Finite-Part Integration of the Generalized Stieltjes Transform and its dominant asymptotic behavior for small values of the parameter Part I: Integer orders

The paper addresses the exact evaluation of the generalized Stieltjes transform $S_{n}[f]=\int_0^{\infty} f(x) (\omega+x)^{-n}\mathrm{d}x$ of integral order $n=1,2, 3,\dots$ about $\omega =0$ from which the asymptotic behavior of $S_{n}[f]$ for small parameters $\omega$ is directly extracted. An attempt to evaluate the integral by expanding the integrand $(\omega+x)^{-n}$ about $\omega=0$ and then naively integrating the resulting infinite series term by term lead to an infinite series whose terms are divergent integrals. Assigning values to the divergent integrals, say, by analytic continuation or by Hadamard's finite part is known to reproduce only some of the correct terms of the expansion but completely misses out a group of terms. Here we evaluate explicitly the generalized Stieltjes transform by means of finite-part integration recently introduced in [E.A. Galapon, {\it Proc. Roy. Soc. A} {\bf 473}, 20160567 (2017)]. It is shown that, when $f(x)$ does not vanish or has zero of order $m$ at the origin such that $(n-m)\geq 1$, the dominant terms of $S_{n}[f]$ as $\omega\rightarrow 0$ come from contributions arising from the poles and branch points of the complex valued function $f(z) (\omega+z)^{-n}$. These dominant terms are precisely the terms missed out by naive term by term integration. Furthermore, it is demonstrated how finite-part integration leads to new series representations of special functions by exploiting their known Stieltjes integral representations. Finally, the application of finite part integration in obtaining asymptotic expansions of the effective diffusivity in the limit of high Peclet number, the Green-Kubo formula for the self-diffusion coefficient and the antisymmetric part of the diffusion tensor in the weak noise limit is discussed

Continuation of the Stieltjes Series to the Large Regime by Finite-part Integration

We devise a prescription to utilize a novel convergent expansion in the strong-asymptotic regime for the Stieltjes integral and its generalizations [Galapon E.A Proc.R.Soc A 473, 20160567(2017)] to sum the associated divergent series of Stieltjes across all asymptotic regimes. The novel expansion makes use of the divergent negative-power moments which we treated as Hadamard's finite part integrals. The result allowed us to compute the ground-state energy of the quartic, sextic anharmonic oscillators as well as the $\mathcal{PT}$ symmetric cubic oscillator, and the funnel potential across all perturbation regimes from a single expansion that is built from the divergent weak-coupling perturbation series and incorporates the known leading-order strong-coupling behavior of the spectra

Exact Evaluation and extrapolation of the divergent expansion for the Heisenberg-Euler Lagrangian II: Non-alternating Case

We applied the method of finite-part integration [Galapon E.A Proc.R.Soc A 473, 20160567(2017)] to evaluate in closed-form the exact one-loop integral representations of the Heisenberg-Euler Lagrangian from QED for a constant electric field and electric-like self-dual background. We also devise a prescription based on the finite-part integration of the Cauchy principal value integral to sum and extrapolate the non-alternating divergent weak-field expansions of the Heisenberg-Euler Lagrangians to recover the non-perturbative Schwinger effect well into the strong field limit.Comment: arXiv admin note: text overlap with arXiv:2310.0819

CANON OR NOT CANON: THE CURIOUS CASE OF MICHAEL FIELD

Although they wrote more than thirty plays, nine volumes of poetry, around thirty volumes of diaries, and thousands of letters, Katharine Bradley and Edith Cooper, an aunt-niece duo, incestuous lovers and poets, more famous under their joint male pseudonym Michael Field, were almost unknown to a wider literary audience until the 1970s, when these two ‘minor’ Victorian women poets were rediscovered mostly by feminist critics. What first drew critics’ attention to this poetic pair was their dual authorial persona and the nature of their relationship, culminating in Lilian Faderman’s groundbreaking book Surpassing the Love of Men (1981). In the 1990s, critics such as Isobel Armstrong, Virginia Blain, Joseph Bristow, Holly Laird, Angela Leighton, Yopie Prins, Martha Vicinus, and Chris White shifted focus to their work, first to poetry and then to dramas and diaries. Finally, the first two decades of the twenty-first century confirmed the resurgence of interest in Michael Field, with critics such as Marion Thain and Ana Parejo Vadillo exploring this poetic duo within the wider context of fin-de-siècle literature. This paper will delve into various possible reasons why Bradley and Cooper, in spite of early positive reviews and the recognition of some literary greats of their age, were never granted a place among canonical British writers. Some of the avenues that will be explored are their fluid, queer identity and their refusal to define their gender; their tendency to write outdated literary genres such as verse historical dramas; the fact that they were female aesthetes, whose poetry was caught between paganism and Catholicism, homosexual and heterosexual love, femininity and masculinity, Victorian age and Modernism, tradition and modernity; their unwillingness to compromise their vision for popularity or commercial success; as well as their determination to create beautiful, well-designed books, published exclusively in rare and limited editions, which made them famous only among a small circle of connoisseurs

Researching and analyzing the features of oil and demand for transporting oil derivates in the area of Belgrade

The paper contains a summary of results obtained when researching demand for transporting dangerous goods of class 3 (oil and oil derivates) in the area of the city of Belgrade in 2007. Considering all transport modes used for carrying oil and oil derivates (road, rail and water transport), we present the total quantities of transported goods on a percentage basis. In light of road transport, the paper presents the features of demand for transport in terms of time and space that are a basis for the management of transporting dangerous goods and input to defining transportation routes to dangerous goods. Among the features of demand for transport in terms of time, the paper presents the quantities of goods transported within different months of the year, different week days and different periods of the day. The features of demand for transport in terms of space presented in the paper are goods flow features depending on their origin and the destination of movement in relation to the defined serviced area and the load of traffic network with goods flows. First published online: 27 Oct 201

Consumer wearable devices for evaluation of heart rate control using digoxin versus beta-blockers: the RATE-AF randomized trial.

Consumer-grade wearable technology has the potential to support clinical research and patient management. Here, we report results from the RATE-AF trial wearables study, which was designed to compare heart rate in older, multimorbid patients with permanent atrial fibrillation and heart failure who were randomized to treatment with either digoxin or beta-blockers. Heart rate (n = 143,379,796) and physical activity (n = 23,704,307) intervals were obtained from 53 participants (mean age 75.6 years (s.d. 8.4), 40% women) using a wrist-worn wearable linked to a smartphone for 20 weeks. Heart rates in participants treated with digoxin versus beta-blockers were not significantly different (regression coefficient 1.22 (95% confidence interval (CI) -2.82 to 5.27; P = 0.55); adjusted 0.66 (95% CI -3.45 to 4.77; P = 0.75)). No difference in heart rate was observed between the two groups of patients after accounting for physical activity (P = 0.74) or patients with high activity levels (≥30,000 steps per week; P = 0.97). Using a convolutional neural network designed to account for missing data, we found that wearable device data could predict New York Heart Association functional class 5 months after baseline assessment similarly to standard clinical measures of electrocardiographic heart rate and 6-minute walk test (F1 score 0.56 (95% CI 0.41 to 0.70) versus 0.55 (95% CI 0.41 to 0.68); P = 0.88 for comparison). The results of this study indicate that digoxin and beta-blockers have equivalent effects on heart rate in atrial fibrillation at rest and on exertion, and suggest that dynamic monitoring of individuals with arrhythmia using wearable technology could be an alternative to in-person assessment. ClinicalTrials.gov identifier: NCT02391337

Redefining β-blocker response in heart failure patients with sinus rhythm and atrial fibrillation: a machine learning cluster analysis

Background: Mortality remains unacceptably high in patients with heart failure and reduced left ventricular ejection fraction (LVEF) despite advances in therapeutics. We hypothesised that a novel artificial intelligence approach could better assess multiple and higher-dimension interactions of comorbidities, and define clusters of β-blocker efficacy in patients with sinus rhythm and atrial fibrillation. Methods: Neural network-based variational autoencoders and hierarchical clustering were applied to pooled individual patient data from nine double-blind, randomised, placebo-controlled trials of β blockers. All-cause mortality during median 1·3 years of follow-up was assessed by intention to treat, stratified by electrocardiographic heart rhythm. The number of clusters and dimensions was determined objectively, with results validated using a leave-one-trial-out approach. This study was prospectively registered with ClinicalTrials.gov (NCT00832442) and the PROSPERO database of systematic reviews (CRD42014010012). Findings: 15 659 patients with heart failure and LVEF of less than 50% were included, with median age 65 years (IQR 56–72) and LVEF 27% (IQR 21–33). 3708 (24%) patients were women. In sinus rhythm (n=12 822), most clusters demonstrated a consistent overall mortality benefit from β blockers, with odds ratios (ORs) ranging from 0·54 to 0·74. One cluster in sinus rhythm of older patients with less severe symptoms showed no significant efficacy (OR 0·86, 95% CI 0·67–1·10; p=0·22). In atrial fibrillation (n=2837), four of five clusters were consistent with the overall neutral effect of β blockers versus placebo (OR 0·92, 0·77–1·10; p=0·37). One cluster of younger atrial fibrillation patients at lower mortality risk but similar LVEF to average had a statistically significant reduction in mortality with β blockers (OR 0·57, 0·35–0·93; p=0·023). The robustness and consistency of clustering was confirmed for all models (p<0·0001 vs random), and cluster membership was externally validated across the nine independent trials. Interpretation: An artificial intelligence-based clustering approach was able to distinguish prognostic response from β blockers in patients with heart failure and reduced LVEF. This included patients in sinus rhythm with suboptimal efficacy, as well as a cluster of patients with atrial fibrillation where β blockers did reduce mortality