Timeline for Existence of solutions to a nonlinear algebraic equation
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| Nov 30, 2015 at 0:51 | comment | added | Amin235 | I know what you mean professor P.Majer because I have been working on Fibonacci sequences for two years, thank you so much for you information. | |
| Nov 29, 2015 at 22:56 | comment | added | Pietro Majer | The latter equation for $u$ also comes from a generalization of the Golden Ratio ($p=1$): dividing a segment into $p+1$ sub-intervals that are in a geometric progression of $p+1$ terms, whose next term is the segment itself. Also, $u_p$ is the limit of the ratio of two consecutive terms $F_m/F_{m+1}$ of a linear recursive sequence $F_m$ where each term is the sum of the preceding $p+1$ terms e.g. starting with $1,1,\dots,1.$ emis.de/journals/JIS/VOL18/Szczyrba/sz3.pdf For instance, 0.5187900635… is the reciprocal of the "Tetranacci constant" oeis.org/A000288 | |
| Nov 29, 2015 at 20:16 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Nov 29, 2015 at 19:59 | vote | accept | Amin235 | ||
| Nov 29, 2015 at 19:06 | comment | added | Amin235 | I would appreciate your help in advance. | |
| Nov 29, 2015 at 19:03 | comment | added | Pietro Majer | That $u_p$ can't be larger than $1$ also follows immediately from the first equation for it, $1/u=1+u+\dots+u^p$. | |
| Nov 29, 2015 at 18:02 | comment | added | Pietro Majer | You're welcome! 1) The series for the solution is a very standard application of the Lagrange inversion formula to $f(u):=u(1- u^{p+1} /2 )$. Use the binomial series to expand $f^{-m}=u^{-m}(1- u^{p+1} /2 )^{-m}$, then find the residue at $0$ extracting the coefficient of $u^{-1}$. 2) Yes, write the equation as $u^p=2u-1$ and compare the values of both LHS and RHS at $0$, and their derivatives at $u=1$. Also, by convexity of $u^p$ there are at most $2$ positive solutions. | |
| Nov 29, 2015 at 17:03 | comment | added | Amin235 | Dear Professor P.Majer Thank you very much for your creative and complete answer . It was amazing for me your attitude to this nonlinear equation. I have two questions , First It is possible to explain how do you get this formula for $u_p$ from Lagrange inversion formula and second would you tell me why $0< u_p<1 $? I mean is there an elementary way to show that for every $p$, The equation $u^{p+2}-2u+1=0$, has a positive real solution between zero and one. Thank you again for your favor to answer to my question. | |
| Nov 29, 2015 at 16:21 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Nov 29, 2015 at 16:04 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Nov 29, 2015 at 14:07 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Nov 29, 2015 at 13:47 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Nov 29, 2015 at 13:35 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Nov 29, 2015 at 13:28 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Nov 29, 2015 at 13:23 | history | answered | Pietro Majer | CC BY-SA 3.0 |