bio  website  go.helmsnet.de 

location  Germany  
age  61  
visits  member for  4 years, 3 months 
seen  1 hour ago  
stats  profile views  1,160 
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 factoranalysis and a matrixorientated calculator MatMate). Around 2002 I came in contact with the internetcommunity in mathnewsgroups and could improve my Collatzdiscussion. Next subject was the Bernoullinumbers, then integer matrices and since 2006 the problem of iterated exponentiation aka tetration. Serving halfterm 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 matrixinverse had to be expressed by $ C^{1} = {I \over IB}$ where $B=IC$ 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

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Semigroup nilpotents and compostional inversion
I've still not finished my reanalysis 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 matrixalgebra 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 FAmodel, 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 itemspecific errorvariance; by manually experimenting with this (when I've studied the method again) I could arrive at higher sums. 
Oct 6 
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The difference between Principal Components Analysis (PCA) and Factor Analysis (FA)
The focus of the first sentence (1) and that of the enhancedformatted 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 Carlemanmatrix for $\sin(x)$ as size $n$ goes to infinity
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Oct 4 
revised 
Trouble with Jordan form of the truncated Carlemanmatrix for $\sin(x)$ as size $n$ goes to infinity
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Oct 4 
revised 
Trouble with Jordan form of the truncated Carlemanmatrix for $\sin(x)$ as size $n$ goes to infinity
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Oct 4 
revised 
Trouble with Jordan form of the truncated Carlemanmatrix for $\sin(x)$ as size $n$ goes to infinity
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Oct 4 
comment 
Trouble with Jordan form of the truncated Carlemanmatrix 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 Carlemanmatrix for $\sin(x)$ as size $n$ goes to infinity
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Oct 4 
revised 
Trouble with Jordan form of the truncated Carlemanmatrix for $\sin(x)$ as size $n$ goes to infinity
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Oct 4 
accepted  Trouble with Jordan form of the truncated Carlemanmatrix for $\sin(x)$ as size $n$ goes to infinity 
Oct 4 
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Trouble with Jordan form of the truncated Carlemanmatrix for $\sin(x)$ as size $n$ goes to infinity
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Oct 1 
comment 
Summation mollifier to ensure a certain alternating series has the correct value
Very well; I use Eulersummation with configurable order. The order must be adapted: when the series to mollify is $11+11...$ order 1 is needed, when it is $13+927+81...+...$ 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 Eulersummation 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 transformationmatrix 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 transformationterms of the Eulersummethod, but I have not yet an idea how to do this... 
Sep 12 
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Inequality of arithmetic, geometric and harmonic means
An even more general solution for the value m for the tuple of $d1$ is: let $\rho_d$ be the root of $g_d(x)= x \cdot ({x^{d}+d1 \over d}+{d \over x^d+d1})2$ 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 $x=\log(m)$ ) 