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Spherical trigonometry was at the heart of astronomy and ocean-going navigation for two millennia. The discipline was a mainstay of mathematics education for centuries, and it was a standard subject in high schools until the 1950s. Today, however, it is rarely taught.Heavenly Mathematicstraces the rich history of this forgotten art, revealing how the cultures of classical Greece, medieval Islam, and the modern West used spherical trigonometry to chart the heavens and the Earth. Glen Van Brummelen explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. He conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation for its elegant proofs and often surprising conclusions. Heavenly Mathematicsis illustrated throughout with stunning historical images and informative drawings and diagrams that have been used to teach the subject in the past. This unique compendium also features easy-to-use appendixes as well as exercises at the end of each chapter that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

The management of inflammatory bowel disease represents a key component of clinical practice for members of the British Society of Gastroenterology (BSG). There has been considerable progress in management strategies affecting all aspects of clinical care since the publication of previous BSG guidelines in 2004, necessitating the present revision.Key components of the present document worthy of attention as having been subject to re-assessment, and revision, and having direct impact on practice include:The data generated by the nationwide audits of inflammatory bowel disease (IBD) management in the UK in 2006, and 2008.The publication of ‘Quality Care: service standards for the healthcare of people with IBD’ in 2009.The introduction of the Montreal classification for Crohn's disease and ulcerative colitis.The revision of recommendations for the use of immunosuppressive therapy.The detailed analysis, guidelines and recommendations for the safe and appropriate use of biological therapies in Crohn's disease and ulcerative colitis.The reassessment of the role of surgery in disease management, with emphasis on the importance of multi-disciplinary decision-making in complex cases.The availablity of new data on the role of reconstructive surgery in ulcerative colitis.The cross-referencing to revised guidelines for colonoscopic surveillance, for the management of metabolic bone disease, and for the care of children with inflammatory bowel disease.Use of the BSG discussion forum available on the BSG website to enable ongoing feedback on the published document http://www.bsg.org.uk/forum (accessed Oct 2010).The present document is intended primarily for the use of clinicians in the United Kingdom, and serves to replace the previous BSG guidelines in IBD, while complementing recent consensus statements published by the European Crohn's and Colitis Organisation (ECCO) https://www.ecco-ibd.eu/index.php (accessed Oct 2010).

Summary Herbivores face various nutritional challenges in their life cycles, challenges that may become increasingly acute under ongoing environmental changes. Here, focusing on calcium, phosphorus and nitrogen, we used nutritional geometry to analyse individual‐based data on foraging and extraction efficiencies, and combined these with data on reproduction and migratory behaviour to understand how a large herbivorous carnivore can complete its life cycle on a narrow and seemingly low quality bamboo diet. Behavioural results showed that pandas during the year switched between four main food categories involving the leaves and shoots of two bamboo species available. Nutritional analysis suggests that these diet shifts are related to the concentrations and balances of calcium, phosphorus and nitrogen. Notably, successive shifts in range use and food type corresponded with a transition to higher concentrations and/or a more balanced intake of these multiple key constituents. Our study suggests that pandas obligatorily synchronize their seasonal migration and reproduction with the disjunct nutritional phenologies of two bamboo species. This finding has potentially important implications for habitat conservation for this species and, more generally, draws attention to the need for understanding the nutritional basis of food selection in devising management plans for endangered species. Lay Summary

This study aimed to determine the impact of using the Cabri II Plus software on students’ achievement and motivation. A quasi-experimental research design was employed to investigate the effects of learning with Cabri II Plus software on the achievement and motivation of second-grade middle school students in the module covering right triangles and the Pythagorean theorem. The study comprised 70 participants, who were divided into two groups: experimental and control. Each group consisted of 35 participants. The study utilized two main tools: a student achievement test and a math motivation questionnaire. The experimental group used interactive activities with Cabri II Plus software, while the control group used existing activities from a textbook provided by the Moroccan Ministry of Education. Thus, the pre- and post-tests on the achievement and motivation scores were carried out for both groups. Analysis of covariance revealed a significant difference in favor of the experimental group in terms of the academic achievement scores (F [1, 67] = 52.12, p = .00 < .05) and the motivation scores (F [1, 67] = 94.08, p = .00 < .05). Thus, the students who used the Cabri II Plus program obtained higher motivation and achievement than their counterparts who did not use Cabri II Plus to learn the test module. The study’s findings also showed a strong, statistically significant positive relationship between academic achievement and motivation among the experimental group students in the post-tests and a moderate, statistically significant relationship between these variables among the control group students. Thus, the teaching method using the Cabri II Plus program has a greater positive impact on the relationship between academic achievement and motivation than traditional teaching methods. Based on the findings, integrating Cabri II Plus into the teaching process effectively improves students’ educational outcomes and fosters a more engaged and motivated learning environment.

One of the most used hydraulic structures for energy dissipation of supercritical flow is the hydraulic jump stilling basin. From dimensional analysis, the sequent flow depth ratio of a hydraulic jump over the right-triangle basin is derived as a function of the inflow Froude number and relative length of the basin front. The proposed structure stabilized the hydraulic jump at the toe of the chute spillway and hydraulic jump characteristics were investigated for the Froude number ranging from 4.4 < F1 < 7. The results obtained from both numerical and experimental simulations yielded increased efficiency in the energy dissipation performance of this novel design. The modeling showed the formation of two large recirculation regions at the jump roller and jump bed at the beginning of the downstream channel, which resulted in intense energy dissipation in the right-triangle basin. The relative energy loss is approximately 37% higher for relative basin front lengths larger than three compared to the classic jump. Practitioners and academia on the usefulness of a right-triangle basin for hydraulic purposes and further experimental tests are needed to estimate the scalability and cost–benefit of this modified system for implementation.

The traditional welding spots detection methods are very sensitive to light pollution, rust and oil stain. In this work, a novel method that combines light-weight deep learning model and circle detection algorithm for welding spots detection is proposed. First, a light-weight model called Modified You Only Look Once version 2 (M-YOLOv2) which substitutes MobileNetv2 for Darknet-19 convolution structure in YOLOv2 and incorporates fine-grained features (FGF) is developed. The proposed M-YOLOv2 is very effective and efficient to capture the position of welding spot compared with YOLOv2, YOLOv3 and Tiny-YOLOv2. Secondly, a novel right-triangle circle detection (RTCD) is developed to refine the shapes of the welding spots. The novel algorithm called M-YOLOv2-RTCD that combines M-YOLOv2 and RTCD is able to accurately and fast detect welding spots. The theoretical analysis and experimental results demonstrate that the proposed M-YOLOv2-RTCD outperforms the methods only using M-YOLOv2 or RTCD, in terms of accuracy and real-time performance.

Let F be a family of sets in Rd (always d≥2). A set M⊂Rd is called F-convex, if for any pair of distinct points x,y∈M, there is a set F∈F, such that x,y∈F and F⊂M. A thin right triangle is the boundary of a non-degenerate right triangle in R2. The aim of this paper is to introduce and begin investigating the thin right triangle convexity for short trt-convexity, which is obtained when F is the family of all thin right triangles. We investigate the trt-convexity of unbounded sets, convex surfaces and planar geometric graphs.

We study a family of substitution tilings with similar right triangles of two sizes which is obtained using the substitution rule introduced in Danzer and van Ophuysen (Res. Bull. Panjab Univ. Sci. 50(1–4), 137–175 (2000)). In that paper, it is proved this family of tilings can be obtained from a local rule using decorated tiles. That is, that this family is sofic. In the present paper, we provide an alternative proof of this fact. We use more decorated tiles than Danzer and van Ophuysen (22 in place of 10). However, our decoration of supertiles is more intuitive and our local rule is simpler.

Food resources are often critical regulating factors affecting individual fitness and population densities. In the Himalayan Mountains, Bharal “blue sheep” (Pseudois nayaur) are the main food resource for the endangered snow leopard (Panthera uncia), as well as being preyed upon by other predators. Blue sheep, however, may face a number of challenges including food resource competition with other wild and domestic ungulates, and hunting pressure. Here, we characterized the diet of blue sheep in the Annapurna Conservation Area (ACA) of Nepal and conducted proximate nutritional analysis on a limited number of plants identified as foods. Furthermore, we investigated the macronutrient and fiber balance of these plants using nutritional geometry which is a state‐space approach to modeling multidimensional and interactive nutritional aspects of foraging. A total of 19 plant species/genera were identified in blue sheep pellets using microhistological analysis. On average, across seasons and regions of the study area, the two most frequently occurring plants in pellets were graminoids: Kobressia sp. and Carex spp. The macronutrient balance of Kobresia sp. was relatively high in carbohydrate and low in protein, while other plants in the diet were generally higher in protein and lipid content. Analysis of fiber balance showed that the two most consumed plants of blue sheep (i.e., Kobresia spp. and Carex spp.) contained the highest concentration of hemicellulose, which is likely digestible by blue sheep. The hemicellulose and lignin balance of plants ranged relatively widely, yet their cellulose contents showed less variation. Foraging by blue sheep may therefore be a balance between consuming highly digestible high‐carbohydrate plants and plants less‐digestible but higher in protein and/or lipid. We characterized the diet of an important herbivore species, Bharal or blue sheep (Pseudois nayaur), in the remote Annapurna Conservation Area of Nepal, including proximate analysis of nutrient content and geometric analysis of macronutrient balance of a limited number of foods. A total of 19 plant species/genera were identified in blue sheep scats, with the most frequently occurring (Kobressia sp.) being high in carbohydrate and hemicellulose, and low in protein, relative to other plants. Our results suggest that foraging by blue sheep may be a balance between consuming highly‐digestible high‐carbohydrate plants and those less‐digestible but higher in protein and/or lipid.

Intuitionistic fuzzy set (IFS) is one of the most robust and trustworthy tools for portraying the imprecise information with the help of the membership degrees. Similarity measure, one of the information measures, plays an important role in treating imperfect and ambiguous information to reach the final decision by determining the degree of similarity between the pairs of the numbers. Motivated by these, this paper aims to present a novel distance/ similarity among the IFSs based on the transformation techniques with their characteristics. To explore the study, the given IFSs are transformed into the right-angled triangle over a unit square area, and hence based on the intersection of the triangles, novel distance and similarity measures are proposed. An algorithm to solve the decision-making problems with the proposed similarity measure is developed and implemented to execute their performance over the numerous examples such as pattern recognition and clustering analysis. The reliability of the developed measure is investigated by applying it in clustering and the pattern recognition problems and their results are compared with some prevailing studies. From the investigation, we conclude that several existing measures fail to give classification results under the different instances such as “division by zero problems” or “counter-intuitive cases” while the proposed measure successfully overcomes this drawback.