Discrete Geometry on Red and Blue Points in the Plane Lattice

Tarski's theory of definability: common themes in descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic

Given a set P of n red and blue points in the plane, a planar bichromatic spanning tree of P is a geometric spanning tree of P, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree T, such that the length of the longest edge in T is minimized. In this paper, we show that this problem is NP-hard for points in general position. Our main contribution is a polynomial-time (8√2)-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck λ can be converted to a planar bichromatic spanning tree of bottleneck at most 8√2 λ.

BackgroundHospitalization rates among persons with HIV (PWH) in the US remain substantial, yet regional variations are understudied. This study analyzed insurance claims data to estimate hospitalization incidence among commercially and Medicare-insured PWH across four US regions.Figure 1.Incidence proportion of persons with more than one hospitalizationThe proportion of PWH with ≥1 hospitalization annually was calculated by dividing the number of individuals with ≥1 hospitalization by the total number of individuals with HIV at the beginning of each year. Rates are shown stratified by region of enrollment (red points and error bars) and admitting hospital region (blue points and error bars). From 2018 to 2020, enrollment-based rates were highest in the Northeast and lowest in the West. The Northeast consistently had the highest incidence (0.09 [95% CI: 0.08–0.10] in 2018 and 2019), while the West remained lowest (0.02 [95% CI: 0.02–0.02] in 2018 and 0.02 [95% CI: 0.02–0.03] in 2019 and 2020). In 2020, a slight decrease was observed in the South (from 0.08 to 0.07), while the Northeast and Midwest remained stable. Hospital-based stratification showed a slightly different pattern: the Northeast remained highest, peaking at 0.12 (95% CI: 0.11–0.13) in 2019, while the South had the lowest proportions, decreasing from 0.05 (95% CI: 0.05–0.06) in 2018 to 0.04 (95% CI: 0.04–0.05) in 2020. The West and Midwest showed modest fluctuations, with the West peaking at 0.10 (95% CI: 0.09–0.11) in 2019 before declining to 0.07 (95% CI: 0.06–0.08) in 2020.Figure 2.Incidence proportion of all hospitalizationsB. Incidence proportion of all hospitalizations.Total hospitalizations per regional person-years was calculated by dividing the number of hospitalizations by the total number of individuals with HIV at the beginning of each year. Results are stratified by enrollment region (black points and error bars). From 2018 to 2020, the highest incidence proportions were observed in the Northeast and South (range: 0.12–0.13). The Midwest increased from 0.12 in 2018 to 0.14 in 2019, then declined slightly in 2020. The West consistently had the lowest incidence, remaining at 0.03 each year with narrow confidence intervals.MethodsThis study used MarketScan data (2017-2020) to determine hospitalization incidence in US PWH (ICD-10-CM B20), using 2017 to identify cases for 2018. To ensure complete annual hospitalization capture and minimize loss-to-follow-up bias, the cohort was limited to continuously enrolled individuals (full 12-month coverage/year). For regional analysis across four US Census regions (South, Northeast, Midwest, West, encompassing all 50 states and D.C.), the population was further restricted to enrollees with available regional data. Annual hospitalization incidence was reported as (1) incidence proportion of ≥1 hospitalization, (2) incidence proportion of all hospitalizations, and (3) incidence density, each with 95% confidence intervals.Figure 3.Incidence density of first hospitalizationIncidence density was defined as the number of individuals with a first hospitalization divided by the total person-time at risk (measured as the sum of days prior to first hospitalization or, if no admission occurred, the total follow-up time). Results are stratified by enrollment region (red points and error bars) and hospital region (blue points and error bars). Incidence density declined across all regions from 2018 to 2020. In enrollment-based analyses, the Northeast had the highest rates, dropping from 0.038 (95% CI: 0.036–0.041) in 2018 to 0.017 (95% CI: 0.016–0.019) in 2020. The West had the lowest rates, declining from 0.007 (95% CI: 0.006–0.008) to 0.004 (95% CI: 0.003–0.005). Hospital-based rates followed similar regional trends, with the Northeast consistently highest. While the South and West showed lower rates overall, West region estimates were slightly higher than those based on enrollment.ResultsAmong 90,836 PWH, 13.5% (n=12,287) had ≥1 hospitalization, a stable proportion. Hospitalized individuals were typically 49–50 years old (median), 73% male, and 95% commercially insured, with most experiencing ≥2 annual hospitalizations (3–4-day median stay). The South (39%) and Northeast (29%) accounted for the largest, stable proportions of all hospitalizations. Hospital-based incidence mirrored enrollment trends but showed greater variation, peaking in the Northeast and declining in the South (Fig). Total hospitalization incidence proportions were higher than ≥1 hospitalization proportions, reflecting multiple admissions. Incidence density revealed temporal declines and regional nuances, suggesting reduced hospitalization frequency in the Northeast and South despite stable proportions, and consistently low rates in the West.ConclusionHospitalization patterns varied regionally, with the highest burden in the Northeast and South. While the proportion of individuals hospitalized remained stable, declining incidence density in these regions suggests reduced hospitalization frequency over time. These findings emphasize the need to explore factors such as access to care and comorbidities driving regional disparities.DisclosuresAll Authors: No reported disclosures

Motivated by applications in computer graphics, visualization, and scientic compu- tation, we study the computational complexity of the following problem: given a set S of n points sampled from a bivariate function f(x;y) and an input parameter > 0, compute a piecewise-linear function (x;y) of minimum complexity (that is, an xy-monotone polyhedral surface, with a mini- mum number of vertices, edges, or faces) such thatj(xp;yp) i zp j for all (xp;yp;zp)2 S.W e give hardness evidence for this problem, by showing that a closely related problem is NP-hard. The main result of our paper is a polynomial-time approximation algorithm that computes a piecewise- linear surface of size O(Ko logKo), where Ko is the complexity of an optimal surface satisfying the constraints of the problem. The technique developed in our paper is more general and applies to several other problems that deal with partitioning of points (or other objects) subject to certain geometric constraints. For instance, we get the same approximation bound for the following problem arising in machine learning: givenn \red andm \blue points in the plane, nd a minimum number of pairwise disjoint triangles such that each blue point is covered by some triangle and no red point lies in any of the triangles.

Suppose that red and blue points occur in Rd according to two simple point processes with finite intensities λR and λB, respectively. Furthermore, let ν and μ be two probability distributions on the strictly positive integers with means ν̅ and μ̅, respectively. Assign independently a random number of stubs (half-edges) to each red (blue) point with law ν (μ). We are interested in translation-invariant schemes for matching stubs between points of different colors in order to obtain random bipartite graphs in which each point has a prescribed degree distribution with law ν or μ depending on its color. For a large class of point processes, we show that such translation-invariant schemes matching almost surely all stubs are possible if and only if λRν̅ = λBμ̅, including the case when ν̅ = μ̅ = ∞ so that both sides are infinite. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage problem. For this scheme, we give sufficient conditions on ν and μ for the presence and absence of infinite components. These results are two-color versions of those obtained by Deijfen, Holroyd and Häggström.

We consider the following problem: Given n red points, m green points and p blue points on a plane, find a trio of red-green-blue points such that the distance between them is smallest. This problem could be solved by an exhaustive search algorithm: test all trios of 3 different colored points and find the closest trio. However, this algorithm has the complexity of O(N 3 );N = max(n;p;q). In our previous research, we already showed a heuristic algorithm with a good complexity to solve this problem based on 2D Voronoi diagram. In this paper, we introduce a better heuristic algorithm for solving this problem. This new heuristic algorithm has the same complexity but it performs much better than the old one.

It is shown that a minimum spanning tree of n points in ℝd can be computed in optimal O(Td(n,n)) time under any fixed Lt−metric, where Td (n, m) denotes the time to find a bichromatic closest pair between n red points and m blue points. The previous bound was O(Td (n, n) log n) and it was proved only for the L2 (Euclidean) metric. Furthermore, for d = 3 it is shown that a minimum spanning tree can be found in optimal O(n log n) time under the L1 and L∞-metric. The previous bound was O(n log n log log n).

Given a point set S = R B , where R is a set of red points and B is a set of blue points, we desire to find T * , a minimum weight spanning tree such that every edge has one red endpoint and one blue endpoint and no two edges cross. We call T * a bichromatic plane minimum spanning tree (MinBPST). We say a point set is semi-collinear when the blue points lie on a line and the red points lie on one side of the line. In this thesis, we present an O(|B | 3 |R| 2 ) running time algorithm for finding T * on a set of semi-collinear points. We also discuss an implementation of this algorithm. Additionally, we describe changes that can be made to the algorithm presented to solve other related problems.

Given some finite point set P in the plane and some real ε, 0<ε<1, we want to colour a minimal subset of P red, such that the following holds: every open halfplane that contains more than ε · ¦P¦ of the points in P also contains at least one red point.It is shown that it always suffices to colour [2/ε]-1 points red (independent of the size of P). If ε<2/3, we can choose these [2/ε]-1 points among the extreme points of P. If all red points must be extreme, our solution is optimal and it can be found in O(nlogn) time. If the red points are allowed to be any elements of P, our result is almost optimal: There are point sets requiring at least 2[1/ε]-2 red points. The both bounds differ at most by one.

Monochromatic geometric k-factors in red-blue sets with white and Steiner points

Central pattern generators (CPGs) are neuronal circuits that control rhythmic movements in animals. The heartbeat of the medicinal leech is controlled by a CPG that is distributed over several ganglia. In ganglia 3 and 4, there are half-center oscillators (HCOs): pairs of reciprocally inhibitory heart interneurons (HNs). These HCOs form the kernel of the heartbeat CPG, and each HCO is composed of a pair of endogenously bursting heart interneurons [1]. A canonical model of the HN accurately reproduces experimental results [1,2]. It also exhibits bistability of bursting and silence in single HNs and the HCO [1,3]. The single HN and HCO model have been expanded into a family of models that satisfy biological constraints [2]. Multistability is highly prevalent in this database [4]. The coexistence of bursting and silent regimes could present a life threatening condition for the animal. If a perturbation were able to trigger a switch in the activity of the CPG to a silent state, circulation of blood in the leech would be crippled. Our hypothesis is that the HCO connectivity protects functional pattern of activity against an external perturbation. Here, we investigated the perturbations that switch the activity of the HN and the HCO from bursting to silence. Over a series of simulations, we applied a 30 ms pulse of current with various amplitudes and phases of application in the period of bursting activity. This protocol provided a set of pulse parameters in the (phase, amplitude) parameter space for which a switch was triggered. In the canonical model of a single HN for which the leak conductance was 9.59 nS, both depolarizing and hyperpolarizing pulses could trigger switches (Figure ​(Figure1).1). We observed contiguous sets of pulse parameters where triggered switches occurred, but there were also fine structures in the parameter space (Figure ​(Figure1).1). We estimated the susceptibility of bursting activity in the HN to perturbations by the proportion of pulse parameter sets that would trigger a switch to the total number of trials. We found that 20.84% of pulses triggered a switch (Figure ​(Figure1).1). We repeated this pulse protocol in the HCO for the same parameter values of the ionic conductances, and found that no pulses elicited a switch from bursting to silence. Our results suggest that in a network consisting of bistable elements, the half-center oscillator connectivity protects the functional regime against external perturbations. Figure 1 Properties of pulses triggering a switch from bursting to silence. Red or blue points indicate a switch or failure to switch. The yellow trajectory is the bursting waveform that defines the phase. The green dashed line indicates 0 nA.

This paper is the continuation of two previous ones devoted to the properties and the derivation of the isomorphic subgroups of p1 ≈ {\bb Z}2, P1 ≈ {\bb Z}3, ..., {\bb Z}n [Billiet (1979). Acta Cryst. A35, 485-496; Billiet & Rolley-Le Coz (1980). Acta Cryst. A36, 242-248]. The preservation of the lattice points, of the lattice directions and of the lattice planes of the group Γ(P1) by an isomorphic subgroup γ of index i are studied here. The typical feature is as follows: every lattice direction (respectively plane) is either unpreserved or preserved by γ, with the same preservation rate as the origin lattice direction (respectively plane). One plane out of p'3 lattice planes is partially preserved; in each preserved lattice plane, one direction out of p'2 lattice directions is partially preserved; along a preserved lattice direction, one point out of p'1 lattice points is preserved; P'1P'2P'3 = i. Algorithms for the identification of preserved lattice points, directions and planes are shown with several practical examples.

Working in the infinite plane R2, consider a Poisson process of black points with intensity 1, and an independent Poisson process of red points with intensity λ. We grow a disc around each black point until it hits the nearest red point, resulting in a random configuration Aλ, which is the union of discs centered at the black points. Next, consider a fixed disc of area n in the plane. What is the probability pλ(n) that this disc is covered by Aλ? We prove that if λ3nlogn = y then, for sufficiently large n, e-8π2y ≤ pλ(n) ≤ e-2π2y/3. The proofs reveal a new and surprising phenomenon, namely, that the obstructions to coverage occur on a wide range of scales.

Geometric planar networks on bichromatic collinear points

Combinatorics Given a set S of n points in the plane, a radial ordering of S with respect to a point p (not in S) is a clockwise circular ordering of the elements in S by angle around p. If S is two-colored, a colored radial ordering is a radial ordering of S in which only the colors of the points are considered. In this paper, we obtain bounds on the number of distinct non-colored and colored radial orderings of S. We assume a strong general position on S, not three points are collinear and not three lines 14;each passing through a pair of points in S 14;intersect in a point of ℝ2 S. In the colored case, S is a set of 2n points partitioned into n red and n blue points, and n is even. We prove that: the number of distinct radial orderings of S is at most O(n4) and at least Ω(n3); the number of colored radial orderings of S is at most O(n4) and at least Ω(n); there exist sets of points with Θ(n4) colored radial orderings and sets of points with only O(n2) colored radial orderings.