SLOWIK R

**023288** SLOWIK R (Silesian Univ of Technology, Poland, Email: roksana.slowik@gmail.com) : **Bilocal automorphisms of T ?(F).** Indian J Pure Appl Math 2017, 48(3), 323-34.

We prove that if F is a field such that |F| > 2, then every bilocal automorphism of T?(F) - the algebra of N × N upper triangular matrices over F, is an automorphism.

18 ref

SARADHA N

**023287** SARADHA N (School of Mathematics, Tata Institute of Fundamental Research, Mumbai, Email: saradha@math.tifr.res.in) : **On the length of the period of a real quadratic irrational.** Indian J Pure Appl Math 2017, 48(3), 311-21.

We review some known and not so well known results on the length of the period of the continued fraction expansion of a quadratic irrational ?D with D > 0. We also show that this length is o((D log D)^{1/2}) for almost all D.

17 ref

CHAKRABARTY A

**023286** CHAKRABARTY A (Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata, Email: arijit.isi@gmail.com) : **Large deviations for truncated heavy-tailed random variables: A boundary.** Indian J Pure Appl Math 2017, 48(4), 671-703.

This paper investigates the decay rate of the probability that the row sum of a triangular array of truncated heavy tailed random variables is larger than an integer (k) times the truncating threshold, as both the number of summands and the threshold go to infinity. The method of attack for this problem is significantly different from the one where k is not an integer, and requires much sharper estimates.

13 ref

BHAMIDI S, BUDHIRAJA A, SEN S

**023285** BHAMIDI S, BUDHIRAJA A, SEN S (Statistics and Operations Research Dep, Univ of North Carolina, Canada, Email: bhamidi@email.unc.edu) : **Critical random graphs and the diffential equations technique.** Indian J Pure Appl Math 2017, 48(4), 633-9.

Over the last few years a wide array of random graph models have been postulated to understand properties of empirically observed networks. Most of these models come with a parameter t (usually related to edge density) and a (model dependent) critical time t_{c} that specifies when a giant component emerges. There is evidence to support that for a wide class of models, under moment conditions, the nature of this emergence is universal and looks like the classical Erd ?os-R ?enyi random graph, in the sense of the critical scaling window and (a) the sizes of the components in this window (all maximal component sizes scaling like n^{2/3}) and (b) the structure of components (rescaled by n^{?1/3}) converge to random fractals related to the continuum random tree. The aim of this note is to give a non-technical overview of recent breakthroughs in this area, emphasizing a particular tool in proving such results called the differential equations technique first developed and used extensively in probabilistic combinatorics in the work of Wormald [52, 53] and developed in the context of critical random graphs by the authors and their collaborators in [10-12].

5 illus, 53 ref

GHOSH S, LEBOWITZ J L

**023284** GHOSH S, LEBOWITZ J L (Mathematics Dep, National Univ of Singapore, Singapore, Email: subhrowork@gmail.com) : **Fluctuations, large deviations and rigidity in hyper uniform systems: A brief survey.** Indian J Pure Appl Math 2017, 48(4), 609-31.

We present a brief survey of fluctuations and large deviations of particle systems with sub extensive growth of the variance. These are called hyper uniform (or super homogeneous) systems. We then discuss the relation between hyper uniformity and rigidity. In particular we give sufficient conditions for rigidity of such systems in d = 1,2.

57 ref

BHATTACHARJEE M, BOSE A

**023283** BHATTACHARJEE M, BOSE A (Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata, Email: monaiidexp.gamma@gmail.com) : **Matrix polynomial generalizations of the sample variance-covariance matrix when pn ^{-1}?Y? (0, ?).** Indian J Pure Appl Math 2017, 48(4), 575-607.

Let {*Z _{u} = ((?_{u,i,j}))_{p×n}*} be random matrices where {

17 ref

CHATTERJEE S, DIACONIS P

**023282** CHATTERJEE S, DIACONIS P (390 Serra Mall, Sequoia Hall, Stanford, Email: souravc@stanford.edu) : **A central limit theorem for a new statistic on permutations.** Indian J Pure Appl Math 2017, 48(4), 561-73.

This paper does three things: It proves a central limit theorem for novel permutation statistics (for example, the number of descents plus the number of descents in the inverse). It provides a clear illustration of a new approach to proving central limit theorems more generally. It gives us an opportunity to acknowledge the work of our teacher and friend B. V. Rao.

29 ref

SINHA K B

**023281** SINHA K B (J. N. Centre for Advanced Scientific Research, Indian Institute of Science, Bangalore, Email: kbs@jncasr.ac.in) : **Brownian bridge in quantum probability.** Indian J Pure Appl Math 2017, 48(4), 551-60.

The Classical Brownian Bridge is constructed in Symmetric Fock space over an appropriate base Hilbert space. While the representation of the classical Ito-Wiener integral with respect to the increments of the Brownian bridge implements the unitary isomorphism between the Fock space and the (classical) L2 space of the Brownian bridge (as is the case with the standard Brownian motion (SBM)), the quantum Ito-integrals with respect to the associated creation and annihilation bridge processes give different left-and right-integrals. This essentially displays the feature that the Brownian Bridge is not a process of independent increments.

3 ref

PAL S

**023280** PAL S (Mathematics Dep, Univ of Washington, Seattle, WA, Email: soumikpal@gmail.com) : **Embedding optimal transports in statistical manifolds 1.** Indian J Pure Appl Math 2017, 48(4), 541-50.

We consider Monge-Kantorovich optimal transport problems on Rd, d ? 1, with a convex cost function given by the cumulant generating function of a probability measure. Examples include the Wasserstein-2 transport whose cost function is the square of the Euclidean distance and corresponds to the cumulant generating function of the multivariate standard normal distribution. The optimal coupling is usually described via an extended notion of convex/concave functions and their gradient maps. These extended notions are nonintuitive and do not satisfy useful inequalities such as Jensen’s inequality. Under mild regularity conditions, we show that all such extended gradient maps can be recovered as the usual supergradients of a nonnegative concave function on the space of probability distributions. This embedding provides a universal geometry for all such optimal transports and an unexpected connection with information geometry of exponential families of distributions.

7 ref

ROY P

**023279** ROY P (Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bengaluru, Email: parthanil.roy@gmail.com) : **Maxima of stable random fields, non singular actions and finitely generated abelian groups: A survey.** Indian J Pure Appl Math 2017, 48(4), 513-40.

This is a self-contained introduction to the applications of ergodic theory of nonsingular (also known as quasi-invariant) group actions and the structure theorem for finitely generated abelian groups on the extreme values of stationary symmetric stable random fields indexed by Zd. It is based on a mini course given in the http://www.isid.ac.in/ lps16/PastWebsites/LPS8/index.html Eighth Lectures on Probability and Stochastic Processes (held in the Bangalore Centre of Indian Statistical Institute during December 6-10, 2013) except that a few recent references have been added in the concluding part. This article is a survey of existing work and the proofs are therefore skipped or briefly outlined.

65 ref

MUKHERJEE N, CASELLA G, KHARE K

**023278** MUKHERJEE N, CASELLA G, KHARE K (Statistical Science Dep, Duke Univ, Durham, Email: nabanita.mukherjee@duke.edu) : **Algorithms for improving efficiency of discrete markov claims.** Indian J Pure Appl Math 2017, 48(4), 495-511.

Sampling from an intractable probability distribution is a common and important problem in scientific computing. A popular approach to solve this problem is to construct a Markov chain which converges to the desired probability distribution, and run this Markov chain to obtain an approximate sample. In this paper, we provide two methods to improve the performance of a given discrete reversible Markov chain. These methods require the knowledge of the stationary distribution only up to a normalizing constant. Each of these methods produces a reversible Markov chain which has the same stationary distribution as the original chain, and dominates the original chain in the ordering introduced by Peskun [11]. We illustrate these methods on two Markov chains, one connected to hidden Markov models and one connected to card shuffling. We also prove a result which shows that the Metropolis-Hastings algorithm preserves the Peskun ordering for Markov transition matrices.

2 tables,15 ref

KARANDIKAR R L

**023277** KARANDIKAR R L (Chennai Mathematical Institute, Chennai, Email: rlk@cmi.ac.in) : **Remarks on the stochatic integral.** Indian J Pure Appl Math 2017, 48(4), 469-93.

In Karandikar-Rao [11], the quadratic variation [M,M] of a (local) martingale was obtained directly using only Doob’s maximal inequality and it was remarked that the stochastic integral can be defined using [M,M], avoiding using the predictable quadratic variation ?M,M? (of a locally square integrable martingale) as is usually done. This is accomplished here- starting with the result proved in [11], we construct ? f dX where X is a semimartingale and f is predictable and prove dominated convergence theorem (DCT) for the stochastic integral. Indeed, we characterize the class of integrands f for this integral as the class L(X) of predictable processes f such that |f| serves as the dominating function in the DCT for the stochastic integral. This observation seems to be new. We then discuss the vector stochastic integral ??f,dY ? where f is Rd valued predictable process, Y is Rd valued semimartingale. This was defined by Jacod [6] starting from vector valued simple functions. Memin [13] proved that for (local) martingales M 1,...,M d: If N n are martingales such that N nt ? Nt for every t and if ?fn such that N nt = ? ?f n, dM?, then ?f such that N = ??f, dM?.Taking a cue from our characterization of L(X), we define the vector integral in terms of the scalar integral and then give a direct proof of the result due to Memin stated above. This completeness result is an important step in the proof of the Jacod-Yor [4] result on martingale representation property and uniqueness of equivalent martingale measure. This result is also known as the second fundamental theorem of asset pricing.

16 ref

Zhao X;Meng F

**022364** Zhao X;Meng F (School of Management, Qufu Normal University, Qufu 273165, P.R. China, Email: qszhaoxueqin@163.com) : **On integral inequalities in n independent variables and their applications.** Far East J mathl Sci 2016, 100(11), 1917-31.

Authors establish some advanced integral inequalities in n independent variables which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain partial differential equations or integral equations. Some applications are gien to illustrate the usefulness of our results.

9 ref

Wiryanto L H;Mungkasi S

**022363** Wiryanto L H;Mungkasi S (NO, Bandung Institute of Technology, Jalan Ganesha 10 Bandung, Indonesia, Email: leo@math.itb.ac.id) : **Numerical solution of waves generated by flow over a bump.** Far East J mathl Sci 2016, 100(10), 1717-26.

When a uniform stream on an open channel is disturbed by an existing bump at the bottom of the channel, the free boundary of the stream forms waves that grow, split and propagate, until a steady formation of a solitary-like wave is achieved. The model of that wave generation can be presented in Boussinesq equations, and its solution if able to simulate those processes. For linear model, the solution is a combination of three functions, each represents a wave ith different amplitude and wave speed, depending on the strength of the incoming flow, presented as the Froude number. The solution is then used to verify a method of forward-time forward-space for a system of transport equations related to the Boussinesq equation. A good agreement can be obtained for supercritical flow.

2 illus, 9 ref

Suryawanshi A M;Bhimani G C

**022362** Suryawanshi A M;Bhimani G C (Statistics Dep, Saurashtra Univ, Rajkot, Gujarat) : **Comparison of three estimation methods for frechet distribution.** Int J multidiscipl Res Dev 2018, 5(1), 38-41.

This paper study the performance of three different estimation methods of scale parameter, namely, least square method, weighted least square method and maximum likelihood method for two parameter Frechet distribution. The Monte- Carlo simulation is used to compare these three methods in terms of bias and mean square error. The simulation study showed that, the maximum likelihood method was the best as compared to least square and weighted least square method in terms of bias as well as mean square error.

2 tables, 4 ref

Sukanya M;Choi J;Rathie A K

**022361** Sukanya M;Choi J;Rathie A K (Mathematics Dep, School of Mathematical and Physical Sciences, Central University of Kerala, Kasaragod-671 316, Email: sukanyam981@gmail.com) : **Note on two results associated with exponential and trigonometric functions.** Far East J mathl Sci 2016, 100(11), 1871-6.

The objective of this note is to show how we can establish Taylor-Maclaurin series expansions of products of exponential and trigonometric functions in an elementary way.

5 ref - 43 -

Srivastava S;Gupta D K;Stanimirovic P;Singh S;Roy F

**022360** Srivastava S;Gupta D K;Stanimirovic P;Singh S;Roy F (Mathematics Dep, School of Arts & Science, Amrita Vishwa Vidyapeetham, Amrita University, Amritapuri, Email: shwetabhiit@gmail.com ) : **Hyperpower iterative method for computing the generalized Drazin inverse of Banach algebra element.** Sadhana 2017, 42(5), 625-30.

A quadratically convergent Newton-type iterative scheme is proposed for approximating the generalized Drazin inverse b^{d} of the Banach algebra element b. Further, its extension into the form of the hyperpower iterative method of arbitrary order p≥2 is presented. Convergence criteria along with the estimation of error bounds in the computation of b^{d} are discussed. Convergence results confirms the high order convergence rate of the proposed iterative scheme.

25 ref

Shatanawi W;Hani M B

**022359** Shatanawi W;Hani M B (Mathematics and General Courses Dep, Prince Sultan University, Riyadh, Saudi Arabia, Email: wshatnawi@psu.edu.sa) : **Fixed point theorem in b-metric spaces with nonlinear contractive condition.** Far East J mathl Sci 2016, 100(11), 1901-8.

In this paper, we utilize the concept of a comparison function to introduce and prove a new fixed point theorem in the setting of b-metric spaces.

18 ref

Sharma D

**022358** Sharma D (Mathematics Dep, Ujjain Engineering College, Ujjain-456 010) : **Common fixed point theorems in uniform spaces.** Int J latest Technol Engng Mgmt appl Sci 2017, 6(5), 4-6.

In the process of generalization of metric spaces to Topological spaces, a few aspects of metric spaces are lost. Therefore, the requirement of generalization of metric spaces leads to the theory of uniform spaces. Uniform spaces stand somewhere in between metric spaces and general topological spaces. Khan[6] extended fixed point theorems due to Hardy and Rogers[2], Jungck[4] and Acharya[1] in uniform space by obtaining some results on common fixed points for a pair of commuting mappings defined on a sequentially complete Hausdorff uniform space. Rhoades et. al.[7] generalized the result of Khan[6] by establishing a general fixed poing theorem for four compatible maps in uniform space. In this paper, a common fixed point theorem in uniform spaces is proved which generalizes the result of Khan[6] and Rhoades et al.[7] by employing the less restrictive condition of weak compatibility for one pair and the condition of compatibility for second pair, the result is proved for six self-mappings.

7 ref

Setiawan R

**022357** Setiawan R (Mathematics Education Dep, Faculty of Teacher Training and Education S, Sebelas Maret University, Ir. Sutami No. 36 A, Kentingan, Jebres, Surakarta, Indonesia, Email: rubono.matematika@staff.uns.ac.id) : **Game theory approach in vendor-buyer probabilistic inventory system with imperfect quality, inspection error, minimum service level constraint and partial backordering.** Far East J mathl Sci 2016, 100(10), 1695-704.

In this paper, we apply game theory concept in probabilistic vendor-buyer inventory model with imperfect quality, inspection error and partial backordering allowed in order to determine the optimal solution of decision variables minimizing a total cost of inventory system. We consider two types of error in buyer's process selection, error may occur when inspector at the - 42 - buyer's process selection, error may occur when the inspector at the buyer's side misclassifies a non-defective item as defective or incorrectly classifies a defective item as non-defective. Several mathematical analyses of optimal solution by partial derivation criteria are organized based on game theory approach and covered in two decision strategies: cooperative and non-cooperative, using Pareto optimal and Stackelberg equilibrium, respectively.

7 ref

Sari I F;Mangku I W;Sumarno H

**022356** Sari I F;Mangku I W;Sumarno H (Mathematics Dep, Bogor Agricultural University, Jalan Meranti, Kampur IPB Dramaga, Bogor 16680, Indonesia, Email: intanfitriasa@yahoo.com) : **Estimating the mean function of a compound cyclic poisson process in the presence of power function trend.** Far East J mathl Sci 2016, 100(11), 1825-40.

An estimator of the mean function of a compound cyclic Poisson process with power function trend is constructed and investigated. We do not assume any parametric form for the cyclic component of the intensity function except that it is periodic. Moreover, we consider the case when only a single realization of the Poisson proces is observed in a bounded interval [0, n]. The proposed estimator is proved to be consistent when the size of the interval indefinitely expands. In addition, asymptotic approximations to the bias and variance of the proposed estimator are computed.

8 ref

Rahman N A;Abdullah L;Ab Ghani T A;Ahmad N A

**022355** Rahman N A;Abdullah L;Ab Ghani T A;Ahmad N A (School of Informatics and Applied Mathematics, Universiti Malaysia Tarengganu, 21030 Kaula Terengganu, Terengganu, Malaysia, Email: hamikahrun@yahoo.com) : **Numerical solution of IT2 FPs by Muller's method.** Far East J mathl Sci 2016, 100(11), 1765-86.

Nowadays, fuzzy polynomials (FPs) are widely used even though they uncertain and imprecise. This is due to the limitations of the method used in explaining and minimizing the effect of uncertainties. For many years, some of the pure and applied mathematicians are attracted in determining the approximated solutions (AS) of FP. This is due to the fact that many problems may be formulated in terms of finding AS. Thus, the iterative method is preferred to use. Therefore, the Muller's method is chosen to fulfil the objective in this study. Interestingly, this method is not presented to find the AS of FP but the interval type-2 fuzzy polynomials (IT2 FP). Then to illustrate the ability of the presented method, two illustrative examples are provided. As a result, it is shown that the Muller's method successfully provides the AS for the IT2 FP with a minimum number of iterations.

2 tables, 52 ref

Prabakaran S

**022354** Prabakaran S (Accounting and Finance Dep, School of Economics and Business Administr, Pontificia Universidad Javeriana Cali, Cali, Colombia, Email: prabakaran@javerianacali.edu.co) : **Construction of risk-neutral measure in a brownian motion with exotic option.** Far East J mathl Sci 2016, 100(10), 1643-74.

Binomial option pricing is a simple but powerful technique that can be used to solve many complex option-pricing problems. In contrast to the Black-Scholes and other complex option-pricing models that require solutions to stochastic differential equations, the binomial option-pricing model is mathematically simple. We are trying to show how to price a derivative security by determining the initial capital which requires hedging a short position in the derivative security. The overriding objective of this research paper is to discover a clever way to solve the partial differential equation using risk-neutral probability measure. The main goal of this study is fourfold: (1) to derive the joint density for a Brownian motion with drift and its maximum to data, (2) to introduce the change of the probability measure with - 41 - risk-neutral approach in the pricing of equity derivatives from real-world to risk-neutral by using Binomial structure model, (3) to extend this approach to treat the price the special type of option called a barrier option, and (4) to compute the risk-neutral price at time zero of the up-and-out call.

76 ref

Pongsumpun P;Changpuek T

**022353** Pongsumpun P;Changpuek T (Mathematics Dep, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang, Chalongkrung Road, Ladkrabang, Email: kppuntan@kmitl.ac.th) : **Analysis of swine flu network model.** Far East J mathl Sci 2016, 100(10), 1627-41.

In this paper, we propose a SEIQR network model incorporating the age structure of human population for swine flu to study an epidemic network distribution. The human population is divided into three groups such as 1-10 years, 11-20 years and more than 20 years, respectively. The SEIQR network model contains people and public places, we suppose that the people move to any public places by random process. The numerical simulation of the disease transmission is presented with assumptions that there is one exposed person and one infected person in each age group at the first day. The results show the time distribution of exposed, infected and quarantined human dealing with different situations.

6 illus, 1 table, 9 ref

Polloshka M;Pasku E

**022352** Polloshka M;Pasku E (NO, RIT Kosovo (A.U.K.), Dr. Shpetim Robaj, p.n., 10,000 Prishtina, Kosovo, Email: mpolloshka@auk.org) : **On permutability of e-unitary clifford semigroups.** Far East J mathl Sci 2016, 100(10), 1705-15.

The aim of this paper is to characterize the permutation property for E-unitary finitely generated Clifford semigroups in terms of finiteness of their maximal subgroups.

11 ref

Munj A Y;Rana B S;Godase S K

**022351** Munj A Y;Rana B S;Godase S K (NO, Regional Fruit Research Station, Vengurle, Maharashtra, Email: aymunj@rediffmail.com) : **Population dynamics of mango hopper, Idioscopus niveosparsus Leth and its natural enemies under coastal Konkan conditions of Maharashtra.** J Agrometeorol 2017, 19(4), 372-4.

^ssc1 table, 5 ref

Matarneh K;Ibrahim H

**022350** Matarneh K;Ibrahim H (School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia, 06010, Sintok, Kedah, Malaysia, Email: matarnehk@yahoo.com) : **Array cyclic (5<.** Far East J mathl Sci 2016, 100(10), 1611-26.

This article introduces a new design that is called - 40 - array cyclic (k^{*}, k^{**}, 4)-cycle design. In the course of developing this design, a combination between a cyclic (k_{1}, k_{2}, k_{3})-cycle system and near-two-factor will be constructed. By doing so, we need to introduce new cycle system, namely array cyclic (5^{*}, 6^{**}, 4)-cycle design of the complete graph with 12n vertices.

1 illus, 2 tables, 9 ref

Ling Z;Li J

**022349** Ling Z;Li J (School of Mathematical Science, Yangzhou University, Yangzhou 225002, P.R. China, Email: zhling@yzu.edu.cn) : **Spatial pattern of a ratio-dependent predator-prey model with cross-diffusion.** Far East J mathl Sci 2016, 100(10), 1551-63.

In this paper, a ratio-dependent predator-prey model with both self-diffusion and cross-diffusion is investigated. We consider the effects of cross-diffusion on pattern formation and obtain the conditions for cross-diffusion-driven instability. Our results show that under certain hypotheses, the positive cross-diffusion can trigger Turning patterns even though the corresponding model without cross-diffusion fails. Finally, the numerical simulations are carried out to provide a better understanding of the results.

3 illus, 17 ref

Li C;Wang W;Ryoo C S

**022348** Li C;Wang W;Ryoo C S (Mathematics Dep, Northeastern University, Shenyang 110004, P.R. China, Email: lichunji@mail.neu.edu.cn) : **Estimating the domain of attraction of tumor-immune system based on moment method.** Far East J mathl Sci 2016, 100(11), 1951-63.

In this article, by employing the moment method, we discuss the domain of attraction of tumor-immune system. Before introducing how to use this method, we provide certain details of the asymptotic stability of the equilibrium point of the system considered. By means of simulation, the effectiveness of the moment method has been illustrated.

1 illus, 10 ref

Janfada A S;Shabani-Solt H

**022347** Janfada A S;Shabani-Solt H (Mathematics Dep, Urmia University, Urmia, Iran, Email: a.sjanfada@urmia.ac.ir) : **On diophantine equation x<.** Far East J mathl Sci 2016, 100(11), 1891-9.

Authors solve the quartic Diophantine equation x^{4} + y^{4} = 2z^{4} + 2n^{2}w^{4}, where n is a congruent number and find integer solutions (x, y, z, w) with xyzw ≠ 0. Authors also show that the rational points of this equation are dense in both Zariski and real analytic topology. Finally, the existence of a steady relation between Kummer surfaces and congruent numbers is asserted.

12 ref

Goo Y H

**022346** Goo Y H (Mathematics Dep, Hanseo University, Seosan, Chungam 356-706, Korea, Email: yhgoo@hanseo.ac.kr) : **Boundness in nonlinear perturbed functional differential systems Via t _{∞}-similarity.** Far East J mathl Sci 2016, 100(10), 1727-44.

In this paper, we show that the solutions to the nonlinear perturbed differential system y' = ƒ(t, y) + ∫^{t}_{t0}g(s, y(s), T_{1}y(s))ds + h(t, y(t), T_{2}y(t)) have the bounded property by imposing conditions on the perturbed part ∫^{t}_{t0}g(s, y(s), T_{1}y(s))ds, h((t, y(t), T_{2}y(t), and on the fundamental matrix of the unperturbed system y' = ƒ(t, y) using the notion of h-stability.

18 ref

Fawzi F A;Senu N;Ismail F;Majid Z A

**022345** Fawzi F A;Senu N;Ismail F;Majid Z A (Mathematics Dep, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia) : **Embedded 6(5) pair of explicit Runge-Kutta method for periodic IVPs.** Far East J mathl Sci 2016, 100(11), 1841-57.

In this paper, an explicit embeded pair Runge-Kutta (RK) method which is denoted by TFRKF6(5) is developed to determine the approximate solution of the first-order IVPs with oscillatory solution. The proposed method solves first order ODEs by first converting higher order ODEs to an equivalent first order system. This embedded scheme has algebraic order six and five. The stability analysis of the new method is also given. The numerical experiment performed shows the efficacy of new embedded pair when compared with some standard and other pairs suggested in the scientific literature for solving oscillatory - 39 - problems.

7 illus, 24 ref

Dauda M K;Mamat M;Waziri M Y;Ahmad F;Mohamad F S

**022344** Dauda M K;Mamat M;Waziri M Y;Ahmad F;Mohamad F S (Mathematics Dep, Universiti Sultan Zainal Abidin, Gong Badak, Tarengganu, Malaysia, Email: mkdfika@gmail.com) : **Inexact CG-method via SR1 update for solving systems of nonlinear equations.** Far East J mathl Sci 2016, 100(11), 1787-804.

In this paper, we propose a CG-method for solving large scale systems of nonlinear equations via memoryless SR1 update. The new method is an enhanced SR1 update which is a matrix and derivative free, thereby requires low memory storage. Under appropriate conditions, the global convergence of the proposed method is established. The preliminary numerical results show the feasibility and effectiveness of the proposed method.

2 illus, 2 tables, 21 ref

Das S

**022343** Das S (Mathematics Dep, Uluberia College, Uluberia, Howrah, West Bengal-711 315) : **Analysis of gene expression using modified relative condon bias strength in Nanoarchaeum equitans.** Biosci Biotechnol Res Asia 2017, 14(2), 793-9.

Genes expressed at higher level exhibit extreme codon bias towards the use of optimal set of codons and based on this hypothesis, several codon usage indices are used to predict gene expressions in different genomes. This study analyzed a codon usage methodology, MRCBS (Modified Relative Codon Bias Strength) to explicate the outlines of gene expression in the Nanoarchaeum equitans kin4-M genome. The analysis of this work revealed that most of the highly expressed genes of this genome involved in protein synthesis, DNA/protein metabolism, protein fate, cellular process, ribosomal / translation and transcription factor, amino acid biosynthesis and many hypothetical proteins. It is also observed that MRCBS correlates better with codon adaptation index than the other widely accepted codon bias measures.

2 illus, 2 tables, 21 ref

Choudhary N;Bora S N

**022342** Choudhary N;Bora S N (Mathematics Dep, Indian Institute of Technology Guwahati, Guwahati-781 039, Email: neelam@iitg.ernet.in) : **Linear sloshing frequencies in the annular region of a circular cylindrical container in the presence of a rigid baffle.** Sadhana 2017, 42(5), 805-15.

Sloshing in any type of container may invite instability to it. If some part of the free liquid surface in the annular region of a specially designed circular cylindrical container is covered with an annular baffle, the natural frequencies and the response of the liquid in the container undergo a drastic change. A partly covered free surface shifts the natural frequency above and away from the control frequency of the vehicle, in which the liquid-filled container is placed, which results in the reduction of sloshing mass participating in the dynamic motion of the system. The fundamental - 38 - natural frequency of an inviscid and incompressible liquid is determined for increasing width of the baffle that is attached to the outer tank wall on the free surface. It is observed that by increasing the width of the baffle, natural frequencies can be significantly increased. Investigations are also carried out for different values of Bond number, which depicts different states of surface tension, and for varying values of the part of the redius in the fluid region. It is also observed that by increasing the fluid height inside the container, the natural frequencies can be increased, which results in reduction of sloshing.

20 illus, 23 ref

Choudhary A;Martha S C

**022341** Choudhary A;Martha S C (Mathematics Dep, Indian Institute of Technology Ropar, Rupnagar-140 001, Email: arunc@iitrpr.ac.in) : **Approximate solutions of some problems of scattering of surface water waves by vertical barriers.** Sadhana 2017, 42(5), 759-68.

A class of mixed boundary value problems (bvps), occurring in the study of scattering of surface water waves by thin vertical rigid barriers placed in water of finite depth, is examined for their approximate solutions. Two different placings of vertical barriers are analyzed, namely, (i) a partially immersed barrier and (ii) a bottom standing barrier. The solutions of the bvps are obtained by utilizing the eigenfunction expansion method, leading to a mathematical problem of solving over-determined systems of linear algebraic equations. The methods of analytical least-square approximation as well as algebraic least-square approximation are employed to solve the corresponding over-determined system of linear algebraic equations and thereby evaluate the physical quantities, namely, the reflection and transmission coefficients. Further, the absolute values of the reflection coefficients are compared to the known results obtained by utilizing a Galerkin type of approximate method after reducing the bvps to integral equations whose complete solutions are difficult to be determined. Various combinations of discretization of the resulting dual series relations obtained in the present analysis are employed to determine the least-square solution.

1 illus, 15 tables, 10 ref

Choi S II

**022340** Choi S II (Mathematics Dep, Hanseo University, Seosan Chungnam 356-706, Republic of Korea, Email: schoi@hanseo.ac.kr) : **h-Stability and boundedness in nonlinear functional perturbed differential systems.** Far East J mathl Sci 2016, 100(11), 1805-24.

For a nonlinear perturbed functional differential system, h-stability and bounded properties are obtained. The primary aim of this paper is to show these properties when the system has two perturbed parts. To do this, we give conditions on the perturbed parts and on the fundamental matrix of the unperturbed system using the notion of t_{∞}-similarity.

21 ref

Chantuk T;Paichit P;Pasunon P

**022339** Chantuk T;Paichit P;Pasunon P (General Business Management Dep, Faculty of Management Science, , Silpakorn University, Email: thirawat.scb@gmail.com) : **Application of econometrics methods in the development of model for maximum profit determination and profit level for the investment decision in Thailand's housing estate industry.** Far East J mathl Sci 2016, 100(10), 1675-93.

This research focuses on the calculation of maximum profit and profit level by using econometrics models to estimate the most appropriate revenue and cost equation. This quantitative research provides cost and revenue assumption using inductive method and develops the total revenue (TR) equation and the total cost (TC) equation by the development of econometric models for non-linear variables in the form of polynomial regression equation, which is adjusted to alleviate the multicollinearity effect. Thereafter, a marginal revenue (MR) equation and a marginal cost (MC) equation can be calculated accordingly by the first derivative of the total revenue and the total cost equation to analyze the maximum profit for the housing estate industry and also to calculate other profit areas in order to facilitate the housing estate developers for the construction decision for sell at the housing estate developers for the construction decision for sell at the maximum profit level, and also providing other profit information to be used as supportive information for an investment decision. The research results show that 240 units of houses constructed in the project earned 94.4 million baht of profit according to the model, and the number of houses that should be constructed to obtain the maximum profit level was 100. The total selling period is at 17.59 weeks. However, the developers had to continue the house construction in the vacant space for sell although the maximum profit was already estimated. Therefore, the profit amount can be segregated into two areas: first, R_{1} was the profit area before reaching the maximum profit level, which was 47.190 million baht; and R_{2} was the profit area after reaching the maximum profit level which was 47.210 million baht. Though R_{2} profit was slightly higher than R_{1} profit, the profit area experienced a diminishing return to profit as the construction of houses went on. This reflected an economic principle with respect to diminishing utility since the profit area was already beyond the maximum profit level. However, R_{2} can be - 37 - considered as an area of profit opportunity according to the model for maximum profit determination and profit level for the investment decision in Thailand's housing estate industry.

1 illus, 4 tables, 26 ref

Asim Z;Sorooshian S

**022338** Asim Z;Sorooshian S (Faculty of Industrial Management, Universiti Malaysia Phang, Malaysia) : **Analytic network process decision making algorithm.** Far East J mathl Sci 2016, 100(10), 1565-78.

To identify an accurate decision model with a number of distinctive criteria and alternatives are considered to be difficult and complex. In this case, analytic network process (ANP) becomes a significant instrument with a character of accuracy among other complicated models which use a number of criteria with complex feedback interrelation. This paper aims at to present a comprehensive description of multi-criteria decision making techniques with a significant feature of ANP. The paper offers a brief analysis regarding the ANP methodology in complex multi-criteria decision making, and highlights some recommendations.

2 illus, 32 ref

Areepong Y

**022337** Areepong Y (Applied Statistics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand, Email: yupaporn.a@sci.kmutnb.ac.th) : **Comparison of the performance of t control charts in short runs processes.** Far East J mathl Sci 2016, 100(11), 1745-54.

The main objective of this paper is to study an efficiency of control charts using median run length (MRL). The performance of control charts for quickest detection of change in mean process with exponentially weighted moving average (EWMA x-bar), t exponentially weighted moving average (EWMA t) and t cumulative sum (CUSUN t) charts when observation is from non-central t distribution is studied. We compare the performance between EWMA x-bar, EWMA t and - 36 - CUSUM t charts by using MRL criterion. The numerical results indicate that EWMA t chart is an effective alternative to be EWMA x-bar and CUSUM t control charts for small shifts.

6 tables, 6 ref

Arbin N;Ramli M F;Abu M S;Amin L M

**022336** Arbin N;Ramli M F;Abu M S;Amin L M (Mathematics Dep, Faculty of Science and Mathematics, Universiti Pendidikan Sultan Idris, 35900 Tanjung Malim, Perak, Malaysia) : **Development of ovate and lanceolate leaf shape model using B-spline generative shape method.** Far East J mathl Sci 2016, 100(11), 1909-15.

The aim of this paper is to develop an overall ovate and lanceolate leaf shape model based on the botanical textual description. For this purpose, the B-spline generative shape method (GSM) has been used. The idea is to assemble the tip, side and base of such leaves. For a given side's geometry, the tip and the base are produced by using the GSM and the leaf shape synthesis system. Expert botanist is referred to assess the drawing results to ensure that the result is in conformity with botanical point of view. The findings show that the use of GSM produced good result except for some minor visual errors. In conclusion, the GSM is practical to model these leaves. The implication is the developed model enables botanists to check whether the textual written descriptions in the botanical text are correct and useful as an aid to determine whether botanists use terminology consistently.

2 illus, 1 tables, 18 ref

Al-Khazaleh A M H;Al Wadi S;Alwadi H

**022335** Al-Khazaleh A M H;Al Wadi S;Alwadi H (Mathematics Dep, Faculty of Science, Al al-Bayt University, Al-Mafraq, Jordan, Email: ahmed_2005kh@yahoo.com) : **Forecasting estimated missing time series data.** Far East J mathl Sci 2016, 100(10), 1579-93.

In 2008, Ahmad Mahir and Al-Khazaleh [11] proposed a new method to estimate the missing data by using the filtering process. However, in this article, we make use of the previous techniques in order to forecast two groups of financial time series data with missing data and without missing data. Real closed price data were collected from Amman Stock Exchange (ASE) in order to improve the forecasting accuracy and in solving missing data problem. The result of the study was ensured with the native test and Theil's statistic which was found that U = 0.535 for the original data and U = 0.6213 for the suggested data with missing data, respectively. Therefore, the suggested model is significant.

3 illus, 7 tables, 12 ref

Al-Hawari M H

**022334** Al-Hawari M H (NO, Irbid National University, Irbid, Jordan, Email: analysis2003@yahoo.com) : **Zeros of polynomials by using some inequalities.** Far East J mathl Sci 2016, 100(10), 1545-50.

Authors applies several trace inequalities to the Frobenius companion matrices of a monic - 35 - polynomial to derive new bounds for the zeros of these polynomials. Our results enable us to obtain the other classical bounds. Inequalities involving differences of absolute values of the zeros of p(z) = z^{n} + a_{n}z^{n-1} + ... + a_{2}z + a_{1} of degree n ≥ 2 with complex coefficients a_{1}, a_{2}, ..., a_{n}, where a_{1} ≠ 0 are also given.

4 ref

Ahmad A;Ong W E;Mt Piah A R

**022333** Ahmad A;Ong W E;Mt Piah A R (School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia, Email: afida.ahmad@gmail.com) : **Choosing shape parameters for multiquadric and Gaussian basis functions to preserve positivity of real data.** Far East J mathl Sci 2016, 100(11), 1755-64.

Preservation of positivity using multiquadric and Gaussian basis functions is out-lined in this paper. The primary focus is to present the analysis of the free shape parameter, ε which is used in preserving positivity for real data sets. The value of ε selected for every data set is between a proposed a lower bound value and an upper bound value. The results gained are then compared with the existing interpolation schemes, namely trigonometric spline and rational cubic Ball. The output from several data sets are presented and the choice of ε varied depending on the data set used. The interpolants are found to be comparable and provide visually pleasing and smooth curves.

4 illus, 3 tables, 14 ref

Agarwal P;Choi J;Jain S

**022332** Agarwal P;Choi J;Jain S (Mathematics Dep, Anand International College of Engineering, Jaipur-303 012, Email: goyal.praveen2011@gmail.com) : **On (k; r)-type fractional integral inequalities.** Far East J mathl Sci 2016, 100(11), 1877-89.

Many inequalities involving Riemann-Liouville, Erdelyi-Kober, and k-fractional integrals are available in literature. In this paper, we present certain inequalities associated with the recently introduced generalized (k; r)- fractional integrals of Riemann-Liouville type.

20 ref

Adiwijaya;Abdurohman M;Erfianto B;Salman A N M

**022331** Adiwijaya;Abdurohman M;Erfianto B;Salman A N M (School of Computing, Telkon University, Jl. Telekomunikasi, 1, Bandung 40257, Indonesia, Email: adiwijaya@telkomuniversity.ac.id) : **Classification of the corona product of a complete graph with some graphs based on its ƒ-chromatic index.** Far East J mathl Sci 2016, 100(11), 1859-70.

Let G (V, E) be a finite and simple graph and let ƒ be a function from V to a positive integer set. An ƒ-coloring of G is a generalized edge-coloring such that every vertex ν ∈ V has at most ƒ(ν) edges colored with same color. The minimum number of colors needed to define an ƒ-coloring of G is called an ƒ-chromatic index of G, denoted by χ'_{ƒ}(G). Based on ƒ-chromatic index, a graph G can be either in the C_{ƒ}1 or C_{ƒ}2. In this paper, we show that the corona product of a path or a star with a complete graph is in C_{ƒ}1, for any ƒ. Because the corona product is not commutative, then we provide a classification of the corona product of a complete graph with a path or a star based on its ƒ-chromatic index.

11 ref

Shalini A J;Karthika C

**021344** Shalini A J;Karthika C (Mathematics PG Dep, Women's Christian College, Chennai-600 006) : **Study on voltage regulation using fuzzy analysis.** Int J latest Technol Engng Mgmt appl Sci 2017, 6(8), 33-5.

A fuzzy logic controller (FLC) [1] with fast reference watts generation to correct and regulate unbalance watts in three-phase system is proposed in [2]. We designed an efficient algorithm to regulate the unbalance watts in three-phase system under different conditions of the utility supply. In our simulation test, the Fuzzy Logic control shows consistent excellence under various operating conditions such as different initial control gains, different load levels, and change of transmission network and consecutive disturbances. As a result we theoretically reduce the electricity bill to minimise the watts and voltage usage.

3 illus, 2 tables, 5 ref

Patel R

**021343** Patel R (Statistics Dep, Veer Narmad South Gujarat Univ, Surat) : **Inventory model for imperfect quality items with different deterioration rates under inflation and permissible delay in payments.** Int J latest Technol Engng Mgmt appl Sci 2017, 6(7), 173-9.

Many times it happens that units produced or ordered are not of 100% good quality. A deterministic inventory model for imperfect quality items is developed when deterioration rate is different during a cycle with permissible delay in payment under inflation. Here it is assumed that holding cost is time dependent. Demand is considered as linear function of time. Numerical example is taken to support the model.

1 illus, 4 tables, 16 ref

Pandey R

**021342** Pandey R (Applied Science Dep, Institute of Engineering & Technology, Sitapur Road, Lucknow-226 021) : **Notes on Bailey's transform.** Int J latest Technol Engng Mgmt appl Sci 2017, 6(7), 80-2.

In this paper transformation formulae for poly-basic hypergeometric functions have been used by making use of Bailey's transform.

5 ref

Majeed A A;Rahi S A

**021341** Majeed A A;Rahi S A (Mathematics Dep, College of Science, Baghdad Univ, Baghdad, Iraq) : **Bifurcation of a food chain across a refuge stage-structure prey-predator model.** Int J Sci Nat 2017, 8(3), 520-35.

In the present paper, the conditions which guarantee the occurrence of local bifurcation (such as saddle-node, transcritical and pitchfork)of all equilibrium points of ecological mathematical model consisting of prey-predator model with two different functional responses incorporating a prey refuge are confirmed, it is observed that there is a transcritical bifurcation near the vanishing equilibrium point E_{0}, while there is either a transcritical or a pitchfork bifurcation near the free top-predator's and free predators' equilibrium points E_{1} and E_{2} respectively, on the other hand there is a saddle-node bifurcation at the coexistence equilibrium point E_{3}. Further investigation with special emphasis on the Hopf bifurcation near the coexistence equilibrium point is established and carried out. Finally, some numerical simulations are used to illustration the occurrence of local bifurcation of this model.

5 illus, 8 ref