12,721 reputation
11656
bio website go.helms-net.de
location Kassel, Germany
age 61
visits member for 4 years
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.


16h
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
16h
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.
1d
comment finding “exp(1)” in the p-adic numbers
Lubin, shouldn't the logarithmic functions have a $+1$ at their arguments? ($L(1+x)$ and $\log(1+x)$)
Sep
12
comment Iteration of $x \to x^x$
If you look at $g(y)=(1+y)^{(1+y)}-1$ instead you can expand it into a power series and study the emerging pattern of coefficients when iterating (however I don't know whether this helps really). After iterating you use $f(x,h)=g(x-1,h)+1$ where $h$ is the iteration-height to evaluate the original iteration
Sep
10
comment GCD of $a+b$ and $\frac{a^p + b^p}{a+b}$
Thanks Gautam - nearly the same as I'd done it but with a different flavour... I'm curious to see what's in it (and perhaps can brush my own text up)
Sep
10
comment GCD of $a+b$ and $\frac{a^p + b^p}{a+b}$
Because you ask for any hint, perhaps this small exercise for myself is helpful/interesting/leading-to-somewhere for you: go.helms-net.de/math/expdioph/CyclicSubgroups_work.pdf It discusses the (prime)-factorization of the numerators expression as well as in relation to that of the denominator.
Sep
10
comment What is $\cdots ((((1/2)/(3/4))/((5/6)/(7/8)))/(((9/10)/(11/12))/((13/14)/(15/16))))/\cdots$?
@ rubik: you're welcome!
Sep
10
comment What is $\cdots ((((1/2)/(3/4))/((5/6)/(7/8)))/(((9/10)/(11/12))/((13/14)/(15/16))))/\cdots$?
The result is already in a paper of 1998 of Shallit/Allouche ("the ubiquituous thue-morse-constant") and was recently available online as pdf-file. See my screenshot of a copy which I've downloaded about 2004.
Sep
7
comment connection between PCA and linear regression
Sorry I m out and have only smartphon e to type. Eric s answer seems to be too restricted as you observe as well pca as well as Regression can be computed outside of the rationale of the Singular value decomposition. Of course the mathematic can be shown to coincide. So far only...it is a martyrium to type on the phone....
Sep
4
comment How is the formal inverse of a power series with constant term developed ( for instance $\cosh^{-1}(x)$)?
Yes, thank you: I'll take this as serious advice! I think now first I'll extend this on my own for the other two g()-functions and then explore a bit the behaves. After that I should then take another ride using expansions on values from the right half plane as you suggest. I'll "accept" your answer because it gave me a usable impulse - again thank you!
Sep
4
comment How is the formal inverse of a power series with constant term developed ( for instance $\cosh^{-1}(x)$)?
In principle I didn't have a specific prevalence; I'm just lucky that I've understood how to do this - at least one viable way. But there is something more: I wanted to play with that g() and h()-functions to try things which are possible with the pairs acosh() and cosh(), and acos() and cos() for instance the impressive behave with iteration of polynomials using $\cosh(2^m \cdot \operatorname{acosh}(-2+x^2))$ giving the m'th composition of the polynomial with itself and allows to generalize this to fractional heights. I thought just to try similar things with $g_0()$ and $h_0(x)$ and so on.
Sep
4
comment How is the formal inverse of a power series with constant term developed ( for instance $\cosh^{-1}(x)$)?
Just an observation: the $z_0 \approx 0.925 - 1.602 I $ which Maple has found is the first complex root and is the cube-root of a purely real negative number (see the first real root $\rho_0$ in my answer above) such that $(\rho_0)^3 = z_0^3 \approx -6.329...$
Sep
3
comment How is the formal inverse of a power series with constant term developed ( for instance $\cosh^{-1}(x)$)?
;-) Now you gave me a riddle to chew on... Let's see tomorrow (it's late here)
Sep
3
comment How is the formal inverse of a power series with constant term developed ( for instance $\cosh^{-1}(x)$)?
Wow, that's bad news... I had already observed, that the range of convergence is small, but I didn't realize, that the range for my version of $h(x)$ and thus the value of the function at all is such limited. Of course a more interesting version were the real axis from $x=1 \to \infty$ (didn't consider complex values so far). So differently from the case of $\operatorname{acosh}(x)$ which gives a series around zero, I need a shifted version $h(1+x)$ like it is for $\operatorname{squareroot}(1+x)$ or $ \log(1+x)$ or similar?
Sep
3
comment Evaluation of a class of continued fractions
Perhaps you find this go.helms-net.de/math/divers/GenContFracRationalE.htm interesting, although the sequences of coefficients in your CF goe in steps of 2 and I've a table of CF's with sequences going in steps of 1. But perhaps the table as such is of interest and gives an idea for the analoguous table for your version
Sep
3
comment How is the formal inverse of a power series with constant term developed ( for instance $\cosh^{-1}(x)$)?
Hi Ionanis, thanks to your hints (and worked example) I was now able to go deeper into this derivative-business and seem to be successful with the understanding of the machine and seem to have reasonable first couple of coefficients. The series arising is not so nice as I'd hoped, but perhaps a recentering might of advantage here, I'll try this later today. I've updated the first few coefficients into my question-box. Thanks for the help!
Sep
2
comment Outline approach to Collatz 3n+1 conjecture / Criticism needed
@ThomasAdrews: it might be instructive to look at the trees which occur when the negative numbers are included. Then we have four separate trees, interwoven in the whole set of integer numbers. The branches having -5 and -7 as elements for instance form a "nest" from which subtrees grow out - but without connection to that of the surrounding tree with root at -1. (A similar situation has the branch which contains -17, the branches of that associated cycle of numbers form a similar nest and are rooting another tree without connection to the tree which is rooted at -1)
Sep
2
comment Is this a valid statement that would imply the Collatz Conjecture?
I think too, that the argument and induction is correct. Whether it "is part of typical description": I think it is the concept of "stopping" time which might be introduced by R.Terras (see Lagarias' overview). T. Oliviera's pages work with it. (I'm formulating weak here because I've never tried to become specifically familiar with this view on the conjecture)
Sep
2
comment How is the formal inverse of a power series with constant term developed ( for instance $\cosh^{-1}(x)$)?
Well, for $a_0$ we have even the $d^{(n-1)}=d^{-1}$ ' th derivative in the above formula. So what I meant was I'm needing a simple, first point where I can step into the whole machinery
Sep
2
comment How is the formal inverse of a power series with constant term developed ( for instance $\cosh^{-1}(x)$)?
Hmm, so what would be the coefficient $a_n$ at $n=0$. The integral of the parenthese to the zeroth power (=1)? But what would that integral-expression be?