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Given a precompact domain $Ω\subseteq\mathbb{R}^2$, the electrostatic skeleton of $Ω$ is defined as a positive measure inside $Ω$, supported on a set with no simple loops, which generates $\partial Ω$ as an equipotential curve. Eremenko conjectured that every convex polygon admits a unique electrostatic skeleton. This conjecture has since been proven for triangles and regular polygons. In this paper, we will prove the conjecture for quadrilaterals with a line of symmetry using arguments from conformal geometry. We will also discuss a natural condition that implies the existence of electrostatic skeletons.

The ultrametric skeleton theorem extracts from every compact metric probability space a subset of ultrametric distortion $O(1/\varepsilon)$ that carries a measure whose balls are controlled by the $(1-\varepsilon)$-power of the original measure on dilated concentric balls. We prove a two-sided version for arbitrary compact metric spaces: for every ball centered on the skeleton, the skeleton measure also has a lower bound in terms of the original measure on a smaller nonconcentric ball contained in it. We also give a short proof of the original skeleton theorem and improve the dilation of its control balls from $\exp(O(1/\varepsilon^2))$ to $O(1/\varepsilon)$.

We introduce the notion of coarse bottlenecking in graphs and coarse skeletons of graphs and show how bottlenecking guarantees that a skeleton resembles (up to quasi-isometry) the original graph. We show how these tools can be used to simplify the structure of graphs upto quasi-isometry that have an excluded asymptotic minor, reducing it to a skeleton of the original containing no $3$-fat minor. We give an example to show that a similar result does not hold for $2$-fat minors. This makes progress towards a Conjecture posed by Georgakopoulos and Papasoglu.

We introduce additively-weighted straight skeletons as a new generalization of straight skeletons. An additively-weighted straight skeleton is the result of a wavefront-propagation process where, unlike in previous variants of straight skeletons, wavefront edges do not necessarily start to move at the begin of the propagation process but at later points in time. We analyze the properties of additively-weighted straight skeletons and show how to compute straight skeletons with both additive and multiplicative weights. We then show how to use additively-weighted and multiplicatively-weighted straight skeletons to generate roofs and terrains for polygonal shapes such as the footprints of buildings or river networks. As a result, we get an automated generation of roofs and terrains where the individual facets have different inclinations and may start at different heights.

As collaborative robots (cobots) continue to gain popularity in industrial manufacturing, effective human-robot collaboration becomes crucial. Cobots should be able to recognize human actions to assist with assembly tasks and act autonomously. To achieve this, skeleton-based approaches are often used due to their ability to generalize across various people and environments. Although body skeleton approaches are widely used for action recognition, they may not be accurate enough for assembly actions where the worker's fingers and hands play a significant role. To address this limitation, we propose a method in which less detailed body skeletons are combined with highly detailed hand skeletons. We investigate CNNs and transformers, the latter of which are particularly adept at extracting and combining important information from both skeleton types using attention. This paper demonstrates the effectiveness of our proposed approach in enhancing action recognition in assembly scenarios.

The unsteady motion of a gas-liquid interface, such as during splashing or atomization, often results in complex liquid structures embedded in the ambient fluid. Here we explore the use of skeletonization to identify the minimum amount of information needed to describe their geometry. We skeletonize a periodic liquid jet by a modification of a recently introduced approach to coarsen multiphase flows while retaining a sharp interface. The process consists of diffusing an index function and at the same time moving the interfaces with it, until they "collapse" into each other and form skeletons. The skeleton represents the basic topology of the jet and we also keep track of how much the interface is moved (or how much volume is "accumulated") during the process, which can be used to approximately reconstruct the jet. We explore various quantitative measures to characterize and distinguish the skeletons. Those include standard morphometrics such as branch length distribution, after segmenting the skeletons into branches, and a more sophisticated representation of the skeleton structures called Topology Morphology Descriptor (TMD), to obtain an "equivalent" description of the skeletons

Algorithmic skeletons are used as building-blocks to ease the task of parallel programming by abstracting the details of parallel implementation from the developer. Most existing libraries provide implementations of skeletons that are defined over flat data types such as lists or arrays. However, skeleton-based parallel programming is still very challenging as it requires intricate analysis of the underlying algorithm and often uses inefficient intermediate data structures. Further, the algorithmic structure of a given program may not match those of list-based skeletons. In this paper, we present a method to automatically transform any given program to one that is defined over a list and is more likely to contain instances of list-based skeletons. This facilitates the parallel execution of a transformed program using existing implementations of list-based parallel skeletons. Further, by using an existing transformation called distillation in conjunction with our method, we produce transformed programs that contain fewer inefficient intermediate data structures.

Gait recognition captures gait patterns from the walking sequence of an individual for identification. Most existing gait recognition methods learn features from silhouettes or skeletons for the robustness to clothing, carrying, and other exterior factors. The combination of the two data modalities, however, is not fully exploited. Previous multimodal gait recognition methods mainly employ the skeleton to assist the local feature extraction where the intrinsic discrimination of the skeleton data is ignored. This paper proposes a simple yet effective Bimodal Fusion (BiFusion) network which mines discriminative gait patterns in skeletons and integrates with silhouette representations to learn rich features for identification. Particularly, the inherent hierarchical semantics of body joints in a skeleton is leveraged to design a novel Multi-Scale Gait Graph (MSGG) network for the feature extraction of skeletons. Extensive experiments on CASIA-B and OUMVLP demonstrate both the superiority of the proposed MSGG network in modeling skeletons and the effectiveness of the bimodal fusion for gait recognition. Under the most challenging condition of walking in different clothes on CASIA-B, ou

Uniquely closable skeletons of lambda terms are Motzkin-trees that predetermine the unique closed lambda term that can be obtained by labeling their leaves with de Bruijn indices. Likewise, uniquely typable skeletons of closed lambda terms predetermine the unique simply-typed lambda term that can be obtained by labeling their leaves with de Bruijn indices. We derive, through a sequence of logic program transformations, efficient code for their combinatorial generation and study their statistical properties. As a result, we obtain context-free grammars describing closable and uniquely closable skeletons of lambda terms, opening the door for their in-depth study with tools from analytic combinatorics. Our empirical study of the more difficult case of (uniquely) typable terms reveals some interesting open problems about their density and asymptotic behavior. As a connection between the two classes of terms, we also show that uniquely typable closed lambda term skeletons of size $3n+1$ are in a bijection with binary trees of size $n$.

Skeletons of branching processes are defined as trees of lineages characterized by an appropriate signature of future reproduction success. In the supercritical case a natural choice is to look for the lineages that survive forever. In the critical case it was earlier suggested to distinguish the particles with the total number of descendants exceeding a certain threshold. These two definitions lead to asymptotic representations of the skeletons as either pure birth process (in the slightly supercritical case) or critical birth-death processes (in the critical case conditioned on the total number of particles exceeding a high threshold value). The limit skeletons reveal typical survival scenarios for the underlying branching processes. In this paper we consider near-critical Bienaymé-Galton-Watson processes and define their skeletons using marking of particles. If marking is rare, such skeletons are approximated by birth and death processes which can be subcritical, critical or supercritical. We obtain the limit skeleton for a sequential mutation model and compute the density distribution function for the time to escape from extinction.

The extension functors between categories of Cartan geometries can be used to define different categories of Cartan geometries with additional morphisms. The Cartan geometries modeled on skeletons can be used for the description of such categories of Cartan geometries and therefore we develop the theory of Cartan geometries modeled on skeletons, in detail. In particular, we show that there are functors from the categories of Cartan geometries with morphisms induced by extension functors to certain categories of Cartan geometries modeled on skeletons with the usual morphisms. We study how to compute (infinitesimal) automorphisms of Cartan geometries modeled on skeletons and how to compute the morphisms induced by the extension functors for particular Cartan geometries. Some examples and applications of the morphisms induced by extension functors are presented in the setting of the Riemannian geometries.

Modern application development allows applications to be composed using lightweight HTTP services. Testing such an application requires the availability of services that the application makes requests to. However, access to dependent services during testing may be restrained. Simulating the behaviour of such services is, therefore, useful to address their absence and move on application testing. This paper examines the appropriateness of Symbolic Machine Learning algorithms to automatically synthesise HTTP services' mock skeletons from network traffic recordings. These skeletons can then be customised to create mocks that can generate service responses suitable for testing. The mock skeletons have human-readable logic for key aspects of service responses, such as headers and status codes, and are highly accurate.