Comments - New Decimal Systems - Great Sandbox for Data Scientists and Mathematicians - Data Science Central2021-09-18T08:26:41Zhttps://www.datasciencecentral.com/profiles/comment/feed?attachedTo=6448529%3ABlogPost%3A714687&xn_auth=noI came across a very technica…tag:www.datasciencecentral.com,2018-04-30:6448529:Comment:7164382018-04-30T17:10:24.520ZVincent Granvillehttps://www.datasciencecentral.com/profile/VincentGranville
<p>I came across a very technical paper entitled <em>Combinatorial and probabilistic properties of systems of numeration</em> (37-pages long, published in 2016 in <em>Ergodic Theory and Dynamical Systems</em>, see <a href="https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/combinatorial-and-probabilistic-properties-of-systems-of-numeration/2CA78270C8B16CEACF9753FA9D6BB8E4" rel="noopener" target="_blank">here</a> -- a PDF version is available…</p>
<p>I came across a very technical paper entitled <em>Combinatorial and probabilistic properties of systems of numeration</em> (37-pages long, published in 2016 in <em>Ergodic Theory and Dynamical Systems</em>, see <a href="https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/combinatorial-and-probabilistic-properties-of-systems-of-numeration/2CA78270C8B16CEACF9753FA9D6BB8E4" target="_blank" rel="noopener">here</a> -- a PDF version is available <a href="https://www.math.tugraz.at/~grabner/Publications/Erda.pdf" target="_blank" rel="noopener">here</a>.) However, despite my PhD in math (I even published in <em>Journal of Number Theory</em>) I could not make sense of it: It is mostly a list of complex theorems with long proofs and many, many equations. I am wondering what is the overlap between that paper, and my findings here. Clearly, they use dynamical systems for their framework, and so do I (the logistic map is usually studied in the context of dynamical systems.) Also they spend a lot of time discussing what their call the <em>invariant measure</em> of these systems, which I assume is equivalent to my "equilibrium distribution." They focus on uniqueness and existence of this measure, and on ergodic properties. I do too here, but from an applied point of view (though I also use the word <em>ergodic</em> at one point.) Some of my systems do not seem to be covered by their general theory (they say the number zero always has all its digits equal to zero in their framework, but my systems do not have this kind of limitation.) They also discuss slow and fast growing systems: My example in section 3 illustrates slow growth, while the example in section 4 illustrates fast growth. Many of my systems (including base-<em>b</em>, even if <em>b</em> is not an integer, continued fractions, or nested square roots) implicitly use what is called the <em>greedy algorithm</em> to compute the digits; they also mention the greedy algorithm throughout their article. Can someone help me understand what they are actually talking about (compared to what I am doing here?)</p>
<p>The same team (or at least researchers benefiting from the same <a href="http://www.agence-nationale-recherche.fr/Project-ANR-12-IS01-0002" target="_blank" rel="noopener">funding source</a>) wrote another interesting yet technical document entitled <em>Fractals arising from numeration and substitutions, </em>see <a href="https://www.math.tugraz.at/~minervino/innsb13.pdf" target="_blank" rel="noopener">here</a>. It is the first time that I see the base-<em>b</em> system, with <em>b</em> not an integer, mentioned, besides my own research. I also mention fractals in my article (see section on Brownian motions) but none of these authors mention the stochastic integral equation, and none of them truly discuss probabilistic features of these systems. Anyway, I could not resist posting their nice fractal from their PDF document, see below:</p>
<p><a href="https://storage.ning.com/topology/rest/1.0/file/get/2773308719?profile=original" target="_self"><img src="https://storage.ning.com/topology/rest/1.0/file/get/2773308719?profile=original" width="300" class="align-center"/></a>See also <a href="https://en.wikipedia.org/wiki/Complex-base_system" target="_blank" rel="noopener">here</a> and <a href="https://en.wikipedia.org/wiki/Positional_notation" target="_blank" rel="noopener">here</a> (Wikipedia.)</p> Hi Emil,
To answer your quest…tag:www.datasciencecentral.com,2018-04-27:6448529:Comment:7160042018-04-27T02:25:39.118ZVincent Granvillehttps://www.datasciencecentral.com/profile/VincentGranville
<p>Hi Emil,</p>
<p>To answer your question, my first reaction is to say that they only have disadvantages. However, I could see them useful in some encryption systems, in the sense that they add some kind of random noise to your message, and each digit provides little information if the base is smaller than 2. It comes with massive auto-correlations, but no matter what your message is, the auto-correlation structure is the same, and thus possibly not a big weakness for hackers trying to…</p>
<p>Hi Emil,</p>
<p>To answer your question, my first reaction is to say that they only have disadvantages. However, I could see them useful in some encryption systems, in the sense that they add some kind of random noise to your message, and each digit provides little information if the base is smaller than 2. It comes with massive auto-correlations, but no matter what your message is, the auto-correlation structure is the same, and thus possibly not a big weakness for hackers trying to decipher your message. It comes with a cost: it requires additional computational power to encrypt / decrypt etc. </p>
<p>If you use a transcendental number for the base, it could make the task of hackers much harder. </p>
<p>Vincent</p> What is the advantage of a nu…tag:www.datasciencecentral.com,2018-04-27:6448529:Comment:7158662018-04-27T01:50:31.998ZEmil M Friedmanhttps://www.datasciencecentral.com/profile/EmilMFriedman
<p>What is the advantage of a number system that uses a non-integer or even a transcendental base?</p>
<p>What is the advantage of a number system that uses a non-integer or even a transcendental base?</p>