A Numerical Method for Computing Double Integrals with Variable Upper Limits

We propose and justify a numerical method for computing the double integral with variable upper limits that leads to the variableness of the region of integration. Imposition of simple variables as functions for upper limits provides the form of triangles of integration region and variable in the external limit of integral leads to a continuous set of similar triangles. A variable grid is overlaid on the integration region. We consider three cases of changes of the grid for the division of the integration region into elementary volumes. The first is only the size of the imposed grid changes with the change of variable of the external upper limit. The second case is the number of division elements changes with the change of the external upper limit variable. In the third case, the grid size and the number of division elements change after fixing their multiplication. In these cases, the formulas for computing double integrals are obtained based on the application of cubatures in the internal region of integration and performing triangulation division along the variable boundary. The error of the method is determined by expanding the double integral into the Taylor series using Barrow’s theorem. Test of efficiency and reliability of the obtained formulas of the numerical method for three cases of ways of the division of integration region is carried out on examples of the double integration of sufficiently simple functions. Analysis of the obtained results shows that the smallest absolute and relative errors are obtained in the case of an increase of the number of division elements changes when the increase of variable of the external upper limit and the grid size is fixed.

Finding Equilibria in the Traffic Assignment Problem with Primal-Dual Gradient Methods for Stable Dynamics Model and Beckmann Model

In this paper, we consider the application of several gradient methods to the traffic assignment problem: we search equilibria in the stable dynamics model (Nesterov and De Palma, 2003) and the Beckmann model. Unlike the celebrated Frank–Wolfe algorithm widely used for the Beckmann model, these gradients methods solve the dual problem and then reconstruct a solution to the primal one. We deal with the universal gradient method, the universal method of similar triangles, and the method of weighted dual averages and estimate their complexity for the problem. Due to the primal-dual nature of these methods, we use a duality gap in a stopping criterion. In particular, we present a novel way to reconstruct admissible flows in the stable dynamics model, which provides us with a computable duality gap.

Automation of Formation Tops Estimation Dramatically Reduces Well Planning Process Duration

Aiming to make the well planning process leaner and agile focusing on duration reduction without compromising quality of deliverables, automation opportunities have been identified within the multi-discipline iterations. The two key criteria considered for the selection of the automation project were: Minimum deployment effort and Maximum value added in efficiency. The initial project objective was to calculate formation tops for a well engineer without requiring the intervention of a geoscientist using commercial software. The methodology utilized is the following: 1. Inputs: Well trajectory and Surfaces. 2. Process: The algorithm finds intersections between surfaces and well trajectory. Surfaces and trajectory are represented as a set of XYZ points. To find the intersection, the software iterates through each point of the trajectory from the top, comparing the depth of the projection to the target surface. The projected depth to the surface is found by 2D interpolation of the surface. Once the trajectory point becomes deeper than the surface projection, the intersection is estimated using geometrical considerations of similar triangles. 3. Deliverables: Estimated formation tops for the given trajectory. 4. Results: Simple in-house developed software enhanced well planning workflow in an Offshore Green Field. The software converted to single executable file and can be run on any device without the open-source software installed. Very accurate results achieved with proposed algorithm with a negligible difference of 0.5 feet with the geoscience traditional software. Well planning duration reduced from average 1 week to 1 or 2 days. The workload for well engineers and the asset team has been dramatically reduced. Reduction of the number of commercial geoscience software licenses required. Way forward: A test with a slightly modified code was used to generate formation tops for more than 400 well in a Long-Term Field Development Plan project for a Brown Field during feasibility study. Upscale to all the Fields within the organization. Improve User Interface for better adoption. Include more formats for both, trajectories, and surfaces. Reduce computing time. This project represents the first initiative in the organization aiming to automate the well planning process. Overall, it represents the beginning of a journey where multiple opportunities for automation can be achieved using an open-source coding software that allows any engineer with little to no experience coding to being able to generate solutions to address daily challenges.

Irregular Tilings of Regular Polygons with Similar Triangles

We say that a triangle T tiles a polygon A, if A can be dissected into finitely many nonoverlapping triangles similar to T. We show that if $$N>42$$ N > 42 , then there are at most three nonsimilar triangles T such that the angles of T are rational multiples of $$\pi $$ π and T tiles the regular N-gon. A tiling into similar triangles is called regular, if the pieces have two angles, $$\alpha $$ α and $$\beta $$ β , such that at each vertex of the tiling the number of angles $$\alpha $$ α is the same as that of $$\beta $$ β . Otherwise the tiling is irregular. It is known that for every regular polygon A there are infinitely many triangles that tile A regularly. We show that if $$N>10$$ N > 10 , then a triangle T tiles the regular N-gon irregularly only if the angles of T are rational multiples of $$\pi $$ π . Therefore, the number of triangles tiling the regular N-gon irregularly is at most three for every $$N>42$$ N > 42 .

Students' difficulties in similar triangle questions

Similar triangles in questions are usually given as separate, adjacent or overlapped. Furthermore, similarity types such as Side-Angle-Side (S.A.S.), Side-Side-Side (S.S.S.) and Angle-Angle (A.A.) are requested in the questions. Students have more trouble in other types of questions. The purpose of this study is to investigate the difficulties of students about similar triangles and the reasons for these difficulties. This research was carried out with the case study method, which is one of the qualitative research approaches. The study was conducted with 55 Science High School 9th grade students and 9 open-ended questions were used to examine students' knowledge about “similarity in triangles”. Furthermore, 5 students were interviewed to find out the reasons for their solutions. Descriptive analysis method was used to analyze the data. As a result, it can be concluded that students have difficulties mostly in overlapped triangles and Angle-Angle type questions. On the other hand, it can be concluded that students are quite successful where similar triangles are given separately. In the light of the findings obtained in this study, it can be advised for lecturers to focus on the questions where similar triangles are overlapped while explaining the similarity in the triangle. Keywords: Similarity, Triangles, Difficulties, High School Students.

Multiple Morphological Constraints-Based Complex Gland Segmentation in Colorectal Cancer Pathology Image Analysis

Histological assessment of glands is one of the major concerns in colon cancer grading. Considering that poorly differentiated colorectal glands cannot be accurately segmented, we propose an approach for segmentation of glands in colon cancer images, based on the characteristics of lumens and rough gland boundaries. First, we use a U-net for stain separation to obtain H-stain, E-stain, and background stain intensity maps. Subsequently, epithelial nucleus is identified on the histopathology images, and the lumen segmentation is performed on the background intensity map. Then, we use the axis of least inertia-based similar triangles as the spatial characteristics of lumens and epithelial nucleus, and a triangle membership is used to select glandular contour candidates from epithelial nucleus. By connecting lumens and epithelial nucleus, more accurate gland segmentation is performed based on the rough gland boundary. The proposed stain separation approach is unsupervised, and the stain separation makes the category information contained in the H&E image easy to identify and deal with the uneven stain intensity and the inconspicuous stain difference. In this project, we use deep learning to achieve stain separation by predicting the stain coefficient. Under the deep learning framework, we design a stain coefficient interval model to improve the stain generalization performance. Another innovation is that we propose the combination of the internal lumen contour of adenoma and the outer contour of epithelial cells to obtain a precise gland contour. We compare the performance of the proposed algorithm against that of several state-of-the-art technologies on publicly available datasets. The results show that the segmentation approach combining the characteristics of lumens and rough gland boundary has better segmentation accuracy.