Combinatorics on number walls and the P ( t ) $P(t)$ ‐adic Littlewood conjecture

Abstract In 2004, de Mathan and Teulié stated the p $p$ ‐adic Littlewood conjecture ( p $p$ ‐LC) in analogy with the classical Littlewood conjecture. LetF q $_q$ be a finite fieldP ( t ) $P(t)$ be an irreducible polynomial with coefficients inF q $_q$ . This paper deals with the analogue of p $p$ ‐LC over the ring of formal Laurent series overF q $_q$ , known as theP ( t ) $P(t)$ ‐adic Littlewood conjecture ( P ( t ) $P(t)$ ‐LC). First, it is shown that any counterexample toP ( t ) $P(t)$ ‐LC for the caseP ( t ) = t $P(t)=t$ induces a counterexample toP ( t ) $P(t)$ ‐LC whenP ( t ) $P(t)$ is any irreducible polynomial. Since Adiceam, Nesharim and Lunnon (2021) disprovedP ( t ) $P(t)$ ‐LC whenP ( t ) = t $P(t)=t$ and whenF q $_q$ is a finite field with characteristic 3, one obtains a disproof ofP ( t ) $P(t)$ ‐LC over any such field in full generality (i.e., for any choice of irreducible polynomialP ( t ) $P(t)$ ). The remainder of the paper is dedicated to proving two metric results on t $t$ ‐LC with an additional monotonic growth function f $f$ over an arbitrary finite field. The first — a Khintchine‐type theorem for t $t$ ‐adic multiplicative approximation — enables one to determine the measure of the set of counterexamples to t $t$ ‐LC for any choice of f $f$ . The second complements this by showing that the Hausdorff dimension of the same set is maximal in the critical case wheref = log 2 $f=\log ^2$ . These results are in agreement with the corresponding theory of multiplicative Diophantine approximation over the reals. Beyond the originality of the results, the main novelty of the work comes from the methodology used. Classically, Diophantine approximation employs methods from either Number Theory or Ergodic Theory. This paper provides a third option: combinatorics. Specifically, an extensive combinatorial theory is developed relatingP ( t ) $P(t)$ ‐LC to the properties of the so‐called number wall of a sequence. This is an infinite array containing the determinant of every finite Toeplitz matrix generated by that sequence. In full generality, the paper creates a dictionary allowing one to transfer statements in Diophantine approximation in positive characteristic to combinatorics through the concept of a number wall, and conversely.

Exercise FITT ‐V in pregnancy with obesity: Preliminary findings for infant adiposity and intergenerational obesity risk

Abstract Prenatal exercise decreases offspring adiposity, but it is uncertain whether this relationship is present in offspring exposed to obesity in utero. We aimed to determine whether exercise during pregnancy reduces infant cellular and whole‐body adiposity in offspring born to women with obesity. This is a sub‐analysis of a randomized controlled trial, where women were randomized to supervised exercise or control for ~24 weeks during pregnancy. Exercise FITT‐V metrics (frequency, intensity, time, type, and volume) were collected. Infant mesenchymal stem cells (MSCs) (healthy weight [ n  = 16], obesity [ n  = 21]) were adipogenically differentiated and stained for lipid content. Infant body composition was measured at 1 month of age via skinfold. Among women randomized to control, maternal BMI influenced infant adiposity; infants exposed to obesity had higher body fat percentage ( p  = 0.02). Birthweight was negatively correlated with infant body fat; offspring with lower birthweight had higher body fat ( R 2  = 0.38, p  = 0.03). Maternal weekly exercise volume trended toward negative association with infant body fat ( R 2  = 0.33, p  = 0.06) and lipid content ( R 2  = 0.21, p  = 0.06). For infants born to women with obesity, exercise during pregnancy helps reduce adiposity.

Pilot investigation into the influence of racial implicit bias on physician clinical reasoning

Deliberate solitude for clinical reasoning

While modern medicine emphasizes teamwork, expert performance in clinical reasoning may require periods of deliberate solitude to refine intuition, enhance diagnostic and/or management accuracy, and mitigate potential cognitive biases. Evidence from cognitive psychology, philosophy, and education suggests that cognitive withdrawal supports deep learning and problem-solving, yet its role in clinical reasoning learning and performance remains underexplored. Medicine often prioritizes speed and real-time collaboration, potentially limiting opportunities for independent time for thought. This article explores whether deliberate solitude could support the development and performance of clinical reasoning. Clinicians might consider engaging in diagnostic rehearsal, independent synthesis, and/or cognitive withdrawal during these solitary moments, but the specific opportunities and benefits remain uncertain. By drawing from research in other disciplines, we consider how solitude might help physicians refine their clinical reasoning, which, in turn, could potentially reduce errors. While no specific course of action can yet be made, this conceptual perspective suggests potential directions for future inquiry.

Problem Solving Criminal Justice

From the mid-2000s to the mid-2010s, the politics of criminal justice took a sudden turn from highly politicized punitiveness, to lower-salience problem solving that led to a range of breakthroughs that reduced the growth in incarceration. This shift was driven by decreasing party competition at the state level, the increasing space for reform produced by lower public salience of crime, and the creative advocacy and political entrepreneurship of advocates. This period of problem solving appears to be over, driven by increasing public concern about crime and repolarization in response to social movement activism. Lessons are offered that advocates and funders can draw from this case, especially concerning the linkage between salience and problem-solving reform.

Osteopathic manipulative treatment vs. standard therapy in the management of acute neck and low back pain in the emergency department

The small mammals of the Maromizaha protected area, central eastern Madagascar: the ability of some Malagasy “forest-dwelling” species to adapt to habitat degradation

All of the native terrestrial small mammals of Madagascar, comprising 59 species (Tenrecidae and Nesomyinae), are endemic and most considered as forest-dependent. However, recent fieldwork in degraded natural forest areas has found that certain species can live in disturbed habitats. Herein based on fieldwork in central eastern Madagascar and employing systematic trapping (live traps and pitfall lines), we test the importance of habitat quality for a range of species in three contiguous moist evergreen forest habitat types: relatively intact, slightly degraded, and secondary. We also employ data from two sites (slightly degraded and secondary) on levels of disturbance, presence of introduced species (Muridae and Soricidae), and soil parameters, to examine possible correlates. The two sites with relatively intact and slightly disturbed forest habitats have similar native small mammal faunas. In contrast, the secondary habitat is depauperate, although with certain tenrecid species that were previously presumed to be “forest-dependent”. Introduced species have not heavily colonized the largely intact or degraded habitats. Our survey data indicate an important level of adaptability of previously presumed “forest-dependent” or “largely forest-dependent” small mammal species to different levels of human disturbance, although at the secondary site, species diversity is notably lower.

‘The mountain has a story to tell’: transhumant production systems in the US Intermountain West

Grazing systems research has taken a notable social-ecological turn in response to recent debates, and integrative work is still needed to address gaps in our understanding of multiscalar dynamics in transhumant systems. Transhumance encompasses diverse cultures, ecological relationships, and traditions across global rangelands. Research illuminating the social-ecological relationships, risks and opportunities for adaptation presented by transhumant systems can inform decision-making for sustainable outcomes. In this paper, we take a multiscalar social-ecological systems approach to explore sheep transhumance in the US Intermountain West. We explore both how and why range sheep producers graze extensive groups (bands, i.e. 500–2 500 head) of dual-purpose sheep across a complex web of economic, bureaucratic and ecological dynamics. We use a four-year social-ecological field study – including participant observation and semi-structured interviews, vegetation and animal monitoring, and grazing records – to describe the intermountain transhumant system in eastern Idaho. We consider how patterns of multi-species interactions inform knowledge of grazing and social-ecological systems and illuminate a more practical list of risks, challenges and opportunities for adaptation presented by the case study. Here, the structure and function of land management governance and food system infrastructure are as important to social-ecological outcomes as are the structure and function of ecosystem processes.

Determining the Scale of Herbarium Sheet Images via Ruler Detection

The digitization and online availability of herbarium specimens have increased the need for automated tools to extract specimen traits at scale. Quantitative trait data, such as leaf area or petal length, are critical for ecology, evolution, and climate change research. In recent years, automated methods such as LeafMachine2 ( Weaver and Smith 2023 ) and Mothra ( Wilson et al. 2022 ) have been developed to measure these traits from images, supporting large-scale studies. However, such image measurements need to be converted from pixels to standardized units (e.g., centimeters) to be useful. Existing methods estimate this conversion factor (CF) by detecting the ruler type and the tick mark locations directly. From these image coordinates, the average pixel distance between them can be calculated to estimate the CF. These methods typically require many annotated samples for each ruler type and tend to perform poorly when applied to unseen or visually distinct rulers.To overcome these limitations, we present a generic pipeline that accurately predicts the CF with few labeled rulers. The approach leverages the fact that many institutions reuse identical rulers, often differing only in institutional logos. Instead of directly classifying or predicting the tick mark locations, the pipeline uses a similarity-based approach. First, rulers on the page are detected using a YOLOv11 object detection model. Second, these rulers are cropped and normalized, ensuring the longest side is horizontal. After detection, we use Contrastive Language-Image Pre-training ( CLIP) to embed each ruler and to retrieve the three most similar ruler candidates from our ground-truth dataset. Next, we use the LightGlue keypoint-matching model to detect corresponding keypoints between the query ruler and each retrieved candidate. This model detects and scores similar keypoint pairs (pixel coordinates) between two input image pairs. The candidate ruler with the highest average score across the 100 best keypoint pairs is selected as the best match. If none of the candidates had at least 10 matching keypoint pairs, we consider the query ruler a “no-match”. Finally, these keypoint pairs are used to estimate a homography , which describes how one image can be transformed to align with another taken from a different viewpoint. This transformation is then applied to the ground truth tick mark locations to calculate the CF for the query ruler.To evaluate the proposed method, we constructed a new dataset by randomly selecting 400 herbarium sheets from the Global Biodiversity Information Facility ( GBIF ), PlantCLEF2020 , and Dillen et al. 2019 . Each ruler was annotated with its bounding polygon and a type. Ruler tick marks were annotated with their coordinate locations and units (e.g., ticks_1cm). The dataset contains a diverse set of 476 rulers from over 40 institutions. We estimated our measurement error at 0.794%, which is in line with the reported 0.8% from LeafMachine2 , by calculating the average standard deviation of pixel distances between each tick mark. We evaluated our approach using a leave-one-out cross-validation, treating each ruler crop as a query and the remaining dataset as candidate matches. We also evaluated our method on the 1131 rulers from LeafMachine2 , which were annotated with a bounding box around the tick marks (see Fig. 1 ).Rulers without a matching type were excluded for evaluation. The method achieved an average CF error of 0.689% using the best-matching ruler. 94.8% of rulers had a CF error less than 2%. On the LeafMachine2 dataset, performance improved with an average CF error of 0.324%. There, 99.0% of rulers had a CF error less than 2%. We plan to investigate how different annotation strategies affect performance. Table 1 details these evaluation results.The proposed method offers several advantages over existing approaches. First, it allows an accurate conversion with fewer annotated rulers of the same type, essentially needing only one. Second, the keypoint matching process is robust to partial detections, occlusions, and low-resolution rulers, as visualized in Fig. 2 . Third, only the detection model needs minimal fine-tuning when new ruler types are added to the dataset, as the matching models remain unchanged. Additionally, the prediction can still be accurate when the best-matching ruler is from a different institution (see Fig. 3 ).The main limitation of our method is that it requires at least one annotated ruler of a similar type to result in a good match. It also assumes the tick marks are on a straight line, resulting in poorer performance for curved rulers. Finally, we found that simple rulers, such as straight lines, tend to perform worse (see Fig. 3 ).This work represents part of our ongoing research to assist trait extraction at scale. We plan to expand the dataset with additional ruler types, including examples from other natural history collections. We will further refine the method to improve its robustness and generalization across institutions and to explore fallback methods when no match is found. The code and dataset are available on GitHub* 1 .

Clarifying clinical findings in Hoffmann syndrome and myxedema