Search results for: membership function

Authors: Takashi Mitsuishi, Koji Saigusa

Abstract:

This paper presents circular polar coordinates transformation of periodic fuzzy membership function. The purpose is identification of domain of periodic membership functions in consequent part of IF-THEN rules. The proposed methods are applied to the simple color construct system.

Keywords: periodic membership function, polar coordinates transformation, defuzzification, circular coordinates

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Authors: Changho Han, Jaemin Lee, Li Hua, Seokkwon Jeong

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Design of membership function ranges in fuzzy logic control (FLC) is presented for robust control of a variable speed refrigeration system (VSRS). The criterion values of the membership function ranges can be carried out from the static experimental data, and two different values are offered to compare control performance. Some simulations and real experiments for the VSRS were conducted to verify the validity of the designed membership functions. The experimental results showed good agreement with the simulation results, and the error change rate and its sampling time strongly affected the control performance at transient state of the VSRS.

Keywords: variable speed refrigeration system, fuzzy logic control, membership function range, control performance

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Authors: O. Poleshchuk, E. Komarov

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This paper presents a regression model for interval type-2 fuzzy sets based on the least squares estimation technique. Unknown coefficients are assumed to be triangular fuzzy numbers. The basic idea is to determine aggregation intervals for type-1 fuzzy sets, membership functions of whose are low membership function and upper membership function of interval type-2 fuzzy set. These aggregation intervals were called weighted intervals. Low and upper membership functions of input and output interval type-2 fuzzy sets for developed regression models are considered as piecewise linear functions.

Keywords: interval type-2 fuzzy sets, fuzzy regression, weighted interval

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Authors: Takashi Mitsuishi

Abstract:

Fuzzy logic has gained acceptance in the recent years in the fields of social sciences and humanities such as psychology and linguistics because it can manage the fuzziness of words and human subjectivity in a logical manner. However, the major field of application of the fuzzy logic is control engineering as it is a part of the set theory and mathematical logic. Mamdani method, which is the most popular technique for approximate reasoning in the field of fuzzy control, is one of the ways to numerically represent the control afforded by human language and sensitivity and has been applied in various practical control plants. Fuzzy logic has been gradually developing as an artificial intelligence in different applications such as neural networks, expert systems, and operations research. The objects of inference vary for different application fields. Some of these include time, angle, color, symptom and medical condition whose fuzzy membership function is a periodic function. In the defuzzification stage, the domain of the membership function should be unique to obtain uniqueness its defuzzified value. However, if the domain of the periodic membership function is determined as unique, an unintuitive defuzzified value may be obtained as the inference result using the center of gravity method. Therefore, the authors propose a method of circular-polar-coordinates transformation and defuzzification of the periodic membership functions in this study. The transformation to circular polar coordinates simplifies the domain of the periodic membership function. Defuzzified value in circular polar coordinates is an argument. Furthermore, it is required that the argument is calculated from a closed plane figure which is a periodic membership function on the circular polar coordinates. If the closed plane figure is continuous with the continuity of the membership function, a significant amount of computation is required. Therefore, to simplify the practice example and significantly reduce the computational complexity, we have discretized the continuous interval and the membership function in this study. In this study, the following three methods are proposed to decide the argument from the discrete polygon which the continuous plane figure is transformed into. The first method provides an argument of a straight line passing through the origin and through the coordinate of the arithmetic mean of each coordinate of the polygon (physical center of gravity). The second one provides an argument of a straight line passing through the origin and the coordinate of the geometric center of gravity of the polygon. The third one provides an argument of a straight line passing through the origin bisecting the perimeter of the polygon (or the closed continuous plane figure).

Keywords: defuzzification, fuzzy membership function, periodic function, polar coordinates transformation

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Authors: M. Kamel El-Sayed

Abstract:

The process of determining the degree of membership for an element to an uncertain concept has been found in many ways, using equivalence and symmetry relations in information systems. In the case of similarity, these methods did not take into account the degree of symmetry between elements. In this paper, we use a new definition for finding the membership based on the degree of symmetry. We provide an example to clarify the suggested methods and compare it with previous methods. This method opens the door to more accurate decisions in information systems.

Keywords: information system, uncertain concept, membership function, similarity relation, degree of similarity

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Authors: Qiuling Xie, Shanliang Zhu, Zongdi Sun

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In this paper, we analyzed the correlation relationship among PM2.5 from other five Air Quality Indices (AQIs) based on the grey relational degree, and built a multivariate nonlinear regression equation model of PM2.5 and the five monitoring indexes. For the optimal control problem of PM2.5, we took the partial large Cauchy distribution of membership equation as satisfaction function. We established a nonlinear programming model with the goal of maximum performance to price ratio. And the optimal control scheme is given.

Keywords: grey relational degree, multiple linear regression, membership function, nonlinear programming

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Authors: Bader Alkandari, Jamal Y. Madouh, Ahmad M. Alkandari, Anwar A. Alnaqi

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A new formulation of fuzzy linear estimation problem is presented. It is formulated as a linear programming problem. The objective is to minimize the spread of the data points, taking into consideration the type of the membership function of the fuzzy parameters to satisfy the constraints on each measurement point and to insure that the original membership is included in the estimated membership. Different models are developed for a fuzzy triangular membership. The proposed models are applied to different examples from the area of fuzzy linear regression and finally to different examples for estimating the electrical load on a busbar. It had been found that the proposed technique is more suited for electrical load estimation, since the nature of the load is characterized by the uncertainty and vagueness.

Keywords: fuzzy regression, load estimation, fuzzy linear parameters, electrical load estimation

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Authors: Jun-Ho Chung, Sung-Hyun Yoo, In-Hwan Choi, Hyun-Kook Lee, Moon-Kyu Song, Choon-Ki Ahn

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The extended finite impulse response (EFIR) filter is addressed to optimize membership functions (MFs) of the fuzzy model that has strong nonlinearity. MFs are important parts of the fuzzy logic system (FLS) and, thus optimizing MFs of FLS is one of approaches to improve the performance of output. We employ the EFIR as an alternative optimization option to nonlinear fuzzy model. The performance of EFIR is demonstrated on a fuzzy cruise control via a numerical example.

Keywords: fuzzy logic system, optimization, membership function, extended FIR filter

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Authors: Eka Yuliana, Siska Shabrina Julyan

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The competitive environment and high customer’s churn rate in telecommunication industries lead Indonesian telecommunication companies become strive to offer products with more value. Offering product with more value can encourage customers to keep using the companies product. One of way to retain customer is give a membership card to the customers as practiced by PT. Telkom by giving Telkom Membership Card to PT. Telkom loyal customer. This study aims to determine the influence of Telkom Membership Card customer perceived value on retaining PT. Telkom Indonesia’s customer in 2013-2014 by using quantitative method with causal study. Analythical technique used in this study is Structural Equation Modelling (SEM) to test the causal relationship with 216 owner of Telkom Membership Card in Indonesia. This study conclude that: (i) Customer perceived value on Telkom Membership Card is located in fair value zone, (ii) PT. Telkom efforts in order to retain the customers is classified as good, (iii) Customer perceived value is influencing the effort to retain the customer with the probability value less than 0.05 and level of influence 69%. Based on result of this study, PT. Telkom should (i) Improve Telkom Membership Card’s promotion because not all customer of PT. Telkom have the membership card. (iia) Adding Telkom Membership Card’s benefit such as discount at various merchant (iib) Making call center for member of Telkom Membership Card (iii) PT. Telkom should be ensure availability of their service. (iv) PT. Telkom should make a priority to customer who have telkom membership card and offers a better service.For future research should be use different variables.

Keywords: customer perceived value, customer retention, marketing, relationship marketing

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Authors: Dhruba Das

Abstract:

In this paper, a method has been developed to construct the membership surfaces of row and column vectors and arithmetic operations of imprecise matrix. A matrix with imprecise elements would be called an imprecise matrix. The membership surface of imprecise vector has been already shown based on Randomness-Impreciseness Consistency Principle. The Randomness- Impreciseness Consistency Principle leads to defining a normal law of impreciseness using two different laws of randomness. In this paper, the author has shown row and column membership surfaces and arithmetic operations of imprecise matrix and demonstrated with the help of numerical example.

Keywords: imprecise number, imprecise vector, membership surface, imprecise matrix

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Authors: Dhruba Das

Abstract:

In this article, the author has put forward the actual mathematical explanation of subnormal imprecise vector. Every subnormal imprecise vector has to be defined with reference to a membership surface. The membership surface of normal imprecise vector has already defined based on Randomness-Impreciseness Consistency Principle. The Randomness- Impreciseness Consistency Principle leads to defining a normal law of impreciseness using two different laws of randomness. A normal imprecise vector is a special case of subnormal imprecise vector. Nothing however is available in the literature about the membership surface when a subnormal imprecise vector is defined. The author has shown here how to construct the membership surface of a subnormal imprecise vector.

Keywords: imprecise vector, membership surface, subnormal imprecise number, subnormal imprecise vector

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Authors: Ouafa Amira, Jiangshe Zhang

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Clustering is an unsupervised machine learning technique; its aim is to extract the data structures, in which similar data objects are grouped in the same cluster, whereas dissimilar objects are grouped in different clusters. Clustering methods are widely utilized in different fields, such as: image processing, computer vision , and pattern recognition, etc. Fuzzy c-means clustering (fcm) is one of the most well known fuzzy clustering methods. It is based on solving an optimization problem, in which a minimization of a given cost function has been studied. This minimization aims to decrease the dissimilarity inside clusters, where the dissimilarity here is measured by the distances between data objects and cluster centers. The degree of belonging of a data point in a cluster is measured by a membership function which is included in the interval [0, 1]. In fcm clustering, the membership degree is constrained with the condition that the sum of a data object’s memberships in all clusters must be equal to one. This constraint can cause several problems, specially when our data objects are included in a noisy space. Regularization approach took a part in fuzzy c-means clustering technique. This process introduces an additional information in order to solve an ill-posed optimization problem. In this study, we focus on regularization by relative entropy approach, where in our optimization problem we aim to minimize the dissimilarity inside clusters. Finding an appropriate membership degree to each data object is our objective, because an appropriate membership degree leads to an accurate clustering result. Our clustering results in synthetic data sets, gaussian based data sets, and real world data sets show that our proposed model achieves a good accuracy.

Keywords: clustering, fuzzy c-means, regularization, relative entropy

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Authors: Elda Maraj, Shkelqim Kuka

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Coronary heart disease causes many deaths in the world. Unfortunately, this problem will continue to increase in the future. In this paper, a fuzzy logic model to predict coronary heart disease is presented. This model has been developed with seven input variables and one output variable that was implemented for 30 patients in Albania. Here fuzzy logic toolbox of MATLAB is used. Fuzzy model inputs are considered as cholesterol, blood pressure, physical activity, age, BMI, smoking, and diabetes, whereas the output is the disease classification. The fuzzy sets and membership functions are chosen in an appropriate manner. Centroid method is used for defuzzification. The database is taken from University Hospital Center "Mother Teresa" in Tirana, Albania.

Keywords: coronary heart disease, fuzzy logic toolbox, membership function, prediction model

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Authors: Kumari Sweta, Ajanta Goswami, Abhilasha Dixit

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Landslide is a geomorphic process that plays an essential role in the evolution of the hill-slope and long-term landscape evolution. But its abrupt nature and the associated catastrophic forces of the process can have undesirable socio-economic impacts, like substantial economic losses, fatalities, ecosystem, geomorphologic and infrastructure disturbances. The estimated fatality rate is approximately 1person /100 sq. Km and the average economic loss is more than 550 crores/year in the Himalayan belt due to landslides. This study presents a comparative performance of a statistical bivariate method and a machine learning technique for landslide susceptibility mapping in and around Dharamshala, Himachal Pradesh. The final produced landslide susceptibility maps (LSMs) with better accuracy could be used for land-use planning to prevent future losses. Dharamshala, a part of North-western Himalaya, is one of the fastest-growing tourism hubs with a total population of 30,764 according to the 2011 census and is amongst one of the hundred Indian cities to be developed as a smart city under PM’s Smart Cities Mission. A total of 209 landslide locations were identified in using high-resolution linear imaging self-scanning (LISS IV) data. The thematic maps of parameters influencing landslide occurrence were generated using remote sensing and other ancillary data in the GIS environment. The landslide causative parameters used in the study are slope angle, slope aspect, elevation, curvature, topographic wetness index, relative relief, distance from lineaments, land use land cover, and geology. LSMs were prepared using information value (Info Val), and Fuzzy Expert System (FES) models. Info Val is a statistical bivariate method, in which information values were calculated as the ratio of the landslide pixels per factor class (Si/Ni) to the total landslide pixel per parameter (S/N). Using this information values all parameters were reclassified and then summed in GIS to obtain the landslide susceptibility index (LSI) map. The FES method is a machine learning technique based on ‘mean and neighbour’ strategy for the construction of fuzzifier (input) and defuzzifier (output) membership function (MF) structure, and the FR method is used for formulating if-then rules. Two types of membership structures were utilized for membership function Bell-Gaussian (BG) and Trapezoidal-Triangular (TT). LSI for BG and TT were obtained applying membership function and if-then rules in MATLAB. The final LSMs were spatially and statistically validated. The validation results showed that in terms of accuracy, Info Val (83.4%) is better than BG (83.0%) and TT (82.6%), whereas, in terms of spatial distribution, BG is best. Hence, considering both statistical and spatial accuracy, BG is the most accurate one.

Keywords: bivariate statistical techniques, BG and TT membership structure, fuzzy expert system, information value method, machine learning technique

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Authors: Mahesh Kothari, K. D. Gharde

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The ANN and fuzzy logic (FL) models were developed to predict the runoff and sediment yield for catchment of Kal river, India using 21 years (1991 to 2011) rainfall and other hydrological data (evaporation, temperature and streamflow lag by one and two day) and 7 years data for sediment yield modelling. The ANN model performance improved with increasing the input vectors. The fuzzy logic model was performing with R value more than 0.95 during developmental stage and validation stage. The comparatively FL model found to be performing well to ANN in prediction of runoff and sediment yield for Kal river.

Keywords: transferred function, sigmoid, backpropagation, membership function, defuzzification

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Authors: Sujeet Kumar Singh, Shiv Prasad Yadav

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This paper develops an approach for solving intuitionistic fuzzy linear fractional programming (IFLFP) problem where the cost of the objective function, the resources, and the technological coefficients are triangular intuitionistic fuzzy numbers. Here, the IFLFP problem is transformed into an equivalent crisp multi-objective linear fractional programming (MOLFP) problem. By using fuzzy mathematical programming approach the transformed MOLFP problem is reduced into a single objective linear programming (LP) problem. The proposed procedure is illustrated through a numerical example.

Keywords: triangular intuitionistic fuzzy number, linear programming problem, multi objective linear programming problem, fuzzy mathematical programming, membership function

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Authors: Said Baadel, Fadi Thabtah, Joan Lu

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Clustering involves the partitioning of n objects into k clusters. Many clustering algorithms use hard-partitioning techniques where each object is assigned to one cluster. In this paper, we propose an overlapping algorithm MCOKE which allows objects to belong to one or more clusters. The algorithm is different from fuzzy clustering techniques because objects that overlap are assigned a membership value of 1 (one) as opposed to a fuzzy membership degree. The algorithm is also different from other overlapping algorithms that require a similarity threshold to be defined as a priority which can be difficult to determine by novice users.

Keywords: data mining, k-means, MCOKE, overlapping

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Authors: Nur Rokhman

Abstract:

A mathematical function gives relationship between the variables composing the function. Interpolation can be viewed as a process of finding mathematical function which goes through some specified points. There are many interpolation methods, namely: Lagrange method, Newton method, Spline method etc. For some specific condition, such as, big amount of interpolation points, the interpolation function can not be written explicitly. This such function consist of computational steps. The solution of equations involving the interpolation function is a problem of solution of non linear equation. Newton method will not work on the interpolation function, for the derivative of the interpolation function cannot be written explicitly. This paper shows the use of Secton method to determine the numerical solution of the function involving the interpolation function. The experiment shows the fact that Secton method works better than Newton method in finding the root of Lagrange interpolation function.

Keywords: Secton method, interpolation, non linear function, numerical solution

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Authors: Faisel Eltuhami Alzaalik, Omar Imhemed Alramli, Ahmed Mohamed Elaieb

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The IEEE 802.11 defines two modes of MAC, distributed coordination function (DCF) and point coordination function (PCF) mode. The first sub-layer of the MAC is the distributed coordination function (DCF). A contention algorithm is used via DCF to provide access to all traffic. The point coordination function (PCF) is the second sub-layer used to provide contention-free service. PCF is upper DCF and it uses features of DCF to establish guarantee access of its users. Some papers and researches that have been published in this technology were reviewed in this paper, as well as talking briefly about the distributed coordination function (DCF) technology. The simulation of the PCF function have been applied by using a simulation program called network simulator (NS2) and have been found out the throughput of a transmitter system by using this function.

Keywords: DCF, PCF, throughput, NS2

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Authors: Allwell Uwazuruike

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With the withdrawal, in 2016, of 3 African states from the ICC, the discourse took an interesting twist. African states, or at least some of them, had now shown their resolve to part ways with the ICC and, by implication, focus on further enthroning regional control and governance through an improved continental justice system. A range of views has been expressed over the years on the allegations of bias by some African states and the continued membership of the ICC. While there may be a split on the merits of the allegations of bias, academic analysts have generally not opposed African states’ membership of the ICC nor been particularly optimistic about the prospects of an African criminal court. There is also a degree of ambivalence on whether there are positives to be taken from African states’ withdrawal from the ICC. This article examines the recent developments with the ICC and analyses whether these could be viewed from the positive (or, at least, alternative) spectrum of the AU’s spirited march towards regional sovereignty or entirely negatively from the point of view of African Heads-of-State seeking to enthrone an era of authoritarianism and non-accountability.

Keywords: international criminal court, Africa, regionalism, criminal justice

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Authors: Alireza Gholami, Amir H. D. Markazi

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In this paper, an observer-based direct adaptive fuzzy sliding mode (OAFSM) algorithm is proposed. In the proposed algorithm, the zero-input dynamics of the plant could be unknown. The input connection matrix is used to combine the sliding surfaces of individual subsystems, and an adaptive fuzzy algorithm is used to estimate an equivalent sliding mode control input directly. The fuzzy membership functions, which were determined by time consuming try and error processes in previous works, are adjusted by adaptive algorithms. The other advantage of the proposed controller is that the input gain matrix is not limited to be diagonal, i.e. the plant could be over/under actuated provided that controllability and observability are preserved. An observer is constructed to directly estimate the state tracking error, and the nonlinear part of the observer is constructed by an adaptive fuzzy algorithm. The main advantage of the proposed observer is that, the measured outputs is not limited to the first entry of a canonical-form state vector. The closed-loop stability of the proposed method is proved using a Lyapunov-based approach. The proposed method is applied numerically on a multi-link robot manipulator, which verifies the performance of the closed-loop control. Moreover, the performance of the proposed algorithm is compared with some conventional control algorithms.

Keywords: adaptive algorithm, fuzzy systems, membership functions, observer

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Authors: O. P. Popoola, O. A. Alawode, M. O. Olayiwola, A. M. Oladele

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This research work is based on using discriminant analysis to forecast crime rate in Nigeria between 1996 and 2008. The work is interested in how gender (male and female) relates to offences committed against the government, against other properties, disturbance in public places, murder/robbery offences and other offences. The data used was collected from the National Bureau of Statistics (NBS). SPSS, the statistical package was used to analyse the data. Time plot was plotted on all the 29 offences gotten from the raw data. Eigenvalues and Multivariate tests, Wilks’ Lambda, standardized canonical discriminant function coefficients and the predicted classifications were estimated. The research shows that the distribution of the scores from each function is standardized to have a mean O and a standard deviation of 1. The magnitudes of the coefficients indicate how strongly the discriminating variable affects the score. In the predicted group membership, 172 cases that were predicted to commit crime against Government group, 66 were correctly predicted and 106 were incorrectly predicted. After going through the predicted classifications, we found out that most groups numbers that were correctly predicted were less than those that were incorrectly predicted.

Keywords: discriminant analysis, DA, multivariate analysis of variance, MANOVA, canonical correlation, and Wilks’ Lambda

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Authors: Shatha S. Alhily

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The purpose of this research paper is to show all the possible values of the pth power of the integrable function which make the integral means of the derivative of univalent function existing and finite.

Keywords: derivative, integral means, self conformal maps, univalent function

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Authors: Mary Banke Iyabo Omoniyi

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Children in the developing countries have complex social, economic, political and environmental contexts that create a wide range of challenges for school children to surmount as they journey through school from childhood to adolescent. Many of these children have little or no personal resources and social support to confront these challenges. This study employed a descriptive research design of survey type to investigate the psychological sense of school membership and coping skills as they relate to the multidimensional life satisfaction of the school children. The sample consists of 835 school children with the age range of 7-11 years who were randomly selected from twenty schools in Ondo state, Nigeria. The instrument for data collection was a questionnaire consisting of 4 sections A, B, C and D. Section A contained items on the children’s bio-data (Age, School, father’s and mother’s educational qualifications), section B is the Multidimensional Children Life Satisfaction Questionnaire (MCLSQ) with a 20 item Likert type scale. The response format range from Never= 1 to Almost always =4. The (MCLSQ) was designed to provide profile of children satisfaction with important domains of (school, family and friends). Section C is the Psychological Sense of School Membership Questionnaire (PSSMQ) with 18 items having response format ranging from Not at true=1 to completely true=5. While section D is the Self-Report Coping Questionnaire (SRCQ) which has 16 items with response ranging from Never =1 to Always=5. The instrument has a test-retest reliability coefficient of r = 0.87 while the sectional reliabilities for MCLSQ, PSSMQ and SRCQ are 0.86, 0.92 and 0.89 respectively. The results indicated that self-report coping skill was significantly correlated with multidimensional life satisfaction (r=592;p<0.05). However, the correlation between multidimensional life satisfaction and psychological sense of school membership was not significant (r=0.038;p>0.05). The regression analysis indicated that the contribution of mother’s education and father’s education to psychological sense of school member of the children were 0.923, Adjusted R2 is 0.440 and 0.730 and Adjusted R2 is 0.446. The results also indicate that contribution of gender to psychological sense of school for male and female has R= 0.782, Adjusted R2 = 0.478 and R = 0.998, Adjusted R2 i= 0.932 respectively. In conclusion, mother’s education qualification was found to contribute more to children psychological sense of membership and multidimensional life satisfaction than father’s. The girl child was also found to have more sense of belonging to the school setting than boy child. The counselling implications and recommendations among others were geared towards positive emotional gender sensitivity with regards to the male folk. Education stakeholders are also encouraged to make the school environment more conducive and gender friendly.

Keywords: multidimensional life satisfaction, psychological sense of school, coping skills, counselling implications

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Authors: Alexandre Olkoski, Marcelo Curth

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Football in Brazil is considered a passion, being the most popular sport in the country, both by the consumer public and by the means of communication that divulge it individually, when compared with other sports modalities. In the last two decades, the soccer teams have given greater importance to the management, since they understood that the same should be managed as a company, but with peculiarities related to the business. In this sense, Brazilian soccer clubs started to make bigger investments for the adhesion of fans in their social frames, allowing a greater need of understanding about the profile of this group of fans/clients. Thus, this work aims to understand the reasons that cause the fans to join the club and identify variables present in the process of intention to join the club. For that, a qualitative exploratory research was conducted, in which thirty-one membership of a soccer club from southern Brazil were interviewed. Based on the interviews, five categories were classified as emotional aspects (passion and love), cognitive aspects (easy access to the stadium and promotional values in tickets), external influences (family and friends), situational aspects (club moment) and aspects related to the event (engagement by modality). As results found in the analysis, it can be highlighted that the motivation of the majority of the respondents to become a member of the analyzed club, is related to the emotional aspects, such as passion and love. Thus, it is perceived that sport, in the case of soccer, generates in the involved ones (fans and leaders) different manifestations, suggesting that the management of this type of business has great complexity and should not be observed only by the spectrum of the club like a business.

Keywords: consumer behavior, marketing, membership, soccer

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Authors: Manju Pandey, Nilay Khare, S. C. Shrivastava

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This paper is the final in a series, which has defined two new classes of aggregation operators for triangular and trapezoidal fuzzy numbers based on the geometrical characteristics of their fuzzy membership functions. In the present paper, a new aggregation operator for trapezoidal fuzzy numbers has been defined. The new operator is based on the geometric mean of the membership lines to the left and right of the maximum possibility interval. The operator is defined and the analytical relationships have been derived. Computation of the aggregate is demonstrated with a numerical example. Corresponding arithmetic and geometric aggregates as well as results from the recent work of the authors on TrFN aggregates have also been computed.

Keywords: LR fuzzy number, interval fuzzy number, triangular fuzzy number, trapezoidal fuzzy number, apex angle, left apex angle, right apex angle, aggregation operator, arithmetic and geometric mean

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Authors: F. U. Rahman, R. Q. Zhang

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This work aims to develop a systematic numerical technique which can be easily extended to many-body problem. The Lippmann Schwinger equation (integral form of the Schrodinger wave equation) is solved for the nonrelativistic radial wave of hydrogen atom using iterative integration scheme. As the unknown wave function appears on both sides of the Lippmann Schwinger equation, therefore an approximate wave function is used in order to solve the equation. The Green’s function is obtained by the method of Laplace transform for the radial wave equation with excluded potential term. Using the Lippmann Schwinger equation, the product of approximate wave function, the Green’s function and the potential term is integrated iteratively. Finally, the wave function is normalized and plotted against the standard radial wave for comparison. The outcome wave function converges to the standard wave function with the increasing number of iteration. Results are verified for the first fifteen states of hydrogen atom. The method is efficient and consistent and can be applied to complex systems in future.

Keywords: Green’s function, hydrogen atom, Lippmann Schwinger equation, radial wave

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Authors: Zhongzhi Hu, Jie Shen, Jiqiang Wang

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A high precision aeroengine model is needed when developing the engine control system. Compared with other main components, the axial compressor is the most challenging component to simulate. In this paper, a compressor map optimizing tool based on the introduction of a modifiable β function is developed for FWorks (FADEC Works). Three parameters (d density, f fitting coefficient, k₀ slope of the line β=0) are introduced to the β function to make it modifiable. The comparison of the traditional β function and the modifiable β function is carried out for a certain type of compressor. The interpolation errors show that both methods meet the modeling requirements, while the modifiable β function can predict compressor performance more accurately for some areas of the compressor map where the users are interested in.

Keywords: beta function, compressor map, interpolation error, map optimization tool

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Authors: Liang-Hsuan Chen, Chia-Jung Chang

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Fuzzy random variables have been introduced as an imprecise concept of numeric values for characterizing the imprecise knowledge. The descriptive parameters can be used to describe the primary features of a set of fuzzy random observations. In fuzzy environments, the expected values are usually represented as fuzzy-valued, interval-valued or numeric-valued descriptive parameters using various metrics. Instead of the concept of area metric that is usually adopted in the relevant studies, the numeric expected value is proposed by the concept of distance metric in this study based on two characters (fuzziness and randomness) of FRVs. Comparing with the existing measures, although the results show that the proposed numeric expected value is same with those using the different metric, if only triangular membership functions are used. However, the proposed approach has the advantages of intuitiveness and computational efficiency, when the membership functions are not triangular types. An example with three datasets is provided for verifying the proposed approach.

Keywords: fuzzy random variables, distance measure, expected value, descriptive parameters

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Authors: Slobodan B. Tričković

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We present the results concerned with trigonometric series that include sine and cosine functions with a parameter appearing in the denominator. We derive two types of closed-form formulas for trigonometric series. At first, for some integer values, as we know that Riemann’s ζ function and Dirichlet η, λ equal zero at negative even integers, whereas Dirichlet’s β function equals zero at negative odd integers, after a certain number of members, the rest of the series vanishes. Thus, a trigonometric series becomes a polynomial with coefficients involving Riemann’s ζ function and Dirichlet η, λ, β functions. On the other hand, in some cases, one cannot immediately replace the parameter with any positive integer because we shall encounter singularities. So it is necessary to take a limit, so in the process, we apply L’Hospital’s rule and, after a series of rearrangements, we bring a trigonometric series to a form suitable for the application of Choi-Srivastava’s theorem dealing with Hurwitz’s zeta function and Harmonic numbers. In this way, we express a trigonometric series as a polynomial over Hurwitz’s zeta function derivative.

Keywords: Dirichlet eta lambda beta functions, Riemann's zeta function, Hurwitz zeta function, Harmonic numbers

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