039156   LOGANATHAN P, DEEPA K (Mathematics Dep, Anna Univ, Chennai- 600 025, Email: logu@annauniv.edu) : Heat and mass transfer analysis of Casson fluid flow on a permeable riga-plate. Indian J Pure Appl Phys 2020, 58(02), 79-86.

Numerical analysis has been carried out for a Casson fluid flow on a riga-plate with temperature dependent thermal conductivity. The physical model which governs the transport properties is solved numerically. This investigation emphasizes the consequence of variable thermal conductivity and electrically conducting magnetic field on the fluid flow. Rate of heat transfer is elevated, while the flow exposed to constructive case of variable thermal conductivity. The flow speed is enhanced for the improved values of modified Hartmann number. Correlation of the results with the similarity solutions declares the accuracy.

11 illus, 33 ref

039155   CHAUDHARY S, CHAUDHARY S, CHAUDHARY M K (Mathematics Dep, Malaviya National Institute of Technology, Jaipur- 302 017, Email: d11.santosh@yahoo.com) : Numerical investigation of unsteady MHD flow and radiation heat transfer past a stretching surface in porous media with viscous dissipation and heat generation/absorption. Indian J Pure Appl Phys 2020, 58(02), 71-8.

Viscous dissipation and radiation effects on unsteady flow of laminar incompressible viscous electrically conducting fluid through stretching surface in a porous media with magnetic field and heat generation/absorption have been investigated. Taking suitable similarity variables, the governing boundary layer equations are converted into a set of ordinary differential equations and solved numerically by Runge-Kutta fourth order method along with shooting technique. The effects of the various physical dimensionless parameters such as unsteadiness parameter, permeability parameter, magnetic parameter, radiation parameter, Prandtl number, heat generation/absorption parameters and Eckert number for velocity and temperature distributions have been analyzed in detail through graphical representations. Further skin friction coefficient and Nusselt number at the surface are numerated and compared with previous researchers.

11 illus, 2 tables, 23 ref

006361   RAMADHAN M K, AL-RAMMAHI A (Mathematics Dep, Kufa Univ, Kufa, Iraq, Email: adilm.hasan@uokufa.edu.iq) : Image cryptography with least squares approximations. J Comput Sci 2019, 15(11), 1659-68.

Security of digital images has become one of the most important issues in network technologies. Different techniques have been evolved over the past few years to provide security for digital images through concealing information, image scrambling and image encryption. The aim of this work is to enhance the network security level by using the cryptographic method based on the Least Square Approximation technique (LSA). This work is one of the first attempts due to the use of the least squares polynomial of fifth degree technique as data interpolation in the image cryptography process. It has been proven in this study that a high level of correlation exists amongst the original and the corresponding decrypted images. The achieved high values of the measured Peak Signal to Noise Ratio (PSNR) for the decrypted images in this study are an indication to the high accuracy of the proposed technique in comparison to other cryptographic techniques. The output of the encrypted technique in this study reveals its reliability and robustness against security attacks.

3 illus, 5 tables, 19 ref

006352   KUMAR B, SINGH B K (Mathematics Dep, VKSU, Arrah, Bihar) : Representation of character group and duality theory of locally compact abelian and non-abelian groups. J Comput Math Sci 2019, 10, 1695-1718.

This paper deals with the study of duality theory of locally compact abelian and non-abelian groups.

6 ref

006351   KUMAR B, SINGH B K (Mathematics Dep, VKSU, Arrah, Bihar) : Representation theory of compact groups and peter-weyl theorem. J Comput Math Sci 2019, 10(12), 1681-94.

This paper deals with representation theory of compact group and Peter– Weyl Theorem.

7 ref

006350   KUMAR B, SINGH B K (Mathematics Dep, VKSU, Arrah, Bihar) : Basic concepts and definitions of compact groups and fourier analysis. J Comput Math Sci 2019, 10(12), 1673-80.

This paper contains all basic concepts and definitions of compact Groups, Locally Compact Groups and Fourier Series.

2 illus, 3 ref

006357   PANDEY T N, SANAT P (Mathematics and Statistics Dep, D. D. U. Gorakhpur Univ, Gorakhpur- 273 009, Email: tnp1952@gmail.com) : Minimum risk estimation of scale parameter of inverse rayleigh distribution under asymmetric loss function. J Comput Math Sci 2019, 10(12), 1665-72.

The inverse Rayleigh distribution is one of the most important distributions from lifetime model. This distribution has been used frequently as a model for survival analysis, life testing and industrial reliability problems. In this paper, the minimum risk estimators of its scale have been obtained under Linex loss, precautionary loss, and the other two loss functions. The relative efficiencies of the estimators have been calculated for comparison.

1 table, 15 ref

006334   AMUTHA S, DEVI M U (Mathematics Dep, Government Arts & Science Coll, Sivakasi– 626 124, Email: amudha.veerasami@gmail.com) : Total edge fibonacci irregular labeling for fan, wheel and umbrella graph. J Comp Math Sci 2019, 10(12), 1654-64.

A total edge Fibonacci irregular labeling f :V(G) $\cup$E(G)$\to$(1, 2,..., k) of a graph G = (V, E) is a labeling of vertices and edges of G in such a way that for any different edges xy and x' y' their weights f(x) + f(xy) +f(y) and f( x') + f( x' y') + f( y') are distinct Fibonacci numbers. The total edge Fibonacci irregularity strength, tefs(G), is defined as the minimum k for which G has a total edge Fibonacci irregular labeling. If a graph has a total edge Fibonacci irregular labeling, then it is called a total edge Fibonacci irregular graph. In this paper, we determined the exact value of the total edge Fibonacci irregularity strength of Fan graph Fn , double Fan graph DFn, umbrella graph U(n, m) and Wheel graph Wn.

2 illus, 6 ref

006354   MAITI S I, SAIKIA S K (Statistics Dep, Visva-Bharati, Santiniketan, West Bengal, Email: surjyasurjya@gmail.com) : Effect size- A magic wand for fickle P-value in experimental biological research. J Adv Sci Res 2019, 10(4), 1-7.

Inappropriate representation of statistical significance as p value ≤0.05 is a convention in experimental biological research. Such convention in Medical Science research led to statistical fallacy with risk to life of people in reality. This convention which often ignores the sample size, compromises with the Type II error that directly confers fallacy to biological findings explained only on p value. One way to overcome such situation is to incorporation of effect size in the analysis. In this article, the need of addressing effect size in experimental research has been explained. A summary of use and interpretation of various measures of effect size has also been outlined.

1 illus, 2 tables, 38 ref

006358   PANGHAL P, KUMAR M, RANI S (Mathematics and Statistics Dep, Chaudhary Charan Singh Hisar Agriculture Univ, Hisar-125 004, Email: m25424553@gmail.com) : Estimation of annual compound growth rates of Guava (Psidium guajava L.) fruit in Haryana using Non linear model. J Appl Nat Sci 2019, 11(4), 778 -84.

Computation of growth rates plays an important role in agricultural and economic research to study growth pattern of a various commodities. Many of the research workers used the parametric approach for computation of annual growth rate but not use the concept of non-linear model. In this paper, an attempt has been made to study growth rates of guava for three districts (Hisar, and Kurukshetra) and Haryana state as a whole using different non-linear models. The time-series data on annual area and production of guava (Psidium guajava L.) in different districts of Haryana from 1990-91 to 2015-16 were collected to fit non linear models. Growth rates were computed through best-fitted non-linear models. It was found that Logistic model could be best fit for computation of growth rates of area for guava fruit in Hisar and Kurukshetra district and Haryana state as a whole whereas Gompertz model was best fit for Yamunanagar district based on high R² and least MSE and RMSE values. It was also observed that monomolecular model was best fit for production of guava fruits in Hisar and Yamunanagar district whereas Logistic model was best fit for production of guava fruit in Kurukshetra and Haryana state as a whole because of high R² and least MSE and RMSE values. R and excel software have been used for fitting the non linear model and computation of growth rates for area and production of guava fruit for the year 1990-91 to 2015-16. None has been used the non linear model growth model for computation of annual growth rate of guava fruit for area and production of Haryana state. But in this work non linear growth model has been used for computation of growth rate instead of parametric approaches.

8 illus, 5 tables, 9 ref

006337   BHATTACHARYYA D, PADHEE S, PAT K C (Mathematics Dep, National Institute of Technology, Rourkela- 769 008) : Modeling of DC–DC converter using exact feedback linearization method: A discussion. IETE J Res 2019, 65(6), 843-54.

DC–DC converter finds widespread use in variety of applications. DC–DC converter inherently possess nonlinear dynamics, therefore different linearization schemes are used to get a linearized model of the converter. The classical state-space averaging method is one of the most widely used linearizing method for DC–DC converter. But the said linearization method has some inherent limitation which limits the accuracy of the developed linearized model. This paper provides a discussion of mathematical modeling of DC–DC converter operating in continuous conduction mode as well as in discontinuous conduction mode using exact feedback linearization (EFL) technique which is based on Lie algebra. A linearized model of the converter is obtained using EFL and the effect of change in duty cycle and change in circuit parameters on the frequency response has been studied. The frequency response of the EFL model and nonlinear model is matching within the range of duty cycle and load resistance considered in this paper.

9 illus, 41 ref

006363   RENUKA J, JESINTHA J J, PADMAVATHI E (PG Mathematics Dep, Women’s Christian Coll, Chennai- 600 008, Email: renukajjm@gmail.com) : K-odd sequential harmonious labeling of some graphs. Bull Pure Appl Sci Sect E Math Stat 2019, 38E(2), 119-26.

A labeling of a graph is the assigning of natural values to the vertices of the graph in some way that induces edge labels according to certain pattern. Interest in graph labeling began in 1967 when A. Rosa (On certain valuations of the vertices of a graph, Theory of Graphs, (International Symposium, Rome, July 1966), (1967), Gordon and Breach, New York and Dunod Pairs, 349–355) published his paper on graph labeling. In this paper, we prove that Ladder, Pyramid and the degree splitting of Star and Bistar graphs are k- odd sequential harmonious graphs.

8 illus, 6 ref

006335   BALA S, DEVI V K, THIRUSANGU K (S. I. V. E. T Coll, Chennai- 600 073, Email: yesbala75@gmail.com) : Some cordial labelings on square graph of a comb. Bull Pure Appl Sci Sect E Math Stat 2019, 38E(2), 112-8.

In this paper, we prove the existence of 3− total cordial edge magic labeling, 3− total sum cordial labeling and total product cordial labeling for the square graph of a comb.

4 illus, 3 tables, 9 ref

006360   PRABHA R, VADIVUKKARASI P R (Mathematics Dep, Ethiraj Coll for Women, Chennai, Email: Prabha75@gmail.com) : Energy of central and middle graph of a regular graph. Bull Pure Appl Sci Sect E Math Stat 2019, 38E(2), 105-11.

An eigenvalue of a graph G is the eigenvalue of its adjacency matrix. The energy E(G) of G is the sum of the absolute values of its eigenvalues. Two graphs having same energy and same number of vertices are called equienergetic graphs. If µ1, . . . , µn are the eigenvalues of the adjacency matrix of G and θ₁, . . . , θ_n are the eigenvalues of the adjacency matrix of all-one matrix (J_n), then the energy of the Central graph of a simple, connected, r-regular graph E(C(G)) and the energy of a Middle graph of a simple, connected, 2-regular graph E(M(G)) is derived in terms of the eigenvalues of the original graph G by us in this paper.

4 illus, 15 ref

006359   PHILOMENA V S, JOICE P T (PG Mathematics Dep, Women’s Christian Coll, Chennai- 600 008, Email: sharonphilo2008@yahoo.com) : Fibonacci divisor cordial labeling for helm and closed helm. Bull Pure Appl Sci Sect E Math Stat 2019, 38E(2), 99-104.

In this paper we prove the Fibonacci divisor cordial labeling for helm and closed helm CH_n for n= 4m and n = 4m− 1 for any positive integer m.

5 illus, 9 ref

006336   BALA S, SUNDARRAJ S, THIRUSANGU K (S.I.V.E.T. Coll, Chennai- 600 073, Email: kthirusangu@gmail.com) : Cordial labelings for comb related graph. Bull Pure Appl Sci Sect E Math Stat 2019, 38E(2), 92-8.

In this paper we investigate the existence of homo-cordial, hetro-cordial and cup(V )-cordial labeling for the extended triplicate graph of a comb.

4 illus, 7 ref

006345   JESINTHA J J, SUBASHINI K (PG Mathematics Dep, Women’s Christian Coll, Chennai- 600 008, Email: jjesintha 75@yahoo.com) : Cordiality in the path union of vertex switching of wheel graphs in increasing order. Bull Pure Appl Sci Sect E Math Stat 2019, 38E(2), 79-84.

The cordial labeling of a graph G is a function f : V (G) → {0, 1} such that every edge uv in G is consigned the label |f (u) − f (v)| with the property |vf (0) − vf (1)| ≤ 1 and |ef∗ (0) − ef∗ (1)| ≤ 1, where the number of vertices is denoted by vf (i) for i = 0,1 and the number of edges is denoted as ef∗ (i) for i = 0,1. The graph which concedes cordial labeling is called a cordial graph. In this paper, we prove that the path union of vertex switching of wheel graphs is cordial.

5 illus, 8 ref

006346   JESINTHA J J, SUBASHINI K, BEULA J R R (PG Mathematics Dep, Women’s Christian Coll, Chennai- 600 008, Email: jjesintha 75@yahoo.com) : Graceful labeling of some new graphs. Bull Pure Appl Sci Sect E Math Stat 2019, 38E(2), 60-4.

A graceful labeling of a graph G with q edges is an injection f : V (G) → {0, 1, 2, . . . , q} with the property that the resulting edge labels are distinct where the edge incident with the vertices u and v is assigned the label |f (u) − f (v)|. A graph which admits a graceful labeling is called a graceful graph. In this paper, we prove that the series of isomorphic copies of Star graph connected between two Ladders are graceful.

3 illus, 11 ref

006367   SUSEENDRAN C, PRIYADHARSHINI S (Mathematics Dep, Ethiraj Coll for Women, Chennai- 600 008, Email: chitrasuseendran@gmail.com) : Secure domination in degree splitting graphs of certain graphs. Bull Pure Appl Sci Sect E Math Stat 2019, 38E(2), 54-9.

A secure dominating set of a graph G = (V, E) is a dominating set S ⊆ V , if for each u ∈ V − S, there exists a v such that v ∈ N (u) ∩ S and (S − {v}) ∪ {u} is dominating. The minimum cardinality of a secure dominating set is the secure domination number, γs (G). In this paper, we find γs (G) for degree splitting graphs of few classes of graphs like paths, complete binary trees, complete graphs and complete bipartite graphs. Further, bounds for γs (G) of degree splitting graphs of regular graphs and few classes of caterpillars are determined.

2 illus, 10 ref

006356   MUBEEN M I, EMERALD J D (Mathematics Dep, Government Arts Coll for Men, Chennai, Tamil Nadu, Email: iffathmalick@hotmail.com) : Splicing regular matrix grammars using parallel communicating grammar systems. Bull Pure Appl Sci Sect E Math Stat 2019, 38E(2), 24-9.

Extensive research has been done on splicing of words. Further, splicing on words has been extended to arrays in Samdanielthompson et al. (Samdanielthompson, G., David, N. Gnanamalar, Nagar, Atulya K. and Subramanian, K.G., Flat splicing array grammar systems generating picture arrays, International Journal of Computer Information System and Industrial Management Applications, (2016) Vol 8, 336–344). In this context, we propose, a grammar system, using queries to splice regular matrix grammars and show that the language generated by this grammar system is incomparable to the language given in Subramanian et al. (Subramanian, K.G., Mary, A. Roslin Sagaya and Dersanambika, K.S., Splicing array grammar systems, Proceedings of the Second International conference on Theoretical Aspects of Computing ICTAC, (2005), 125–135.) and has more generative power than in Samdanielthompson et al. (op. cit.).

2 illus, 5 ref

006340   EMERALD PRINCESS SHEELA J D, PARIMALAM C K, SHARON PHILOMENA V, JENCY A (Mathematics Dep, Queen Mary’s Coll, Chennai, Tamil Nadu) : Generating pictures using P system on matrix grammars. Bull Pure Appl Sci Sect E Math Stat 2019, 38E(2), 18-23.

The study of image or picture generation has been of great interest for researchers as they find a variety of applications such as character recognition, pictorial information system design, pattern recognition and so on. Matrix grammars are an extension of string grammars to two dimensions in formal language theory, to generate picture array languages. P system is an area of membrane computing which was proposed by P˘aun inspired by the structure and functioning of living cells. P systems are useful in dealing with different problems, including the problem of handling picture array generation. In this paper we define P systems for matrix grammars with regular and context free rules which brings out the suitability of P system model for picture array generation.

3 illus, 5 ref

006341   EMERALD PRINCESS SHEELA J D, THANGA MURUGESHWARI V, J. JEBA JESINTHA, DEVADHARSHIN C D (Mathematics Dep, Queen Mary’s Coll, Chennai, Tamil Nadu, Email: jjesintha 75 @yahoo.com) : Generation of C_m · P ^l _n graphs using P system. Bull Pure Appl Sci Sect E Math Stat 2019, 38E(2), 12-7.

Graph grammars, used for parsing and generating graphs, provide a useful formalism for describing structural manipulation of multi-dimensional data. Hence graph grammars are used as a synonym for graph rewriting system, especially in the context of languages. Hyperedge replacement graph grammars, edge replacement graph grammars and node replacement graph grammars are the three types of graph replacement techniques in which hyperedge replacement graph grammar is used in this paper. P system is a membrane structure consisting of objects and evolution rules in it. The objects can be transformed from on membrane to the other or can dissolve the membrane. In this paper we generate C_m · P ^l _n graphs for m ≥ 3, n ≥ 2, l ≥ 2 using the P system.

6 illus, 9 ref

006344   JESINTHA J J, GLORY R J, BHABITA G (P.G. Mathematics Dep, Women’s Christian Coll, Chennai, Tamil Nadu, Email: godblessglory@gmail.com) : Odd gracefulness of the chain of trim kite graph and bipartite graph. Bull Pure Appl Sci Sect E Math Stat 2019, 38E(2), 7-11.

In graph theory, graph labeling technique was first introduced in the mid 1990’s. Most of the graph labeling methods had their origin from the graceful labeling which was first introduced by Rosa (A. Rosa (1967). On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Symposium, Rome, July 1966), Gordon and Breach, New York and Dunod Paris, 349–355) in the year 1967. Odd graceful labeling is one such method introduced by Gnanjothi (R.B. Gnanajothi, Topics in Graph Theory, Ph. D. Thesis, Madurai Kamaraj University, Tamil Nadu, India (1991)) in the year 1991. In this paper we prove that the chain of a trim kite graph and a bipartite graph is odd graceful.

7 illus, 8 ref

006338   CHAKRABARTY D (Statistics Dep, Handique Girls' Coll, Guwahati- 781 001, Email: dhritikesh.c@rediffmail.com) : A general method of defining average of function of a set of values. Aryabhatta J Math Inform 2019, 11(2), 269-84.

A general method has been derived, in this study, for defining the average of a function of a set (or a list) of values which can describe/yield most of the definitions/formulations of average. This paper is based on the derivation of this general method of defining average along with the derivations of various definitions/formulations of average from the method obtained.

31 ref

006342   GOWRI R, RAJAYAL A K R (Mathematics Dep, Government Coll for Women(Autonomous), Kumbakonam, Tamil Nadu, Email: gowrigck@rediffmail.com) : Supra semi alpha open sets in supra bitopological spaces. Aryabhatta J Math Inform 2019, 11(2), 263-8.

In this paper, we define and study the notion of S_(τ_ij )-Sα-open and closed sets in supra bitopological spaces. Also, we analysed the properties of these sets. Mathematics Subject Classification: 54D05, 54D10, 54D08, 54D20.

11 ref

006366   SREEDHAR C (NBKR Institute of Science & Technology, Vidyanagar- 524 413, Email: csreedhar.nainu@gmail.co) : Extension of linear 2-functionals in complex 2-normed linear space. Aryabhatta J Math Inform 2019, 11(2), 259-62.

In this paper we extended the theorem of Das in more general way over the field of K where K is the field R of real numbers or field C of complex numbers using norm technique of Hahn Banach theorem in normed linear spaces.

4 ref

006365   SINGH T P, YADAV H K (Mathematics Dep, Baba Mast Nath Univ, Asthal Bohar Rohtak, Email: hrkyadav12@gmail.com) : Fuzzy inventory model with weibull distributed deterioration along variable demand and time varying holding cost. Aryabhatta J Math Inform 2019, 11(2), 249-58.

The present paper investigates a fuzzy continuous inventory model for deteriorating items with variable demand. The deteriorating rate follows weibull distribution. In Real world situation it has been observed that the cycle time of every supply chain system is uncertain hence, it has described with triangular fuzzy parameters. The Yager's formula and signed distance method is applied to de-fuzzify the cost function. A numerical illustration has been proposed to validate the model. The sensitivity analysis has been carried out to explore the effect of minor changes in parameters with the optimal solution associated to different parameters.

4 tables, 19 ref

006339   DEVI M, GUPTA V K, KUMAR V (Govt. Coll Behrampur, Panipat, Haryana, Email: ghalyan.manju@gmail.com) : Determination of MTSF and availability of a three unit standby stochastic system by using base state. Aryabhatta J Math Inform 2019, 11(2), 237-48.

To evaluate easily and quickly, the key parameters of a stochastic system for its profit/cost analysis, has always remained the need of the hour. While using the Regenerative Point Technique, many state equations are to be written and solved recursively after taking Laplace/Stieltjes transforms of the state equations and then applying the concepts of limits, it takes a lot of time. Also while using Regenerative Point Graphical Technique (RPGT), introduced by Gupta [4], the presence of many circuits make the calculations lengthy. To overcome this difficulty, Gupta et al [3] introduced a new concept of 'base state' for doing reliability analysis to find the key parameters of the system (under steady state conditions) by using modified formulae of RPGT. In the present paper, the path analysis of a three unit stand-by stochastic system, in which initially, two units are in operative mode and a similar unit in cold stand-by mode, has been done to determine the 'base state' of the system, which is further used to determine the MTSF and Availability of the system using modified RPGT (under steady state conditions).

1 illus, 11 tables, 9 ref

006368   TOMAR S, HOODA N (Mathematics Dep, Kanya Mahavidyalaya, Sonepat, Haryana, Email: s_saroha30@yahoo.com) : Non-Archimedean stability of system of AQ reciprocal functional equations. Aryabhatta J Math Inform 2019, 11(2), 227-36.

In this paper, we will prove generalised Hyers-Ulam-Rassias stability of a system of additive and quadratic reciprocal functional equations in non - Archimedean normed spaces.

11 ref

006348   KAUR A R, ARORA B S, JHA C P C (Institute of Information Technology & Management, Janakpuri, New Delhi, Email: rrdk_07@yahoo.com) : Optimal component selection for embedded systems with mandatory redundancy in critical modules based on reuse-build-buy decision. Aryabhatta J Math Inform 2019, 11(2), 213-26.

Embedded systems are developed in the form of modular, distributed components connected in serial or parallel. The modularized software components are a result of CBSD, wherein the components can be either obtained as COTS components, developed in-house or re-used after modification. This decision is based upon a number of internal and external factors and is known as Reuse-Build-Buy Decision. Software industries are majorly involved in reusing the components in order to save development& testing time and associated cost. Reusability of components depends upon several parameters such as cost and time for modifications & testing, reliability etc. In life-critical embedded systems, rendering the system to run in the external, uncontrolled environment will be unwise. It becomes obligatory to identify the critical modules and incorporate fault tolerance, which further implies additional cost. In this paper, RB/1/1 fault tolerant architecture has been implemented on embedded systems with mandatory redundancy for critical modules to make the system more reliable. The aim of the paper is to select hardware and software components for the modular system to maximize the overall reliability and minimizing the overall cost and SLOC with a constraint of mandatory redundancy for critical modules under RB/1/1 architecture incorporating Reuse-build-or-buy decision.

2 illus, 4 tables, 27 ref

006362   RENU, BHATIA P (Mathematics Dep, Baba Mastnath Univ, Rohtak, Haryana, Email: renushiv85@gmail.com) : Reliability analysis of die casting machine system having two types repair facility with condition of rest. Aryabhatta J Math Inform 2019, 11(2), 205-12.

We are taking two high pressure die casting machines case, initially the primary unit is working while the second unit is at cold standby, non priority is standby mode the unit cannot fail in standby mode. All the state possibilities for the system have been discussed in up states and down states. On failure of the primary unit, standby and secondary unit is operative and failed unit is put under repair immediately. The secondary unit may be working to certain time called maximum allowable operative time. The maximum allowable operative time is over the second unit is put under rest if the primary unit is in progress. In this mode two types of repair facility is considered one ordinary repairman and other expert repairman. The expert repairman is only need based when the ordinary repairman fails to repair the failed primary unit. Secondary unit is always repaired by ordinary repairman. The Failure time of primary as well as secondary unit are exponentially distributed but with different parameters. The repair time distribution of primary unit is arbitrary for both expert repairman and ordinary repairman with different probability distributions. The repair time of standby unit follows exponential distribution .The distribution of rest time, maximum allowable operative time and patience time of ordinary repairman are also assumed to follow exponential distribution distributions with different parameters. Using mean time to system failure, steady state, availabilities, expected no of visit of the system, system behavior & profit function have been discussed in all possible cases.

4 illus, 9 ref

006343   JANAKI G, SARANYA C (PG & Research Mathematics Dep, Cauvery Coll for Women, Trichy-18, Email: janakikarun@rediffmail.com) : Correlation between pyramidal numbers. Aryabhatta J Math Inform 2019, 11(2), 197-204.

A pyramidal number is a figurate number that represents to a pyramid with a polygonal base and a given number of triangular sides. In this paper, we have made an attempt to obtain various interesting relations among pyramidal numbers and other special numbers.

12 ref

006349   KUMAR A, PRUTHI M (Mathematics Dep, Indira Gandhi Univ, Meerpur-Rewari, Haryana, Email: ajay61286@gmail.com) : Applications of Eulerian graph for mathematical and real situation. Aryabhatta J Math Inform 2019, 11(2), 193-6.

Graph Theory is an important topic of discrete mathematics and computer science having applications in modeling a variety of real life situations in many disciplines. This article is intended for the attention of young researchers and provides an introductory discussion of some well known application problems in which role of eulerian graph and eulerian circuit is significant. In this paper, we have made an attempt to use the eulerian graph to solve the mathematical and real life situations.

1 illus, 10 ref

006353   KUSUM, SINGH T P (B.M.N Univ, Asthal Bohar, Rohtak, Email: kusumsansanwal123@gmail.com) : Performance measure of 2 - stage fuzzy scheduling with single trasport agent using robust ranking technique. Aryabhatta J Math Inform 2019, 11(2), 185-92.

Scheduling problems are concerned in searching an optimal or near optimal sequence in which a number of tasks can be performed subject to the number of constraints. In most of these problems, the performance measures are a function of the order or sequence of jobs. Though a number of heuristic techniques have been developed to solve the scheduling problems, yet many of these are often not practical in dynamic real world scenario due to complex constraints & unexpected disruptions. This paper put forward an algorithm to find the performance measure for fuzzy scheduling in which the transport agent returns on first machine after delivering the task on second machine. The processing time has been considered with triangular fuzzy parameters. The basic idea is to convert the fuzzy processing time in the crisp value by applying Robust Ranking technique. Robust ranking index is easier to apply and helpful tool in fuzzy decision making environment. The algorithm is based on a theorem which has been verified. Numerical illustration is also given to justify the study.

11 ref

006364   SINGH A P (Mathematics Dep, Jubail Industrial Coll, Jubail, Saudi Arabia, Email: arungeit@yahoo.co.in) : Mathematical model on stationary information system in a distributed service network. Aryabhatta J Math Inform 2019, 11(2), 181-4.

In Distributed Service Network, the main concern of the user is mostly focused on the immediate response time of a call for service. The user evaluates the efficiency of a service network in terms of how long he or she has been waiting until a service unit arrives at the scene of a call. Short-term policies such as dispatching, repositioning or routing depend critically on the information available and the communications system operated by the dispatcher at the moment a service request arises. Thus, information has great value in distributed service network. In this paper the mathematical model is proposed for the expected response time under stationary information system.

13 ref

006347   KALANOV T Z (Home of Physical Problems, Tashkent- 100 128 , Uzbekistan, Email: tzk_uz@yahoo.com) : Definition of derivative function: Logical error in mathematics. Aryabhatta J Math Inform 2019, 11(2), 173-80.

The critical analysis of the foundations of the differential calculus is proposed. Methodological basis of the analysis is the unity of formal logic and of rational dialectics. It is shown that differential calculus is fictitious mathematical theory because the concept of the limiting process is the starting point for definition of the derivative function. The passage to the limit "zero" in the definition of the derivative function signifies that the variable quantity takes the only essential value "zero". This fact leads to the following errors.The definition of the derivative function is based on the violation of the necessary and sufficient condition for the validity of the relationship between the increment of the function argument and the increment of the function because the increment of the function is divided by the zero increment of the argument in the case of the limiting process. The definition of the derivative function is based on the contradiction which is that the increment of the argument is both zero and not zero in the same relationship. This contradiction represents a violation of the formal-logical law of identity and of the formal-logical law of the lack of contradiction. The definition of the differential of function is based on two contradictory (mutually exclusive) features: the differential of the argument is not zero while the increment of the argument in the definition of the derivative function is zero. Keywords: general mathematics, foundations of mathematics, differential calculus, integral calculus, methodology of mathematics, philosophy of mathematics, philosophy and mathematics, mathematics education, logic, physics.

16 ref

006355   MISHRA P, SINGH U, PANDEY C M, MISHRA P, PANDEY G (Biostatistics and Health Informatics Dep, Sanjay Gandhi Post Graduate Institute of Medical Sciences, Lucknow, Uttar Pradesh, Email: mishrapk79@gmail.com) : Application of student’s t-test, analysis of variance, and covariance. Ann Card Anaesth 2019, 22(4), 407-11.

Student’s t test (t test), analysis of variance (ANOVA), and analysis of covariance (ANCOVA) are statistical methods used in the testing of hypothesis for comparison of means between the groups. The Student’s t test is used to compare the means between two groups, whereas ANOVA is used to compare the means among three or more groups. In ANOVA, first gets a common P value. A significant P value of the ANOVA test indicates for at least one pair, between which the mean difference was statistically significant. To identify that significant pair(s), we use multiple comparisons. In ANOVA, when using one categorical independent variable, it is called one‑way ANOVA, whereas for two categorical independent variables, it is called two‑way ANOVA. When using at least one covariate to adjust with dependent variable, ANOVA becomes ANCOVA. When the size of the sample is small, mean is very much affected by the outliers, so it is necessary to keep sufficient sample size while using these methods.

1 table, 24 ref

005108   UFUKTEPE U, KULAHCIOGLU B, AKMAN O (Mathematics Dep, Illinois State Univ, Normal, USA, Email: oakman@ilstu.edu) : Stability analysis of a prey refuge predator–prey model with allee effects. J Biosci 2019, 44(4), 85.

We constructed a discrete-time predator–prey model by adding prey refuge and Allee effects (predator saturation on prey and mate limitation on predator) to an earlier prey–predator model and examined its dynamics. We show the existence of positive fixed points and study the stability properties. The numerical simulations and bifurcation diagrams verify the impact of refuge and the Allee mechanism on the system.

13 illus, 3 tables, 15 ref

005102   PANGHAL P, KUMAR M, RANI S (Mathematics and Statistics Dep, Chaudhary Charan Singh Hisar Agriculture Univ, Hisar- 125 004, Email: m25424553@gmail.com) : Estimation of annual compound growth rates of guava (Psidium guajava L.) fruit in Haryana using non linear model. J Appl Nat Sci 2019, 11(4), 778 -84.

Computation  of  growth  rates  plays  an  important  role  in  agricultural  and  economic research to study growth pattern of a various commodities. Many of the research workers used the parametric approach for computation of annual growth rate but not use the concept of non-linear model.  In this paper, an attempt has been made to study growth rates of guava for three districts (Hisar, and Kurukshetra) and Haryana state as a whole using different non-linear models. The time series data on annual area and production of guava (Psidium guajava L.) in different districts of Haryana from 1990-91 to 2015-16 were collected to fit non linear models. Growth rates were computed through best fitted non-linear models. It was found that Logistic model could be best fit for computation of growth rates of area for guava fruit in Hisar and Kurukshetra district and Haryana state as a whole whereas  Gompertz  model  was  best  fit for  Yamunanagar  district  based  on high R² and least MSE and RMSE values. It was also observed that monomolecular model was best fit for production of guava fruits in Hisar and Yamunanagar district whereas Logistic model was best fit for production of guava fruit in Kurukshetra and Haryana state as a whole because of high R² and least MSE and RMSE values. R and excel software have been used for fitting the non linear model and computation of growth rates for area and production of guava fruit for the year 1990-91 to 2015-16. None has been used the non linear model growth model for computation of annual growth rate of guava fruit for area and production of Haryana state. But in this work non linear growth model has been used for computation of growth rate instead of parametric approaches.

8 illus, 5 tables, 9 ref

005094   CHATTERJEE N, YADAV N (Mathematics Dep, Indian Institute of Technology Delhi, Hauz Khas- 110 016) : Fuzzy rough set-based sentence similarity measure and its application to text summarization. IETE Tech Rev 2019, 36(5), 517-25.

Fuzzy Rough Sets are designed for decision-making with uncertainty, imprecision, and incompleteness in data. We propose to use Fuzzy Rough Sets for the task of sentence similarity-based Text Summarization. Text data inherently possess uncertainty, imprecision, and incompleteness for data representation. Two sentences may be equivalent in their meanings despite having different vector space representation while Fuzzy Rough Sets incorporates the meanings of sentences. Fuzzy Rough Set-based sentence similarity for Text Summarization has not been proposed in literature before the present work. The contribution of the research is two-fold, namely (i) Fuzzy Rough Set-based sentence similarities has been proposed and validated on SICK2014 dataset. (ii) The proposed similarities between the sentences are thereby proposed for Single document Text Summarization and evaluated for DUC2002 dataset. Experimental results confirm the applicability and efficiency of using the proposed models for both sentence similarity computations as well as for summarization.

1 illus, 4 tables, 30 ref

005093   CHAKRABARTY D (Statistics Dep, Handique Girls' Coll, Guwahati- 781 001, Email: dhritikesh.c@rediffmail.com) : A general method of defining average of function of a set of values. Aryabhatta J Math Inform 2019, 11(2), 269-84.

A general method has been derived, in this study, for defining the average of a function of a set (or a list) of values which can describe/yield most of the definitions/formulations of average. This paper is based on the derivation of this general method of defining average along with the derivations of various definitions/formulations of average from the method obtained.

31 ref

005096   GOWRI R, RAJAYAL A K R (Mathematics Dep, Government Coll for Women, Kumbakonam, Tamil Nadu, Email: gowrigck@rediffmail.com) : Supra semi alpha open sets in supra bitopological spaces. Aryabhatta J Math Inform 2019, 11(2), 263-8.

In this paper, we define and study the notion of S_(τ_ij )-Sα-open and closed sets in supra bitopological spaces. Also, we analysed the properties of these sets.

11 ref

005106   SREEDHAR C (NBKR Institute of Science & Technology, Vidyanagar- 524 413, Email: csreedhar.nainu@gmail.co) : Extension of linear 2-functionals in complex 2-normed linear space. Aryabhatta J Math Inform 2019, 11(2), 259-62.

In this paper we extended the theorem of Das in more general way over the field of K where K is the field R of real numbers or field C of complex numbers using norm technique of Hahn Banach theorem in normed linear spaces.

4 ref

005105   SINGH T P, YADAV H K (Mathematics Dep, Baba Mast Nath Univ, Asthal Bohar, Rohtak, Email: hrkyadav12@gmail.com) : Fuzzy inventory model with weibull distributed deterioration along variable demand and time varying holding cost. Aryabhatta J Math Inform 2019, 11(2), 249-58.

The present paper investigates a fuzzy continuous inventory model for deteriorating items with variable demand. The deteriorating rate follows weibull distribution. In Real world situation it has been observed that the cycle time of every supply chain system is uncertain hence, it has described with triangular fuzzy parameters. The Yager's formula and signed distance method is applied to de-fuzzify the cost function. A numerical illustration has been proposed to validate the model. The sensitivity analysis has been carried out to explore the effect of minor changes in parameters with the optimal solution associated to different parameters.

4 tables, 19 ref

005095   DEVI M, GUPTA V K, KUMAR V (Govt. Coll Behrampur, Panipat, Haryana, Email: ghalyan.manju@gmail.com) : Determination of MTSF and availability of a three unit standby stochastic system by using base state. Aryabhatta J Math Inform 2019, 11(2), 237-48.

To evaluate easily and quickly, the key parameters of a stochastic system for its profit/cost analysis, has always remained the need of the hour. While using the Regenerative Point Technique, many state equations are to be written and solved recursively after taking Laplace/Stieltjes transforms of the state equations and then applying the concepts of limits, it takes a lot of time. Also while using Regenerative Point Graphical Technique (RPGT), introduced by Gupta, the presence of many circuits make the calculations lengthy. To overcome this difficulty, Gupta et al introduced a new concept of 'base state' for doing reliability analysis to find the key parameters of the system (under steady state conditions) by using modified formulae of RPGT. In the present paper, the path analysis of a three unit stand-by stochastic system, in which initially, two units are in operative mode and a similar unit in cold stand-by mode, has been done to determine the 'base state' of the system, which is further used to determine the MTSF and Availability of the system using modified RPGT (under steady state conditions).

11 tables, 9 ref

005107   TOMAR S, HOODA N (Mathematics Dep, Kanya Mahavidyalaya, Sonepat, Haryana, Email: s_saroha30@yahoo.com) : Non-archimedean stability of system of AQ reciprocal functional equations. Aryabhatta J Math Inform 2019, 11(2), 227-36.

In this paper, we will prove generalised Hyers-Ulam-Rassias stability of a system of additive and quadratic reciprocal functional equations in non-Archimedean normed spaces.

11 ref

005099   KAUR A R, ARORA B S, JHA C P C (Institute of Information Technology & Management, Janakpuri, New Delhi, Email: rrdk_07@yahoo.com) : Optimal component selection for embedded systems with mandatory redundancy in critical modules based on reuse-build-buy decision. Aryabhatta J Math Inform 2019, 11(2), 213-26.

Embedded systems are developed in the form of modular, distributed components connected in serial or parallel. The modularized software components are a result of CBSD, wherein the components can be either obtained as COTS components, developed in-house or re-used after modification. This decision is based upon a number of internal and external factors and is known as Reuse-Build-Buy Decision. Software industries are majorly involved in reusing the components in order to save development& testing time and associated cost. Reusability of components depends upon several parameters such as cost and time for modifications & testing, reliability etc. In life-critical embedded systems, rendering the system to run in the external, uncontrolled environment will be unwise. It becomes obligatory to identify the critical modules and incorporate fault tolerance, which further implies additional cost. In this paper, RB/1/1 fault tolerant architecture has been implemented on embedded systems with mandatory redundancy for critical modules to make the system more reliable. The aim of the paper is to select hardware and software components for the modular system to maximize the overall reliability and minimizing the overall cost and SLOC with a constraint of mandatory redundancy for critical modules under RB/1/1 architecture incorporating Reuse-build-or-buy decision.

2 illus, 5 tables, 27 ref

005103   RENU, BHATIA P (Mathematics Dep, Baba Mastnath Univ, Rohtak, Haryana, Email: renushiv85@gmail.com) : Reliability analysis of die casting machine system having two types repair facility with condition of rest. Aryabhatta J Math Inform 2019, 11(2), 205-12.

We are taking two high pressure die casting machines case, initially the primary unit is working while the second unit is at cold standby, non priority is standby mode the unit cannot fail in standby mode. All the state possibilities for the system have been discussed. in up states and down states. On failure of the primary unit, standby and secondary unit is operative and failed unit is put under repair immediately. The secondary unit may be working to certain time called maximum allowable operative time. The maximum allowable operative time is over the second unit is put under rest if the primary unit is in progress. In this mode two types of repair facility is considered one ordinary repairman and other expert repairman. The expert repairman is only need based when the ordinary repairman fails to repair the failed primary unit .Secondary unit is always repaired by ordinary repairman. The Failure time of primary as well as secondary unit are exponentially distributed but with different parameters. The repair time distribution of primary unit is arbitrary for both expert repairman and ordinary repairman with different probability distributions. The repair time of standby unit follows exponential distribution.The distribution of rest time, maximum allowable operative time and patience time of ordinary repairman are also assumed to follow exponential distribution distributions with different parameters. Using mean time to system failure, steady state, availabilities, expected no of visit of the system, system behavior & profit function have been discussed in all possible cases.

6 illus, 9 ref

005097   JANAKI G, SARANYA C (PG & Research Mathematics Dep, Cauvery Coll for Women, Trichy-18, Email: janakikarun@rediffmail.com) : Correlation between pyramidal numbers. Aryabhatta J Math Inform 2019, 11(2), 197-204.

A pyramidal number is a figurate number that represents to a pyramid with a polygonal base and a given number of triangular sides. In this paper, we have made an attempt to obtain various interesting relations among pyramidal numbers and other special numbers.

12 ref

005100   KUMAR A, PRUTHI M (Mathematics Dep, Indira Gandhi Univ, Rewari, Haryana, Email: ajay61286@gmail.com) : Applications of eulerian graph for mathematical and real situation. Aryabhatta J Math Inform 2019, 11(2), 193-6.

Graph Theory is an important topic of discrete mathematics and computer science having applications in modeling a variety of real life situations in many disciplines. This article is intended for the attention of young researchers and provides an introductory discussion of some well known application problems in which role of eulerian graph and eulerian circuit is significant. In this paper, we have made an attempt to use the eulerian graph to solve the mathematical and real life situations.

10 ref