Stratifying systems, which have been defined for module, triangulated and exact categories previously, were developed to produce examples of standardly stratified algebras. A stratifying system $\Phi$ is a finite set of objects satisfying some orthogonality conditions. One very interesting property is that the subcategory $\mathcal{F}(\Phi )$ of objects admitting a composition series-like filtration with factors in $\Phi$ has the Jordan-Hölder property on these filtrations. This article has two main aims. First, we introduce notions of subobjects, simple objects and composition series for an extriangulated category, in order to define a Jordan-Hölder extriangulated category. Moreover, we characterise Jordan-Hölder, length, weakly idempotent complete extriangulated categories in terms of the associated Grothendieck monoid and Grothendieck group. Second, we develop a theory of stratifying systems in extriangulated categories. We define projective stratifying systems and show that every stratifying system $\Phi$ in an extriangulated category is part of a minimal projective one $(\Phi ,Q)$. We prove that $\mathcal{F}(\Phi )$ is a length, Jordan-Hölder extriangulated category when $(\Phi ,Q)$ satisfies a left exactness condition. We give several examples and answer a recent question of Enomoto–Saito in the negative.

Maintaining optimal growth of preterm infants after hospital discharge remains a challenge. There has been no data on the long-term growth trajectory of preterm infants in Indonesia. We aimed to describe the growth trajectory of preterm infants up to 24 months of corrected age and its variation among gestational age groups. A longitudinal study was conducted in Cipto Mangunkusumo Hospital, Jakarta, from 2018 to 2020. All preterm infants who were discharged during the study period were included. Growth trajectory analysis used weight-for-age, length-for-age and weight-for-length z-score of 3-month time intervals across gestational age groups using repeated measure ANOVA and generalised estimating equation regression. Length trajectory was specifically reported as a stunted proportion. Among 306 preterm infants included, most were moderate preterm (49·67 %) and low birth weight (69·93 %). Overall weight-for-age at 0 months was in the median of the curve, then decreased at 3 months but consistently increased slowly until 24 months. The weight-for-age trends were unique across gestational age groups but statistically similar (P= 0·263). The proportion of stunted gradually decreases to 13·40 % at 24 months, mostly among the moderate preterm group in the first 6 months (P< 0·001), but then becomes similar at 24 months. All subjects were in the normal range for weight-for-length but had variations in trends across gestational age groups (P< 0·001). Growth trajectory differed between weight, length and weight-for-length in the first 24 months and varied among gestational age groups. Close follow-up is crucial to ensure optimal growth after neonatal intensive care unit discharge.

The ocean quahog, Arctica islandica, is a commercially important bivalve in the eastern USA but very little is known about the recruitment frequency and rebuilding capacity of this species. As the longest-living bivalve on Earth, A. islandica can achieve lifespans in excess of 200 y; however, age determinations are difficult to estimate and age variability at size is extreme. Objectives for this study included the creation of an extremely large age-composition dataset to constrain age at length variability, development of reliable age-length keys (ALK), and descriptions of sex-based population dynamics for the quasi-virgin A. islandica population at Georges Bank (GB) within the greater US Mid-Atlantic stock. Sexually dimorphic characteristics are clearly present, as females are larger than males within age classes and males tend to dominate the oldest age classes. A male represented the maximum age of 261 years and is older than the maximum age previously documented for this region. Sex-specific ALKs were robust and reliable but not interchangeable. This population had higher estimated natural mortality rates than presumed for other regions in the Mid-Atlantic, and females have the highest mortality rate. However, recruitment expansion was also occurring which would affect the age-frequency data used to derive mortality estimates and result in higher mortality. Age frequencies at GB suggest effective recruitment to the population each year since 1867 CE. Reduced recruitment periods are documented and likely attributed to fluctuating environmental conditions. Sex-based demographics are clearly divergent in regard to growth rate, maximum size, longevity and mortality rates.

We prove that most permutations of degree $n$ have some power which is a cycle of prime length approximately $\log n$. Explicitly, we show that for $n$ sufficiently large, the proportion of such elements is at least $1-5/\log \log n$ with the prime between $\log n$ and $(\log n)^{\log \log n}$. The proportion of even permutations with this property is at least $1-7/\log \log n$.

Let $\Omega $ be a connected open set in the plane and $\gamma : [0,1] \to \overline {\Omega }$ a path such that $\gamma ((0,1)) \subset \Omega $. We show that the path $\gamma $ can be “pulled tight” to a unique shortest path which is homotopic to $\gamma $, via a homotopy h with endpoints fixed whose intermediate paths $h_t$, for $t \in [0,1)$, satisfy $h_t((0,1)) \subset \Omega $. We prove this result even in the case when there is no path of finite Euclidean length homotopic to $\gamma $ under such a homotopy. For this purpose, we offer three other natural, equivalent notions of a “shortest” path. This work generalizes previous results for simply connected domains with simple closed curve boundaries.

Twenty one samples of relatively pure tubular halloysites (HNTs) from localities in Australia, China, New Zealand, Scotland, Turkey and the USA have been investigated by X-ray diffraction (XRD), infrared spectroscopy (IR) and electron microscopy. The halloysites occur in cylindrical tubular forms with circular or elliptical cross sections and curved layers and also as prismatic tubular forms with polygonal cross sections and flat faces. Measurements of particle size indicate a range from 40 to 12,700 nm for tube lengths and from 20 to 600 nm for diameters. Size distributions are positively skewed with mean lengths ranging from 170 to 950 nm and mean diameters from 50 to 160 nm. Cylindrical tubes are systematically smaller than prismatic ones. Features related to order/ disorder in XRD patterns e.g. as measured by a ‘cylindrical/prismatic’ (CP) index and IR spectra as measured by an ‘OH-stretching band ratio’ are related to the proportions of cylindrical vs. prismatic tubes and correlated with other physical measurements such as specific surface area and cation exchange capacity. The relationships of size to geometric form, along with evidence for the existence of the prismatic form in the hydrated state and the same 2M 1 stacking sequence irrespective of hydration state (i.e. 10 vs. 7 Å) or form, suggests that prismatic halloysites are the result of continued growth of cylindrical forms.

The aims of this study were to develop and validate a prediction equation of fat-free mass (FFM) based on bioelectrical impedance analysis (BIA) and anthropometry using air-displacement plethysmography (ADP) as a reference in Asian neonates and to test the applicability of the prediction equations in an independent Western cohort. A total of 173 neonates at birth and 140 at two weeks of age were included. Multiple linear regression analysis was performed to develop the prediction equations in a two-third randomly selected subset and validated on the remaining one-third subset at each time point and in an independent Queensland cohort. FFM measured by ADP was the dependent variable, and anthropometric measures, sex and impedance quotient (L2/R50) were independent variables in the model. Accuracy of prediction equations was assessed using intra-class correlation and Bland–Altman analyses. L2/R50 was the significant predictor of FFM at week two but not at birth. Compared with the model using weight, sex and length, including L2/R50 slightly improved the prediction with a bias of 0·01 kg with 2 sd limits of agreement (LOA) (0·18, −0·20). Prediction explained 88·9 % of variation but not beyond that of anthropometry. Applying these equations to the Queensland cohort provided similar performance at the appropriate age. However, when the Queensland equations were applied to our cohort, the bias increased slightly but with similar LOA. BIA appears to have limited use in predicting FFM in the first few weeks of life compared with simple anthropometry in Asian populations. There is a need for population- and age-appropriate FFM prediction equations.