13,551 reputation
11859
bio website go.helms-net.de
location Kassel, Germany
age 62
visits member for 4 years, 4 months
seen 1 hour ago

Mathematics is only a hobby - although I have done undergrad courses in the 70s. But part of my job was doing statistics and this kept me near my favourite subject "linear algebra" (programmed factor-analysis and a matrix-orientated calculator MatMate). Around 2002 I came in contact with the internet community in math newsgroups and could improve my Collatz discussion. Next subject was the Bernoulli numbers, then integer matrices and since 2006 the problem of iterated exponentiation aka tetration. Serving half-time jobs in teaching here at the university I found time to fiddle with that subjects in depth - and found love with the exploratory approach and impulse of the 18'th century numbertheory, namely L. Euler, "the master of us all"... Due to lack of formal education I've to do my "research" widely on my own - but that's what I just like: to find structure, pattern, laws from the ground.


1d
comment How can I solve this equation $x^{x^{x^{x^{.^{.^{.}}}}}}-a=0$
Perhaps it is of interest to do the generalization to complex numbers. Then your mentioned interval becomes a region of convergence called "Shell-Thron-region" (see wikipedia)
Jan
21
comment Finding matrix exponential
One should possibly add in the introductory remark, that the block-composition is one of the diagonal form, so the blocks do not interfere when -for instance- the power series of A is evaluated (which we thus do not need to do actually, but only that blocks on the diagonal, as you've done it)
Jan
21
comment Efficient Method/Algorithm to Compute the P-adic Ordinal of a Positive Integer
Hmm, more concrete to your concern of actual computation: if n is "small", as you say, why not simply do repeated trial division by the interesting $p$? In a spreadsheet like excel you can assign a visual-basic routine to any cell, and in visual basic it is easy to program such a function for n less than, say, 2^32 or the like. Also, what size has $a$ usually in your problem?
Jan
21
comment Efficient Method/Algorithm to Compute the P-adic Ordinal of a Positive Integer
I don't know whether this is of help: I've worked with that problem myself and made a little "algebra" with it in the pdf-file linked at "Mersenne numbers, cyclic subgroups ... " at my math-site go.helms-net.de/math/index.htm Perhaps it gives at least some more idea/suggestion for the formalization of your problem and then a better implementation of the computation (I do things always in Pari/GP and sometimes in Excel)
Jan
18
comment Why is it so difficult to obtain the spectral properties related to infinite matrices, especially when they are not symmetric?
@let-me-out: True, and even more: it seems that the diagonalization need not be unique - we had a short discussion about this in the tetration-forum, but I couldn't follow that discussion properly for a longer thread then (so I also don't have an example at hand) I think it was something about the relation of the trace of a matrix to its eigenvalues for the case of finite dimension and this was taken as expectation for the same relation with an infinite-sized Carleman-matrix. I only remember something ambiguous due to this concept...
Jan
14
comment Why we cannot ascribe values to behavior of functions at poles?
If one rewrites $(-2)! = (-1)!/(-1)$, $(-3)! = (-1)!/(-1)/(-2)$, etc., and thus reduces the formal expressions of the factorials/gammas at the negative integers to the basis of the formal entity $\psi=(-1)!$ then the above regularizations give easily expressions in terms of $\psi$ with the contents, that $\psi$ can be seen alternatively as multivalued form requiring $\gamma$ and the set of scaled Stirling-numbers first kind $s1_k \cdot 2! / k! $ which we can find by the generating function $\log(1+x)^2$ . (I don't know/claim whether this adds anything useful to your observations)
Jan
12
comment Fixed point and fractional iteration: if $F(k)=k$ then $F^{1\over n}(k)$ is another fixed point of $F$
(...) possibly stop now from such repeated "caveat caveat" warnings. Because I even don't know whether there is real substance in it... :-/
Jan
12
comment Fixed point and fractional iteration: if $F(k)=k$ then $F^{1\over n}(k)$ is another fixed point of $F$
As at the notation for tetration: yes, that's a correct notation I think. With the "fractal" and "equator" curve the aspect which triggered me, was, that seemingly between two integer iterates of the $\Psi$-function (a fractional iteration of the tetration) the curve might get some break out to a singularity which in turn would force us to make sure, that no anormality like this (or possibly others) would destroy our nice formula. Of course, such an "equator"-curve might only occur with such special bases $b$ in tetration on the border of the Shell-Thron-region. But I think I should (...)
Jan
11
comment Proving that $1+2^m+3^m+…n^m$ is divisible by $(n+1)$
Isn't that very closely related to the (unsolved) Agoh/Giuga-conjecture?
Jan
11
comment Fixed point and fractional iteration: if $F(k)=k$ then $F^{1\over n}(k)$ is another fixed point of $F$
(...) your mentioned/assumed properties possibly do no more hold. I'd marked that discussion with the term "equator" and made some pictures for the tetration-forum, where also Sheldon made some nice comments. Perhaps this is interesting as well. See math.eretrandre.org/tetrationforum/… for instance, but all posts around it are related. Closing remark: sorry that the discussion of tetration comes so dominant here, I found it surely interesting for the more general case of iterated functions as properly adressed by your question.
Jan
11
comment Fixed point and fractional iteration: if $F(k)=k$ then $F^{1\over n}(k)$ is another fixed point of $F$
One more reminder. Of course, if we have $F$ has a fixpoint, say $F(t_0)=t_0$ and $\Psi$ as some $1/n$ iterate of $F$ with the property that $t_1 = \Psi(t_0) \ne t_0$ but $\Psi^{on}(t_0)=t_0$ then what you say should be obvious, and a set of points $t_0,t_1,...,t_n=t_0$ occurs. But even more should be expected for the generalization of $\Psi$ for continuous iterations: then instead of a set of points $t_0,t_1,...,t_n=t_0$ we should have a continuous curve (if in the complex plane). I considered one time a bizarre effect when then curve became fractal (and possibly some of (....)
Jan
11
comment Fixed point and fractional iteration: if $F(k)=k$ then $F^{1\over n}(k)$ is another fixed point of $F$
MphLee, sorry, it's just over my head at this days; I just wanted to give that reminder of an older discussion (which also had let me headscratching until today... ;-)) I hope someone else shall help out here.
Jan
11
comment Proof of Andrica when Assuming Oppermann
@daniel: ok, thank you! (I'm getting away for now)
Jan
10
comment Is this property of the Collatz sequence interesting?
I've just deleted my answer because it does not more than to restate your question. However, to make the focus of your question a bit more obvious, it might be useful to introduce the known term "total stopping time" into your text, and formulate something like: a) assume $s(n_0)$ being the "total stopping time" for $n_0$ under the original Collatz-map and consider the iteration $n_{k+1} = s(n_k)$ . b) Is my conjecture true that the sequence of $n_0,n_1,n_2,...$ converges to $1$ for all $n_0 \gt 1$ ? (Or some other wording of your conjecture in this spirit)
Jan
10
comment Proof of Andrica when Assuming Oppermann
@daniel: not necessarily. Consider there is a modular argument which tells us that between $m=n^2$ and $m-\sqrt m = n^2-n $ there is a prime, but that this does not need to hold if $m$ not a square. Then by Andrica, we need a smaller range than $m+\sqrt m$ for the next prime to occur. But, well, I've definitely not yet stepped into this sufficiently deep...
Jan
10
comment Proof of Andrica when Assuming Oppermann
I have now one concern. Oppermann's conjecture focuses on the interval around a square-number $n^2$ . So this conjectures only for the case, that the center $m=n^2$ of the interval [$m-\sqrt m$ .. $m+\sqrt m$] is a square. But what if Oppermann is true for such cases, but untrue for cases, where the center $m$ is not a square-number? (I don't know actually whether this is possibly a non-issue, didn't step into the conceptual part of question too deeply)
Jan
10
comment Proof of Andrica when Assuming Oppermann
I don't know why @daniel s answer was deleted? My "hmm" was just a keyboard-reflex, that I was thinking... Just - I would have had expanded the answer by including, that by the truth of the square-root-inequality for $n \ge 1$ the OP's question was completely resolved.
Jan
9
comment Converting a matrix to the nearest positive definite matrix
What I've sometimes done was simply to do the Cholesky-algorithm and to truncate possible tiny negative eigenvalues to zero (but that was only possible because I myself had the hands on the programming of the Cholesky-routine...) and possibly Robert Israel's proposal is a better one.
Jan
9
comment Is composition associative?
At least additive operations in the exponent of complex numbers (as bases) are not associative (in the terminology of that addition as functional composition)
Jan
9
comment Fixed point and fractional iteration: if $F(k)=k$ then $F^{1\over n}(k)$ is another fixed point of $F$
I don't know whether this is relevant, but I've one time stumbled on the problem of self-intersecting flows in the iteration of the exponentiation, which means in some mysterious way, that the same complex point could be arrived by different iteration-heights (and so should be a fixed point) but also been left on different pathes. I do not yet understand that phenomen (if it were one at all, I still scratch my head). It was for instance discussed on this post in the tetration-forum math.eretrandre.org/tetrationforum/… and other places.