Search results for: Generalized module homomorphism
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 721

Search results for: Generalized module homomorphism

691 On Fourier Type Integral Transform for a Class of Generalized Quotients

Authors: A. S. Issa, S. K. Q. AL-Omari

Abstract:

In this paper, we investigate certain spaces of generalized functions for the Fourier and Fourier type integral transforms. We discuss convolution theorems and establish certain spaces of distributions for the considered integrals. The new Fourier type integral is well-defined, linear, one-to-one and continuous with respect to certain types of convergences. Many properties and an inverse problem are also discussed in some details.

Keywords: Fourier type integral, Fourier integral, generalized quotient, Boehmian, distribution.

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690 A Unified Framework for a Robust Conflict-Free Robot Navigation

Authors: S. Veera Ragavan, V. Ganapathy

Abstract:

Many environment specific methods and systems for Robot Navigation exist. However vast strides in the evolution of navigation technologies and system techniques create the need for a general unified framework that is scalable, modular and dynamic. In this paper a Unified Framework for a Robust Conflict-free Robot Navigation System that can be used for either a structured or unstructured and indoor or outdoor environments has been proposed. The fundamental design aspects and implementation issues encountered during the development of the module are discussed. The results of the deployment of three major peripheral modules of the framework namely the GSM based communication module, GIS Module and GPS module are reported in this paper.

Keywords: Localization, Sensor Fusion, Mapping, GIS, GPS, and Autonomous Mobile Robot Navigation.

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689 The Induced Generalized Hybrid Averaging Operator and its Application in Financial Decision Making

Authors: José M. Merigó, Montserrat Casanovas

Abstract:

We present the induced generalized hybrid averaging (IGHA) operator. It is a new aggregation operator that generalizes the hybrid averaging (HA) by using generalized means and order inducing variables. With this formulation, we get a wide range of mean operators such as the induced HA (IHA), the induced hybrid quadratic averaging (IHQA), the HA, etc. The ordered weighted averaging (OWA) operator and the weighted average (WA) are included as special cases of the HA operator. Therefore, with this generalization we can obtain a wide range of aggregation operators such as the induced generalized OWA (IGOWA), the generalized OWA (GOWA), etc. We further generalize the IGHA operator by using quasi-arithmetic means. Then, we get the Quasi-IHA operator. Finally, we also develop an illustrative example of the new approach in a financial decision making problem. The main advantage of the IGHA is that it gives a more complete view of the decision problem to the decision maker because it considers a wide range of situations depending on the operator used.

Keywords: Decision making, Aggregation operators, OWA operator, Generalized means, Selection of investments.

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688 Enhancing the Performance of a Photovoltaic Module Using Different Cooling Methods

Authors: Ahmed Amine Hachicha, Chaouki Ghenai, Abdul Kadir Hamid

Abstract:

Temperature effect on the performance of a photovoltaic module is one of the main concerns that face this renewable energy, especially in hot arid region, e.g. United Arab Emirates. Overheating of the PV modules reduces the open circuit voltage and the efficiency of the modules dramatically. In this work, water-cooling is developed to enhance the performance of PV modules. Different scenarios are tested under UAE weather conditions: front, back and double cooling. A spraying system is used for the front cooling whether a direct contact water system is used for the back cooling. The experimental results are compared to non-cooling module and the performance of the PV module is determined for different situations. The experimental results show that the front cooling is more effective than the back cooling and may decrease the temperature of the PV module significantly. 

Keywords: PV cooling, solar energy, cooling methods, electrical efficiency, temperature effect.

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687 Project and Module Based Teaching and Learning

Authors: Jingyu Hou

Abstract:

This paper proposes a new teaching and learning approach-project and module based teaching and learning (PMBTL). The PMBTL approach incorporates the merits of project/problem based and module based learning methods, and overcomes the limitations of these methods. The correlation between teaching, learning, practice and assessment is emphasized in this approach, and new methods have been proposed accordingly. The distinct features of these new methods differentiate the PMBTL approach from conventional teaching approaches. Evaluation of this approach on practical teaching and learning activities demonstrates the effectiveness and stability of the approach in improving the performance and quality of teaching and learning. The approach proposed in this paper is also intuitive to the design of other teaching units. 

Keywords: Computer science education, project and module based, software engineering.

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686 Behrens-Fisher Problem with One Variance Unknown

Authors: Sa-aat Niwitpong, Rada Somkhuean, Suparat Niwitpong

Abstract:

This paper presents the generalized p-values for testing the Behrens-Fisher problem when one variance is unknown. We also derive a closed form expression of the upper bound of the proposed generalized p-value.

Keywords: Generalized p-value, hypothesis testing, upper bound.

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685 Evolving Knowledge Extraction from Online Resources

Authors: Zhibo Xiao, Tharini Nayanika de Silva, Kezhi Mao

Abstract:

In this paper, we present an evolving knowledge extraction system named AKEOS (Automatic Knowledge Extraction from Online Sources). AKEOS consists of two modules, including a one-time learning module and an evolving learning module. The one-time learning module takes in user input query, and automatically harvests knowledge from online unstructured resources in an unsupervised way. The output of the one-time learning is a structured vector representing the harvested knowledge. The evolving learning module automatically schedules and performs repeated one-time learning to extract the newest information and track the development of an event. In addition, the evolving learning module summarizes the knowledge learned at different time points to produce a final knowledge vector about the event. With the evolving learning, we are able to visualize the key information of the event, discover the trends, and track the development of an event.

Keywords: Evolving learning, knowledge extraction, knowledge graph, text mining.

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684 Developing Islamic Module Project for Preschool Teachers Using Modified Delphi Technique

Authors: Mazeni Ismail, Nurul Aliah, Hasmadi Hassan

Abstract:

The purpose of this study is to gather the consensus of experts regarding the use of moral guidance amongst preschool teachers vis-a-vis the Islamic Project module (I-Project Module). This I-Project Module seeks to provide pertinent data on the assimilation of noble values in subject-matter teaching. To obtain consensus for the various components of the module, the Modified Delphi technique was used to develop the module. 12 subject experts from various educational fields of Islamic education, early childhood education, counselling and language fully participated in the development of this module. The Modified Delphi technique was administered in two mean cycles. The standard deviation value derived from questionnaires completed by the participating panel of experts provided the value of expert consensus reached. This was subsequently analyzed using SPSS version 22. Findings revealed that the panel of experts reached a discernible degree of agreement on five topics outlined in the module, viz; content (mean value 3.36), teaching strategy (mean value 3.28), programme duration (mean value 3.0), staff involved and attention-grabbing strategy of target group participating in the value program (mean value 3.5), and strategy to attract attention of target group to utilize i-project (mean value 3.0). With regard to the strategy to attract the attention of the target group, the experts proposed for creative activities to be added in order to enhance teachers’ creativity.

Keywords: Islamic project, modified Delphi technique, project approach, teacher moral guidance.

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683 Development and Acceptance of a Proposed Module for Enhancing the Reading and Writing Skills in Baybayin: The Traditional Writing System in the Philippines

Authors: Maria Venus G. Solares

Abstract:

The ancient Filipinos had their own spelling or alphabet that differed from the modern Roman alphabet brought by the Spaniards. It consists of seventeen letters, three vowels, and fourteen consonants and is called Baybayin. The Baybayin, a traditional writing system, is composed of characters that represent syllables. A proposal in the Philippine Congress to declare Baybayin as the national writing system inspired this study. The main objective of this study was to develop and assess the proposed module for enhancing the reading and writing skills in Baybayin of the students. The aim was to ensure the acceptability of the Baybayin using the proposed module and to meet the needs of students in developing their ability to read and write Baybayin through the module. A quasi-experimental research design was used in this study.  The data were collected through the initial and final analysis of the students of Adamson University's ABM 1102 using convenient sampling techniques. Based on statistical analysis of data using weighted mean, standard deviation, and paired t-tests, the proposed module helped improve the students' literacy skills, and the response exercises in the proposed module changed the acceptability of the Baybayin in their minds. The study showed that there was an important difference in the scores of students before and after the use of the module. The students' response to the assessment of their reading and writing skills on Baybayin was highly acceptable. This study will help to develop the reading and writing skills of the students in Baybayin and to teach the Baybayin in response to the revival of a part of Philippine culture that has been long forgotten.

Keywords: Baybayin, proposed module, ancient writing, acceptability.

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682 Adomian Method for Second-order Fuzzy Differential Equation

Authors: Lei Wang, Sizong Guo

Abstract:

In this paper, we study the numerical method for solving second-order fuzzy differential equations using Adomian method under strongly generalized differentiability. And, we present an example with initial condition having four different solutions to illustrate the efficiency of the proposed method under strongly generalized differentiability.

Keywords: Fuzzy-valued function, fuzzy initial value problem, strongly generalized differentiability, adomian decomposition method.

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681 Decomposition of Homeomorphism on Topological Spaces

Authors: Ahmet Z. Ozcelik, Serkan Narli

Abstract:

In this study, two new classes of generalized homeomorphisms are introduced and shown that one of these classes has a group structure. Moreover, some properties of these two homeomorphisms are obtained.

Keywords: Generalized closed set, homeomorphism, gsghomeomorphism, sgs-homeomorphism.

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680 Low Power Circuit Architecture of AES Crypto Module for Wireless Sensor Network

Authors: MooSeop Kim, Juhan Kim, Yongje Choi

Abstract:

Recently, much research has been conducted for security for wireless sensor networks and ubiquitous computing. Security issues such as authentication and data integrity are major requirements to construct sensor network systems. Advanced Encryption Standard (AES) is considered as one of candidate algorithms for data encryption in wireless sensor networks. In this paper, we will present the hardware architecture to implement low power AES crypto module. Our low power AES crypto module has optimized architecture of data encryption unit and key schedule unit which could be applicable to wireless sensor networks. We also details low power design methods used to design our low power AES crypto module.

Keywords: Algorithm, Low Power Crypto Circuit, AES, Security.

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679 Complex-Valued Neural Network in Signal Processing: A Study on the Effectiveness of Complex Valued Generalized Mean Neuron Model

Authors: Anupama Pande, Ashok Kumar Thakur, Swapnoneel Roy

Abstract:

A complex valued neural network is a neural network which consists of complex valued input and/or weights and/or thresholds and/or activation functions. Complex-valued neural networks have been widening the scope of applications not only in electronics and informatics, but also in social systems. One of the most important applications of the complex valued neural network is in signal processing. In Neural networks, generalized mean neuron model (GMN) is often discussed and studied. The GMN includes a new aggregation function based on the concept of generalized mean of all the inputs to the neuron. This paper aims to present exhaustive results of using Generalized Mean Neuron model in a complex-valued neural network model that uses the back-propagation algorithm (called -Complex-BP-) for learning. Our experiments results demonstrate the effectiveness of a Generalized Mean Neuron Model in a complex plane for signal processing over a real valued neural network. We have studied and stated various observations like effect of learning rates, ranges of the initial weights randomly selected, error functions used and number of iterations for the convergence of error required on a Generalized Mean neural network model. Some inherent properties of this complex back propagation algorithm are also studied and discussed.

Keywords: Complex valued neural network, Generalized Meanneuron model, Signal processing.

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678 Upper Bound of the Generalize p-Value for the Behrens-Fisher Problem with a Known Ratio of Variances

Authors: Rada Somkhuean, Suparat Niwitpong, Sa-aat Niwitpong

Abstract:

This paper presents the generalized p-values for testing the Behrens-Fisher problem when a ratio of variance is known. We also derive a closed form expression of the upper bound of the proposed generalized p-value.

Keywords: Generalized p-value, hypothesis testing, ratio of variances, upper bound.

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677 Mathematical Modelling for Separation of Binary Aqueous Solution using Hollow Fiber Reverse Osmosis Module

Authors: Anil Kumar, S. Deswal

Abstract:

The mathematical equation for Separation of the binary aqueous solution is developed by using the Spiegler- Kedem theory. The characteristics of a B-9 hollow fibre module of Du Pont are determined by using these equations and their results are compared with the experimental results of Ohya et al. The agreement between these results is found to be excellent.

Keywords: Binary aqueous solution, modeling, reverse osmosis module, Spiegler-Kedem theory.

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676 Solving of the Fourth Order Differential Equations with the Neumann Problem

Authors: Marziyeh Halimi, Roushanak Lotfikar, Simin Mansouri Borojeni

Abstract:

In this paper we considered the Neumann problem for the fourth order differential equation. First we define the weighted Sobolev space 2 Wα and generalized solution for this equation. Then we consider the existence and uniqueness of the generalized solution, as well as give the description of the spectrum and of the domain of definition of the corresponding operator.

Keywords: Neumann problem, weighted Sobolev spaces, generalized solution, spectrum of linear operators.2000 mathematic subject classification: 34A05, 34A30.

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675 Bio-Inspired Generalized Global Shape Approach for Writer Identification

Authors: Azah Kamilah Muda, Siti Mariyam Shamsuddin, Maslina Darus

Abstract:

Writer identification is one of the areas in pattern recognition that attract many researchers to work in, particularly in forensic and biometric application, where the writing style can be used as biometric features for authenticating an identity. The challenging task in writer identification is the extraction of unique features, in which the individualistic of such handwriting styles can be adopted into bio-inspired generalized global shape for writer identification. In this paper, the feasibility of generalized global shape concept of complimentary binding in Artificial Immune System (AIS) for writer identification is explored. An experiment based on the proposed framework has been conducted to proof the validity and feasibility of the proposed approach for off-line writer identification.

Keywords: Writer identification, generalized global shape, individualistic, pattern recognition.

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674 The Extraction and Stripping of Hg (II) from Produced Water via Hollow Fiber Contactor

Authors: Dolapop Sribudda, Ura Pancharoen

Abstract:

The separation of Hg (II) from produced water by hollow fiber contactors (HFC) was investigation. This system included of two hollow fiber modules in the series connecting. The first module used for the extraction reaction and the second module for stripping reaction. Aliquat336 extractant was fed from the organic reservoirs into the shell side of the first hollow fiber module and continuous to the shell side of the second module. The organic liquid was continuously feed recirculate and back to the reservoirs. The feed solution was pumped into the lumen (tube side) of the first hollow fiber module. Simultaneously, the stripping solution was pumped in the same way in tube side of the second module. The feed and stripping solution was fed which had a countercurrent flow. Samples were kept in the outlet of feed and stripping solution at 1 hour and characterized concentration of Hg (II) by Inductively Couple Plasma Atomic Emission Spectroscopy (ICP-AES). Feed solution was produced water from natural gulf of Thailand. The extractant was Aliquat336 dissolved in kerosene diluent. Stripping solution used was nitric acid (HNO3) and thiourea (NH2CSNH2). The effect of carrier concentration and type of stripping solution were investigated. Results showed that the best condition were 10 % (v/v) Aliquat336 and 1.0 M NH2CSNH2. At the optimum condition, the extraction and stripping of Hg (II) were 98% and 44.2%, respectively.

Keywords: Hg (II), hollow fiber contactor, produced water, wastewater treatment.

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673 The BGMRES Method for Generalized Sylvester Matrix Equation AXB − X = C and Preconditioning

Authors: Azita Tajaddini, Ramleh Shamsi

Abstract:

In this paper, we present the block generalized minimal residual (BGMRES) method in order to solve the generalized Sylvester matrix equation. However, this method may not be converged in some problems. We construct a polynomial preconditioner based on BGMRES which shows why polynomial preconditioner is superior to some block solvers. Finally, numerical experiments report the effectiveness of this method.

Keywords: Linear matrix equation, Block GMRES, matrix Krylov subspace, polynomial preconditioner.

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672 Preliminary Tests on the Buffer Tank for the Vented Liquid Nitrogen Flow of an SRF Module

Authors: Ming-Hsun Tsai, Ming-Chyuan Lin, Fu-Tsai Chung, Ling-Jhen Chen, Yu-Hang Lin, Meng-Shu Yeh, Lee-Long Han

Abstract:

Since 2005, an SRF module of CESR type serves as the accelerating cavity at the Taiwan Light Source in the National Synchrotron Radiation Research Center. A 500-MHz niobium cavity is immersed in liquid helium inside this SRF module. To reduce heat load, the liquid helium vessel is thermally shielded by liquid-nitrogen-cooled copper layer, and the beam chambers are also anchored with pipes of the liquid nitrogen flow in middle of the liquid helium vessel and the vacuum vessel. A strong correlation of the movement of the cavity-s frequency tuner with the temperature variation of parts cooled with liquid nitrogen was observed. A previous study on a spare SRF module with the niobium cavity cooled by liquid nitrogen instead of liquid helium, satisfactory suppression of the thermal oscillation was achieved by attaching a temporary buffer tank for the vented shielding nitrogen flow from the SRF module. In this study, a home-made buffer tank is designed and integrated to the spare SRF module with cavity cooled by liquid helium. Design, construction, integration, and preliminary test results of this buffer tank are presented.

Keywords: Cryogenics, flow control, oscillation.

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671 Crank-Nicolson Difference Scheme for the Generalized Rosenau-Burgers Equation

Authors: Kelong Zheng, Jinsong Hu,

Abstract:

In this paper, numerical solution for the generalized Rosenau-Burgers equation is considered and Crank-Nicolson finite difference scheme is proposed. Existence of the solutions for the difference scheme has been shown. Stability, convergence and priori error estimate of the scheme are proved. Numerical results demonstrate that the scheme is efficient and reliable.

Keywords: Generalized Rosenau-Burgers equation, difference scheme, stability, convergence.

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670 A Modified Inexact Uzawa Algorithm for Generalized Saddle Point Problems

Authors: Shu-Xin Miao

Abstract:

In this note, we discuss the convergence behavior of a modified inexact Uzawa algorithm for solving generalized saddle point problems, which is an extension of the result obtained in a recent paper [Z.H. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math., 46 (2003) 157-171].

Keywords: Saddle point problem, inexact Uzawa algorithm, convergence behavior.

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669 2D Numerical Analysis of Sao Paulo Tunnel

Authors: A.H. Akhaveissy

Abstract:

Nonlinear finite element method and Serendipity eight nodes element are used for determining of ground surface settlement due to tunneling. Linear element with elastic behavior is used for modeling of lining. Modified Generalized plasticity model with nonassociated flow rule is applied for analysis of a tunnel in Sao Paulo – Brazil. The tunnel had analyzed by Lades- model with 16 parameters. In this work modified Generalized Plasticity is used with 10 parameters, also Mohr-Coulomb model is used to analysis the tunnel. The results show good agreement with observed results of field data by modified Generalized Plasticity model than other models. The obtained result by Mohr-Coulomb model shows less settlement than other model due to excavation.

Keywords: Non-associated flow rule, Generalized plasticity, tunnel excavation, Excavation method.

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668 Merging and Comparing Ontologies Generically

Authors: Xiuzhan Guo, Arthur Berrill, Ajinkya Kulkarni, Kostya Belezko, Min Luo

Abstract:

Ontology operations, e.g., aligning and merging, were studied and implemented extensively in different settings, such as, categorical operations, relation algebras, typed graph grammars, with different concerns. However, aligning and merging operations in the settings share some generic properties, e.g., idempotence, commutativity, associativity, and representativity, which are defined on an ontology merging system, given by a nonempty set of the ontologies concerned, a binary relation on the set of the ontologies modeling ontology aligning, and a partial binary operation on the set of the ontologies modeling ontology merging. Given an ontology repository, a finite subset of the set of the ontologies, its merging closure is the smallest subset of the set of the ontologies, which contains the repository and is closed with respect to merging. If idempotence, commutativity, associativity, and representativity properties are satisfied, then both the set of the ontologies and the merging closure of the ontology repository are partially ordered naturally by merging, the merging closure of the ontology repository is finite and can be computed, compared, and sorted efficiently, including sorting, selecting, and querying some specific elements, e.g., maximal ontologies and minimal ontologies. An ontology Valignment pair is a pair of ontology homomorphisms with a common domain. We also show that the ontology merging system, given by ontology V-alignment pairs and pushouts, satisfies idempotence, commutativity, associativity, and representativity properties so that the merging system is partially ordered and the merging closure of a given repository with respect to pushouts can be computed efficiently.

Keywords: Ontology aligning, ontology merging, merging system, poset, merging closure, ontology V-alignment pair, ontology homomorphism, ontology V-alignment pair homomorphism, pushout.

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667 Comparative Study of Intuitionistic and Generalized Neutrosophic Soft Sets

Authors: Debabrata Mandal

Abstract:

The aim of this paper is to define several operations such as Intersection, Union, OR, AND operations of intuitionistic (resp. generalized) neutrosophic soft sets in the sense of Maji and compare these with intuitionistic (resp. generalized) neutrosophic soft sets in the sense of Said et al via examples. At the end of the paper, a new concept - extension is introduced, which can be used to refine our choices in case of decision making.

Keywords: AND, OR, Union, Intersection, Extension, Decision making.

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666 An Attribute Based Access Control Model with POL Module for Dynamically Granting and Revoking Authorizations

Authors: Gang Liu, Huimin Song, Can Wang, Runnan Zhang, Lu Fang

Abstract:

Currently, resource sharing and system security are critical issues. This paper proposes a POL module composed of PRIV ILEGE attribute (PA), obligation and log which improves attribute based access control (ABAC) model in dynamically granting authorizations and revoking authorizations. The following describes the new model termed PABAC in terms of the POL module structure, attribute definitions, policy formulation and authorization architecture, which demonstrate the advantages of it. The POL module addresses the problems which are not predicted before and not described by access control policy. It can be one of the subject attributes or resource attributes according to the practical application, which enhances the flexibility of the model compared with ABAC. A scenario that illustrates how this model is applied to the real world is provided.

Keywords: Access control, attribute based access control, granting authorizations, privilege, revoking authorizations, system security.

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665 Fuzzy Ideals in Near-subtraction Semigroups

Authors: D.R Prince Williams

Abstract:

In this paper,we introduce a notion of fuzzy ideals in near-subtraction semigroups and study their related properties.

Keywords: subtraction algebra, subtraction semigroup, an ideal, near-subtraction semigroup, fuzzy level set, fuzzy ideal, fuzzy homomorphism.

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664 A New Approach For Ranking Of Generalized Trapezoidal Fuzzy Numbers

Authors: Amit Kumar, Pushpinder Singh, Parampreet Kaur, Amarpreet Kaur

Abstract:

Ranking of fuzzy numbers play an important role in decision making, optimization, forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. In this paper, with the help of several counter examples it is proved that ranking method proposed by Chen and Chen (Expert Systems with Applications 36 (2009) 6833-6842) is incorrect. The main aim of this paper is to propose a new approach for the ranking of generalized trapezoidal fuzzy numbers. The main advantage of the proposed approach is that the proposed approach provide the correct ordering of generalized and normal trapezoidal fuzzy numbers and also the proposed approach is very simple and easy to apply in the real life problems. It is shown that proposed ranking function satisfies all the reasonable properties of fuzzy quantities proposed by Wang and Kerre (Fuzzy Sets and Systems 118 (2001) 375-385).

Keywords: Ranking function, Generalized trapezoidal fuzzy numbers

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663 Evolutionary Techniques Based Combined Artificial Neural Networks for Peak Load Forecasting

Authors: P. Subbaraj, V. Rajasekaran

Abstract:

This paper presents a new approach using Combined Artificial Neural Network (CANN) module for daily peak load forecasting. Five different computational techniques –Constrained method, Unconstrained method, Evolutionary Programming (EP), Particle Swarm Optimization (PSO), and Genetic Algorithm (GA) – have been used to identify the CANN module for peak load forecasting. In this paper, a set of neural networks has been trained with different architecture and training parameters. The networks are trained and tested for the actual load data of Chennai city (India). A set of better trained conventional ANNs are selected to develop a CANN module using different algorithms instead of using one best conventional ANN. Obtained results using CANN module confirm its validity.

Keywords: Combined ANN, Evolutionary Programming, Particle Swarm Optimization, Genetic Algorithm and Peak load forecasting.

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662 A Note on the Convergence of the Generalized AOR Iterative Method for Linear Systems

Authors: Zhong-xi Gao, Hou-biao Li

Abstract:

Recently, some convergent results of the generalized AOR iterative (GAOR) method for solving linear systems with strictly diagonally dominant matrices are presented in [Darvishi, M.T., Hessari, P.: On convergence of the generalized AOR method for linear systems with diagonally dominant cofficient matrices. Appl. Math. Comput. 176, 128-133 (2006)] and [Tian, G.X., Huang, T.Z., Cui, S.Y.: Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant cofficient matrices. J. Comp. Appl. Math. 213, 240-247 (2008)]. In this paper, we give the convergence of the GAOR method for linear systems with strictly doubly diagonally dominant matrix, which improves these corresponding results.

Keywords: Diagonally dominant matrix, GAOR method, Linear system, Convergence

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