Triangular Numbers Research Articles

A new graph labeling with Tribonacci, Fibonacci and Triangular numbers

In this paper, a new graph labeling technique using three sequences of numbers, namely Tribonacci, Fibonacci and Triangular is introduced. They are named as Tribo-Fibo-Triangular labeling and denoted as TFΔ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document} labeling. Assignment of the labeling is done based on the families of generalized Petersen graphs GP (n,2), GP(n,3) and a generalization of these graph are also presented. Applications related to recruitment of employees in a health care sector and the selection of teams in a school ambience are arrived by using of these three sequences of graphs along with computational intelligence. A board game is developed and presented using these three sequences.

An Inventory Model for Deteriorating Items with Stock-Dependent Demand under Fuzzy Lead Time and Partial Backlogging

For inventory management purposes, lead time refers to the amount of time it takes for a purchase order to be completed. Its effect is an important phenomenon in inventory management system. It also plays a significant role when lead time unknown to the decision makers. This paper deals with an inventory model for deteriorating items under fuzzy lead time. The stock dependent demand with partially backlogged shortages are considered in the proposed model. The total inventory costs for both crisp model and fuzzy model are derived. The fuzzy lead time is assumed to be triangular and trapezoidal numbers. The signed distance methods (SD) is used for defuzzification purpose. The developed model is validated with the help of numerical illustration under both crisp and fuzzy scenario. A pictorial presentation is furnished to explain the behaviour of the total inventory costs towards lead-crisp, lead-triangular and lead-trapezoidal values. Lastly a sensitivity analysis is performed to judge the sensitive behaviour of the total cost towards changes of the cost parameters.

Nearly Ideal Random Tiling with Dodecagonal Symmetry from Pentablock Quarterpolymers of the AB1CB2D Type.

Two-dimensional tiling manners as cross-sectional views of cylindrical domain assembly formed by pentablock quarterpolymers of the AB1CB2D type in bulk were investigated. Several binary and ternary blends from three mother polymers having different ϕB1/ϕB2 ratios (ϕB1 and ϕB2 are the volume fractions of the B1 and B2 blocks, respectively) represent nonperiodic but ordered triangle/square tilings, where the N3/N4 ratios (N3 and N4 are the numbers of triangles and squares in the observed area, respectively) are all close enough to the theoretical value of 4/√3 ≑ 2.31 for the dodecagonal quasicrystalline (DDQC) state, irrespective of the total number of polygons. The TEM images, having almost the same N3/N4 ratios, were proved to show 4- and 6-fold symmetries in terms of the angular appearance of equilateral polygon sides via image analyses. Among them, a ternary blend showed a nearly ideal random tiling pattern that is almost equivalent to the theoretically predicted tiling by SCFT. Moreover, the magnitude of phason strain estimated for a TEM image from the ternary blend was proved to be quite small when the observing area is narrow, while it deviates from the ideal quasicrystalline tiling with an increasing number of vertices in the observing area.

Corporate network formation in Kuwait: board interlocks network analysis using exponential random graph models

PurposeInterlocking directorates are a common phenomenon across several markets around the world. Yet, the institutional environment and the role of corporate elites in forming board networks promote some developing markets as a unique setting to understand the corporate boards network structure. This study aims to first explore the board directors’ network of all publicly listed companies in Kuwait. This paper then evaluates the effects of exogenous factors and endogenous network structural processes on the likelihood of board interlock.Design/methodology/approachThis study analyzes the interlocks network structures of 167 listed companies in 13 different market sectors in Kuwait relying on hand-collected directors’ data and using four measures of network centrality: betweenness, degree, closeness and eigenvector. The authors predict board interlocks using exponential random graph models (ERGM) and firm-level information from the Bloomberg database.FindingsThis study observes that both the firms and directors’ networks consist of 55 components, with the largest component containing about half of the total number of firms/directors. The firm’s network consists of one giant component of 85 firms, including all but one bank. This study shows the importance of endogenous network variables, such as the number of edges, centralization and triangles on the estimation of the factors that promote the board interlocks. Highly centralized firms are less likely to interlock with other firms, while two firms that are interlocked with a common third firm are more likely to interlock.Originality/valueThis paper is the first to analyze in-depth the structure of the directors network of companies in Kuwait. This study illustrates the complex map of interfirm and directors social networks in Kuwait. To the best of the authors’ knowledge, this study is among the first to exploit ERGM in the context of board interlocks to account for potential cross-dependencies and emergent network structures. Managers can identify the director interlock with other firms in the network and take advantage of the connection as a source of external knowledge and influence.

Sum of Two Triangular Numbers

Sum of Two Triangular Numbers

Advances in the study of nuclear collective rotation using pairing theory in the framework of the shell model

Advances in the study of nuclear collective rotation using pairing theory in the framework of the shell model

Balancing performance and comfort in virtual reality: A study of FPS, latency, and batch values

AbstractThis manuscript investigates the relationships among various performance metrics in a virtual reality (VR), namely frames per second (FPS), latency, batches, and the number of triangles (tris) and vertices (verts). The study aims to uncover correlations and directional associations between these metrics, shedding light on their impact on VR performance. The findings reveal a significant correlation between FPS and latency, albeit in opposite directions. Higher FPS values are associated with reduced latency, indicating that a smoother visual experience is accompanied by shorter delays in the VR. Conversely, lower FPS values are linked to increased latency, suggesting a potential degradation in overall system responsiveness. Additionally, a strong correlation is observed between latency and batches processed. This finding implies that latency has a direct impact on the system's ability to efficiently process and render objects within VR. Furthermore, a positive correlation is identified between the number of batches and the values of tris and verts. This relationship suggests that higher batch counts are associated with larger quantities of triangles and vertices, reflecting a more complex scene rendering process. Consequently, the performance of VR may be influenced by the density and intricacy of the virtual environments, as indicated by these metrics.

Faster Counting and Sampling Algorithms using Colorful Decision Oracle

In this work, we consider d - Hyperedge Estimation and d - Hyperedge Sample problems that deal with estimation and uniform sampling of hyperedges in a hypergraph ℋ( U (ℋ), ℱ(ℋ) in the query complexity framework, where U (ℋ) denotes the set of vertices and ℱ(ℋ) denotes the set of hyperedges. The oracle access to the hypergraph is called Colorful Independence Oracle ( CID ), which takes d (non-empty) pairwise disjoint subsets of vertices A 1 ,..., A d ⊆ U (ℋ) as input and answers whether there exists a hyperedge in ℋ having exactly one vertex in each A i for all i ∈ {1, 2, ..., d }. Apart from the fact that d - Hyperedge Estimation and d - Hyperedge Sample problems with CID oracle access seem to be nice combinatorial problems, Dell et al. [SODA’20 & SICOMP’22] established that decision vs. counting complexities of a number of combinatorial optimization problems can be abstracted out as d - Hyperedge Estimation problem with a CID oracle access. The main technical contribution of this article is an algorithm that estimates m = |ℱ(ℋ)| with \(\widehat{m}\) such that \(\begin{equation*} \frac{1}{C_{d}\log ^{d-1} n} \;\le \; \frac{\widehat{m}}{m} \;\le \; C_{d} \log ^{d-1} n \end{equation*}\) by using at most C d log d +2 n CID queries, where n denotes the number of vertices in the hypergraph ℋ and C d is a constant that depends only on d . Our result, when coupled with the framework proposed by Dell et al. (SODA’20 & SICOMP’22), leads to implies improved bounds for (1 ± ε)-approximation (where ε ∈ (0,1)) for the following fundamental problems: Edge Estimation using the Bipartite Independent Set ( BIS ) query. We improve the bound obtained by Beame et al. (ITCS’18 & TALG’20). Triangle Estimation using the Tripartite Independent Set ( TIS ) query. Currently, Dell et al.’s result gives the best bound for the case of triangle estimation in general graphs (SODA’20 & SICOMP’22). The previous best bound for the case of graphs with low co-degree (co-degree of a graph is the maximum number of triangles incident over any edge of the graph) was due to Bhattacharya et al. (ISAAC’19 & TOCS’21). We improve both of these bounds. Hyperedge Estimation & Sampling using Colorful Independence Oracle ( CID ). We give an improvement over the bounds obtained by Dell et al. (SODA’20 & SICOMP’22).

Asymmetric 1,4-Addition of Diarylphosphine Oxides to α, β-Unsaturated 2-Acyl Imidazoles.

Here we introduce a metal-free, catalytic and enantioselective strategy from α,β-unsaturated 2-acyl imidazoles to the chiral phosphorous 2-acyl imidazoles. Interestingly, this methodology was catalyzed by the classical and commercial oxazaborolidine under mild conditions. This strategy features a wide range of substrates scope with good yields and excellent enantioselectivities. The possible mechanism further suggests the key of this reaction through the cleavage of diarylphosphine oxides using Frustrated Lewis Pairs theory.

No selection lemma for empty triangles

Let P be a set of n points in general position in the plane. The Second Selection Lemma states that for any family of Θ(n3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta(n^3)$$\\end{document} triangles spanned by P, there exists a point of the plane that lies in a constant fraction of them.For families of Θ(n3-α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta(n^{3-\\alpha})$$\\end{document} triangles, with 0≤α≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0\\le \\alpha \\le 1$$\\end{document}, there might not be a point in more than Θ(n3-2α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta(n^{3-2\\alpha})$$\\end{document} of those triangles.An empty triangle of P is a triangle spanned by Pnot containing any point of P in its interior. Bárány conjectured that there exists an edgespanned by P that is incident to a super-constant number of empty triangles of P. The number of empty trianglesof P might be as low as Θ(n2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta(n^2)$$\\end{document}; in such a case, on average, every edge spanned by P is incident to a constant numberof empty triangles. The conjecture of Bárány suggests that for the class of empty triangles the above upper boundmight not hold. In this paper we show that, somewhat surprisingly,the above upper bound does in fact hold for empty triangles. Specifically, we show that for any integer n and real number 0≤α≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0\\leq \\alpha \\leq 1$$\\end{document} there exists a point set of size n with Θ(n3-α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta(n^{3-\\alpha})$$\\end{document} empty triangles such that any point of the plane is only in O(n3-2α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n^{3-2\\alpha})$$\\end{document} empty triangles.

Neutrosophic linear fractional optimization problem using fuzzy goal programming approach

This research paper develops an algorithm to solve a neutrosophic linear fractional optimization problem, where the cost of the objective functions, the technology coefficients, and the resources are neutrosophic numbers (triangular and trapezoidal numbers). Two defuzzification methods are used here to convert the neutrosophic linear fractional optimization problem into a multi-objective linear fractional optimization problem (MOLFOP). The fractional objective function is defuzzified using the component-wise optimization method, and the linear constraints are defuzzified using the aggregation ranking function strategy. Hence, the MOLFOP is then solved using the fuzzy goal programming method. In order to further explain our suggested algorithm, two numerical examples are solved at the end, and the outcomes obtained by our method are compared with the outcomes obtained using the other algorithm.

Valuation of Internet of Energy (IoE) Platforms in Smart Cities: A Hybrid Multi-Criteria Decision Making Approach

A smart city is a place where people live using computing, sensing, and networking technology in a variety of ways, including smart buildings, smart homes, smart transportation, smart workplaces, smart offices, smart industries, and smart agriculture. These smart technologies need to connect so; we need to use the Internet of Things (IoT) that has applications in many areas of our smart lives. IoT has been used due to the growing usage of renewable energy sources and the challenges of network management. Internet of Energy (IoE) integrates the IoT and smart grid capabilities. The basic role of smart grids or IoE is an essential element in creating smart cities. In this work, we present some IoE platforms that represent the alternatives and their criteria. The proposed study treats the uncertainty of multi-criteria decision-making (MCDM) modified by single value triangular neutrosophic number (SVTrN) by choosing the optimal IoE platform based on the criteria importance through inter-criteria correlation (CRITIC) method and multi-attributive ideal-real comparative analysis (MAIRCA). The proposed model proved its effectiveness by applying it to an experimental case. The results show that the DCC (A3) platform is the best one.

Alternating size field optimizing and parameterization domain CAD model remeshing

Alternating size field optimizing and parameterization domain CAD model remeshing

Omega Invariant of Complement Graphs and Nordhaus-Gaddum Type Results.

The study aimed to obtain relationships between the omega invariants of a graph and its complement. We used some graph parameters, including the cyclomatic numbers, number of components, maximum number of components, order, and size of both graphs G and G. Also, we used triangular numbers to obtain our results related to the cyclomatic numbers and omega invariants of G and G. Several bounds for the above graph parameters have been obtained by the direct application of the omega invariant. We used combinatorial and graph theoretical methods to study formulae, relations, and bounds on the omega invariant, the number of faces, and the number of compo-nents of all realizations of a given degree sequence. Especially so-called Nordhaus-Gaddum type resulted in our calculations. In these calculations, the triangular numbers less than a given number play an important role. Quadratic equations and inequalities are intensively used. Several relations between the size and order of the graph have been utilized in this study. In this paper, we have obtained relationships between the omega invariants of a graph and its complement in terms of several graph parameters, such as the cyclomatic numbers, number of components, maximum number of components, order, and size of G and G, and triangular numbers. Some relationships between the omega invariants of a graph and its complement have been obtained.

New Relations Between Zagreb Indices and Omega Invariant.

In this work, we studied the problem of determining the values of the Zagreb indices of all the realizations of a given degree sequence. We first obtained some new relations between the first and second Zagreb indices and the forgotten index sometimes called the third Zagreb index. These relations also include the triangular numbers, order, size, and the biggest vertex degree of a given graph. As the first Zagreb index and the forgotten index of all the realizations of a given degree sequence are fixed, we concentrated on the values of the second Zagreb index and studied several properties including the effect of vertex addition. In our calculations, we make use of a new graph invariant, called omega invariant, to reach numerical and topological values claimed in the theorems. This invariant is closely related to Euler characteristic and the cyclomatic number of graphs. Therefore this invariant is used in the calculation of some parameters of the molecular structure under review in terms of vertex degrees, eccentricity, and distance.

A Direct Method to Solve Double Refined Indeterminate Triangular Neutrosophic Linear Programming Problem

Double Refined Indeterminate Triangular Neutrosophic Number (DRITrNN) provides the additional possibility to represent with sensitivity and accuracy the uncertain, imprecise, incomplete and inconsistent information. In this article a Linear programming problem is considered in DRITrNN situation and an efficient novel algorithm is proposed to solve the LPP with variables in the form of DRITrNN and its efficiency is validated with Numerical illustration. Keywords: Single valued triangular neutrosophic number , Double Refined Indeterminate Triangular Neutrosophic Number , Linear Programming Problem, Algorithm.

Exploring the pyro-phototronic effect for giant lateral photoresponse in an ITO/CdS/Si heterojunction position-sensitive detector

The demand for a high-performance position sensitive detector (PSD), a novel type of photoelectric sensor, is increasing due to advancements in digitization and automation technology. Cadmium sulfide (CdS), a non-centrosymmetric material, holds significant potential in photoelectric devices. However, the pyroelectric effect of CdS in PSDs and its influence on lateral photoresponse are still unknown. In this work, we fabricated an ITO/CdS/Si heterojunction using chemical bath deposition (CBD) and investigated the pyro-phototronic effect under nonuniform illumination. The theory of electron-hole pairs’ generation, separation, and carrier diffusion was carefully considered to understand the underlying mechanisms. Our experimental findings revealed that the device exhibited an exceptionally high position sensitivity (PS) of 1061.3 mV/mm, surpassing the generally observed PS of 655.1 mV/mm induced by single photovoltaic effect by 160.5%. Meanwhile, the PSD demonstrated rapid response times of 0.01 and 0.04 ms, respectively. Moreover, the influence of ambient temperature and electrode distance on the pyro-phototronic effect was well analyzed. Notably, the PSD exhibited remarkable stability even at ambient temperatures up to 150 °C. Despite the considerable working distance of 11 mm, the PS of the PSD remained at 128.99 mV/mm. These findings provide valuable theoretical and experimental foundations for optimizing the design and implementation of high-performance large working distance PSDs.

A Finite Element Analysis-Unascertained Measure Theory-Based Hybrid Approach to Safety Assessment for Pipelines Subject to Landslide Disasters

Abstract Pipeline safety faces a prevalent threat in mountainous areas due to landslides. The advent of landslides introduces the risk of pipeline leaks or ruptures, posing a significant threat to the environment, with the potential for casualties. Throughout the occurrence of landslides, uncertainties abound, yet few studies have addressed the incorporation of uncertainties in assessing pipeline safety. This work proposes a novel hybrid approach to the safety assessment for pipelines under landslides. The use of finite element analysis (FEA) models the pipeline under the action of landslides. The numerical outcomes, combined with unascertained measure theory (UMT), develop a multi-indicator unascertained measure (UM) matrix. Random forest (RF) algorithm is employed to determine the weight of indicators in the matrix. The hybrid application of set pair theory and the UM evaluation vector finally determine the pipeline safety degree and level. The proposed methodology has been well-validated through a case study on an in-service pipeline. The results indicate that the case pipeline safety degree is 2.777, 2.132, 3.132, 3.904, and 2.240, respectively. The corresponding safety level is III, II, III, IV, and II, respectively, which is consistent with the pipeline's actual condition. Different from the conventional safety assessment approach, the proposed methodology demonstrates the enhanced effectiveness, facilitating a more precise evaluation of the pipeline's safety condition.

A case study of the impact of carbon emissions and inflation on nonlinear dense neutrosophic fuzzy inventory system of varying demand with delayed deterioration

In a comprehensive consensus, human should reduce the carbon emission on the way to reduce the adverse effect of global warming. Manufacturing firms contributes significant amount of carbon emissions in the environment. In many countries, regulatory authorities are taking actions to reduce emissions. Thus, in this paper we have analysed the effect of carbon emissionfor non-instantaneous deteriorating items having advertisement and price dependent demand. The paper also focused on the effect of inflation where shortages are partially backlogged and partially lost in sales. In this paper a new concept of non-linear triangular dense neutrosophic number with its basic properties are developed. Further the classifications of symmetry, asymmetry is introduced and thereafter De-neutrosophication technique has been applied for crispification. Since the effect of some parameters like carbon emissions, advertising, and inflation are uncertain so we have considered it in this new aforesaid form to grab the uncertain characters of the underlying parameters. The classification of the uncertain parameter based on the symmetric and asymmetric nature and linear-nonlinear nature of triangular dense neutrosophic number have been also investigated here. Additionally, the effect of the model is examined under different situations for both linear and non-linear triangular dense fuzzy, dense intuitionistic, dense neutrosophic. Finally, a numerical example is considered to illustrate the model and it is observed that the model is optimum when the parameters are considered in Asymmetric Convex Non-Linear number for shorter replenishment cycle. Also, comparative study has been done by performing sensitivity analysis through the case study and provides managerial insight into this outcome.

Upper Tail Behavior of the Number of Triangles in Random Graphs with Constant Average Degree

Upper Tail Behavior of the Number of Triangles in Random Graphs with Constant Average Degree