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Lecture سخنرانی

Iterations and Limits, Complexity and Surveyability

Mojtaba Moniri  
Western Illinois University, USA  

Thursday, August 3, 2017,   14:00 - 16:00

VENUE   Lecture Hall 1, Niavaran Bldg.



The first theme, complexity reduction on large sets, is mainly based on part of a post-conference contribution made to the Proceedings of Frontiers in Mathematical Sciences, the Third Conference, at IPM, and the extensions in the journal version in Complex Systems. Examples of computationally simplifying some sequences of nonnegative integers are presented. The reduction might be at the cost of leaving out a set of exceptional inputs of zero or rather small density. Iterations \((a_m)_{m\in\mathbb{N}}\) of \(\sqrt{2+x}\) with specific initial values \(x\in[-2,2]\) are considered. Modulo base-4 normality of \(\frac{1}{\pi^2}\), when \(x=0\) and \(m\) is outside a set of density about \(\frac{1}{12}\), \(\lfloor \frac{1}{2-a_m}\rfloor\) equals \(\lfloor\frac{4^{m+1}}{\pi^2}\rfloor\); plus 1 on the exceptional set. Adding the second term of a series for \(\frac{1}{2-a_m}=\frac{1}{4}\csc^2(\frac{\pi}{2^{m+2}})\), it is asked whether any exceptions remain. Next, Wolfram sequences \(c\), of iterated \(\lfloor\frac{3}{2}x\rfloor\) starting at 2, \(s\) of their base-2 lengths and \(r_m=\min \{k |s_{k}\geq m\}\) are discussed. Under some conditions, including \(c\) not achieving a power of 2 greater than 4, \(r_m=\lfloor \frac{m}{\log_2(\frac{3}{2})}+\gamma \rfloor -1\) with \(\gamma\approx 0.0972\cdots\) expressible via an Odlyzko-Wilf constant. Unconditionally, \(\gamma\) can be removed if outside a set of density between 0.9027 and 0.9028, so is \(-1\). The next topic is on observance of a proven 5-periodicity, and is based on recent joint work with Saman Moniri. The logistic map with parameter $r\in (0,4]$ is defined on $[0,1]$ as $f_{r}(x)=rx(1-x)$. Although most parameters $r\in (3.570,4)$ lead to a chaotic orbit for an arbitrary $x_0\in (0,1)$, calculations involving Groebner bases show the first onset of 5-cycles to be $\approx 3.7382$ with first bifurcation at $\approx 3.7411$. For $r=3.74$ (resp. $r=3.742$), the existence of a $5$- (resp. $10$-) cycle can also be proved using Brouwer's fixed point theorem. When one tries to calculate the orbit for 3.74 using codes in Mathematica (or MATLAB, etc), there is a wrong impression of 10-periodicity. Lyapunov exponents might shed some light here. A negative value would indicate convergence of orbits of nearby points to a stable cycle. For 3.74, this is $\approx -0.1118$. The exact rational values have rapidly growing numerators. That of the 26th term has more than 134 million digits, by which no more than four digits are stabilized. Such indirect computer-assisted proofs are of philosophical interest too. In lieu of a current way of calculating the limiting cycles to two dozen digits, the mere 5-periodicity is surveyable.


تهران، ضلع‌ جنوبی ميدان شهيد باهنر (نياوران)، پژوهشگاه دانش‌های بنيادی، پژوهشکده رياضيات
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Niavaran Bldg., Niavaran Square, Tehran
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