Triangular Numbers Research Articles

Assessing Creative Awareness and Literary Writing Games in the Norwegian Context

This paper reports on a survey and teaching intervention done with Norwegian students in their first (VG1), second (VG2), and third year (VG3) of high school English (16-19 years old). The goals of the project: 1. To examine Norwegian student/teacher attitudes toward creativity; 2. To design, play-test, and asses creative reading games. The study found students surprisingly open to creativity and games in education, however, there was a general wariness of assessment. As the students progressed through high school, they increasingly saw writing as something one is forced to do, and resisted sharing their writing. Almost all students saw themselves as outside the community of writers. While the novel nature of the games made for initial difficulties, this was also a motivator. The key to effectiveness was pairing theory with practice to give students a clear sense of the pedagogical goal: to develop and experience the use of creativity (divergent thinking). The use of multimodality in games (drawing) was immensely effective for student engagement and collaboration; however, it was less effective in terms of maintaining clear pedagogical goals in the time given. Finally, the most effective approach in terms of engagement was to involve students as game designers. This paper will focus on a presentation of student views on creativity followed by an assessment of the game results in light of these views.

A novel method for solving “Entrepreneurship and Entrepreneurship” career ability evaluation of local college students with triangular neutrosophic numbers

With the proposal of the “Double First Class” construction concept in Chinese universities, cultivating high-quality talents has become the main direction of university education. At present, local universities in China are actively adapting to the changes of the times, taking the construction of the Double First Class as an opportunity, continuously adjusting and deepening the reform of the double innovation and innovation talent cultivation mode, further improving the curriculum system, increasing investment in professional practical teaching, stimulating students’ enthusiasm for innovation and entrepreneurship (IAE), and focusing on improving students’ comprehensive abilities, achieving significant results in talent cultivation. The “Entrepreneurship and Entrepreneurship” career ability evaluation of local college students could be considered as multiple attribute decision-making (MADM). Recently, the Combined Compromise Solution (CoCoSo) method and information entropy method was employed to deal with MADM. The triangular fuzzy neutrosophic sets (TFNSs) are employed as a better tool for expressing uncertain information during the “Entrepreneurship and Entrepreneurship” career ability evaluation of local college students. In this paper, the triangular fuzzy neutrosophic number CoCoSo (TFNN-CoCoSo) based on the Hamming distance and Euclid distance is constructed to cope with the MADM under TFNSs. The information entropy method is employed to compute the weight values based on the Hamming distance and Euclid distance under TFNSs. Finally, a numerical example of “Entrepreneurship and Entrepreneurship” career ability evaluation of local college students is constructed and some decision comparisons are constructed to verify the TFNN-CoCoSo method.

ПАРАМЕТРИЧЕСКИЙ МЕТОД РАСЧЕТА ТРЕУГОЛЬНЫХ ВЕСОВЫХ КОЭФФИЦИЕНТОВ В МАТРИЦЕ ПОПАРНЫХ СРАВНЕНИЙ

The article deals with development of the method for calculation of triangular weight coefficients in fuzzy problem of decision making. The study objective is to improve the accuracy of triangular weight coefficient determination. In order to achieve the above-mentioned objective, a set of methods of scientific and empirical investigation: analysis and splitting of the problem into two tasks, methods of mathematical programming and fuzzy logic, method of comparison. The required weight coefficients are represented in a parametrical form with unlike fuzzy parameters which allowed for two tasks to be separated and formalized: determination of modal values and the task to determine fuzziness parameters. Both tasks have been formulated in terms of non-linear mathematical programming. The proposed method differs from the known ones by using the parameters of triangular numbers fuzziness as optimized variables which are assumed to be different for the left and right borders in all weight coefficients.
 The proposed approach allowed for the accuracy of weight coefficients to be improved. The accuracy validation was conducted comparing with the known LLSM method on two matrices. The calculations have demonstrated that the integral absolute error of the proposed method more than twice smaller than that of the know method for each of the matrices.

Superrosiness and dense pairs of geometric structures

Let T be a complete geometric theory and let TP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_P$$\\end{document} be the theory of dense pairs of models of T. We show that if T is superrosy with -rank 1 then TP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_P$$\\end{document} is superrosy with -rank at most ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document}.

A Fermatean fuzzy MCDM method for selection and ranking Problems: Case studies

The fuzzy set theory has been evolving to represent the uncertainty in the real-world decision-making environment. Literature has been steadily expanding to incorporate subjective judgments and ambiguous information in the decision-making process, aiming to enhance the reliability and flexibility of data representation. Fermatean Fuzzy Sets (FFSs), a recent extension of intuitionistic fuzzy sets, address the limitations associated with membership functions and the representation of hesitation in multi-criteria decision-making (MCDM) methods. The aim of this study is to examine the performance of FFSs in exploiting uncertainty in selection and ranking decisions in MCDM problems. The performance of FFSs is investigated for Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) which is extended with Stepwise Weight Assessment Ratio Analysis (SWARA) method for criteria evaluations. This study is designed to present a comparative analysis for three different types of fuzzy set definitions: classical fuzzy sets with fuzzy triangular numbers, Intuitionistic Fuzzy Sets (IFSs), and FFSs, for MCDM problems under uncertainty. The proposed methodology is applied to two real case studies: (i) to rank Turkish research universities for performance assessment and (ii) to select the optimal facility location for a company in the beverage industry. To assess the effectiveness of the FFSs in multiple criteria selection and ranking decisions, a comparative analysis is conducted on two real-world problems. The results show that FFSs provide valuable insight especially for multiple criteria ranking problems. The comparative analysis highlights the effectiveness of Fermatean Fuzzy SWARA-TOPSIS method and its potential for practical ranking applications.

Triangle-degree and triangle-distinct graphs

Let G be a simple graph and v be a vertex of G. The triangle-degree of v in G is the number of triangles that contain v. While every graph has at least two vertices with the same degree, there are graphs in which every vertex has a distinct triangle-degree. In this paper, we construct an infinite family of graphs with this property. We also study the vertex degrees and size of graphs with this property.

Path integrals formulations leading to propagator evaluation for coupled linear physics in large geometric models

Reformulating linear physics using second kind Fredholm equations is very standard practice. One of the straightforward consequences is that the resulting integrals can be expanded (when the Neumann expansion converges) and probabilized, leading to path statistics and Monte Carlo estimations. An essential feature of these algorithms is that they also allow to estimate propagators for all types of sources, including initial conditions. The resulting practice is a single Monte Carlo run, for one given set of sources, producing propagators that can later be used with any other set of sources for fast simulations, typically as parts of optimization, inversion, sensitivity analysis and command control algorithms. The present paper illustrates how this practice can be extended to problems involving several interacting physics, provided that their coupling is only at the boundary of the system or at interfaces between sub-parts, and may itself be given the form of a second kind Fredholm equation. A full practical implementation is described as part of the Stardis code, with the example of transferring heat via the coupling of radiation, reaction-diffusion and convection as typically expected in the multidisciplinary context of urban climate modeling. Besides, we show how recent advances in computer graphics indicate that these algorithms can be made numerically extremely efficient when facing large CAD geometries: computing the propagator becomes strictly independent of the geometry refinement, i.e. is identical whatever the number of triangles and tetraedra used to numerize the surface and volume descriptions. To the best of our knowledge this is the first report of propagator computations that remains practical for coupled physics in large CAD geometries. Program summaryProgram Title: Stardis 0.7.2 (built on stardis-solver 0.12.3)CPC Library link to program files:https://doi.org/10.17632/k76zrx4n6b.1Developer's repository link:https://www.meso-star.com/projects/stardis/stardis.htmlLicensing provisions: GPLv3Programming language: ANSI C and PythonNature of problem: Estimating temperatures in coupled heat transfer systems involving large CAD and/or large numbers of spatially distributed sources.Solution method: Stardis uses the Monte Carlo Method. Each temperature estimate constructs a propagator of each of the energy sources within the system: initial temperature, temperature boundary conditions, volume powers and surface fluxes. The propagator can be stored for further use outside the Monte Carlo code.Additional comments including restrictions and unusual features: The Stardis estimates of propagators are only reliable when radiative transfer can be linearized around a reference temperature. Stardis can deal with the nonlinearity of radiation when computing temperatures, but then nothing can be interpreted as a meaningfull propagator for any external usage.Python scripts were used to execute the different configurations, post-process the results and generate the different graphs.

An exponential Diophantine equation on triangular numbers

An exponential Diophantine equation on triangular numbers

An Exploration of Very Triangular Numbers

We present a collection of results concerning the location and distribution of very triangular numbers among triangular numbers, including the twin very triangular number theorem, the existence of arbitrarily long gaps between – and an analog of Bertrand's postulate for – very triangular numbers.

An improved quantum particle swarm algorithm for solving multi-objective fuzzy flexible job shop scheduling problem

Due to the dynamic nature of work conditions in the manufacturing plant, it is difficult to obtain accurate information on process processing time and energy consumption, affecting the implementation of scheduling solutions. The fuzzy flexible job shop scheduling problem with uncertain production parameters has not yet been well studied. In this paper, a scheduling optimization model with the objectives of maximum completion time, production cost and delivery satisfaction loss is developed using fuzzy triangular numbers to characterize the time parameters, and an improved quantum particle swarm algorithm is proposed to solve it. The innovations of this paper lie in designing a neighborhood search strategy based on machine code variation for deep search; using cross-maintaining the diversity of elite individuals, and combining it with a simulated annealing strategy for local search. Based on giving full play to the global search capability of the quantum particle swarm algorithm, the comprehensive search capability of the algorithm is enhanced by improving the average optimal position of particles. In addition, a gray target decision model is introduced to make the optimal decision on the scheduling scheme by comprehensively considering the fuzzy production cost. Finally, simulation experiments are conducted for test and engineering cases and compared with various advanced algorithms. The experimental results show that the proposed algorithm significantly outperforms the compared ones regarding convergence speed and precision in optimal-searching. The method provides a more reliable solution to the problem and has some application value.

Fatigue life analysis of a slightly curved hydraulic pipe based on Pairs theory

Fatigue life analysis of a slightly curved hydraulic pipe based on Pairs theory

An Efficient Technique for Algebraic System of Linear Equations Based on Neutrosophic Structured Element

Neutrosophic logic is frequently applied to the engineering technology, scientific administration, and financial matters, among other fields. In addition, neutrosophic linear systems can be used to illustrate various practical problems. Due to the complexity of neutrosophic operators, however, solving linear neutrosophic systems is challenging. This work proposes a new straightforward method for solving the neutrosophic system of linear equations based on the neutrosophic structured element (NSE). Here unknown and right-hand side vectors are considered as triangular neutrosophic numbers. Based on the NSE, analytical expressions of the solution to this equation and its degrees are also provided. Finally, several examples of the methodology are provided.

Auto Mesh generation algorithm for the convex domain with the triangular elements

Mesh creation is one of the primary tasks in implementing the Finite Element Method (FEM) to solve two- and three-dimensional boundary value problems. However, the whole procedure becomes tedious and problematic when higher-order finite elements are employed to construct mesh and prepare corresponding element data. In this study, we strive to develop a versatile algorithm for discretizing the two-dimensional domain using linear, quadratic, and cubic triangular finite elements. The algorithm is developed based on n vertices (actual or more in number) that constitute the boundary of the domain, along with a computed (n+1)-th point such as the centroid. The algorithm, using this input, generates an initial mesh consisting of n triangular elements. Then, in every subsequent step, the algorithm increases the number of triangles in the mesh fourfold compared to the previous step. The algorithm we present here can employ (a) straight-sided (linear, quadratic, and cubic) triangular elements to generate the mesh for domains with polygonal boundaries and (b) both straight-sided and curved (quadratic and cubic) triangular elements with two straight sides and one curved side to generate meshes for domains with curved boundaries. Thus, the algorithm generates the desired meshes if the vertices are specified once at the initial step. Additionally, the inclusion of the mathematical expression of the curved boundary enables the algorithm to generate the fine mesh for the curved domain utilizing higher-order curved and straight-sided triangular elements. We also present a computer code in MATLAB incorporating the algorithm to create the mesh, prepare the element's data, and determine the element's connectivity.
 GANIT J. Bangladesh Math. Soc. 43.1 (2023) 017- 035

SUM OF DIVISORS FUNCTION

We show that the recurrence relation deduced by Robbins and Osler et al for the sum of divisors function can be solved in terms of the complete Bell polynomials. Besides, the connection between and the number of representations of n as the sum of four triangular numbers allows obtain arecurrence relation where only participate the values of with m odd.

An extended ISM and MICMAC method under neutrosophic environment

PurposeISM technique is one of the tools of interest in soft operations research. The soft nature of this technique has made inevitable use of indeterminacy theories. The present research attempts to develop ISM technique and MICMAC analysis in a neutrosophic space due to the complexity and uncertainty of the decision-making environment.Design/methodology/approachIn this study, single-valued triangular neutrosophic numbers is used to develop Neutrosophic ISM (NISM) and Neutrosophic MICMAC (NMICMAC). First, the general algorithm of NISM and NMICMAC is provided. In the following, the complete description of NISM steps including level value determination, Factor Leveling Algorithm and NISM digraph algorithm are presented. Finally, a description of the NMICMAC steps is described.FindingsAn illustrative example – supplier selection problem – is given to verify the effectiveness of the proposed method and in the discussion section; the comparison and analysis of different aspects of the NISM with the previous methods are discussed.Originality/valueIn this study, NISM and NMICMAC are presented for the first time, so that each pairwise comparison judgment is provided as single valued triangular neutrosophic numbers. The development of the model is continued until the final stages of calculations with neutrosophic numbers, and only in the final stage the results are crispy presented. In addition, not only the factors of process are leveled, but at each level the factors are lined up and their importance is determined.

Multi-Objective Reliability Optimization using Fuzzy Nonlinear Programming with Interval Membership Functions and Bias Functions: A Comparison of Particle Swarm Optimization and Genetic Algorithm

Multi-objective reliability optimization is a complex problem that involves simultaneously optimizing multiple objectives while ensuring that the system meets certain reliability requirements. In this paper, we present a methodology for solving multi-objective reliability optimization problems using fuzzy nonlinear programming. The methodology involves representing the reliability of each component as a triangular interval number and each objective function as an interval membership function. Conflicts between objectives are resolved using linear and nonlinear membership functions, and exponential and quadratic membership functions are used to obtain definite biases towards the objective. The proposed methodology employs Particle Swarm Optimization (PSO) or Genetic Algorithm (GA) to solve the problem, and the approach is compared with GA for linear and nonlinear membership functions. The results indicate the effectiveness of the methodology in addressing multi-objective reliability optimization problems

A novel approach based on neutrosophic Bonferroni mean operator of trapezoidal and triangular neutrosophic interval environments in multi-attribute group decision making

Neutrosophic multicriteria is a method of decision-making that uses indeterminacy to combine several criteria or elements, frequently with incomplete or ambiguous information, to find a solution. The neutrosophic multicriteria analysis enables the assessment of qualitative and subjective aspects and can assist in resolving conflicting goals and preferences. In the Neutrosophic Multi-Attribute Group Decision Making (NMAGDM) problems, all the information provided by the decision makers (DMs) is expressed as single value neutrosophic triangular and trapezoidal numbers examined in this study which can provide more flexibility and accuracy in capturing uncertainty and aggregating preferences. We offer a novel approach for determining the neutrosophic possibility degree of two and three trapezoidal and triangular neutrosophic sets and the concepts of neutrosophic possibility mean value. The trapezoidal and triangular neutrosophic Bonferroni mean (TITRNBM) operator and the trapezoidal and triangular neutrosophic weighted Bonferroni mean (TITRNWBM) operator are two aggregation methods we then create. Further, we examine the TITRNBM and TITRNWBM attributes and their uniqueness. The NMAGDM approach with trapezoidal and triangular information is suggested based on the TITRNWBM operator and possibility degree. Finally, a concrete example of manufacturing companies searching for the best supplier for assembling the critical parts is provided to validate the established strategies and show their practical applicability and efficacy.

Complete Bell Polynomials and Recurrence Relations for Arithmetic Functions

We use the Z-transform to solve a type of recurrence relation satisfied by the number of representations of an integer n as the sum of squares and as the sum of triangular numbers, and also by the color partitions of n; the corresponding solution is in terms of the complete Bell polynomials.

The effects of gamification on engagement and response accuracy in discriminatory sensory testing

Gamification, a concept that has been increasingly leveraged by different sectors, aims to increase engagement and performance in regards to a given activity. Few studies have applied gamification principles to sensory methods to improve panelist engagement and test performance. The objective of the present study was to determine whether gamification has an effect on panelist engagement and performance in discriminatory sensory testing. Specifically, this study applied three individual game elements (progress bar, feedback, narrative), and their combination, to a series of triangle tests. For each gamified condition, subject engagement and discriminatory performance was compared to a control condition with no gaming elements. This research demonstrates that a combination of game elements can increase subject engagement during a discrimination sensory task. Despite this, the average number of correct triangles did not increase. The findings also show the relationship between panelist engagement and performance is nuanced, and task difficulty plays an important role. Overall, this study provides a baseline understanding of the methodical application of gamification in a discrimination sensory task.

Extremal numbers of disjoint triangles in r-partite graphs

For two graphs G and F, the extremal number of F in G, denoted by ex(G,F), is the maximum number of edges in a spanning subgraph of G not containing F as a subgraph. Determining ex(Kn,F) for a given graph F is a classical extremal problem in graph theory. In 1962, Erdős determined ex(Kn,kK3), which generalized Mantel's Theorem. On the other hand, in 1974, Bollobás, Erdős, and Straus determined ex(Kn1,n2,…,nr,Kt), which extended Turán's Theorem to complete multipartite graphs. In this paper, we determine ex(Kn1,n2,…,nr,kK3) for r≥4 and 10k−4≤n1+4k≤n2≤n3≤⋯≤nr.