User markus redeker - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T20:40:39Z http://mathoverflow.net/feeds/user/22882 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/125148/are-there-results-in-digit-theory Are there results in "Digit Theory"? Markus Redeker 2013-03-21T10:46:41Z 2013-03-27T11:08:40Z <p>Results about numbers that are related to their decimal representation are usually confined to recreational mathematics. There I have seen mainly questions about individual numbers, like finding a number that is the sum of the cubes of its digits. But I have not yet seen a systematic study of such questions.</p> <p>A result that prompted me to asked this question was the following from Kurt Hensel's "<a href="http://www.gutenberg.org/ebooks/38986" rel="nofollow">Zahlentheorie</a>". There he shows that the exponent of highest power of a prime $p$ that divides the number $n$ is $$\frac{s_{n-1} - s_n + 1}{p - 1},$$ where $s_n$ is the sum of the digits in the representation of $n$ with base $p$. From this he derives that the highest power of $p$ that divides $n!$ is $$\mu_n = \frac{n - s_n}{p - 1}.$$ This is a different and much simpler formula than the representation $\mu_n = \sum_{k = 0}^\infty \left\lfloor \frac{n}{p^k} \right\rfloor$ by Legendre, which I usually see in the literature.</p> <p>I would like to know whether there are more results of this type. To be more precise, I define the not-yet-existing field of "Digit Theory" as the study of number-theoretic questions with help of the positional representation of numbers, and the study of the positional representation itself.</p> <blockquote> <p>So are there already systematic treatments of "Digit Theory" or of parts of it? Are there other interesting results like that one above?</p> </blockquote> http://mathoverflow.net/questions/123034/visualizing-a-graph/124407#124407 Answer by Markus Redeker for Visualizing a graph Markus Redeker 2013-03-13T12:24:29Z 2013-03-13T12:24:29Z <p>You could try <a href="http://www.yworks.com/en/products_yed_about.html" rel="nofollow">yEd</a>. It can import data in various formats, and there is an <a href="http://www.yworks.com/en/imageviewer.php?img=../img/ygucd/spc.png&amp;album=ygucd&amp;fs=1" rel="nofollow">image</a> on their web site of a graph with 13,500 nodes and 26,000 edges, which is much larger than your graph.</p> http://mathoverflow.net/questions/123081/should-one-attack-hard-problems/123088#123088 Answer by Markus Redeker for Should one attack hard problems? Markus Redeker 2013-02-27T10:00:24Z 2013-02-27T10:00:24Z <p>When thinking about attacking a hard problem one should ask: Do I have a tool or an idea that other people who attacked the problem did not have? If so, one should give it a try. If not, one will be most probably not better than others.</p> http://mathoverflow.net/questions/63321/what-are-some-examples-of-chimeras-in-mathematics/98180#98180 Answer by Markus Redeker for What are some examples of "chimeras" in mathematics? Markus Redeker 2012-05-28T09:16:56Z 2012-05-28T09:16:56Z <p>In quantum mechanics, the set of possible energies for a system of two attracting particles, say an electron and a proton, consists of a discrete part (the bound states) and a continuous part (the unbound states).</p> <p>So this is quite a "natural" example. To make it more "mathematical" one can express it as an eigenvalue problem.</p> http://mathoverflow.net/questions/97949/generalised-de-bruijn-graph Generalised de Bruijn Graph Markus Redeker 2012-05-25T14:41:45Z 2012-05-25T14:41:45Z <p>I have encountered sets of the following type, consisting of words over a finite aphabet $A$.</p> <p>If $S$ is such a set, then</p> <ol> <li><p>$S$ is finite,</p></li> <li><p>No word in $S$ is part of another element of $S$, and</p></li> <li><p>every infinite sequence $\dots a_{-2} a_{-1} a_0 a_1 a_2 \dots$ of elements of $A$ can be <em>covered</em> by elements of $S$ in the following sense: for every $k \in \mathbb Z$ there are numbers $i$, $j$ with $i \leq k \leq j$ such that $a_i a_{i + 1} \dots a_j \in S$.</p></li> </ol> <p>Now I take the elements of $S$ as vertices of a directed graph $G$ with loops; an edge between two words $w_1$, $w_2 \in S$ exists exactly then when there is a sequence of elements of $A$ in which $w_1$ is followed directly by $w_2$.</p> <p>If $S = A^n$, the set of all words of length $n$, then $G$ is a de Bruijn graph. But what about the general case? Has anyone written about it, or do you know anything interesting about such graphs?</p> http://mathoverflow.net/questions/93947/notation-for-ends-of-a-string Notation for ends of a string Markus Redeker 2012-04-13T09:34:09Z 2012-04-14T08:20:32Z <p>I work now a lot with strings of characters and other finite sequences and found that I need many times a good notation for "cutting the end" a string. If $a$ is a finite sequence and $a'$ is its right end, then there is a sequence $x$ such that $a = x a'$ (where the product is defined by concatenation). What I would like to have is a suggestive notation for $x$: $x$ might be $a$ "minus" $a'$, or $a$ "divided by $a'$, if we view the set of all strings as a semigroup. I have therefore thought of writing $x$ as $a / a'$ but am not completely happy with it. Obvoiusly I need also a notation for "cutting at the left end"; by continuing with the previous idea the notation $a' \setminus a$ could stand for the solution of $a = a' x$, but it collides a bit with the notation for set difference.</p> <p>So far I have thought, and my question is: Is there a good notation for these concepts already in use in some areas of mathematics or computer science, or has someone else already defined a suggestive notation?</p> <p><em>P.S.</em> And I also would like to have a notation for the "overlapping concatenation" of strings: If $a = a'x$ and $b = xb'$, what is $a'xb'$?</p> http://mathoverflow.net/questions/28758/uppercase-point-labels-in-high-school-diagrams-from-euclid/28855#28855 Comment by Markus Redeker Markus Redeker 2013-04-06T10:46:43Z 2013-04-06T10:46:43Z This is because in complex analysis and differential geometry a point is also a variable (in $\mathbb C$ or $\mathbb R^n$, respectively), and such variables are by convention lowercase. http://mathoverflow.net/questions/125148/are-there-results-in-digit-theory/125466#125466 Comment by Markus Redeker Markus Redeker 2013-03-25T09:45:54Z 2013-03-25T09:45:54Z Thanks for bringing Bunjakovskiĭ to our attention. Do you refer to the first or the second formula? It seems that &quot;digit theory&quot; has existed for a long time in the mathematical underground, without ever becoming really respectable. http://mathoverflow.net/questions/125148/are-there-results-in-digit-theory Comment by Markus Redeker Markus Redeker 2013-03-22T09:57:01Z 2013-03-22T09:57:01Z This of course requires a simple direct proof for the formula for $\mathrm{ord}_p(n!)$. Do you know one? http://mathoverflow.net/questions/123081/should-one-attack-hard-problems/123088#123088 Comment by Markus Redeker Markus Redeker 2013-03-11T11:11:33Z 2013-03-11T11:11:33Z Yes, but he started to work at the problem when it became known that there was a connection. At this point he knew that he had an approach that earlier researchers had not. http://mathoverflow.net/questions/123081/should-one-attack-hard-problems/123088#123088 Comment by Markus Redeker Markus Redeker 2013-02-27T13:18:02Z 2013-02-27T13:18:02Z @Frank: But you can often see that a method is new. Examples are Andrew Wiles' use of elliptic curves for Fermat's problem or Heinrich Heesch's idea of using computers to attack the Four Colour Problem. http://mathoverflow.net/questions/42929/suggestions-for-good-notation/43011#43011 Comment by Markus Redeker Markus Redeker 2013-01-24T18:38:23Z 2013-01-24T18:38:23Z @Rasmus: Your notation will become a bit less useful once $a$ and $b$ are replaced by more complex terms. In that case Dijkstra's notation is better. http://mathoverflow.net/questions/42929/suggestions-for-good-notation/109747#109747 Comment by Markus Redeker Markus Redeker 2013-01-24T18:32:22Z 2013-01-24T18:32:22Z This can also be done by writing $\int_x^{2\pi} dx\, \sin x$ etc., which is a little bit shorter. I have often seen this in physics. http://mathoverflow.net/questions/119455/visualizing-polyhedra-from-their-1-skeletons Comment by Markus Redeker Markus Redeker 2013-01-21T11:44:05Z 2013-01-21T11:44:05Z Hexahedral graph 5: Take a tetrahedron, glue on one of its faces another tetrahedron and on that one a third tetrahedron, such that the three tetrahedra have one edge in common. Visualisation worked for me because I recognised the graph of the tetrahedron in the heahedral graph. This may be part of a technique. http://mathoverflow.net/questions/116744/statements-which-were-given-as-axioms-which-later-turned-out-to-be-false/116759#116759 Comment by Markus Redeker Markus Redeker 2012-12-22T13:53:50Z 2012-12-22T13:53:50Z &quot;Condensation of Singularities&quot; was invented by Hermann Hankel. Basically it is the following: You have a function $f$ that has a singularity at 0. Make a periodic function $g$ from it that has this singularity at every integer. Then form the sum $G(x) = \sum_{n=0}^\infty a_n g(n x)$, with the $a_n$ chosen such that the sum converges. Then $G$ has this singluarity at every rational point. This can be done with mny types of singularities, provided that they are well-behaved enough under addition. http://mathoverflow.net/questions/116649/on-similar-concepts-in-mathematics-whose-similarity-is-a-non-trivial-fact/116776#116776 Comment by Markus Redeker Markus Redeker 2012-12-19T13:59:15Z 2012-12-19T13:59:15Z What are the conditions on $p$ and $q$ in (4)? http://mathoverflow.net/questions/115947/at-what-point-does-number-theory-stop-playing-with-finite-rings/115950#115950 Comment by Markus Redeker Markus Redeker 2012-12-10T16:33:05Z 2012-12-10T16:33:05Z This does not really qualify, because Euler re-proved an older result. But didn't he get more results from his proof than Euclid's theorem? http://mathoverflow.net/questions/93947/notation-for-ends-of-a-string Comment by Markus Redeker Markus Redeker 2012-04-14T08:24:42Z 2012-04-14T08:24:42Z @Joel: Thanks for the hint - I have added the tag.