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Exchangeable partitions of the integers and their corresponding mass partitions on 
$\mathcal{P}_{\infty} = \{\mathbf{s} = (s_{1},s_{2},\ldots)\colon s_{1} \geq s_{2} \geq \cdots \geq 0$ and 
$\sum_{k=1}^{\infty}s_{k} = 1\}$ play a vital role in combinatorial stochastic processes and their applications. In this work, we continue our focus on the class of Gibbs partitions of the integers and the corresponding stable Poisson–Kingman-distributed mass partitions generated by the normalized jumps of a stable subordinator with an index 
$\alpha \in (0,1)$, subject to further mixing. This remarkable class of infinitely exchangeable random partitions is characterized by probabilities that have Gibbs (product) form. These partitions have practical applications in combinatorial stochastic processes, random tree/graph growth models, and Bayesian statistics. The most notable class consists of random partitions generated from the two-parameter Poisson–Dirichlet distribution 
$\mathrm{PD}(\alpha,\theta)$. While the utility of Gibbs partitions is recognized, there is limited understanding of the broader class. Here, as a continuation of our work, we address this gap by extending the dual coagulation/fragmentation results of Pitman (1999), developed for the Poisson–Dirichlet family, to all Gibbs models and their corresponding Poisson–Kingman mass partitions, creating nested families of Gibbs partitions and mass partitions. We focus primarily on fragmentation operations, identifying which classes correspond to these operations and providing significant calculations for the resulting Gibbs partitions. Furthermore, for completion, we provide definitive results for dual coagulation operations using dependent processes. We demonstrate the applicability of our results by establishing new findings for Brownian excursion partitions, Mittag-Leffler, and size-biased generalized gamma models.
The outer solar system is theoretically predicted to harbour an undiscovered planet, often referred to as Planet Nine. Simulations suggest that its gravitational influence could explain the unusual clustering of minor bodies in the Kuiper Belt. However, no observational evidence for Planet Nine has been found so far, as its predicted orbit lies far beyond Neptune, where it reflects only a faint amount of Sunlight. This work aims to find Planet Nine candidates by taking advantage of two far-infrared all-sky surveys, which are IRAS and AKARI. The epochs of these two surveys were separated by 23 years, which is large enough to detect Planet Nine’s 
$\sim3'$/year orbital motion. We use a dedicated AKARI Far-Infrared point source list for the purpose of our Planet Nine search — AKARI-FIS Monthly Unconfirmed Source List (AKARI-MUSL), which includes sources detected repeatedly only in hours timescale, but not after months. AKARI-MUSL is more advantageous than the AKARI Bright Source Catalogue (AKARI-BSC) for detecting moving and faint objects like Planet Nine with a twice-deeper flux detection limit. We search for objects that moved slowly between IRAS and AKARI detections given in the catalogues. First, we estimated the expected flux and orbital motion of Planet Nine by assuming its mass, distance, and effective temperature to ensure it can be detected by IRAS and AKARI, then applied the positional and flux selection criteria to narrow down the number of sources from the catalogues. Next, we produced all possible candidate pairs including one IRAS source and one AKARI source whose angular separations were limited between 42′ and 
$69.6'$, corresponding to the heliocentric distance range of 500 – 700 AU and the mass range of 7 – 17M
$_{\oplus}$. There are 13 candidate pairs obtained after the selection criteria. After image inspection, we found one good candidate, of which the IRAS source is absent from the same coordinate in the AKARI image after 23 years and vice versa. However, AKARI and IRAS detections are not enough to determine the full orbit of this candidate. This issue leads to the need for follow-up observations, which will determine the Keplerian motion of our Planet Nine candidate.
An unusual orbital element clustering of Kuiper belt objects (KBOs) has been observed. The most promising dynamic solution is the presence of a giant planet in the outer Solar system, Planet Nine. However, due to its extreme distance, intensive searches in optical have not been successful. We aim to find Planet Nine in the far-infrared, where it has the peak of the black body radiation, using the most sensitive all-sky far-infrared survey to date, AKARI. In contrast to optical searches, where the energy of reflected sunlight decreases by 
$d^{4}$, thermal radiation in the infrared decreases with the square of the heliocentric distance 
$d^{2}$. We search for moving objects in the AKARI Single Scan Detection List. We select sources from a promising region suggested by an N-body simulation from Millholland and Laughlin 2017: 
$30^{\circ}\lt$ R.A. 
$\lt50^{\circ}$ and 
$-20^{\circ}\lt$ Dec. 
$\lt20^{\circ}$. Known sources are excluded by cross-matching AKARI sources with 9 optical and infrared catalogues. Furthermore, we select sources with small background strength to avoid sources in the cirrus. Since Planet Nine is stationary in a timescale of hours but moves on a monthly scale, our primary strategy is to select slowly moving objects that are stationary in 24 h but not in six months, using multiple single scans by AKARI. The selected slowly moving AKARI sources are scrutinised for potential contamination from cosmic rays. Our analysis reveals two possible Planet Nine candidates whose positions and flux are within the theoretical prediction ranges. These candidates warrant further investigation through follow-up observations to confirm the existence and properties of Planet Nine.
Brown dwarfs are failed stars with very low mass (13–75 Jupiter mass) and an effective temperature lower than 2 500 K. Their mass range is between Jupiter and red dwarfs. Thus, they play a key role in understanding the gap in the mass function between stars and planets. However, due to their faint nature, previous searches are inevitably limited to the solar neighbourhood (20 pc). To improve our knowledge of the low mass part of the initial stellar mass function and the star formation history of the Milky Way, it is crucial to find more distant brown dwarfs. Using James Webb Space Telescope (JWST) COSMOS-Web data, this study seeks to enhance our comprehension of the physical characteristics of brown dwarfs situated at a distance of kpc scale. The exceptional sensitivity of the JWST enables the detection of brown dwarfs that are up to 100 times more distant than those discovered in the earlier all-sky infrared surveys. The large area coverage of the JWST COSMOS-Web survey allows us to find more distant brown dwarfs than earlier JWST studies with smaller area coverages. To capture prominent water absorption features around 2.7 
${\unicode{x03BC}}$m, we apply two colour criteria, 
$\text{F115W}-\text{F277W}+1\lt\text{F277W}-\text{F444W}$ and 
$\text{F277W}-\text{F444W}\gt\,0.9$. We then select point sources by CLASS_STAR, FLUX_RADIUS, and SPREAD_MODEL criteria. Faint sources are visually checked to exclude possibly extended sources. We conduct SED fitting and MCMC simulations to determine their physical properties and associated uncertainties. Our search reveals 25 T-dwarf candidates and 2 Y-dwarf candidates, more than any previous JWST brown dwarf searches. They are located from 0.3 to 4 kpc away from the Earth. The spatial number density of 900–1 050 K dwarf is 
$(2.0\pm0.9) \times10^{-6}\text{ pc}^{-3}$, 1 050–1 200 K dwarf is 
$(1.2\pm0.7) \times10^{-6}\text{ pc}^{-3}$, and 1 200–1 350 K dwarf is 
$(4.4\pm1.3) \times10^{-6}\text{ pc}^{-3}$. The cumulative number count of our brown dwarf candidates is consistent with the prediction from a standard double exponential model. Three of our brown dwarf candidates were detected by HST, with transverse velocities 
$12\pm5$, 
$12\pm4$, and 
$17\pm6$ km s
$^{-1}$. Along with earlier studies, the JWST has opened a new window of brown dwarf research in the Milky Way thick disk and halo.
The negative impact of adverse childhood experiences (ACEs) on mental health has been well documented. While most of the evidence comes from samples of adolescents and young adults, few studies have investigated whether ACEs contribute to poorer mental health among older adults. In particular, depressive symptoms are common in old age, and they display heterogeneous patterns of development across individuals. Therefore, it is important to examine if ACEs are predictive of distinct trajectories of depressive symptoms among older adults.
Using longitudinal data from the English Longitudinal Study of Ageing (ELSA), we aimed to examine if ACEs could differentiate between distinct trajectories of depressive symptoms over eight years in community-dwelling older adults.
Participants from ELSA aged 60 or above who reported no psychiatric diagnoses and completed the items of ACEs at baseline (wave 3) were included in the current study. Nine items of ACEs were subject to a principal component analysis to identify the underlying subtypes. Data of depressive symptoms from waves 3 to 7 (2-year apart), assessed with the 8-item Centre for Epidemiological Studies Depression Scale, were extracted for modelling the distinct trajectories using latent class growth analysis. The trajectories were predicted by subtypes of ACEs using multinomial logistic regression, adjusting for childhood socioeconomic status, sex, age and ethnicity.
The final sample consisted of 4057 participants (54.4% female, mean age= 71.34 (SD= 8.14)). We identified five trajectories of depressive symptoms (Figure 1): ‘low stable’ (73.4%), ‘increasing then decreasing’ (9.9%), ‘high decreasing’ (7.1%), ‘high stable’ (5.7%) and ‘moderate increasing’ (4.0%). Four subtypes of ACEs (i.e., sexual abuse, separation from natural parents, family dysfunction and physical assault) were evident. Compared to the ‘low stable’ group, higher levels of family dysfunction were reported in the ‘increasing then decreasing’ (aOR = 1.35, 95% CI [1.10 - 1.66], p = .012), ‘high stable’ (aOR = 1.59, 95% CI [1.30 - 1.96], p < .001) and ‘moderate increasing’ (aOR = 1.55, 95% CI [1.18 - 2.04], p = .011) groups. The ‘high stable’ group also reported a higher level of separation from natural parents than the ‘low stable’ group (aOR = 1.34, 95% CI [1.04 - 1.72], p = .047). Sexual abuse and physical assault did not predict any group differences.
Image:
Distinct trajectories of depressive symptoms among older adults were predicted by family dysfunction in childhood. Our findings suggested that the negative impact of ACEs on mental health may extend beyond adolescence and young adulthood into the old age.
None Declared
