YADAV R N, SRIVASTAVA A K, BANIK B K

**047645** YADAV R N, SRIVASTAVA A K, BANIK B K (f Mathematics and Natural Sciences, Prince Mohammad Bin
Fahd Univ, Alkhobar, Kingdom of Saudi Arabia, Email: bimalbanik10@gmail.com; bbanik@pmu.edu.sa) : **Microwave - Induced Bismuth Nitrate - Catalyzed Michael
Reaction of 3 - Amino ? - Lactams with Enones
.** Asian J. Chem 2020, 32 (2), 233 - 6.

Microwave - induced bismuth nitrate - catalyzed reaction of 3 - amino ? - lactams with unsaturated ketones is performed in order to obtain substituted amino ? - lactams. Amino ? - lactams were obtained through the strategy of [2+2] ketene - imine cycloaddition followed by deprotection of phthalimido ? - lactams with ethylene diamine.

3 tables, 20 ref

YADAV H K , SINGH T P, KUMAR V

**047590** YADAV H K , SINGH T P, KUMAR V (Baba Mast Nath Univ, Rohtak, Haryana, Email: hrkyadav12@gmail.com, tpsingh78@yahoo.com) : **Time horizon inventory cost model having weibull distributed deterioration with shortages and partial backlogging.** AJMI 2020, 12 (1), 85 - 92.

In the present study, an inventory cost models for weibull rate of deteriorating items with constant holding cost and quadratic demand rate, partial backlogging over a time horizon is proposed. The demand rate is considered as quadratic function of time. Shortages are allowed and partially backlogging case is discussed. The main objective of this paper is to minimize the total cost per unit time in an inventory system. Numerical problems are presented to demonstrate the result. Also, the effect of changes in different parameters on optimal total cost is graphically presented.

2 tables, 17 ref

SARANYA P, JANAKI G

**047589** SARANYA P, JANAKI G (Mathematics Dep, Cauvery Coll for Women, Tiruchirappalli, Tamilnadu, Email: saranyap.maths@cauverycollege.ac.in) : **On some non - extendable gaussian triples involving mersenne and gnomonic number.** AJMI 2020, 12 (1), 81 - 4.

In this paper we made an attempt to generate some non - extendable Gaussian triples involving Mersenne number, Gnomonic number under three sections with the properties D (1), D (9) and D (64).

29 ref

GUPTA D, VANDANA, KAUR M

**047588** GUPTA D, VANDANA, KAUR M (Mathematics Dep, Maharishi Markandeshwar (Deemed to be Univ), Mullana, Ambala, Email: vandanarana452@gmail.com, kaurm5092@gmail.com ) : **Minimizing total waiting time of j0bs in specially structured two stage flowshop with disjoint job block concept.** AJMI 2020, 12 (1), 65 - 72.

The present paper is aimed to provide algorithm for minimizing the total waiting time of jobs for specially structured two stage flow - shop scheduling. The model includes the processing times associated with probabilities and transportation time with disjoint job block criteria. The algorithm is made clear by numerical illustration. The lemma has been provided on which study is based.

9 tables, 14 ref

BHATIA P, RANI V

**047587** BHATIA P, RANI V (Mathematics Dep, G.N.Coll, BMU Rohtak, Email: raheja.veena@yahoo.com) : **Performance analysis of a paper machine considering its minor/major faults and power failure.** AJMI 2020, 12 (1), 57 - 64.

The purpose of present paper is to analyze the performance of a paper machine using stochastic modeling of a single unit considering various types of minor/major faults and removal thereof by repair/replacement of different parts as well as power failure during operation of the machine. On occurrence of minor faults, operation of the machine is partially stopped while due to major faults, machine stopped working completely. A real data related to different failures is collected from a famous paper mill in Northern Haryana namely Sri Jagdumbe Paper Mills Ltd. located at Sirsa (Haryana). Various measures of system - effectiveness such as MTSF, Reliability, Availability and Busy period etc. are derived by using Semi - Markov process and Regenerative Point technique. Using collected data, various numerical results and graphs are drawn and performance of the system along with profit is analyzed, which may be helpful for the maintenance team of the paper mill.

4 illus, 11 ref

KUSUM D

**047586** KUSUM D (GC Rithoj, Gurugram, Haryana, Email: kusum.aug13@gmail.com) : **Behaviorial analysis of a heterogeneous feedback queue system.** AJMI 2020, 12 (1), 49 - 56.

The paper studies the behavior analysis of a network of queue system having two heterogeneous service channels wherefeed - backing are allowed for defective items. The arrivals and departures at each channel follows poissonlaw. Connectingthe various probabilities, the differential difference equations have been formed. The marginal mean queue size for each channel and for the entire system has been explored. On the basis of graphical study the behavior of the system has been studied. An effort has been made to analyze how the system behaves on changing the parameter.

8 illus, 2 tables, 13 ref

LAL D R

**047585** LAL D R (DAV Institute of Management, Sector - 10, Chandigarh, Email: drroshanlal999@gmail.com
) : **Impact of global recession on modern business - A critical study.** AJMI 2020, 12 (1), 35 - 48.

Entrepreneurship in India is rightly defined as a rich man's monopoly & a poor man's golden dream. With the end of the license raj, a good amount of opportunity was given to upcoming entrepreneurs in India but without a proper policy & sufficient financial support from the government, the result is a haphazard growth of industries & business. With oil prices reaching a high, inflation touching Himalayan peaks it is a make or break for the interested entrepreneurs. During the recession time, many banks including international ones having gone bust, there is no financial support in the form of loans, etc to be given to the prospective entrepreneurs. The outbreak of pandemic Corona Virus (Covid - 19) all over the world has disturbed the political, social, economic, religious and financial structures of the whole world. The situation of the World has become worst. As a result, the normal process of establishment & nurturing, the young entrepreneurs are in danger and on the other hand, institutions which depend upon industries & business people for fulfilling their placement commitments to their students are increasingly at disadvantageous position. It appears that both the modern entrepreneurs & the technical institutions are at the cross roads, whether to meet for a healthier growth & development or depart on different tunes in different directions leaving the scenario unchanged for the worse to come. In this situation , what should be the role of budding entrepreneurs coming out of deemed universities & other higher institutions and that of industry leaders whether they are MNC'S or SSI'S is the topic of discussion in this paper.

31 ref

SONI S, MISHRA D R

**047584** SONI S, MISHRA D R (Mathematics Dep, Pt. Rajpati Degree Coll Patti, Pratapgarh, (U.P.), Email: rag2010maths4u@gmail.com) : **Summation formulae involving generalized I?Function of two variables.** AJMI 2020, 12 (1), 31 - 4.

The aim of this paper is to derive some summation formulae involving generalized I?function of two variables.

3 ref

SINGH N J, MEDIRATTA D S R

**047583** SINGH N J, MEDIRATTA D S R (Principal Yamuna Polytechnic of Engg, Yamuna Nagar, Haryana, Email: nirmal_jit@yahoo.com) : **Simulation and analysis of quadratic assignment problems (QAP) using ant colony optimization.** AJMI 2020, 12 (1), 025 - 30.

It is the human nature to look for proficient and effectual optimization of any problem. This paper presents optimization of quadratic assignment problem. This article describes ant colony optimization for better results in QAP. The algorithm is based on the development of a mathematical model to tackle the problems of allocation of distributors and wholesalers/retailers. The Simulation is done on 15 distributors and 15 wholesalers/retailers and a matrix of 15 X 15 has been prepared for the distances between them. MATLAB is used for the design of GUI which is helpful for the Assignment Problem. Both situations of Assignment Problem i.e. Maximization or Minimization have been covered in this paper. The results have been tested for Laxmi Plywood data.

2 illus, 25 ref

CHAUHAN M S, SHRIVASTAVA R, VERMA R, KABIR Q A

**047582** CHAUHAN M S, SHRIVASTAVA R, VERMA R, KABIR Q A (Mathematics Dep, Saifia Science Coll, Bhopal , M.P, Email: dr.msc@rediffmail.com, rajeshraju0101@rediffmail.com, rupali.varma1989@gmail.com, qaziaftabkabir@gmail.com) : **Some non - unique fixed point theorems of ?IRI? type on parametric b - metric spaces.** AJMI 2020, 12 (1), 019 - 24.

In this paper, we establish some fixed point theorem for some non - unique fixed point theorems of ?iri? type on parametric b - metric space, inspired by the interesting results of ?iri? [2]. We investigate some existing non unique fixed points of certain operators in the context of parametric b - metric spaces. Our main result generalizes and unifies several existing results in the literature.

17 ref

GUPTA D, KAUR M, VANDANA

**047581** GUPTA D, KAUR M, VANDANA (Mathematics Dep, Maharishi Markandeshwar (Deemed to be Univ), Mullana, Ambala, Email: kaurm5092@gmail.com, vandanarana452@gmail.com) : **Two stage specially structured flowshop scheduling including transportation time, probabilities associated with processing time and job block criteria : A review paper.** AJMI 2020, 12 (1), 011 - 8.

The present paper is aimed to provide algorithm minimizing the total waiting time of jobs for specially structured two stage flowshop scheduling. The model includes the transportation time , job block criteria , probabilities are associated with processing time of machines. The algorithm is made clear by numerical illustration. The paper is a review study made by various researchers of the related field. The lemma has been provided on which study is based.

9 tables, 13 ref

SARANYA C, JANAKI G

**047580** SARANYA C, JANAKI G (Mathematics Dep, Cauvery Coll for Women, Trichy - 18, Email: c.saranyavinoth@gmail.com, janakikarun@rediffmail.com) : **Ramanujan - type diophantine equation involving jarasandha numbers.** AJMI 2020, 12 (1), 007 - 10.

In this paper, we accomplish integer solutions to the Ramanujan - type diophantine equation involving jarasandha numbers.

13 ref

SAINI D S L

**047579** SAINI D S L (Maths Dep, MLN (PG) Coll, Yamuna Nagar, Email: hesynr.org@gmail.com) : **Physical significance of ? and ? in the ? - ? definition of continuity of a function.** AJMI 2020, 12 (1), 001 - 6.

The ? - ? definition of continuity of a function is theoretically taught in classroom but generally the teachers fail to clarify the concept & subject matter in real world situation. In this paper we have made an attempt to explore the physical significance of ? & ? occurring in the definition to make it more clear and understandable. On the basis of graphical study and explaining the meaning of definition through practical situation the concept is made self explanatory. Even the layman can understand the meaning and concept easily.

2 illus, 1 ref

Davis S

**000867** Davis S (Research Foundation of Southern California, USA, Email: sbdavis@resfdnsca.org) : **The classical limit of quantum commutation relations.** Bull Pure Appl Sci Sect E Math Stat 2021, 40E(1), 14-7.

The commutation relations of quantum mechanics have a classical limit equal to the Poisson brackets if the coefficient is generalized to be a complex rather than purely imaginary. The effect on the uncertainty relations is described. The complex number may be identified with a modular variable and the quantization is derived from topology

6 ref

Pankajam V and Gowri S

**000866** Pankajam V and Gowri S (Mathematics Dep, Sri G.V.G. Visalakshi Coll for Women, Tamil Nadu-642 128, Email: pankajamgurusamy@gmail.com) : **On generalized star ωαI-closed sets in ideal topological spaces.** Bull Pure Appl Sci Sect E Math Stat 2021, 40E(1), 121-26.

In this paper, we introduce a new class of sets named as generalized star ωαI-closed sets in ideal topological spaces and study some of their properties. Further we also define and study the concept of ωαI-open sets in ideal topological spaces and discuss their properties.

15 ref

Myithili K K and Beulah R D

**000865** Myithili K K and Beulah R D (Mathematics Dep, Vellalar Coll for Women, Tamil Nadu-642 128, Email: mathsmyth@gmail.com) : **Recognizing emotions using elementary hyperedges in intuitionistic fuzzy soft hypergraphs.** Bull Pure Appl Sci Sect E Math Stat 2021, 40E(1), 108-20.

In this paper, we introduce the notion of intuitionistic fuzzy soft hypergraphs(IFSHGs), certain types of IFSHGs including core, simple, elementary, sectionally elementary, (µ, ν)-tempered (IFSHGs) are dealt with examples. Also this concept is analyzed in emotion recognition as an application too.

8 illus, 6 tables, 19 ref

KALAVATHI A, KOHILA T, UPADHYAYA L M

**000864** KALAVATHI A, KOHILA T, UPADHYAYA L M (Mathematics Dep, Sri G.V.G. Visalakshi Coll, Tamil Nadu-642 128, Email: kohilathangaraj99@gmail.com) : **On the degenerate Elzaki transform.** Bull Pure Appl Sci Sect E Math Stat 2021, 40E(1), 99-107.

Based on the Fourier transform, Elzaki in 2011 (Tarig M.Elzaki, The new integral transform Elzaki transform, Global Journal of Pure and Applied Mathematics, 7(1), 57–64 (2011)) defined the Elzaki transform. Motivated by the work Kim and Kim (Taekyun Kim and Dae San Kim, Degenerate Laplace transform and degenerate gamma function, Russian Journal of Mathematical Physics, 24(2), 241–248 (2017)) on the intro-duction of the degenerate Laplace transform, in this paper, we introduce the degenerate ELzaki transform and investigate some of its properties and relations. In particular we find the degenerate Elzaki transforms of the degenerate sine, the degenerate cosine, the degenerate hyperbolic sine and the degenerate hyperbolic cosine functions. Moreover, we investigate a scale preserving theorem for the degenerate Elzaki transform and establish that the degenerate Elzaki transform is a theoretical dual transform to the degenerate Laplace transform.

20 ref

GHADLE K P, MOHAMMED M A, GHADLE A K

**000863** GHADLE K P, MOHAMMED M A, GHADLE A K (Mathematics Dep, Dr. Babasaheb Ambedkar Marathwada Univ, Maharashtra-431 004, Email: drkp.ghadle@gmail.com) : **Usage of the finite difference method for solving one-dimensional heat equations.** Bull Pure Appl Sci Sect E Math Stat 2021, 40E(1), 87-98.

In this paper the one-dimensional heat equations with the heat generation arising in the associated fractal transient conduction is investigated. Analytical solutions are obtained by using the finite difference method (FDM). The method, in general, is easy to implement and it yields good results. In addition, the uniqueness and stability results are also discussed. Some illustrative examples are included to demonstrate the validity and applicability of the technique

3 illus, 5 tables, 25 ref

BRAHMA J, BEHERA D K, RATH P

**000861** BRAHMA J, BEHERA D K, RATH P (Mathematics Dep, Ravenshaw Univ, Cuttack, Odisha, Email: dwiti78@gmail.com) : **Solution of some multi-objective nonlinear programming problems.** Bull Pure Appl Sci Sect E Math Stat 2021, 40E(1), 70-4.

In this paper a method for solving bi-objective nonlinear programming problem is derived. Here, the bi-objective nonlinear programming problem is converted into a crisp problem using Zimmermann’s fuzzy programming technique with linear membership function and after that the crisp problem is solved by separable programming technique to find the optimal compromise solution. A numerical example is given for the sake of illustration.

12 ref

BARUG F, SERENBAY S K

**000860** BARUG F, SERENBAY S K (Mathematics Dep, Harran Univ, Turkey, Email: barugfahri@gmail.com) : **Approximation of modified favard szasz-mirakyan operators of maximum-product type.** Bull Pure Appl Sci Sect E Math Stat 2021, 40E(1), 60-9.

Nonlinear positive operators by means of maximum and product were intro-duced by B. Bede. In this paper, we introduce nonlinear maximum-product type modified Favard Szasz–Mirakyan operators. Our main purpose is to give a theorem on the rate of convergence.

10 ref

Prasad R S, Kumar R, Prasad B G

**000859** Prasad R S, Kumar R, Prasad B G (Mathematics Dep, Government Engineering Coll, Bihar, Email: shekharraj.2010@gmail.com) : **Study of couple stresses on MHD poiseuille flow through a porous medium past an accelerated plate.** Bull Pure Appl Sci Sect E Math Stat 2021, 40E(1), 45-59.

In this paper, we discuss an analytical study of the unsteady MHD flowof an incompressible electrically conducting couple stress fluid in the presence of porous medium between two parallel plates when the lower plate is accelerating while the upper plate is at rest, taking into account the pulsation of the pressure gradient effect and under the influence of a uniform magnetic field of strength B0. The solution to the problem is obtained with the help of the perturbation technique. Analytical expressions are given for the velocity field, the shear stresses on the boundaries and the discharge between the plates and the effects of the various governing parameters entering into the problem are reported graphically to depict the interesting aspects of the solution.

15 illus, 27 ref

UPADHYAYA L M, SHEHATA A, KAMAL A

**000858** UPADHYAYA L M, SHEHATA A, KAMAL A (Mathematics Dep, Municipal Post Graduate Coll, Uttarakhand, Email: lmupadhyaya@rediffmail.com) : **An update on the Upadhyaya transform.** Bull Pure Appl Sci Sect E Math Stat 2021, 40E(1), 26-44.

It is almost two years now when the Upadhyaya transform was introduced by the first author (Upadhyaya, Lalit Mohan, Introducing the Upadhyaya integral transform, Bull. Pure Appl. Sci. Sect. E Math. Stat., 38(E)(1), 471–510, doi 10.5958/2320- 3226.2019.00051.1 https://www.researchgate.net/publication/334033797)) as the most powerful, versatile and robust generalization and unification of a number of variants of the classical Laplace transform which have appeared in the mathematics research literature during the years 1993 to 2019. In this paper we provide an update on the Upadhyaya transform, where we explain the definition the one-dimensional Upadhyaya transform and its n-dimensional generalization in more detail and we show that how many other various variants of the classical Laplace transform, that have come to our notice since then and most of which are introduced into the mathematics research literature during the past two years by a number of authors after the advent of the Upadhyaya transform, follow as a special case of the Upadhyaya transform. We find the Upadhyaya transform of some trigonometric and hyperbolic functions, the sine integral, the generalized hypergeometric function and the Bessel function of the first kind in order to exemplify the vast power of the Upadhyaya transform and we also correct a minor typo in the aforementioned paper of the first author.

40 ref

RAM P, KUMAR A

**003426** RAM P, KUMAR A (Mathematics Dep, National Institute of Technology, Kurukshetra - 136 119, Email: anilkumar6150013@gmail.com) : **Hematite suspension based absorbent pad inclined slider influenced by slip and squeeze velocity with altering film ratio.** Def Sci J 2021, 71 (2), 185 - 91.

The effects of various entities like slip and squeeze velocities, inlet-outlet film ratio, and the material parameter have been fairly explored in a hematite suspension based absorbent (porous) pad inclined slider. Mathematical expressions for pressure, load capacity (lifting force), friction, friction coefficient, and position of centre of pressure (COP) in terms of the above physical parameters have been acquired. Jenkins model has been employed as a mathematical set of governing equations. It has been found that an increase in the squeeze velocity has enhanced the load capacity and diminished the friction coefficient whereas the escalating values of slip velocity and material properties have reversed the trends. Besides, the optimum value of the inlet-outlet film ratio for maximum load capacity has reduced with a rise in the squeeze velocity. Improvement in material parameters shifted the position of COP slightly towards the inlet while an enhancement in the squeeze velocity and film ratio shifted the same slightly towards the outlet. The results acquired in the present paper will be helpful in designing and modifying the various types of fluid dynamic slider bearings.

5 illus, 4 tables, 30 ref

ARBEL J, FAVARO S

**000746** ARBEL J, FAVARO S (Grenoble Alpes Univ, Grenoble INP, France) : **Approximating predictive probabilities of Gibbs-type priors.** Sankhya A 2021, 83-A(1), 496-519.

Gibbs-type random probability measures, or Gibbs-type priors, are arguably the most “natural” generalization of the celebrated Dirichlet prior. Among them the two parameter Poisson–Dirichlet prior certainly stands out in terms of mathematical tractability and interpretability of its predictive probabilities, which made it the natural candidate in a plethora of applications. Given a random sample of size n from an arbitrary Gibbs-type prior, we show that the corresponding predictive probabilities admit a large n approximation, with an error term vanishing as o(1/n), which maintains the same desirable features as the predictive probabilities of the two parameter Poisson–Dirichlet prior. Our result is illustrated through an extensive simulation study, which includes an application in the context of Bayesian nonparametric mixture modeling.

6 illus, 44 ref

LIN G D, HU C-Y

**000745** LIN G D, HU C-Y (Hwa-Kang Xing-Ye Foundation, Taipei, Taiwan) : **Formulas of absolute moments.** Sankhya A 2021, 83-A(1), 476-95.

The absolute moments of probability distributions are much more complicated than conventional ones. By using a direct and simpler approach, we retreat Hsu’s (J. Chinese Math. Soc. N.S. 1, 257–280, 1951) formulas in terms of the characteristic function (which have been ignored in the literature) and provide some new results as well. The case of nonnegative random variables is also investigated through both characteristic function and Laplace–Stieltjes transform. Besides, we prove that the distribution of a nonnegative random variable with a finite fractional moment can be completely determined by a proper subset of the translated fractional moments. This improves significantly Hall’s (Z. Wahrsch. Verw. Gebiete 62, 355–359, 1983) result for distributions on the right half-line.

37 ref

EGHWERIDO J T

**000744** EGHWERIDO J T (Federal Univ of Petroleum Resources, Effurun, Nigeria) : **The alpha power gompertz distribution: characterization, properties, and applications.** Sankhya A 2021, 83-A(1), 449-75.

A new three-parameter model called the Alpha power Gompertz is derived, studied and proposed for modeling lifetime Poisson processes. The advantage of the new model is that, it has left skew, decreasing, unimodal density with a bathtub shaped hazard rate function. The statistical structural properties of the proposed model such as probability weighted moments, moments, order statistics, entropies, hazard rate, survival, quantile, odd, reversed hazard, moment generating and cumulative functions are investigated. The new proposed model is expressed as a linear mixture of Gompertz densities. The parameters of the proposed model were obtained using maximum likelihood method. The behaviour of the new density is examined through simulation. The proposed model was applied to two real-life data sets to demonstrate its flexibility. The new density proposes provides a better fit when compared with other existing models and can serve as an alternative model in the literature.

5 illus, 4 tables, 31 ref

BAHRAOUI T, KOLEV N

**000743** BAHRAOUI T, KOLEV N (Sao Paulo Univ, Sao Paulo, Brazil) : **New measure of the bivariate asymmetry.** Sankhya A 2021, 83(A), 421-48.

A new measure of the bivariate asymmetry of a dependence structure between two random variables is introduced based on copula characteristic function. The proposed measure is represented as the discrepancy between the rank–based distance correlation computed over two complementary orderpreserved sets. Generalproperties of the measure are established, as well as an explicit expression for the empirical version. It is shown that the proposed measure is asymptotically equivalent to a fourth–order degenerate V -statistics and that the limit distributions have representations in terms of weighted sums of an independent chi-square random variables. Under dependent random variables, the asymptotic behavior of bivariate distance covariance and variance process is demonstrated. Numerical examples illustrate the properties of the measures.

1 illus, 2 tables, 55 ref

BAUMANN M H

**000742** BAUMANN M H (Bayreuth Univ, Bayreuth, Germany) : **A new stochastic Fubini-type theorem on interchanging expectations and Ito integrals.** Sankhya A 2021, 83(A), 408-20.

When a stochastic process is given through an It¯o integral, i.e. a stochastic integral, or a stochastic differential equation (SDE), an analytical solution does not have to exist—and even if there is a closed-form solution, the derivation of this solution can be very complex. When the solution of the stochastic process is not needed but only the expected value as a function of time, the question arises whether it is possible to use the expectation operator directly on the stochastic integral or on the SDE and to somehow calculate the expectation of the process as a Riemann integral over the expectation of the integrands and integrators. In this paper, we show that if the integrator is linear in expectation, the expectation operator and an It¯o integral can be interchanged. Additionally, we state how this can be used on SDEs and provide an application from the field of technical trading, i.e. from the field of mathematical finance.

11 ref

LONE S A

**000741** LONE S A (LONE S A, Saudi Electronic Univ, Riyadh, Kingdom of Saudi Arabia) : **Design and analysis of accelerated life testing and its application under rebate warranty.** Sankhya A 2021, 83(A), 393-407.

The study deals with the advancement of accelerated life testing in the field of product warranty. The expected total cost and expected cost rate for age replacement is estimated under warranty policy using accelerated life testing (ALT) plans. Under constant stress, the lifetimes of the units are assumed to follow generalised exponential distribution. The estimation process is carried through maximum likelihood estimation method. Also, the Age Replacement Policy under Pro-rate Rebate Warranty is discussed. Finally, an application example is presented to illustrate the theoretical results.

4 tables, 20 ref

JENSEN J L

**000740** JENSEN J L (Aarhus Univ, Aarhus, Denmark) : **On the use of saddlepoint approximations in high dimensional inference.** Sankhya A 2021, 83(A), 379-92.

Inference in high dimensional parameter space poses many challenges. One of these is the possible use of saddlepoint approximations. Motivated by a recent use of the saddlepoint approximation to construct a conditional test, we argue that the precision is questionable. We illustrate this by an example giving a 50% relative error in the calculation of the p-value. A power study of the underlying test reveals a low power in many situations. As an alternative it is suggested to use the likelihood ratio test.

4 illus, 1 table, 7 ref

QIN Y, LEI Q

**000739** QIN Y, LEI Q (Guangxi Normal Univ, Guilin, China) : **Empirical likelihood for mixed regressive, spatial autoregressive model based on GMM.** Sankhya A 2021, 83(A), 353-78.

For the parameters in a mixed regressive, spatial autoregressive (MRSAR) model, it is shown, in this article, that for every generalized method of moments (GMM) estimator, there is an empirical likelihood (EL) estimator which has the same asymptotic variance as the GMM estimator. Specifically, we show that there exists an EL estimator which is asymptotically efficient as the best GMM estimator and the EL confidence region for the parameters in the MRSAR model is constructed without estimating the asymptotic variance.

14 tables, 27 ref

SLAOUI Y

**000738** SLAOUI Y (Poitiers Univ, Chasseneuil, France) : **Data-driven deconvolution recursive kernel density estimators defined by stochastic approximation method.** Sankhya A 2021, 83(A), 312-52.

In this paper we show how one canimplement in practice the bandwidth selection in deconvolution recursive kernel estimators of a probability density function defined by the stochastic approximation algorithm. We consider the so called super smooth case where the characteristic function of the known distribution decreases exponentially. We show that, using the proposed bandwidth selection and some special stepsizes, the proposed recursive estimator will be very competitive to the nonrecursive one in terms of estimation error and much better in terms of computational costs. We corroborate these theoretical results through simulations and a real dataset.

6 illus, 10 tables, 40 ref

DOU X, KURIKI S, LIN G D, RICHARDS D

**000737** DOU X, KURIKI S, LIN G D, RICHARDS D (Waseda Univ, Tokyo, Japan) : **Dependence properties of B-Spline copulas.** Sankhya A 2021, 83(A), 283-311.

We construct by using B-spline functions a class of copulas that includes the Bernstein copulas arising in Baker’s distributions. The range of correlation of the B-spline copulas is examined, and the Fr´echet–Hoeffding upper bound is proved to be attained when the number of B-spline functions goes to infinity. As the B-spline functions are well-known to be an order-complete weak Tchebycheff system from which the property of total positivity of any order follows for the maximum correlation case, the results given here extend classical results for the Bernstein copulas. In addition, we derive in terms of the Stirling numbers of the second kind an explicit formula for the moments of the related B-spline functions on the right half-line.

1 table, 27 ref

FALK M, STUPFLER G

**000736** FALK M, STUPFLER G (Wurzburg Univ, Wurzburg, Germany) : **The min-characteristic function: Characterizing distributions by their min-linear projections.** Sankhya A 2021, 83(A), 254-82.

Motivated by a (seemingly previously unnoticed) result stating that d−dimensional distributions on (0, ∞) d are characterized by the collection of their min-linear projections, we introduce and study a notion of min-characteristic function (min-CF) of a random vector with strictly positive components. Unlike the related notion of max-characteristic function which has been studied recently, the existence of the min-CF does not hinge on any integrability conditions. It is itself a multivariate distribution function, which is continuous and concave, no matter which properties the initial distribution function has. We show the equivalence between convergence in distribution and pointwise convergence of min-CFs, and we also study the functional convergence of the min-CF of the empirical distribution function of a sample of independent and identically distributed random vectors. We provide some further insight into the structure of the set of min-CFs, and we conclude by showing how transforming the components of an arbitrary random vector by a suitable one-to-one transformation such as the exponential function allows the construction of a notion of min-CF for arbitrary random vectors.

23 ref

LEFEBVRE M

**000735** LEFEBVRE M (Polytechnique Montreal, Montreal, Canada) : **Moments of first-passage places for jump-diffusion processes.** Sankhya A 2021, 83(A), 245-53.

Let T be the first time that the time-homogeneous jump-diffusion process X(t) leaves the interval (a, b). The jump size is assumed to be a positive constant. We derive exact formulas for E[Xk(T)] in special cases. For the general case, an approximate analytical expression can be obtained for E[Xk(T)]. This expression is precise when the jump size is small.

3 illus, 8 ref

SUTRADHAR B C

**000734** SUTRADHAR B C (Carleton Univ, Ottawa, Canada) : **An overview on econometric models for linear spatial panel data.** Sankhya A 2021, 83(A), 206-44.

When spatial data are repeatedly collected from the same spatial locations over a short period of time, a spatial panel/longitudinal data set is generated. Thus, this type of spatial longitudinal data must exhibit both spatial and longitudinal correlations, which are not easy to model. This work is motivated by existing studies in statistics and econometrics literature but the proposed model and inference procedures should be applicable to the spatial panel data encountered in other fields as well such as environmental and/or ecological setups. Specifically, unlike the existing studies, we propose a new dynamic mixed model to accommodate both spatial and panel correlations. A complete theoretical analysis is given for the estimation of regression effects, and spatial and panel correlations by exploiting second and higher order moments based quasi-likelihood methods. Asymptotic properties are also studied in details. The step by step estimation results developed in the paper should be useful to the practitioners dealing with spatial panel data..

18 ref

FOTOPOULOS S B, JANDHYALA V K, PAPARAS A

**000733** FOTOPOULOS S B, JANDHYALA V K, PAPARAS A (Washington State Univ, Pullman, USA) : **Some properties of the multivariate generalized hyperbolic laws.** Sankhya A 2021, 83(A), 187-205.

The purpose of this study is to characterize multivariate generalized hyperbolic (MGH) distributions and their conditionals by considering the MGH as a subclass of the mean-variance mixing of the multivariate normal law. The essential contribution here lies in expressing MGH densities by utilizing various integral representations of the Bessel function. Moreover, in a more convenient form these modified density representations are more advantageous for deriving limiting results. The forms are also convenient for studying the transient as well as tail behavior of MGH distributions. The results include the normal distribution as a limiting form for the MGH distribution. To support the MGH model an empirical study is conducted to demonstrate the applicability of the MGH distribution for modeling not only high frequency data but also for modeling low frequency data. This is against the currently prevailing notion that the MGH modelis relevant for modeling only high frequency data.

3 tables, 32 ref

RATTIHALLI R N, PATIL S B

**000732** RATTIHALLI R N, PATIL S B (Central Univ of Rajasthan, Ajmer, Rajasthan) : **Data dependent asymmetric kernels for estimating the density function.** Sankhya A 2021, 83(A), 155-86.

In the nonparametric setup, smooth density estimators are obtained by using a kernel function and it is used to account for contribution of each observation to the estimator. Generally, these kernel functions are symmetric about zero. We propose an estimator by using data dependent asymmetric kernels being generated by an unimodal symmetric kernel. To account for the contribution of the i th order statistic to the estimator, a suitable asymmetric kernel is used. Asymptotic Bias, Mean Square Error (MSE) and Mean Integrated Square Error (MISE) of the estimator are obtained with higher accuracies by using some properties of the densities of order statistics. The convergence rates depend on both n, the sample size and hn, the band width. Expressions for the optimum band width minimizing the asymptotic MISE and the rate of convergence of the optimum MISE are obtained. Simulation study and data analysis indicate that the proposed estimator not only reduces bias outside the support significantly but also it is closer to the true density function as compared to that of the usual estimator based on a common symmetric kernel. Scope for the multivariate generalization of the method is given. Some methods of generating multivariate skew symmetric kernels are described. As an illustration one of them has been used in the simulation study and estimation of density in bivariate case

11 illus, 3 tables, 26 ref

KIM S, PARK C J

**000731** KIM S, PARK C J (Louisiana Univ, Lafayette, USA) : **An asymptotic conditional test of independence in Bernoulli sequences using the number of runs given the number of successes.** Sankhya A 2021, 83(A), 143-54.

In this paper, we prove the asymptotic normality of the conditional distribution of the number of runs given the number of successes for a sequence of independent Bernoulli random variables. In our proof, the Frobenius-Harper technique is used to represent the number of runs as the sum of independent and not necessarily identically distributed Bernoulli random variables. Then, an asymptotic conditional test for independence is provided. Our simulation results exhibit that the test based on conditional distribution performs better that one based on unconditional distribution, over the entire range of success probability and first order correlation. In addition, the UMVUEs of the factorial moments and the probabilities of the number of runs are presented in this paper.

4 tables, 10 ref

MUKHERJEE A, SINGH A

**000730** MUKHERJEE A, SINGH A (Pondicherry Univ, Puducherry- 605 014) : **Optimal systematic sampling when sample size is odd in the presence of linear trend and two-way linear trend.** Sankhya A 2021, 83(A), 128-42.

In the present paper, a new sampling scheme is proposed for sample selection for odd sample size where population size is a multiple of sample size for a population characteristic having a linear or a two-way linear trend. On comparison of efficiencies of the proposed estimator with the simple random sampling without replacement, linear systematic sampling, stratified sampling and balanced systematic sampling, it is observed from theory as well as real life data, that the proposed scheme performs better.

11 tables, 9 ref

ODA R, YANAGIHARA H, FUJIKOSHI Y

**000729** ODA R, YANAGIHARA H, FUJIKOSHI Y (Hiroshima Univ, Hiroshima, Japan) : **Strong consistency of log-likelihood-based information criterion in high-dimensional canonical correlation analysis.** Sankhya A 2021, 83(A), 109-27.

We consider the strong consistency of a log-likelihood-based information criterion in a normality-assumed canonical correlation analysis between q- and p-dimensional random vectors for a high-dimensional case such that the sample size n and number of dimensions p are large but p/n is less than 1. In general, strong consistency is a stricter property than weak consistency; thus, sufficient conditions for the former do not always coincide with those for the latter. We derive the sufficient conditions for the strong consistency of this log-likelihood-based information criterion for the high-dimensional case. It is shown that the sufficient conditions for strong consistency of several criteria are the same as those for weak consistency obtained by Yanagihara et al.

1 table, 17 ref

DIERCKX G, GOEGEBEUR Y, GUILLOU A

**000728** DIERCKX G, GOEGEBEUR Y, GUILLOU A (Southern Univ, Odense, Denmark) : **Local robust estimation of pareto-type tails with random right censoring.** Sankhya A 2021, 83(A), 70-108.

We propose a nonparametric robust estimator for the tail index of a conditional Pareto-type distribution in the presence of censoring and random covariates. The censored distribution is also of Pareto-type and the index is estimated locally within a narrow neighbourhood of the point of interest in the covariate space using the minimum density power divergence method. The main asymptotic properties of our robust estimator are derived under mild regularity conditions and its finite sample performance is illustrated on a small simulation study. A real data example is included to illustrate the practical applicability of the estimator.

10 illus, 35 ref

SUTRADHAR B C

**000727** SUTRADHAR B C (Carleton Univ, Ottawa, Canada) : **Two stage cluster sampling based asymptotic inferences in survey population models for longitudinal count and categorical data.** Sankhya A 2021, 83-A, 26-69.

Sutradhar (2008, Sankhya B, 18-33) has studied a GLMM (generalized linear mixed model) for count data in a finite population setup where a sample of families/clusters were chosen from a finite population using PPS (probability proportional to size) sampling scheme. The properties of the estimators were studied through a simulation study. In this paper we consider (1) an auto-regressive type dynamic (ARTD) model for longitudinal count data, and (2) a MDL (multinomial dynamic logits) model for longitudinal categorical/multinomial data; and develop a practically important two-stage cluster sampling (TSCS) design weights based estimating equations for the regression and dynamic dependence parameters involved in both ARTD and MDL models in a finite population setup. Specifically, the survey weighted generalized quasi-likelihood (WGQL) and the survey weighted maximum likelihood (WML) estimation approaches are used for the count and multinomial models, respectively. We also demonstrate in the paper that the proposed TSCS based WGQL and WML estimators specially for the regression parameters (of main interest) are consistent, whereas the existing ‘working’ longitudinal correlations based GEE (generalized estimating equations) fails to produce consistent estimates and/or consistent variance estimates. The variances of the regression estimates along with their unbiased estimates are derived for both count and multinomial models. Furthermore, the asymptotic normality of the regression estimators are developed based on a large number of independent but non-identical clusters under non-overlapping stratums.

32 ref

QIN Y

**000726** QIN Y (Guangxi Normal Univ, Guilin, China) : **Empirical likelihood for spatial autoregressive models with spatial autoregressive disturbances.** Sankhya A 2021, 83(A), 1-25.

The empirical likelihood ratio statistics are constructed for the parameters in spatial autoregressive models with spatial autoregressive disturbances. It is shown that the limiting distributions of the empirical likelihood ratio statistics are chi-squared distributions, which are used to construct confidence regions for the parameters in the models.

3 tables, 16 ref

YEPUGANTI K, REDDY G R

**000427** YEPUGANTI K, REDDY G R (Vellore Institute of Technology, Vellore- 632 014) : **A novel strong decorrelation approach for image subband coding using polynomial evd algorithms.** Proc Natl Acad Sci India Sect A Phys Sci 2021, 91(1), 147-57.

Subband coding is a popular technique to achieve multichannel data compression and efficient data transmission in image and video communications. In this paper, we focus on designing a data-dependent image subband coder which divides the image into strongly (total) decorrelated and spectrally majorized subbands, which seeks to reduce the redundancy to achieve better compression and efficient transmission. To achieve strong decorrelation and spectral majorization between the subbands, we adopt a set of new iterative polynomial eigenvalue decomposition (PEVD) algorithms: sequential matrix diagonalization (SMD) and maximum element sequential matrix diagonalization (ME-SMD). Using this SMD-based PEVD approach, we design the data dependent subband coder (DDSC) for image subband coding. We compare the performance of the proposed SMD/ME-SMD algorithms with the existing DDSC methods like SBR2, SBR2C, K–L transform (KLT) coder and data independent subband coder (DISC)-based discrete wavelet transform (DWT) technique. To measure the performance, we use the parameters like coding gain, correlation coefficient, MSE (mean square error) and peak signal-to-noise ratio (PSNR). The presented simulation results for standard images in the absence of quantization show that the proposed SMD-based PEVD technique performs far better than the existing techniques.

25 ref

RANA K K, TRIPATHI S, RAW R S

**000426** RANA K K, TRIPATHI S, RAW R S (Indian Institute of Technology (Indian School of Mines), Dhanbad) : **Fuzzy logic-based directional location routing in vehicular ad hoc network.** Proc Natl Acad Sci India Sect A Phys Sci 2021, 91(1), 135-46.

The vehicular ad hoc network (VANET) is an autonomous system of mobile vehicles where mobile vehicles can transmit or receive data packet over the wireless link. Due to the movable nature and limited communication, range of vehicles causes breaking the link between them very frequently, which in turn causes failure in the data delivery. Therefore, a stable path should be established from the source to the destination node to deliver data packets at the intended destination node. Therefore, for a stable route from source to the destination node selection of the next hop should be appropriate. The proposed work in this paper has been made to discuss how the best next hop node can be selected to deliver data packet successfully at the destination node *D*. A mathematical model fuzzy logic-based directional location routing (FLDLR) has been proposed to select a remarkable next hop node in the VANET. To select next hop, FLDLR has considered fuzzy logic sets of the routing metrics, i.e., next hop distance, node speed, closeness, the data transmission rate of the node, and node movement direction. To investigate the performance of FLDLR, the simulation work has been done through network simulator-2 and compared with existing LAR and D-LAR protocols. Through simulated results has shown that the FLDLR outperforms existing protocols from the standpoints routing overhead and packet delivery delay.

22 ref

NOOR-UL-AMIN M, TAYYAB M

**000424** NOOR-UL-AMIN M, TAYYAB M (COMSATS Univ Islamabad-Lahore Campus, Lahore, Pakistan) : **Influence on estimation of mean by using quartile paired double ranked set sampling.** Proc Natl Acad Sci India Sect A Phys Sci 2021, 91(1), 117-21.

In this paper, we present an efficient quartile paired double ranked set sampling (QPDRSS) scheme to estimate the population mean. QPDRSS provides an unbiased improved population mean estimator when the distribution is symmetric and bears negligible biasness for non-symmetric distribution. The performance of the proposed mean estimator is compared to its competitors in simple random sampling, double ranked set sampling, extreme paired double ranked set sampling and paired double ranked set sampling by manipulating simulation process for various (symmetric and non-symmetric) distributions. A real dataset is also studied to exemplify the achievement of the proposal.

24 ref

MISHRA H K, PANDEY R K

**000423** MISHRA H K, PANDEY R K (Applied sciences and Humanities Dep, KNIT, Sultanpur- 228118, U.P) : **Time-fractional nonlinear dispersive type of the zakharov–kuznetsov equation via HAFSTM.** Proc Natl Acad Sci India Sect A Phys Sci 2021, 91(1), 97-110.

In this paper, a semi-analytic solution of time-fractional nonlinear dispersive type of the Zakharov–Kuznetsov (ZK) equations is obtained by the means of homotopy analysis fractional Sumudu transform method. Several examples are evaluated and compared with exact solutions to show the effectiveness of arbitrary order of ZK equations. The results demonstrate that the method is highly reliable and convenient to apply and evaluate without restrictions like finite domain, finite discretization or perturbation of small and large physical parameters.

47 ref

PARASURAMAN L, MOHANAVEL D

**000422** PARASURAMAN L, MOHANAVEL D (Mathematics Dep, Anna Univ, Chennai, Tamilnadu) : **Time-Dependent nonlinear finite difference analysis of buoyancy-driven convective flow over an oscillating porous moving vertical cylinder.** Proc Natl Acad Sci India Sect A Phys Sci 2021, 91(1), 89-96.

The thermal and mass boundary layer flow of a time-dependent isochoric viscous fluid over an oscillating cylinder entrenched in a porous medium is investigated numerically. The governing problem contains a set of coupled nonlinear partial differential equations with significant initial and boundary conditions and are solved using an implicit finite difference scheme, namely Crank–Nicholson method. The importance of physical parameters on velocity profiles, temperature profiles, concentration profiles, local skin-friction coefficient, mean skin-friction coefficient, local Nusselt number, mean Nusselt number, local Sherwood number and mean Sherwood number is explained in detail and is depicted graphically. The validation of these results is verified with the existing results, and it is correlated well.

21 ref

CHALUVARAJU B, BOREGOWDA H S, CANGUL I N

**000421** CHALUVARAJU B, BOREGOWDA H S, CANGUL I N (Mathematics Dep, Bursa Uludag Univ, Bursa- 16059, Turkey) : **Some inequalities for the first general zagreb index of graphs and line graphs.** Proc Natl Acad Sci India Sect A Phys Sci 2021, 91(1), 79-88.

The first general Zagreb index Mα1(G)M1α(G) of a graph *G* is equal to the sum of the ααth powers of the vertex degrees of *G*. For α≥0α≥0 and k≥1k≥1, we obtain the lower and upper bounds for Mα1(G)M1α(G) and Mα1(L(G))M1α(L(G)) in terms of order, size, minimum/maximum vertex degrees and minimal non-pendant vertex degree using some classical inequalities and majorization technique, where *L*(*G*) is the line graph of *G*. Also, we obtain some bounds and exact values of Mα1(J(G))M1α(J(G)) and Mα1(Lk(G))M1α(Lk(G)), where *J*(*G*) is a jump graph (complement of a line graph) and Lk(G)Lk(G) is an iterated line graph of a graph *G*.

33 ref