Differential Equation of Appell Polynomials Via the Factorization Method

Let {Pn(x)}∞n=0 be a sequence of polynomials of degree n. We define two sequences of differential operators Φn and Ψn satisfying the following properties: By constructing these two operators for Appell polynomials, we determine their differential equations via the factorization method introduced by Infeld and Hull (Rev. Mod. Phys. 23 (1951) 21). The differential equations for both Bernoulli and Euler polynomials are given as special cases of the Appell polynomials

Asymptotics of Zeros of Relativistic Hermite Polynomials

The relativistic Hermite polynomial (RHP) is a class of orthogonal polynomials associated with varying weights. We study the asymptotics of the zeros of the RHP when both degree $n$ of polynomials and relativistic parameter $N$ approach infinity

On Quadrature Rules Associated with Appell Polynomials

A quadrature rule using Appell polynomials and generalizing both the Euler-MacLaurin quadrature formula and a similar quadrature rule, obtained in Bretti et al [15], which makes use of Euler (instead of Bernoulli) numbers and even (instead of odd) derivatives of the given function at the extrema of the considered interval, is derived. An expression of the remainder term and a numerical example are also enclosed

Quantifying the challenges of detecting unseen planetary companions with transit timing variations

Both ground and space-based transit observatories are poised to significantly increase the number of known transiting planets and the number of precisely measured transit times. The variation in a planet's transit times may be used to infer the presence of additional planets. Deducing the masses and orbital parameters of such planets from transit time variations (TTVs) alone is a rich and increasingly relevant dynamical problem. In this work, we evaluate the extent of the degeneracies in this process, systematically explore the dependence of TTV signals on several parameters and provide phase space plots that could aid observers in planning future observations. Our explorations are focused on a likely-to-be prevalent situation: a known transiting short-period Neptune or Jupiter-sized planet and a suspected external low-mass perturber on a nearly-coplanar orbit. Through approximately 10^7 N-body simulations, we demonstrate how TTV signal amplitudes may vary by orders of magnitude due to slight variations in any one orbital parameter (0.001 AU in semimajor axis, 0.005 in eccentricity, or a few degrees in orbital angles), and quantify the number of consecutive transit observations necessary in order to obtain a reasonable opportunity to characterize the unseen planet (approximately greater or equal to 50 observations). Planets in or near period commensurabilities of the form p:q, where p < 21 and q < 4, produce distinct TTV signatures, regardless of whether the planets are actually locked in a mean motion resonance. We distinguish these systems from the secular systems in our explorations. Additionally, we find that computing the autocorrelation function of a TTV signal can provide a useful diagnostic for identifying possible orbits for additional planets and suggest that this method could aid integration of TTV signals in future studies of particular exosystems.Comment: 53 pages total, including 18 figures, 1 table, and 1 appendix. Accepted for publication in ApJ. Better resolution plots will appear in online journa

Exploring Stakeholder Value Models via Interactive Visualization

AbstractIn the design of complex systems, models can be used to predict a future system's performance as well as approximate stakeholder preferences on performance. This paper examines the problem of model truthfulness and the challenge of trusting models, with a focus on value models and how they are used to predict stakeholder preferences. A framework is proposed for the analysis of these issues (truthfulness and trust), which is used to discuss the relationship between models and decision outcomes. Interactive visualization is proposed as an efficient and effective method for increasing model truthfulness and model trust, and hence making better decisions. An interactive visualization tool is also presented, and an application of the tool to a complex decision case is discussed

Co-evolution and Nuclear Structure in the Dwarf Galaxy POX 52 Studied by Multi-wavelength Data From Radio to X-ray

The nearby dwarf galaxy POX 52 at $z = 0.021$ hosts an active galactic nucleus (AGN) with a black-hole (BH) mass of $M_{\rm BH} \sim 10^{5-6} M_\odot$ and an Eddington ratio of $\sim$ 0.1-1. This object provides the rare opportunity to study both AGN and host-galaxy properties in a low-mass highly accreting system. To do so, we collected its multi-wavelength data from X-ray to radio. First, we construct a spectral energy distribution, and by fitting it with AGN and host-galaxy components, we constrain AGN-disk and dust-torus components. Then, while considering the AGN-disk emission, we decompose optical HST images. As a result, it is found that a classical bulge component is probably present, and its mass ($M_{\rm bulge}$) is consistent with an expected value from a local relation. Lastly, we analyze new quasi-simultaneous X-ray (0.2-30 keV) data obtained by NuSTAR and XMM-Newton. The X-ray spectrum can be reproduced by multi-color blackbody, warm and hot coronae, and disk and torus reflection components. Based on this, the spin is estimated to be $a_{\rm spin} = 0.998_{-0.814}$, which could suggest that most of the current BH mass was achieved by prolonged mass accretion. Given the presence of the bulge, POX 52 would have undergone a galaxy merger, while the $M_{\rm BH}$-$M_{\rm bulge}$ relation and the inferred prolonged accretion could suggest that AGN feedback occurred. Regarding the AGN structure, the spectral slope of the hot corona, its relative strength to the bolometric emission, and the torus structure are found to be consistent with Eddington-ratio dependencies found for nearby AGNs.Comment: 26 pages, 10 figures, 4 tables, accepted for publication in ApJ. The data in Table 4 are available from http://www.kusastro.kyoto-u.ac.jp/~kawamuro/arxiv/SEDplusComponents_u.tx

The Brave study: promoting active breaks in secondary school from students’ point of view

Background: According to the WHO recommendations, children and adolescents should perform at least 60 minutes per day of moderate to vigorous physical activity (PA). Active Breaks (ABs) interventions, short physical activity breaks of 5- 15 minutes during school hours, have been examined in primary school children as a potential strategy to counteract a sedentary lifestyle, with minimal disruption to school learning activities. The aim of the BRAVE STUDY is to investigate the feasibility of ABs in a secondary school setting from the students’ point of view. Methods: In December 2020, 10 students (age 12-13, 6 females and 4 males) attending the second and third grade of secondary schools located in Bologna province (Italy) were involved in a focus group (FG). The FG was conducted online because of COVID-19 and the answers transcribed for a later analysis. Students’ opinions were probed on the role of PA in school and possible facilitators/barriers to implementation of ABs in the school. Results: Students reported they wanted to be more active as a consequence of time spent in class sitting at a desk. Students also reported that an organized activity like ABs conducted in the classroom setting provides an excellent opportunity to improve social relations with classmates. Students also highlighted the possible psycho-physical well-being benefits arising from PA. Among the potentially negative aspects reported, students underlined the possible confusion that would be created in classroom and the time subtracted from academic learning activities. Conclusions: The FG represents an ideal mean to obtain in-depth information on how people feel about a pending program or a change in their routine. The current FG reinforces positive outcomes from exposure to a PA program that can be intertwined with their daily classroom activities. ABs programs can help to reconcile the needs of students that arise during the day with the PA objectives recommended by the WHO. Key messages: ABs can be a zero-cost intervention strategy to achieve WHO recommendations and would create conditions for a greater psycho-physical benefits in classrooms. A qualitative approach, such as FGs, provides a mean to collect information not obtainable with quantitative methods, that could be useful to co-design interventions for children and adolescents

A pilot feasibility study of daily rTMS to modify corticospinal excitability during lower limb immobilization

Short term immobilization of the lower limb is associated with increased corticospinal excitability at 24 hours post cast removal. We wondered whether daily stimulation of the motor cortex might decrease brain reorganization during casting. We tested the feasibility of this approach. Using transcranial magnetic stimulation (TMS), resting motor threshold and recruitment curves were obtained at baseline in 6 healthy participants who then had leg casts placed for 10 days. On 7 of the 10 days subjects received 20 minutes of 1 Hz repetitive TMS (rTMS). TMS measures were then recorded immediately after and 24 hours post cast removal. Four of 6 subjects completed the study. At the group level there were no changes in excitability following cast removal. At the individual level, two participants did not show any change, 1 participant had higher and one lower excitability 24 hours after cast removal. Daily rTMS over motor cortex is feasible during casting and may modify neuroplastic changes occurring during limb disuse. A prospective double blind study is warranted to test whether daily rTMS might improve outcome in subjects undergoing casting, and perhaps in other forms of limb disuse such as those following brain injury or weightlessness in space flight