Nonlinear stability analysis of the Emden-Fowler equation

In this paper we qualitatively study radial solutions of the semilinear elliptic equation $\Delta u + u^n = 0$ with $u(0)=1$ and $u'(0)=0$ on the positive real line, called the Emden-Fowler or Lane-Emden equation. This equation is of great importance in Newtonian astrophysics and the constant $n$ is called the polytropic index. By introducing a set of new variables, the Emden-Fowler equation can be written as an autonomous system of two ordinary differential equations which can be analyzed using linear and nonlinear stability analysis. We perform the study of stability by using linear stability analysis, the Jacobi stability analysis (Kosambi-Cartan-Chern theory) and the Lyapunov function method. Depending on the values of $n$ these different methods yield different results. We identify a parameter range for $n$ where all three methods imply stability.Comment: 12 pages; new reference added; 3 new references added; fully revised versio

Algebraic structure of gravity in Ashtekar variables

The BRST transformations for gravity in Ashtekar variables are obtained by using the Maurer-Cartan horizontality conditions. The BRST cohomology in Ashtekar variables is calculated with the help of an operator $\delta$ introduced by S.P. Sorella, which allows to decompose the exterior derivative as a BRST commutator. This BRST cohomology leads to the differential invariants for four-dimensional manifolds.Comment: 19 pages, report REF. TUW 94-1

The Effect of Polyphenols, Minerals, Fibers, and Fruits on Irritable Bowel Syndrome: A Systematic Review

Background: Irritable bowel syndrome (IBS) is a chronic gastrointestinal disorder characterized by abdominal pain, bloating, and changes in bowel habits. Various dietary factors have been implicated in the pathogenesis and management of IBS symptoms. This systematic review aims to evaluate the effects of polyphenols, minerals, fibers, and fruits on the symptoms and overall well-being of individuals with IBS. Materials and Methods: A comprehensive literature search was conducted in several electronic databases, including PubMed, Scopus, and Web of Science. Studies published up until July 2023 were included. Results: The selected studies varied in terms of study design, participant characteristics, intervention duration, and outcome measures. Overall, the findings suggest that dietary interventions involving polyphenols, minerals, fibers, and fruits can have a positive impact on IBS symptoms. Dietary fiber supplementation, particularly soluble fiber, has been associated with reduced bloating and enhanced stool consistency. Conclusions: This systematic review provides evidence supporting the beneficial effects of polyphenols, minerals, fibers, and fruits in IBS patients. These dietary components hold promise as complementary approaches for managing IBS symptoms. However, due to the heterogeneity of the included studies and the limited number of high-quality randomized controlled trials, further well-designed trials are warranted to establish the optimal dosages, duration, and long-term effects of these interventions. Understanding the role of specific dietary components in IBS management may pave the way for personalized dietary recommendations and improve the quality of life for individuals suffering from this complex disorder

Clocking Auger Electrons

Intense X-ray free-electron lasers (XFELs) can rapidly excite matter, leaving it in inherently unstable states that decay on femtosecond timescales. As the relaxation occurs primarily via Auger emission, excited state observations are constrained by Auger decay. In situ measurement of this process is therefore crucial, yet it has thus far remained elusive at XFELs due to inherent timing and phase jitter, which can be orders of magnitude larger than the timescale of Auger decay. Here, we develop a new approach termed self-referenced attosecond streaking, based upon simultaneous measurements of streaked photo- and Auger electrons. Our technique enables sub-femtosecond resolution in spite of jitter. We exploit this method to make the first XFEL time-domain measurement of the Auger decay lifetime in atomic neon, and, by using a fully quantum-mechanical description, retrieve a lifetime of $2.2^{ + 0.2}_{ - 0.3}$ fs for the KLL decay channel. Importantly, our technique can be generalised to permit the extension of attosecond time-resolved experiments to all current and future FEL facilities.Comment: Main text: 20 pages, 3 figures. Supplementary information: 17 pages, 6 figure

Clocking Auger electrons

Intense X-ray free-electron lasers (XFELs) can rapidly excite matter, leaving it in inherently unstable states that decay on femtosecond timescales. The relaxation occurs primarily via Auger emission, so excited-state observations are constrained by Auger decay. In situ measurement of this process is therefore crucial, yet it has thus far remained elusive in XFELs owing to inherent timing and phase jitter, which can be orders of magnitude larger than the timescale of Auger decay. Here we develop an approach termed ‘self-referenced attosecond streaking’ that provides subfemtosecond resolution in spite of jitter, enabling time-domain measurement of the delay between photoemission and Auger emission in atomic neon excited by intense, femtosecond pulses from an XFEL. Using a fully quantum-mechanical description that treats the ionization, core-hole formation and Auger emission as a single process, the observed delay yields an Auger decay lifetime of 2.2_−0.3^+0.2 fs for the KLL decay channel