficiency and scale efficiency in the Italian banking sector: a non-parametric approach, Applied Economics, 27 (1995), 385-395.  doi: 10.1080/00036849500000123.  Google Scholar [23] L. Gardini, F. Tramontana, V. Avrutin and M. Schanz, Border-collision bifurcations in 1d piecewise-linear maps and Leonov's approach, International Journal of Bifurcation and Chaos, 20 (2010), 3085-3104.  doi: 10.1142/S021812741002757X.  Google Scholar [24] A. Giannola, A. Lopes, C. Ricci and G. Scarfiglieri, Divari territoriali ed efficienza del sistema bancario italiano, Quintieri B. (a cura di), Finanza Istituzioni e Sviluppo Regionale, il Mulino, Bologna, 219-253. Google Scholar [25] L. Giordano and A. Lopes, Preferenza al rischio e qualità degli impieghi come determinanti dell'efficienza del sistema bancario italiano, Giannola, A. (a cura di), Riforme istituzionali e Mutamento Strutturale, Mercati, imprese e istituzioni in un sistema dualisticoi, Carocci, Roma. Google Scholar [26] L. Giordano and A. Lopes, Le banche italiane tra consolidamento e recuperi di efficienza (1999-2008). una promessa mantenuta?, Giannola, A. Lopes, A. Sarno, D. (a cura di), I Problemi Dello Sviluppo Economico e del suo Finanziamento Nelle Aree Deboli, Carocci, Roma, 169-202. Google Scholar [27] C. Girardone, P. Molyneux and E. P. Gardener, Analysing the determinants of bank efficiency: The case of Italian banks, Applied Economics, 36 (2004), 215-227.  doi: 10.1080/0003684042000175334.  Google Scholar [28] M. A. Klein, A theory of the banking firm, Journal of Money, Credit and Banking, 3 (1971), 205-218.  doi: 10.2307/1991279.  Google Scholar [29] F. Lamantia, M. Pezzino and F. Tramontana, Dynamic analysis of discontinuous best response with innovation, Journal of Economic Dynamics and Control, 91 (2018), 120-133.  doi: 10.1016/j.jedc.2018.01.024.  Google Scholar [30] A. Medio and M. Lines, Nonlinear Dynamics: A Primer, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511754050.  Google Scholar [31] M. Monti et al., Deposit, Credit and Interest Rate Determination Under Alternative Bank Objective Function, . Karl Shell, Giorgio P. SzegÃ¶ (Eds. ), Mathematical methods in investment and finance, North-Holland/American Elsevier, 1972, 431-454. Google Scholar [32] A. Resti, Evaluating the cost-efficiency of the Italian banking system: What can be learned from the joint application of parametric and non-parametric techniques, Journal of Banking & Finance, 21 (1997), 221-250.  doi: 10.1016/S0378-4266(96)00036-2.  Google Scholar [33] E. Saltari and G. Travaglini, L'economia Italiana Del Nuovo Millennio, (Vol. 282). Carocci, Roma, 2009. Google Scholar [34] M. L. Stefani, V. Vacca, D. Coin, S. Del Prete, C. Demma, M. Galardo, I. Garri, S. Mocetti and D. Pellegrino, Le Banche Locali e il Finanziamento Dei Territori: Evidenze Per l'Italia (2007-2014), 324, Banca d'Italia, 2016. Google Scholar [35] F. Tramontana, Heterogeneous duopoly with isoelastic demand function, Economic Modelling, 27 (2010), 350-357.  doi: 10.1016/j.econmod.2009.09.014.  Google Scholar [36] F. Tramontana, L. Gardini, V. Avrutin and M. Schanz, Period adding in piecewise linear maps with two discontinuities, International Journal of Bifurcation and Chaos, 22 (2012), 1250068, 1-30. doi: 10.1142/S021812741250068X.  Google Scholar [37] F. Tramontana, L. Gardini, R. Dieci and F. Westerhoff, The emergence of bull and bear dynamics in a nonlinear model of interacting markets, Discrete Dynamics in Nature and Society, 2009 (2009), Art. ID 310471, 1-30. doi: 10.1155/2009/310471.  Google Scholar [38] F. Tramontana, F. Westerhoff and L. Gardini, On the complicated price dynamics of a simple one-dimensional discontinuous financial market model with heterogeneous interacting traders, Journal of Economic Behavior & Organization, 74 (2010), 187-205.  doi: 10.1016/j.jebo.2010.02.008.  Google Scholar [39] F. Tramontana, F. Westerhoff and L. Gardini, A simple financial market model with chartists and fundamentalists: Market entry levels and discontinuities, Mathematics and Computers in Simulation, 108 (2015), 16-40.  doi: 10.1016/j.matcom.2013.06.002.  Google Scholar
In the $(c_1,c_2)-$plane the set of points between the red and blue curve ensures positive equilibrium loans, while the yellow curve represents the parameter values $(c_1,c_2)$ for which a flip bifurcation occurs. In Panel (a) the parameters are: $a = 3.2$, $\alpha = 1.5$, $b = 1.5$, $\gamma = 0.7$, $r_k = 2.8$. In panel (b) $\gamma = 0.1$ while the other parameters as in Panel (a)
Two interesting economic scenarios through bifurcation diagrams. In Panel (a) the parameters are: $a = 3$, $\alpha = 1.38$, $b = 0.12$, $\gamma = 0.2$, $r_k = 2.8$, $c_{1} = 0.15$ and $c_{2}\in (0.01,0.6)$. In Panel (b) $c_{1} = 0.1$ and $c_{2}\in (0.01,0.6)$ while the other parameters as in panel (a)
On the left is depicted the attractor of the 4-cycle for $\alpha = 1.25$, $\gamma = 0.35$, $r_{k} = 2.8$, $a = 3.2$, $b = 0.1$, $c_{1} = 0.08$, $c_{2} = 0.7$. On the right panel a bifurcation diagram for $c_{1}\in (1.4,2.1)$ showing the period-doubling of the 2-cycle for $\alpha = 2$, $\gamma = 0.25$, $r_{k} = 2.3$, $a = 3.2$, $b = 0.4$, $c_{2} = 1.1$
On the left, basins of attraction of the two-cycle, given the following values of the parameters, $\alpha = 1.35$, $\gamma = 0.7$, $r_{k} = 1.5$, $a = 3$, $b = 1.9$, $c_{1} = 0.01$, $c_{2} = 0.27$. On the right, basins of attraction showing the birth and the stability of a 4-cycle for $a = 3.27$ and the other parameters as in the left panel
In (a), four pieces chaotic attractor for $\alpha = 1.38$, $\gamma = 0.35$, $r_{k} = 2.8$, $a = 3$, $b = 0.12$, $c_{1} = 0.13$, $c_{2} = 0.78$. In (b), two pieces chaotic attractor for $c_{2} = 0.8$ and the other parameters as in Panel (a)
In (a), connected chaotic attractor for $c_{2} = 0.84$ and the other parameters as in Panel (a) of Figure 5. On the right, cycle cartogram in the ($c_{1},c_{2}$)-plane for parameter values as in Panel (a) of Figure 5
The time series of the loans of large (Panel (a)) and small (Panel (b)) Italian banks. The real time series are depicted in blue, while in red the time series of the loans resulting from the stochastic model
Monte Carlo simulations for analyse differences in loans' demand of the two banks for several combinations of costs and regulation parameters. In (a), $c_{1},c_{2}\in [0.01,0.4]$ and the other parameters as in Table 1. In (b) and (c), joint effect of the costs and regulation keeping fixed all other parameters as in Table 1. In (b), the loans' demand are plotted with respect to the costs of the two banks when $c_{1},c_{2}\in [0.01,0.3]$ and $\gamma\in[0.05,0.4]$. In (c) the loans' demand are plotted with respect to the parameter of regulation $\gamma$ when $c_{1},c_{2}\in [0.01,0.3]$ and $\gamma\in[0.05,0.4]$. In (d), the effects of regulation on the loans' demand when $\gamma\in [0.05,0.4]$ and the other parameters as in Table 1
Parameter setting and initial values
 $\alpha$ $\gamma$ $r_{k}$ $a$ $b$ $c_{1}$ $c_{2}$ $\sigma_{1}$ $\sigma_{1}$ 1.04 0.18 3.6 1.68 0.043 0.08 0.38 0.12 0.06
 $\alpha$ $\gamma$ $r_{k}$ $a$ $b$ $c_{1}$ $c_{2}$ $\sigma_{1}$ $\sigma_{1}$ 1.04 0.18 3.6 1.68 0.043 0.08 0.38 0.12 0.06
Summary statistics of banks' loans including mean, standard deviation (sd), skewness, minimum and maximum value for real time series of the loans of large banks ($L_{1}^{real}$), real time series of the loans of small banks ($L_{2}^{real}$) and the simulated stochastic time series of the loans for large ($L_{1}^{SS}$) and small ($L_{2}^{SS}$) banks
 Mean sd Min Max Skewness Kurtosis $L_{1}^{real}$ 291, 440, 000 64, 858, 000 192, 020, 000 356, 490, 000 -0.4471 1.6507 $L_{2}^{real}$ 224, 430, 000 25, 598, 000 186, 240, 000 263, 480, 000 -0.0867 1.7266 $L_{1}^{SS}$ 345, 330, 000 7, 315, 000 331, 320, 000 355, 280, 000 -0.4075 2.2085 $L_{2}^{SS}$ 224, 080, 000 3, 652, 000 216, 660, 000 230, 890, 000 -0.1410 2.9930
 Mean sd Min Max Skewness Kurtosis $L_{1}^{real}$ 291, 440, 000 64, 858, 000 192, 020, 000 356, 490, 000 -0.4471 1.6507 $L_{2}^{real}$ 224, 430, 000 25, 598, 000 186, 240, 000 263, 480, 000 -0.0867 1.7266 $L_{1}^{SS}$ 345, 330, 000 7, 315, 000 331, 320, 000 355, 280, 000 -0.4075 2.2085 $L_{2}^{SS}$ 224, 080, 000 3, 652, 000 216, 660, 000 230, 890, 000 -0.1410 2.9930
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### Lightning Released for Mainnet

The anticipated Lightning Network has finally launched on the bitcoin and litecoin main net as a beta version, Lightning Network is an open source protocol that allows people to send a series of bitcoin and litecoin payments without having them on the block chain, and only having a small fraction of them on the main block chain with full security, Lightning Labs, one of the main companies working on Lightning Network and the creator of this beta software has raised 2.5$Million earlier this year from different investors including litecoin creator Charlie Lee and twitter CEO Jack Dorsey. Currently there is a limit of 1400$ per channel or about 400\$ per payment, as this version is aimed towards developers and users who are able to run the Lightning Network command line along with a bitcoin or litecoin full node. but with almost 1000 nodes currently running Lightning and thousands of more developers building applications on it, Lighting maybe one of the biggest innovations to hit the block chain as it allows users to send payments with a much lower fraction of fees compared to the current methods, it also allows developers to build applications on the block chain. A more stable and accessible software would allow more audience to use Lightning, but until that, we will take what we can get.

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Innhold:


text:
# On order automorphisms group of a finite group

Let $G$ be finite group and $Aut(G)$ denote the automorphisms group of $G$.

1) If $\mid G\mid>2$ is even, then do $\mid Aut(G)\mid$ is even?

2) Do we can determine structure all finite groups such that the order of the automorphisms group is even?

-
If $C_2$ is the cyclic group of order two then $\operatorname{Aut}(C_2)$ is trivial, and so has odd order. –  user1729 Apr 10 '13 at 15:59
Ok. I forgot to write that $\mid G\mid >2$. I must edit it. –  maryam Apr 10 '13 at 16:02
Do we need the condition $|G|$ is even? (Interestingly, GAP says we do. Without this condition, the smallest non-trivial counterexample is of order $3^6=729$, and there's many of this order.) –  Douglas S. Stones Apr 10 '13 at 16:03
More seriously, $G/Z(G)$ is a subgroup of $\operatorname{Aut}(G)$. Thus, if $G/Z(G)$ is even then so is $\operatorname{Aut}(G)$. For example, if $G$ is a non-abelian group of order $2^n$ then $\operatorname{Aut}(G)$ is even. –  user1729 Apr 10 '13 at 16:03
@DouglasS.Stones: No. Take $C_3\times C_3$. –  user1729 Apr 10 '13 at 16:04

a group $H$ of order $3 \times 19 \times 7^{12}$ is constructed, which has trivial centre and is isomorphic to its own automorphism group. So $H \times C_2$ has even order but with odd order automorphism group.
Note that it $|G|$ is even and $|G/Z(G)|$ is odd, then $G$ is a direct product of a group of odd order and an abelian 2-group $T$, and if $|{\rm Aut}(G)|$ is odd then we must have $|T|=2$. So all examples must be similar this one (although the odd order direct factor would not necessarily have to be complete).
The smallest examples of groups with nontrivial automorphism groups of odd order are of order $3^6$. For example, $G = \operatorname{SmallGroup}(3^6, 90)$ has an automorphism group of order $3^7$. So by your remark, $C_2 \times G$ is a smallest possible example of even order. Reference: D. MacHale and R. Sheehy, Finite groups with odd order automorphism groups, Math. Proc. R. Ir. Acad. 95A (1995), no. 2, 113–116. JSTOR –  Mikko Korhonen Apr 10 '13 at 17:19


text:
# Indefinite Integrals

#### john-1

Can someone show me the steps to finding the integral of (x+1)/(x^2+2x) [ln(x^2+2x)] dx

I'm not sure where to begin because I don't have any integration rules with the lnx

#### pickslides

MHF Helper
Can someone show me the steps to finding the integral of (x+1)/(x^2+2x) [ln(x^2+2x)] dx
Do you mean 'show that' $$\displaystyle \int \frac{x+1}{x^2+2x}~dx = \ln(x^2+2x)$$ or 'find' $$\displaystyle \int \frac{x+1}{x^2+2x} \ln(x^2+2x)~dx$$

??

#### john-1

Do you mean 'show that' $$\displaystyle \int \frac{x+1}{x^2+2x}~dx = \ln(x^2+2x)$$ or 'find' $$\displaystyle \int \frac{x+1}{x^2+2x} \ln(x^2+2x)~dx$$

??

#### Also sprach Zarathustra

The second one, the first on is not so true...

in second integral use integration by parts...

#### john-1

could you show me the steps please? I only have the answer with me in my text but no steps

#### TheCoffeeMachine

Using the substitution $$\displaystyle u = \ln(x^2+2x)$$, we have:

$$\displaystyle \displaystyle \int\frac{(x+1)\ln(x^2+2x)}{x^2+2x}\;{dx} = \frac{1}{2}\int\left(\frac{x+1}{x^2+2x}\right)\left(\frac{x^2+2x}{x+1}\right)\cdot u \;{du} = \frac{1}{2}\int{u}\;{du} = \frac{1}{4}u^2+k.$$

Therefore $$\displaystyle \displaystyle \int\frac{(x+1)\ln(x^2+2x)}{x^2+2x}\;{dx} = \frac{1}{4}\ln^2(x^2+2x)+k$$.

Last edited:
john-1

#### john-1

TheCoffeeMachine, in your second step, why did you put (x^2 + 2x) / (x+1)? Where did that come from? mostly confused about 2nd step

#### AllanCuz

If $$\displaystyle u = ln(x^2 + 2x)$$ then $$\displaystyle \frac{du}{dx} = \frac{2x +2}{x^2+2x} \to dx = (\frac{1}{2}) \frac{x^2 + 2x}{x+1}du$$

He then subbed this into the integral for dx!


text:
# I update my fenics version (2017) and now I have problems with the nonlinearsolver parameters.

311
views
0
13 months ago by
ffc_options={"quadrature_degree":4,"optimize":True,"eliminate_zeros":False}
problem=NonlinearVariationalProblem(F,w,bcs,J,ffc_options)
solver=NonlinearVariationalSolver(problem)

error

KeyError: "'eliminate_zeros' is not a parameter"

Thanks for all.


text:
# How to open a dynamic view in clear case with a given config spects using command prompt?

I have config spects for a view and I need to create a view in Clear Case using command prompt. Please let me about the commands I can use with some example.

-

That would be using:

For creating a dynamic view:

cleartool mkview -tag yourDynView  \\shared\path\to\viewStorage\yourDynView.vws

That will create a synamic view with a default config spec:

element * CHECKEDOUT
element * /main/LATEST

You will then be able to replace that config spec with any other config spec stored in a file, with the setcs command.

-
Hi, Thanks for the reply..Can you tell me what should be the directory location for the text file ? –  savalakh Feb 21 '12 at 6:51
@savalakh: the text file with the right config spec can be... anywhere you want. Its full path will be the parameter 'pname' of the 'setcs' command. –  VonC Feb 21 '12 at 7:22
Hi, Thanks again... I am not able to mount a vob in it .. I am using base clear case . –  savalakh Feb 21 '12 at 13:45
@savalakh can you post in a comment the error message you have when trying to mount a vob? –  VonC Feb 21 '12 at 14:51
@VonC: in mkview, what is a good location to set your "file whose contents are to become the view's new config spec" to? thanks –  amphibient Sep 20 '12 at 15:07


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# High tower

## Main Question or Discussion Point

The earth rotates around its own axis within 24 hours. Theoretically, if at the equator perform a tower with a height - H, ignoring its effect on the slowdown of the Earth's rotation (we assume that the material of low density), objects and other complications, at a certain height, the linear velocities at the top of the tower would be very large compared to the stationary observer in relation to the axis of rotation of the Earth and was in the outer space at the height of the top. You can consider the view of the observers on the Earth on the equator and the one on the top of the tower, and in space next to the top of the tower passing him. You can increase the H value and try to calculate the linear velocity of the top of the tower relative to theoretical observer in space next to the top of the tower passing him.

The second thing can be if the tower is non-zero width along the meridian and its height would be theoretically 24 light hours, would it cover You some of the space view surrounding the equator by the width of the tower? Will You see the tower spreading from east to west? Would photones go from space be swept away by the tower if it did not reflect light? What would see the observer in space next to the top of the tower passing him? Will he see the meridian on the Earth?

You can also create a new questions, about it.
Thank You so much :)
and wish You Fascinating considerations :)

Related Special and General Relativity News on Phys.org
mfb
Mentor
and its height would be theoretically 24 light hours
That doesn't work. The tip cannot move faster than the speed of light. If you try to build 24 light hours / (2 pi) tall, no material will be able to hold the tower together. This is a fundamental limit.

Assuming a smaller tower:
Would photones go from space be swept away by the tower if it did not reflect light?
Sort of. The tower would appear bent.
Will he see the meridian on the Earth?
After sufficient time: Sure.

mfb said:
After sufficient time: Sure.
For theoretical 24 light hours tower height? I am not sure that the meridian will be seen from space (in the same plane and omiiting the phenomena of changing light trajectory, etc.) due to meeting the tower on the way of photons.
I do not have an unambiguous answer because there are phenomena typical of the speed of light near the speed.
Unless you meant the shorter tower, than it will be "less bright" maybe. :)

Ibix
You can't build a tower more than $24/2\pi\simeq 3.8$ light hours tall because its tip speed would exceed the speed of light and it must disintegrate at or before that point (a lot before, in practice). So what you'd see from a 24 light hour tall one isn't really possible to describe.


text:
User:
• Matrices
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Line graphs

A Line graph (chart) is a way to summarize series of quantities.

• The horizontal scale across the bottom and the vertical scale along the side provides information on how much or how many.
• The points on the graph show the facts.
• The lines connecting the points give estimates the values between the points.

The following line graph represents the value of Robert's baseball card collection.

1. How much has the value of Robert's collection increased from January to June? $10.00. 2. If this trend continues, what would be a reasonable estimate for the value of this collection in July?$132.50

The car salesman made a graph to show the number of cars he sold during the past few months

How many cars did he sold during April?

Solution:


text:
# Measuring Output Impedance of an RF (HF/VHF) Active Circuit with Hobby-level Tools

Is there a simple (by simple, I mean a hobby friendly method that trades some accuracy for expense) method of measuring the "actual" output impedance of an active or "hot" RF (HF to VHF) circuit?

As an amateur radio enthusiast, I often find myself in a situation where I'd like to measure the actual output impedance of some amplifier or oscillator (type of amplifier). Right now, I have an oscillator operating in the low VHF region and I'd like to match its output to a mixer that has a well defined input impedance. But that's just one example, I've had plenty others in the past. I can calculate the output impedance theoretically, but it would be nice to confirm the theory with a practical measurement.

I have some hobby or entry grade tools to hand, including an oscilloscope and a NanoVNA. Is there a reasonably accurate measurement method using these tools? One that doesn't damage the measurement device (especially the VNA).

One I read somewhere that made some kind of sense was to match the output with a (passive) device that varied the resistance, inductance and capacitance (a type of complex load). You could use a power meter to find the match point. Then measure the input impedance of the device/load with a VNA and use the conjugate as the output impedance.

Sounds simple enough, but I'm guessing the devil is in the construction of the complex load. What other methods are there?

UPDATE: Since asking this on EE SE, I've solved my immediate problem by using the two measurement technique (wonderfully explained in this video by W2AEW). However, I believe this approach becomes less accurate at RF frequencies and I have a continuing interest in discovering a practical, simple and inexpensive solution - if it exists!?

• At HF and VHF, a simple matching network, like an L-, $\pi$, or T-section might effect the match, as indicated on an SWR meter. Then, using a spreadsheet or Smith chart program (e.g., SimSmith), the impedance looking into the matching network is easily calculated. The source impedance is the complex conjugate of this value; i.e., same (series) resistance and opposite (series) reactance. – Brian K1LI Jun 20 '20 at 8:58


text:
Problem: According to the following reaction, how many moles of sodium hydroxide will be formed upon the complete reaction of 29.6 grams of sodium with excess water? sodium (s) + water (l) → sodium hydroxide (aq) + hydrogen (g)

FREE Expert Solution
87% (312 ratings)
Problem Details

According to the following reaction, how many moles of sodium hydroxide will be formed upon the complete reaction of 29.6 grams of sodium with excess water?

sodium (s) + water (l) → sodium hydroxide (aq) + hydrogen (g)


text:
# Is The Grand Asteroid Challenge plausible or just a PR stunt?

The Grand Asteroid Challenge is a project to capture an incoming asteroid, enter it into a stable Lunar orbit and send a manned mission to its surface.

That sounds... implausible at best. Do we have even hope of acquiring necessary technological means - like a vehicle capable of delivering enough delta-V to an asteroid to get it into an orbit and then prepare and send a manned mission (and this all until 2017)?

How do the scientists of NASA see the plausibility of this idea?

Yes, if you find a suitable asteroid in a suitable orbit, it can be done. The asteroid would be meters in size, requiring a few hundred m/s $\Delta V$. Calculations on asteroid populations indicate that we can find such an asteroid.

From the paper referenced in the comment below (ERO = "easily retrievable object", where "easily" must be meant in relative terms):

Indeed, the paper presents a list of 12 EROs, with a total of 25 trajectories to periodic orbits near L2 and 6 near L1 below a cost of 500 m/s, and the number of these objects is expected to grow considerably in the coming years. The lowest cost is of 58 m/s to transfer asteroid 2006 RH120 to a halo southern family with a single burn on 1st February 2021. All the capture transfer opportunities to Earth’s vicinity have been identified for the currently catalogued NEOs during the next 30 years, and enable capture of bodies within 2-5 meters diameter with low propellant costs.

As for your question on schedule, 2017 is extremely aggressive even just for the launch of the capture mission, so probably not. I did not see that date in your link, so I don't know what it is referring to.

• By the way, the "or" in the question title is not exclusive. It could be plausible and a PR stunt. – Mark Adler Sep 1 '13 at 2:19

According to this reference

NASA associate administrator for human exploration and operations Bill Gerstenmaier said the use of “redirect” in the mission concept is deliberate, since they’re interested in objects that already would pass close to the Earth. “We take advantage of the fact that it’s already coming back so we don’t have to change the velocity in that direction,” he said. “All we need to do is essentially redirect it” into a lunar orbit, taking advantage of the Moon’s gravity. Left unstated was that redirection would also be useful if an object is passing too close to the Earth.

So I think is that it is possible since the asteroid is heading towards the earth the only thing we need to do is to catch it and redirect it to an orbit ( which may sometimes require adding or reducing velocity which require fuel )


text:
474
Q:

How many squares do you see?

 A) 44 B) 45 C) 46 D) 50

Explanation:

Number of squares in the given figure is

16 + 9 + 19 + 1 = 45.

Q:

Solve the following equations and find the value of abc?

 A) 3 B) 6 C) 9 D) 12

Explanation:

From the given equations,

$(a+b+c)2=a2+b2+c2+2(ab+bc+ca)$

36 - 14 = 2(ab + bc + ca)
22/2 = ab + bc + ca
ab + bc + ca = 11

= (6) (14 - 11)
=(6)(3)
=18

Since,

36 - 3abc = 18

-3abc = 18 - 36

-3abc = -18

abc = 6.

Filed Under: Math Puzzles
Exam Prep: Bank Exams , CAT

0 1
Q:

Find the missing number in the following puzzle?

 A) 86 B) 98 C) 104 D) 112

Explanation:

The logic in the given puzzle:

2+3 = 8
3+7 = (7+2)x3 = 27
4+5 = (5+3)x4 = 32
5+8 = (8+4)x5 = 60
6+7 = (7+5)x6 = 72

Similarly,

7+8 = (8+6)x7 = 98.

Filed Under: Math Puzzles
Exam Prep: Bank Exams

22 3373
Q:

Following the same logic, crack this mathematical puzzle ?

 A) 4 B) 2 C) 1 D) 3

Explanation:

Logic Followed in the given puzzle ::

The logic here is the number of circles in each 4 digit number mentioned.

8801 has 5

5261 has 1

and so on

Filed Under: Math Puzzles
Exam Prep: Bank Exams , CAT , GATE

7 5637
Q:

What value will replace the question mark in the puzzle?

 A) 64 B) 56 C) 46 D) 36

Explanation:

Logic ::

First number on the right hand side is the second number in the left side of equation ( and ) second number on the right side is the sum of two numbers on the left side of the equation.

Hence, 2 + 4 = 4(2+4) = 46.

Filed Under: Math Puzzles
Exam Prep: AIEEE , Bank Exams , CAT

6 673
Q:

Solve the value of D?

A + B = 15 .... (1)

B x C = 84 .... (2)

C ÷ 4 = 3  .... (3)

B + D = 18 .... (4)

From (3), we get C = 12

From (2), we get B = 84/12 = 7

From (1), we get A = 15 - 7 = 8

From (4), we get D = 18 - 7 = 11

Hence, D = 11.

Subject: Math Puzzles Exam Prep: Bank Exams
Job Role: Bank Clerk , Bank PO

654
Q:

Can you solve this maths puzzle?

 A) 10 B) 12 C) 14 D) 18

Explanation:

The logic in this puzzle is summing the result of [subtraction of upper two digits] & [addition of lower two digits]

Hence, the result is

(5 - 3) + (2 + 8) = 2 + 10 = 12.

Filed Under: Math Puzzles
Exam Prep: Bank Exams

13 853
Q:

90% of people fail to solve this math puzzle?

 A) 45 B) -20 C) 35 D) -60

Explanation:

A - A - A = 20
B - B - B = 15
C - C - C = 25

A = -20
B = -15
C = -25

=> A + B + C = (-20)+(-15)+(-25) = -20 - 15 - 25 = -60.

Filed Under: Math Puzzles
Exam Prep: CAT , Bank Exams
Job Role: Bank Clerk , Bank PO

7 841
Q:

Can you solve this riddle?

 A) 1 B) 3 C) 5 D) 0

Explanation:

5 + 5 = 10

5x 2 + 2 = 12

5 x 2 - 1 x 5 = 5

Hence, $∆$ = 1.


text:
# Revision history [back]

I agree, at least in part. I would use the term "array" in deference to "multiple output". The entire manner in which arrays are handled in Calc is inelegant, including the CTRL+SHIFT+ENTER entry method, the non-editable / hidden bracket / control characters nature, and non-conformant notation. I would have thought:

=TRANSPOSE({A1:B2})
=TRANSPOSE({name_of_range})

... to be more sensible and conforming with existing array notation in other languages than the current:

{=TRANSPOSE(A1:B2)}
{=TRANSPOSE(name_of_range)}

... which has to be entered in a non-standard manner. The proposed examples should be able to be entered in the usual manner i.e., using ENTER.

I'm not sure what the cell itself should show, maybe something to indicate whether the procedure completed successfully.

I think is the single biggest drawback to the method / notation indicated in the question. All current functions are entered in the cell they output to. I think so long as the main entry cell (i.e., top left) is somehow marked and the formula remains editable, that it is better to keep the required parameters as simple as possible.

Please post a link to any bugs you file in a comment below using the format "fdo#123456".

Thanks!

I agree, at least in part. I would use the term "array" in deference to "multiple output". It is also worth noting that this type of change would likely be dependent upon a change to the ODF specification, so it may not happen quickly.

The entire manner in which arrays are handled in Calc is inelegant, including the CTRL+SHIFT+ENTER entry method, the non-editable / hidden bracket / control characters nature, and non-conformant notation. I would have thought:

=TRANSPOSE({A1:B2})
=TRANSPOSE({name_of_range})

... to be more sensible and conforming with existing array notation in other languages than the current:

{=TRANSPOSE(A1:B2)}
{=TRANSPOSE(name_of_range)}

... which has to be entered in a non-standard manner. The proposed examples should be able to be entered in the usual manner i.e., using ENTER.

I'm not sure what the cell itself should show, maybe something to indicate whether the procedure completed successfully.

I think is the single biggest drawback to the method / notation indicated in the question. All current functions are entered in the cell they output to. I think so long as the main entry cell (i.e., top left) is somehow marked and the formula remains editable, that it is better to keep the required parameters as simple as possible.

Please post a link to any bugs you file in a comment below using the format "fdo#123456".

Thanks!


text:
# Rock, paper and scissors

I started reading a book about python two days ago and today I wrote my first code. It's supposed to be a rock, paper, scissors game and from the tests I've done so far it works quite well.

However since I'm new I had to google a lot stuff and I'm 100% sure that there are a lot of things I could have done better, especially regarding the amount of elif-statements that I needed in order to cover every possible outcome.

#Imports the random module
import random

#Prints the introduction to the game
print("Welcome! This program allows you to play the famous game: Rock, paper, scissors! \n")

a = input("Now it's your time to choose, what will you take? Rock, paper or scissors? ")

#Repeats the users choice for aesthetic reasons
print ("You chose", a, "!")

#Turns the users input into a lowercase word for an easier use later on
a = a.lower()

#Creates a list of choices for the computer to "choose" from
game_list = ["Rock", "Paper", "Scissors"]

#Lets the computer choose a random option
b = random.choice(game_list)

#Shows the user which option the computer choose
print("The computer chose", b, "!")

#Turns the computers choice into a lowercase word for an easier use later on
b = b.lower()

#All the possible outcomes and conditions the computer has to check
if a == "rock" and b == "rock":
print("It's a draw!")
elif a == "rock" and b == "paper":
print("You lost!")
elif a == "rock" and b == "scissors":
print("You won!")
elif a == "paper" and b == "rock":
print("You won!")
elif a == "paper" and b == "paper":
print("It's a draw!")
elif a == "paper" and b == "scissors":
print("You lost!")
elif a == "scissors" and b == "rock":
print("You lost!")
elif a == "scissors" and b == "paper":
print("You won!")
elif a == "rock" and b == "scissors":
print("It's a draw!")
else:
print("Something went wrong.")
print("Have you typed in your choice correctly?")

#The computer says goodbye
print("Thank you for playing, I hope you had fun!")

I'll also add my "worksheet" or whatever you want to call it. I've seen a video on Youtube of a guy saying that it's important to first think about what the program has to do, before you start coding.

So here is what I wrote before I started and what I wrote after I was done (corrections) and after I had to add more things that I didn't think of before:

1. The program introduces the game to the player.
2. The program has to ask the player for input (Rock, paper or scissors).
3. It has to save the input as a variable to use it later on.
4. The program itself has to choose an option by itself (randomly).
5. The program needs to create conditions for outcomes for every possible combination of decisions the player and the computer can make.
6. The program has to test the human input and computer choice under the conditions.
7. The computer has to tell the player what the outcome is (output).
8. The computer should say goodbye.

Corrections:

1. I had to import the random module first.
2. I decided that it looks better if the program repeats the user input.
3. I decided that it was smarter to turn the user input into an lowercase word, so that the program wouldn't run into trouble later on (during the conditions).
4. Creating a list made the random choice easier.
5. I decided that it looks better if the program outputs the computer choice.
6. Lowercase word again.

Advice here is appreciated too, especially regarding my way of writing down what the program has to do.

Organize code in commented paragraphs or small sections. It's good that you are being methodical in your approach and that you have embraced code comments. However, every good thing can be taken to unhelpful extremes. A comment-per-line or a comment-per-step is probably too much. Comments do not help much if they merely duplicate what's already clear from the code. They work much better as organizational devices, as mechanisms to provide a high-level summary of the code's purpose and main points, or to clarify tricky details. Think of comments the way you might think of subheadings in an long email message (organizational) or as ways to elaborate on complex portions of the code (explanatory). Since nothing in this program is very complex algorithmically, your comments can be mostly organizational.

Normalize your internal data. Your code needs a list of the choices. Store them either lowercase or uppercase -- whichever makes computation easier. If you need them capitalized differently for presentation to the user, Python has many ways to do that easily. The idea is to standardize your data so that the code logic is simpler.

Use data structures to eliminate repetition. Your code has a big if-else structure than can be reduced to a simple dict-lookup. This game has a small number of cases to handle: put them in a dict that maps every pair of choices to the correct outcome.

Aim for variable names that communicate well. Naming is never easy, but it is important for writing clear, easy-to-read, easy-to-maintain code. Your current names strike me as too abstract (a and b) and less specific and direct than they could be (game_list).

Favor less chatty messages. This is a bit of a stylistic point, but most humans don't want to slog through long messages from their computer overlords. Also, code is easier to maintain and edit if your messages are concise. Get to the point in as few words as possible.

Validate user input. Rather than having a mystery outcome (something went wrong), just tell the user they goofed up.

Why limit the fun? Wrap a while True loop around your code and let the user play as long as they like.

import random

# Set up the choices and outcomes.

CHOICES = ['rock', 'paper', 'scissors']

DRAW = 'Draw.'
WON = 'You won!'
LOST = 'You lost!'

OUTCOMES = {
('rock', 'rock'): DRAW,
('rock', 'paper'): LOST,
('rock', 'scissors'): WON,
('paper', 'rock'): WON,
('paper', 'paper'): DRAW,
('paper', 'scissors'): LOST,
('scissors', 'rock'): LOST,
('scissors', 'paper'): WON,
('scissors', 'scissors'): DRAW,
}

# Play game.

while True:
# Get human choice and validate.
human = input('Rock, paper, or scissors? ').lower().strip()
if not human:
break
elif human not in CHOICES:
print("Invalid choice.")
continue

# Computer choice.
computer = random.choice(CHOICES)
print(f"The computer chose {computer}.")

# Game outcome.
print(OUTCOMES[human, computer])

Future improvements you might consider. Keep track of wins and losses. Let the user freely abbreviate their choice (eg, entering just r for rock, etc).

• Oh wow. So simple. Very nice. Jun 17 at 19:57
• I saw this and couldn't resist eliminating the manually defined OUTCOMES array. I'm not sure it's the neatest style, but I'd offer the following: ENDGAMES = ['Draw.', 'You lost!', 'You won!'] OUTCOMES = {(first, second): ENDGAMES[(second_i-first_i) % len(CHOICES)] for first_i, first in enumerate(CHOICES) for second_i, second in enumerate(CHOICES)} Jun 18 at 0:16
• @yoniLavi Yes, I was briefly tempted as well.
– FMc
Jun 18 at 0:35
• Wow, thank you a lot! I'll need some time to understand the code you wrote, but it's incredible that you can create the same game with less code while also adding stuff to the game itself. I wrote down every advice you gave me and I'll try to apply it in my next project that I'll post in a few hours. :) Thanks again! Jun 18 at 6:45


text:
# RBSE Solutions for Class 11 Maths Chapter 7 Binomial Theorem Ex 7.4

## Rajasthan Board RBSE Class 11 Maths Chapter 7 Binomial Theorem Ex 7.4

Question 1.
Expand the following Binomials upto fourth term:

Solution:
Expansion upto four terms of (1 + x2)-2

Question 2.
Find the required terms in the following expansions:
(i) Fourth term of (1 – 3x)-1/3
(ii) Seventh term of (1 + x)5/2
(iii) Eighth term of (1 + 2x)-1/2
Solution:
(i) Fourth term of (1 – 3x)-1/3
= Fourth term of (1 – 3x)-1/3
= Fourth term of {1 + (- 3x)}1/3

(ii) Seventh term of (1 + x)5/2

(iii) Eighth term of (1 + 2x)-1/2

Question 3.
Find the general term of the following expansions:
(i) (a3 – x3)2/3
(ii) (1 – 2x)-3/2
(iii) (1 – x)-p/q
Solution:
(i) General term in the expansion of (a3 – x3)2/3

(ii) General term in the expansion of (1 – 2x)-3/2

(iii) General term of in the expansion of (1 – x)-p/q

Question 4.
If x < 3, find the coefficient of (3 – x)-8 in the expansion of (3 – x)– 5.
Solution:
(x + 1)th term in the expansion of (3 – x)-8

Question 5.
Find the coefficient of x6 in the expansion of (a + 2bx2)-3
Solution :
(r + 1)th term in the expansion of (a + 2bx2)-3

For the coefficient of x6 in the this term
2r = 6 ⇒ r = 3
Hence, from equation (i), coefficient of x6 in the expansion of (a + 2bx2)-3

Question 6.
Find the coefficient of x10 in the expansion of

Solution:

Question 7.
Find the coefficient of xr in the expansion of (1 – 2x + 3x2 – 4x3 + …)n and if x = $$\frac { 1 }{ 2 }$$ and n = 1, then find the value of the expression.
Solution:
we know that (1 + x)-2 = 1 – 2x + 3x2 – 4x3 +…+ (-1)r (r+1)xr+ ….. (i)
Given progression (1 – 2x + 3x2 – 4x3 +…)n
(1 – 2x + 3x2 – 4x3 +…)n
= {(1 + x)-2}n  [From equation (i)]
= (1 + x)-2n

Question 8.
Prove that (1 + x + x2 + x3 + …)2 = 1 + 2x + 3x2 + …
Solution:
We know that
1 + x + x2 + x3 + … = (1 – x)-1  …. (i)
(1 + x + x2 + x3 + …)2
= {(1 – x)-1}2  [From equation (i)]
= (1 – x)2

Question 9.
Prove that (1 + x + x2 + x3 +….)  (1 + 3x + 6x2+ …)
= (1 + 2x + 3x2 + …)2
Solution:
We know that
(1 – x)-1 = 1 + x + x2 + x3 +…           … (i)
and (1 – x)-3 = 1 + 3x + 6x2 + 10x3 + … (ii)
L.H.S. = (1 + x + x2 + x3 +…)(1 + 3x + 6x2 + …)
= {(1 – x)-1} {(1 – x)-3}
= {(1 – x)2}-2
= {(1 – x)-2}2
= (1 + 2x + 3x2 +…)2
= R.H.S.
Hence Proved.

Question 10.
If x = 2y + 3y2 + 4y3 + …then express y in series of as ascending powers of x.
Solution:
⇒ x = 2y + 3y2 + 4y2 + …
⇒ x = (1 + 2y + 3y2 + 4y3 + …)- 1
[∵ (1 – x)-2 = 1 + 2x + 3x2 + …]
⇒ (x+ 1) = (1 -y)-2


text:
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# A Giant Refracting Telescope Has an Objective Lens of Focal Length 15 m. If an Eye Piece of Focal Length 1.0 cm is Used, What is the Angular Magnification of the Telescope ? - CBSE (Science) Class 12 - Physics

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ConceptOptical Instruments The Microscope

#### Question

A giant refracting telescope has an objective lens of focal length 15 m. If an eye piece of focal length 1.0 cm is used, what is the angular magnification of the telescope ?

#### Solution

Let:

f0 = Focal length of the objective lens = 15 m = 1500 cm

fe = Focal length of the eye lens = 1.0 cm

Angular magnification of the giant refracting telescope is given by

m_0=|f_0/(f_e)|

:.m_0=|1500/1|=1500

Is there an error in this question or solution?

#### Video TutorialsVIEW ALL [2]

Solution A Giant Refracting Telescope Has an Objective Lens of Focal Length 15 m. If an Eye Piece of Focal Length 1.0 cm is Used, What is the Angular Magnification of the Telescope ? Concept: Optical Instruments - The Microscope.
S


text:
# Formalizing Ordinal Partition Relations Using Isabelle/HOL

@article{Damonja2021FormalizingOP,
title={Formalizing Ordinal Partition Relations Using Isabelle/HOL},
author={Mirna D{\vz}amonja and Angeliki Koutsoukou-Argyraki and Lawrence Charles Paulson},
journal={Experimental Mathematics},
year={2021}
}
• Published 26 November 2020
• Computer Science
• Experimental Mathematics
This is an overview of a formalisation project in the proof assistant Isabelle/HOL of a number of research results in infinitary combinatorics and set theory (more specifically in ordinal partition relations) by Erd\H{o}s--Milner, Specker, Larson and Nash-Williams, leading to Larson's proof of the unpublished result by E.C. Milner asserting that for all $m \in \mathbb{N}$, $\omega^\omega\arrows(\omega^\omega, m)$. This material has been recently formalised by Paulson and is available on the…
2 Citations
Wetzel: Formalisation of an Undecidable Problem Linked to the Continuum Hypothesis
. In 1964, Paul Erdős published a paper [5] settling a question about function spaces that he had seen in a problem book. Erdős proved that the answer was yes if and only if the continuum hypothesis
Towards Formalising Schutz' Axioms for Minkowski Spacetime in Isabelle/HOL
• Computer Science
ArXiv
• 2021
A mechanisation in Isabelle/HOL of the system of axioms as well as theorems relating to temporal order is presented, particularly where the formal work required additional steps, alternative approaches, or corrections to Schutz’ prose.

## References

SHOWING 1-10 OF 59 REFERENCES
Zermelo Fraenkel Set Theory in Higher-Order Logic
The theory provides two type classes with the aim of facilitating developments that combine V with other Isabelle/HOL types: embeddable, the class of types that can be injected into V (including V itself as well as V*V, V list, etc.), and small, theclasses that correspond to some ZF set.
From LCF to Isabelle/HOL
• Computer Science
Formal Aspects of Computing
• 2019
This work focuses on Isabelle/HOL and its distinctive strengths, which include automatic proof search, borrowing techniques from the world of first order theorem proving, but also the automatic search for counterexamples.
Nitpick: A Counterexample Generator for Higher-Order Logic Based on a Relational Model Finder
• Computer Science, Mathematics
ITP
• 2010
Nitpick is a counterexample generator for Isabelle/HOL that builds on Kodkod, a SAT-based first-order relational model finder. Nitpick supports unbounded quantification, (co)inductive predicates and
On Fraisse's order type conjecture
ing from the embeddability relation between order types, define a quasi-order to be a reflexive, transitive relation. Throughout this paper, the letters Q and R will range over quasi-ordered sets and
The use of machines to assist in rigorous proof
• R. Milner
• Computer Science
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
• 1984
A method of tactic composition is presented in the form of an elementary theory of goal-seeking, by which simple elementary tactics are combined into more complex tactics, which may even be strategies complete for a class of problems.
A Machine-Checked Proof of the Odd Order Theorem
• Mathematics
ITP
• 2013
This paper reports on a six-year collaborative effort that culminated in a complete formalization of a proof of the Feit-Thompson Odd Order Theorem in the Coq proof assistant, using a comprehensive set of reusable libraries of formalized mathematics.
SOME PROBLEMS ON PINITR AND INFINITE GRAPHS
Many of the problems Hajnal and I posed 15 years ago have been solved positively or negatively or shown to be undecidable. I state soae of the remaining ones and add a few new ones. I do not give a
Isabelle/HOL: A Proof Assistant for Higher-Order Logic
• Computer Science
• 2002
This presentation discusses Functional Programming in HOL, which aims to provide students with an understanding of the programming language through the lens of Haskell.
Borel Sets and Ramsey's Theorem
• Mathematics
J. Symb. Log.
• 1973
The main result is that all Borei sets are Ramsey, which was discovered independently by Andrzej Ehrenfeucht, Paul Cohen, and probably many others, but no proof has been published.
On the logical strength of Nash-Williams' theorem on transfinite sequences
We show that Nash-Williams' theorem asserting that the countable transfinite sequences of elements of a better-quasi-ordering ordered by embeddability form a better-quasi-ordering is provable in the


text:
## College Algebra 7th Edition

(a) shift $f(x)$ left 2 units and down 2 units (b) shift $f(x)$ right 2 units and up 2 units
(a) The graph $g(x)=|x+2|-2$ is obtained by shifting $f(x)=|x|$ left 2 units and down 2 units. (b) The graph $g(x)=g(x)=|x-2|+2$ is obtained by shifting $f(x)=|x|$ right 2 units and up 2 units.


text:
# Health Reports Added, free and total sugar content and consumption of foods and beverages in Canada

by Siyuan Liu, Lalani L. Munasinghe, Arto Ohinmaa and Paul J. Veugelers

Release date: October 21, 2020

In Canada, chronic diseases account for 89% of all deathsNote 1 and more than \$80 billion in annual health care costs.Note 2 Adopting healthy lifestyle behaviours, such as healthy eating, has the potential to prevent 80% of type 2 diabetes and cardiovascular disease, 40% of cancers, and other chronic diseases.Note 1Note 3 Despite healthy eating recommendations issued by Health Canada,Note 4 eating habits continue to deteriorate, and overweight prevalence rates continue to increase.Note 5

Since sugars contribute to the overall energy composition of diet, excess sugar intake may lead to a positive energy balance and weight gain. Considerable epidemiological and experimental evidence has emerged suggesting an association between sugar consumption and obesity, high blood lipid levels, visceral adiposity, fatty liver disease, insulin resistance, type 2 diabetes, cardiovascular disease, metabolic syndrome, and dental caries.Note 6Note 7Note 8Note 9Note 10Note 11Note 12Note 13 While nutritious foods like fruits and milk contain naturally occurring sugarsNote 14 many other foods and beverages contain added and free sugars that, when consumed in excess amounts, increase calorie intake and displace consumption of nutritious foods.Note 15

The definitions of added, free and total sugars vary. This study uses the World Health Organization (WHO) definition of free sugars: “all monosaccharides and disaccharides added to foods by the manufacturer, cook or consumer, and sugars naturally present in honey, syrups, fruit juices and fruit juice concentrates.”Note 6 Added sugars are defined as refined sugars added during cooking or manufacturing, not including naturally occurring sugars and fruit juice.Note 16Note 17 Total sugars are defined as the sum of all naturally occurring and added sugars.

In 2015, the WHO released new recommendations to reduce the consumption of free sugars to less than 10% of daily total energy intake (TEI), and ideally to less than 5% of TEI to achieve additional health benefits.Note 6 The 2015-to-2020 edition of Dietary Guidelines for Americans, released by the U.S. Department of Agriculture (USDA), recommends that Americans consume less than 10% of calories from added sugars.

Various countries have documented added and free sugar consumption. For example, in the United States, added sugars were estimated to comprise 13.1% to 17.5% >of TEI among children and 11.2% to 14.5% of TEI among adults.Note 18Note 19 Children in Greece were estimated to obtain 11.2% of TEI from free sugars.Note 20 These population averages exceed the sugar recommendations of the WHO and the USDA and call for interventions to reduce added and free sugar intake. Estimates of added and free sugar consumption are not available for Canada because the added and free sugar content of foods and beverages in Canada is not systematically documented.

In 2004, the average total sugar intake of Canadians was estimated to be 21.4% of daily TEI,Note 21 an estimate that includes both naturally occurring and added sugars. While subsequent work was done to quantify added sugar intake specifically, these estimates did not consider all sources of added sugars.Note 22 Research showed that 66% of 40,000 packaged food products in Canada contained at least one added sugar.Note 23 In a recent study, Bernstein et al.Note 24 estimated the free sugar content of prepackaged foods in Canada using the Food Label Information Program. Although the authors were able to estimate the free sugar content for a large number of prepackaged foods, their work did not include all dietary sources. Therefore, to date, there are no accurate population-based estimates of the added and free sugar content of the Canadian diet.

Given the increasing concerns about the health consequences of excess added and free sugar intake, there is growing interest in applying policies and programs to reduce this consumption. Various jurisdictions have introduced such policies and programs to reduce consumption at the population level.Note 25Note 26 For Canada, scientific research into the health effects of excess added and free sugar consumption and thepotential benefits of interventions starts with a good understanding of added and free sugar consumption at the population level. Therefore, to provide accurate sugar consumption information for researchers and policy makers, this study aims (1) to document the added, free and total sugar content of foods and beverages consumed in Canada; and (2) to estimate the consumption and dietary sources of added, free and total sugars in Canada.

## Data and methods

### Data sources

The 2015 Canadian Community Health Survey (CCHS) – Nutrition is a cross-sectional national survey that was conducted by Statistics Canada and Health Canada from January to December 2015. The survey recruited individuals aged 1 and older living in private dwellings in the 10 Canadian provinces using a multi-stage, clustered design.Note 27 A national response rate of 61.6% was achieved. The survey’s 24-hour dietary recall collected the food intakes of 20,487 respondents, 7,608 of whom completed a second 24-hour dietary recall. These 24-hour dietary recalls were administered using an Automated Multiple-Pass Method.Note 28 Respondents reported the consumption of 2,784 ingredient-level foods and 2,590 recipe-level foods in the 2015 CCHS – Nutrition. Recipe-level foods are foods that consist of two or more ingredients. A sandwich and Caesar salad are examples of recipe-level foods, and an apple and noodles are examples of ingredient-level foods.Note 27 The nutrient content of ingredient-level foods was used to calculate the nutrient content of recipe-level foods. The Canada Food Guide file was used to determine the serving sizes.Note 28

The University of Alberta Research Ethics Board approved this study (Pro00073295). The 2015 CCHS – Nutrition data were accessed through Statistics Canada’s Research Data Centres Program.

### Estimation of the added and free sugar content of foods and beverages

Statistics Canada calculated the nutrient intake, including total sugar intake, for respondents based on the consumption and nutrient content of the reported foods and beverages. However, this calculation does not include added and free sugar intake, as the added and free sugar content of foods and beverages is not systematically documented in Canada.

In 2015, Louie et al.Note 17 published a decision tree with a systematic methodology to estimate the added sugar content of foods in Australia. Their 10-step standardized approach had high inter-rater repeatability and was designed for both national (Australia) and international use.Note 17 The approach by Louie et al.Note 17 was adapted for use in Canada. Figure 1 illustrates the decision algorithm of the nine-step approach. Note that 1 of the 10 steps from the Louie et al. approach (step number 4) was omitted because recipe-level foods in the 2015 CCHS – Nutrition dataset used more than one recipe. A brief description of each of the nine steps is provided below. More details on each of the steps are provided elsewhere.Note 29

Step 1: Assign 0 g of added and free sugars for ingredient-level foods that contain 0 g in total sugars.

Step 2: Assign 0 g of added and free sugars for ingredient-level foods that are unprocessed or processed without added or free sugars.

Assign 0 g of added sugars for food types such as unsweetened fruit or vegetable juice (including concentrate), unsweetened dairy products, and all fats and oils. Assign 0 g of free sugars for these food types, except for fruit juice and foods with fruit juice.

Step 3: Assign 100% of total sugars as added and free sugars for ingredient-level foods with very little naturally occurring sugars.

Assign 100% of total sugars as added sugars for food types such as sugar-sweetened beverages, confectionery, sugars, syrups and sweetener without added fruits, chocolate, and dairy products. Assign 100% of total sugars as free sugars for these food types and for 100% fruit juice and foods that contain fruits.

Step 4: Calculate added and free sugar content by comparing the total sugar content of a food or beverage with the sugar content of an unsweetened version of this food or beverage.

Added sugars and free sugars per 100 g (AS100g and FS100g) were calculated using the following formulas:

, and

,

where ${\text{TS}}_{\text{s}}$ is the total sugar content per 100 g of the sweetened food or beverage, and ${\text{TS}}_{\text{us}}$ is the total sugar content per 100 g of the unsweetened food or beverage.

Step 5: Calculate added and free sugar content based on lactose and maltose content.

Where data for lactose and maltose were available in the Canadian Nutrient File or the USDA Food Composition Database, added sugar and free sugar content of ingredient-level foods was calculated using the following formulas:

, and

.

Step 6: Calculate added sugar and free sugar content using the content values of similar foods from steps 1 to 5 or other nutrient databases.

Similar foods captured in steps 1 to 5 and in the Australian Food, Supplement and Nutrient Database were searched for information on added and free sugars. Foods were considered to be similar if they differed only in water content, if they contained similar ingredients (such as similar vegetables in soup), or if they were reduced in calories or energy or fat. Where a similar food was identified (matching), the added and free sugar content of the target food was estimated using the following formulas:

,

and .

Step 7: Estimate added and free sugar content for ingredient-level foods subjectively based on common recipes and ingredient lists.

Step 8: Assign 50% of total sugars as added and free sugar content for all remaining ingredient-level foods.

Step 9: Calculate added and free sugar content for recipe-level foods using respondent-specific recipes and the above estimated added and free sugar content of ingredient-level foods.

After all ingredient-level foods in the 2015 CCHS – Nutrition were estimated (steps 1 to 8), the added and free sugar content of recipe-level foods was calculated using the following formulas:

and , where $R{W}_{i}$ is the raw weight of the ith ingredient in the recipe, ${W}_{i}$ is the weight of the $i$th ingredient in the recipe after cooking, $A{S}_{i}$ is the added sugar content per 100 g of the $i$th ingredient and $F{S}_{i}$ is the free sugar content per 100 g of the $i$th ingredient.

The average added and free sugar content of ingredient-level foods was calculated for 18 food groups (displayed in Figure 2). These food groups were based on the Bureau of Nutritional Sciences Food Group Codes and DescriptionsNote 30 and the Canadian Nutrient File food groups.Note 31

### Estimation of usual intake of added, free and total sugar

The National Cancer Institute (NCI) method was applied to estimate the distribution of usual intake of added, free and total sugar by Canadians.Note 32 The NCI method assumes that usual intake is equal to the probability of consumption on a given day times the average amount consumed on a “consumption day”Note 33 and uses repeated 24-hour recalls. Depending on whether certain nutrients are commonly consumed (i.e., daily and by most respondents), this method uses either a one-part or a two-part model.Note 34 The one-part model considers only the amount of a nutrient consumed and is to be applied to nutrients that are commonly consumed. The two-part model considers both the proportion of the population consuming the nutrient and the amount of the nutrient consumed, and it is to be applied to nutrients that are not commonly consumed. This study assumed that added, free and total sugars were consumed daily and by all members of the population. For the usual intake of added, free and total sugars for each of the 18 food groups, this study considered the proportion of the population that consumed no sugar. A one-part model was applied for food groups where less than 5% of the population consumed no sugar. A two-part model was applied for food groups where more than 20% of the population consumed no sugar. For food groups where 5% to 20% of the population consumed no sugar, both models were fitted, and the model with the best fit was used.Note 34 For the usual intake of added and free sugars expressed as a percentage of TEI, an extension of the NCI method was used.Note 34Note 35

All analyses included age, sex, the sequence of 24-hour recalls (first recall versus second recall), and the 24-hour recall collection day of the week (weekdays, or Monday to Thursday, versus weekends, or Friday to Sunday) as covariates when accommodating individual-level and within-individual variation. Sampling weights provided by Statistics Canada were used to ensure study estimates apply to residents of Canada (for ease of presentation referred to as Canadians). All statistical analyses were performed using SAS (version 9.4, SAS Institute) software. SAS Macros of the NCI method were available online.Note 36

## Results

### Estimated added, free and total sugar content of foods and beverages

The added and free sugar content of 2,784 ingredient-level foods and 2,590 recipe-level foods consumed in Canada was calculated using the nine-step method. Table 1 shows the sugar content of selected ingredient-level food items recorded in the 2015 CCHS – Nutrition. The table provides one example for each of the 18 food groups. A table with the added, free and total sugar content of all ingredient-level foods is available elsewhereNote 29 for public use. Recipes (and recipe-level foods) are respondent-specific and subject to restricted release for confidentiality reasons, but they can be calculated from the ingredient-level foods.

### Estimated added, free and total sugar content of food groups

For each of the 18 food groups, Figure 2 shows the average added, free and total sugar content of all foods and beverages reported by respondents. Desserts and sweets had the highest added, free and total sugar content, at 37.0, 38.1 and 42.9 g per 100 g, respectively. Except for fruits, all food groups high in total sugar also had high added and free sugar content. Breakfast cereals, baked products, beverages, baby foods and snacks were the food groups that contained high free and added sugar content. Fruit juice had high free sugar content (12.5 g / 100 g) but low added sugar content (1.9 g / 100 g). Fats and oils, meats, sausages and luncheon meats, pasta, grains and flours, eggs, vegetables, spices, soups, sauces and gravies, nuts and seeds, and dairy products had relatively low sugar content.

### Estimated added, free and total sugar consumption and adherence to recommendations

Table 2 shows the mean and percentiles of usual added, free and total sugar intake in Canada. Canadians consume daily, on average, 57.1 g of added sugars, 67.1 g of free sugars and 105.6 g of total sugars. On average, added sugars constitute approximately 54.1% of total sugars, and free sugars make up approximately 63.5% of total sugars. The average estimated added, free and total sugar intake contributed to 11.1%, 13.3% and 21.6% of TEI, respectively.

### Contribution of food groups to added, free and total sugar intake

The contribution of each of the 18 food groups to added, free and total sugar intake is summarized in Figure 3. Desserts and sweets and beverages were the two food groups that contributed the most to added, free and total sugar intake in the Canadian diet. An estimated 67.3%, 57.5% and 41.1% of added, free and total sugar intake, respectively, came from desserts and sweets, and an estimated 17.4%, 17.5% and 12.6% of added, free and total sugar intake, respectively, came from beverages. Other food groups, such as baked products, soups, sauces and gravies, and breakfast cereals, also contributed to added, free and total sugar intake.

There were some differences in the relative contribution of food groups to the intake of added, free and total sugars (Figure 3). Fruits (12.9%) and dairy products (11.6%) were significant sources of total sugar intake, but not primary sources of added and free sugar intake.

## Discussion

This study revealed the added, free and total sugar content and consumption of foods and beverages in Canada. For the majority of Canadians, consumption of added and free sugars exceeded USDA and WHO recommendations. Desserts and sweets and beverages contributed to most of the added, free and total sugar intake of Canadians.

Both the USDA’s added sugar definition and the WHO’s free sugar definition were applied in the present study. Added sugars differ from free sugars in their exclusion of naturally occurring sugars in fruit juices and fruit juice concentrates. Various studies have not clearly defined added and free sugars,Note 17Note 18Note 19Note 24 whereas the present study and othersNote 37Note 38 have demonstrated substantial differences in the consumption of added versus free sugars.

Total sugar intake in Canada was estimated at 110.0 g/day using the 2004 CCHS – Nutrition.Note 21 This seems higher than the 105.6 g/day that was estimated in this study using the 2015 CCHS – Nutrition. Langlois et al.,Note 39 who studied the temporal changes in total sugar consumption, concluded that the apparent reduction between 2004 and 2015 may actually be caused by misreporting as a result of changes in survey methodology. They also reported that total sugar consumption from foods actually increased between 2004 and 2015, while total sugar intake from beverages decreased.Note 39 Studies from other countries and of earlier time periods concluded that total sugar intake either decreased or levelled off, both in absolute terms (g/day) and in relative terms (percentage of total sugar in TEI).Note 40Note 41Note 42

To date, very few studies of added and free sugar consumption have been population-based, and very few have reported on compliance with USDA and WHO recommendations. Louie et al. found that 1% of Australian children and youth (aged 2 to 16) consumed added sugars below 5% of TEI, and 18.1% consumed added sugars below 10% of TEI.Note 42 Sluik et al. revealed that 29% of Dutch residents (aged 7 and older) met the recommendation for added sugars (less than 10% of TEI) and 19% met the recommendation for free sugars (less than 10% of TEI).Note 38 The present study’s estimates that 49.0% and 33.8% of Canadians met the existing recommendations for added and free sugars, respectively, are notably higher than these Australian and Dutch estimates.

This study revealed that desserts and sweets, breakfast cereals, baked products, beverages, and snacks are food groups with high added and free sugar content. This ranking seems consistent with the ranking by Bernstein et al. for prepackaged foods and beverages.Note 24 This study also showed that the intake of added, free and total sugars through desserts and sweets was substantially higher than the intake of these sugars through beverages. This appears to be consistent with observations from a review of 11 European studies that concluded that sweet products contributed to a higher proportion of added sugar intake (40% to 50% for children and 36% to 61% for adults) than beverages did (20% to 34% for children and 12% to 31% for adults).Note 43

Various countries and jurisdictions have adopted strategies to improve diet quality or reduce energy intake. These include food guides, school nutrition policies, front-of-package labelling, social marketing and various forms of taxation.Note 4Note 25Note 44Note 45 Several studies have reported on the effectiveness of these interventions and the extent to which they resulted in a decrease in added and free sugar consumption.Note 26Note 46Note 47 However, other studies have shown that there remains plenty of room for improvement. For example, economic evaluations revealed that Canada would have avoided health care costs of approximately \$863 million in 2014 and \$830 million in 2018 if Canadians had avoided consuming sugar-sweetened beverages.Note 48Note 49 The present study’s added and free sugar value and consumption estimates may support future research into the health and economic costs associated with the consumption of these sugars. They may also inform potential intervention targets for added and free sugar consumption.

The present study has several strengths. It uses the 2015 CCHS – Nutrition, which is a large population-based survey that included the 24-hour recall data of more than 20,000 respondents, with repeat measures, to allow for the estimation of usual intake. A study limitation relates to the subjective judgment used in steps 6, 7 and 8, as this may have introduced error in the estimates of added and free sugar content. However, these steps did involve a limited number of foods and beverages: Figure 1 shows that they involved 475 (8.8%) of the 5,374 recorded foods and beverages when estimating added sugars and 397 (7.4%) of the foods and beverages when estimating free sugars. In addition, Louie et al.Note 17 reported high inter-rater repeatability for this methodology. Another limitation relates to the use of 24-hour recalls, which are prone to error, as is every dietary assessment method.

## Acknowledgments

This research was funded through an operating grant by the Canadian Institutes of Health Research (grant number 384559) to Paul J. Veugelers. This research was also supported by funds to the Canadian Research Data Centres Network (CRDCN) from the Social Sciences and Humanities Research Council, the Canada Foundation for Innovation, and Statistics Canada. Although the research and analysis are based on data from Statistics Canada, the opinions expressed do not represent the views of Statistics Canada or the CRDCN.

References

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# How to check if function is Lipschitz continuously differentiable?

I do have a problem with achieving convergence in Newton method (using Armijo rule) for system of algebraic non-linear equations. I suspect that my function is not continuously differentiable, however I'd like to be sure if that is so. How do I test it if my F( x ) is Lipschitz continuously differentiable?

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Are you referring to Lipschitz Continuity? –  Han Altae-Tran May 7 '12 at 8:32
Yes, I do. In general I know what is contuinity etc. But I cannot find an example with Lipschitz continuity in any of my math handbooks. –  Misery May 7 '12 at 8:42
Are you asking for a definition of Lipschitz continuity? –  Did May 7 '12 at 10:45
Yes, and how to testif my function is continous (after googling a little i know it isn't, but I need to prove it) –  Misery May 7 '12 at 10:59
If your function ${\bf x}\mapsto {\bf F}({\bf x})$ is some sort of explicit expression in terms of elementary functions it is continuously differentiable wherever bona fide defined; i.e., excepted in points where $\log0$, $\sqrt{0}$ and such things occur. –  Christian Blatter May 7 '12 at 12:52
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1. A function $f$ (either scalar or vector valued) is Lipschitz if there is a constant $L$ such that $\|f(x)-f(y)\|\le L\|x-y\|$ for all $x,y$. This is denoted by $f\in\mathrm{Lip}(U)$ or $f\in C^{0,1}(U)$ where $U$ is the domain of $f$.
2. A function $f$ is Lipschitz smooth or Lipschitz continuously differentiable if there is a constant $L$ such that $\|\nabla f(x)-\nabla f(y)\|\le L\|x-y\|$ for all $x,y$. This is denoted by $\nabla f\in\mathrm{Lip}(U)$ or $f\in C^{1,1}(U)$ where $U$ is the domain of $f$.
That is, the second condition is the Lipschitz continuity of first-order derivatives. A sufficient condition for $f\in C^{1,1}$ is that the second derivative of $f$ is bounded: see $C^{1,1}$ regularity.