13,561 reputation
11859
bio website go.helms-net.de
location Kassel, Germany
age 62
visits member for 4 years, 4 months
seen 58 mins 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.


3h
revised Collatz conjecture: Largest number in sequence with starting number n
added 744 characters in body
3h
answered Collatz conjecture: Largest number in sequence with starting number n
22h
answered “Orthogonal” Rectangular Matrix
22h
comment “Orthogonal” Rectangular Matrix
Only if $m \ge n$, otherwise some entries on the diagonal of $I_n$ are zero... So surely this condition should be included in the definition of the problem.
Jan
24
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
23
revised Is there a systematic relation between the generating functions for the rows vs that for columns of infinite sized Carleman-matrices?
added background, reference
Jan
23
revised Is there a systematic relation between the generating functions for the rows vs that for columns of infinite sized Carleman-matrices?
added background, reference
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
20
revised Find all matrices where the matrix is its own inverse and the determinant is 1
added 127 characters in body
Jan
20
answered Find all matrices where the matrix is its own inverse and the determinant is 1
Jan
19
revised Is there a systematic relation between the generating functions for the rows vs that for columns of infinite sized Carleman-matrices?
added tag "reference request"
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
15
revised Find $f \in B[0,2] $ and $g$ Riemann-Stieltjes integrable on $[0,1]$ and $[1,2]$ but not in $[0,2]$
typo
Jan
14
revised a doubt with the series $ \sum_{n=0}^{\infty}e^{-nx} $
added tag
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
14
answered Understanding infinity
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 (...)