Triangular Numbers Research Articles

Delving Deeper: What Else Comes after 1 + 2 = 3?

The simplest of prekindergarten equations, 1 + 2 = 3, is the basis for an investigation involving much of high school mathematics, including triangular numbers, arithmetic sequences, and algebraic proofs.

ПРО ПІДХІД ДО РОЗВ’ЯЗАННЯ НЕЧІТКОЇ ЗАДАЧІ КОМІВОЯЖЕРА НА ОСНОВІ МЕТОДУ ІМІТАЦІЇ ВІДПАЛУ

This article discusses the annealing simulation method for solving the fuzzy traveling salesman problem, which is formulated as the problem of finding a route to visit a given number of cities without repetitions with a minimum travel time. The content of the annealing simulation method is presented, the method formalization algorithm is described. The axiomatics of fuzzy triangular numbers are given. A fuzzy traveling salesman problem is formulated, in which the time parameters of movement between cities are given in the form of right fuzzy numbers, the carrier value in which depends on various external conditions and factors. The results of calculations for solving the traveling salesman problem in clear and fuzzy forms with different parameters of slices of fuzzy numbers are presented. Key words: traveling salesman problem, simulated annealing method, algorithm, fuzzy numbers, level set, formalization of time intervals.

The Cheeger Constants of Random Belyi Surfaces

Abstract Brooks and Makover developed a combinatorial model of random hyperbolic surfaces by gluing certain hyperbolic ideal triangles. In this paper, we show that for any $\epsilon>0$, as the number of ideal triangles goes to infinity, a generic hyperbolic surface in Brooks–Makover’s model has Cheeger constant less than $\frac {3}{2\pi }+\epsilon $.

Topological fields with a generic derivation

We study a class of tame L-theories T of topological fields and their Lδ-extension Tδ⁎ by a generic derivation δ. The topological fields under consideration include henselian valued fields of characteristic 0 and real closed fields. We show that the associated expansion by a generic derivation has L-open core (i.e., every Lδ-definable open set is L-definable) and derive both a cell decomposition theorem and a transfer result of elimination of imaginaries. Other tame properties of T such as relative elimination of field sort quantifiers, NIP and distality also transfer to Tδ⁎. As an application, we derive consequences for the corresponding theories of dense pairs. In particular, we show that the theory of pairs of real closed fields (resp. of p-adically closed fields and real closed valued fields) admits a distal expansion. This gives a partial answer to a question of P. Simon.

Neutrosophic multi-item inventory control models with constraints

Inventory managers are expected to handle a large number of items in their inventory while adhering to budgetary and space limits, as well as the number of items bought from vendors. Multi-item inventory models with one or more resource constraints, such as budget, space, or number of orders. This paper talks about an EOQ model in neutrosophic multi-item inventory control models with constraints. The ordering costs, the holding costs, demands, storage area, investment amount, and the maximum average number of units are considered as triangular neutrosophic numbers, as opposed to crisp values, to make the inventory model more realistic. This idea is used to decide the neutrosophic optimal order quantities with the assistance of the Lagrange multiplier. Eventually, the proposed method is delineated with a numerical instance and the results are analysed briefly.

Sizes of flats of cycle matroids of complete graphs

We show that the problem of counting the number of flats of size k for a cycle matroid of a complete graph is equivalent to the problem of counting the number of partitions of an integer k into triangular numbers. In addition, we give some values of k such that there is no flat of size k in a cycle matroid of a complete graph of order k. Finally, we give a minimum bound for the number of values, k, for which there are no flats of size k in the given cycle matroid.

Oblate Distorted Harmonic Oscillator Symmetry

Abstract: Harmonic oscillator states have shown advantageous for nuclear structure. Nuclear physics experts have developed sophisticated group theory-based mathematical techniques to handle n-particle states in the harmonic oscillator (ho) potential as a result of this. The deformed harmonic oscillator (HO)'s oblate shell closure has been studied, and the results demonstrate that the accompanying magic numbers are connected to a universal sequence of the triangular numbers that identically characterize oblate, spherical, and prolate. This offers a distinct framework for comprehending the distorted HO. There is discussion of the effects on oblate nuclei in the present study. The current work examines the symmetries of the deformed HO on the oblate side of deformation and demonstrates that these spherical degeneracies are equally important on this side of deformation as they are on the prolate side. The significance of these symmetries for comprehending nuclear structure is highlighted.

Local detour centrality: a novel local centrality measure for weighted networks

Centrality, in some sense, captures the extent to which a vertex controls the flow of information in a network. Here, we propose Local Detour Centrality as a novel centrality-based betweenness measure that captures the extent to which a vertex shortens paths between neighboring vertices as compared to alternative paths. After presenting our measure, we demonstrate empirically that it differs from other leading central measures, such as betweenness, degree, closeness, and the number of triangles. Through an empirical case study, we provide a possible interpretation for Local Detour Centrality as a measure that captures the extent to which a word is characterized by contextual diversity within a semantic network. We then examine the relationship between our measure and the accessibility to knowledge stored in memory. To do so, we show that words that occur in several different and distinct contexts are significantly more effective in facilitating the retrieval of subsequent words than are words that lack this contextual diversity. Contextually diverse words themselves, however, are not retrieved significantly faster than non-contextually diverse words. These results were obtained for a serial semantic memory task, where the word’s location constitutes a significant mediator in the relationship between the proposed measure and accessibility to knowledge stored in memory.

Assessment of Sustainable Maintenance Strategy for Manufacturing Industry

This study creates a framework to aid in the sustainability of maintenance strategies. The framework was created using expertise from the industry and academia. Using this knowledge, three multi-criteria tools were chosen for the maintenance strategies evaluation. The tools include grey relational analysis (GRA) techniques, additive ratio assessment (ARAS), and step-wise weight assessment ratio analysis (SWARA). In a production system, they were used to assess four planned maintenance strategies. The strategies are periodic maintenance (S1), meter-based maintenance (S2), predictive maintenance (S3) and prescriptive maintenance (S4). The ARAS approach was used to obtain the strategy rating for the various requirements. This study used the SWARA method to determine the requirements’ importance using an intuitionistic fuzzy triangular number. The ARAS results were combined using the GRA method. This study observed that the criteria utilised to choose a maintenance strategy for equipment depend on the information collected from six specialists in a manufacturing organisation. For instance, it was discovered that S3 was the maintenance approach that best suited the system’s technical needs. At the same time, S2 was found to be less effective. The economic needs analysis showed that S1 is the maintenance strategy that is most appropriate for the system, while S3 is the least appropriate. S1 is the most appropriate maintenance method for the system, given the social requirements, whereas S2 is the least effective. According to the results of the environmental requirements, S2 is the best maintenance plan for the system, while S4 is the worst. According to the GRA approach, the system’s best and least appropriate maintenance strategies are S2 and S4, respectively.

Multiplicity of Hexagon Numbers

Summary Imagine you have an unlimited supply of congruent equilateral triangles. Polygon numbers are the number of these triangles used to tile a convex polygon. For example, triangle numbers are square integers n 2, where the positive integer n is the side length of a tiled equilateral triangle. Hexagon numbers are the number of triangles used to tile a convex hexagon, and can be realized by removing corners from a tiled equilateral triangle. The number of ways (multiplicity up to congruence) that a given hexagon number can be constructed geometrically from such tiles is the subject of our paper.

Character sheaves for classical symmetric pairs

We establish a Springer theory for classical symmetric pairs. We give an explicit description of character sheaves in this setting. In particular we determine the cuspidal character sheaves.

An IPA Approach towards Including Citizens’ Perceptions into Strategic Decisions for Smart Cities in Romania

City administrators have begun to consider citizens’ perception as an essential enabler of smart and sustainable city development. In response to their need, our study mainly aimed to analyze the citizens’ perceptions related to the attributes of a smart city and to offer an evidence-based importance–performance analysis (IPA) technique, revised with triangular number fuzzy, to set priorities to develop it further successfully, taking Oradea, Romania, as an example. Specifically, we investigated varying the citizens’ perceptions regarding the attributes of a smart city across the thematic content analysis followed by an IPA approach to identify the attributes of a smart city that could be expected to advance its future development. The results show that most of our respondents appreciated the city administrators’ efforts and expressed relatively higher preferences for the fifteen smart city attributes of Oradea on which the local authorities should focus. The research brings valuable insight into the strategic decision making related to smart city development, which can serve academics and practitioners, complementing existing literature on the citizens’ perception of sustainable smart city development.

A Relationship of Triangular and Star Numbers

Summary We give a visual proof for an identity relating triangular and star numbers.

Using 2D-Phthalocyanine Metal Organic Framework-Based Catalysts for Oxygen Reduction Reaction in Alkaline Media

Developing high performance electrocatalysts to improve the sluggish kinetics of oxygen reduction reaction (ORR) at the cathode is essential in expanding fuel cell applications. Metal organic frameworks (MOFs), given their high surface area, tunable pore size, and diverse metal choices, are promising candidates to catalyze such a reaction. Herein, we report a family of nine 2D-phthalocyanine MOF-based catalysts for alkaline ORR, where M = Co, Ni, Cu. Structurally, these bimetallic materials have two distinct metal sites, where M1 is a phthalocyanine moiety and M2 is a node, existing as M-N4 moiety. X-ray Diffraction (XRD) shows that all as-prepared MOFs have similar structures and inter-layer spacing. X-ray Photoelectron Spectroscopy (XPS) shows slight shifts in binding energy when the same element is placed in a M1 or M2 site, which indicates that the M1 and M2 sites are chemically distinct.Through cyclic voltammetry we evaluate the role of the metal sites for ORR activity and selectivity in 0.1 M KOH. All materials are active for alkaline ORR, while Ni (M1) – Co (M2) has the best overall kinetic activity performance of - 5 mA cm-2 at 0.7 V vs. RHE. Cu – Cu, on the other hand, is the least active combination under the same testing conditions. Within composition series where M2 = Co, the activity trend for M1 is Ni > Cu ~ Co. From a 2-electron selectivity perspective, M1 = Ni > Cu > Co. From these general trends we hypothesize that Co M2 and Ni M1 sites are essential in promoting 4- and 2-electron selectively, respectively. On average we find a nominal ORR performance trend where M2 = Co > Cu > Ni. Interestingly, we note that the M1 identity also plays a non-linear role in activity and selectivity modulation. Though we identify one of the metal sites (M1 or M2) as the dominating site for activity and/or selectivity, the other site also contributes to the total performance. The density functional theory (DFT) calculation supports our experimental results and finds out that Ni as M1 is highly selective for H2O2 formation whereas Co as M2 shows excellent activity for ORR. Our predicted volcano plot shows that MOF with Ni – Co and Cu – Co combinations at Co active site offering lowest ORR overpotential among all other combinations, which is also in good agreement with experimental findings.By evaluating different metal sites performance through electrochemical testing, we can better understand the active species in the MOF catalytic system. By pairing theory with experiment, we gain active site insight to design new specific enhanced active motifs. We believe that these performance trends can assist in better designing efficient MOF-based ORR catalysts that can lead to advances in clean energy technology for sustainable development.

Three-Dimensional Engine-Based Geometric Model Optimization Algorithm for BIM Visualization with Augmented Reality.

Building information modeling (BIM), a common technology contributing to information processing, is extensively applied in construction fields. BIM integration with augmented reality (AR) is flourishing in the construction industry, as it provides an effective solution for the lifecycle of a project. However, when applying BIM to AR data transfer, large and complicated models require large storage spaces, increase the model transfer time and data processing workload during rendering, and reduce visualization efficiency when using AR devices. The geometric optimization of the model using mesh reconstruction is a potential solution that can reduce the required storage while maintaining the shape of the components. In this study, a 3D engine-based mesh reconstruction algorithm that can pre-process BIM shape data and implement an AR-based full-size model is proposed, which is likely to increase the efficiency of decision making and project processing for construction management. As shown in the experimental validation, the proposed algorithm significantly reduces the number of vertices, triangles, and storage for geometric models while maintaining the overall shape. Moreover, the model elements and components of the optimized model have the same visual quality as the original model; thus, a high performance can be expected for BIM visualization in AR devices.

The Series of Reciprocals of Triangular Numbers

We provide a visual proof that the infinite sum of the reciprocals of the triangular numbers is 1. Such visual proofs usually stack line segments or rectangular blocks. Here we stack angles.

Bi-Objective Flow Shop Scheduling with Equipotential Parallel Machines

Scheduling is very important concept in each and every field of life especially in case of manufacturing process. Actually, a production schedule is the time table that tells the time at which an assignment will be processed on various machines. The Schedule also gives the information about starting and finishing of a work on one machine. This paper also deals with the theory of Scheduling. The main attraction of this study is the optimization done on like Parallel machines with the help of Fuzzy Processing Times. Here the problem of optimization on Two Stage Flow Shop Model has been taken into consideration. This paper reveals an algorithm using Branch and bound method for scheduling on three like parallel machines available at initial stage and solo machine at next stage having processing period of all works as fuzzy triangular numbers involving transportation time from first stage to second stage. Algorithm provides an optimal sequence of jobs for minimizing make span as well as the unit operational cost of each job on all three parallel machines. Numerical example has also been discussed for elaborating this situation. The proposed model is the extension of model presented by Deepak Gupta and Sonia Goel [18].

Two distinctive temperature dependences of the London penetration depth in high-Tc cuprate superconductors as support for the theory of Bose-liquid superconductivity

In the framework of the theory of Bose-liquid superconductivity, we show that in high-Tc cuprate superconductors the magnetic field penetration depth (known as the London penetration depth λL(T)) decreases first exponentially below the superconducting transition temperature Tc with decreasing the temperature T and then λL(T) decreases like a certain power of T at lower temperatures due to the vanishing of the energy gap Δg(T) in the excitation spectrum of a superfluid Bose-liquid at a characteristic temperature Tc⁎ lower than Tc. Below Tc, the London penetration depth λL(T) has an exponential temperature dependence down to Tc⁎, which is different from a BCS- like exponential temperature dependence, and then λL(T) will follow a power law temperature dependence down to T=0, which is also different from power-law temperature dependences predicted by some BCS-like (d-wave) pairing theories. In unconventional (bosonic) cuprate superconductors characterized by the strong interboson coupling constant (γB≳1), λL(T) exhibits the distinctly different temperature dependences in different temperature intervals 0≤T≤Tc⁎<<Tc and Tc⁎<T<Tc when the energy gap Δg(T) vanishes at a temperature Tc⁎ well below Tc. In the weak interboson coupling (γB<<1) the energy gap Δg(T) vanishes at a temperature Tc⁎ close to Tc and the reduced London penetration depth λL(T)/λL(0) has an unusual Tn power-law temperature dependence in a wide temperature range 0<T<Tc⁎ below Tc, i.e. the expression λL(T)/λL(0)≃[1−aL(T)(T/Tc)n]−1/2 (where n≳4 and aL(T)≲1) is valid only below Tc⁎ and is similar to the well-known empirical Gorter-Casimir law. Our theoretical predictions of the distinctive power-law and exponential temperature dependences of λL(T)/λL(0) are in good agreement with the experimental results reported for GdBa2Cu3O7−δ ceramics and YBa2Cu3O7−δ films. The peculiar temperature dependences of λL(T)/λL(0) observed in these high-Tc materials at low and high temperatures do not follow the predictions of the s- and d-wave BCS-like theories of Fermi-liquid superconductivity and give evidence for the predictions of the theory of Bose-liquid superconductivity.

A Newly Proposed and Comparative Study on Multi - Criteria Decision Making in Interval Valued Bipolar Triangular Neutrosophic Set by PROMETHEE II

Objectives: To propose the operational laws and operators in interval valued bipolar triangular neutrosophic sets and also to resolve multi criteria decision making problem using PROMETHEE II. Methods: Based on the combination of bipolar neutrosophic sets and interval valued triangular neutrosophic sets, an interval valued bipolar triangular neutrosophic set is proposed followed by its operational laws, score function, accuracy and certainty function. Findings: The arithmetic operators, operational laws and also the score function, accuracy and certainty functions have been proposed for interval valued bipolar triangular neutrosophic sets. The PROMETHEE II technique discovered and employed an aggregated weighted arithmetic operator to address the multi-criterion decision-making problem. Novelty: By the concept of interval valued bipolar triangular neutrosophic set, the results can be viewed with more accuracy for the linguistic variables rather than the crisp values. An illustrative example has been explained for the interval valued bipolar triangular neutrosophic numbers by which the effectiveness is tested with the comparative approach by using PROMETHEE II method. Keywords : Interval valued neutrosophic numbers, Bipolar neutrosophic numbers, Interval valued bipolar triangular neutrosophic numbers, Score function, PROMETHEE II method.

Cracking Ion Pairs in the Electrical Double Layer of Ionic Liquids

Here we investigate a limiting case of the theory for aggregation and gelation in the electrical double layer (EDL) of ionic liquids (ILs). The limiting case investigated only accounts for ion pairs, ignoring the possibility of larger clusters and a percolating ionic network. This simplification, permits analytical solutions for the properties of the EDL. The resulting equations demonstrate the competition between the free energy of an association and the electrostatic potential in the EDL. For small electrostatic potentials and large negative free energies of associations, ion pairs dominate in the EDL. Whereas, for electrostatic potential energies larger than the free energy of an association, electric-field-induced cracking of ion pairs occurs. The differential capacitance for this consistent ion pairing theory has a propensity to have a “double hump camel” shape. We compare this theory against previous free ion approaches, which do not consistently treat the reversible associations in the EDL.