Leaves shed by deciduous trees contain 40\% of the annually sequestered carbon, and include nutrients vital to the expansion and health of forest ecosystems. To achieve this, leaves must fall quickly to land near the parent tree -- otherwise, they are lost to the wind, like pollen or gliding seeds. However, the link between leaf shape and sedimentation speed remains unclear. To gauge the relative performance of extant leaves, we developed an automated sedimentation apparatus (ASAP) capable of performing $\sim100$ free fall experiments per day on biomimetic paper leaves. The majority of 25 representative leaves settle at rates similar to our control (a circular disc). Strikingly, the Arabidopsid mutant asymmetric leaves1 (as1) fell 15\% slower than the wild type. Applying the as1-digital mutation to deciduous tree leaves revealed a similar speed reduction. Data correlating shape and settling across a broad range of natural, mutated, and artificial leaves support thefast-leaf-hypothesis: Deciduous leaves are symmetric and relatively unlobed in part because this maximizes their settling speed and concomitant nutrient retention.
The human visual environment is comprised of different surfaces that are distributed in space. The parts of a scene that are visible at any one time are governed by the occlusion of overlapping objects. In this work we consider "dead leaves" models, which replicate these occlusions when generating images by layering objects on top of each other. A dead leaves model is a generative model comprised of distributions for object position, shape, color and texture. An image is generated from a dead leaves model by sampling objects ("leaves") from these distributions until a stopping criterion is reached, usually when the image is fully covered or until a given number of leaves was sampled. Here, we describe a theoretical approach, based on previous work, to derive a Bayesian ideal observer for the partition of a given set of pixels based on independent dead leaves model distributions. Extending previous work, we provide step-by-step explanations for the computation of the posterior probability as well as describe factors that determine the feasibility of practically applying this computation. The dead leaves image model and the associated ideal observer can be applied to study segmentati
We study the extreme local structure of plane binary trees through the distribution of leaves at maximum depth. We first address two basic questions: (i) the asymptotic probability that exactly two leaves occur at the deepest level, and (ii) the asymptotic mean number of leaves at that level. These problems lead to generating functions coupled with the Catalan iteration $I_{k+1}(z)=1+zI_k(z)^2$ through quasi-logistic recurrences. We show that both associated series have dominant singularity $ρ=1/4$ and admit square-root singular expansions. The singular terms are obtained through a three-zone dominated-convergence analysis of the critical scaling regime of the truncation error. We then extend the framework to derive the full limiting distribution of the number of deepest leaves. Enumerating trees with exactly $2m$ deepest leaves yields a hierarchy of differential equations that reduces to successive polynomial integrations. Encoding these parameters into a bivariate generating function transforms the nonlinear dynamics back into the Catalan recurrence. Using continuous iteration theory and the Fatou coordinate associated with an Abel equation, we obtain a functional equation charac
Current vision-language benchmarks predominantly feature well-structured questions with clear, explicit prompts. However, real user queries are often informal and underspecified. Users naturally leave much unsaid, relying on images to convey context. We introduce HAERAE-Vision, a benchmark of 653 real-world visual questions from Korean online communities (0.76% survival from 86K candidates), each paired with an explicit rewrite, yielding 1,306 query variants in total. Evaluating 39 VLMs, we find that even state-of-the-art models (GPT-5, Gemini 2.5 Pro) achieve under 50% on the original queries. Crucially, query explicitation alone yields 8 to 22 point improvements, with smaller models benefiting most. We further show that even with web search, under-specified queries underperform explicit queries without search, revealing that current retrieval cannot compensate for what users leave unsaid. Our findings demonstrate that a substantial portion of VLM difficulty stem from natural query under-specification instead of model capability, highlighting a critical gap between benchmark evaluation and real-world deployment.
Using newly-assembled data from 1980 through 2024, we show that 25% of scientifically-active, US-trained STEM PhD graduates leave the US within 15 years of graduating. Leave rates are lower in the life sciences and higher in AI and quantum science but overall have been stable for decades. Contrary to common perceptions, US technology benefits from these graduates' work even if they leave: though the US share of global patent citations to graduates' science drops from 70% to 50% after migrating, it remains five times larger than the destination country share, and as large as all other countries combined. These results highlight the value that the US derives from training foreign scientists - not only when they stay, but even when they leave.
For a Poisson manifold endowed with a pseudo-Riemannian metric, we investigate degeneracies arising when the metric is restricted to symplectic leaves. Central to this work is the generalized double bracket (GDB) vector field-a geometric construct introduced in our earlier work-which generalizes gradient dynamics to indefinite metric settings. We identify admissible regions where the so-called double bracket metric remains non-degenerate on symplectic leaves, enabling the GDB vector field to function as a gradient flow on the admissible regions with respect to this metric. We illustrate these concepts with a variety of examples and carefully discuss the complications that arise when the pseudo-Riemannian metric fails to induce a non-degenerate metric on certain regions of the symplectic leaves.
We prove that if the leaves of a minimal Lie foliation are locally isometric to a symmetric space of non-compact type without a Poincare disk factor, then the foliation is smoothly conjugate to a homogeneous Lie foliation up to finite covering. This result generalizes and strengthens Zimmer's theorem, which characterizes minimal Lie foliations with leaves isometric to a symmetric space of non-compact type without real rank one factors as pullbacks of homogeneous foliations. As applications, we extend Zimmer's arithmeticity theorem for holonomy groups and establish a rigidity theorem for Riemannian foliations with locally symmetric leaves.
The main goal of this present paper is to bring the results proved by Barbosa, Kenmotsu and Oshikiri (1991) and its ideas to a perspective where the Ricci curvature is bounded from below. For instance, for a foliation by CMC hypersurfaces on a compact (without boundary) Riemannian manifold $M^{n+1}$ with Ricci curvature bounded from below by $-nK_0\leq 0$ and such that the mean curvature $H$ of the leaves of the foliation satisfies $|H|\geq \sqrt{K_0}$, we prove that $|H|\equiv \sqrt{K_0}$ and all the leaves are totally umbilical. This gives, in particular, a generalization for the result proved by Barbosa, Kenmotsu and Oshikiri (1991), where the above result was proved in the case $K_0=0$. We also obtain a proof of the following: for a foliation by CMC hypersurfaces on a compact (without boundary) Riemannian manifold $M$ with Ricci curvature bounded from below by $-nK_0\leq 0$, the mean curvature $H$ of the leaves of the foliation satisfies $|H|\leq \sqrt{K_0}$. Furthermore, if the foliation contains a leaf $L$ whose absolute mean curvature is $|H_L|=\sqrt{K_0}$, then either $K_0=0$ and all the leaves of $\mathfrak{F}$ are totally geodesic, or $K_0>0$ and there is a totally umb
A compact Polish foliated space is considered. Part of this work studies coarsely quasi-isometric invariants of leaves in some residual saturated subset when the foliated space is transitive. In fact, we also use "equi-" versions of this kind of invariants, which means that the definition is satisfied with the same constants by some given set of leaves. For instance, the following properties are proved. Either all dense leaves without holonomy are equi-coarsely quasi-isometric to each other, or else there exist residually many dense leaves without holonomy such that each of them is coarsely quasi-isometric to meagerly many leaves. Assuming that the foliated space is minimal, the first of the above alternatives holds if and if the leaves without holonomy satisfy a condition called coarse quasi-symmetry. A similar dichotomy holds for the growth type of the leaves, as well as an analogous characterization of the first alternative in the minimal case, involving a property called growth symmetry. Moreover some classes of growth are shared, either by residually many leaves, or by meagerly many leaves. If some leaf without holonomy is amenable, then all dense leaves without holonomy are e
In this work we exhibit examples of $5$-manifolds that are not homeomorphic to any leaf of any $C^2$ codimension one foliation of any compact $6$-manifold but are homeomorphic to (proper) leaves of some $C^1$ codimension one foliations and also to (proper) leaves of some $C^\inf$ codimension $2$ foliations. As far as we know, this is the first example of this nature. In addition, it is shown examples of $C^{r}$ codimension one foliations, $r\in[0,2)$, with a minimal invariant set whose leaves are pairwise nonhomeomorphic.
We present a generalization of a combinatorial result by Aggarwal, Guibas, Saxe and Shor [Discrete & Computational Geometry, 1989] on a linear-time algorithm that selects a constant fraction of leaves, with pairwise disjoint neighborhoods, from a binary tree embedded in the plane. This result of Aggarwal et al. is essential to the linear-time framework, which they also introduced, that computes certain Voronoi diagrams of points with a tree structure in linear time. An example is the diagram computed while updating the Voronoi diagram of points after deletion of one site. Our generalization allows that only a fraction of the tree leaves is considered, and it is motivated by linear-time Voronoi constructions for non-point sites. We are given a plane tree $T$ of $n$ leaves, $m$ of which have been marked, and each marked leaf is associated with a neighborhood (a subtree of $T$) such that any two topologically consecutive marked leaves have disjoint neighborhoods. We show how to select in linear time a constant fraction of the marked leaves having pairwise disjoint neighborhoods.
Lesion detection on plant leaves is a critical task in plant pathology and agricultural research. Identifying lesions enables assessing the severity of plant diseases and making informed decisions regarding disease control measures and treatment strategies. To detect lesions, there are studies that propose well-known object detectors. However, training object detectors to detect small objects such as lesions can be problematic. In this study, we propose a method for lesion detection on plant leaves utilizing class activation maps generated by a ResNet-18 classifier. In the test set, we achieved a 0.45 success rate in predicting the locations of lesions in leaves. Our study presents a novel approach for lesion detection on plant leaves by utilizing CAMs generated by a ResNet classifier while eliminating the need for a lesion annotation process.
We introduce leave-one-out unfairness, which characterizes how likely a model's prediction for an individual will change due to the inclusion or removal of a single other person in the model's training data. Leave-one-out unfairness appeals to the idea that fair decisions are not arbitrary: they should not be based on the chance event of any one person's inclusion in the training data. Leave-one-out unfairness is closely related to algorithmic stability, but it focuses on the consistency of an individual point's prediction outcome over unit changes to the training data, rather than the error of the model in aggregate. Beyond formalizing leave-one-out unfairness, we characterize the extent to which deep models behave leave-one-out unfairly on real data, including in cases where the generalization error is small. Further, we demonstrate that adversarial training and randomized smoothing techniques have opposite effects on leave-one-out fairness, which sheds light on the relationships between robustness, memorization, individual fairness, and leave-one-out fairness in deep models. Finally, we discuss salient practical applications that may be negatively affected by leave-one-out unfai
Let $G$ be a connected graph and $L(G)$ the set of all integers $k$ such that $G$ contains a spanning tree with exactly $k$ leaves. We show that for a connected graph $G$, the set $L(G)$ is contiguous. It follows from work of Chen, Ren, and Shan that every connected and locally connected $n$-vertex graph -- this includes triangulations -- has a spanning tree with at least $n/2 + 1$ leaves, so by a classic theorem of Whitney and our result, in any plane $4$-connected $n$-vertex triangulation one can find for any integer $k$ which is at least $2$ and at most $n/2 + 1$ a spanning tree with exactly $k$ leaves (and each of these trees can be constructed in polynomial time). We also prove that there exist infinitely many $n$ such that there is a plane $4$-connected $n$-vertex triangulation containing a spanning tree with $2n/3$ leaves, but no spanning tree with more than $2n/3$ leaves.
For any Coxeter system we introduce the concept of singular light leaves, answering a question of Williamson raised in 2008. They provide a combinatorial basis for Hom spaces between singular Soergel bimodules.
A long-term study of the elements Mg, Al, Si, P, Ca, S, Cl, Fe and Mn in leaves is in progress. The objective of this study is to develop a week-by-week profile of these elements in leaves during several growing seasons. The profile includes the following information: (1) Which elements each tree collects in its leaves. (2) The location in the leaf with the highest concentration, top side, under side or interior. (3) The week during the growing season when each element first appears in the leaves of each tree. (4) The change in the relative concentration from week to week. (5) The source of the element i.e., deposition from the atmosphere or the root system of the tree. This information is profile for each year and will be correlated with environmental conditions for that year. Leaves are collected weekly from first unfolding in early spring until leaf drop in the fall. They are from the 31 trees and 26 species in Broome County, NY. From time to time leaves from most of the 26 species are being randomly collected from trees growing throughout the northeastern US.
The $\text{PSL}(4,\mathbb{R})$ Hitchin component of a closed surface group $π_1(S)$ consists of holonomies of properly convex foliated projective structures on the unit tangent bundle of $S$. We prove that the leaves of the codimension-$1$ foliation of any such projective structure are all projectively equivalent if and only if its holonomy is Fuchsian. This implies constraints on the symmetries and shapes of these leaves. We also give an application to the topology of the non-${\rm T}_0$ space $\mathfrak{C}(\mathbb{RP}^n)$ of projective classes of properly convex domains in $\mathbb{RP}^n$. Namely, Benzécri asked in 1960 if every closed subset of $\mathfrak{C}(\mathbb{RP}^n)$ that contains no proper nonempty closed subset is a point. Our results imply a negative resolution for $n \geq 2$.
Standard techniques such as leave-one-out cross-validation (LOOCV) might not be suitable for evaluating the predictive performance of models incorporating structured random effects. In such cases, the correlation between the training and test sets could have a notable impact on the model's prediction error. To overcome this issue, an automatic group construction procedure for leave-group-out cross validation (LGOCV) has recently emerged as a valuable tool for enhancing predictive performance measurement in structured models. The purpose of this paper is (i) to compare LOOCV and LGOCV within structured models, emphasizing model selection and predictive performance, and (ii) to provide real data applications in spatial statistics using complex structured models fitted with INLA, showcasing the utility of the automatic LGOCV method. First, we briefly review the key aspects of the recently proposed LGOCV method for automatic group construction in latent Gaussian models. We also demonstrate the effectiveness of this method for selecting the model with the highest predictive performance by simulating extrapolation tasks in both temporal and spatial data analyses. Finally, we provide insi
In this paper, we establish a number of results about the topology of the leaves of a closed singular Riemannian foliation $(M,\fol)$. If $M$ is simply connected, we prove that the leaves are finitely covered by nilpotent spaces, and characterize the fundamental group of the generic leaves. If $M$ has virtually nilpotent fundamental group, we prove that the leaves have virtually nilpotent fundamental group as well.
We investigate the coarse homology of leaves in foliations of compact manifolds. This is motivated by the observation that the non-leaves constructed by Schweitzer and by Zeghib all have non-finitely generated coarse homology. This led us to ask whether the coarse homology of leaves in a compact manifold always has to be finitely generated. We show that this is not the case by proving that there exist many leaves with non-finitely generated coarse homology. Moreover, we improve Schweitzer's non-leaf construction and produce non-leaves with trivial coarse homology.