## Journal of Combinatorics, Information & System Sciences : (A Quarterly International Scientific Journal)

**Published in Association with Forum for Interdisciplinary Mathematics**

**Current Volume: **46 (2021 )

**ISSN: **0250-9628

**e-ISSN: **0976-3473

**Periodicity: **Quarterly

**Month(s) of Publication: **March, June, September & December

**Subject: ** Mathematics

**DOI: **10.32381/JCISS

**Online Access is free for all life members of JCISS.**

**Journal of Combinatorics & System Sciences (JCISS)** publishes significant original papers and survey articles on important advances, carrying theoretical and practical results of Mathematical orientation in any one of the following major areas:

**Combinatorics** (e.g., Graph Theory, finite Geometries, Combinatorial Geometries, Combinatorial Problems in Number Theory and Abstract Algebra, Coding Theory, Block/Factorial Designs, Search and Combinatorial Problems in Statistical Inference).

**Information Theory** (e.g., The Shannon Theory, Rate-Distortion Theory, Theory of Error-Correcting Codes, Detection and Estimation, Communication, Sampling Theorems, and Stochastic Processes, Pattern Recognition, Measures of Information and their Applications in Physical Sciences, Social Sciences, Commerce and Business Management, Humanities, and Statistics)

**Mathematical System Theory** (e.g., Topological Dynamics, Theory and Dynamical Poly systems, General System Theory, Formal System Theory, Reliability Theory and Practice, Mathematical Theory of Automation and Algorithms, Algebraic System Theory, Ergodic Theory, and Mathematical Linguistics)

**Computational Intelligence Techniques** (e.g., Fuzzy set theory and its generalizations, Soft set theory, Computing with Words, Neutrosophic Sets, and Systems, Interval Analysis, Rough Sets, Algorithmic Developments and Applications in Real-Life Problems.

Journal of Combinatorics & System Sciences is published four times a year. As a result, the journal serves as a forum for all researchers and practitioners engaged in research and development in these fields.

Journal of Combinatorics & System Sciences also presents refereed surveys, book reviews, tutorial articles, and comments on already published papers in JCISS. Special issues devoted exclusively to any important subarea of the journal may also be published.

**Zentralblatt Math
Mathematical Reviews
EBSCO **

**Editor -in -chief**

Viswanath Ramakrishna

Dept. of Mathematical Sciences,

University of Texas at Dallas, 2601

N. Floyd Rd, Richardson,

TX 75080, USA.

**Associate Editor**

Raj Kumar Verma

University of Chile

8330015 Santiago, Chile

**Managing Editor**

Deepankar Sharma

Dr. K.N.Modi Institute of Engg.& Tech. Modi Nagar, U.P.

**Editorial Board**

Toby Berger

Dept. of Electrical anbd Computer Engineering

University of Virginia

Charlottesville, VA USA

Peter Horak

Dept. of Maths

University of Washington,

Tacoma, USA

Ram P. Sharma

Himachal Pradesh University

Shimla, HP-171005, India

Pascale Charpihn

Institut National De Rercherche en

Informatique en Automatique

Paris, France

Michail Ch. Klin

Deptt. of Maths and Computer Science

Ben Gurion University of Nagev,

PO Box 653, Beez- Sheva 84105, Israel

K. P. Shum

Institute of Mathematics

Yannan University,

Kunmimg, Yunnan 650091

P. R. China

D. K. Ray-Chaudhuri

Dept. of Mathematics

The Ohio State University,

Columbus, OH, USA

Jose M. Merigo Lindahi

Faculty of Engineering & Information Technology

University of Technology

Sydney, Australia

Deepa Sinha

Department of Mathematics

South Asian University,

Akbar Bhavan, Chanakyapuri

New Delhi, India

Charles J. Colbourn

School of Computing,

Information & Decision System Engineering

Arizona State University

Tempe, USA

W. Mader

Instut Fur Mathematica Dept of Maths

Universitat Hannover,

3000, Hannover, Germany

Henk Van Tilborg

Technische Universiteit Eindhoven

5600 MB, Eindhoven

The Netherlands

B. K. Dass

Former Professor

University of Delhi,

Delhi-110007, India

Takis Papaioannou

Dept of Maths, Prob- Stat & O. R.

University of Ioannian

45110, Ioannian, Greece

Junzo Watada

Dept of Computer & Information Sciences

University Technology PETRONS

32610, Seri Iskandar

Perek Darul Ridzum, Malaysia

Michael J. Ferrara

University of Colorado Denver,

Co-80204, USA

C. R. Rao

Center for Statistical Analysis

Pennsylvania State University,

University Park, PA-16802, USA

D.J.A. Welsh

Mathematical Institute

Oxford University,

OX1 4JD, England

Robert G. Gallager

Dept. of Elect. Engineering

M.I.T., Cambridge

Mass 021139, USA

Alexander Rosa

Dept. of Maths & Statistics

McMaster University,

Hamilton, Ontario

L85 4K1, Canada

Dong Ye

Middle Tennessee State University,

Murfreesbovo, TN-37132,USA

Ron Gould

Goodrich C. White Professor

Dept. of Mathematics & Computer Science

Emory University, Atlanta

GA 30332, USA

Alexander Schrijver

Stichting Mathematisch Centrum

Kruislaara 413, 1098 SJ Amsterdam

The Netherlands

Moshe Zakai

Faculty of Elect. Engineering

Technion- israel Inst of Tech.

Halfa, Israel

### Volume 46 Issue 1-4 , (Jan-2021 to Dec-2021)

**System of Caputo Fractional Differential Equations with Applications to Predator and Prey Model**

**By 1:**
Govinda Pageni ,
Aghalaya S. Vatsala

**Page No **: 1-18

**Abstract**

In this research article, we provide a methodology to solve the three systems of qth order linear Caputo fractional differential equations, where 0 < q < 1. Since the Caputo derivative is in the convolution form, we can apply the Laplace transform technique. The solution of the two linear system can be used as a tool to study the stability of the equilibrium solution of the Lotka-Volterra predator-prey model. We have referenced three system SIR model in this work. Due to the global nature of the Caputo derivative, the solution obtained is closer to the real data than the integer derivative.

**Authors :**

**Aghalaya S. Vatsala :** University of Louisiana at Lafayette, Lafayette, LA-70504

**Govinda Pageni : **University of Louisiana at Lafayette, Lafayette, LA-70504

**DOI : https://doi.org/10.32381/JCISS.2021.46.1-4.1**

#### Price: 251

**Bilevel Optimization and Conjectural Equilibrium: Theoretical Results and Numerical Algorithms (An Invited Tutorial Paper)**

**By 1:**
Nataliya Kalashnykova ,
Viacheslav Kalashnikov ,
José Guadalupe Flores- Muñiz

**Page No **: 19-113

**Abstract**

We study a mixed oligopoly model, including a nonprivate firm that maximizes the convex combination of domestic social surplus and net profit, under the concepts of consistent conjectural variations (CCVE), Cournot–Nash, and perfect competition equilibria and analyze the behavior of these equilibria. We compare the results to develop an optimality criterion for a parameter, decided by the nonprivate firm, known as socialization level. To work with the CCVE, we need to define the concept of consistency; hence, we also consider a classic oligopoly, which only includes private firms that maximize net profit, and reformulated it as a non-cooperative game, named meta-game, where the players (the private firms) select a conjecture as their strategies; then, we prove that the Nash equilibrium for the meta-game implies the CCVE. After that, due to its similarities with the meta-game, we study the Tolls Optimization Problem (TOP). Taking advantage of the sensitivity analysis for convex quadratic optimization and the filled function method; then, we present an efficient algorithm to solve the TOP. Finally, we apply these results to a financial model in which a group of sectors buy, sell, or exchange a set of instruments; each sector faces some uncertainty when selecting its assets and liabilities to optimize its portfolio structure. The problem is modeled using the concepts of conjectural variations and meta-game and we present results concerning the CCVE.

** Authors :
Nataliya Kalashnykova : **Universidad Autónoma de Nuevo León (UANL), Av. Universidad S/N, Ciudad Universitaria, San Nicolás de los Garza, Nuevo León, 66455, Mexico

**Viacheslav Kalashnikov :**Instituto Tecnológico y de Estudios Superiores de Monterrey (ITESM), Campus Monterrey, Ave. Eugenio Garza Sada 2501 Sur, Monterrey, Nuevo León, 64849, Mexico

**José Guadalupe Flores-Muñiz :**Universidad Autónoma de Nuevo León (UANL), Av. Universidad S/N, Ciudad Universitaria, San Nicolás de los Garza, Nuevo León, 66455, Mexico.

**DOI : https://doi.org/10.32381/JCISS.2021.46.1-4.2**

#### Price: 251

**Studies on Association Rule-based Table Data Analysis and its Applications - New Mathematics for Data Sciences - (An Invited Tutorial Paper)**

**By 1:**
Hiroshi Sakai

**Page No **: 115-230

**Abstract**

This tutorial paper will survey the research on association rule-based table data analysis and its application. The proposed methods obtain helpful information from discrete table data through rules, which differs from statistical table data analysis by mean or variance. Regression expressions in continuous value table data can characterize past data and predict future data values. The same will hold for rules in discrete value table data.

This tutorial paper consists of the following three parts.

Part I: Background and Examples of Rule Generation from Tables

Part II: Mathematical Research of Rule Generation from Tables

Part III: Realization of Software Tools and Applications. We introduce the research trends in this field widely through three parts.

** Author :
Hiroshi Sakai :** Department of Basic Sciences, Graduate School of Engineering, Kyushu Institute of Technology, Tobata, Kitakyushu 804-8550, Japan

**DOI : DOI:-https://doi.org/10.32381/JCISS.2021.46.1-4.3**

#### Price: 251

**Algebras and Logics Emerging Out of Rough Sets (An Invited Review Paper)**

**By 1:**
Masiur Rahaman Sardar ,
Mihir Kumar Chakraborty

**Page No **: 231-313

**Abstract **

This is basically a survey work though some new results have also been incorporated. The contents presented are from the research done by Chakraborty and his co-workers for last three decades. A major part consists in presentation of set-models of abstract algebraic structures generated from rough sets. Three Types of logics developed during the course of research are categorized. Some foundational issues are raised at the end and open questions are mentioned.

** Authors :
Mihir Kumar Chakraborty : **School of Cognitive Science, Jadavpur University, Kolkata, 700032, West Bengal, India.

**Masiur Rahaman Sardar :**Department of Mathematics, City College, Kolkata, 700009, West Bengal, India.

**DOI : https://doi.org/10.32381/JCISS.2021.46.1-4.4**

#### Price: 251

Jan-2019 to Dec-2019

**Algorithms for the Polar Decomposition in Certain Groups and the Quaternion Tensor Square**

By: Brandon Whiteley , Deep Desai , Marcus Cisneros , Samreen Khan , Francis Adjei , Viswanath Ramakrishna

Page No : 1-34

**Abstract:**

Constructive algorithms, not even requiring 2×2 eigencalculations, are provided for finding the entries of the positive definite factor in the polar decomposition of matrices in many groups. These groups include the indefinite orthogonal groups of signature (1, n – 1) nd (n – 1, 1) and fifteen groups preserving certain bilinear forms in dimension four. The Lorentz group belongs to both classes. Some of these algorithms extend to the indefinite orthogonal groups of arbitrary signature with nominal additional work. These procedures are then used to find quaternionic representations for the four dimensional groups mentioned above, analogous to the representation of the special orthogonal group via a pair of unit quaternions. A key ingredient is a characterization of positive definite matrices in these groups. Two algorithms are proposed for the Lorentz group. The former also works for the group whose signature is (n – 1, 1) and (1, n – 1). The second enables (and is aided by) the inversion of the double covering of the Lorentz group by SL(2, C). A key observation is that the inversion of the covering map, when the target is a positive definite matrix, can be achieved essentially by inspection as we demonstrate. For the group whose signature matrix is I2,2 a completion procedure based on the aforementioned characterization of positivity leads to yet another algorithm for the computation of the polar decomposition. For the other four dimensional groups, explicit isomorphisms provided by quaternion algebra lead to methods for the polar decomposition. As byproducts we give a simple proof of the fact that positive definite matrices in each of these groups belong to the connected component of the identity, find an explicit expression for their logarithm, and provide a characterization of the symmetric matrices in the connected component of the identity of two of these groups in terms of their preimages in the corresponding covering group.

**Authors :**

**Francis Adjei**

Department of Mathematics and Statistics,Virginia Polytechnic and State University,Blacksburg, VA.

**Marcus Cisneros**

Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX.

**Deep Desai**

Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX.

**Samreen Khan**

Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX.

**Viswanath Ramakrishna**

Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX.

**Brandon Whiteley**

Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.1**

Price: 251

**Subclass of Analytic Functions in Conic Domains Associated with Symmetric q– Derivative Operator**

By: G. Murugusundaramoorthy , S. Sivasubramanian , M. Govindaraj

Page No : 35-56

**Abstract:** Motivated by the works of Kanas and R|ducanu [18] and Kanas and Yaguchi [16], we define and the new class k-ST (?), parabolic starlike functions associated with symmetric q-calculus operator and investigated several interesting properties of the function class related to the domains bounded by conical sections.

**Authors :**

**M. Govindaraj**

Department of Mathematics, Sona College of Technology, Salem 636005 Tamil Nadu, India.

**S. Sivasubramanian**

Department of Mathematics, University College of Engineering, Tindivanam Anna University, Tindivanam 604001, Tamil Nadu, India.

**G. Murugusundaramoorthy**

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore-632014,India.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.2**

Price: 251

**Convexities, Centroids in Graphs and their Application in Mobile Ad hoc Networks**

By: Prakash Veeraraghavan

Page No : 57-70

**Abstract:**

Mobile ad hoc networks (MANET) are continuously self-configuring, infrastructureless network of mobile devices connected wirelessly. They found their application everywhere from a military tactical application to a casual video gaming. In MANET, every mobile node is expected to route packets on-behalf of other nodes. Depending on the position in the network, some nodes are expected to route more packets than other nodes. This process will deplete more of their battery power than other nodes. In this paper, we model this problem through the classical g-convexity in graphs. We also prove that the g-centroid location problem for any arbitrary connected graph is NP-hard.

**Author :**

**Prakash Veeraraghavan**

Department of Computer Science and Information Technology, La Trobe University, Bundoora, 3086 VIC, Australia.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.3**

Price: 251

**A Note on Monotone Algorithms for Catalytic Converter Model**

By: V. Antony Vijesh , Rupsha Roy

Page No : 71-82

**Abstract:**

Recently, a few researchers studied a coupled system of first order partial differential equation arising from a catalytic converter and interesting existence and uniqueness results have been proved. Later, the same system has been solved using finite difference method after linearizing it using a monotone iterative scheme. Even though all the iterative schemes considered were proved to have linear convergence, numerical examples suggest that certain iterative scheme converges faster than the others. The theoretical justification for the above behavior is missing in these works. This short note intents to bridge this gap by providing a mathematical justification for the faster convergence.

**Authors :**

**Rupsha Roy**

School of Basic Sciences, Indian Institute of Technology Indore, Indore-453552, India.

**V. Antony Vijesh**

School of Basic Sciences, Indian Institute of Technology Indore, Indore-453552, India.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.4**

Price: 251

**Limit Formulas Related to q-Gamma and q-Digamma Functions at their Singularities**

By: A. Swaminathan , Sourav Das

Page No : 83-90

**Abstract:**

In this work we present some limit formulas of ratios between q-gamma functions, q-digamma functions and their derivatives at their singularities. The recent methods developed for gamma functions are used in order to obtain main results.

**Authors :**

**Sourav Das**

Department of Mathematics, National Institute of Technology Jamshedpur, Jamshedpur-831014, Jharkhand, India.

**A. Swaminathan**

Department of Mathematics, Indian Institute of Technology, Roorkee Roorkee-247667, Uttarakhand, India.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.4**

Price: 251

**One Dimensional Sub-hyperbolic Equation via Sequential Caputo Fractional Derivative**

By: Yunxiang Bai , Aghalaya S. Vatsala

Page No : 91-102

**Abstract:**

It is well known that the fractional derivative is global in nature compared with integer derivatives. In addition, the fractional dynamic equations represent as better and as more economical models compared with dynamic systems with integer derivative. In this work, we obtain a representation formula for the one dimensional sequential sub hyperbolic linear nonhomogeneous problem with initial and Dirichlet boundary conditions. This is achieved by using the eigenfunction expansion method and Green’s formula. In addition, we have used the Laplace transform method to solve the sequential time derivative dynamic equation. Our result yields the well known integer result as a special case. AMS (MOS) Subject Classification. 34A08, 35C15, 35L04.

**Authors :**

**Aghalaya S. Vatsala**

Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504, USA.

**Yunxiang Bai**

Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504, USA.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.6**

Price: 251

**Computations in Ancient Indian Mathematics**

By: V. Ramakalyani

Page No : 103-134

**Abstract:** Since the early times, methods of computation were developed and perfected in India. The development of the decimal place value system, computations on numbers, fractions and zero, first order indeterminate equations, their applications to astronomy and combinatorics as noticed in early Indian mathematics are discussed briefly in this paper. Attempt is made to show that computer programs can be written for the algorithms given by the ancient Indian mathematicians. Though these works were written several centuries ago they are suitable for application even today. 2010 Mathematics Subject Classification.

**Authors :**

**V. Ramakalyani**

The Kuppuswami Sastri Research Institute, 84, Thiru. Vi. Ka. Road, Mylapore, Chennai- 600004.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.7**

Price: 251

**Legendre Wavelet based Solution to Burgers’ and Coupled Burgers’ Equation**

By: Ram Jiwari , K. Harish Kumar

Page No : 135-150

**Abstract:** In this paper a quasilinearization based Legendre wavelet numerical scheme is pro posed for time dependent one dimensional Burgers’ and coupled Burgers’ equation. The present method is completely independent of time disctrization contrast to the recent literature for time disctrization based on wavelet scheme for Burgers’ and coupled Burgers’ equation. One of the advantages of the proposed scheme is that they capture the physical behavior of Burgers’ and coupled Burgers’ equations efficiently for large Reynolds numbers with less number of grid points. Further, straightforward numerical implementation makes them an effective alternative to the existing schemes.

**Authors :**

**K. Harish Kumar**

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee - 247667, India.

**Ram Jiwari**

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee - 247667, India.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.8**

Price: 251

**Cyber – Physical Systems: Mathematics, Logic and Computation**

By: T V Gopal

Page No : 151-162

**Abstract:**

Cyber-Physical Systems [CPS] are evolving and dynamic systems comprising digital, analog, physical, and human components engineered to collaborate through a feasible integration of physics and logic with the notions of time and free will. The Bergson’s view of time is within the ambit of the Greek metaphysics. Cybernetics provides an interdisciplinary systemic view of collaborating components. This paper is an essay on the emerging Cyber – Physical Systems and the role of Cybernetics, Software Development and Computational Mathematics in making them function to a purpose.

**Author :**

**T V Gopal**

Professor, Department of Computer Science and Engineering, CEG Campus, Anna University, Chennai – 600 025, India.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.9**

Price: 251

**Evaluation of the Discounted Warranty Cost of Minimally Repaired Series Systems under Gamma and Mixed Exponential Failure Laws**

By: Jafreezal Jaafar , Mohd Hilmi Hasan , Naga Sunil Kumar Gandikota

Page No : 163-183

**Abstract:** Various factors should be considered in the modeling of DWC of repairable systems or products, including system structure, component failure processes, discounting methods, and the warranty policy itself. This paper presents discounted warranty cost (DWC) models for repairable series systems subject to Gamma and Mixed Exponential Failure Laws. In particular, a policy on a free repair warranty and a policy on a pro-rata guarantee are being studied. The impact of repair actions on the failure times of the components is assumed to be minimal, hence NHPPs is used to describe the failure processes. This paper considers two types of discounting methods: a continuous discount function and a discrete discount function. Expressions for both the expected value and the variance of DWC are obtained for Gamma and Mixed Exponential Failure Laws. The WRD was calculated for Gamma Failure Laws under application section. Further extension of this research is provided in the conclusion section.

**Authors :**

**Naga Sunil Kumar Gandikota**

Centre for Research in Data Science, Computer & Information Sciences Department, Universiti Teknologi PETRONAS, 32610 Seri Iskandar, Perak, Malaysia.

**Mohd Hilmi Hasan**

Centre for Research in Data Science, Computer & Information Sciences Department, Universiti Teknologi PETRONAS, 32610 Seri Iskandar, Perak, Malaysia.

**Jafreezal Jaafar**

Centre for Research in Data Science, Computer & Information Sciences Department, Universiti Teknologi PETRONAS, 32610 Seri Iskandar, Perak, Malaysia.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.10**

Price: 251

**On Use of Range Rate Information in Target Tracking under Dense Clutter Background in Airborne Radars**

By: TV Prakash Rao , Narasimhan RS

Page No : 185-215

**Abstract:**

Data association techniques in radar target tracking problem play an important role in tracking targets of interest with high accuracy during both target non-maneuvering and maneuvering phases in a multiple target scenario. The data association problem further gets complicated due to the presence of detections from clutter leaks and false alarms in addition to detections from targets of interest. These techniques have to cater for uncertainty in the origin of measurement and low to moderate probability of detection of targets of interest. In this paper, we propose to utilize range rate information extensively to reduce plot to track miss-correlation, detect target maneuvers early, estimate target manoeuver intensity and choose optimum revisit rate as well as waveform to maximize target visibility. Tracking range rate is essential to exploit this information in tracking functions. This poses a problem due to its highly nonlinear relation to Cartesian target states. Use of nonlinear tracking filters have been shown to suffer from stability issues in addition to being sub-optimum. In this paper, we propose to decouple the tracking of range rate and position by using two parallel Interacting Multiple Model (IMM) filters. The range rate filter consists of four models in IMM framework to track range and range rate of targets in non-maneuvering and maneuvering phases. Since range rate is highly sensitive to target kinematic changes, a reliable and early maneuver detection is possible in range rate dimension and this is exploited in the design of tracking filter. We propose an efficient maneuver tracking algorithm using the electronic beam steering feature of phased array radars to detect and track evasive target maneuvers. In addition, the range rate filter is used to estimate the maneuver intensity and choose optimum update rate for tracking. Finally, the maneuver tracking performance of position tracking filter is improved by adapting mode transition probability matrix. We establish the performance of range rate and position tracking filters for non-maneuvering and maneuvering targets through simulations.

**Authors :**

**Narasimhan RS**

Scientist F, LRDE, DRDO, CV Raman Nagar, Bengaluru-560093.

**TV Prakash Rao**

Member, Institute of Defence Scientists and Technologists, Defence Research and Development Organisation [DRDO], Bengaluru-560075.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.11**

Price: 251

**Computational Mathematics for Software Reliability Engineering**

By: Subburaj Ramasamy

Page No : 217-244

**Abstract:**

This is an extensive tutorial introducing the state of art in Software Reliability Engineering and associated metrics. Software Reliability Engineering (SRE) encompasses many tasks such as Reliability Growth Testing (RGT) based on software reliability growth modeling, assessment of operational reliability, software release time determination and determination of estimated testing time or required testing efforts and resources. The foremost activity in SRE is Software Reliability growth modeling to plan and execute RGT which is carried out as final testing activity in the software development life cycle (SDLC) and is known in different names such as system testing. This phase in SDLC requires extensive use of Computational Mathematics to estimate parameters of the software reliability growth model (SRGM). In this paper, we will briefly discuss the various SRGM parameter estimation methodologies and conclude that the Least Squares methodology provides a satisfactory solution in all situations.

**Author :**

**Subburaj Ramasamy**

Research Professor, Machine Learning for AI and Software Reliability Engineering, 24/11 Dev Apartment, 5th Avenue, Besant Nagar, Chennai – 600090, India.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.12**

Price: 251

**Experimental Study of Intermediate Filters applied to Robust Optical Flow Estimation in Digital Images**

By: Vanel Lazcano

Page No : 245-268

**Abstract:**

Optical flow (OF) is the motion of pixels between two consecutive images. Variational models are used to estimate the OF. The flow that minimizes this model is the OF of the image sequence. The minimization procedure performs a coarse-to-fine approach. In each iteration, the estimated OF is processed with an intermediate filter (IF) to eliminate noise. We considered a robust OF model, and we tested it with different IF to improve OF estimation. Using the best IF and best parameters, we have evaluated the model in state-of-the-art dataset obtaining position 113 of 192 methods. Our approach outperforms similar techniques, such that SPARSEFLOW.

**Author :**

**Vanel Lazcano**

Universidad Mayor, Núcleo de Matemática, Físicay Estadística.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.13**

Price: 251

**Cryptanalysis of a Group Key Transfer Protocol: Generalization and Countermeasures**

By: V. Ch. Venkaiah , Kallepu Raju , Appala Naidu Tentu

Page No : 269-283

**Abstract:**

In Cryptography, group key distribution protocol is a mechanism in which a group key is generated and distributed by Key Generation Centre (KGC) to a set of communicating parties in a group. This group key generally ensures secure communication among communicating parties on an insecure channel. Key establishment protocols allow two or more communicating parties to establish their common secret key called a session key. Harn and Lin protocol(HL) is one such protocol and it is based on Shamir’s threshold secret setting. Nam et al. exposed the vulnerability in HL protocol through their replay attack and proposed a countermeasure using a nonce mechanism. In this paper, we are general- izing the replay attack presented by Nam et al. and proposing an alterna- tive countermeasure without using nonce mechanism. The novelty of our countermeasure is that KGC is not required to detect replay messages and hence each user doesn’t need to compute authentication messages as in Nam et al. Presented countermeasure thereby brings down the computational complexity of the scheme. Also, we propose an improved version of HL protocol a version that uses short signatures for authentication and overcomes drawbacks, and is resistant to existing attacks.

**Authors :**

**Appala Naidu Tentu**

CR Rao Advanced Institute of Mathematics, Statistics, and Computer Science, University of Hyderabad Campus, Hyderabad-500046, India.

**Kallepu Raju**

School of Computer and Information Sciences, University of Hyderabad, Hyderabad-500046, India, Hyderabad-500046, India.

**V. Ch. Venkaiah**

School of Computer and Information Sciences, University of Hyderabad, Hyderabad-500046.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.14**

Price: 251

**Compliant Ternary Matrices based on Restricted Isometry Property: Construction and Application to Image Retrieval**

By: M. Srinivas , C. S. Sastry , R. Ramu Naidu

Page No : 285-298

**Abstract:**

The class of matrices possessing compliance with the Restricted Isometry Property (RIP) provides data of certain types with sparse representation. The ternary matrices with 0, ±1 as their elements have the potential to result in fast algorithms. The existing ternary constructions combine binary and bipolar (Hadamard) matrices. In the present work, however, we construct deterministic and sparse ternary matrices from the known binary matrices. The resulting ternary matrices are shown to be better-behaved, in terms of their sparse recovery properties, than the input binary matrices. As our construction does not use bipolar matrices, it avoids the need for bipolar matrices. This saves memory space and computational cost when our method is implemented on computer. In addition, our matrices are very sparse compared to their generating binary matrices. We demonstrate that the ternary matrices constructed herein work well for dimensionality reduction in wavelet domain. In particular, through experimental work, we show that the wavelet weighted ternary matrices are capable of being used for applications like content based image retrieval.

**Authors :**

**R. Ramu Naidu**

Department of Mathematics, Indian Institute of Petroleum & Energy,Visakhapatnam-530003, India.

**C. S. Sastry**

Department of Mathematics, Indian Institute of Technology, Hyderabad, Medak, Telangana, 502205, India.

**M. Srinivas**

Department of Computer Science and Engineering, National Institute of Technology, Warangal, 506004, India.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.15**

Price: 251

**Some Interesting Properties of the Matrix Classes (?,c) and (?, c; p’)**

By: P N Natarajan

Page No : 299-309

**Abstract:**

In this paper, entries of sequences and innite matrices are real or complex numbers. We prove some nice properties of the innite matrix classes (l,c) and (l,c;P?) 2000 Mathematics Subject Classication. 40C05, 40D05, 40H05.

**Author :**

**P N Natarajan**

Old No. 2/3, New No. 3/3, Second Main Road, R.A. Puram, Chennai 600028, India.

**DOI : https://doi.org/10.32381/JCISS.2019.44.1-4.16**

Price: 251

Jan-2018 to Dec-2018

**Construction of Normal Bases over Finite Fields**

By: Arun Kumar , Kiran Devi , P. L. Sharma

Page No : 1-11

**Abstract**

Normal bases in finite fields are of much importance in many applications of cryptography and coding theory. The arithmetic computations of normal bases in finite fields are fast. In the extensions of finite fields there exist normal elements. A normal element generates the normal basis. In the present paper, we use a root of an irreducible polynomial and the trace function associated with it to obtain the normal element of the normal polynomial. Further, we find the normal basis whose elements are exactly the roots of the normal polynomial.

** Authors :
P. L. Sharma : **Department of Mathematics & Statistics Himachal Pradesh University, Shimla 171 005, India

**Kiran Devi :**Assistant Professor Mahatma Gandhi Govt Engineering College Jeori Camp at Sundernagar, Mandi 175018, (H.P)

**Arun Kumar :**Assistant Professor Mathematics Swami Vivekanand Govt. College Ghumarwin (H.P.)

**DOI : https://doi.org/10.32381/JCISS.2018.43.1-4.1**

Price: 251

**Maximum Distance Separable Hankel Rhotrices over Finite Fields**

By: Shalini Gupta , Arun Kumar , P. L. Sharma

Page No : 13-45

**Abstract**

Maximum Distance Separable (MDS) matrices are important components of the diffusion layer in the cryptographic ciphers. Rhotrices are represented by coupled matrices. Therefore, the use of rhotrices in cryptographic ciphers increase the security of the data in cryptosystems. We define Hankel rhotrix and construct the MDS rhotrices using Hankel rhotrices over finite fields.

** Authors :
P. L. Sharma : **Department of Mathematics and Statistics, Himachal Pradesh University, Shimla-171005, India

**Arun Kumar :**Research Scholar, Department of Mathematics and Statistics, Himachal Pradesh University, Shimla-171005, India

**Shalini Gupta :**Assistant Professor, Bahra University, Solan, H. P., India

**DOI : https://doi.org/10.32381/JCISS.2018.43.1-4.2**

Price: 251

**Partitioning Ideals of a Semiring R and its Skew Group Semiring R * G**

By: Madhu , Richa Sharma

Page No : 47-56

**Abstract**

Let G be a finite group acting on a semiring R and R G its skew group semiring. Then the quotient semiring RQ(I) is isomorphic to , for each g ? G. Further, if I is a G–invariant, Q–ideal of R, then I G is a Q G – ideal of R G. Assume that R is additively cancellative, then RQ(I) is additively cancellative (see [6]) and hence (RQ(I)) ? exists which is isomorphic to , where Q is an additive subgroupoid of R. In this case, ((R G)Q G(I G))? ? ((R? G)Q ? G(I ? G)).

** Authors :
Richa Sharma : **Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, India

**Madhu :**Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, India.

**DOI : https://doi.org/10.32381/JCISS.2018.43.1-4.3**

Price: 251

**Morita Equivalence Between H–Fixed Semialgebra and Smash Product Semialgebra**

By: M. Devendran , V. Selvan

Page No : 57-78

**Abstract**

In this paper, we study Hopf algebra(H) action on a semialgebra A and its smash product semialgebra A#H. We study the semimodule theoretic connection between the semialgebra A and its smash product semialgebra A#H. In this paper, we prove the Masches-type theorem for the smash product A#H. Further, we establish the Morita equivalence between H–fixed semialgebra AH and the smash product semialgebra A#H.

** Authors :
V. Selvan : **Department of Mathematics RKM Vivekananda College Chennai-600004, India

**M. Devendran :**Department of Mathematics RKM Vivekananda College Chennai-600004, India

**DOI : https://doi.org/10.32381/JCISS.2018.43.1-4.4**

Price: 251

**Packing Copies of Cartesian Product of Certain Classes of Paths and Cycles into Power graphs**

By: Sriram. V. , Medha Itagi Huilgol

Page No : 79-91

**Abstract**

An H -packing of a graph G is a set L = {H1,H2,...,Ht} of edge-disjoint subgraphs of G, where each subgraph Hi is isomorphic to H. In this paper, we have proved (CmCn)|(CmCn)k for m, n ? 2k – 1, with m, n, relatively prime to all numbers less than or equal to k and for the particular cases of m, n ? 0, 2 or 4(mod 6), n ? 8, we have (CmCn)|(CmCn)3 . Also we have found the maximum possible packing of PnCm in (PnCm)3 for n ? 4 and m ? 3 when m is even.

** Authors :
Medha Itagi Huilgol :** Department of Mathematics Bangalore University, Bengaluru, India

**Sriram. V :**Department of Mathematics, M.E.S. Degree College, Bengaluru, India

**DOI : https://doi.org/10.32381/JCISS.2018.43.1-4.5**

Price: 251

**A Formula for Pentanacci-Like Sequence**

By: Pankaj

Page No : 93-98

**Abstract**

Many papers are concerning a variety of generalizations of the Fibonacci sequence. In this paper, we define a Pentanacci-Like sequence in terms of first five terms and then derive the general formula for term of the Pentanacci -Like sequence.

** Author :
Pankaj : **Department of Mathematics, Indira Gandhi University, Meerpur (Rewari)-122502, Haryana, India

**DOI : https://doi.org/10.32381/JCISS.2018.43.1-4.6**

Price: 251

**Ordered Power Product of Polynomials and Product Quasi Cyclic Codes**

By: Ankita Gaur

Page No : 99-110

**Abstract**

Quasi cyclic codes, special type of cyclic codes, are an important class of implementable codes. In previous paper a new type of product of polynomial called, “Ordered Power Product of Polynomials” (OPP) was introduced. By means of this new kind of product of polynomials we defined OPP of two cyclic codes and developed a scheme of getting a cyclic code from the OPP of two cyclic codes. Also, it was shown that the code obtained by OPP of polynomials is more efficient than the code obtained by Kronecker type of product of same constituent codes. In this paper we apply the Ordered Power Product on quasi cyclic codes.

** Author :
Ankita Gaur : **Department of Mathematics Manav Rachna University, Sector-43, Aravalli Hills, Delhi-Surajkund Road, Faridabad-121004(Haryana), India,

**DOI : https://doi.org/10.32381/JCISS.2018.43.1-4.7**

Price: 251

**Effectiveness Analysis of Two Unit Cold Standby System with Fault Detection, Inspection and Priority**

By: Darpandeep Kour , J P Singh Joorel , Neha Sharma

Page No : 111-129

**Abstract**

This paper deals with a two unit (non-identical) system in which first unit is operative and other is kept as cold standby. On failure upon of first unit (main unit), the standby unit (secondary unit) becomes operative while failed unit puts first for fault detection and then for its repair. If no fault is detected then it becomes again operative. There is also a inspection policy for main unit to check whether the repair is perfect or not. In case the repair is not perfect it is again put under post repair. If both the units stop working the system will shut down. The main unit gets priority for both operation and repair. There is a single repair facility which is available with the system for repair as well as for fault detection, inspection and post repair. The distributions associated to different times including repair time of main unit, random variables, are assumed to be exponential but with different parameters while the post repair time of main unit and repair time of secondary unit is taken as general. Using regenerative point technique the various characteristics of the system have been obtained which help in studying the effectiveness of the system such as transition probabilities, mean time to system failure, availability, busy period of repairman and expected profit incurred.

** Authors :
Neha Sharma : **Department of Statistics University of Jammu, Jammu-180006, India

**J P Singh Joorel :**Department of Statistics University of Jammu, Jammu-180006, India

**Darpandeep Kour :**Department of Statistics University of Jammu, Jammu-180006, India

**DOI : https://doi.org/10.32381/JCISS.2018.43.1-4.8**

Price: 251

**Modified ?-graph G(?m) of some special graphs**

By: R. Kala , V. Anusuya

Page No : 131-143

**Abstract**

Gerd H. Frickle et al. [1] introduced ? -graph of a graph. Consider the family of all ? - sets in a graph G and we define G(? ) = (V (? ), E(? )) to be the graph whose vertices correspond 1 to 1 with the ? -sets of G and two ? -sets say S1 and S2 are adjacent in G(? ) if there exist a vertex v ? S1 and a vertex w ? S2 such that v is adjacent to w and S1 = S2 – {w} ? {v} or equivalently S2 = S1 – {v} ? {w}. The concept of ? -graph inspired us to define Modified ? -graph of a graph. Consider the family of all ? -sets of a graph G and define the modified ? -graph G(? m) = (V (? m), E(? m)) of G to be the graph whose vertices V (? m) correspond 1-1 with the ? -sets of G and two ? -sets S1 and S2 form an edge in G(? m) if there exists a vertex v ? S1 and w ? S2 such that S1 = S2 – {w} ? {v} and S1 = S2 – {v} ? {w}. In this paper we determine G(? m) of some special graphs.

** Authors :
V. Anusuya :** Department of Mathematics, S.T.Hindu College, Nagercoil- 629 002, Tamil Nadu, India

**R. Kala :**Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli 629 012, Tamil Nadu, India.

Price: 251

**On Certain Types of Product Set-Labeling of Graphs**

By: Sudev Naduvath

Page No : 145-154

**Abstract**

The product set of two sets A and B of integers, denoted by A*B, is the set A*B = {ab : a ? A, b ? B}. For X ? N, a product set-labeling (PS-labeling) of a graph G is an injective function ƒ : V (G) ?P(X) such that the induced function f* : V (G) ?P(X) is defined as ƒ*(uv) = ƒ(u)*ƒ(v)? uv ? E(G), ƒ(u) *ƒ(v) being the product set of ƒ(u) and ƒ(v). The PS-labeling of a graph can be classified into certain types in two ways: in accordance with the cardinalities of the set-labels and according to the nature of the collection of set-labels of elements of the graph G. This paper discusses different types of PS-labeling of graphs.

** Author :
Sudev Naduvath :** Department of Mathematics CHRIST (Deemed to be University) Bangalore, India

**DOI : https://doi.org/10.32381/JCISS.2018.43.1-4.10**

Price: 251

**A New Class of Fuzzy Implication on LI**

By: Balasubramaniam Jayaram , Satya Prakash Singh

Page No : 155-166

**Abstract**

In multi-valued logic, different domains of truth values are considered and hence it is imperative to define logical operations that respect the underlying structure. One such important logical connective is the implication. The lattice LI is used to capture the semantics of interval-valued fuzzy sets. In this work, we propose a new class of fuzzy implications on the lattice LI and discuss the different properties possessed by them.

** Authors :
Satya Prakash Singh : **Department of Mathematics, Indian Institute of Technology Hyderabad, Hyderabad, Telangana -502 285.

**Balasubramaniam Jayaram :**Department of Mathematics, Indian Institute of Technology Hyderabad, Hyderabad, Telangana -502 285.

**DOI : https://doi.org/10.32381/JCISS.2018.43.1-4.11**

Price: 251

**The Study of Near-Ring by Some Standard Groups**

By: H. K. Paliwal , R. Purohit

Page No : 167-176

**Abstract**

In the present paper it is shown that every Near-Ring satisfies some identity properties. Moreover, some examples are given which prove the necessity of identity properties for Near-Rings.

** Authors :
R. Purohit & H. K. Paliwal : **Department of Mathematics and Statistics M. L. Sukhadia University, Udaipur, India

**DOI : https://doi.org/10.32381/JCISS.2018.43.1-4.12**

Price: 251

**The Study of Derivational Commuting on a Near-Ring**

By: H. K. Paliwal , R. Purohit

Page No : 177-182

**Abstract**

In this paper we introduce the notion of Semiprime Near-Ring with various derivations commuting on it. The main purpose of this paper is to study and investigate some results on derivations which are commuting on Semiprime Near-Ring Nr.

**Authors :**

**R. Purohit & H.K. Paliwal : **Department of Mathematics and Statistics M. L. Sukhadia University Udaipur

**DOI : https://doi.org/10.32381/JCISS.2018.43.1-4.13**

Price: 251

**Sequential Estimation of the Lomax Distribution**

By: Sumit Koul

Page No : 183-190

**Abstract**

Sequential procedure is developed for minimum risk point estimation of the Lomax distribution under quadratic plus linear cost of sampling of the shape parameter. Secondorder approximations are derived for the expected sample size and regret associated with sequential procedure.

** Author :
Sumit Koul : **Assistant Professor Department of Mathematics (School of Chemical Engineering and Physical Sciences) Lovely Professional University Jalandhar- Delhi G.T. Road (NH-1) Phagwara, Punjab-144411, India

**DOI : https://doi.org/10.32381/JCISS.2018.43.1-4.14**

Price: 251

**Some Enumerative Properties of a Class of Fibonacci-like Cubes**

By: Haiyuan Yao , Xu Wang , Xuxu Zhao

Page No : 191-209

**Abstract**

A filter lattice is a distributive lattice formed by all filters of a poset in the antiinclusion order. We study the combinatorial properties of the Hasse diagrams of filter lattices of certain posets, so called Fibonacci-like cubes, in this paper. Several enumerative polynomials, e.g. rank generating function, cube polynomials and degree sequence polynomials are obtained. Some of these results relate to Fibonacci sequence and Padovan sequence.

** Authors :
Xuxu Zhao, Xu Wang & Haiyuan Yao : **College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, PR China

**DOI : https://doi.org/10.32381/JCISS.2018.43.1-4.15**

Price: 251

Jan-2017 to Dec-2017

**Fuzzy Network Flow Problems**

By: S.U. Malini , Felbin C. Kennedy

Page No : 1-21

**Abstract**

The concept of octagonal fuzzy numbers was introduced by the authors in an earlier paper [14]. In this paper, solution to network flow problems using octagonal fuzzy numbers are dealt with. viz, finding the shortest path of the network and project completion time. These problems were dealt with by several authors under different circumstances for different types of fuzzy numbers (to cite a few, [2],[3],[6],[7],[10]). In this paper the duration of activities are defined by octagonal fuzzy numbers which are more realistic and general in nature. Ranking using ?–cut [14] is used to compare octagonal fuzzy numbers. An algorithm is proposed to find fuzzy shortest path and fuzzy critical path of a project. Fuzzy shortest path, fuzzy critical path, fuzzy project completion time, fuzzy earliest start time, fuzzy earliest finish time, fuzzy latest start time, fuzzy latest finish time and fuzzy slack time for all activities are obtained as octagonal fuzzy numbers that captures the problem in a more realistic way.

**Authors :**

**Felbin C. Kennedy : **Associate Professor, Department of Mathematics, Stella Maris College (Autonomous), Chennai, Tamilnadu, India.

**S.U. Malini :** Research Scholar, Department of Mathematics, Stella Maris College Faculty, D.G. Vaishnav College, Chennai, Tamilnadu, India

Price: 251

**A Note on the Sparing Number of Generalised Petersen Graphs**

By: Germina Augustine , Chithra Kaithavalappil , Sudev Naduvath

Page No : 23-31

**Abstract**

Let X denote a set of non-negative integers and P(X) be its power set. An integer additive set-labeling (IASL) of a graph G is an injective set-valued function ƒ : V (G) ? P(X) – {?} where induced function ƒ+ : E (G) ? P(X) – {?} is defined by ƒ+(uv) = ƒ(u) + ƒ(v), whereƒ(u) + ƒ(v) is the sumset of ƒ(u) and ƒ(v). The cardinality of the set-label of an element of G is said to be the set-indexing number of that element. A weak integer additive set-labeling is an IASL, in which the set-indexing number of every edge is equal to the setindexing number of at least one of its end vertices. The sparing number of a graph G is the minimum number of edges having set indexing number 1 required for G to admit weak IASL. In this paper, we discuss the sparing number of certain generalised Petersen graphs.

** Authors :
Sudev Naduvath : **Department of Mathematics Vidya Academy of Science & Technology Thrissur, Kerala, India.

**Chithra Kaithavalappil :**Naduvath Mana, Nandikkara Thrissur, Kerala, India.

**Germina Augustine :**Department of Mathematics Central University of Kerala Kasaragod, India.

Price: 251

**Hopf Algebra Actions On Semialgebras**

By: M. Devendran , V. Selvan

Page No : 33-50

**Abstract**

Let A be a k – semialgebra and let H be a k – Hopf algebra. In this paper we introduce the action of the Hopf algebra H on the semialgebra A. Also we study some ideal theoretic connection between the semialgebra A, its H – fixed subsemialgebra AH, by using the connection between ideals of the algebras of differences A? of A, (AH)? of AH.

**Authors :**

**M. Devendran and V. Selvan : **Department of Mathematics, RKM Vivekananda College, Mylapore, Chennai-600004, India.

Price: 251

**Estimation and Testing Procedures for the Reliability Functions of the Moore and Bilikam Family of Lifetime Distributions under Progressive type II right censoring**

By: Narender Kumar , Ajit Chaturvedi

Page No : 51-80

**Abstract**

In this article the Moore and Bilikam family of lifetime distributions is considered which covers many lifetime distributions as specific cases. Two measures of reliability R(t) = P(X>t) and P=P(X >X*) are considered. Point estimation and testing procedures are developed for R t( ) and P under progressive type II right censoring. Three types of point estimators namely, uniformly minimum variance unbiased, maximum likelihood and invariantly optimal estimators are considered. Asymptotic confidence intervals are developed for the parameters ? and ?. Testing procedures are also developed for the hypotheses related to different parametric functions. A comparative study of different methods of estimation is done through Monte Carlo simulation study.

** Authors :
Ajit Chaturvedi & Narender Kumar : **Department of Statistics, University of Delhi, Delhi-110007, India

Price: 251

**Disjoint Union Metric Spaces**

By: Meenakshi

Page No : 81-87

**Abstract**

From the given metrics di on disjoint sets Xi , i = 1,2,3..., we define a new metric d on their union X so that it becomes a metric space. Further, we relate open spheres of the given sets to the open spheres of X so as to determine the structure of an open set in X. Finally, by using the inclusion mappings, we observe that the disjoint union topology of the topologies generated by di ’s on Xi ’s is the same as the topology generated by metric d on X.

**Author :**

**Meenakshi : **Research Scholar, Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005

Price: 251

**Constructions of Maximum Distance Separable Toeplitz Rhotrices over Finite Fields**

By: Shalini Gupta , P. L. Sharma

Page No : 89-110

**Abstract**

Maximum Distance Separable (MDS) Matrices are used in block ciphers to achieve the maximum diffusion. Use of rhotrices increase the security as these are the representations of coupled matrices. We define Toeplitz rhotrix and construct the MDS rhotrices using Toeplitz rhotrices over finite fields.

** Authors :
P. L. Sharma : **Professor, Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005

**Shalini Gupta :**Assistant Professor, Department of Mathematics, Bahra University, Solan, H. P., India

Price: 251

**Constructions of Maximum Distance Separable Toeplitz Rhotrices over Finite Fields**

By: Pankaj

Page No : 111-117

**Abstract**

In this paper, some new properties of k-Jacobsthal and k-Jacobsthal-Lucas numbers are derived and proved by using matrices. The identities, which have been proved in this paper, are not encountered in the k-Jacobsthal and k-Jacobsthal-Lucas numbers literature.

**Author :**

**Pankaj :** Department of Mathematics, Indira Gandhi University, Meerpur (Rewari)-122502, Haryana India

Price: 251

**Prioritized Information Fusion Method for Uncertain Linguistic Information and its Application to Multiple Attribute Group Decision Making**

By: Rajkumar Verma

Page No : 119-143

**Abstract**

In this paper, we investigate the uncertain linguistic multiple attribute group decision making (MAGDM) problems in which the attributes and decision-makers are in different priority levels. By combining the idea of quasi-arithmetic mean and prioritized weighted average (PWA) operator, we first propose a new aggregation operator called quasi-uncertain linguistic prioritized weighted average (QULPWA) operator for aggregating uncertain linguistic information. The properties of the new aggregation operator are studied in detail and their special cases are examined. Furthermore, based on the QULPWA operator, an approach to deal with multiple attribute decision-making problems under uncertain linguistic environments is developed. Finally, a practical example about the selection of university faculty is provided to illustrate the developed multiple attribute decision-making process.

** Author :
Rajkumar Verma : **Department of Applied Sciences Delhi Technical Campus (Affiliated to Guru Gobind Singh Indraprastha University) 28/1, Knowledge Park-III, Greater Noida, Uttar Pradesh 201306, India

Price: 251

**Restricted 2-Burst-Correcting Non-Binary Optimal Codes**

By: Tarun Lata , Vinod Tyagi

Page No : 145-154

**Abstract **

Burst is the most common error in almost all communication channels and burst error correcting codes are developed to deal with such errors. According to Fire [7], a burst of length b or less is a vector whose all the non-zero components are confined to some b consecutive positions, the first and the last of which is non-zero. In this paper, we have introduced the idea of “restricted burst”, which is the modification of the definition of a burst. By a restricted burst of length b or less we mean a vector whose all the non-zero components are confined to some b consecutive positions, the first and last of which is nonzero with a restriction that all the non-zero consecutive positions contain same field element. For example in non-binary case for q = 3, n = 3 and b = 2, we have the following vectors of restricted burst of length 2 or less 110, 220, 011, 022, 100, 200, 010, 020, 001, 002. This idea may look hypothetical from practical point of view but we know it is true for binary case. Also the work may not have any practical utility at present but it is interesting from theoretical point of view. In this work, we present two construction techniques for optimal codes with respect to restricted burst of length 2 or less for linear as well as for byte oriented codes with redundancy r ?3.

**Authors :**

**Vinod Tyagi :** Department of Mathematics, Shyam Lal College (Evening), University of Delhi, Delhi-110032, India.

**Tarun Lata : **Research Scholar, Department of Mathematics, University of Delhi, Delhi-110007, India

**DOI : Authors :**

Price: 251

**Properties of Generalized k-Jacobsthal like Sequence Using Matrices**

By: Pankaj

Page No : 155-163

**Abstract**

The Jacobsthal sequence has been generalized in many ways, some by preserving the initial conditions, others by preserving the recurrence relation. In this paper, we define a new generalization {Mk,n}? n = 1,with initial conditionsMk,o = 2Mk,1= m+k, which is generated by the recurrence relation Mk,n+1 = kMk,n + 2Mk,n–1 for n ? 1, where k, m are integer numbers. Some new properties concerning this new generalization have been described using matrix algebra.

**Author :**

**Pankaj : **Department of Mathematics, Indira Gandhi University, Meerpur (Rewari)-122502, Haryana India.

Price: 251

**Ideals in Gamma Semirings**

By: Shweta Gupta , Tilak Raj Sharma

Page No : 165-177

**Abstract**

In this paper, the concept of ideals, k –ideals are introduced and their properties are studied. These results will be used to obtain some other new results such as the operations on ideals, the inclusion preserving bijection between the set of all ideals of a ??? semiring R and the set of all ideals of Rn,n and other properties are studied. Moreover, ideals of Notherian and regular ??? semirings are generalized.

** Authors :
Tilak Raj Sharma and Shweta Gupta : **Department of Mathematics, H. P. University Regional Centre Mohli, Khaniyara, Dharamshala (H.P.) -176218, INDIA

Price: 251

**Local Metric Dimension of Families of Certain Graphs**

By: V Jude Annie Cynthia , Fancy V F

Page No : 179-186

**Abstract**

Let G(V, E) be a graph with vertex set V and edge set E. Then a subset of V namely W is said to be a local metric basis of G, if for any two adjacent vertices u, v ??V\W, there exists a vertex w ??W, such that d(u, w) ? d(v,w). The minimum cardinality of local metric basis is called the local metric dimension of G and is denoted by ?l (G). In this paper we discuss the local metric basis of some simple graphs and investigate the local metric dimension of some families of Generalised Petersen graph. Also we discuss the local metric basis of Circulant graphs in this paper.

** Authors :
V Jude Annie Cynthia and Fancy V F : **Department of Mathematics, Stella Maris College, Chennai-86, INDIA

Price: 251

**Group {1,–1,i,–i} Cordial Labeling of Some New Graphs**

By: R.Ponraj , S. Athisayanathan , M.K.Karthik Chidambaram

Page No : 187-196

**Abstract**

Let G be a (p,q)graph and A be a group. Let f :V (G) ? A be a function. The order of u ? A is the least positive integer n such that un = e. We denote the order of u by o(u). For each edge uv assign the label 1 if (o(f(u)),o(f(v))) = 1 or 0 otherwise. f is called a group A Cordial labeling if |vf (a) –vf (b)| ? 1 and |ef (0) –ef (1)| ? 1, where vf (x) and ef (n) respectively denote the number of vertices labeled with an element x and number of edges labeled with n(n = 0,1). A graph which admits a group A Cordial labeling is called a group A Cordial graph. In this paper we define group {1,–1,i,–i} Cordial graphs and prove that the Sunflower graph SFn , the graph Lotus inside a circle LCn , the Umbrella graph Un,n are all group {1,–1,i,–i} Cordial for all n. We further characterize Jewel graphs that are group {1,–1,i,–i} Cordial.

** Authors :
M.K.Karthik Chidambaram, S. Athisayanathan & R.Ponraj : **Department of Mathematics, St. Xavier’s College , Palayamkottai 627 002, Tamil Nadu, India.

Price: 251

Jan-2016 to Sep-2016

**Metric Dimensions of Crystals**

By: Ajay Kumar

Page No : 1-5

**Abstract**

Solids that possess long-range order are called crystals. The arrangement of the atoms at one point in a crystal is identical to that any other part of the crystal. In this paper we found metric dimensions of closed packing of constituent particles (atoms, ions or molecules) in crystals.

** Author :
Ajay Kumar :** Department of Mathematics Uttaranchal University Arcadia Grant PO - Chandanwari, Prem Nagar, Dehradun Uttarakhand - 248007, India

Price: 251

**On (k, d)-Hooked Skolem Graceful Graphs**

By: S. Arumugam , Tarkeshwar Singh , Jessica Pereira

Page No : 7-15

**Abstract**

A (p, q) graphG = (V, E) is said to be (k, d)-hooked Skolem graceful if there exists a bijection f : V (G)? {1, 2, . . . , p ? 1, p + 1} such that the induced edge labeling gf : E ? {k, k + d, . . . , k + (q ? 1)d} defined by gf (uv)= |f (u)? f (v)|?uv ? E is also bijective, where k and d are positive integers. Such a labeling f is called (k, d)-hooked Skolem graceful labeling of G. Note that when k = d = 1, this notion coincides with that of hooked Skolem (HS) graceful labeling of the graph G. In this paper, we present some preliminary results on (k, d)-hooked Skolem graceful graphs and prove that nK2 is (2, 1)-hooked Skolem graceful if and only if n ? 1 or 2 (mod 4).

** Authors :
Jessica Pereira and Tarkeshwar Singh : **Department of Mathematics, Birla Institute of Technology and Science Pilani K K Birla, Goa Campus, NH-17B, Zuarinagar, Goa, India

Price: 251

**On the Harmonious Coloring of Certain Class of Graphs**

By: V. Sriram , Medha Itagi Huilgol

Page No : 17-29

**Abstract **

Given a simple graph G a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. In this paper the harmonious chromatic number of distance degree regular graphs of diameter 3 is given. In this paper we also provide the harmonious chromatic number of some class of graphs viz., Parachute graph, Gear graph, Hn,n, Helm graph, Jelly fish. If G is a simple graph with n vertices, we give a constructive technique to harmoniously color G K ? mi , for 1 ? i ? n given that the graph G with n vertices has harmonious chromatic number k.

** Authors :
Medha Itagi Huilgol and V. Sriram : **Department of Mathematics, Bangalore University, Bengaluru, India.

Price: 251

**Radio Number of 1, 2 - Cartesian Product of Paths**

By: D. Rambabu , P. Venkata Subba Reddy ,

Page No : 31-38

**Abstract**

Let G be a simple, connected and undirected graph. A radio labeling f of G is an assignment of positive integers, called labels to the vertices of G such that if u, v ? V (G) are distinct then ?f(u) ? f(v)? ? D(G) + 1 ? d (u, v), where D(G) is the diameter of the graph G and d (u, v) is the distance between the vertices u and v in G. The maximum label (positive integer) assigned by f to some vertex of G is called the span of f. The radio number of G denoted by rn(G) is the minimum span over all radio labelings of G. In this paper, we prove a lower bound for the 1 , 2 -Cartesian product of paths.

** Authors :
Raaghave Sood, P. Venkata Subba Reddy and D. Rambabu : **Department of Computer Science and Engineering National Institute of Technology Warangal - 506 004, India.

Price: 251

**Enquˆete on the Existence of Hamiltonian Graph**

By: Ajay Kumar

Page No : 39-46

**Abstract**

This paper is a collection of conditions for the existence of Hamiltonian graphs buildout time by time. Researchers are assiduously involved for finding new genre of the conditions of the existence of Hamiltonian graphs. In this paper, I try to accumulate conditions for a graph to be Hamiltonian.

**Author :**

Ajay Kumar : Department of Mathematics Uttaranchal University, Arcadia Grant PO -Chandanwari, Prem Nagar, Dehradun Uttarakhand -248007, India

Price: 251

**A Characterization Theorem for gbent Function Over Z16**

By: N. Dhiman , P. L. Sharma

Page No : 47-55

**Abstract **

The characterization of gbent function in Zq for q = 23 has been discussed in the literature. We have extended this characterization of gbent functions for q = 24 and obtained some conditions for the bentness of these functions.

** Authors :
P. L. Sharma and N. Dhiman : **Professor, Department of Mathematics and Statistics Himachal Pradesh University, Summer Hill, Shimla – 171005.

Price: 251

**On Bounds for Fourth Central Moment**

By: R. Saini , Rajesh Sharma

Page No : 57-64

**Abstract **

The bounds for the variance in terms of the range of the random variable are well known in literature. We discuss here bounds for the fourth central moment recently obtained by Sharma et al. (2015).

** Authors :
Rajesh Sharma : **Professor Department of Mathematics and Statistics Himachal Pradesh University, Shimla 171005, India

**R. Saini :**Research Scholar Department of Mathematics and Statistics Himachal Pradesh University, Shimla 171005, India.

Price: 251

**Morita Equivalence between Partial Actions and Global Actions for Semialgebras**

By: Ram Parkash Sharma , Meenakshi

Page No : 65-77

**Abstract**

Given a strong partial action(?,A) of a finite group G on an additively cancellative yoked semialgebra A with each Dg as a semialgebra, there exists a unique global action (?,B) of it [11]. In this case, the partial skew group ring A Y?G and the skew group ring B Y?G are associative semirings. Here, it is our goal to exhibit a Morita context between them.

**Authors :**

**R.P. Sharma :** Professor, Department of Mathematics & Statistics Himachal Pradesh University Shimla-171005, India

**Meenakshi : **Research Scholar, Department of Mathematics & Statistics Himachal Pradesh University Shimla-171005, India

Price: 251

**Some Conditions on Gamma Semirings**

By: Shweta Gupta , Tilak Raj Sharma

Page No : 79-87

**Abstract**

In this paper, we introduce the concept of a commutative, simple, additive idempotent and multiplicative ? – idempotent ? – semiring. Here we study the consequences of imposing some conditions on a ? – semiring. In particular we will look at the conditions of being additive idempotent and multiplicative ? – idempotent ? – semiring and of being a simple ? – semiring.

**Authors :**

**Tilak Raj Sharma : **Assistant Professor, Department of Mathematics, H. P. University Regional Centre Mohli, Khaniyara, Dharamshala (H.P.) -176218- INDIA

**Shweta Gupta : **Department of Mathematics, H. P. University Regional Centre Khaniyara, Dharamshala (H.P.) -176218- INDIA.

Price: 251

**Generalized Uncertain Linguistic Prioritized Weighted Average Operator with Application in Multiple Attribute Group Decision-Making**

By: Rajkumar Verma

Page No : 89-122

**Abstract**

The prioritized weighted average (PWA) operator was introduced by Yager in 2008. An important characteristic of the PWA operator is that it takes into consideration prioritization among attributes and the decision makers. Most evaluation processes employ linguistic variables. We propose a new prioritized weighted aggregation operator called ‘generalized uncertain linguistic prioritized weighted average (GULPWA) operator’ for aggregating uncertain linguistic variables. Properties of the new aggregation operator are studied and their special cases examined. Based on the GULPWA operator, an approach to deal with multiple attribute group decision-making problems under uncertain linguistic environments is developed. A real-life problem is included to illustrate the use of GULPWA in multiple attribute group decision-making process. The prioritized weighted average (PWA) operator was introduced by Yager in 2008. An important characteristic of the PWA operator is that it takes into consideration prioritization among attributes and the decision makers. Most evaluation processes employ linguistic variables. We propose a new prioritized weighted aggregation operator called ‘generalized uncertain linguistic prioritized weighted average (GULPWA) operator’ for aggregating uncertain linguistic variables. Properties of the new aggregation operator are studied and their special cases examined. Based on the GULPWA operator, an approach to deal with multiple attribute group decision-making problems under uncertain linguistic environments is developed. A real-life problem is included to illustrate the use of GULPWA in multiple attribute group decision-making process.

** Author :
Rajkumar Verma :** Department of Applied Sciences Delhi Technical Campus (Affiliated to Guru Govind Singh Indraprastha University, Delhi) 28/1, Knowledge Park-III, Greater Noida – 201306 Uttar Pradesh, India

Price: 251

**Characterization of Disconnected Two-cyclic Graphs with a Self Vertex Switching**

By: G. Sumathy

Page No : 123-140

**Abstract **

A vertex v ? V(G) is said to be a self vertex switching of G if G is isomorphic to Gv , where Gv is the graph obtained from G by deleting all edges of G incident to v and adding all edges incident to v which are not in G. A graph G is called a two-cyclic graph if it has exactly two cycles. Trees, forests, both connected and disconnected unicyclic graphs and connected two-cyclic graphs are characterized, each with a self vertex switching. In this paper, we characterize disconnected two-cyclic graphs with a self vertex switching.

** Authors :
C. Jayasekaran :** Department of Mathematics Pioneer Kumaraswamy College, Nagercoil – 629 003

**G. Sumathy :**Department of Mathematics, Sree Ayyappa College for Women, Nagercoil – 629 003, India

Price: 251

**Generalised Altering Distance Functions For Six Mappings Satisfying Subcompatible Condition**

By: Manjusha P. Gandhi , Anushri A. Aserkar

Page No : 141-160

**Abstract **

In the present paper a theorem is established for two generalised altering distance functions defined for four variables and six mappings which in pairs commutes and satisfy sub compatibility condition. The concept of sub compatibility is very recent and more generalised condition than other compatibility conditions like occasionally weakly compatibility, weakly compatible. Thus the result in this paper generalises the earlier results available in the literature.

** Authors :
Anushri A. Aserkar :** Rajiv Gandhi College of Engineering and Research, Nagpur, India

**Manjusha P. Gandhi**: Yashwantrao Chavan College of Engineering, Nagpur, India.

Price: 251

Sep-2016 to Dec-2016

**Cordial Labeling of Arbitrary Supersubdivision of Wheel, Helm, Closed Helm**

By: Smita A. Bhatavadekar , Ujwala Deshmukh

Page No : 161-179

**Abstract**

In this paper, we prove that supersubdivision of wheel with certain conditions is cordial. We also prove that with some criterion super subdivision of helm and closed helm are cordial. We give few illustrations.

** Authors :
Ujwala Deshmukh :** Department of Mathematics Mithibai College, Vile Parle(West), Mumbai, India

**Smita A. Bhatavadekar :**Department of Applied Mathematics Lokmanya Tilak College of Engineering University of Mumbai, Mumbai, India

Price: 251

**A Computational Method for Minimum of Octagonal Fuzzy Numbers and its Application to Decision making**

By: Dhanalakshmi V

Page No : 181-202

**Abstract**

In this article, we introduce the minimum of octagonal fuzzy numbers in two forms. The former involves ?-cuts and the latter involves co-ordinate wise operations and Hausdorff metric is used to compare them. As an application, we present a multi-attribute decision making (MADM) problem, in which we use octagonal fuzzy numbers to describe linguistic variables. Finally MATHCAD 14 (a mathematical software) programs are presented, as a template to solve the MADM problem, so that the problem can be extended to any finite number.

** Authors :
Dhanalakshmi V. : **Research Scholar & Assistant Professor, Department of Mathematics, Stella Maris College (Autonomous), Chennai, Tamil Nadu, India

**Felbin C. Kennedy :**Research Guide & Associate Professor, Department of Mathematics, Stella Maris College (Autonomous), Chennai, Tamil Nadu, India

Price: 251

**Remark On Variance Bounds**

By: R. Bhandari

Page No : 203-206

**Abstract**

It is shown that the formula for the variance of combined series yields surprisingly simple proofs of some well known variance bounds.

** Authors :
R. Sharma : **Professor Department of Mathematics & Statistics Himachal Pradesh University Shimla-5, India.

**R. Bhandari :**Assistant Professor Department of Mathematics Govt. Degree College, Solan, India

Price: 251

**Some New Determinantal Identities of k-Pell Sequences**

By: Pankaj

Page No : 207-213

**Abstract** In this paper, we find some new determinantal identities using generalized k-Pell sequences which are defined as: P = {Pk,n+t , Pk,n+i , Pk,n+i+j , Pk,n+m, Pk,n+m+j}, where n, t, i, j, m are positive integers with 0.

** Author :
Pankaj :** Department of Mathematics, Indira Gandhi University, Meerpur (Rewari)-122502, Haryana, India

Price: 251

**Cryptanalysis and Improvement of Two Provably Secure Certificateless Signature Schemes**

By: Vishnu Sharma , Pankaj Kumar

Page No : 215-222

**Abstract **

Certificateless signature scheme provide the solution of certificate management problem and key escrow problem. Recently, Malhi and Batra construct an efficient certificateless signature scheme for adhoc networks and Liu et al. proposed a certificateless signature scheme. Malhi and Batra, Liu et al. both proved that their scheme is secure under the adaptive chosen message attacks by providing security analysis. In this paper, we analyzed the Malhi and Batra CLS scheme and found it insecure against with the malicious but passive attack type 2 adversary whereas Liu et al. CLS scheme is found insecure against the honest but curious attack type 2 adversary.

** Authors :
Pankaj Kumar and Vishnu Sharma :** School of Computing Science and Engineering, Galgotias University Greater Noida, India

Price: 251

**Relationships between a Semiring R and Quotient Semiring RQ(I)**

By: Ram Parkash Sharma , Madhu , Richa Sharma

Page No : 227-234

**Abstract** The main aim of this paper is to investigate relationships between the ideals of semirings R and RQ(I), where RQ(I) is the quotient semiring and I is the partitioning ideal of R. It is shown that if R is additively cancellative and yoked, then RQ(I) is additively cancellative and yoked. If R is additively cancellative, then RQ(I) is isomorphic to R/?I (R/?I is the factor semiring factored by Bourne relation induced by partitioning ideal I) and (RQ(I))? (the ring of dfferences of (RQ(I))?), is isomorphic to R?/I? .

** Authors :
Ram Parkash Sharma, Madhu and Richa Sharma : **Department of Mathematics and Statistics Himachal Pradesh University Shimla 171005, India.

Price: 251

**The Explicit Factorization of x 2n –? over Finite Fields**

By: Sudhir Batra , Manjit Singh

Page No : 235-243

**Abstract**

Let Fq be a finite field of odd order q and ? be a nonzero element in Fq of the order 2k , where k ? 2. In this paper, the irreducible factorization of x 2n – ? over Fq is given in very explicit form, where integer n ? 1. It is shown that all the irreducible factors of x2n – ? over Fq are binomials.

** Authors :
Manjit Singh and Sudhir Batra :** Department of Mathematics, Deenbandhu Chhotu Ram University of Science and Technology, Murthal-131039, Sonepat, Haryana, India

Price: 251

**Convection Driven by Surface Tension and Buoyancy in a Relatively Hotter or Cooler Layer of Liquid with Free slip Bottom**

By: A.K. Gupta , D. Surya

Page No : 245-255

**Abstract **

Rayleigh-Bénard Marangoni convection in a relatively hotter or cooler layer of liquid is studied theoretically by means of modified linear stability theory. The upper surface of the layer is considered to be non-deformable free where surface tension gradients arise on account of variation of temperature and the lower boundary surface is also free, each subject to constant heat flux condition. The Galerkin technique is used to obtain the eigenvalue equation analytically which is then computed numerically. This analysis predicts that the onset of convection in a relatively hotter layer of liquid is more stable than a cooler one under identical conditions, irrespective of whether the two mechanisms causing instability act individually or simultaneously, and that the coupling between the two agencies causing instability remains perfect.

** Authors :
A.K. Gupta :** Department of Mathematics, Himachal Pradesh University Centre for Evening Studies, Shimla, India

**D. Surya :**Department of Mathematics & Statistics Himachal Pradesh University Summer Hill, Shimla, India

Price: 251

Jan-2015 to Dec-2015

**A Dually Flat Geometry of the Manifold of F-Escort Probability Distributions**

By: Subrahamanian Moosath, K.S , Harsha, K.V.

Page No : 1-12

**Abstract**

In this paper, we consider a deformed exponential family called F-exponential family. Then on a F-exponential family, we describe a dually flat structure called the c-geometry and we obtain that this c-geometry is the conformal flattening of the (F, G)-geometry for suitable choices of F and G. The potential functions and the dual coordinates are given for the dually flat structure. The dual coordinates are defined using the F-escort probability distributions. Thus one can say that the manifold of F-escort probability distributions is dually flat by the conformal flattening of the (F, G)-geometry defined on the manifold of usual probability distributions. Further a maximum likelihood estimator is defined using the F-escort probability distribution.

Price: 251

**Positive Linear Functionals and Geršgorin Disks of Matrices**

By: R. Kumar , R. Sharma

Page No : 13-19

**Abstract**

We show that positive unital linear functionals provide several inequalities related to Geršgorin disk theorem.

**Authors :**

**R. Sharma and R. Kumar : **Department of Mathematics Himachal Pradesh University Shimla 171005, India.

Price: 251

**Super Geometric Mean Labeling**

By: B. Shiny , E. Ebin Raja Merly , S. S. Sandhya

Page No : 21-31

**Abstract **

Let f:V(G) ? {1,2,…,p+q} be an injective function. For a vertex labeling “f ” the induced edge labeling f * (e = uv) is defined by, f * (e) = or . Then “f” is called a Super Geometric mean labeling if {f(V(G))} ? {f(e):e ? E(G)} = {1,2,3,…,p+q}. A graph which admits Super Geometric mean labeling is called Super Geometric mean graph. In this paper, we prove Super Geometric mean labeling behaviour for some standard graph

** Authors :
S. S. Sandhya : **Assistant Professor of Mathematics, Sree Ayyappa College for Women, Chunkankadai – 629003 Kanyakumari District.

**E. Ebin Raja :**Merly Assistant Professor of Mathematics, Nesamony Memorial Christian College, Marthandam – 629165.

**B. Shiny :**Assistant Professor of Mathematics, DMI Engineering College Aralvaimozhi – 629 301

Price: 251

**Convergence of Best Proximity Point Theorems**

By: A. Anthony Eldred , S. Arul Ravi

Page No : 33-39

**Abstract**

We shall consider a self map f defined on the union of two subsets A and B of a metric space and satisfying f(A) Í B and f (B) Í A. We give some contractive type result for a best proximity point, that is a point x in A È B such that d(x, f(x)) = d(A, B), based on the conditions studied in fixed point theorems for non-expansive mappings by Chi Song Wong.

** Authors :
S. Arul Ravi and A. Anthony Eldred : **P.G. and Research Department of Mathematics,, St. Joseph’s College (Autonomous) Tiruchirappalli, India.

Price: 251

**Prime Correspondence Between a Graded Semiring R And Its Identity Component R1**

By: Rosy Joseph , Ram Parkash Sharma

Page No : 41-52

**Abstract**

The generalization of the results of group theory and ring theory to semirings is a very desirable feature in the domain of mathematics. The analogy between rings graded by a finite group G and rings on which G acts as automorphism has been observed by a number of mathematicians. The results of M. Lorenz and D.S. Passman [3], M. Lorenz, S. Montgomery and L.W. Small [4] proved for the rings with finite groups acting on them were extended by M. Cohen and S. Montgomery [1] for the group graded rings. Motivated by the analogy between the rings graded by a finite group G and rings on which G acts as automorphisms, we have derived certain results for a group graded semiring R, its ring of differences R? , and the smash product R # K[G]*, where R is a Ksemialgebra over a commutative semiring K in [7]. In this paper, we study some fundamental properties of subtractive, prime ideals of a Group graded semiring R and its identity component R1. We establish the existence of fuzzy ideals of R1 corresponding to the fuzzy ideals of R which enables us to settle many results for fuzzy ideals of R and R1.

** Authors :
Rosy Joseph : **Department of Mathematics Stella Maris College, 17, Cathedral Road, Chennai

**Ram Parkash Sharma :**Department of Mathematics, Himachal Pradesh University, Summer Hill, Shimla, India

Price: 251

**Series of generalized incomplete trojan-type designs**

By: Eldho Varghese , Cini Varghese , Seema Jaggi

Page No : 53-60

**Abstract**

Generalized incomplete Trojan-type designs are row-column designs in which each cell, corresponding to the intersection of row and column, contains more than one treatment and the rows are incomplete. Here, a series of generalized incomplete Trojan-type designs has been proposed for treatments of the form v = sn + 1 with flexibility in choosing the cell size (s) and the number of columns (n)

** Authors :
Seema Jaggi, Cini Varghese and Eldho Varghese :** Indian Agricultural Statistics Research Institute, Library Avenue, New Delhi-110 012.

Price: 251

**An algorithm to detect s-consistency in line sigraph**

By: Anshu Sethi , Deepa Sinha

Page No : 61-91

**Abstract **

Given a sigraph S, H = L(S) called the line sigraph of S is that sigraph in which edges of S are represented as vertices, two of these vertices are defined to be adjacent whenever the corresponding edges in S have a vertex in common and any such edge ef is defined to be negative whenever both e and f are negative edges in S. Here S is called root sigraph of H. For a given sigraph H, H is sigraph marked if the vertices of H are marked with the signs of the distinct edges of some sigraph M, called a marker of H, in a one-to-one manner such that adjacent vertices of H receive the signs of adjacent edges of M; (often, it is convenient to say that H is M-Marked when M is to be specified); H is then said to be S-Consistent whenever a marker S is to be specified, if there is an S-Marking ? of H with respect to which the marked sigraph H? is consistent. In this paper, we give an algorithm to obtain a line sigraph and detect whether it is S-Consistent or not and determine its complexity. In the end we will propose a technique that will use adjacency matrix of S and adjacency matrix of L(S) which is S-Consistent as a parameter to encrypt an image and forward the data in the form of S-Consistent L(S) and will decrypt it by applying inverse image and matrix operations.

** Authors :
Deepa Sinha : **South Asian University Akbar Bhawan, Chanakyapuri New Delhi- 110 021, India.

**Anshu Sethi :**Center For Mathematical Sciences Banasthali University Banasthali- 304 022 Rajasthan, India.

Price: 251

**Population Selection Based On Count Data**

By: Madhuri S. Mulekar

Page No : 93-105

**Abstract **

The count data are popularly described using binomial, Poisson and negative binomial distributions in managerial applications to study consumer behavior; in entomological and ecological applications to study species’ behavior; and in quality control applications to study systems’ behavior. For each of these distributions, several solutions under different configurations are available for the problem of selecting a best population among the k available populations. In this study, fixed-sample-size procedures with the indifference zone approach are considered. Interesting differences and similarities exhibited by these three distributions in the existence of selection procedures and the consistency of selection procedures are explored.

** Author :
Madhuri S. Mulekar :** Department of Mathematics & Statistics University of South Alabama 411 University Blvd N, ILB 325 Mobile, Alabama 36688-0002

Price: 251

**Signed Domination in Signed Graphs**

By: B. D. Acharya , Satish V. Motammanavar , H.B. Walikar

Page No : 107-128

**Abstract **Let S = (V,E, ?) be a signed graph, a function f : V ? {?1, +1} is a signed domination function (SDF) of S if f(N[v]) ? 1 where f(N[v]) = f (v)+ ?(uv) f(u), for every v ? V. Every graph can be viewed as a signed graph in which every edge is positive and every graph admits a signed dominating function but it is not true in case of heterogeneous signed graphs (i.e, signed graph possessing at least one negative edge). In this paper we characterize some classes of signed graphs which admit the signed domination function and we have also shown that signed graph can be embedded as an induced subgraph of a signed graph that admits an SDF

** Authors :
H.B. Walikar : **Former Professor and Head Department of Computer Science, Karnatak University Dharwad, Karnataka, India.

**Satish V. Motammanavar :**Department of Mathematics, R. V. College of Engineering, Bangalore, Karnataka, India.

**B. D. Acharya :**Ex-Director PIMSci Karnatak University, Dharwad, Karnataka, India.

Price: 251

**Some New Results on Mean Graphs**

By: S. Anusa , S. Somasundaram , S. S. Sandhya

Page No : 120-143

**Abstract**

A graph G = (V, E) with p vertices and q edges is said to be a mean graph if it is possible to label the vertices x ? V with distinct elements f(x) from 0, 1,2,…,q in such a way that when each edge e = uv is labeled with if f(u) + f(v) is even and if f(u) + f(v) is odd, then the resulting edge labels are distinct. In this case f is called a mean labeling of G. In this paper we prove that Triple Triangular, Triple Quadrilateral snake and Four Triangular snake and Four Quadrilateral snake graphs are mean graphs. Also we generalize that k ? T triangular snakes are mean graphs.

** Authors :
S. S. Sandhya : **Department of Mathematics Sree Ayyappa College for Women Chunkankadai-629003

**S. Somasundaram :**Department of Mathematics M.S. University Tirunelveli-627012

**S. Anusa :**Department of Mathematics Arunachala College of Engineering for Women Vellichanthai-629203

Price: 251

**Geometric Mean Labeling of Graphs Obtained By Subdividing Quadrilateral Snakes**

By: S. P. Viji , S. Somasundaram , S. S. Sandhya

Page No : 145-171

**Abstract**

A Graph G = (V,E) with p vertices and q edges is said to be a Geometric mean graph if it possible to label the vertices x ? V with distinct labels f(x) from 1,2…..q+1 in such a way that when each edge e=uv is labeled with f(e=uv) (or) , then the resulting edge labels are distinct. In this case f is called Geometric mean labeling of G. In this paper we prove that Geometric mean labeling behaviour of S(G) for some Quadrilateral snakes.

** Authors :
S. Somasundaram : **Department of Mathematics M.S. University Tirunelveli - 627012

**S. S. Sandhya :**Department of Mathematics Sree Ayyappa College for Women Chunkankadai - 629003

**S. P. Viji :**Department of Mathematics K.N.S.K. College of Engineering Therekalputhoor - 629004

Price: 251

**On Weakly Indexable Graphs**

By: P. Shankaran , S.M. Hegde

Page No : 173-209

**Abstract**

A (p,q)-graph G is said to be strongly k-indexable, if its vertices can be assigned distinct nonnegative integers 0, 1, 2..., p?1 so that the values of the edges, obtained as the sum of the numbers assigned to their end vertices, can be arranged as an arithmetic progression k, k+1, k + 2,..., k + (q?1). In this paper we introduce and study weakly indexable graphs. A graph G is said to be weakly k-indexable, if its vertices can be assigned distinct non-negative integers 0, 1, 2,...,p?1 so that the values of the edges, obtained as the sum of the numbers assigned to their end vertices form the multiset of numbers k, k +1, k +2,..., k + t, where t is a positive integer less than q?1. In this paper, we obtain some necessary conditions on weakly k-indexable graphs and investigate classes of graphs which admit weak indexer.

** Authors :
S.M. Hegde : **Dept. of Mathematical and Computational Sciences, N.I.T.K., Surathkal, Srinivasanagar-575025, India.

**P. Shankaran :**N.M.A.M. Institute of Technology, Nitte-574110, India

Price: 251

**A Class of Nonparametric Tests Based on Sample Quantiles**

By: Narinder Kumar

Page No : 211-233

**Abstract **

A general class of distribution-free tests for the two-sample location problem is proposed. The tests are based on two-sample U-statistics. This class generalizes the tests proposed by Kumar et al (2003) [Kumar, N., Singh, R.S. and Ozturk, O. (2003), A new class of distribution-free tests for location parameters, Sequential Analysis, 22, 107-128] and includes many other tests as its members. The proposed class of tests is compared with other known tests already existing in literature in the Pitman asymptotic relative efficiency sense for different underlying distributions.

** Author :
Narinder Kumar : **Department of Statstics Panjab University Chandigarh - 160014 INDIA

Price: 251

**Wavelet Based Numerical Simulation of Non Linear Klein/Sine Gordon Equation**

By: K. Harish Kumar , V. Antony Vijesh

Page No : 225-244

**Abstract **

This paper propose a new numerical techniques for solving nonlinear Klein / Sine Gordon equation with initial and boundary condition. In this approach Chebyshev and Legendre wavelet based collocation methods have been combined with quasi-linearization. Further, to produce better accuracy, even the time derivative has also been approximated by wavelets, in contrast to the recent wavelet based methods for nonlinear partial differential equations. The presented numerical scheme has been illustrated using appropriate examples and obtained results shows that the they are better than the methods available in the recent literature.

** Authors :
V. Antony Vijesh : **School of Basic Sciences Indian Institute of Technology Indore Indore - 452017, India

**K. Harish Kumar :**School of Basic Sciences Indian Institute of Technology Indore Indore - 452017, India

Price: 251

**On A Dual of a Theorem Due to Chvátal-Komlós**

By: Sriraman Sridharan

Page No : 245-251

**Abstract**

Consider a directed graph G and r integers p1, p2, . . . , pr such that p1 + p2 + · · · + pr < B(G), the minimum number of directed paths partitioning the vertex set of the graph G. Let X1, X2, · · · , Xr be any partition of the vertex set X of G. Then there exists an index i such that the stability number (independence number) of the induced subgraph G[Xi ] satisfies the inequality ?(G[Xi ]) > pi . This can be considered as a dual of the theorem of Chvátal-Komlós (see [3]). A generalization of this result is also considered.

** Author :
Sriraman Sridharan :** Département de Mathématiques et d’Informatique Université de Perpignan Via Domitia 52 Avenue Paul Alduy 66100 Perpignan France

Price: 251

**Construction of Graceful Directed Graphs**

By: Kumudakshi , S.M. Hegde

Page No : 253-264

**Abstract **

Dulmage and Mendelsohn while working on matrix reducibility observed that, there is a one-to-one correspondence between bigraphs and digraphs using the concept of adjacency matrices. In this paper we construct a relation between bigraphs and digraphs in terms of labelings. It is known that a graceful graph always gives rise to a graceful digraph. Start with any gracefully labeled undirected graph G with vertex labeling ?(x) for vertex x. Simply orienting the edges of G to point towards the larger vertex value produces a graceful digraph D with G as its underlying graph. Using the construction of bigraphs and digraphs, we show that the gracefulness of some classes of digraphs also gives rise to the gracefulness of its associated bigraph and vice versa.

** Authors :
S.M. Hegde : **Department of Mathematical and Computational Sciences National Institute of Technology, Karnataka, Surathkal Srinivasnagar - 575 025 Mangalore, India

**Kumudakshi :**Department of Mathematics, NMAM Institute of Technology, Nitte - 574110. Udupi Dist. Karnataka, India

Price: 251

Jan-2014 to Dec-2014

**The M/M/1/K Interdependent Retrial Queueing Model with Controllable Arrival Rates**

By: M. Thiagarajan , B. Antline Nisha , A. Srinivasan

Page No : 1-22

**Abstract **

In this paper we introduce an M/M/1/K interdependent retrial queueing model with controllable arrival rates. The steady state solutions and the system characteristics are derived and analysed for this model. Numerical results are given for better understanding.

** Authors :
B. Antline Nisha and M. Thiagarajan : **P. G. and Research Department of Mathematics, St. Joseph’s College (Autonomous), Tiruchirappalli - 620 002, Tamilnadu, India.

**A. Srinivasan :**P. G. and Research Department of Mathematics, Bishop Heber College (Autonomous), Tiruchirappalli - 620 017, Tamilnadu, India.

Price: 251

**Some Results On the Splitting Signed Graphs S(S)**

By: Mukti Acharya , Sangita Kansal , Rashmi Jain

Page No : 23-32

**Abstract**

A signed graph (or, in short, sigraph) S = (Su, s) consists of an underlying graph Su := G = (V, E) and a function s : E(Su) ? {+, -}, called the signature of S. A marking of S is a function m: V(S) ? {+, -}. The canonical marking of a signed graph S, denoted ms, is given as The splitting signed graph of a signed graph S, denoted S(S), is formed as follows: Take a copy of S and for each vertex v of S, take a new vertex v ¢. Join v ¢ to all vertices u Î N(v) by negative edge, if ms(u) = ms(v) = - in S and by positive edge otherwise. Here, N(v) is the set of all adjacent vertices to v. In this paper, we give structural characterization of signed graph S so that S(S) is balanced and ?-consistent. We also establish a characterization of splitting signed graphs S(S).

** Authors :
Rashmi Jain and Sangita Kansal : **Department of Applied Mathematics Delhi Technological University Shahbad Daulatpur, Main Bawana Road Delhi- 110 042, India.

**Mukti Acharya :**House No 22, 10th Cross, 5th Main MalleshPalya, Bangaluru-560 075, India.

Price: 251

**Planarity of Eccentric Digraph of Graphs**

By: Syed Asif Ulla S , Medha Itagi Huilgol

Page No : 33-47

**Abstract **

The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G. A vertex v is an eccentric vertex of vertex u if the distance from u to v is equal to e(u). The eccentric digraph ED(G) of a graph(digraph) G is the digraph that has the same vertex set as G and an arc from u to v exists in ED(G) if and only if v is an eccentric vertex of u in G. In this paper, we consider planarity of eccentric digraph of a graph.

** Authors :
Medha Itagi Huilgol and Syed Asif Ulla S. :** Department Of Mathematics, Bangalore University, Central College Campus, Bangalore - 560 001, India.

Price: 251

**Relations for Moments of Generalized Order Statistics from Chen Distribution and Its Characterization**

By: Taruna Kumari , Anupam Pathak

Page No : 49-56

**Abstract **

Chen distribution was introduced by Chen (2000). In this article recurrence relations for single and productmoments of generalized order statistics (gos) for Chen distribution have been driven. Moments of order statistics and k-upper records are discussed as special cases. Characterization of Chen distribution through conditional expectation is also presented.

** Authors :
Taruna Kumari and Anupam Pathak : **Department of Statistics, University of Delhi, Delhi-110007, India

Price: 251

**Construction of Fractional Factorial Designs with Some Linear Trend Free Effects Through Finite Field**

By: Veena Budhraja , Puja Thapliyal , Poonam Singh

Page No : 57-76

**Abstract **

Sometimes in experimental situations, experimental units exhibit a smooth trend over time or space. In that situation randomization may lead to an undesirable run order in the sense that observations obtained may be affected by uncontrollable variables which are correlated with the time or position in which they occur. Instead systematic run order may have to be used to eliminate the effects of such a trend. Such type of ordering of treatments is referred to as trend free designs. This article presents a simple methodology to construct the fractional factorial designs with s, s ? 2 levels, having some effects (main and two factor interactions) linear trend free using finite field. Further this method is used to generate the blocked fractional factorial designs and mixed fractional factorial designs with some linear trend free effects.

** Authors :
Poonam Singh : **Associate Professor in Department of Statistics, University of Delhi.

**Puja Thapliyal :**Research scholar in Department of Statistics, University of Delhi.

**Veena Budhraja :**Associate Professor in Department of Statistics, Sri Venkateswara College, University of Delhi.

Price: 251

**Some Properties of the Euler Summability Method in Complete Ultrametric Fields**

By: V. Srinivasan , R. Deepa , P N Natarajan

Page No : 77-86

**Abstract **

In this paper, we study some properties of the Euler method of summability in complete, non-trivially valued, ultrametric fields of characteristic zero, introduced earlier by Natarajan [9]

** Authors :
R. Deepa and V. Srinivasan :** Department of Mathematics, Faculty of Engineering and Technology, SRM University, Kattankulathur - 603 203, India

**P. N. Natarajan :**Old No. 2/3, New No. 3/3, Second Main Road, R.A. Puram, Chennai-600 028, India

Price: 251

**Multisample Testing Problem Related to Lorenz Curves**

By: Preety Rattan , Kalpana K. Mahajan , Sangeeta Arora

Page No : 87-101

**Abstract**

Estimation of the income inequality in the empirical set up is quite important and can be done using variety of methods involving either the comparison of income shares across different quantiles of income distribution or through employing different indices of inequality. One approach of studying inequality in income distribution is Lorenz dominance method. In this paper asymptotic distribution free test has been developed for comparing Multisample Lorenz curves in the interval of interest [ p1 , p2 ] where 0 < p1 < p2 <1. Simulation study is carried out giving out an illustration and power of the proposed test.

** Authors :
Sangeeta Arora and Kalpana K. Mahajan :** Professor Department of Statistics, Panjab University Chandigarh- 160014, India.

Price: 251

**Balanced Treatment Block Designs with Nested Rows And Columns**

By: B.N. Mandal , P.R. Sreenath , Niti Agarawal , Rajender Parsad

Page No : 103-116

**Abstract **

This article provides some methods of construction of block designs with nested rows and columns for comparing several test treatments with a control treatment. The methods are based on the balanced incomplete block (BIB) designs, initial block solutions and orthogonal Latin squares.

** Authors :
Rajender Parsad, Niti Agarawal, P.R. Sreenath and B.N. Mandal : ** ICAR - I.A.S.R.I., Library Avenue, New Delhi – 110 012

Price: 251

**Trend Free Block Designs Balanced for Interference Effects from Neighbouring Experimental Units**

By: Arpan Bhowmik , Eldho Varghese , Cini Varghese , Seema Jaggi

Page No : 117-133

**Abstract**

In this paper, we have considered block model with interference effects arising from the immediate neighbouring units on both sides and incorporating spatial trend component. The case of one-sided (say left) interference effects has been considered as a particular case. The information matrices for estimating direct as well as interference effects incorporating trend component have been derived. Further, the conditions for a block design with interference effects to be trend free have been obtained. Method of constructing trend free block design balanced for interference effects have been discussed with reference to both complete and incomplete blocks. The trend free block designs so obtained are totally balanced for estimating direct and interference effects of treatments.

** Authors :
Arpan Bhowmik, Seema Jaggi, Cini Varghese and Eldho Varghese :** Indian Agricultural Statistics Research Institute, Library Avenue, New Delhi – 110 012, India

Price: 251

**Degree Associated Reconstruction Number of Graphs Obtained by Certain Graph Operations**

By: S. Monikandan , A. Anu

Page No : 135-148

**Abstract**

A vertex-deleted subgraph of a graph G is called a card of G. A card of G with which the degree of the deleted vertex is also given is called a degree associated card (or dacard) of G. The degree associated reconstruction number of a graph G, drn (G), is the size of the smallest collection of dacards of G that uniquely determines G. We show that if G is a disconnected graph whose components are complete graphs, star graphs, cycles or paths, then drn (G) is 2. We also prove that if a hamiltonian graph G of order n contains m set of k nonadjacent vertices of degree n - k, then drn (G) is 1, 2 or 3. Finally, we prove that if G is K2 Pm or Pm[K2] for m > 6, then drn (G) is 2.

** Authors :
A. Anu and S. Monikandan : **Department of Mathematics Manonmaniam Sundaranar University Tirunelveli - 627 012 Tamilnadu, India.

Price: 251

**Analysing Censored Data On First Birth Interval for Heterogeneous Group of Females**

By: R. C. Yadava

Page No : 149-165

**Abstract **

A model for the time of first conception after marriage with censored data and accounting for heterogeneity in the population with respect to conception rate is proposed. For comparison, the model without accounting for heterogeneity is also discussed. The maximum likelihood estimation procedures are also given. To test the adequacy of the models, these have been applied to real data taken from National Family Health Survey -3 (NFHS-3)

** Authors :
Anup Kumar and R. C. Yadava :** DST-CIMS, Department of Statistics, Banaras Hindu University, Uttar Pradesh, India.

Price: 251

Finite Abelian Group Labeling of Graphs

By: Purnima Gupta , Pranjali , Mukti Acharya

Page No : 167-174

**Abstract **

In this paper, we have introduced a new type of labeling called an abelian group labeling, in which the operation on vertex labels vary according to the nature of the group. Viewed in this way, the unitary addition Cayley graph turns out to be maximal with respect to an abelian group labeling. Further, we have shown that every finite graph admits an abelian group labeling. Furthermore, we found that representation labeling can be obtained from abelian group labeling for certain graphs. Several new directions for further research are also indicated through problems.

** Authors :
Pranjali : **Department of Mathematics, University of Delhi, Chhatra Marg, Delhi-110007, India.

**Former Professor and Head, Department of Applied Mathematics, Delhi Technological University Delhi-110042, India.**

Mukti Acharya :

Mukti Acharya :

**Purnima Gupta :**Department of Mathematics, Sri Venkateswara College, University of Delhi-110021, India.

Price: 251

**Assessing Causal Effects with Truncation Due to Death and Missing Mortality Status**

By: David Wright , Vicki S. Hertzberg , Megan Price

Page No : 175-190

**Abstract **

Public health research is susceptible to many different sources of missing data. There is a growing body of literature specifically addressing the kind of missing data that results from interference by a post-treatment covariate - a problem commonly referred to as truncation due to death. Most of this research, using the method of principal stratification, assumes complete data except missing outcome measures truncated by death. Specifically, most research assumes complete data for the post-treatment covariate on which principal stratification is based. The authors investigate the affect of competing sources of missing data - due to both truncation and loss to follow-up, and involving both the primary outcome measure and the post-treatment covariate - on an estimated causal effect. This paper presents the results of a sensitivity analysis to determine the influence of different assumptions regarding the different sources of missing data.

** Authors :
Megan Price : **Director of Research at the Human Rights Data Analysis Group.

**Vicki S. Hertzberg :**Professor in the Department of Biostatistics and Bioinformatics at the Rollins School of Public Health at Emory University.

**David Wright :**Professor at the Emory University School of Medicine

Price: 251

**On Generalized Fuzzy Divergence Measure and Their Application to Multi-Criteria Decision Making**

By: Rajkumar Verma

Page No : 191-213

**Abstract**

Divergence is an important measure in information theory as well as in fuzzy set theory which has widely used by researchers in many application areas. Generalized divergence measures provide flexibility to the users and enhance their applicability range. In this paper, we introduce a new divergence measure, a kind of a discrimination measure, in the setting of fuzzy set theory. This measure is a generalized version of fuzzy divergence measure proposed by Shang and Jiang (1997), having a flexibility parameter l. Some properties of this measure and its applications bringing out the crucial role of the parameter in decision making problems under multi-criteria are demonstrated.

**Author :**

**Rajkumar Verma : **Department of Applied Sciences, HMR Institute of Technology and Management, Hamidpur, Delhi-110036, India.

Price: 251

**A New Method of Construction of a Series of Nested Balanced Incomplete Block Designs**

By: Vineeta Singh , Keerti Jain

Page No : 215-221

**Abstract**

In this paper a new method of construction of a series of Nested Balanced Incomplete Block Designs (NBIBDs) is introduced in which the inner blocks are constructed using Latin square. The construction is presented in the form of a theorem. The values of the parameters are satisfying the conditions required for NBIBD. The paper concludes with a numerical illustration to obtain a NBIBD with parameters v = 16, r = 15, b1 = 60, k1 = 4, b2 = 120, m = 2, k2 =2, l1 = 3, l2 =1. The appendix consists of parameters of some NBIBDs with their efficiency factors.

** Authors :
Keerti Jain : **Assistant Professor, School of Engineering, G D Goenka University, Sohna, Gurgaon, Haryana, India.

**Vineeta Singh :**Professor & Head, Department of Statistics, Institute of Social Science, Dr. B.R.Ambedkar University, Agra, Uttar Pradesh, India.

Price: 251

**Statistical Inference and Confidence Intervals for Absolute Partial Mean Dominance**

By: K.K. Mahajan , S. Arora , P. Bansal , Sangeeta Arora

Page No : 223-238

**Abstract **

Absolute Partial Mean (APM) is related to Bonferroni curve in the same way as Absolute Lorenz curve with Lorenz curve. The curve gives the idea of absolute inequality and can be shown to have relation with all the main curves of inequality viz. Lorenz curve, Generalized Lorenz curve, Bonferroni curve, Absolute Lorenz curve and Comic curve. The present paper considers the asymptotic variance – covariance structure of APM curve ordinates at some pre-selected points. Asymptotically non – parametric joint confidence intervals for APM curve ordinates are derived. Testing for APM dominance is carried out through two different approaches viz. Confidence Interval Approach and Specified Interval Approach and a comparison of these two testing procedures is done by comparing their corresponding powers. A real life illustration is also included.

** Authors :
P. Bansal, S. Arora and K.K. Mahajan : **Department of Statistics, Panjab University, Chandigarh – 160014, India

**Sangeeta Arora :**Professor, Department of Statistics, Panjab University, Chandigarh-160014, India

Price: 251

**Prediction Interval for the Smallest Value in a Future Sample Based on Imprecise Data**

By: Natasa Dasgupta , S. P. Mukherjee

Page No : 239-248

**Abstract **

A multiplicative error model involving observable variable Z, true variable X and error E viz. Z=XE, with the error distributed over a finite interval, is suggested for a non-negative random variable. Exact and approximate distributions of Z are worked out. Prediction interval for the smallest in a set of k future observations has been derived and some numerical illustrations have been provided.

** Authors :
S.P. Mukherjee :** University of Calcutta Kolkata

**Natasa Dasgupta :**Lady Brabourne College, Kolkata

Price: 251

**On a Bivariate Version of Zero-Inflated Logarithmic Series Distribution and its Properties**

By: A. Riyaz , C. Satheesh Kumar

Page No : 249-262

**Abstract**

In this paper we consider a bivariate version of the zero-inflated logarithmic series distribution of Kumar and Riyaz (Statistica, 2013) through its probability generating function (PGF) and derive expressions for its probability mass function, factorial moments, marginal and conditional PGFs. Recursion formulae for probabilities, raw moments and factorial moments of the bivariate distribution are also obtained. Moreover, the maximum likelihood estimation of the parameters of the bivariate distribution is attempted and the distribution has been fitted to a real life data set.

** Authors :
C. Satheesh Kumar and A. Riyaz :** Department of Statistics University of Kerala, Trivandrum-695 581, India

Price: 251

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