Pentagonal Number Research Articles

Truncated versions of three identities of Euler and Gauss

AbstractIn 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Motivated by the works of Andrews and Merca, Guo and Zeng deduced truncated versions for two other classical theta series identities of Gauss. Very recently, Xia et al. proved new truncated theorems of the three classical theta series identities by taking different truncated series than the ones chosen by Andrews–Merca and Guo–Zeng. In this paper, we provide a unified treatment to establish new truncated versions for the three identities of Euler and Gauss based on a Bailey pair due to Lovejoy. These new truncated identities imply the results proved by Andrews–Merca, Wang–Yee, and Xia–Yee–Zhao.

Dividing a Sphere Hierarchically into a Large Number of Spherical Pentagons Using Equal Area or Equal Length Optimization

Dividing a 2-dimensional sphere uniformly into a large number of spherical polygons is a challenging mathematical problem, which has been studied across many disciplines due to its important practical applications. Most sphere subdivisions are achieved using spherical triangles, quadrangles, or a combination of hexagons and pentagons. However, spherical pentagons, which may create elegant configurations, remain under-explored. This study presents a new sphere subdivision method to generate a large number of spherical pentagons based on successively subdividing a module of an initial spherical dodecahedron. The new method can conveniently control the shapes of generated spherical pentagons through specified design parameters. Two optimization problems have been investigated: (I) dividing a sphere into spherical pentagons of equal area; (II) minimizing the number of different arc lengths used in the pentagonal subdivision. A variety of examples are presented to demonstrate the effectiveness of the new method. This study shows that treating the mathematical challenge of dividing a sphere uniformly into a large number of spherical polygons as an optimization problem can effectively obtain equal area or equal length sphere subdivisions. Furthermore, considering additional constraints on the optimization problem may achieve sphere subdivisions of specific characteristics. • A new sphere subdivision method is developed to generate a large number of spherical pentagons. • The subdivision approach is performed locally and hierarchically on a module of a spherical dodecahedron. • The mathematical challenge of dividing a sphere uniformly is treated as an optimization problem. • Optimization algorithms are used to achieve sphere subdivisions of specific characteristics.

Close Interval Approximation of Pentagonal Fuzzy Numbers for Interval Data-Based Transportation Problems

Due to globalization in this modern age of technology and other uncontrollable influences, transportation parameters can differ within a certain range of a given period. In this situation, a managerial position’s objective is to make appropriate decisions for the decision-makers. However, in general, the determination of an exact solution to the interval data-based transportation problem (IDTP) becomes an NP-hard problem as the number of choices within their respective ranges increases enormously when the number of suppliers and buyers increases. So, in practice, it is difficult for an exact method to find the exact solution to the IDTP in a reasonable time, specifically the large-sized problems with large interval sizes. This paper introduces solutions to the IDTP where supply, demand, and cost are all in interval numbers. One of the best interval approximations, namely the closed interval approximation of pentagonal fuzzy number, is proposed for solving the IDTP. First, in the proposed closed interval approximation method (Method-1), the pentagonal fuzzification method converts the IDTP to a fuzzy transportation problem (FTP). Subsequently, two new ranking methods based on centroid and in-center triangle concepts are presented to transfer the pentagonal fuzzy number into the corresponding crisp (non-fuzzy) value. Thereafter, the optimal solution was obtained using Vogel’s approximation method coupled with the modified distribution method. The proposed Method-1 is reported against a recent method and shows superior performance over the aforementioned and a proposed Method-2 via benchmark instances and new instances.

Toward the Nash Equilibrium Solutions for Large-Scale Pentagonal Fuzzy Continuous Static Games

This study aims to characterize and clarify a pentagonal fuzzy continuous static game (PF-CSG) that constraints and cost functions are fuzzy numbers. Pentagonal fuzzy numbers characterize their fuzzy parameters. The α − Pareto optimal solution concept has specified. The decomposition approach has applied to decompose the problem into subproblems each of them having smaller and independent subproblems. In addition, the Nash equilibrium solution concept was used to obtain the solutions of these subproblems. The advantages of this study are the players independently without collaboration with any of the others and that each player seeks to minimize the cost function. Also, the information available to each player consists of the cost function and constraints. An illustrated numerical example has discussed for proper understanding and interpretation of the proposed concept.

Infinite families of Hecke-Rogers type series and their truncated representations

Recently, Andrews and Merca studied the truncated version of Euler's pentagonal number theorem. Their work opened up a new study on truncated series. Since then, many truncated versions of theta series have been discovered. Very recently, Wang and Yee generalized the study on truncated series to Hecke-Rogers type series and obtained three new truncated Hecke-Rogers series. Inspired by these works, we not only prove several infinite families of Hecke-Rogers type series and their truncated versions, but also establish some inequalities on some interesting partition functions in this paper. In particular, we generalize some results due to Wang and Yee, and He. Our work mainly relies on some formulas on the evaluation of certain terminating basic hypergeometric ϕ23 series.

Generalized Guglielmo Numbers: An Investigation of Properties of Triangular, Oblong and Pentagonal Numbers via Their Third Order Linear Recurrence Relations

In this paper, we investigate the generalized Guglielmo sequences and we deal with, in detail, four special cases, namely, triangular, triangular-Lucas, oblong and pentagonal sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

Generalizations of the truncated pentagonal number theorem results

Generalizations of the truncated pentagonal number theorem results

Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they conjectured that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobi's triple product identity in a different manner.

Implementation of Neutrosophic Bipolar Pentagonal Fuzzy Set on Multi-Criteria Decision-Making Scenario

Several others have already discussed various types of new techniques in multi-criteria decision-making problems. De-fuzzification, De-eutrophication, and De-Bipolarisation all of these are new approaches that have been established. But this work depicts the concept and features of the Neutrosophic Bipolar Pentagonal Fuzzy number. The relationships and products of the picture fuzzy set, as well as their associated results, are investigated. The different kinds of neutrosophic single-typed bipolar pentagonal numbers (nbpnum) were also discussed. A multi-criteria choice is also done in a triangular Bipolar Neutrosophic Fuzzy Set to identify the ideal output depending on several attributes. A multi-criteria choice is also done in a triangular Bipolar Neutrosophic Fuzzy Set to identify the ideal output depending on several attributes.

Nonlinear pentagonal intuitionistic fuzzy number and its application in EPQ model under learning and forgetting

In this current era, the concept of nonlinearity plays an important and essential role in intuitionistic fuzzy arena. This article portrays an impression of different representation of nonlinear pentagonal intuitionistic fuzzy number (PIFN) and its classification under different scenarios. A new de-intuitification technique of non-linear PIFN is addressed in this article along with its various graphical representations. Additionally, in this paper, we have observed this by applying it in an economic production quantity model where the production is not perfect and defective items are produced which are reworked. The model is considered under learning and forgetting, where learning is considered as linear, nonlinear PIFN and crisps arena. It is observed from the numerical study that high learning effects in rework lead to decrease in production of defective item, which, besides an economic advantage, may have a positive effect on the environment. Even though forgetting has an adverse effect, the average total cost is much less than that of the basic model which ignores worker learning and forgetting. Finally, comparative and sensitivity analysis result shows the utility of this noble work.

New truncated theorems for three classical theta function identities

In 2012, Andrews and Merca derived a truncated version of Euler’s pentagonal number theorem. Their work inspired several mathematicians to work on truncated theta series including Guo and Zeng, who examined two other classical theta series identities of Gauss. In this paper, revisiting these three theta series identities of Euler and Gauss, we obtain new truncated theorems. As corollaries of our results, we obtain infinite families of linear inequalities involving the partition function, the overpartition function and the pod function. These inequalities yield the positivity result of Andrews and Merca on the partition function as well as a conjecture on the overpartition function, which was posed by Andrews–Merca and Guo–Zeng, and proved independently by Mao and Yee. We will also provide a unified combinatorial treatment for our results.

Connections Between Partitions and Divisors Related to the Parity of the Partition Function

Summary In this paper, the author gives two linear homogeneous recurrence relations to compute the values of Euler’s partition function p(n) when n is odd. These recurrences do not involve the generalized pentagonal numbers and provide new connections between the partition function p(n) and the positive divisor function for x = 0 and x = 1. Some relations involving the divisor functions . and the parity of the partition function p(n) are presented in this context.

On the maximum number of odd cycles in graphs without smaller odd cycles

AbstractWe prove that for each odd integer , every graph on vertices without odd cycles of length less than contains at most cycles of length . This extends the previous results on the maximum number of pentagons in triangle‐free graphs, conjectured by Erdős in 1984, and asymptotically determines the generalized Turán number for odd . In contrary to the previous results on the pentagon case, our proof is not computer‐assisted.

IMPLICATION - BASED FUZZY TERNARY SEMIGROUPS FOR ISOLATED SYSTOLIC HYPERTENSION USING PENTAGONAL FUZZY NUMBER MATRIX

In this paper, medical diagnosis concerning isolated systolic hypertension and a study on implication operators is introduced, but fuzzy pentagonal number matrix concepts developed to give a numerical example and graphical representation are also established.

ANALYSIS OF FUZZY QUEUES BY USING PENTAGONAL FUZZY NUMBER

This paper deals with the fact thatpentagonal fuzzy numbers are pre-owned and systematic outcomes are discussed in real-life situations. The fuzzy set supposition is combined with well-established classical queuing theory but the classical queuing theory is far away from real-life situations. In this approach, we can use both fuzzy and probability theory to make this work more realistic with the help of the α-cut technique. Symmetric pentagonal fuzzy numbers are used to elaborate on the situation of the queue in linguistic terms.

Computing and Analysis of Multi Server Queueing Model Using Pentagonal Fuzzy Numbers

In the present study, analysed the multiple server queue miniature using Pentagonal Fuzzy number. To cope with a real-world scenario, the input parameters are converted from crisp to fuzzy type by taking into account. The system parameters, rate of arrival and rate of service are taken fuzzy. The system characteristics like expected number of clients waiting in queue and expected number of clients waiting in system are calculated using DSW algorithm and using alpha-cut at various levels, in interval [0,1]. Using pentagonal number arithmetic, the performance measures are calculated directly as PFNs. Different numerical examples with arrival rate and service rate as PFN are demonstrated to show the authenticity implementation of the performance measures. Finally effect of different parameters are shown for validation purpose.

A formula for the number of partitions of n in terms of the partial Bell polynomials

We derive a formula for $p(n)$ (the number of partitions of $n$) in terms of the partial Bell polynomials using Fa\`{a} di Bruno's formula and Euler's pentagonal number theorem.

Bio-structure on the rachis outer surface of a bird feather

From the perspective of biomechanics, this paper studies the outer surface of the rachis. The bio-structure was determined by microscopy. Finite element analysis was conducted on this bio-structure. The results show that, in the feather, the bio-structure of the outer surface of the rachis consists of the basic unit structure of the hexagonal shape of the edge protrusion and a small number of pentagons. The results of the finite element analysis for the basic geometry structure and flange were the edges of which the main bearing stress sites. The practical reference value was for the determination one of safety-structures.

Generalized Lambert Series and Euler’s Pentagonal Number Theorem

We consider the number of k’s in all the partitions of n, and provide new connections between partitions and functions from multiplicative number theory. In this context, we obtain a new generalization of Euler’s pentagonal number theorem and provide combinatorial interpretations for some special cases of this general result.

English

English