12,943 reputation
11657
bio website go.helms-net.de
location Kassel, Germany
age 61
visits member for 4 years, 1 month
seen 7 hours 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.


Oct
26
comment Software, techniques and tricks of experimental mathematics to conjecture possible closed forms
I don't know whether this technique is not meant: I've often related values gotten by approximations, and the first step is then to look at the arithmetic or geometric progression and remove parts of the numbers to come nearer to the common core. This can help much to enter the correct path to go.
Oct
22
comment Is there a such a thing as a “log root” or perhaps a “power root”?
Perhaps you like this: there was a somehow "epical" article online, called "wexzal", where the authors discussed this number, its computation and its application/occurence in technical problems in width and length... I think a google search should be sufficient
Oct
19
comment Do these series converge to logarithms?
Well, I'll play a bit with it. If I find something I'll let you know here. I've seen things going interesting with matrices whose design reflects the primefactorization/divisors of the integers.
Oct
19
comment Do these series converge to logarithms?
Coming late to this question by your related one today from where you linked to here. Did you ever try to interpolate this to fractional indexes, say $\log(2.5)$ ?
Oct
16
comment Representing $i$ with divergent series
The powerseries for $ \sqrt{1-2x} = 1 - x - 1/2 x^2 - 1/2 x^3 - 5/8 x^4 - 7/8 x^5 - 21/16 x^6 - 33/16 x^7 - 429/128 x^8 + O(x^9) $ evaluated at $x=1$ ? It is divergent and "represents" the imaginary unit...
Oct
11
comment Number of possible cycles in collatz conjecture
Well - so it deserves a +1 now. :-)
Oct
11
comment Number of possible cycles in collatz conjecture
@Arthur: it is a well known approach to difficult proofs to establish first the question whether it is possible that infinitely or only finitely many solutions can exist, and then if it is -for instance- known that there cannot be infinitely many solutions to reduce the upper bound of possible solutions to arrive at zero. The early attempts to solve FLT show a similar structure of proceedings: to exclude specific exponents (3,5,7,...), then to exclude whole classes of exponents (possibly of infinite cardinality, Sophie Germain, Kummer) and so on.
Oct
11
comment Number of possible cycles in collatz conjecture
Well the link to the cycle-section in wikipedia deals with the proof, that cycles with certain structures cannot exist - the structures discussed have nothing to do with length: actually the proofs say, that such cycle-structures cannot exist for any length, for instance cycles of the structure "first only increases - then only decreases" without regard of how many increases preceed how many decreases.
Oct
5
comment Is there a problem for which it is known that the only solution is “iterative”?
Isn't the evaluation of a powerseries on argument $x$ also inherently an iterative process by increasing the powers of $x$?
Sep
28
comment Outline approach to Collatz 3n+1 conjecture / Criticism needed
Well, I think, if the separate trees each have a cycle at the root, may the cycle have 1,2,6 or more elements (in the "odd-numbers-only" version of counting steps, so we have the cycles $(1)$ $(-1)$ $(-5,-7)$ and one more including $(-13,-33...,-83)$ they have disjunct trees because the trajectories downwards are unique (but that's only reasoning, I've never had even the idea to contemplate along this way. As I write this, I recall that iterating some values along the sinh() as well as along its inverse asinh() it is possible to have the same attracting fixpoint, so I do not really know)
Sep
28
comment Outline approach to Collatz 3n+1 conjecture / Criticism needed
Yes; for instance if you include the negative numbers, we find three additional trees (not only one more, so in the negative numbers we find multiple cycles). If I recall it correctly, for the $5x+1$ problem we find three or four separate trees only in the positive numbers (didn't check the nagatives in that problem).
Sep
27
comment Inverting the infinite matrix $+\mathbf{I}$ with entries $\mathbf{P}_{ij}={i-1\choose j-1}$
@semiclassical: I mixed the adress for my comment making it pinging at g.kov; sorry. Rereading the thread I see it was you who wrote about polylogarithms. So please see my previous comment.
Sep
27
comment Inverting the infinite matrix $+\mathbf{I}$ with entries $\mathbf{P}_{ij}={i-1\choose j-1}$
@g.kov : the pattern is not difficult to recognize. for $r \ge c$ it is $(-1)^{r-c} \cdot \binom{r}{c} \cdot \eta(-(r-c)) $ where $\eta(x) $ is the alternating zeta-function and we find it for the nonpositve arguments having $1/2,1/4,0,-1/8,0,1/4,...$. The matrix-indices go from zero to infinity and $r$ is the row and $c$ is the column index.
Sep
27
comment Inverting the infinite matrix $+\mathbf{I}$ with entries $\mathbf{P}_{ij}={i-1\choose j-1}$
The series begins with $ 1/2 + 1/4 D - 1/48 D^3 + 1/480 D^5 - 17/80640 D^7 + 31/1451520 D^9 $ $ - 691/319334400 D^{11} +... $. The numbers $17,31,691$ in connection with the index in steps of 2 should ring some bell in direction of Bernoulli-numbers, or even better, of Euler numbers (see wikipedia). The remaining seems to be not too difficult...
Sep
21
comment Lambert- W -Function calculation?
wikipedia has a derivate containing code ("wikicode"? "Rosettastone"?). I found it easy to find a recursive programming-example, translatable into Pari/GP
Sep
19
comment Why is variance squared?
@Did: true. Anyway: for me this was something like "enlightement" when I realized this way of viewing things. Perhaps it is possible to even enhance the aspects which give it a "natural meaning" - like the identifying of the resultant of (physical) forces with that of (for instance:psychometric) items in factor analysis and regression et al. As I said in the first sentence: a completion/an additional view...
Sep
18
comment Convergence of power towers
Also the number $w = -s/4 \approx −0.316049330525... $ occurs in my answer mathoverflow.net/questions/71429/… in which I explained the application of the schröder-function a bit more.
Sep
18
comment Convergence of power towers
Let $s=2 \dot \exp(c_{\sqrt2}) \approx 1.26419732$. I've worked with this number some time ago; it occurs by the method which I mentioned in my earlier comment. Let $\sigma(x)$ be the schröder-function for the exponential to base $\sqrt 2$ centered around the fixpoint $t=2$. Then the same value occurs for $w=\sigma (x/t-1)$ at $x=1$ and (for some reason I don't remember) I looked for the value $-4 \cdot w$ which is equal to the number $s$ to the number of digits which you've provided. So your number $c_{\sqrt 2}$ has in some sense "canonical" relevance (and can be generalized to other bases)
Sep
17
comment Inverting the infinite matrix $+\mathbf{I}$ with entries $\mathbf{P}_{ij}={i-1\choose j-1}$
The most appropriate search might be the link go.helms-net.de/math/pascal/bernoulli_en.pdf
Sep
17
comment Inverting the infinite matrix $+\mathbf{I}$ with entries $\mathbf{P}_{ij}={i-1\choose j-1}$
If you might be able to observe that $P*V(x)=V(x+1)$ where $V(x)=$columnvector$(1,x,x^2,x^3,...)$ (for instance by simple pattern-detection, or even more clever, by the binomial theorem), then you're getting the key. Consider $(P+I)^ {-1}=I-P+P^2-P^3+P^4-...+...$ then $(P+I)^{-1} \cdot V(x)=V(x)-V(x+1)+V(x+2)-...+...$ .This leads you to the alternating (infinite) sum of powers of x, and after that to the Hurwitz-zeta and Bernoulli-polynomials and much more. Goggle for "helms" and "pascalmatrix" for more to find a couple of articles of mine about it.