This paper studies a scheduling problem related to organizing twenty-five students into six separate work sessions. In each session, the students are divided into five groups, with five students in each group. One important condition in this problem is that any two students are allowed to work together only once during all sessions. This condition helps ensure fairness and prevents repeated grouping. To solve this problem, the affine plane of order five, , is used as a mathematical model. The affine plane is constructed algebraically from the projective plane over the finite field . In the proposed model, students are represented by points, while work groups are represented by lines. The use of parallel classes in makes it possible to organize the groups without conflicts, so that no student pair is repeated. Although this study focuses on the case of twenty-five students, the results indicate that the combination of finite geometry and companion matrix methods can also be useful for solving similar scheduling and allocation problems.

Distinctive electrocardiographic pattern in acute myocardial infarction.

We consider a nodal curve C in the complex projective plane whose irreducible components C_{i} are smooth. A minimal set of generators G for the first and second syzygy modules of the Jacobian ideal of C are described, using recent results by Th. Kahle, H. Schenck, B. Sturmfels and M. Wiesmann on the likelihood correspondence. The elements of G have explicit formulas in terms of the equations f_{i}=0 of the irreducible components C_{i} of C . Similar results, including extensions to hypersurfaces arrangements in \mathbb{P}^{n} were obtained by R. Burity, Z. Ramos, A. Simis and St. Tohăneanu with a genericity assumption which may not be easy to test in practice.

In the search and rescue operation of the submersible, to better search for the missing or faulty submersible, taking the marine environment simulated by the HYCOM model as the sample, it is necessary to use the Kalman filter model to predict the time location of the submersible and provide information support for the follow-up search and rescue operations according to the position information transmitted to the main ship when the submersible is running normally. Monte Carlo simulation is used to quantitatively analyze the probability of the possible area of the submersible in four possible cases after the fault, to obtain the location of the initial search deployment point, that is, the minimum plane projection area of the covering sample. Python software was used to quantitatively analyze the probability of finding the submersible with the passage of time and cumulative search results. Moreover, we conducted a comparative analysis of the method proposed in this paper with previous methods to illustrate the advancement of the method proposed in this paper. By introducing the nearest neighbor correlation algorithm into the multi-target tracking algorithm, the motion position of multiple submersibles in the same area can be predicted.

In order to solve the difficult problem of feeding process in the process of disorderly logistics package stacking entering automatic sorting equipment in logistics package sorting industry, an algorithm based on 3D vision to determine the normal vector of three points-projection to determine the position of the central point is proposed, which is used to obtain the position information of the physical and geometric central point on the projection plane of spatial logistics package. The mathematical model of the algorithm is established. The method of calculating the normal vector and the attitude of the center point of the algorithm and the workflow of the algorithm are expounded. In order to verify the accuracy of the proposed algorithm, an experimental method of determining the center point with dual lasers and the spatial pose with dual probes is proposed, and an experimental scene based on 3D vision, industrial robot, dual lasers and PC (Personal Computer) is constructed. The experimental results show that the algorithm can solve the position and attitude information of the physical and geometric center of the projection plane of the target logistics package in the disorderly logistics package stacking, and the maximum positioning error is [Formula: see text], and the average error is [Formula: see text]. The average positioning time is 17.55ms. The algorithm of determining the normal vector of three points and determining the position of the center point by projection can solve the position information of the physical geometric center of the projection plane of spatial logistics packages, which provides reference for the research of disorderly logistics package sorting technology.

This paper presents a geometric difficulty analysis framework for Formula 1 racing lines based on telemetry data from the 2024 season. To ensure geometric consistency across multiple laps, a representative racing line is identified using the discrete Fréchet distance, and corner segments are modeled using biarc approximation to estimate stable curvature. Based on the resulting geometric representation, we introduce three curvature-based difficulty metrics—the Curvature Exposure Index (CEI), Maximum Curvature Severity (MCS), and Curvature Variation Index (CVI)—to quantify both local and global track characteristics. This approach establishes a strictly geometric definition of difficulty based on the planar projection of the trajectory, purposely decoupling structural complexity from 3D terrain features, vehicle dynamics, and race context. Experimental results across 24 tracks demonstrate that these metrics effectively capture distinct track characteristics: CEI ranged from 1.97 rad/km (Italian) to 8.44 rad/km (Monaco), MCS from 230.54 km−1 (Spanish) to 1689.54 km−1 (Monaco), and CVI from 7.60 (British) to 9.33 (Monaco and Qatar). Although this framework focuses on planar geometry, it provides a compact, extensible foundation for geometric analysis and future applications incorporating elevation profiles and dynamic variables.

ABSTRACTAdditive manufacturing is transforming how microfluidic devices are prototyped and fabricated. Among various 3D printing methods, stereolithography (SLA) has become a dominant technique for microfluidics due to its high resolution and design flexibility, with widespread use in lab‐on‐a‐chip applications. However, intrinsic limitations of SLA printing, such as challenges related to multi‐material integration and microstructure fabrication in enclosed channels, continue to hinder the development of more complex microsystems, especially for analytical separation and tissue engineering applications. In this paper, we present a multiphase flow‐assisted in situ 3D printing method to address these challenges, developed based on our previously reported in situ 3D polymerization (IS‐3DP) concept. Our method utilizes an aqueous two‐phase system (ATPS) to generate sequential printing layers through controlled fluidic confinement and integrates an image‐guided alignment system to enable precise projection of printing patterns in microchannels. We demonstrate that viscosity tuning of the ATPS printing and blocking phases enables dynamic control of layer thickness, allowing customized and adaptive design of the 3D structure slicing. The image‐guided alignment system employs a homography transformation mechanism to map the projection and printing planes via image feedback, providing real‐time mask alignment with microchannel geometries. We characterize the mapping accuracy and projection fidelity and demonstrate the capability of this method by direct in‐channel fabrication of complex 3D microstructures such as pyramids, cuboids, bridge‐like void structures, as well as multi‐material patterns. We envision the multiphase flow‐assisted in situ 3D printing to offer a versatile tool for spatially controlled, high‐fidelity, and multi‐material microfabrication within confined microchannels in novel lab‐on‐a‐chip applications.

Abstract We consider a finite analytic morphism $$\varphi =(f,g)$$ φ = ( f , g ) defined from a complex analytic normal surface ( Z , z ) to $$\mathbb {C}^2$$ C 2 . We describe the topology of the image by $$\varphi $$ φ of a reduced curve on ( Z , z ) by means of iterated pencils defined recursively for each branch of the curve from the initial one $$\langle f,g \rangle $$ ⟨ f , g ⟩ . This result generalizes the one obtained in a previous paper for the case in which ( Z , z ) is smooth and the curve irreducible. The methods we use also permit us to describe the topological type of the discriminant curve of $$\varphi $$ φ , in particular, the topological type of each branch of the discriminant can be obtained from the map without previous knowledge of the critical locus.

We prove that the expected area of the amoeba of a complex plane curve of degree d d is less than 3 ln ⁡ ( d ) 2 / 2 + 9 ln ⁡ ( d ) + 9 \displaystyle {3\ln (d)^2/2+9\ln (d)+9} and once rescaled by ln ⁡ ( d ) 2 \ln (d)^2 , is asymptotically bounded from below by 3 / 4 3/4 . In order to get this lower bound, given disjoint isometric embeddings of a bidisc of size 1 / d 1/\sqrt {d} in the complex projective plane, we lower estimate the probability that one of them is a submanifold chart of a complex plane curve. It exponentially converges to one as the number of bidiscs grows to + ∞ +\infty .

Monitoring and management of power line corridors are essential for ensuring the safe and reliable operation of power transmission systems. Traditional manual inspection methods are not only inefficient but also pose significant safety risks, while certain existing automated approaches suffer from limited effectiveness in complex terrains or in the presence of discontinuities in point cloud data, resulting in insufficient accuracy in power line extraction and frequent reconstruction failures. To address these challenges, this study proposes a novel power line reconstruction method termed Weighted Multi-feature & Multi-plane Projection Geometric Fusion (WM-MPGF). The proposed method comprises two sequential stages: Weighted Multi-Feature SVM (WMF-SVM) and Multi-plane Projection and Geometric Joint Reconstruction (MPG-Recon). Specifically, WMF-SVM introduces a weighted multi-feature support vector machine framework that integrates elevation data derived from the Digital Elevation Model (DEM) with spatial features extracted via Principal Component Analysis (PCA), while optimizing feature weights through the entropy weight method to enhance the accuracy of power line identification. Subsequently, MPG-Recon performs geometric analysis to construct directional projection planes and applies the Hough transform to project power line points and determine their dominant orientations on the XOY and YOZ planes. The Davies-Bouldin index is employed to determine the optimal number of clusters, thereby enabling accurate estimation of the number of power lines. By integrating the K-means clustering algorithm, the method achieves effective separation of multiple power lines and ensures high-precision fitting of individual conductors. Experimental results indicate that the proposed approach achieves average fitting errors of 5.41 cm on the XOY plane and 5.68 cm in the vertical direction, successfully capturing the three-dimensional structural characteristics of power lines. The method constructs a robust 3D model and provides critical technical support for advanced applications in power line corridor monitoring and maintenance.

A fake projective plane is a complex surface with the same Betti numbers as $\mathbb{C} P^2$ but not biholomorphic to it. We study the fake projective plane $\mathbb{P}_{\operatorname{fake}}^2 = (a = 7, p = 2, \emptyset, D_3 2_7)$ in the Cartwright-Steger classification. In this paper, we exploit the large symmetries given by $\operatorname{Aut}(\mathbb{P}_{\operatorname{fake}}^2) = C_7 \rtimes C_3$ to construct an embedding of this surface into $\mathbb{C} P^5$ as a system of $56$ sextics with coefficients in $\mathbb{Q}(\sqrt{-7})$. For each torsion line bundle $T \in \operatorname{Pic}(\mathbb{P}_{\operatorname{fake}}^2)$, we also compute and study the linear systems $|nH + T|$ with small $n$, where $H$ is an ample generator of the Néron-Severi group.

A novel six-bar tetrahedral tensioned stringed cylindrical lattice shell was proposed based on the structural configuration of the six-bar tetrahedral cylindrical lattice shell which can be assembled by rhombic projection plane six-bar tetrahedral units. A self-balancing structural system is formed with the counterbalance between the horizontal thrust at the support and the tension from the added tensioned stringed part. The internal force distribution of the overall structure is more reasonable. Meanwhile, the advantages of standardized design, industrial production and prefabricated construction are inherited. The configuration of six-bar tetrahedral tensioned stringed cylindrical lattice shell was analyzed. A parameterized modeling program with visualization was developed. Completeness and effectiveness of the parameters were verified. The construction path of the structure was discussed. Based on the test model of six-bar tetrahedral cylindrical lattice shell, a tensioned stringed test model was built by the addition of struts and steel tie rods and the change of bearing properties along span direction from fixed bearing to sliding bearing. Static tests of the model were conducted. Reliability of the structure was verified by the experimental research.

Abstract In this paper, we examine the combinatorial properties of conic arrangements in the complex projective plane that possess certain quasi–homogeneous singularities. First, we introduce a new tool that enables us to characterize the property of being plus–one generated within the class of conic arrangements with some naturally chosen quasi–homogeneous singularities. Next, we present a classification result on plus–one generated conic arrangements admitting only nodes and tacnodes as singularities. Building on results regarding conic arrangements with nodes and tacnodes, we present new examples of strong Ziegler pairs of conic-line arrangements – that is, arrangements having the same strong combinatorics but distinct derivation modules.

ABSTRACT An untouchable set in a projective plane is a set of points such that no line of the plane meets the set in exactly one point. Recently, Héger and Nagy (Avoiding Secants of Given Size in Finite Projective Planes, J. Combin. Des . 33:83–93, 2024.) provided a generalization of untouchable sets to ‐avoiding sets, and addressed the issue of the spectrum of sizes that such sets can attain in finite planes. In the case of untouchable sets, the authors state as an open question the existence of untouchable sets of size and . We answer this question in the affirmative for Desarguesian planes of even order, and provide a construction of untouchable sets of size in for .

Free gases of spinless fermions moving on a lattice-symmetric geometric background are considered. Their topological properties at zero temperature can be used to classify their Fermi seas and associated boundaries. The flat orbifolds Rd/Γ, where Γ is the crystallographic group of symmetry in d-dimensional momentum space, are used to accomplish this task. Two topological classes exist for d=1: an interval, which is identified as a conductor, and a circumference, which corresponds to an insulator. The number of topological classes increases to 17 for d=2: 8 have the topology of a disk, that are generally recognized as conductors, and 4 correspond to a two-sphere, matching insulators. Both sets eventually contain a finite number of conical singularities and reflection corners at the boundaries. The remaining cases in the listing relate to conductors (annulus, Möbius strip) and insulators (two-torus, real projective plane, Klein bottle). Examples that fall under this list are given, along with physical interpretations of the singularities. It is anticipated that the findings of this classification will be robust under perturbative interactions due to its topological character.

The Rauzy gasket is the attractor of a parabolic, nonconformal iterated function system on the projective plane which describes an exceptional parameter set in various important topological and dynamical problems. Since 2009, there have been several attempts to calculate the Hausdorff dimension of the Rauzy gasket, which is a challenging problem due to the combination of the parabolicity and nonconformality in the geometry. In this paper, we prove that the Hausdorff dimension of the Rauzy gasket equals the affinity dimension. The key technical result underpinning this is a partial generalisation of the work of Hochman and Solomyak to the \mathrm{SL}(3,\mathbb{R}) setting, where we establish the exact dimension of stationary (Furstenberg) measures supported on the Rauzy gasket. The dimension results for both stationary measures and attractors are established in broader generality and extend recent work on projective iterated function systems to higher dimensions.

To compare the effects of quad helix (QH) anchored on permanent molars versus rapid maxillary expansion (RME) anchored on deciduous teeth on palatal morphology in early mixed dentition patients. A two-arm randomized controlled trial, together with a non-randomized normal bite data for comparison. Seventy-one patients (mean age: QH=9.3years; RME=9.4years) with unilateral posterior crossbite were analysed. The QH group (n=36) and RME group (n=35) were evaluated at baseline (T0), post-retention (T2), and one-year post-treatment (T3). A third age- and sex-matched control group (n=22; mean age=9.1years) served as a normative reference. Evaluated outcomes were 3D palatal measurements, as well as treatment success rate and total treatment duration. Both treatment groups showed significant increases in palatal surface area, projection plane area, and volume from T0 to T3. The RME group experienced a greater increase in palatal surface area (7.0%) compared to the QH group (4.2%) over the same period (P=0.045). Palatal volume increased notably more in the RME group during active treatment (T0-T2), with an 11.2% gain versus 6.8% in the QH group (P=0.046). By T3, palatal vault dimensions had normalized in both groups compared to the control group. The RME group completed treatment 97days earlier than the QH group. Treatment with either QH or RME resulted in normalized palatal vaults compared to the control group. RME had a significantly shorter treatment time but achieved similar success in correcting posterior crossbite as QH. This trial was registered at ClinicalTrials.gov (ID NCT04458506) and Researchweb.org (project number 260581).

We propose a new method to construct a finite projective plane. Its incidence matrix is expressed in the special Paige-Wexler normal form whose lower right part is a circulant block matrix.

Online adaptive radiation therapy necessitates independent dose verification systems for QA processes. Advanced irradiation techniques like IMRT, VMAT, and DWA require accurate Monte Carlo-based dose calculations, where intra-MLC transport plays a significant role due to complex beam-shaping demands. This study develops an efficient ray-tracing-based intra-MLC transport algorithm for independent dose verification in patient QA for online adaptive therapy. We developed a ray-tracing-based quasi-analytical algorithm for particle transport in a multi-leaf collimator (MLC) that considers photon attenuation and primary Compton scattering within the MLC. To achieve both efficiency and high accuracy in calculating the transparency within the MLC, the algorithm employs an intersection-based ray-tracing technique to calculate the path length of photons by determining intersections with MLC components on projection planes corresponding to the leaf motion and arrangement planes. The transmittance of photons is calculated based on the path length in the MLC and assigned as the weight of the photon, following Sieber's approach. To accelerate the method, two types of Russian roulette techniques were implemented. The first is a geometrical Russian roulette method based on the MLC aperture, which aims at reducing the computational burden of photons transported outside the irradiation field. The second applies the Russian roulette method to the generation of Compton photons, which suppresses the impact of large-angle scattered Compton photons and further improves computational efficiency. We implemented the proposed transport calculation method for the MLC with Geant4 to calculate dose distributions in both digital phantoms and patient geometries. The proposed method was verified using the geometry of the Vero4DRT radiation therapy system (Hitachi High-Tech Corporation., Tokyo, Japan). The fluence and dose distributions obtained from the method were compared with the results of full Geant4 simulations, where all stages of the calculation process were performed using Geant4. The computational efficiency of the MLC transport calculation was also analyzed to confirm its efficiency. The calculation results obtained using the proposed method reasonably reproduced the fluence and dose distributions calculated by full Geant4 simulations for both digital phantoms and patient geometries. The 95th percentile of the absolute point-dose differences was within 2% for three-dimensional voxels with dose exceeding 10% of the maximum dose obtained from the full Geant4 simulation, in the presence of propagated statistical uncertainties. For the MLC transport calculation efficiency, the introduction of the two Russian roulette techniques successfully suppressed the variation in computation time caused by field size and achieved a speed-up of 56-160 times compared to Geant4. In this study, we proposed an efficient quasi-analytical MLC transport calculation method using an intersection-based ray-tracing technique. The developed code was integrated with Geant4, and comparisons with full Geant4 simulations confirmed that the calculation accuracy was sufficient. The algorithm can be integrated with GPU-accelerated Monte Carlo-based dose calculation codes designed for voxel geometries, which have been actively developed in recent years, making it a useful tool for independent dose verification in online adaptive radiotherapy.

Cave mapping represents one of the most complex challenges in geomorphological cartography, as it must convey the true three-dimensional geometry of subterranean spaces such as overlapping passages, irregular cross-sections, and variable ceiling and floor morphologies, within a two-dimensional framework. This study examines the methodological and interpretive challenges of cave mapping, utilizing the Gamssteighöhle cave in the Austrian Alps as a case study. During the 11 years of exploration, over 10 km of passages were surveyed using the DistoX vector survey method. Yet, the complex morphological forms necessitated deviations from the standard symbology recommended by the International Union of Speleology. Several key visualization challenges are analyzed, including subvertical pits, overlapping passages, and 3D maze-like networks. Solutions such as multiple projection planes, transparency effects, perpendicular cross-sections, and splitting maps into separate sheets are proposed to maintain readability and spatial context. We evaluate traditional vs. LiDAR-based mapping, concluding that while dense 3D point clouds offer exceptional precision, they do not inherently yield readable or informative maps. Cartographic generalization, with its interpretative input of the cartographer, remains indispensable for transforming spatial data into coherent and communicative cave maps. LiDAR and photogrammetry greatly enhance visualization and quantitative analysis but complement rather than replace traditional mapping.