The object of study is a transversely bent triangular plate made of orthotropic material fixed along the edges of the plate under the action of a uniformly distributed load. Fourth-order differential equilibrium equations with variable orthotropy parameters were used. The equations were approximated by finite differences for a grid of scalene triangles. This type of mesh accurately describes the boundary contour of the triangular-shaped plates. The boundary conditions for the mesh consider the orthotropy of the plate material. Seven standard finite-difference equations have been developed considering the boundary conditions along the three edges of the plate and the presence of three angles of an irregularly shaped triangle. A finite difference matrix is obtained. The matrix structure makes calculating a triangular plate at different base angles possible. The boundary conditions were varied in the form of rigid or hinged support of triangular plates. The calculation method considers the orthotropic parameters of the material in two mutually perpendicular planes. The adaptation of the numerical method to the calculation of orthotropic plates of arbitrary shape is described. Relationships are given for determining the rigidity characteristics of orthotropic materials. An algorithm for simple engineering calculation of triangular orthotropic plates is proposed, which makes it possible to perform accurate calculations in variant design.

The surgical extraction of the impacted third molar is frequently associated with several complications. The purpose of this study is to assess how two different surgical protocols affect post-operative complications during the extraction of the lower third molars. In order to compare and evaluate two different techniques (triangular flap vs envelope flap), and the relative post-extraction complications, two groups of 150 patients each, underwent to surgical third molar extraction and 60 days of follow-up. A complication rate founded in two groups was respectively 14,00% in group A and 17,33% in group B. There was a strong association between smoking (OR:2.8) and the use of oral contraceptives (OR:1.75) with complications. The age and sex-related incidence of complications in hard tissue healing finds great variability in the literature; the analysis performed on our data did not show a statistically significant association between them. Even though related to a higher incidence of transient changes in sensitivity, it was found that the envelope flap saw a lower percentage of complications. There is still no clarity on which is the best protocol for the extraction of the lower third molar, and the choice often depends on the surgeon’s experience.

A new generation of universal tools and languages for modeling and simulation multi-physical domain applications emerged and became widely accepted, which generate large-scale systems of differential algebraic equations (DAEs) automatically. Motivated by the characteristics of DAEs systems with large dimension, high index or block structures, we first propose a modified Pantelides’ algorithm (MPA) for any high order DAEs based on its Σ matrix, which is similar to Pryce’s Σ method. By introducing a vital parameter vector, a modified Pantelides’ algorithm with parameter has been presented.It leads to a block Pantelides’ algorithm (BPA) naturally which can immediately compute the crucial canonical offsets for whole (coupled) systems with block-triangular form. We illustrate these algorithms by some examples. And numerical experiments show that the time complexity of BPA can be reduced by at least O(ℓ) compared to the MPA, which is mainly consistent with the results of our analysis.

The main ideas of F-transform came from representing expert rules. It would be therefore re reasonable to expect that the more accurately the membership functions describe human reasoning, the more efficient will be the corresponding F-transform formulas. We know that an adequate description of our reasoning corresponds to complicated membership functions -- however, somewhat surprisingly, most efficient applications of F-transform use the simplest possible triangular membership functions. There exist some explanations for this phenomenon which are based on local behavior of the signal. In this paper, we supplement this local explanation by a global one: namely, we prove that triangular membership functions are the only one that provide the accurate description of appropriate global characteristics of the signal.

Intuitionistic fuzzy set, which can be represented using the triangular intuitionistic fuzzy number (TIFN), is a more generalized platform for expressing imprecise, incomplete and inconsistent information when solving multi-criteria decision-making problems, as well as for reflecting the evaluation information exactly in different dimensions. In this paper, the TIFN has been applied for solving some multi-criteria decision-making problems by developing a new triangular intuitionistic fuzzy geometric aggregation operator, that is the generalized triangular intuitionistic fuzzy ordered weighted geometric averaging (GTIFOWGA) operator, and defining some triangular intuitionistic fuzzy geometric aggregation operators including the triangular intuitionistic fuzzy weighted geometric averaging (TIFWGA) operator, the ordered weighted geometric averaging (TIFOWGA) operator and the hybrid geometric averaging (TIFHWGA) operator. Based on these operators, a new approach for solving multicriteria decision-making problems when the weight information is fixed has been proposed. Finally, the proposed method has been compared with some similar existing computational approaches by virtue of a numerical example to verify its feasibility and rationality.

In India, the divorce rate is rising year after year. Divorces are thought to have increased by more than a factor of two during the past two decades. India has great cultural diversity, and each of its states has something unique to offer. The divorce rate in India, which differs from state to state, is among the most frequently discussed subjects. More than 30% of marriages end in divorce in big cities like Delhi, Mumbai, Chennai, and Bangalore. Divorce applications have nearly tripled in the past few years in cities like Delhi, Bangalore, Mumbai, Kolkata, and Lucknow. In India, the divorce rate has increased, especially in small cities. Hence, the reasons behind divorce in India were explored in this article. In order to analyze the causes of divorce, the current study aimed to combine the critical components of TOPSIS and COPRAS methodologies in a fuzzy environment. Additionally, it is shown that a comparison study may pinpoint the characteristics that influence divorces the most.

Under the same seismic load, different local topography will have different effects, which is an important research field in seismic engineering. Hills are connected with valleys is a very typical terrain. Taking this terrain as an example, its response to the most destructive SH waves in the earthquake is studied. By simplifying and dividing the model, the wave functions satisfying the wave equation and boundary conditions in each region are constructed by using the complex function and moving coordinate method. The equations satisfying the boundary continuity conditions are established. With the help of calculation software, the numerical results are obtained. The influence of different parameters on ground motion is discussed.

Background: Neonatal cholestasis (NC) initiates in the first trimester of a newborn, comprising extra and intrahepatic medical conditions, with a high risk of fatality warranting early diagnosis and treatment to prevent morbimortality. The differential diagnosis of NC is a challenge demanding an accurate diagnosis for disease detection. The current study evaluates NC's clinical symptoms and differential diagnosis using ultrasound, liver biopsy, histopathology, and biochemistry. Methods: Infant registry data from Hevi Pediatric Teaching Hospital (January 2016 - January 2022) were obtained and screened for subject selection. The inclusion criteria include infants with direct hyperbilirubinemia within an onset of 1 to 90 days of birth. As indicated, the recruited subjects underwent liver ultrasound, blood biochemistry, and liver biopsy. Results: Seventy-two children presented with the criteria for inclusion in the study. The study found that ultrasound helped diagnose 43.1% of subjects for BA compared to 34.7% through histopathology. The histopathology confirmed 13 children (18.1%) having neonatal hepatitis. Test sensitivity of the ultrasound method for BA and NH was 60% (40.74, 76.6) and 38.46% (17.71, 64.48), respectively. Conclusion: The study found both ultrasound and liver biopsy to be critical diagnostic methods to differentiate the etiology of NC. Ultrasound has a higher specificity and sensitivity for diagnosing BA than NH. Histopathology and blood biochemistry should be considered, too, for effective diagnosis. In the future, larger sample and multicenter studies should be conducted to develop practically implementable strategies.

This paper proposed a triangular inequality-based rewiring method for the Rapidly exploring Random Tree (RRT)-Connect robot path-planning algorithm that guarantees the planning time compared to the RRT algorithm, to bring it closer to the optimum. To check the proposed algorithm’s performance, this paper compared the RRT and RRT-Connect algorithms in various environments through simulation. From these experimental results, the proposed algorithm shows both quicker planning time and shorter path length than the RRT algorithm and shorter path length than the RRT-Connect algorithm with a similar number of samples and planning time.

Level index was introduced in 2017 for rooted trees which is a component of Gini index. In the origin, Gini index is a tool for economical investigations but Balaji and Mahmoud defined the graph theoretical applications of this index for statistical analysis of graphs. Level index is an important component of Gini index. In this paper we define a new graph polynomial which is called level polynomial and calculate the level polynomial of some classes of trees. We obtain some interesting relations between the level polynomials and some integer sequences.

To solve the problem that sampling-based Rapidly-exploring Random Tree (RRT) method is difficult to guarantee optimality. This paper proposed the Post Triangular Processing of Midpoint Interpolation method minimized the planning time and shorter path length of the sampling-based algorithm. The proposed Post Triangular Processing of Midpoint Interpolation method makes a closer to the optimal path and somewhat solves the sharp path problem through the interpolation process. The experiments were conducted to verify the performance of the proposed method. Applying the method proposed in this paper to the RRT algorithm increases the efficiency of optimization compared to the planning time.

This paper studied the natural convection heat transfer from two equal-sized cylinders of triangular cross-section confined in triangular enclosure. The inner cylinders have hot surfaces and the outer enclosure has a cold surfaces. The annular space between inner cylinders and the outer one was filled with incompressible Newtonian fluid. The work examined the behavior of fluid flow in the process of transferring heat energy. The work was achieved numerically using the commercial code ANSYS-CFX. The adopted initial conditions were selected for these ranges: Prantl number (Pr = 1 to 10³), Rayleigh number (Ra = 10³ to 10⁵). It was found that the increment of Ra number increases the rate of heat transfer. However, the effect of Pr number on heat transfer is almost negligible.

High-precision ground point cloud data has a wide range of applications in various fields, and the separation of ground points from non-ground points is a crucial preprocessing step. Therefore, designing an efficient, accurate, and stable ground extraction algorithm is of great significance for improving the processing efficiency and analysis accuracy of point cloud data. The study area in this article was a Park in Guilin, Guangxi, China. The point cloud was obtained by utilizing the UAV platform. In order to improve the stability and accuracy of the Filter algorithm, this article proposed a triangular grid filter based on Slope Filter, found violation points by the spatial position relationship within each point in the triangulation network, improved KD-Tree-Based Euclidean Clustering, and applied it to the non-ground points extraction, this method has good accuracy, stability and achieves good results in separating ground points from non-ground points. At first, using Slope Filter to remove some non-ground points, to reduce the error of taking ground points as non-ground points; Secondly, established a triangular grid based on the triangular relationship between each point, the violation-triangle can determin through the grid, and then the corresponding violation points were found in the violation-triangle; Thirdly, according to the three-point collinear method to extract the regular points, used these points to extract the regular landmarks by KD-Tree-Based Euclidean Clustering and Convex Hull Algorithm; Finally, removed disperse points and irregular landmarks by Clustering Algorithm. In order to confirm the superiority of this algorithm, this article compared the filter effects of various algorithms on the study area and filtered the 15 sample data provided by ISPRS, and obtained an average error of 3. 46%. The results showed that the algorithm in this article have a high processing efficiency and accuracy, which can greatly improve the processing efficiency of point cloud data in practical applications.

We provide representations for the generalized Drazin inverse of an anti-triangular matrix of the form $\left( \begin{array}{cc} a&b\\ 1&0 \end{array} \right)$ in a Banach algebra $\mathcal{A}$, under the condition that $ab=ba$. Specifically, we present the representation of Drazin inverse for these types of anti-triangular matrices in Banach algebras.

We study various properties of the polygonal numbers; such as their recurrence relations; fundamental identities; weighted binomial and ordinary sums; partial sums and generating functions of their powers; and a continued fraction representation for them. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers; unlike what obtains in most literature.

Background: Post-operative pain and swelling are frequently observed after sinus lift procedures. The aim of the present study was a clinical evaluation of swelling and pain of two sinus lift different flaps technique measured through visual analogue scale (VAS), verbal rating scale (VRS) and infrared thermal imaging (i.e., thermography). Methods: A randomized controlled trial was conducted in fifteen patients treated for a total of 30 sinus randomly allotted into two groups. In patients of Group I, a triangular flap was performed, while in Group II modified triangular flap without anterior release was performed. Postoperative pain was scored by VAS and inflammation was recorded by VRS at 2, 4, 6 and 14 days. The facial temperature was recorded before and after sinus augmentation, at 2, 4, 6 day intervals for the first week and at 14 day, to check the course of healing. Results: In group I the pain intensity was recorded at 2 days after the surgery with a score of 38.67 ± 6.4 mm. Swelling was higher at 2 and 4 day, and it was absent at day 6. Facial temperature difference before and after the procedure was 3 (4.737 °C ± 0.37). In Group II the pain was lower than in Group I (p < 0,05). Swelling scored 2 on the first and second day, and it was reduced at day 4. After the second day the difference of temperature reduced significantly compared to day of surgery (0.77 °C), at 2 and 4 days no difference was registered. Conclusion: In conclusion based on the outcome of the present study, modified triangular flap can be used with success in sinus augmentation procedures and is to be preferred to trapezoidal flap.

With the continuous improvement of people's quality of life and aesthetics, furniture products have higher requirements. Excellent solid wood chairs are the most representative products in furniture to enhance space and taste and whose design may significantly impact consumers' emo-tional experience and purchase decisions. This study aims to evaluate how the modeling imagery of solid wood chairs affects user preferences and emotional experiences. The development of the current era is inseparable from the analysis of big data. Firstly, a representative sample was ob-tained by multidimensional scaling (MDS), and the sample was analyzed and evaluated by fac-tor analysis. Moreover, five groups of adjective vocabulary were selected to describe the model-ing imagery of solid wood chairs, such as "balanced and coordinated", "unique and novel", "prac-tical and simple", "quality and detailed" and "traditional and plain". Further, the triangular fuzzy theory was applied to analyze and discuss the 12 types of solid wood chairs in the five groups of adjective vocabulary. Then the study verified that the differences in the evaluations of the 12 sam-ples in the "unique and novel" and "quality and detailed" groups were significant, but more minor differences in the groups of "traditional and plain", "balanced and coordinated" and "practical and simple". Through comprehensive comparison, five groups with similar modeling imagery were created, and solid wood chairs with different modeling imagery should be placed in suita-ble spaces. According to the results of this study, the evaluation of the modeling imagery of solid wood chairs can not only rely on subjective judgments but can be reasonably described through data analysis and mathematical algorithms. It can scientifically and effectively reflect the poten-tial perception needs of consumers on the modeling imagery of solid wood chairs, as well as help to improve the design efficiency of the furniture product development stage.

This work presents a linear smoothing scheme over high-order triangular elements in the framework of a cell-based strain smoothed finite element method for two-dimensional nonlinear problems. The main idea behind the proposed linear smoothing scheme for strain-smoothed finite element method (S-FEM) is no subdivision of finite element cells to sub-cells while the classical S-FEM needs sub-cells. Since the linear smoothing function is employed, S-FEM is able to use quadratic triangular or quadrilateral elements. The modified smoothed matrix obtained node-wise is evaluated. In the same manner with the computation of the strain-displacement matrix, the smoothed stiffness matrix and deformation graident are obtained over smoothing domains. A series of benchmark tests are investigated to demonstrate validity and stability of the proposed scheme. The validity and accuracy are confirmed by comparing the obtained numerical results with the standard FEM using 2nd-order triangular element and the exact solutions.

Abstract: New paradigms have emerged in mapping since the 20th century due to theoretical and practical advances in computer technology, communications, and cartography. For the design of the Expert-Topographic Map Service in Mexico, we adopted a theoretical framework of Cybercartography and expanded to incorporate expert systems and eventually other artificial intelligence resources. Using a transdisciplinary approach, our efforts could focus on the interconnection between knowledge and geotechnology to offer solutions to societal complex problems. This paper explores the theoretical pillars supporting the design of an Expert-Topographic Map Service, including cybercartography, knowledge-based systems and surface mathematical modeling. For the first time, cybercartography is applied in the realm of topographic mapping. As shown, a map conceived solely as an information system has been transformed, in the digital era, into a container of knowledge through the abstract representation of features like relief via the Triangular Irregular Network Model. Thereby, we can bridge from digital topographic maps to other knowledge-based models such as expert systems. The societal demand and empirical context for this research demonstrate that topographic maps and services are still at the core of cartographic initiatives in the 21st century.

Management practices must be developed to improve yam production sustainability. Image-based phenotyping techniques could help developing such practices based on non-destructive analyses of important plant traits. Our objective was to determine the potential of image-based phenotyping methods to assess traits relevant for tuber yield formation in yam grown in glasshouse and field. We took plant and leaf pictures with consumer cameras. We used the numbers of image pixels to derive the shoot biomass and the total leaf surface and calculated the ‘triangular greenness index’ (TGI) which is an indicator of the plant nitrogen (N) nutritional status. Under glasshouse conditions, the number of pixels obtained from nadir view (image taken top down) was positively correlated to the shoot biomass, and the total leaf surface, while the TGI was negatively correlated to the N content of diagnostic leaves. Under field conditions, pictures taken from the nadir view showed an increase in soil surface cover and a decrease in TGI with time. TGI was negatively correlated to SPAD measured on specific leaves but was not correlated to the N content of these leaves. In conclusion, these phenotyping techniques deliver relevant results but need to be further developed and validated for application in yam.

This research work is an extension of a previous work in which the authors created an innovative family of fuzzy implication using as t-conorm probor, as fuzzy negation the 1-x and the type of Newton's binomial. The purpose of this paper is to compare the type of fuzzy implication probor with three other constructed fuzzy implications produced by max, Einstein and Lukasiewicz t-conorms maintaining in each case the same negation. T-conorms perform the fuzzy logical operation "or" of the corresponding fuzzy sets. The main methods used for the comparisons are the basic axioms of fuzzy logic. The authors performed six combinations of t-conorm comparisons, in order to find the rank order of the five fuzzy implications. Although the general type of fuzzy implication probor came out smaller than Hamacher, Einstein and Lukasiewicz and the type of Einstein smaller than Lukasiewicz authors make extensive tests using four different membership functions and different values to the parameter m so as to examine which out of the four fuzzy implications receive the best results. The application of isosceles trapezium to the fuzzy implications probor and Einstein give the best values (most values greater than or equal to 0.9 and equal to 1).

We investigate the existence and representation of the generalized core-EP inverse of some triangular matrices over a Banach algebra. Further, the general representations of the generalized core-EP inverse of a triangular matrix over a $C^*$-algebra are presented. As applications, the generalized core-EP inverses of some block operator matrices over Hilbert spaces are given.

We investigate emergent conductive phenomena triggered by collinear antiferromagnetic orderings. We show that an up-down-zero spin configuration in a triangle cluster leads to linear and nonlinear spin conductivities even without the relativistic spin–orbit coupling; the linear spin conductivity is characterized by the Drude-type one, while the nonlinear spin conductivity is characterized by the Hall-type one. We demonstrate the emergence of both spin conductivities in the breathing kagome system consisting of a triangle cluster. The nonlinear spin conductivity becomes larger than the linear one when the Fermi level lies near the region where a small partial band gap opens. Our results indicate that collinear antiferromagnets with the triangle geometry give rise to rich spin conductive phenomena.

A triangular lattice model for pattern formation by core-shell particles at fluid interfaces is introduced and studied for the particle to core diameter ratio equal to 3. Repulsion for overlapping shells and attraction at larger distances due to capillary forces are assumed. Ground states and thermodynamic properties are determined analytically and by Monte Carlo simulations for soft outer- and stiffer inner shells, with different decay rates of the interparticle repulsion. We find that thermodynamic properties are qualitatively the same for slow and for fast decay of the repulsive potential, but the ordered phases are stable for temperature range depending strongly on the shape of the repulsive potential. More importantly, there are two types of patterns formed for fixed chemical potential - one for a slow and another one for a fast decay of the repulsion at small distances. In the first case two different patterns - for example clusters or stripes - occur with the same probability for some range of the chemical potential. For fixed concentration, an interface is formed between two ordered phases with the closest concentration, and the surface tension takes the same value for all stable interfaces. In the case of degeneracy, a stable interface cannot be formed for one out of four combinations of the coexisting phases, because of a larger surface tension. Our results show that by tuning the architecture of a thick polymeric shell, many different patterns can be obtained for sufficiently low temperature.

In this research paper an algorithm for the derivation of methods based on theorems and axioms of fuzzy logic is presented, analyzed and applied. This new proposed procedure generates fuzzy methods and evaluates the value of fuzzy propositions through fuzzy implication. The new implications of the family should satisfy a number of axioms. Also, we denote the conditions in order to satisfy the maximum number of axioms. Moreover, authors state and prove theorems focusing on fuzzy implication, observing the rule: the fuzzy function of fuzzy implication is strong and that lead to fuzzy negation. In addition, the optimal number of repetitions is calculated according to the expected truth value. The formulas produced are verified through the use of temperature and humidity values over a certain period of time and in a certain geographical area in Greece. The basic steps of the methodology are the fuzzification of the data given ​​using four membership degree functions and the application of the membership degrees of temperature and humidity values on a new type of fuzzy implication. Finally, we observed that the isosceles trapezium gave the best results and the type of fuzzy implication can be effectively applied.

Triangles are an ever present feature in nature, which the building construction industry duly echoes. However, an exact expression intended to offer the radiant field due to any triangle in an upright or inclined position had not been identified by previous research. In this case, the author has been able to solve by direct integration, the canonical expression of radiative transfer. This result alone confers a myriad of possibilities that had been inconceivable before, for studying in detail the three-dimensional heat transfer behavior of volumes and figures in which triangles do manifest, such as fins, windows, roof-gables and louvers of various kinds. Typically, shading devices, when tilted, give rise in their extremes to rhomboidal shapes which were difficult to take into account or had to be subject to discretization and subsequent Monte Carlo methods in order to perform an approximate calculation of their emission. This implied a lengthy and inexact procedure that induced many errors and consumed computing time. We can now avoid all the former downsides due to the advances hereby presented. As the novel expression can be converted into an algorithm it will be advantageously employed for simulation. This significant finding dovetails in the intricate puzzle of radiated heat and we believe that their consequences will greatly affect the conception and design of HVAC devices, aircraft manufacturing and specifically building or lighting industries among others.

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.

In the paper, by the Faά di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients of two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers and discover inverses of fifteen closely related lower triangular integer matrices.