Skip to main Communities My dashboard Log in Sign up Published March 14, 2026 | Version v1 Preprint Open Inverse Singularity Detection from Renormalized Tail Orbits Authors/Creators Mohammad Abu-Ghuwaleh 1 Show affiliations 1.
Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan Description We continue the renormalized-tail program initiated in \cite{AbuGhuwaleh2026} by addressing an inverse problem: what dominant singular data can be recovered from the orbit of normalized Taylor tails \[ T_n^f(w):=\sum_{k\ge 0}\frac{a_{n+k}}{a_n}w^k \] of an analytic power series $f(z)=\sum_{n\ge 0} a_n z^n$ with no vanishing coefficients? We prove a universal inverse transport principle. If \[ \frac{a_{n+1}}{a_n}=\rho\bigl(1+c u_n+o(u_n)\bigr) \] on an admissible logarithmically flat scale $u_n$, then uniformly on each closed disk $\{ |\rho w|\le s\}$ with $0<s<1$, \[ T_n^f(w)=\frac{1}{1-\rho w}+c u_n\frac{\rho w}{(1-\rho w)^2}+o(u_n), \] and conversely the same tail expansion forces the same ratio expansion. Thus the first deviation of the orbit from the geometric attractor has a rigid universal shape, and its scalar amplitude is exactly the ratio perturbation. We then combine this transport law with standard transfer theorems of singularity analysis to build an inverse dictionary for three dominant model classes: algebraic singularities, algebraic-logarithmic singularities, and confluent algebraic singularities. In particular, the orbit determines the dominant singularity location $\zeta=\rho^{-1}$, the algebraic exponent $\alpha$, the logarithmic power $\mu$, and for confluent terms $b(1-z/\zeta)^\lambda$ with $0<\lambda<1$, both the exponent $\lambda$ and amplitude $b$. We also prove that first-order orbit data alone cannot distinguish algebraic, algebraic-logarithmic, and confluent models sharing the same principal exponent; secondary scales are genuinely necessary. The results provide a concrete inverse diagnostic procedure for renormalized tail dynamics. Files 004_inverse_singularity_detection.pdf Files (415.8 kB) Name Size Download all 004_inverse_singularity_detection.pdf md5:13fe0b960cb432d232c5a651f2d1fd95 415.8 kB Preview Download 18 Views 15 Downloads Show more details All versions This version Views Total views 18 18 Downloads Total downloads 15 15 Data volume Total data volume 7.1 MB 7.1 MB More info on how stats are collected.... Versions External resources Indexed in OpenAIRE Communities Keywords and subjects Keywords analytic functions Taylor series normalized tails renormalization singularity analysis inverse problems Details DOI DOI Badge DOI 10.5281/zenodo.19573092 Markdown [](https://doi.org/10.5281/zenodo.19573092) reStructuredText .. image:: https://zenodo.org/badge/DOI/10.5281/zenodo.19573092.svg :target: https://doi.org/10.5281/zenodo.19573092 HTML <a href="https://doi.org/10.5281/zenodo.19573092"><img src="https://zenodo.org/badge/DOI/10.5281/zenodo.19573092.svg" alt="DOI"></a> Image URL https://zenodo.org/badge/DOI/10.5281/zenodo.19573092.svg Target URL https://doi.org/10.5281/zenodo.19573092 Resource type Preprint Publisher Zenodo Languages English Rights License Creative Commons Attribution 4.0 International The Creative Commons Attribution license allows re-distribution and re-use of a licensed work on the condition that the creator is appropriately credited. Read more Citation Export Technical metadata Created April 14, 2026 Modified April 14, 2026 Jump up About About Policies Infrastructure Principles Projects Roadmap Contact Blog Blog Support Help FAQ Developers REST API OAI-PMH Contribute GitHub Donate Funded by Powered by CERN Data Centre & InvenioRDM Status Privacy policy Cookie policy Terms of Use This site uses cookies. Find out more on how we use cookies Accept all cookies Accept only essential cookies