Pairs Of Points Research Articles

Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles

AbstractHardy conjectured that the error term arising from approximating the number of lattice points lying in a radius‐ disc by its area is . One source of support for this conjecture is a folklore heuristic that uses i.i.d. random variables to model the lattice points lying near the boundary and square root cancellation of sums of these random variables. We examine this heuristic by studying how these lattice points interact with one another and prove that their autocorrelation is determined in terms of a random model. Additionally, it is shown that lattice points near the boundary which are “well separated” behave independently. We also formulate a conjecture concerning the distribution of pairs of these lattice points.

Feasibility of Robotic Transorbital Surgery.

The transorbital approach (TOA) facilitates access to pathologies lateral to the optic nerve, a region that is difficult to access with an endonasal approach. In this study, we sought to investigate the feasibility of robotic-assisted surgery in lateral TOA. Six colored-silicon-injected human postmortem heads were prepared for dissection. The DaVinci Xi model was used with a 0-degree camera, 8 mm in diameter. A black diamond microforceps with an 8-mm diameter and 10-mm jaw length was used. The entry point of V1 (superior orbital fissure), V3 (foramen ovale), and posterior root of the trigeminal ganglion were chosen as the surgical targets. The length from the entry opening to each target point was measured. The angles formed between pairs of target points were measured to obtain the horizontal angle (root of the trigeminal ganglion-entry-V1) and the vertical angle (root of the trigeminal ganglion-entry-V3). Dissection was performed on 12 sides (6 specimens). The median distance from the entry point was 55 mm (range 50-58 mm) to the entry point of V1 (superior orbital fissure), 65 mm (range 57-70 mm) to the entry point of V3 (foramen ovale), and 76 mm (range 70-87 mm) to the root of the trigeminal ganglion. Meanwhile, the median of surgical angle between the entry point and the target was 19.1° (range 11.8-30.4°) on the horizontal angle and 16.5° (range 6.2-21.6°) on the vertical angle. This study found that application of lateral TOA in robotic-assisted surgery is premature because of the large size of the tool. However, although the entrance in lateral TOA is narrow, the internal surgical space is wide; this offers potential for design of appropriate surgical tools to allow increase tool usage.

Probing the Shape of the Weyl Fermi Surface of NbP Using Transverse Electron Focusing.

Weyl semimetals are defined by their unique Fermi surface, comprising pairs of Weyl points of opposite chirality, connected through topological surface states. Angle-resolved photoemission spectroscopy (ARPES) has been used to verify the existence of the Weyl points and the Fermi arcs. However, ARPES is limited in resolution, leading to significant uncertainty when characterizing the shape of the Fermi surface of semimetals and measuring features such as the distance between the Weyl points. Here, to surpass the resolution of ARPES, we combine quantum oscillation measurements with transverse electron focusing experiments. These techniques offer complementary information, enabling the reconstruction of the distinctive peanut-shaped cross section of the Weyl Fermi surface and accurately determining the separation between Weyl points in the Weyl semimetal NbP. Our Letter showcases the integration of quantum oscillations and transverse electron focusing, allowing for the measurements of complex Fermi surface geometries, concurrently with carriers' transport properties, in high-mobility quantum materials.

Development of Multigroup Monte Carlo Neutron Transport Method with Regionwise Even-Parity Discontinuity Factor

The multigroup Monte Carlo (MC) neutron transport method with a regionwise even-parity discontinuity factor (REPDF), i.e. the discontinuity factor (DF)–MC method, is developed with the aim to provide a reference solution for deterministic transport calculations with DF. Applying the analogy with optics, neutrons are transmitted or reflected at a region surface during random walks. The probability of transmission or reflection is determined by REPDFs in adjacent regions. The DF is traditionally used in deterministic neutron transport methods to reduce the discretization error due to spatial homogenization and energy condensation. The DF-MC method can treat DF in the framework of the multigroup MC method. In this paper, the weight cancellation technique based on the closest pair of points using the divide-and-conquer algorithm is used because negative weights appear due to the neutron reflection. The REPDF is calculated by the method of characteristics (MOC). The verification calculations are carried out in the pin-by-pin homogenized and assembly homogenized KAIST-2A core geometry. The DF-MC calculation can reproduce the results of the MOC with the REPDF. These results demonstrate the principle of the DF-MC method and extend the application of the DF to the probabilistic neutron transport method.

Interplay between depth and width for interpolation in neural ODEs

Neural ordinary differential equations have emerged as a natural tool for supervised learning from a control perspective, yet a complete understanding of the role played by their architecture remains elusive. In this work, we examine the interplay between the width p and the number of transitions between layers L (corresponding to a depth of L+1). Specifically, we construct explicit controls interpolating either a finite dataset D, comprising N pairs of points in Rd, or two probability measures within a Wasserstein error margin ɛ>0. Our findings reveal a balancing trade-off between p and L, with L scaling as 1+O(N/p) for data interpolation, and as 1+Op−1+(1+p)−1ɛ−d for measures.In the high-dimensional and wide setting where d,p>N, our result can be refined to achieve L=0. This naturally raises the problem of data interpolation in the autonomous regime, characterized by L=0. We adopt two alternative approaches: either controlling in a probabilistic sense, or by relaxing the target condition. In the first case, when p=N we develop an inductive control strategy based on a separability assumption whose probability increases with d. In the second one, we establish an explicit error decay rate with respect to p which results from applying a universal approximation theorem to a custom-built Lipschitz vector field interpolating D.

Simulation of soil-structure interaction using inelasticity-separated finite element method

The soil–structure interaction (SSI) effect is nonnegligible during the nonlinear seismic response analysis of large-scale or underground structures. However, the computation of nonlinear response of a structure considering SSI effect usually is a time-consuming process because the numerical model should involve the structure itself, a wide-range soil domain and the soil–structure interface, especially for large-scale structure. This study aims to incorporate the inelasticity-separated finite element method (IS-FEM), which is an algorithm recently developed for efficient structural nonlinear analysis, to improve the efficiency of the soil-structure interaction system. To this end, an inelasticity-separated contact element model for modeling the nonlinear behavior of soil and structure interface is developed in this study. To derive the inelasticity-separated governing equation of this model, a reference elastic stiffness is defined so that the normal and shear deformation of any point pair in the presented contact element is decomposed into reference linear and reference nonlinear parts according to this reference elastic stiffness. As a result, the problem of inconsistency between the requirement of IS-FEM for constant initial linear elastic stiffness and the existence of inflection point for the stress versus relative displacement relationship of contact behavior can be solved. Then, the Goodman element formulation for depicting the nonlinear contact behavior is introduced and an interpolation scheme to establish the reference nonlinear deformation field in an element is incorporated. By combining the proposed model with existing inelasticity-separated elements, a global analytical model of the soil–structure interaction system can be established within the framework of IS-FEM. Because the nonlinearity usually occur in some local regions of the global model and the use of the IS-FEM only require performing these operations for a small scale matrix representing local nonlinearity, the computational efficiency of nonlinear structural response analysis considering SSI effect can be improved significantly. The applicability and efficiency of the proposed method is finally verified by two numerical examples.

Furstenberg Family and Chaos for Time-Varying Discrete Dynamical Systems.

Assume that (Y,ρ) is a nontrivial complete metric space, and that (Y,g1,∞) is a time-varying discrete dynamical system (T-VDDS), which is given by sequences (gl)l=1∞ of continuous selfmaps gl:Y→Y. In this paper, for a given Furstenberg family G and a given T-VDDS (Y,g1,∞), G-scrambled pairs of points of the system (Y,g1,∞) (which contains the well-known scrambled pairs) are provided. Some properties of the set of G-scrambled pairs of a given T-VDDS (Y,g1,∞) are studied. Moreover, the generically G-chaotic T-VDDS and the generically strongly G-chaotic T-VDDS are defined. A sufficient condition for a given T-VDDS to be generically strongly G-chaotic is also presented.

The weighted Euclidean one-center problem in $${\mathbb {R}}^n$$

AbstractGiven a finite set of distinct points in $${\mathbb {R}}^n$$ R n and a positive weight for each point, primal and dual algorithms are developed for finding the Euclidean ball of minimum radius so that the weighted Euclidean distance between the center of the ball and each point is less than or equal to the radius of the ball. Each algorithm is based on a directional search method in which the search path at each iteration is either a ray or a two-dimensional circular arc in $${\mathbb {R}}^n$$ R n . At each iteration, a search path is constructed by intersecting bisectors of pairs of points, where the bisectors are either hyperplanes or n-dimensional spheres. Each search path preserves complementary slackness and primal (dual) feasibility for the primal (dual) algorithm. The step size along each search path is determined explicitly. Test problems up to 1000 dimensions and 10,000 points were solved to optimality by both primal and dual algorithms. Computational results also show that these algorithms outperform several open-source SOCP codes.

Marker points 3D matching based on Delaunay triangfulation structure and error dispersion

Abstract Point cloud registration techniques based on marker points are widely used in optical 3D industrial measurements. However, in this process, marker points 3D matching methods are often haunted by low efficiency and accuracy. To improve the performance of marker points 3D matching, we propose a two-step method of ‘matching-verification’. In the matching process, Delaunay triangulation is introduced to extract the 3D structure of the marker points set, and then the 3D structure is deconstructed into 2D units for matching, which simplifies complexity and improves the efficiency of the algorithm. In the verification process, the mismatched pairs of points are located and removed by the method that is based on the error dispersion of initial matched results, and the initial transformation results are iteratively verified to obtain the optimal transformation matrix. The experimental results show that our method takes an average of 2.2 s for each matching, the average error of coarse registration point cloud is 0.075 mm and the root mean square is 0.219 mm, which effectively solves the problem of the low efficiency and accuracy of marker points 3D matching methods.

Neural network algorithm for optical-SAR image registration based on a uniform grid of points

The paper considers the problem of satellite multimodal image registration, in particular, optical and SAR (Synthetic Aperture Radar). Such algorithms are used in object detection, change detection, navigation. The paper considers algorithms for optical-to-SAR image registration in conditions of rough image pre-alignment. It is known that optical and SAR images have an inaccuracy in registration with georeference (up to 100 pixels with a spatial resolution of 10 m/pixel). This paper presents a neural network algorithm for optical-to-SAR image registration based on descriptors calculated for a uniform grid of points. First, algorithm find uniform grid of points for both images. Next, the neural network calculates descriptors for each point and finds descriptor distances between all possible pairs of points between optical and SAR images. Using obtained descriptor distances, a matching is made between the points on the optical and SAR images. The found matches between points are used to calculate the geometric transformation between images using the RANSAC algorithm with a limited (to combinations of translation, rotation and uniform scaling) affine transformation model. The accuracy of the proposed algorithm for optical-to-SAR image registration was investigated with different distortions in rotation and scale.

Uni-variable cross-spectral densities

We introduce a novel class of planar, scalar sources whose cross-spectral density is a function of a single complex variable. For a typical pair of source points, the modulus and argument of such a variable equal the product of their radial coordinates and the difference of their angular coordinates, respectively. As functions of a single complex variable, the corresponding cross-spectral densities (CSD) are expandable in power series in their convergence domain, and a virtually infinite number of source models can be devised. All such sources are shown to have vortex fields as modes. The closed analytical form of these uni-variable CSDs makes it possible to evaluate in a simple way the significant quantities characterizing the sources and the ones of the fields they radiate. The basic tool leading to the definition of the uni-variable CSDs are the reproducing-kernel Hilbert spaces. As an example, the CSD derived from the so called Szegö kernel is studied in some detail, and its features are derived, together with of those of the radiated field in the far zone.

Implementation of the Shortest Path Method with Excel Solver to Optimize Goods Delivery Routes

The shortest path solver is a program that aims to find the route with the lowest total edge weight between two points in a graph. Commonly used algorithms include Dijkstra for graphs with non-negative edge weights, Bellman-Ford for graphs with negative edge weights, and Floyd-Warshall for finding the shortest path between all point pairs. Its application is wide, ranging from navigation systems, computer networks, to logistics and games. The process of using it involves creating a graph model , selecting the appropriate algorithm, running a solver, and analyzing the results. A practical example shows how the Dijkstra algorithm can be used to determine the shortest route between cities in a road network , with effective and accurate results . The shortest path solver proves to be a versatile and essential tool for solving a wide range of problems in a variety of fields. This research uses quantitative methods with an experimental approach to test the effectiveness and efficiency of using the shortest path method with Solver Excel in optimizing goods delivery routes. The research object is the delivery route from the warehouse to several delivery destinations, with a sample of 10 routes that are most frequently used and have the highest delivery volume. Primary data was obtained through direct observation and interviews with company logistics managers. The results of this study show that the use of shortest path method with Excel Solver is effective in optimising the route of delivery of goods, reducing the cost and time of delivery by 15% and 10%. Although there are limitations for large networks, this tool remains useful and flexible. This implementation can be a reference for other companies to improve their logistics efficiency.

Dynamic Euclidean bottleneck matching

A fundamental question in computational geometry is for a set of input points in the Euclidean space, that is subject to discrete changes (insertion/deletion of points at each time step), whether it is possible to maintain an exact/approximate minimum weight perfect matching and/or bottleneck matching (a perfect matching that minimizes the length of the longest matched edge), in sublinear update time. In this work, we answer this question in the affirmative for points on a real line and for points in the plane with a bounded geometric spread.For a set P of n points on a line, we show that there exists a dynamic algorithm that maintains an exact bottleneck matching of P and supports insertion and deletion in O(log⁡n) time. Moreover, we show that a modified version of this algorithm maintains an exact minimum-weight perfect matching with O(log⁡n) update (insertion and deletion) time. Next, for a set P of n points in the plane, we show that a (62)-factor approximate bottleneck matching of Pk, at each time step k, can be maintained in O(log⁡Δ) amortized time per insertion and O(log⁡Δ+|Pk|) amortized time per deletion, where Δ is the geometric spread of P (the ratio between the diameter of P and the distance between the closest pair of points in P).

Accuracy evaluation of dental CBCT and scanned model registration method based on pulp horn mapping surface: an in vitro proof-of-concept

Background and aim3D fusion model of cone-beam computed tomography (CBCT) and oral scanned data can be used for the accurate design of root canal access and guide plates in root canal therapy (RCT). However, the pose accuracy of the dental pulp and crown in data registration has not been investigated, which affects the precise implementation of clinical planning goals. We aimed to establish a novel registration method based on pulp horn mapping surface (PHMSR), to evaluate the accuracy of PHMSR versus traditional methods for crown-pulp registration of CBCT and oral scan data.Materials and methodsThis vitro study collected 8 groups of oral scanned and CBCT data in which the left mandibular teeth were not missing, No. 35 and No. 36 teeth were selected as the target teeth. The CBCT and scanned model were processed to generate equivalent point clouds. For the PHMSR method, the similarity between the feature directions of the pulp horn and the surface normal vectors of the crown were used to determine the mapping points in the CBCT point cloud that have a great influence on the pulp pose. The small surface with adjustable parameters is reconstructed near the mapping point of the crown, and the new matching point pairs between the point and the mapping surface are searched. The sparse iterative closest point (ICP) algorithm is used to solve the new matching point pairs. Then, in the C + + programming environment with a point cloud library (PCL), the PHMSR, the traditional sparse ICP, ICP, and coherent point drift (CPD) algorithms are used to register the point clouds under two different initial deviations. The root square mean error (RSME) of the crown, crown-pulp orientation deviation (CPOD), and position deviation (CPPD) were calculated to evaluate the registration accuracy. The significance between the groups was tested by a two-tailed paired t-test (p < 0.05).ResultsThe crown RSME values of the sparse ICP method (0.257), the ICP method (0.217), and the CPD method (0.209) were not significantly different from the PHMSR method (0.250). The CPOD and CPPD values of the sparse ICP method (4.089 and 0.133), the ICP method (1.787 and 0.700), and the CPD method (1.665 and 0.718) than for the PHMSR method, which suggests that the accuracy of crown-pulp registration is higher with the PHMSR method.ConclusionCompared with the traditional method, the PHMSR method has a smaller crown-pulp registration accuracy and a clinically acceptable deviation range, these results support the use of PHMSR method instead of the traditional method for clinical planning of root canal therapy.

Enumeration of non-nodal real plane rational curves

Welschinger invariants enumerate real nodal rational curves in the plane or in another real rational surface. We analyze the existence of similar enumerative invariants that count real rational plane curves having prescribed non-nodal singularities and passing through a generic conjugation-invariant configuration of appropriately many points in the plane. We show that an invariant like this is unique: it enumerates real rational three-cuspidal quartics that pass through generically chosen four pairs of complex conjugate points. As a consequence, we show that through any generic configuration of four pairs of complex conjugate points, one can always trace a pair of real rational three-cuspidal quartics.

Frequency shifting of plasmon polariton defect modes in a non-Hermitian finite heterostructure

A finite photonic layered system alternating usual media with a dispersive metamaterial with two central layers in a PT- symmetric arrangement is investigated under the oblique incidence of an electromagnetic wave. We have found a defect mode within the gap, and we have studied its optical properties under the influence of gain and loss distributed in these central layers. As a consequence of PT-symmetry in the system, we find that exceptional points can be attracted to or repelled from each other depending on which side of the frequency band under study. Below the plasmon frequency, repulsion occurs between pairs of exceptional points, while above it they attract each other.

Real-Time Path Guiding Using Bounding Voxel Sampling

We propose a real-time path guiding method, Voxel Path Guiding (VXPG), that significantly improves fitting efficiency under limited sampling budget. Our key idea is to use a spatial irradiance voxel data structure across all shading points to guide the location of path vertices. For each frame, we first populate the voxel data structure with irradiance and geometry information. To sample from the data structure for a shading point, we need to select a voxel with high contribution to that point. To importance sample the voxels while taking visibility into consideration, we adapt techniques from offline many-lights rendering by clustering pairs of shading points and voxels. Finally, we unbiasedly sample within the selected voxel while taking the geometry inside into consideration. Our experiments show that VXPG achieves significantly lower perceptual error compared to other real-time path guiding and virtual point light methods under equal-time comparison. Furthermore, our method does not rely on temporal information, but can be used together with other temporal reuse sampling techniques such as ReSTIR to further improve sampling efficiency.

Topological properties of superconductive transition metal phosphorus compounds

Based on the first-principles calculations and effective k⋅p model, we propose the topological nodal-line (NL) semimetal state in the family of non-centro-symmetric ternary transition metal phosphorus compounds TRuX (T=Zr, Hf, Nb; X = P, As), when the spin-orbit coupling (SOC) is ignored. A K-point centered nodal line on the kz = 0 plane is protected by the mirror symmetry Mz. When SOC is considered, the gap opens along the NL in the kz=0 plane. But, at the kz = ±0.015*2πc planes, there are 12 pairs of Weyl points. The Berry phase, Berry curvature and Fermi Arc of those Weyl points are found. In these compounds, the Fu-Kane Z2 invariant is trivial (0;000), but the mirror Chern number is 2. Since these compounds show superconductivity at about 11 K, they might be mirror symmetry protected topological superconductors.

Longitudinal relations between child emotional difficulties and parent-child closeness: a stability and malleability analysis using the STARTS model

BackgroundPast empirical evidence on the longitudinal relations between emotional mental health symptoms and parent-child close relationships has produced mixed and inconclusive results. Some studies suggest a unidirectional relation, whereas other studies point toward a bidirectional association. Additionally, most of the past research has been carried out with adolescent samples, rather than children. Hence, this study aimed to estimate the longitudinal relations between children’s trait emotional difficulties and trait parent-child closeness, accounting for the time-invariant and time-varying state components of each factor.MethodsParticipants were 7,507 children (ages 3 years, 5 years, 7 years, and 9 years) from the Growing Up in Ireland cohort. Α bivariate stable trait, autoregressive trait, and state (STARTS) model was estimated using Bayesian structural equation modelling.ResultsThe STARTS model revealed that children’s emotional difficulties and parent-child closeness were relatively stable across time, and these overarching traits were strongly negatively correlated. Children’s earlier trait emotional difficulties predicted later trait parent-child closeness and vice versa between 3 years and 5 years, and between 5 years and 7 years, but these effects disappeared between 7 years and 9 years. At all pairs of time points, state emotional difficulties and state parent-child closeness were weakly negatively correlated.ConclusionsOverall, the results suggest that early and middle childhood are critical stages for improving parent-child relationships and reducing children’s emotional difficulties. Developing close parent-child relationships in childhood appears to be a key factor in reducing children’s subsequent emotional difficulties. Children who face greater than usual emotional difficulties tend to be more withdrawn and less receptive to close parent-child relationships and this could serve as an important screening indicator.

Search for an antiferromagnetic Weyl semimetal in (MnTe) m (Sb2Te3) n and (MnTe) m (Bi2Te3) n superlattices

The interaction between topology and magnetism can lead to novel topological materials including Chern insulators, axion insulators, and Dirac and Weyl semimetals. In this work, a family of van der Waals layered materials using MnTe and Sb2Te3 or Bi2Te3 superlattices as building blocks are systematically examined in a search for antiferromagnetic Weyl semimetals, preferably with a simple node structure. The approach is based on controlling the strength of the exchange interaction as a function of layer composition to induce the phase transition between the topological and the normal insulators. Our calculations, utilizing a combination of first-principles density functional theory and tight-binding analyses based on maximally localized Wannier functions, clearly indicate a promising candidate for a type-I magnetic Weyl semimetal. This centrosymmetric material, Mn10Sb8Te22 (or (MnTe) m (Sb2Te3) n with m = 10 and n = 4), shows ferromagnetic intralayer and antiferromagnetic interlayer interactions in the antiferromagnetic ground state. The obtained electronic bandstructure also exhibits a single pair of Weyl points in the spin-split bands consistent with a Weyl semimetal. The presence of Weyl nodes is further verified with Berry curvature, Wannier charge center, and surface state (i.e. Fermi arc) calculations. Other combinations of the MnSbTe-family materials are found to be antiferromagnetic topological or normal insulators on either side of the Mn:Sb ratio, respectively, illustrating the topological phase transition as anticipated. A similar investigation in the homologous (MnTe) m (Bi2Te3) n system produces mostly nontrivial antiferromagnetic insulators due to the strong spin–orbit coupling. When realized, the antiferromagnetic Weyl semimetals in the simplest form (i.e. a single pair of Weyl nodes) are expected to provide a promising candidate for low-power spintronic applications.