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Presbyopia is often the first sign of ageing experienced by humans. Standardising terminology and adopting it across the BCLA CLEAR Presbyopia reports, improves consistency in the communication of the evidence-based understanding of this universal physiological process. Presbyopia can be functionally and psychologically debilitating, especially for those with poor access to eyecare. Presbyopia was defined as occurring when the physiologically normal age-related reduction in the eye's focusing range reaches a point that, when optimally corrected for far vision, the clarity of vision at near is insufficient to satisfy an individual's requirements. Accommodation is the change in optical power of the eye due to a change in crystalline lens shape and position, whereas pseudo-accommodation is the attainment of functional near vision in an emmetropic or far-corrected eye without changing the refractive power of the eye. Other definitions specific to vision and lenses for presbyopia were also defined. It is recommended that these definitions be consistently adopted in order to standardise future research, clinical evaluations and education.

Cataract surgery including intraocular lens (IOL) insertion, has been refined extensively since the first such procedure by Sir Harold Ridley in 1949. The intentional creation of monovision with IOLs using monofocal IOL designs has been reported since 1984. The first reported implantation of multifocal IOLs was published in 1987. Since then, various refractive and or diffractive multifocal IOLs have been commercialised. Most are concentric, but segmented IOLs are also available. The most popular are trifocal designs (overlaying two diffractive patterns to achieve additional focal planes at intermediate and near distances) and extended depth of focus designs which leave the patient largely spectacle independent with the reduced risk of bothersome contrast reduction and glare. As well as mini-monovision, surgical strategies to minimise the impact of presbyopia with IOLs includes mixing and matching lenses between the eyes and using IOLs whose power can be adjusted post-implantation. Various IOL designs to mimic the accommodative process have been tried including hinge optics, dual optics, lateral shifts lenses with cubic-type surfaces, lens refilling and curvature changing approaches, but issues in maintaining the active mechanism with post-surgical fibrosis, without causing ocular inflammation, remain a challenge. With careful patient selection, satisfaction rates with IOLs to manage presbyopia are high and anatomical or physiological complications rates are no higher than with monofocal IOLs.

Cataract surgery including intraocular lens (IOL) insertion, has been refined extensively since the first such procedure by Sir Harold Ridley in 1949. The intentional creation of monovision with IOLs using monofocal IOL designs has been reported since 1984. The first reported implantation of multifocal IOLs was published in 1987. Since then, various refractive and or diffractive multifocal IOLs have been commercialised. Most are concentric, but segmented IOLs are also available. The most popular are trifocal designs (overlaying two diffractive patterns to achieve additional focal planes at intermediate and near distances) and extended depth of focus designs which leave the patient largely spectacle independent with the reduced risk of bothersome contrast reduction and glare. As well as mini-monovision, surgical strategies to minimise the impact of presbyopia with IOLs includes mixing and matching lenses between the eyes and using IOLs whose power can be adjusted post-implantation. Various IOL designs to mimic the accommodative process have been tried including hinge optics, dual optics, lateral shifts lenses with cubic-type surfaces, lens refilling and curvature changing approaches, but issues in maintaining the active mechanism with post-surgical fibrosis, without causing ocular inflammation, remain a challenge. With careful patient selection, satisfaction rates with IOLs to manage presbyopia are high and anatomical or physiological complications rates are no higher than with monofocal IOLs.

Presbyopia is often the first sign of ageing experienced by humans. Standardising terminology and adopting it across the BCLA CLEAR Presbyopia reports, improves consistency in the communication of the evidence-based understanding of this universal physiological process. Presbyopia can be functionally and psychologically debilitating, especially for those with poor access to eyecare. Presbyopia was defined as occurring when the physiologically normal age-related reduction in the eye's focusing range reaches a point that, when optimally corrected for far vision, the clarity of vision at near is insufficient to satisfy an individual's requirements. Accommodation is the change in optical power of the eye due to a change in crystalline lens shape and position, whereas pseudo-accommodation is the attainment of functional near vision in an emmetropic or far-corrected eye without changing the refractive power of the eye. Other definitions specific to vision and lenses for presbyopia were also defined. It is recommended that these definitions be consistently adopted in order to standardise future research, clinical evaluations and education.

Background Inhibition abilities are known to have impact on self-regulation, behavior, and academic success, and they are frequently impaired in children born preterm. We investigated the possible contributions of resting-state functional brain connectivity to inhibition following preterm birth. Methods Forty-four preterm and 59 term-born participants aged 8–13 years were administered two inhibition tasks and resting-state functional MRI was performed. Functional connectivity (FC) networks were compared between groups using network-based statistics. Associations of FCNs and inhibition abilities were investigated through multivariate linear regression models accounting for the interaction between birth status and inhibition. Results NBS revealed weaker FC in children born preterm compared to term-born peers in connections between motor and supplementary motor regions, frontal lobe, precuneus, and insula. Irrespective of birth status, connections between the cerebellum, frontal, and occipital lobes and inter-lobar, subcortical, intra-hemispheric long-range connections were positively correlated with one of the two inhibition tasks. Conclusions Preterm birth results in long-term alterations of FC at network level but these FCN alterations do not specifically account for inhibition problems in children born very preterm. Impact Irrespective of birth status, significant associations were found between the subdomain of response inhibition and functional connectivity in some subnetworks. A group comparisons of functional brain connectivity measured by rsfMRI in school-aged children born very preterm and at term. The investigation of network-level functional connectivity at rest does not appear adequate to explain differences in inhibition abilities between children born very preterm and at term, hence other imaging techniques might be more suited to explore the underlying neural mechanisms of inhibition abilities in school-aged children born very preterm.

Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P. We say two points a, b∈ P can see each other if the line segment seg (a, b) is contained in P. We denote by V(P) the family of all minimum guard placements. The Hausdorff distance makes V(P) a metric space and thus a topological space. We show homotopy-universality, that is, for every semi-algebraic set S there is a polygon P such that V(P) is homotopy equivalent to S. Furthermore, for various concrete topological spaces T, we describe instances I of the art gallery problem such that V(I) is homeomorphic to T.

For sets of n points, n even, in general position in the plane, we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least Cn/2 different plane perfect matchings, where Cn/2 is the n/2-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings. We show the following results. (1) For every k≤164n2-3532nn+122564n , any set with n points, n sufficiently large, admits a perfect matching with exactly k crossings. (2) There exist sets of n points where every perfect matching has at most 572n2-n4 crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n. (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for k= 0 , 1 , 2 , and maximize the number of perfect matchings with (n/22) crossings and with (n/22)-1 crossings.

Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P. We say two points a, b∈ P can see each other if the line segment seg (a, b) is contained in P. We denote by V(P) the family of all minimum guard placements. The Hausdorff distance makes V(P) a metric space and thus a topological space. We show homotopy-universality, that is, for every semi-algebraic set S there is a polygon P such that V(P) is homotopy equivalent to S. Furthermore, for various concrete topological spaces T, we describe instances I of the art gallery problem such that V(I) is homeomorphic to T.

For sets of n points, n even, in general position in the plane, we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least Cn/2 different plane perfect matchings, where Cn/2 is the n/2-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings. We show the following results. (1) For every k≤164n2-3532nn+122564n , any set with n points, n sufficiently large, admits a perfect matching with exactly k crossings. (2) There exist sets of n points where every perfect matching has at most 572n2-n4 crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n. (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for k= 0 , 1 , 2 , and maximize the number of perfect matchings with (n/22) crossings and with (n/22)-1 crossings.

Enclosing depth is a recently introduced depth measure which gives a lower bound to many depth measures studied in the literature. So far, enclosing depth has only been studied from a combinatorial perspective. In this work, we give the first algorithms to compute the enclosing depth of a query point with respect to a data point set in any dimension. In the plane we are able to optimize the algorithm to get a runtime of O(n log n). In constant dimension, our algorithms still run in polynomial time.

Imagine you are a dog behind a fence $Q$ and a hiker is passing by at constant speed along the hiking path $P$. In order to fulfil your duties as a watchdog, you desire to bark as long as possible at the human. However, your barks can only be heard in a fixed radius $\rho$ and, as a dog, you have bounded speed $s$. Can you optimize your route along the fence $Q$ in order to maximize the barking time with radius $\rho$, assuming you can run backwards and forward at speed at most $s$? We define the barking distance from a polyline $P$ on $n$ vertices to a polyline $Q$ on $m$ vertices as the time that the hiker stays in your barking radius if you run optimally along $Q$. This asymmetric similarity measure between two curves can be used to detect outliers in $Q$ compared to $P$ that other established measures like the Fr\'echet distance and Dynamic Time Warping fail to capture at times. We consider this measure in three different settings. In the discrete setting, the traversals of $P$ and $Q$ are both discrete. For this case we show that the barking distance from $P$ to $Q$ can be computed in $O(nm\log s)$ time. In the semi-discrete setting, the traversal of $Q$ is continuous while the one of $P$ is again discrete. Here, we show how to compute the barking distance in time $O(nm\log (nm))$. Finally, in the continuous setting in which both traversals are continuous, we show that the problem can be solved in polynomial time. For all the settings we show that, assuming SETH, no truly subquadratic algorithm can exist.

We provide polynomial-time reductions between three search problems from three distinct areas: the P-matrix linear complementarity problem (P-LCP), finding the sink of a unique sink orientation (USO), and a variant of the $\alpha$-Ham Sandwich problem. For all three settings, we show that "two choices are enough", meaning that the general non-binary version of the problem can be reduced in polynomial time to the binary version. This specifically means that generalized P-LCPs are equivalent to P-LCPs, and grid USOs are equivalent to cube USOs. These results are obtained by showing that both the P-LCP and our $\alpha$-Ham Sandwich variant are equivalent to a new problem we introduce, P-Lin-Bellman. This problem can be seen as a new tool for formulating problems as P-LCPs., Comment: 29 pages, 9 figures

During the diagnostic process, doctors incorporate multimodal information including imaging and the medical history - and similarly medical AI development has increasingly become multimodal. In this paper we tackle a more subtle challenge: doctors take a targeted medical history to obtain only the most pertinent pieces of information; how do we enable AI to do the same? We develop a wrapper method named MINT (Make your model INTeractive) that automatically determines what pieces of information are most valuable at each step, and ask for only the most useful information. We demonstrate the efficacy of MINT wrapping a skin disease prediction model, where multiple images and a set of optional answers to $25$ standard metadata questions (i.e., structured medical history) are used by a multi-modal deep network to provide a differential diagnosis. We show that MINT can identify whether metadata inputs are needed and if so, which question to ask next. We also demonstrate that when collecting multiple images, MINT can identify if an additional image would be beneficial, and if so, which type of image to capture. We showed that MINT reduces the number of metadata and image inputs needed by 82% and 36.2% respectively, while maintaining predictive performance. Using real-world AI dermatology system data, we show that needing fewer inputs can retain users that may otherwise fail to complete the system submission and drop off without a diagnosis. Qualitative examples show MINT can closely mimic the step-by-step decision making process of a clinical workflow and how this is different for straight forward cases versus more difficult, ambiguous cases. Finally we demonstrate how MINT is robust to different underlying multi-model classifiers and can be easily adapted to user requirements without significant model re-training., Comment: 15 pages, 7 figures

Regression depth, introduced by Rousseeuw and Hubert in 1999, is a notion that measures how good of a regression hyperplane a given query hyperplane is with respect to a set of data points. Under projective duality, this can be interpreted as a depth measure for query points with respect to an arrangement of data hyperplanes. The study of depth measures for query points with respect to a set of data points has a long history, and many such depth measures have natural counterparts in the setting of hyperplane arrangements. For example, regression depth is the counterpart of Tukey depth. Motivated by this, we study general families of depth measures for hyperplane arrangements and show that all of them must have a deep point. Along the way we prove a Tverberg-type theorem for hyperplane arrangements, giving a positive answer to a conjecture by Rousseeuw and Hubert from 1999. We also get three new proofs of the centerpoint theorem for regression depth, all of which are either stronger or more general than the original proof by Amenta, Bern, Eppstein, and Teng. Finally, we prove a version of the center transversal theorem for regression depth.

In this paper, we disprove the long-standing conjecture that any complete geometric graph on 2n vertices can be partitioned into n plane spanning trees. Our construction is based on so-called bumpy wheel sets. We fully characterize which bumpy wheels can and in particular which cannot be partitioned into plane spanning trees (or even into arbitrary plane subgraphs). Furthermore, we show a sufficient condition for generalized wheels to not admit a partition into plane spanning trees, and give a complete characterization when they admit a partition into plane spanning double stars. Finally, we initiate the study of partitions into beyond planar subgraphs, namely into k-planar and k-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.

For sets of n points, n even, in general position in the plane, we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least Cn/2 different plane perfect matchings, where Cn/2 is the n/2-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings. We show the following results. (1) For every k=164n2-3532nn--v+122564n , any set with n points, n sufficiently large, admits a perfect matching with exactly k crossings. (2) There exist sets of n points where every perfect matching has at most 572n2-n4 crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n. (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for k=0,1,2 , and maximize the number of perfect matchings with (n/22) crossings and with (n/22)-1 crossings., Research on this work was initiated at the 16th European Research Week on Geometric Graphs which was held from November 18 to 22, 2019, near Strobl (Austria). We thank all participants for the good atmosphere as well as for discussions on the topic. Further, we thank Clemens Huemer for bringing this problem to our attention in the course of a meeting of the H2020-MSCA-RISE Project 734922 - CONNECT. O. A. and R. P. supported by FWF Grant W1230. P. S. supported by ERC Grant ERC StG 716424-CASe. D. P., I. P., and B. V. supported by FWF Project I 3340-N35. I. P. supported by the Independent Research Fund Denmark Grant 2020-2023 (9131-00044B) “Dynamic Network Analysis” and by the Margarita Salas Fellowship funded by the Ministry of Universities of Spain and the European Union (NextGenerationEU)., Peer Reviewed, Postprint (published version)

We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the Cascade conjecture, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy our most restrictive set of axioms, which includes Tukey depth. Along the way, we introduce and study a new depth measure called enclosing depth, which we believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets.

We study the problem of recognizing whether a given abstract simplicial complex $K$ is the $k$-skeleton of the nerve of $j$-dimensional convex sets in $\mathbb{R}^d$. We denote this problem by $R(k,j,d)$. As a main contribution, we unify the results of many previous works under this framework and show that many of these works in fact imply stronger results than explicitly stated. This allows us to settle the complexity status of $R(1,j,d)$, which is equivalent to the problem of recognizing intersection graphs of $j$-dimensional convex sets in $\mathbb{R}^d$, for any $j$ and $d$. Furthermore, we point out some trivial cases of $R(k,j,d)$, and demonstrate that $R(k,j,d)$ is ER-complete for $j\in\{d-1,d\}$ and $k\geq d$., Comment: 6 pages, 1 figure, presented at EuroCG'23

Regression depth, introduced by Rousseeuw and Hubert in 1999, is a notion that measures how good of a regression hyperplane a given query hyperplane is with respect to a set of data points. Under projective duality, this can be interpreted as a depth measure for query points with respect to an arrangement of data hyperplanes. The study of depth measures for query points with respect to a set of data points has a long history, and many such depth measures have natural counterparts in the setting of hyperplane arrangements. For example, regression depth is the counterpart of Tukey depth. Motivated by this, we study general families of depth measures for hyperplane arrangements and show that all of them must have a deep point. Along the way we prove a Tverberg-type theorem for hyperplane arrangements, giving a positive answer to a conjecture by Rousseeuw and Hubert from 1999. We also get three new proofs of the centerpoint theorem for regression depth, all of which are either stronger or more general than the original proof by Amenta, Bern, Eppstein, and Teng. Finally, we prove a version of the center transversal theorem for regression depth., Comment: To be presented at the 39th International Symposium on Computational Geometry (SoCG 2023)

Purpose: Automated distinct bone segmentation from CT scans is widely used in planning and navigation workflows. U-Net variants are known to provide excellent results in supervised semantic segmentation. However, in distinct bone segmentation from upper body CTs a large field of view and a computationally taxing 3D architecture are required. This leads to low-resolution results lacking detail or localisation errors due to missing spatial context when using high-resolution inputs. Methods: We propose to solve this problem by using end-to-end trainable segmentation networks that combine several 3D U-Nets working at different resolutions. Our approach, which extends and generalizes HookNet and MRN, captures spatial information at a lower resolution and skips the encoded information to the target network, which operates on smaller high-resolution inputs. We evaluated our proposed architecture against single resolution networks and performed an ablation study on information concatenation and the number of context networks. Results: Our proposed best network achieves a median DSC of 0.86 taken over all 125 segmented bone classes and reduces the confusion among similar-looking bones in different locations. These results outperform our previously published 3D U-Net baseline results on the task and distinct-bone segmentation results reported by other groups. Conclusion: The presented multi-resolution 3D U-Nets address current shortcomings in bone segmentation from upper-body CT scans by allowing for capturing a larger field of view while avoiding the cubic growth of the input pixels and intermediate computations that quickly outgrow the computational capacities in 3D. The approach thus improves the accuracy and efficiency of distinct bone segmentation from upper-body CT., Comment: Under submission

Optical flow provides information on relative motion that is an important component in many computer vision pipelines. Neural networks provide high accuracy optical flow, yet their complexity is often prohibitive for application at the edge or in robots, where efficiency and latency play crucial role. To address this challenge, we build on the latest developments in event-based vision and spiking neural networks. We propose a new network architecture, inspired by Timelens, that improves the state-of-the-art self-supervised optical flow accuracy when operated both in spiking and non-spiking mode. To implement a real-time pipeline with a physical event camera, we propose a methodology for principled model simplification based on activity and latency analysis. We demonstrate high speed optical flow prediction with almost two orders of magnitude reduced complexity while maintaining the accuracy, opening the path for real-time deployments., Comment: Accepted for IEEE CVPRW, Vancouver 2023. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media. Copyright 2023 IEEE

Optical flow provides information on relative motion that is an important component in many computer vision pipelines. Neural networks provide high accuracy optical flow, yet their complexity is often prohibitive for application at the edge or in robots, where efficiency and latency play crucial role. To address this challenge, we build on the latest developments in event-based vision and spiking neural networks. We propose a new network architecture, inspired by Timelens, that improves the state-of-the-art self-supervised optical flow accuracy when operated both in spiking and non-spiking mode. To implement a real-time pipeline with a physical event camera, we propose a methodology for principled model simplification based on activity and latency analysis. We demonstrate high speed optical flow prediction with almost two orders of magnitude reduced complexity while maintaining the accuracy, opening the path for real-time deployments.

Gamma-hydroxybutyrate (GHB) remains a challenging clinical/forensic toxicology drug. Its rapid elimination to endogenous levels mainly causes this. Especially in drug-facilitated sexual assaults, sample collection often occurs later than the detection window for GHB. We aimed to investigate new GHB conjugates with amino acids (AA), fatty acids, and its organic acid metabolites for their suitability as ingestion/application markers in urine following controlled GHB administration to humans. We used LC–MS/MS for validated quantification of human urine samples collected within two randomized, double-blinded, placebo-controlled crossover studies (GHB 50 mg/kg, 79 participants) at approximately 4.5, 8, 11, and 28 h after intake. We found significant differences (placebo vs. GHB) for all but two analytes at 4.5 h. Eleven hours post GHB administration, GHB, GHB-AAs, 3,4-dihydroxybutyric acid, and glycolic acid still showed significantly higher concentrations; at 28 h only GHB-glycine. Three different discrimination strategies were evaluated: (a) GHB-glycine cut-off concentration (1 µg/mL), (b) metabolite ratios of GHB-glycine/GHB (2.5), and (c) elevation threshold between two urine samples (> 5). Sensitivities were 0.1, 0.3, or 0.5, respectively. Only GHB-glycine showed prolonged detection over GHB, mainly when compared to a second time- and subject-matched urine sample (strategy c).

We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the Cascade conjecture, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy our most restrictive set of axioms, which includes Tukey depth. Along the way, we introduce and study a new depth measure called enclosing depth, which we believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets.

Schaefer's dichotomy theorem [Schaefer, STOC'78] states that a boolean constraint satisfaction problem (CSP) is polynomial-time solvable if one of six given conditions holds for every type of constraint allowed in its instances. Otherwise, it is NP-complete. In this paper, we analyze boolean CSPs in terms of their topological complexity, instead of their computational complexity. We attach a natural topological space to the set of solutions of a boolean CSP and introduce the notion of projection-universality. We prove that a boolean CSP is projection-universal if and only if it is categorized as NP-complete by Schaefer's dichotomy theorem, showing that the dichotomy translates exactly from computational to topological complexity. We show a similar dichotomy for SAT variants and homotopy-universality., Comment: 18 pages, 1 figure

It is well-known that the 2-Thief-Necklace-Splitting problem reduces to the discrete Ham Sandwich problem. In fact, this reduction was crucial in the proof of the PPA-completeness of the Ham Sandwich problem [Filos-Ratsikas and Goldberg, STOC'19]. Recently, a variant of the Ham Sandwich problem called $\alpha$-Ham Sandwich has been studied, in which the point sets are guaranteed to be well-separated [Steiger and Zhao, DCG'10]. The complexity of this search problem remains unknown, but it is known to lie in the complexity class UEOPL [Chiu, Choudhary and Mulzer, ICALP'20]. We define the analogue of this well-separability condition in the necklace splitting problem -- a necklace is $n$-separable, if every subset $A$ of the $n$ types of jewels can be separated from the types $[n]\setminus A$ by at most $n$ separator points. By the reduction to the Ham Sandwich problem it follows that this version of necklace splitting has a unique solution. We furthermore provide two FPT algorithms: The first FPT algorithm solves 2-Thief-Necklace-Splitting on $(n-1+\ell)$-separable necklaces with $n$ types of jewels and $m$ total jewels in time $2^{O(\ell\log\ell)}+m^2$. In particular, this shows that 2-Thief-Necklace-Splitting is polynomial-time solvable on $n$-separable necklaces. Thus, attempts to show hardness of $\alpha$-Ham Sandwich through reduction from the 2-Thief-Necklace-Splitting problem cannot work. The second FPT algorithm tests $(n-1+\ell)$-separability of a given necklace with $n$ types of jewels in time $2^{O(\ell^2)}\cdot n^4$. In particular, $n$-separability can thus be tested in polynomial time, even though testing well-separation of point sets is coNP-complete [Bergold et al., SWAT'22]., Comment: 16 pages, 7 figures

We solve a problem of Dujmovi\'c and Wood (2007) by showing that a complete convex geometric graph on $n$ vertices cannot be decomposed into fewer than $n-1$ star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs., Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023)

We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the Cascade conjecture, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy our most restrictive set of axioms, which includes Tukey depth. Along the way, we introduce and study a new depth measure called enclosing depth, which we believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets.

Assume you have a 2-dimensional pizza with 2n ingredients that you want to share with your friend. For this you are allowed to cut the pizza using several straight cuts, and then give every second piece to your friend. You want to do this fairly, that is, your friend and you should each get exactly half of each ingredient. How many cuts do you need? It was recently shown using topological methods that n cuts always suffice. In this work, we study the computational complexity of finding such n cuts. Our main result is that this problem is PPA-complete when the ingredients are represented as point sets. For this, we give a new proof that for point sets n cuts suffice, which does not use any topological methods. We further prove several hardness results as well as a higher-dimensional variant for the case where the ingredients are well-separated.

In this paper, we disprove the long-standing conjecture that any complete geometric graph on 2n vertices can be partitioned into n plane spanning trees. Our construction is based on so-called bumpy wheel sets. We fully characterize which bumpy wheels can and in particular which cannot be partitioned into plane spanning trees (or even into arbitrary plane subgraphs). Furthermore, we show a sufficient condition for generalized wheels to not admit a partition into plane spanning trees, and give a complete characterization when they admit a partition into plane spanning double stars. Finally, we initiate the study of partitions into beyond planar subgraphs, namely into k-planar and k-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.

A family of k point sets in d dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of generalized ham-sandwich cuts for the point sets exist. But how hard is it to check if a given family of high-dimensional point sets has this property? Starting from this question, we study several algorithmic aspects of the existence of transversals and separations in high-dimensions. First, we give an explicit proof that k point sets are well-separated if and only if their convex hulls admit no (k - 2)-transversal, i.e., if there exists no (k - 2)-dimensional flat that intersects the convex hulls of all k sets. It follows that the task of checking well-separation lies in the complexity class coNP. Next, we show that it is NP-hard to decide whether there is a hyperplane-transversal (that is, a (d - 1)-transversal) of a family of d + 1 line segments in ?^d, where d is part of the input. As a consequence, it follows that the general problem of testing well-separation is coNP-complete. Furthermore, we show that finding a hyperplane that maximizes the number of intersected sets is NP-hard, but allows for an ?((log k)/(k log log k))-approximation algorithm that is polynomial in d and k, when each set consists of a single point. When all point sets are finite, we show that checking whether there exists a (k - 2)-transversal is in fact strongly NP-complete. Finally, we take the viewpoint of parametrized complexity, using the dimension d as a parameter: given k convex sets in ?^d, checking whether there is a (k-2)-transversal is FPT with respect to d. On the other hand, for k ? d+1 finite point sets in ?^d, it turns out that checking whether there is a (d-1)-transversal is W[1]-hard with respect to d.

Traumatic events during childhood/early adolescence can cause long-lasting physiological and behavioral changes with increasing risk for psychiatric conditions including psychosis. Genetic factors and trauma (and their type, degree of repetition, time of occurrence) are believed to influence how traumatic experiences affect an individual. Here, we compared long-lasting behavioral effects of repeated social defeat stress (SD) applied during either peripuberty or late adolescence in adult male WT and Gclm-KO mice, a model of redox dysregulation relevant to schizophrenia. As SD disrupts redox homeostasis and causes oxidative stress, we hypothesized that KO mice would be particularly vulnerable to such stress. We first found that peripubertal and late adolescent SD led to different behavioral outcomes. Peripubertal SD induced anxiety-like behavior in anxiogenic environments, potentiated startle reflex, and increased sensitivity to the NMDA-receptor antagonist, MK-801. In contrast, late adolescent SD led to increased exploration in novel environments. Second, the long-lasting impact of peripubertal but not late adolescent SD differed in KO and WT mice. Peripubertal SD increased anxiety-like behavior in anxiogenic environments and MK-801-sensitivity mostly in KO mice, while it increased startle reflex in WT mice. These suggest that a redox dysregulation during peripuberty interacts with SD to remodel the trajectory of brain maturation, but does not play a significant role during later SD. As peripubertal SD induced persisting anxiety- and fear-related behaviors in male mice, we then investigated anxiety in a cohort of 89 early psychosis male patients for whom we had information about past abuse and clinical assessment during the first year of psychosis. We found that a first exposure to physical/sexual abuse (analogous to SD) before age 12, but not after, was associated with higher anxiety at 6-12 months after psychosis onset. This supports that childhood/peripuberty is a

A family of k point sets in d dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of generalized ham-sandwich cuts for the point sets exist. But how hard is it to check if a given family of high-dimensional point sets has this property? Starting from this question, we study several algorithmic aspects of the existence of transversals and separations in high-dimensions. First, we give an explicit proof that k point sets are well-separated if and only if their convex hulls admit no (k - 2)-transversal, i.e., if there exists no (k - 2)-dimensional flat that intersects the convex hulls of all k sets. It follows that the task of checking well-separation lies in the complexity class coNP. Next, we show that it is NP-hard to decide whether there is a hyperplane-transversal (that is, a (d - 1)-transversal) of a family of d + 1 line segments in ?^d, where d is part of the input. As a consequence, it follows that the general problem of testing well-separation is coNP-complete. Furthermore, we show that finding a hyperplane that maximizes the number of intersected sets is NP-hard, but allows for an ?((log k)/(k log log k))-approximation algorithm that is polynomial in d and k, when each set consists of a single point. When all point sets are finite, we show that checking whether there exists a (k - 2)-transversal is in fact strongly NP-complete. Finally, we take the viewpoint of parametrized complexity, using the dimension d as a parameter: given k convex sets in ?^d, checking whether there is a (k-2)-transversal is FPT with respect to d. On the other hand, for k ? d+1 finite point sets in ?^d, it turns out that checking whether there is a (d-1)-transversal is W[1]-hard with respect to d.

For sets of n= 2 m points in general position in the plane we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least Cm different plane perfect matchings, where Cm is the m-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings. We show the following results. (1) For every k≤164n2-O(nn), any set of n points, n sufficiently large, admits a perfect matching with exactly k crossings. (2) There exist sets of n points where every perfect matching has fewer than 572n2 crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n. (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for k= 0, 1, 2, and maximize the number of perfect matchings with (n/22) crossings and with (n/22)-1 crossings.

A family of $k$ point sets in $d$ dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of generalized ham-sandwich cuts for the point sets exist. But how hard is it to check if a given family of high-dimensional point sets has this property? Starting from this question, we study several algorithmic aspects of the existence of transversals and separations in high-dimensions. First, we give an explicit proof that $k$ point sets are well-separated if and only if their convex hulls admit no $(k - 2)$-transversal, i.e., if there exists no $(k - 2)$-dimensional flat that intersects the convex hulls of all $k$ sets. It follows that the task of checking well-separation lies in the complexity class coNP. Next, we show that it is NP-hard to decide whether there is a hyperplane-transversal (that is, a $(d - 1)$-transversal) of a family of $d + 1$ line segments in $\mathbb{R}^d$, where $d$ is part of the input. As a consequence, it follows that the general problem of testing well-separation is coNP-complete. Furthermore, we show that finding a hyperplane that maximizes the number of intersected sets is NP-hard, but allows for an $\Omega\left(\frac{\log k}{k \log \log k}\right)$-approximation algorithm that is polynomial in $d$ and $k$, when each set consists of a single point. When all point sets are finite, we show that checking whether there exists a $(k - 2)$-transversal is in fact strongly NP-complete., Comment: 14 pages, 1 figure; a preliminary version appeared in SWAT 2022

Purpose: The localisation and segmentation of individual bones is an important preprocessing step in many planning and navigation applications. It is, however, a time-consuming and repetitive task if done manually. This is true not only for clinical practice but also for the acquisition of training data. We therefore not only present an end-to-end learnt algorithm that is capable of segmenting 125 distinct bones in an upper-body CT, but also provide an ensemble-based uncertainty measure that helps to single out scans to enlarge the training dataset with. Methods We create fully automated end-to-end learnt segmentations using a neural network architecture inspired by the 3D-Unet and fully supervised training. The results are improved using ensembles and inference-time augmentation. We examine the relationship of ensemble-uncertainty to an unlabelled scan's prospective usefulness as part of the training dataset. Results: Our methods are evaluated on an in-house dataset of 16 upper-body CT scans with a resolution of \SI{2}{\milli\meter} per dimension. Taking into account all 125 bones in our label set, our most successful ensemble achieves a median dice score coefficient of 0.83. We find a lack of correlation between a scan's ensemble uncertainty and its prospective influence on the accuracies achieved within an enlarged training set. At the same time, we show that the ensemble uncertainty correlates to the number of voxels that need manual correction after an initial automated segmentation, thus minimising the time required to finalise a new ground truth segmentation. Conclusion: In combination, scans with low ensemble uncertainty need less annotator time while yielding similar future DSC improvements. They are thus ideal candidates to enlarge a training set for upper-body distinct bone segmentation from CT scans. }

Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P. We say two points a, b ∊ P can see each other if the line segment seg(a, b) is contained in P. We denote by V (P) the family of all minimum guard placements. The Hausdorff distance makes V(P) a metric space and thus a topological space. We show homotopy-universality, that is for every semi-algebraic set S there is a polygon P such that V(P) is homotopy equivalent to S. Furthermore, for various concrete topological spaces T, we describe instances I of the art gallery problem such that V(I) is homeomorphic to T.

Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P. We say two points a, b ∊ P can see each other if the line segment seg(a, b) is contained in P. We denote by V (P) the family of all minimum guard placements. The Hausdorff distance makes V(P) a metric space and thus a topological space. We show homotopy-universality, that is for every semi-algebraic set S there is a polygon P such that V(P) is homotopy equivalent to S. Furthermore, for various concrete topological spaces T, we describe instances I of the art gallery problem such that V(I) is homeomorphic to T.

Combinatorial representations of point sets play an important role in discrete and computational geometry. In this work, we investigate a new combinatorial quantity of a point set, called Tukey depth histogram. The Tukey depth histogram of k-flats in Rd with respect to a point set P, is a vector Dk,d(P), whose i’th entry Dik,d(P) denotes the number of k-flats spanned by k+ 1 points of P that have Tukey depth i with respect to P. It turns out that several problems in discrete and computational geometry can be phrased in terms of such depth histograms. As our main result, we give a complete characterization of the depth histograms of points, that is, for any dimension d we give a description of all possible histograms D0,d(P). This then allows us to compute the exact number of different histograms of points.

For sets of n= 2 m points in general position in the plane we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least Cm different plane perfect matchings, where Cm is the m-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings. We show the following results. (1) For every k≤164n2-O(nn), any set of n points, n sufficiently large, admits a perfect matching with exactly k crossings. (2) There exist sets of n points where every perfect matching has fewer than 572n2 crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n. (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for k= 0, 1, 2, and maximize the number of perfect matchings with (n/22) crossings and with (n/22)-1 crossings.

In this paper, we disprove the long-standing conjecture that any complete geometric graph on 2n vertices can be partitioned into n plane spanning trees. Our construction is based on so-called bumpy wheel sets. We fully characterize which bumpy wheels can and in particular which cannot be partitioned into plane spanning trees (or even into arbitrary plane subgraphs). Furthermore, we show a sufficient condition for generalized wheels to not admit a partition into plane spanning trees, and give a complete characterization when they admit a partition into plane spanning double stars. Finally, we initiate the study of partitions into beyond planar subgraphs, namely into k-planar and k-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.

Diagnosing and mitigating changes in model fairness under distribution shift is an important component of the safe deployment of machine learning in healthcare settings. Importantly, the success of any mitigation strategy strongly depends on the structure of the shift. Despite this, there has been little discussion of how to empirically assess the structure of a distribution shift that one is encountering in practice. In this work, we adopt a causal framing to motivate conditional independence tests as a key tool for characterizing distribution shifts. Using our approach in two medical applications, we show that this knowledge can help diagnose failures of fairness transfer, including cases where real-world shifts are more complex than is often assumed in the literature. Based on these results, we discuss potential remedies at each step of the machine learning pipeline.

The Ham-Sandwich theorem is a well-known result in geometry. It states that any d mass distributions in R^d can be simultaneously bisected by a hyperplane. The result is tight, that is, there are examples of d+1 mass distributions that cannot be simultaneously bisected by a single hyperplane. In this abstract we will study the following question: given a continuous assignment of mass distributions to certain subsets of R^d, is there a subset on which we can bisect more masses than what is guaranteed by the Ham-Sandwich theorem? We investigate two types of subsets. The first type are linear subspaces of R^d, i.e., k-dimensional flats containing the origin. We show that for any continuous assignment of d mass distributions to the k-dimensional linear subspaces of R^d, there is always a subspace on which we can simultaneously bisect the images of all d assignments. We extend this result to center transversals, a generalization of Ham-Sandwich cuts. As for Ham-Sandwich cuts, we further show that for d-k+2 masses, we can choose k-1 of the vectors defining the k-dimensional subspace in which the solution lies. The second type of subsets we consider are subsets that are determined by families of n hyperplanes in R^d. Also in this case, we find a Ham-Sandwich-type result. In an attempt to solve a conjecture by Langerman about bisections with several cuts, we show that our underlying topological result can be used to prove this conjecture in a relaxed setting.

Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P. We say two points a,b∈P can see each other if the line segment seg(a,b) is contained in P. We denote by V(P) the family of all minimum guard placements. The Hausdorff distance makes V(P) a metric space and thus a topological space. We show homotopy-universality, that is for every semi-algebraic set S there is a polygon P such that V(P) is homotopy equivalent to S. Furthermore, for various concrete topological spaces T, we describe instances I of the art gallery problem such that V(I) is homeomorphic to T.

Contact lenses in the future will likely have functions other than correction of refractive error. Lenses designed to control the development of myopia are already commercially available. Contact lenses as drug delivery devices and powered through advancements in nanotechnology will open up further opportunities for unique uses of contact lenses. This review examines the use, or potential use, of contact lenses aside from their role to correct refractive error. Contact lenses can be used to detect systemic and ocular surface diseases, treat and manage various ocular conditions and as devices that can correct presbyopia, control the development of myopia or be used for augmented vision. There is also discussion of new developments in contact lens packaging and storage cases. The use of contact lenses as devices to detect systemic disease has mostly focussed on detecting changes to glucose levels in tears for monitoring diabetic control. Glucose can be detected using changes in colour, fluorescence or generation of electric signals by embedded sensors such as boronic acid, concanavalin A or glucose oxidase. Contact lenses that have gained regulatory approval can measure changes in intraocular pressure to monitor glaucoma by measuring small changes in corneal shape. Challenges include integrating sensors into contact lenses and detecting the signals generated. Various techniques are used to optimise uptake and release of the drugs to the ocular surface to treat diseases such as dry eye, glaucoma, infection and allergy. Contact lenses that either mechanically or electronically change their shape are being investigated for the management of presbyopia. Contact lenses that slow the development of myopia are based upon incorporating concentric rings of plus power, peripheral optical zone(s) with add power or non-monotonic variations in power. Various forms of these lenses have shown a reduction in myopia in clinical trials and are available in various markets.