1,926 reputation
1518
bio website go.helms-net.de
location Germany
age 61
visits member for 4 years, 3 months
seen 1 hour ago
Mathematics is only a hobby - although I have done undergrad courses in the 70ties. 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-term 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 Semigroup nilpotents and compostional inversion
(...contd) I've just recently had the case, that the matrix-inverse had to be expressed by $ C^{-1} = {I \over I-B}$ where $B=I-C$ and thus $C^{-1}= I + B + B^2 + B^3 + ... $ and in the case that $C$ is triangular with unit diagonal, $B$ is nilpotent and thus the nilpotency might appear in your problem. However - this is just an idea along which I'll try to reformulate and analyze your problem at the moment. (But likely it'll take some more time until I've more and, of course, if that ansatz shall show up as helpful at all...)
1d
comment Semigroup nilpotents and compostional inversion
I've still not finished my re-analysis of the problem. But maybe this observation helps already. I'm used to try to express the functional whereabouts in terms of Carlemanmantrices, which provide the framework of matrix-algebra for the analysis of composition of functions, especially in the case which you describe here (so this might be fruitful). Now we look at the inverse of a function which also means the inverse of a carlemanmatrix. ... (cont)
Oct
21
comment How did the summation operation come into use?
Hmm, isn't that question (and that answer as well) more appropriate designed for wikipedia?
Oct
6
comment The difference between Principal Components Analysis (PCA) and Factor Analysis (FA)
(...) After we've chosen the FA-model, and decided to respect the itemspecific variance: then we have the option to define (from theory or some earlier study) the amount of that part of variance -as large as possible, or possibly even zero or also slightly correlated. In fact (and is little known) the programmed estimation of the itemspecific error (based on the inversion of the covariance matrix) does not give the maximal possible sum of item-specific error-variance; by manually experimenting with this (when I've studied the method again) I could arrive at higher sums.
Oct
6
comment The difference between Principal Components Analysis (PCA) and Factor Analysis (FA)
The focus of the first sentence (1) and that of the enhanced-formatted one (2) are diametral. In fact it is the statistical model which gives the basis for the correct choice. In (2) it is correctly added that the covariance matrix admits both models. But then proceeds with a "rank"-argument (a purely mathematical one) instead of the statistical argument. (...)
Oct
4
revised Trouble with Jordan form of the truncated Carleman-matrix for $\sin(x)$ as size $n$ goes to infinity
edited body
Oct
4
revised Trouble with Jordan form of the truncated Carleman-matrix for $\sin(x)$ as size $n$ goes to infinity
edited body
Oct
4
revised Trouble with Jordan form of the truncated Carleman-matrix for $\sin(x)$ as size $n$ goes to infinity
added 650 characters in body
Oct
4
revised Trouble with Jordan form of the truncated Carleman-matrix for $\sin(x)$ as size $n$ goes to infinity
added 650 characters in body
Oct
4
comment Trouble with Jordan form of the truncated Carleman-matrix for $\sin(x)$ as size $n$ goes to infinity
@ChristianRemling : When I asked this I was relatively new to the method of Jordan decomposition and the question "how to explain" should have better been asked as "how to understand". After much reading and exercising I think I've understood this now.
Oct
4
revised Trouble with Jordan form of the truncated Carleman-matrix for $\sin(x)$ as size $n$ goes to infinity
added 1051 characters in body
Oct
4
revised Trouble with Jordan form of the truncated Carleman-matrix for $\sin(x)$ as size $n$ goes to infinity
added 1051 characters in body
Oct
4
accepted Trouble with Jordan form of the truncated Carleman-matrix for $\sin(x)$ as size $n$ goes to infinity
Oct
4
revised Trouble with Jordan form of the truncated Carleman-matrix for $\sin(x)$ as size $n$ goes to infinity
added 722 characters in body
Oct
1
comment Summation mollifier to ensure a certain alternating series has the correct value
Very well; I use Euler-summation with configurable order. The order must be adapted: when the series to mollify is $1-1+1-1...$ order 1 is needed, when it is $1-3+9-27+81-...+...$ order 3 is needed and so on. If I see it correctly the current proposal is the matrix (or a matrix very near to that) for Euler-summation of order 1. But after your last formula it seems that you can circumvent quotients of absolute values $ \gt 1$, so that matrix should sufffice for all cases and no additional consideration is needed.
Oct
1
comment Summation mollifier to ensure a certain alternating series has the correct value
Hmm, if I understand things correctly it should be of vital interest how much the growth rate of the coefficients is which should be mollified (arithmetic/geometric/hypergeometric growth). The matrix given by Gerry Myerson can mollify geometric series with q=-2 at most. I think what you ask for is the transformation-matrix for Eulersummation (optimally: with configurable order). Is that correct so far? Second question: do you want to mollify the sequence of partial sums or the coefficients themselves?
Sep
26
awarded  Quorum
Sep
15
comment Prove that the Dirichlet eta function is monotonic
Your focused variation in the behave of the function $g(p)$ might be overcome if we apply Eulersum to the terms of the series $\eta(p)$ for $ p \gt 0$. Possibly this is not too difficult to prove by the construction of the transformation-terms of the Eulersum-method, but I have not yet an idea how to do this...
Sep
12
comment Inequality of arithmetic, geometric and harmonic means
An even more general solution for the value m for the tuple of $d-1$ is: let $\rho_d$ be the root of $g_d(x)= x \cdot ({x^{-d}+d-1 \over d}+{d \over x^d+d-1})-2$ then simply $m_d=\rho_d^d$. It gives for $m_3 \approx 6.638$ and $m_4 \approx 22.59$ and so on.
Sep
12
comment Inequality of arithmetic, geometric and harmonic means
I've found another representation for the value of around $m \approx 6.64$: let $w$ be the cubic root of complex unity $w=1/2 - 0.866... î$ then $m = (w^{1/3}+w^{-1/3})^3$ (in my answer I had just $x=\log(m)$ )