Random Polygons Research Articles

Modelling of Chloride Concentration Profiles in Concrete by the Consideration of Concrete Material Factors under Marine Tidal Environment

The corrosion of reinforcement induced by chloride ions is one of the most significant causes of durability deterioration for reinforced concrete (RC) buildings. The concrete material factors, including the water-to-cement ratio (w/c) of concrete, as well as the content, shape, particle grading, and random distribution of coarse aggregate embedded in mortar, have a marked effect on chloride transport performance within concrete. However, comprehensive consideration for the effects of both w/c and coarse aggregate performances on chloride diffusion characteristics in concrete is scarce, especially regarding the chloride diffusion model of concrete. In this paper, an indoor exposure experiment exploring chloride ions intruding into mortar and concrete specimens with w/c = 0.4, 0.5 and 0.6 was carried out, in order to acquire the chloride diffusion parameters for concrete three-phases composites. Based on the numerical algorithm of random generation and placement of two-dimensional random convex polygon coarse aggregate, mesoscopic numerical models for concrete, considering various coarse aggregate contents as well as grading, were established. Using the numerical simulation method of finite element analysis for chloride transport in cement-based materials, which can replace some of the exposure tests, the influences of w/c, coarse aggregate content and grading on chloride diffusion performance in concrete mesoscopic models were systematically probed. According to the Fick’s second law, a chloride diffusion model by the consideration of w/c, volume fraction of coarse aggregate (VFCA), and maximum size of coarse aggregate (MSCA) was developed to assess the chloride concentration profiles in concrete under arbitrary w/c, coarse aggregate content, and coarse aggregate grading conditions. Certainly, the precision accuracy for this proposed chloride diffusion model was validated. The research results can provide theoretical support for chloride erosion behavior and structural durability assessment of concrete with different mix proportions.

Investigations of polygonal patterned ground in continuous Antarctic permafrost by means of ground penetrating radar and electrical resistivity tomography: Some unexpected correlations

AbstractThe results of a combined geophysical and geomorphological investigation of thermal‐contraction‐crack polygons near Gondwana station (Germany) in northern Victoria Land (Antarctica) are reported. An area of about 20,000 m2 characterized by random orthogonal polygons was investigated using integrated ground penetrating radar, electrical resistivity tomography, geomorphological surveys, and two trench excavations. The polygons are well developed only at elevations higher than 6–7 m above current sea level on Holocene‐age raised beaches. It is concluded that the polygons are composite in nature because the shallow linear depressions that outline the polygons are underlain by fissures that can contain both sandy gravel and foliated ice (i.e., ice wedges) even in the same polygon network and at distances of just a few meters. Unexpectedly, most of the polygons follow the border of the raised beaches and develop in correspondence with stratigraphic layers dipping toward the sea, imaged by ground penetrating radar (GPR) profiles and interpreted as prograding layers toward the present‐day shoreline.

2-Opt Moves and Flips for Area-optimal Polygonizations

Our work on the Computational Geometry Challenge 2019 on area-optimal polygonizations is based on two key components: (1) sampling the search space to obtain initial polygonizations and (2) optimizing such a polygonizations. Among other heuristics for obtaining polygonizations for a given set P of input points, we discuss how to combine 2-opt moves with a line sweep to convert an initial random (non-simple) polygon whose vertices are given by P into a polygonization P . The actual optimization relies on a constrained triangulation of the interior and exterior of a polygonization to speed-up local modifications of the polygonization to increase or decrease its area.

The second Vassiliev measure of uniform random walks and polygons in confined space

Biopolymers, like chromatin, are often confined in small volumes. Confinement has a great effect on polymer conformations, including polymer entanglement. Polymer chains and other filamentous structures can be represented by polygonal curves in three-space. In this manuscript, we examine the topological complexity of polygonal chains in three-space and in confinement as a function of their length. We model polygonal chains by equilateral random walks in three-space and by uniform random walks (URWs) in confinement. For the topological characterization, we use the second Vassiliev measure. This is an integer topological invariant for polygons and a continuous functions over the real numbers, as a function of the chain coordinates for open polygonal chains. For URWs in confined space, we prove that the average value of the Vassiliev measure in the space of configurations increases as O(n 2) with the length of the walks or polygons. We verify this result numerically and our numerical results also show that the mean value of the second Vassiliev measure of equilateral random walks in three-space increases as O(n). These results reveal the rate at which knotting of open curves and not simply entanglement are affected by confinement.

Controllability Robustness of Henneberg-Growth Complex Networks

It is important for a networked-system to have both good controllability and robustness, where the former measures the ability that the networked system can be properly steered via control to any target state form any initial state within a finite time duration, while the later reflects how well this system can regain its controllability after being destructively attacked. Empirical observations suggest that multi-loop structures are beneficial for enhancing the controllability robustness. The Henneberg-growth mechanism in social networks offers a natural manner to generate multi-clique structures that can be developed to multi-loops by assigning proper edge directions. In this paper, a series of random polygon networks are generated, forming random triangle, rectangle, pentagon and hexagon networks. Then, their controllability robustness is investigated. A realistic measure is designed for characterizing their controllability robustness, which can be used to filter out trivial performances after the network is severely destructed, so that the computation and analysis become much more efficient. Extensive simulation results suggest that, for random polygon networks, 1) non-loop polygon structures are inferior to the loop polygons, confirming that the multi-loop structures are indeed beneficial for controllability robustness; 2) polygons with more sides possess better controllability robustness; and 3) the correlation between controllability robustness and connectivity robustness is weak, implying that the two objectives cannot be enhanced in the same way.

Knot probabilities in equilateral random polygons

We consider the probability of knotting in equilateral random polygons in Euclidean three-dimensional space, which model, for instance, random polymers. Results from an extensive Monte Carlo dataset of random polygons indicate a universal scaling formula for the knotting probability with the number of edges. This scaling formula involves an exponential function, independent of knot type, with a power law factor that depends on the number of prime components of the knot. The unknot, appearing as a composite knot with zero components, scales with a small negative power law, contrasting with previous studies that indicated a purely exponential scaling. The methodology incorporates several improvements over previous investigations: our random polygon data set is generated using a fast, unbiased algorithm, and knotting is detected using an optimised set of knot invariants based on the Alexander polynomial.

New Stick Number Bounds from Random Sampling of Confined Polygons

The stick number of a knot is the minimum number of segments needed to build a polygonal version of the knot. Despite its elementary definition and relevance to physical knots, the stick number is poorly understood: for most knots we only know bounds on the stick number. We adopt a Monte Carlo approach to finding better bounds, producing very large ensembles of random polygons in tight confinement to look for new examples of knots constructed from few segments. We generated a total of 220 billion random polygons, yielding either the exact stick number or an improved upper bound for more than 40% of the knots with 10 or fewer crossings for which the stick number was not previously known. We summarize the current state of the art in Appendix A, which gives the best known bounds on stick number for all knots up to 10 crossings.

Knotting spectrum of polygonal knots in extreme confinement

Random knot models are often used to measure the types of entanglements one would expect to observe in an unbiased system with some given physical property or set of properties. In nature, macromolecular chains often exist in extreme confinement. Current techniques for sampling random polygons in confinement are limited. In this paper, we gain insight into the knotting behavior of random polygons in extreme confinement by studying random polygons restricted to cylinders, where each edge connects the top and bottom disks of the cylinder. The knot spectrum generated by this model is compared to the knot spectrum of rooted equilateral random polygons in spherical confinement. Due to the rooting, such polygons require a radius of confinement R ⩾ 1. We present numerical evidence that the polygons generated by this simple cylindrical model generate knot probabilities that are equivalent to spherical confinement at a radius of R ≈ 0.62. We then show how knot complexity and the relative probability of different classes of knot types change as the length of the polygon increases in the cylindrical polygons.

Learning hierarchical and efficient Person re-identification for robotic navigation

Recent works in the person re-identification task mainly focus on the model accuracy while ignoring factors related to efficiency, e.g., model size and latency, which are critical for practical application. In this paper, we propose a novel Hierarchical and Efficient Network (HENet) that learns hierarchical global, partial, and recovery features ensemble under the supervision of multiple loss combinations. To further improve the robustness against the irregular occlusion, we propose a new dataset augmentation approach, dubbed random polygon erasing, to random erase the input image’s irregular area imitating the body part missing. We also propose an Efficiency Score (ES) metric to evaluate the model efficiency. Extensive experiments on Market1501, DukeMTMC-ReID, and CUHK03 datasets show the efficiency and superiority of our approach compared with epoch-making methods. We further deploy HENet on a robotic car, and the experimental result demonstrates the effectiveness of our method for robotic navigation.

An extended numerical manifold method for unsaturated soil‐water interaction analysis at micro‐scale

AbstractTo investigate unsaturated soil‐water interaction at micro‐scale, this study extends the numerical manifold method (NMM) by incorporating a soil‐water coupling model considering specific capillary water distribution and capillary force calculation. The soil skeleton is constructed by a soil skeleton generation algorithm with random polygons. To more realistically capture the interaction between soil grains and capillary water, a capillary mechanics‐based geometric algorithm is proposed to iteratively calculate the capillary water distribution. The capillary forces corresponding to the capillary water distribution are calculated based on the Young‐Laplace equation. The proposed capillary water solving framework is first verified by reproducing the soil‐water characteristic curve and the capillary water distribution of an ideal contact‐disk model against analytical solutions. To further validate the ability of the capillary water solving framework to predict hydraulic behavior of the real soil, a laboratory test on the Toyoura sand is reproduced numerically. Then an ideal direct shear test is performed to further validate the two‐way soil‐water coupling procedure, in which a comparison between the numerical and analytical results regarding the shear strength and matric suction is presented. Finally, microscopic hydraulic and compression tests are conducted on two soil specimens with the same porosity and mean grain diameter but different uniformity coefficients. The results elucidate that the extended method is a potential tool to explore unsaturated soil behaviors at micro‐scale.

Random Polygons and Optimal Extrapolation Estimates of pi

We construct optimal extrapolation estimates of π based on random polygons generated by n independent points uniformly distributed on a unit circle in R2. While the semiperimeters and areas of these random n-gons converge to π almost surely and are asymptotically normal as n → ∞, in this paper we develop various extrapolation processes to further accelerate such convergence. By simultaneously considering the random n-gons and suitably constructed random 2n-gons and then optimizing over functionals of the semiperimeters and areas of these random polygons, we derive several new estimates of π with faster convergence rates. These extrapolation improvements are also shown to be asymptotically normal as n → ∞.

Limit theorems for generalized perimeters of random inscribed polygons. II

Lao and Mayer (2008) recently developed the theory of U-max statistics, where instead of the usual sums over subsets, the maximum of the kernel is considered. Such statistics often appear in stochastic geometry. Examples include the greatest distance between random points in a ball, the maximum diameter of a random polygon, the largest scalar product in a sample of points, etc. Their limit distributions are related to distribution of extreme values. This is the second article devoted to the study of the generalized perimeter of a polygon and the limit behavior of the U-max statistics associated with the generalized perimeter. Here we consider the case when the parameter y, arising in the definition of the generalized perimeter, is greater than 1. The problems that arise in the applied method in this case are described. The results of theorems on limit behavior in the case of a triangle are refined.

Pancharatnam-Berry phase algorithm to calculate the area of arbitrary polygons on the Poincaré sphere.

We introduce a very efficient noniterative algorithm to calculate the signed area of a spherical polygon with arbitrary shape on the Poincaré sphere. The method is based on the concept of the geometric Berry phase. It can handle diverse scenarios like convex and concave angles, multiply connected domains, overlapped vertices, sides and areas, self-intersecting polygons, holes, islands, cogeodesic vertices, random polygons, and vertices connected with long segments of great circles. A set of MATLAB routines of the algorithm is included. The main benefits of the algorithm are the ability to handle all manner of degenerate shapes, the shortness of the program code, and the running time.

On the Average Perimeter of the Inscribed Random Polygon

Assume that n independent uniformly distributed random points are set on the unit circle. Construct the convex random polygon with vertices in these points. What are the average area and the average perimeter of this polygon? Brown computed the average area several years ago. We compute the average perimeter and obtain quite similar expressions. We also discuss the rate of the convergence of this value to the limit and evaluate the average value of the sum of squares for the sides of the inscribed triangle.

Limit theorems for generalized perimeters of random inscribed polygons. I

Lao and Mayer (2008) recently developed the theory of U-max-statistics, where instead of the usual averaging the values of the kernel over subsets, the maximum of the kernel is considered. Such statistics often appear in stochastic geometry. Their limit distributions are related to distributions of extreme values. This is the first article devoted to the study of the generalized perimeter (the sum of side powers) of an inscribed random polygon, and of U-max-statistics associated with it. It describes the limiting behavior for the extreme values of the generalized perimeter. This problem has not been studied in the literature so far. One obtains some limit theorems in the case when the parameter y, arising in the definition of the generalized perimeter does not exceed 1.

On the average perimeter of the inscribed random polygon

On the average perimeter of the inscribed random polygon

Meso-Scale Simulation of Concrete Uniaxial Behavior Based on Numerical Modeling of CT Images.

It is important to study the failure mechanism of concrete by observing the crack expansion and capturing key structures at the mesoscale. This manuscript proposed a method for efficiently identifying aggregate boundary information by X-ray computed tomography technology (CT) and a discrete element modeling method (DEM) for equivalent random polygon aggregates. This method overcomes the shortcomings of the Grain Based Model (GBM) which is impossible to establish a mesoscopic model with a large difference in grain radius. Through the above two methods, the CT slice images were processed in batches, and the numbers of edges, axial length, elongation of the aggregate were identified. The feasibility of the method was verified by the comparison between experimental and simulating results. Three mesoscopic models for different porosities were established. Based on aggregate statistics, this manuscript achieved the meso-model recovery to the maximum extent. The test results show that the crack appeared at the tip of the aggregate firstly, and then the broken boundary was applied in the direction of the applied load and around the pores. Finally, the crack was selectively expanded under the axial force. During the loading process, the minor principal stress was normally distributed. As the porosity and loading time increased, the heterogeneity increased.

Random Polygons and Estimations of π

Abstract In this paper, we study the approximation of π through the semiperimeter or area of a random n-sided polygon inscribed in a unit circle in ℝ2. We show that, with probability 1, the approximation error goes to 0 as n → ∞, and is roughly sextupled when compared with the classical Archimedean approach of using a regular n-sided polygon. By combining both the semiperimeter and area of these random inscribed polygons, we also construct extrapolation improvements that can significantly speed up the convergence of these approximations.

Robust Histopathology Image Analysis: to Label or to Synthesize?

Detection, segmentation and classification of nuclei are fundamental analysis operations in digital pathology. Existing state-of-the-art approaches demand extensive amount of supervised training data from pathologists and may still perform poorly in images from unseen tissue types. We propose an unsupervised approach for histopathology image segmentation that synthesizes heterogeneous sets of training image patches, of every tissue type. Although our synthetic patches are not always of high quality, we harness the motley crew of generated samples through a generally applicable importance sampling method. This proposed approach, for the first time, re-weighs the training loss over synthetic data so that the ideal (unbiased) generalization loss over the true data distribution is minimized. This enables us to use a random polygon generator to synthesize approximate cellular structures (i.e., nuclear masks) for which no real examples are given in many tissue types, and hence, GAN-based methods are not suited. In addition, we propose a hybrid synthesis pipeline that utilizes textures in real histopathology patches and GAN models, to tackle heterogeneity in tissue textures. Compared with existing state-of-the-art supervised models, our approach generalizes significantly better on cancer types without training data. Even in cancer types with training data, our approach achieves the same performance without supervision cost. We release code and segmentation results on over 5000 Whole Slide Images (WSI) in The Cancer Genome Atlas (TCGA) repository, a dataset that would be orders of magnitude larger than what is available today.

Random Triangles and Polygons in the Plane

In 1997, Jean-Claude Hausmann and Allen Knutson introduced a natural and beautiful correspondence between planar n-gons and the Grassmann manifold of 2-planes in real n-space. This construction leads to a natural probability distribution and a natural metric on polygons which has been used in shape classification and computer vision. In this paper, we provide an accessible introduction to this circle of ideas by explaining the Grassmannian geometry of triangles. We use this to find the probability that a random triangle is obtuse, which was a question raised by Lewis Carroll. We then explore the Grassmannian geometry of planar quadrilaterals, providing an answer to Sylvester’s four-point problem, and describing explicitly the moduli space of unordered quadrilaterals.