12,564 reputation
11555
bio website go.helms-net.de
location Kassel, Germany
age 61
visits member for 3 years, 11 months
seen 33 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.


1d
comment Is there an algebra of summable series?
You should explain how you define multiplication $X=D_1 \cdot D_2$ of series $D_1$ and $D_2$: the termwise multiplication $x_k=d_{1,k} \cdot d_{2,k}$ ? The cauchy-product? Or the product of the partial evaluations up to some index $k$? And if the latter: of the original terms? Or of the terms after the transformation by the divergent summation-procedure?
2d
comment Is there an algebra of summable series?
ABC, please see my comment which I put in an answerbox because it is too long for the commentfunction
2d
comment How to sum arithmetic combinations of divergent zeta-values?
@Max: perhaps this small treatize on the representation of the Euler-MacLaurin-formula in terms of a matrix-operation might be interesting, because the Euler-MacLaurin-formula is at the heart of the Ramanujan-summation. (Having this in a matrix form one might say that the "Ramanujan-summation is a linear operator" .) see: go.helms-net.de/math/binomial_new/EulerMacLaurin.pdf
2d
comment If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?
Max, just after my email I searched for the link to H.W.Gould here in MSE and here it is (didn't check whether the referenced homepage is still alive) see: math.stackexchange.com/questions/742083/…
2d
comment If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?
ad bonus-question: do you know the collection of combinatorical and series identities of H.W. Gould? I've about 8 pdf's but didn't evaluate them so far.
Aug
20
comment Does the inverse of this matrix of size $n \times n$ approach the zero-matrix in the limit as $\small n \to \infty$?
I've got an answer in MO: see mathoverflow.net/questions/104370 which satisfies my needs. I'm accepting the answer below to "close the case"
Aug
16
comment Iteration of $x \to x^x$
If I recall correctly then Heryk Trappmann had made his PhD thesis (in german) about symmetric algebraic operations, and $x \to x^x$ was one of its subjects. Try to find it via "Tetration Forum" (Sorry, I'm lazy to search it for you)
Aug
15
comment Can $p^{q-1}\equiv 1 \pmod {q^3}$ for primes $p<q$?
For the infinitude of solutions when $p \lt q$ is removed and for the general case where p is not prime this heuristical article might be interesting go.helms-net.de/math/expdioph/fermatquot_ge2_table1.htm
Aug
13
comment Integer partitions without rotated solutions?
@hardmath: Possibly I should say better "cyclic shift" instead "order" which does not matter. So [1,2,0,3] is different from [2,1,0,3] but not from [3,1,2,0]
Aug
13
comment Integer partitions without rotated solutions?
@hardmath: hmm, a quick view into your link let's it ambiguous for me. Compositions seem to give longer lists, while my list is based on partitions and then furtherly reduced. Don't know yet whether I can make compositions properly restricted. But thanks for the hint!
Aug
11
comment Plotting an Ellipse after an Ellipse Fit
I have a problem with the reconstruction of the procedure in the mentioned article. In (eq 28) there is the explicite condition that $a_1^T \cdot C_1 \cdot a_1 = 1$ which requires a correction by division by about $288443.686$ and then the first resulting coefficient $a \ne 1$. However, to reproduce your data/function, it seems to be required that $a=1$ instead, such that the condition involving $C_1$ does not hold. What's going on here?
Aug
9
comment Weighing correlation by sample size
Hm, I don't know whether standard dev is the best. I'm toying with the idea to use the likelihood of finding "such a result" in "such a big sample" (where "sample" is the number of coins or gems in a tomb). Then if there are only 2 gems there, then by the binomial distribution we had p=0.25 that 2 of the 2 gems are saphire. So the null-hypothesis for each sample/tomb is, that the number of saphires is just the mean category of the binomially expected frequency etc. Perhaps then even the stderror is a useful measure/interval...
Aug
9
comment Limit of differences of truncated series and integrals give Euler-gamma, zeta and logs. Why?
@David: corrected, thanks!
Aug
9
comment Limit of differences of truncated series and integrals give Euler-gamma, zeta and logs. Why?
@DavidH: hmm. True ;-) after I've realized your remark. Well, for m=1 the separate limits were divergent, but well, your argument shows I should have thought over it a longer time. Anyway, I should look again at some graphic representation for the integral/series difference, which I've seen elsewhere to refresh my intuition for this
Aug
9
comment In calculus, which questions can the naive ask that the learned cannot answer?
@MichaelHardy: Your identity looked a bit familiar but made me curious whether this could be generalized, and I toyed around a bit... with again curious heuristics. See my new question math.stackexchange.com/questions/891918/…
Jul
30
comment General method to “naturally interpolate” to a complex map?
"...overkill..." true, and it introduces also new difficulties (represention of the function as power series, range of convergence and so). So I think we could leave it with the ansatz that you've chosen.
Jul
29
comment General method to “naturally interpolate” to a complex map?
...fairly general concept, but before I start explaining it you should make sure, that I've got your problem correctly at all. (If this is correct, then instead of "interpolation" a better tag would possibly be "dynamical systems" and "fractional iteration")
Jul
29
comment General method to “naturally interpolate” to a complex map?
If you look for a "more general" way to interpolate an operation between so to say: 0 times applied ... 1 time applied or in other words: "fractional iteration" then the use of Carleman-matrices might be of fundamental help: fractional iterates of an analytical function can sometimes be performed/approximated by fractional powers of a Carleman-matrix for the specific operation. Here the Carleman-matrix contains the coefficients of the power series of your function (and of its powers), and powers of the Carleman-matrix the coefficients of the iterates of the function. It's a ...
Jul
16
comment Calculating a Factorial Base Representation
very cool idea. It needed two times reading for me to understand that and why this is so easy. Wow.
Jul
16
comment Example of two convergent series whose product is not convergent.
Hmm, the two series are only conditionally convergent. Are there also examples where they are absolutely convergent?