Miscellaneous
Here I share some particular stuff that you might find interesting: my photography achievements, program codes, notes for my students. Also, here I share some some theorems, and mathematical observations that I am not intending to publish in a journal.
To my students at ÓE.
Please send me an email to get the slides for the course Multivariate statistical methods.
To my students at BGE. Here I share some videos that you may find useful:
Graphs of functions, and their transformations (1st week):
https://www.youtube.com/watch?v=Tmdrjs9xufc
https://www.youtube.com/watch?v=sTCRB6hMsC4
Composite function, inverse function (2nd week):
https://www.youtube.com/watch?v=ZFPkQkURSxk (composite function)
https://www.youtube.com/watch?v=2zeYEx4eTdc (examples for inverse function calculation)
Sequences and their properties (3rd and 4th weeks):
https://www.youtube.com/watch?v=tHy3TXmZpF0 (Monotone and Bounded Sequences)
https://www.youtube.com/watch?v=XdkoTb8PEG0 (Converging and Diverging Sequences)
Photography
In my free time I do street and abstract photography. I have some achievements on international photography competitions that you can see below.
Minimalist Photography Awards 2023 - Architecture category, honourable mention.
A photo of mine titled “Dishu at Xuanwu lake” was exhibited at Barabás Villa, Budapest, on 2022. 12. 06. This photo is part of the project presented on the webpage of the Hungarian-Chinese Friendship Association.
My photograph "Abstract in Calais" received an honorable mention in Street Photography category in the Minimalist Photography Awards 2022.
My photo album “Kínai Életképek” was selected as one of the best works among the finalists in the photo competition China – Close Sight (中国近观) by the Hungarian Chinese Friendship Association (2021).
My photo-essay entitled Candid Surveillance received honourable mention in the International Photography Awards 2020 in Editorial / Press, Photo Essay / Feature Story category.
The same set was awarded with honourable mention in the Budapest International Photography Awards in 2020 in Photo Essay category.
I received honorable mention in amateur street category in the 2019 International Black & White Photography Contest.
Trieste PhotoDays 2019 exhibited a photo of mine between 2019. 10. 22 and 11. 10 on the International Festival of Urban Photography with Martin Parr jury president.
Another of my works, titled Building Abstract II, received honourable mention on the Minimalist Photography Awards 2019 in Abstract category.
The same photo was published in the 20th issue of the Black & White Minimalism Photography Magazine.
My photo Waiting for the opportunity received honourable mention on the International Photography Awards 2019 in Street Photography, Lifestyle (urban scenery) category.
This work was also selected by the Black & White Minimalism Photography Magazine in 2019.
If you are interested, you can find some of my photos on Instagram.
Mathematics
Two new integral representations for the Lambert W function
I found an integral representation for the principal branch of the W function.
Another integral representation is published on arXiv. This representation is not new, however. It was discovered earlier in a paper by G. A. Kalugin et al.
The p-adic Lambert W function
It is possible to define the p-adic version of the classical Lambert W function. I did this in this short note, where it was proven, among others, that the p-adic W function is not an analytic element, i.e., it is not a (uniform) limit of a sequence of rational functions.
The p-adic infinite power tower
The curious infinite power tower is expressible in terms of the Lambert function. Once we have the p-adic Lambert function defined, the question naturally arises whether some infinite p-adic power tower can be defined such that it is similarly expressible with the p-adic Lambert function as in the classical archimedean case. It turns out, that the tesselation operation, although can be defined up to arbitrary depth, does not give rise to a meaningful infinite p-adic power tower.
The situation is somewhat similar to the p-adic versus classical Mandelbrot set: the second is a remarkably complex structure, while the first is just the closed unit disk of C_p (see Proposition 2.1 here).
The r-Lambert function
I worked on a new generalization of the Lambert W function which helps to solve some transcendental equations arising in physics and combinatorics. I wrote a C code which calculates this function in the real case. You can either build this code on your computer if you have a C compiler, or just copy the code text directly to OnlineGDB.com
Special thanks to Keith Briggs, who wrote a code to calculate W.
This C code was translated into Maple by Abdelkaddous Taha. Many thanks to him for authorizing the sharing of the code here.
The r-Lambert function has infinitely many branches over the complex plane, the whole description of the branches is recently done.
A p-adic number library in C++
To refresh and update my object oriented coding knowledge (what I have not used for 10 years or so), I wrote a C++ library for basic p-adic number calculations, using some new features of C++11. The code is deposited on GitHub.
Lower and upper estimation for the Mahler measure of the Bell polynomials
With a tiny effort a rather good estimation can be given for the Mahler measure of the Bell polynomials. The result is contained in this pdf.
The Furstenberg topological space
There is a special topology on the set of integers that enables one to prove the infinitude of primes. This topological space has some non-trivial features. My colleague, Rezső Lovas and I gave a metric completion of this space that, interestingly enough, involves infinitely big 'integers' and involves the so-called factorial number system.
These results are published in the Elemente der Mathematik 69: 1-14.
The above partition relation was proven by Sierpiński. Here I offer another proof that uses a useful analytic theorem.
I found a proof for another partition relation which comes from the book Introduction to Cardinal Arithmetic.
An alternative form of the Uehling potential
The classical Coulomb potential ~1/r describes the static electric field at distance r around a point charge placed at the origin. From Quantum Electrodynamics (QED) point of view, this corresponds to virtual photon emission from the charge towards its ambient. It is known from QED that virtual photons can emit and absorb virtual electron-positron pairs during their traveling, so the Coulomb electric field around a point charge must be corrected accordingly. Considering no other effects, this virtual e-e+ emission and absorption (the so-called one-loop correction or vacuum polarization effect) results in a modified potential first described by Uehling in 1935. I noted that the Uehling potential can be written in a different but equivalent form by using the hyperbolic cosine and sine integrals.
The boundary of the convergence domain of the Kapteyn series
The Kapteyn series of a complex function f is convergent if f is analytic on a given domain. This domain is quite involved, but it can fully be described by the r-Lambert function. Once this discovery is made, the visualization of the domain is quite simple. Here is a Mathematica worksheet for this task.