147

Planery Lecture 1

Aesthetics and Science

Madabusi Raghunathan Professor, DAE-UM Centre for Excellence in Basic Sciences, Mumbai

madabusisr@gmail.com

Abstract

Art, it is readily agreed is a quest for beauty. That the aesthetic drive is a major force behind science is

much less appreciated. In fact in popular perception science is a dry pursuit far removed from beauty. In

this talk we will explore the role of aesthetics in Science and especially its role as motivator behind

science.

148

Planery Lecture 2 Frontiers in Black Hole Physics and Cosmoly

Pankaj S Joshi Vice-Chancellor, Charusat University, and Founding Director, International Center for

Cosmology (ICC)

Abstract

Astronomers call any region of space a `Black hole', if very large quantity of matter

is compacted in a rather small volume. The Einstein theory of gravity has a more strict

requirement, namely that an Event Horizon must form covering a central singularity, for a black

hole to exist. Major recent observational missions, such as the Event Horizon Telescope and LIGO

have recently probed such regimes in the faraway universe, and exciting results have been

reported. We review the current scenario and discuss the fascinating implications, when the Event

Horizon is present, or else when the central singularity is visible. Emerging future perspectives and

opportunities are indicated, including connections to a future Quantum Theory of Gravity.

Bio-sketch: Pankaj S Joshi, was a Senior Professor in Astronomy and Astrophysics at the Tata

Institute of Fundamental Research, Mumbai, before joining CHARUSAT University as their

Provost, and Founding Director for the International Center for Cosmology (ICC). He works on

black hole physics and cosmology and has made fundamental contributions in

gravitation theory. His extensive analysis of general relativistic gravitational collapse is widely

recognized as providing significant insights into the final fate of massive collapsing stars in the

universe, formation of space-time singularities, and cosmic censorship, reported in more than 200

research papers, and in monographs from OUP (Oxford) and CUP (Cambridge),

and other books. He has also contributed a large number of books and articles towards science

outreach, and has given many public lectures, and is awarded with several prizes.

149

IT-1

Some Fascinating Consequences of General Theory of Relativity

Zafar Ahsan

Department of Mathematics,

Maulana Azad National Urdu University

Hyderabad (India)

zafar.ahsan@rediffmail.com

Abstract

Albert Einstein is still an icon for many. This is not just due to the immense impact of his theories but

also because of his wisdom and the social world around him. Einstein had once remarked, “most of the

fundamental ideas of science are essentially simple, and may, as a rule be expressed in a language

comprehensible to everyone.” Einstein has given two theories (i) Special Theory of Relativity (1905) and

(ii) General Theory of Relativity (1916). Day by day the hold of Einstein’s general relativity on rest of the

physics is strengthening as the theory is successfully tested on each and every of its predictions. The

general theory of relativity changed our perception of the universe in which we live and thus

revolutionised the whole physics. This theory links the gravitation and the structure of the spacetime.

There are many appreciable connections between general relativity, particle physics and string theory and

at the same time there are number of exciting astrophysical applications of general relativity such as black

holes, gravitational lensing, the production and detection of gravitational waves, the big bang, the early

universe, the late universe and the cosmological constant, etc. In this talk, some of the consequences of

general relativity have been discussed.

150

IT-2

Bouncing Cosmology in Modified Gravity

P. K. Sahoo Department of Mathematics,

Birla Institute of Technology and Science-Pilani,Hyderabad Campus, Hyderabad

pksahoo@hyderabad

Abstract

The present discussion is devoted to the study of bouncing cosmology in f(R,T) modified gravity where

we presume f(R,T) = R+2λT, being the model parameter. We will discuss here a novel parameterization

of Hubble parameter which is apt in representing a successful bouncing scenario and simultaneously

generate viable estimates of a0(t), q0(t), H0(t)and t0. We proceed to present a complete analysis of the

proposed bouncing model by studying the Hubble slow roll parameters, energy conditions and stability

against linear homogeneous perturbations in flat space-time. We also delineate bouncing cosmology for

the proposed model by employing Quintom matter. The present article further communicate for the first

time that violation of energy-momentum materialize for both the contracting and expanding universes

except for the bouncing epoch with energy flow directed away and into the matter fields for the

contracting and expanding universe respectively. We further present a thorough investigation about the

feasibility of the proposed bouncing scenario against first and generalized second law of thermodynamics.

We found that the proposed bouncing scenario obeys the laws of thermodynamics for the constrained

parameter space of λ. The manuscript conclude after investigating the viability of the proposed bouncing

model in non minimal f(R,T) gravity where f(R,T) = R + χ RT.

151

IT-3

Curvature Properties of Some Spacetimes

Absos Ali Shaikh

Department of Mathematics,

The University of Burdwan,Burdwan, West Bengal, India

aashaikh@math.buruniv.ac.in

Abstract

In the literature of differential geometry, there are various themes of research. The recent trends of

research of modern mathematics are abstraction, generalization, existence, characterization and

application. A spacetime is a connected 4-dimensional Lorentzian manifold. Curvature plays a crucial

role to determine the shape of a space and curvature of a space is determined by its metric tensor. A

manifold is said to be of constant curvature if it is of constant sectional curvature and its Riemannian-

Christo_el curvature R is in the form R = kG, where k is a constant and G is the Gaussian curvature

(where G = 1/2 (g ^ g) with Kulkarni-Nomizu product `^'). During the last eight decades the notion of

manifold of constant curvature has been weakened by many researchers throughout the globe in different

directions such as locally symmetric spaces by Cartan1, semisymmetric spaces by Cartan2, recurrent

manifolds by Ruse3 weakly symmetric spaces by Selberg4, generalized recurrent manifolds by Dubey5 ,

pseudosymmetric spaces by Adamow and Deszcz6, pseudosymmetric spaces by Chaki7, weakly

symmetric spaces by Tam_assy and Binh 8, hyper generalized recurrent manifolds by Shaikh and Patra9,

quasi-generalized recurrent manifolds by Shaikh and Roy10, weakly generalized recurrent manifolds by

Shaikh and Roy11 etc. The above processes of generalization are concerned with the first order or second

order covariant derivatives of various curvature tensors and such generalized structures are called

“Curvature restricted geometric structures". We note that pseudo-symmetric manifold by Chaki is

different from pseudosymmetric manifold by Deszcz and weakly symmetric manifold by Selberg is

different from that by Tamassy and Binh. As a generalization of semi-symmetric spaces, the notion of

pseudosymmetric spaces arose during the study of geodesic mappings by Adamow and Deszcz. This is

called pseudosymmetric spaces by Deszcz or Deszcz-symmetric spaces. The geometric interpretation of

such space is also given by Haesen and Verstraelen12. Hence during the last three decades Deszcz-

symmetric space is an important topic of research in differential geometry. The main objective of this

lecture is to investigate various curvature restricted geometric structures of various spacetimes such as

Som-Raychaudhuri spacetime, pure radiation spacetime, Robinson-Trautman spacetime, Nariai

Spacetime, Melvin Spacetime etc.

References: 1 Bull. Soc. Math. Fr, 1926. 2 Lecons sur la geometrie des espaces de Riemann, 1946. 3 Proc. London Math. Soc., 1949. 4 Indian J. of Math., 1956. 5 Indian J. Pure Appl. Math., 1979. 6Demonstr. Math.,1983.

152

IT-4

Cosmological Study of Particle Creation in Higher Derivative Theory

G. P. Singh Department of Mathematics

Visvesvaraya National Institute of Technology, Nagpur

gpsingh@mth.vnit.ac.in

Abstract

Considering the universe as an open thermodynamic system, the creation of matter particles out of

gravitational energy is interesting area of research. A new class of homogeneous and isotropic

cosmological models with particle creation have been obtained and their dynamical properties are also

examined. The state finder diagnostic pair {r, s} are also scrutinized which characterize the evolution of

these cosmological models.

153

IT-5

A Model Based on Fuzzy Inference System to

Analyze the Trends of Financial Market

Sanjeev Kumar

Dr. Bhimrao Ambedkar University

IBS, Khandari Campus, Agra, India

sanjeevibs@yahoo.co.in

Abstract

Stock investment has become an important investment activity and the internet makes it easier to

exchange stock information and to make stock transactions. Trading system in stock market is full of

uncertainty therefore nobody can make accurate decision for investing their money and therefore

investors often lose money due to unclear investment objective. Predicting the stock market is very

difficult since it depends on several unknown factors. Technical analysis is sometimes used in financial

markets to assist traders to make buying and selling decision. This work will examine a trading model

that combines fuzzy logic and technical analysis to find patterns and trends in financial market. To

accomplish this goal, the daily data of a financial institute from July 2012 to June 2013 is used. Here take

four input factors and use fuzzy logic to find the output. For fuzzifying these input data, trapezoidal

membership function is used and center of gravity method is used for defuzzification of fuzzy output.

Finally, observed that this fuzzy logic model gives best result to put on hold with degree of precision

37.587%.

154

IT-6

Predictive and System Biology: The Role of Mathematics

Jomar Fajardo Rabajante Institute of Mathematical Sciences and Physics,

University of the Philippines Los Baños

jfrabajante@up.deu.ph

Abstract

Mathematics is very useful in predicting scenarios in biology and medicine. Mathematics also plays a big

role in studying complex biological systems using a systems approach. In this talk, I will present two

examples of my researches – one at the molecular level and the other one at the ecological level. The first

study that I will discuss is on predicting glycan-associated biomarkers of lung adenocarcinoma for

diagnostics and drug discovery purposes. Then I will present my study on predicting and controlling

disease epidemics in agriculture using an ecosystems approach.

155

IT-7

Modelling and Understanding Heat Transport and Temperature Variations

within Biological Tissues and Body Organs

M. Kanoria

Formally Department of Applied Mathematics, University of Calcutta, India

Department of Mathematics, Sister Nivedita University, India

k_mri@yahoo.com, mridula.k@snuniv.ac.in

Abstract

In the present analysis, the bioheat equation is studied in the context of memory responses. The heat

transport equation for this problem involving the memory-dependent derivative (MDD) on a slipping

interval in the context of three-phase (3P) lag model under two-temperature theory is formulated and is

then used to study the thermal damage within the skin tissue during the thermal therapy. Laplace

transform technique is implemented to solve the governing equations. The influences of the MDD and

moving heat source velocity on the temperature of skin tissues are precisely investigated. The numerical

inversion of the Laplace transform is carried out using Zakian method. The numerical outcomes of

temperatures are represented graphically. Excellent predictive capability is demonstrated for identification

of an appropriate procedure to select different kernel functions to reach effective heating in hyperthermia

treatment. Significant effect of thermal therapy is reported due to the effect of delay time and the velocity

of moving heat source as well.

IT-8

Mathematics of Life

Anuradha Devi

Department of Mathematics and Dean, Royal School of Applied & Pure Sciences,

The Assam Royal Global University, BetKuchi, Guwahati, Assam

anuradha.devi@rgi.edu.in

Abstract

In this talk, an attempt will be made to discuss the contribution of Mathematics to understand Life. Life

represents duration between birth and death. For any living being, be it human, plant, microorganism,

bacteria etc. within an environment are considered as complex dynamical system. Here, we will try to

understand how Mathematics may be used as useful tool to define such complex dynamical systems.

Starting from a single variable life to interactive living being anything and everything can be described

beautifully by Mathematical tools. It is also important to understand limitation of deriving such natural

system.

156

IT-9

Study on charged Strange Stars in f(R,T) Gravity

Saibal Ray*, D. Deb, S.V. Ketov, M. Khlopov Department of Physics, Government College of Engineering and Ceramic Technology,

Kolkata, West Bengal, India

saibal@associates.iucaa.in

Abstract

We investigate the effects of the modified f(R, T) gravity on the charged strange quark stars with the

standard choice of f(R, T) = R + 2χT. Those types of stars are supposed to be made of strange quark

matter (SQM) whose distribution is governed by the phenomenological MIT bag EOS as p = 1/3 (ρ − 4B),

where B is the bag constant, while the form of charge distribution is chosen to be q (r) = Q(r/R)3 = αr3

with α as a constant. We derive the values of the unknown parameters by matching the interior spacetime

to the exterior Reissner-Nordstrom metric followed by the appropriate choice of the values of the

parameters χ and α. Our study reveals that besides SQM, a new kind of matter distribution originates due

to the interaction between the matter and the extra geometric term, while the modification of the Tolman-

Oppenheimer-Volkoff (TOV) equation invokes the presence of a new force Fc. The accumulation of the

electric charge distribution reaches its maximum at the surface, and the predicted values of the

corresponding electric charge and electric field are of the order of 10 19 - 20 C and 1021 - 22 V/cm,

respectively. To examine the physical validity of our solutions, we perform tests of the energy conditions,

stability against the equilibrium of the forces, the adiabatic index, etc., and find that the proposed f(R, T)

model survives all these critical tests. Therefore, our model can describe the non-singular charged strange

stars and justify the supermassive compact stellar objects having their masses beyond the maximum mass

limit for the compact stars in the standard scenario. Our model also supports the existence of several

exotic astrophysical objects like super-Chandrasekhar white dwarfs, massive pulsars, and even magnetars,

which remain unexplained in the framework of General Relativity (GR).

157

IT-10

Anisotropic Cosmological Model in f(R,T) Gravity

Aroonkumar Beesham1*, Vijay Singh2

1Department of Mathematical Sciences, University of Zululand, South Africa, abeesham@yahoo.com 2Department of Mathematical Sciences, University of Zululand, South Africa, gtrcosmo@gmail.com

Abstract

A plane symmetric locally rotationally symmetric cosmological model is studied in the modified f(R,T)

theory, as opposed to Einstein’s general relativity. Solutions are found by considering a specific form for

the Hubble parameter and by imposing the weak energy condition. Both decelerating as well as

accelerating universes are possible.

Corresponding solutions in general relativity obey the weak energy condition only for late times and can

describe only decelerating and late time accelerating phases. In contrast, we find that our solutions are

valid for the entire evolution and obey the weak energy condition for all times. In other words, our

solutions can admit, in addition to the usual decelerating and accelerating phases, the early inflationary

phase as well. This is one of the most pleasing features of our model.

Keywords: Cosmology, dark energy, f(R,T) theory

158

IT-11

Probing Holographic Ricci Dark energy model with bulk viscosity

C. P. Singh

Department of Applied Mathematics, Delhi Technological University, Delhi, India

cpsphd@rediffmail.com

Abstract

In this talk, we present Ricci dark energy (RDE) model with bulk viscosity to observe the cosmic

accelerating expansion phenomena. It is thought that the negative pressure caused by bulk viscosity can

play the role of a dark energy component. We assume that the total bulk viscosity coefficient is

proportional to the velocity and acceleration of the expansion of the universe in the form, ξ= ξ0 + ξ1H + ξ2

qH, where ξ0, ξ1 and ξ2 are the constants. We show that the model corresponds to early deceleration and

then a smooth transition into an accelerated epoch. We analyze the model with statefinder and Om(z)

diagnostics and find that the model is different from standard ΛCDM model at present but approaches to

ΛCDM in late time. We constrain the model using latest observational data namely Ia Supernovae data

(SN Ia), observed Hubble parameter dataset (OHD) and baryon acoustic oscillations (BAO) measurement

to evaluate the best estimated values of all bulk viscous parameters. It is claimed that the non-viscous

RDE model suffers the age problem. However, we find that viscous RDE alleviates the age problem. The

viscous Ricci dark energy model is compatible to explain the present accelerated expansion of the

universe.

159

IT-12

Some Cosmological and astrophysical features

in the back ground of Finsler geometry

Farook Rahaman

Department of Mathematics,

Jadavpur University, Kolkata, India

farookrahaman@gmail.com

Abstract

Finsler geometry is a generalization of Riemannian geometry. At first we write the self-consistent

gravitational field equation in Finsler spacetime. Using the above modified field equations, we will

study several astrophysical as well as cosmological phenomena.

IT-13

Optimal Assignment of Tasks to the Distributed Computing System

Prof. Avanish Kumar

Chairman, Commission for Scientific & Technical Terminology,

MHRD, New Delhi,

dravanishkumar@yahoo.com

Abstract

In many real-life application domains, (e.g. meteorology, cryptography, signal processing, solar and radar

surveillance, simulation of VLSI circuits, image processing, space science, military science and

engineering systems, astronomy, genetic engineering, and Industrial process monitoring etc.), in which

increased complexity and scale has led to the need for more powerful computation resources; distributed

computing systems have emerged as a powerful platform for addressing such complex problems.

Distributed Computing System consists of a set of cooperating nodes (either homogeneous or

heterogeneous) communicating over the communication links. Distributed Real Time Computing Systems

provides enormous platform for large number of research problems. An application running in a

distributed system could be partitioned into a number of tasks and executed concurrently on different

nodes of the system, referred to as the task assignment problem. Here assigning tasks to any node of the

distributed computing system plays the important role towards the improving the performance of the

distributed system. However, the issue related to the assignment of tasks for such systems is an NP-

Complete problem. The assignment of tasks can be done by statically and dynamically. The load

balancing and load sharing also becomes the important parameters for improving the performance of the

systems. The present talk discusses the optimal ways of the statically and dynamically distribution of

tasks to any distributed system. Further load balancing and load sharing schemes are also taken care off

while discussing the criterion for distribution of tasks to any node of the distributed system. To improve

the performance of distributed system, several studies have been devoted to the distribution of tasks with

160

the main concern on the performance measures such as minimizing the execution and communication

costs, minimizing the application turnaround time, and maximizing the total throughput of the distributed

system. The present study includes various algorithms that have been discussed for variety of different

cases. It is noted that these algorithms help a lot in improving the performance of the distributed

computing systems. Further in order to support the results comparisons have been made with the other

existing algorithms. The proposed model has been simulated in Mat Lab 7.11.0. Some examples have

been taken to test and verify the suggested algorithms.

Keywords: Task Assignment, Distributed Computing Systems, Optimal Policies

IT-14

Reducibility in Heyting Algebras using forbidden structures

Vilas Kharat Center for Advanced Study(CAS-II),

Dept. of Mathematics

SP Pune Univ. (SPPU), India

laddoo1@yahoo.com

Abstract

It is known that many important classes of lattices have been characterized by means of the non-

existence of certain sublattice structures, viz, distributive, modular, semidistributive, semimodular, etc.

Also, the concept of reducibility in finite lattices and posets has been studied by researchers and

characterized deletable elements. Here, we consider the classes of Heyting Algebras including Stone

Algebras, Relatively Stone Algebras and Boolean Algebras. Recently, we have established the

characterizations in some classes and sufficient conditions in some other classes for deletable elements in

terms of forbidden structures that contain the deletable element (joint work with Manish Agalave).

161

IT-15

On Graph of a Finite Group

K. L. Bondar

Department of Mathematics,

Government Vidarbha Institute of Scienceand Humanities, Amravati

klbondar_75@rediffmail.com

Abstract

In this talk the definition of a graph structure R(G) on a finite group G called the graph of a finite group is

introduced and some properties of this graph using the different properties of group are discussed. An

attempt has been made to convert group into graph and tried to study various properties of group by using

graph theory. Moreover some results on graph of finite group have been discussed.

162

OP-1

Bianchi Type-III Cosmic Strings and Domain Walls in 𝐟(𝐑, 𝐓) Gravity

S. D. Tade1, Anisa M. Ahmad2*

1Jawaharlal Nehru Arts, Commerce and Science College, Nagpur, Nagpur University, India 2G.H. Raisoni Institute of Engineering and Technology, Nagpur

email: drstade25@gmail.com, ahmad.anisa20@gmail.com

Abstract:

In this paper, we have considered Bianchi type-III cosmological model in the context of cosmic strings

and domain walls in the framework of f(R, T) gravity proposed by Harko et al. (2011). To obtain a

determinate solution, a special law of variation for Hubble’s parameter proposed by Berman (Nuovo

Ciment B, 74, 182, 1983) is used. Some physical and kinematical properties of the model are also

discussed.

Keywords: Bianchi type-III, cosmic strings, domain walls, f(R, T) gravity

OP-2

Comparative Study of Minimal Spanning Tree Problem by Prim’s, Dijkstra’s

and Kruskals Algorithm In Fuzzy Environment

A. A. Deshpande1*, O. K. Chaudhari1

1G.H. Raisoni College of Engineering, Nagpur, Maharashtra, India

email: archana.deshpande@raisoni.net, onkar.chaudhary@raisoni.net

Abstract:

In the field of graph theory, the minimum spanning tree is a special kind of tree that minimizes the length

of the edges of the tree which is then symmetrically used for finding the shortest path. This is widely used

in the field of transportation for minimizing the cost through minimization of the path. For network flow

and salesman travel plan also the use of solving minimal spanning tree is useful. There are various

algorithms to find minimal spanning tree for the shortest path. When the data or edges of the graph are of

uncertain nature then the available algorithms have limitations. Hence now a days the minimal spanning

tree with edges having fuzzy values is most recent problem in fuzzy area to find the solutions. In this

paper, the minimum spanning tree problem on the graph with fuzzy edge weights is considered. A fuzzy

version of the three algorithms Prim’s, Dijkstra’s and Kruskals used to find the minimal spanning tree.

The results of these algorithms are compared to find the best minimal spanning tree in Fuzzy

environment.

Keywords: Minimal Spanning tree, Prim’s algorithm, Dijkstra’s algorithm, Kruskal’s algorithm Fuzzy

graph

163

OP-3

Tautological Algebra of the Moduli Space of Semistable

Bundles on an Elliptic Curve

Arijit Mukherjee1*

1 School of Mathematics and Statistics, University of Hyderabad

Hyderabad – 500046, India

email: mukherjee90.arijit@gmail.com

Abstract:

By tautological algebra, one means a subalgebra of the cohomology ring (or Chow ring) of a suitable

moduli space, generated by the cohomology class (or respectively cycles) of some geometrically defined

subvarieties of that moduli space. We choose the moduli space to be the space of semistable bundles over

an elliptic curve. Also, we take the tautological algebra to be the subalgebra of the cohomology ring of

this moduli space, generated by cohomology classes of the Brill-Noether subvarieties as these are

geometric by definition.

In 2003, P. E. Newstead launched a project named “The Brill-Noether Project”. In that project, he

proposed a question regarding the discussion of the detailed geometry of the Brill-Noether loci and about

their classes in the Chow ring or cohomology ring of the moduli space. We have worked on very similar

problems. We completely describe the tautological algebra of the moduli space of semistable bundles

over an elliptic curve and the same for fixed determinant moduli space. Moreover, we obtain results

similar to the Poincar´e relations on a Jacobian variety.

As an immediate generalisation, one can investigate if similar relations hold in higher genus case. We

have made some progress in that direction.

Keywords: Jacobian varieties, Elliptic curves, vector bundles, moduli spaces

References:

1. E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris: Geometry of Algebraic Curves, I (1) (1985)

2. M. F. Atiyah: Proc. London Math. Soc., 3 (7), 414-452 (1957)

OP-4

Fixed Point Theorems in Partial Metric Space Using Some New Contractive

Conditions

Ashish Prabhakar Nawghare1*, HareshGambhir Chaudhari2

1Bhusawal Arts, Science and P. O. Nahata Commerce college, Bhusawal

email: ashishnawghare@gmail.com , hareshgchaudhari@gmail.com

Abstract:

The fixed-point theory is one of the important fields in non-linear analysis. The Banach Contraction

Theorem is milestone in the study of non-linear analysis, given by Banach [1992] in 1922. Since that,

many authors have given different versions of Banach Contraction theorem in different spaces using a

variety of contraction mappings. The study of denotational semantics of dataflow networks by Matthews

[1994] introduced the partialmetric space. He generalized the Banach contraction principle for application

in program verification in the context of partial metric spaces. After Matthews many authors studied the

Bancah contraction in partial metric space with a range of contractive conditions. In this study we present

164

a new contractive condition to prove contraction theorem of Banach in partial metric space where we

have the concept of non-zero self-distances. One can find need of such study in computer science, to

certain problems of denotational semantics [1994].

Keywords: Metric space, Fixed point, Cauchy Sequence, contraction mapping

References:

1. S. Banach: Surles operations dans les ensembles abstraitsetleur application aux equations integrales,

Fund. Math. 3, 133-181, (1922)

2. S.G. Matthews: Partial metric topology, Proc. 8th Summer Conference on General Topology and

Applications, Ann. New York Acad. Sci., 728, 183-197, (1994)

OP-5

Generalization of Two-Dimensional Fourier-Finite

Mellin Transform and its Applications

A. N. Rangari1*, V.D. Sharma2 1Adarsh College, Dhamangaon Rly- 444709 (M.S), India

2Department of Mathematics, Arts, Commerce and Science College, Amravati- 444606(M.S), India

email: ashwinarangari@gmail.com, vdsharma@hotmail.co.in

Abstract:

Integral Transforms are being applied for solving ordinary differential equations, Partial differential

equations, Integral equations and many more. Integral transform is playing an ever more important role in

the physical science, engineering science as well as in biological science. Fourier transform and Finite

Mellin transform have applications in signal processing, image processing, pattern recognition, and

analysis of algorithm. In the proposed work, we extend two dimensional Fourier-Finite Mellin Transform

in the distributional manner and discussed its Analytical structure, Inversion theorem and Applications of

Two-dimensional Fourier-Finite Mellin Transform.

Keywords: Fourier Transform, Finite Mellin Transform, Generalized function, Testing functionspace

References:

1. LokenathDebnath.Dambaru Bhatta, “Integral Transforms and their Applications”. Chapman and

Hall/CRC Taylor and Francis Group Boca Raton London, New York, 2007

2. A.H. Zemanian, “Distribution theory and transform analysis”, McGraw Hill, New York, 1965

3. A. H Zemanian, “Generalized integral transform”, Inter science publisher, New York, 1968

165

OP-6

On Generalized Almost Statistical Convergence of

Bounded Sequences of Real Numbers

Absos Ali Shaikh1, Biswa Ranjan Datta1*

1Department of Mathematics, The University of Burdwan, Burdwan-713104, West Bengal, India

email: aask2003@yahoo.co.in, aashaikh@math.buruniv.ac.in, biswaranjandatta2019@gmail.com

Abstract:

There are several divergent real sequences in usual sense which are “nearly to be” convergent. Therefore,

it is always better to have larger class of convergent sequences in more generalized concept under

research. Almost convergence1 and statistical convergence2,3 are two crucial generalization of usual

convergence. But such concepts of convergence are incomparable4.

The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS)

convergence of bounded real sequences, which generalizes the notion of almost convergence as well as

statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also

introduce the concept of Banach statistical limit functional and prove the existence of Banach statistical

limit functional. The notion of GAS convergence mainly depends on the existence of Banach statistical

limit functional. Then we have shown the existence of a GAS convergent sequence, which is neither

statistical convergent nor almost convergent. Also, we have shown that the space of all GAS convergent

sequences is a closed, non-separable subspace of the space of all bounded real sequences.

Keywords: Banach limit functional, almost convergence, statistical convergence.

References:

1. G. G. Lorentz, A contribution to the theory of divergent sequences. Acta. Math., 80, 167-190 (1948)

2. H. Fast, Sur la convergencestatistique. Colloq. Math., 2, 241-244, (1951)

3. H. Steinhaus, Sur la convergence ordinaireet la asymptotique, Colloq. Math., 2, 73-74 (1951)

4. H. I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans.

Amer. Math. Soc., 347, 1811-1819 (1995)

OP-7

Ricci Soliton and Critical Points of Scalar Curvature Absos Ali Shaikh1, Chandan Kumar Mondal2*

1Department of Mathematics, The University of Burdwan, Burdwan-713104, West Bengal, India 2Netaji Subhas Open University, Kolkata, West Bengal, India

email: aask2003@yahoo.co.in, chan.alge@gmail.com

Abstract:

In this paper we consider the Ricci curvature of a Ricci solitons. In particular, we have showed that a

complete non-compact gradient Ricci soliton with non-negative Ricci curvature possesses a convex

function with finite weighted Dirichlet integral satisfying an integral condition is Ricci flat. We have also

proved that in a complete non-compact gradient Ricci soliton having convex potential with bounded Ricci

curvature, the scalar curvature has at most one critical point.

Keywords: Ricci soliton, scalar curvature, Ricci flat, Convex function, Riemannian manifold

166

OP-8

Kaluza-Klein Cosmological models with General Relativistic Hydrodynamics

in )(Rf Theory of Gravitation

S. D. Katore1, C.D. Wadale2*, H.G. Parlikar3

1Department of Mathematics, Sant Gadge Baba Amravati University, Amravati Pincode- 444602, India 2Department of Mathematics, Smt. Radhabai Sarda Arts, Commerce and Science College, Anjangaon

Surji, Dist. Amravati-444705, India. 3Department of Mathematics, Brijlal Biyani Science College, Amravati, Dist. Amravati- 444602, India

email: katoresd@rediffmail.com, wadalecd@gmail.com, hparlikar05@gmail.com

Abstract:

We have investigated Kaluza-Klein cosmological models for General Relativistic Hydrodynamics in the

context of )(Rf theory of gravitation. Exact solutions of the field equations are obtained for exponential

and power law volumetric expansion. The physical and kinematical behaviours of the investigated models

are also discussed.

Keywords: Kaluza-Klein space-time, daGeneral Relativistic Hydrodynamics, )(Rf theory of

gravitation

OP-9

Jordan Ideals in Prime Rings and Generalized Derivations

A.R. Gotmare1, D.H. Arekar1, S. N. Patil1

1NMU, Jalgaon

email: gotmarear@gmail.com, arekardh@gmail.com, patilsumit2603@gmail.com

Abstract:

Let R be a 2− torsion free prime ring and J be a nonzero Jordan ideal of R . Let F and G be two

generalized derivations with associated derivations f and g , respectively. Our main objects in this

paper is to shows that if, ( ) ( ) 0F x x xG x− = for all x J , then R is commutative and F G= or G

is a left multiplier and F G f= + .

Keywords: Prime ring, Jordan Ideal, generalized derivativedee

167

OP-10

Heat Source/ Sink Effects on Unsteady MHD Free Convective Flow of Rivlin-

Ericksen Fluid with Slip Conditions

Deepti1*, B. P. Garg2

1Department of Mathematics, Shivaji College (University of Delhi), Raja Garden, New Delhi-110027 2Adesh Institute of Engineering and Technology, Faridkot, Punjab

email: deepti_math@yahoo.com

Abstract:

In the present analysis, an investigation on MHD flow of viscoelastic fluid of Rivlin Ericksen type across

a semi-infinite plate in the presence of heat source/sink under free convection is conducted. The plate is

given a constant velocity having exponentially varying suction along with slip conditions. Further, the

free stream velocity is also taken oscillatory in nature. The transport equations governing the fluid flow

are solved under perturbation method. Hence, the solutions obtained for the fields which characterize the

flow viz. velocity, temperature and concentration are deliberated through graphs for varying situational

parameters appeared in the flow. Further, the drag at the plate, Nusselt number and Sherwood number are

assessed numerically for situational variables which are compiled in the tabular format.

Keywords: Free convection, MHD, viscoelastic fluid (Rivlin Ericksen), heat source/sink, slip conditions.

References:

1. S. Banyal and D. K. Sharma, Journal of Mechanical Engineering and Sciences. 4, 462-471, (2013)

2. B. P. Garg, K. D. Singh and A. K. Bansal, Journal of Rajasthan Academy of Physical Sciences, 13

(3), 289 – 304, (2014)

3. D.D Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer, New York, (1990)

OP-11

Higher Dimensional Viscous Cosmology with Inhomogeneous Equation of

State and Future Singularity

Deepti Raut 1*, Arti Ghogre2, N. V. Gharad3

1 Department of Mathematics, Rajiv Gandhi College of Engineering and Research, Hingna Road

Wanadogri, Nagpur 441110, Maharashtra, India 2 Department of Mathematics, Yeshwantaro College of Engineering, Hingna Road, Wanadogri, Nagpur

441110, Maharashtra, India 3Jawaharlal Nehru Arts, Commerce and Science College, Wadi, Nagpur, India

email: diptigudadhe0587@gmail.com, arti_ghogre@rediffmail.com, nvgharad@gmail.com

Abstract :

A universe media is considered as a bulk viscosity described by inhomogeneous equation of state (EOS)

of the form p = (γ − 1)ρ + Λ(t), where Λ(t) is a time dependent parameter. A generalized dynamical

equation for the scale factor of the universe is proposed to describe the cosmological evolution, in which

we assume the bulk viscosity and time dependent parameter Λ are linear combination of two terms of the

form: ζ = ζ0 + ζ1H, Λ(t) = Λ0 + Λ1H, i.e. one is constant and other is proportional to Hubble

parameter H =a

a . In this framework, we demonstrate that higher dimensional model can be used to

explain the dark energy dominated universe, and the inhomogeneous term of specific form introduced in

168

EOS, may lead to three kinds of fates of cosmological evolution: no future singularity, big rip or Type III

singularity as presented by S. Nojiri and S. D. Odintsov (2005).

Keywords : Higher dimensional cosmology, inhomogeneous equation of state, bulk viscosity, dark

energy, future singularity

References :

1. S. Nojiri and S. D. Odintsov, Phys. Rep. 505, 59 (2011).

2. M. Li, X. D. Li, S. Wang and Y. Wang, Commun. Theor. Phys. 56, 525 (2011).

3. X. H. Meng and P. Wang, Class. Quantum Grav. 20, 4949 (2003).

OP-12

Magnetized Strange Quark Cosmological Model in

Modified Theory of Gravitation

S. D. Katore1, D. P. Tadas2*, S. M. Shingne 3

1Department of Mathematics, Sant Gadge Baba Amravati University

Amravati-446602. (M. S.) India. 2*Department of Mathematics, Toshniwal Arts, Commerce and Science College, Sengaon,

Hingoli- 431542, (M. S.) India. 3Department of Mathematics, G. S. Science, Arts and Commerce College Khamgaon,

Buldana- 444303, (M. S.) India

email: katoresd@rediffmail.com, dtadas144@rediffmail.com

Abstract:

The present article devoted to study of hypersurface homogeneous cosmological models with magnetized

strange quark matter in ),( TRf theory of gravity. The exact solutions of the field equations are obtained

using the equation of state (EoS) for strange quark matter ( )

,3

4 CBp

−=

where CB bag is constant.

Furthermore, we have discussed the behaviour of the investigated model for physical concern.

Keywords: Hypersurface Homogeneous, Magnetized Strange Quark Matter, f(R, T) gravity

reddy

169

OP-13

Inventory Model for Deteriorating Items with Weibull Distribution and

Generalized Pareto Decay Having Demand as Function of A Quadratic

Demand with Shortages

M. Srinivasa Reddy1*, R. Venkateswarlu2

1Department of Mathematics, IIIT – Ongole, RGUKT – A.P, Idupulapaya – 516330, India 2GITAM School of International Business, GITAM University, Visakhapatnam – 530045, India

email: naveenasrinu@gmail.com, rangavajhala_v@yahoo.co.in

Abstract:

In this paper, an EPQ model for deteriorating items is developed and analyzed with the assumption that

the replenishment is random and follows a Weibull distribution. It is further assumed that the life time of

a commodity is random and follows a generalized Pareto distribution and demand is a function of a

quadratic demand. Using the differential equations, the instantaneous state of inventory is derived. With

suitable cost considerations, the total cost function is obtained. By minimizing the total cost function, the

optimal ordering policies are derived. Through numerical illustrations, the sensitivity analysis is carried.

The sensitivity analysis of the model reveals that the random replenishment has significant influence on

the ordering and pricing policies of the model. This model also includes some of the earlier models as

particular cases for specific values of the parameters.

Keywords: EPQ model, Weibull distribution, Pareto distribution, Quadratic demand.

OP-14

Magnetized String Cosmological Model with Bulk Viscous Fluid in Rosen’s

Bimetric Gravity

N. P. Gaikwad1*, M. S. Borkar2

1Department of Mathematics, Dharampeth M. P. Deo Memorial Science College, Nagpur – 440033, India 2Department of Mathematics, R. T. M. Nagpur University, Nagpur – 440 033, India

email: np_gaikwad@rediffmail.com, borkar.mukund@rediffmail.com

Abstract:

We have presented the solution of LRS Bianchi type II space-time with magnetic field and with string

viscous fluid by solving the field equations of Rosen’s bimetric theory of gravitation. It is observed that

the magnetic field could have the cosmological origin of the model and it is agreed with Harrison (1973).

The small value of magnetic field originated the universe and starts evolving it with maximum density

and ending with zero density. The strong magnetic field ruled out the existence of the universe. Other

geometrical and physical behavior of the model have been studied in the evolution of universe.

Keywords: Gravitation theory, Magnetic field and Cosmology.

References: 1. Harrison E R: Origin of Magnetic Fields in the Early Universe, Phys. Rev. Lett. 301973 188.

http://dx.doi.org/10.1103/PhysRevLett.30.188

2. Tyagi A and Sharma K: Locally Rotationally Symmetric Bianchi Type-II Magnetized String Cosmological

Model with Bulk Viscous Fluid in General Theory of Gravitation, Chin. Phys. Lett. 28(8) 2011089802.

3. Gaikwad N P, Borkar M S and Charjan S S: Bianchi Type I Massive String Magnetized Baratropic Perfect

Fluid Cosmological Model in Bimetric Theory of Gravitation1 Chi. Phys. Lett. 28(8)2011 089803.

170

OP-15

Fractional Mellin -Fractional Double Laplace Transform and it’s Properties

R.V. Kene1

1Department of Mathematics, Rajarshee Shahu Science College, Chandur Rly, Amravati

email: rekhakorpe9@gmail.com

Abstract:

In this article we define fractional mellin transform and fractional double laplace transform of order ,

0< 1, for fractional differentiable function . some main properties for fractional double laplace -

fractional mellin transform are established.

Keywords: Fractional Mellin transform and fractional double Laplace transform, fractional differentiable

function

References:

1. Adem Kilicman, Mryyam Omran: Springer Plus, 5, 100, (2016)

2. Butzer PL, Jansche S, Mellin transform theory and the role of its differential and integral operators.

In, Rusev P, (1998)

3. R.V. Kene & A.S. Gudadhe: Generalized Fractional Mellin-Whittaker Transform International

Journal of Mathematics Research, 4(5), 535-540 (2012)

OP-16

Bianchi Type -III Perfect Fluid Cosmological Model with Electromagnetic

Field in Brans-Dicke Theory of Gravitation

K.R. Mule1*

, V.G. Mete2, V.M. Ingle2

1*Department of Mathematics, S.D.M.B. Science& Arts College, Shegaon, Dist. Buldana 2Department of Mathematics, R.D.I.K. & K.D. College, Badnera-Amravati, India

email: kailasmule@rediffmail.com, vmete5622@gmail.com, vishalinglemi@gmail.com

Abstract:

In this paper, we investigate the roll of relativistic charged perfect fluid in Bianchi type-III cosmological

model in Brans-Dicke theory of gravitation. Solutions of the models are obtained by volumetric

exponential expansion, power law expansion and power law relation between scalar field and the scalar

factor. Some physical features and kinematical properties of the model are also discussed.

Keywords: Bianchi type-III universe, Brans-Dicketheory of gravitation, electromagnetic field, perfect

fluid, constant vector potentials

References:

1. H. Brans and R. H Dicke, Physical Review A, 124(3), 925-935, 1961

2. C. Mathiazhagan and V. B. Johri, Classical and Quantum Gravity, 1, .2, 1984

3. L. O. Pimentel: Modern Physics Letters A. 12, .25, 1865-1870 (1997)

171

OP-17

Modified Cosmic Chaplygin Gas with FRW Bulk Viscous Cosmology in (2+1)-

Dimensional Spacetime

Praveen Kumar1, Safiqul Islam2, Kashika Srivastava3*, G. S. Khadekar4

1,4Department of Mathematics, Rashtrasant Tukadoji Maharaj Nagpur University, Nagpur-440033, India

2Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad-211019, India 3Sherwood College of Engineering Research& TechnologyLucknow, India

email: pkumar6743@gmail.com, gkhadekar@yahoo.com, sofiqul001@yahoo.co.in

Kashikasrivastava10@gmail.com

Abstract:

In this paper we study FRW bulk viscous cosmology by considering modified cosmic Chaplygin gas in

the framework of (2+1)-dimensional spacetime. we obtain generalized Friedmann equations due to bulk

viscosityand modified cosmic Chaplygin gas in (2+1)-dimensional spacetime. For this, we calculate

various form of bulk viscosity coefficient ζ and then obtain the physical parameters like time-dependent

energy densityρ, Hubble expansion parameter H and deceleration parameter q. Finally, we discuss the

stability of the model by using the speed of sound.

Keywords: modified cosmic Chaplygin gas, bulk viscous cosmology, bulk viscosity

OP-18

Kantowski - Sachs Viscous String Cosmological Model with

Varying Cosmological Term

Shilpa Samdurkar1*, G. S. Khadekar2, Shoma Sen3

1Department of Mathematics, VidyaVikas Arts Commerce and Science College, Samudrapur

Dist. Wardha, India 2Department of Mathematics, Rashtrasant Tukadoji Maharaj Nagpur University, Mahatma Jyotiba Phule

Educational Campus, Amravati Road, Nagpur-440033 3Department of Applied Mathematics, Priyadarshini Institute of Engineering and Technology, Hingna

Road, Nagpur, India

email: rpahade2008@rediffmail.com, gkhadekar@yahoo.com, shomasen0808@gmail.com

Abstract:

This paper deals with the Kantowski Sachs cosmological model of the universe filled with a bulk viscous

cosmic fluid by considering bulk viscosity and time dependent parameter as a linear combination of

two terms in the forms: Ht 21)( += and Ht 10)( += . The physical and dynamical properties of

the model have been discussed and plotted the graphs. The model is found to be compatible with the

results of astronomical observations.

Keywords: Kantowski Sachs spacetime, bulk viscosity, cosmological term

172

OP-19

A Fuzzy Logic Approach To Calculate The Risk of Cancellation of Policies

Gaurav Sharma1*, Sanjeev Kumar1

1 Department of Mathematics, Dr. B.R. Ambedkar University, IBS, khandari Campus, Agra-282002

email: garv13sharma@gmail.com, sanjeevibs@yahoo.co.in

Abstract :

The preventive avoidance of cancellation is a key problem facing insurance companies. A conversation

with the client held prior to latter’s decision to cancel a contract increases the likelihood of contract

continuity. So companies are in need of reliable expert system that can help them to evaluate the risk of

cancellation of the policies in future. With the help of fuzzy system it is possible to identify clients who

may potentially cancel and take timely measure to safeguard the portfolio. Here a model is presented,

which is designed by using fuzzy mathematics and expert system to provide indicative results on the risk

of cancellation of the policies in future.

Keywords : Fuzzy logic, Insurance, Risk classification, Inference system, MATLAB, Index of

vagueness.

References :

1. Arora, N. and Arora : P“Insurance Premium optimizatiom : Perspective of Insurance Seeker and

insurance provider” Journal of management and Science. (2014)

2. Kumar, S. and Jain, H : “Indicative results on the risk of cancellation of policies ; A fuzzy approach”,

Workshop on Opt. and Info. Theory with their Appl ; 24th -26th March, 60-66 (2011)

3. Kumar, S. and Pathak, P : “Fuzzy based bonus-malus system for premium decision in car insurance”,

Int. Rev. Of Pure & Applied Math, 1, 77-81 (2010)

4. Kumar, S. and Pathak, P : “Premium allocation- fuzzy approach in insurance business”, Proceeding of

the 3rd National Conference ; INDIACom, 703-705 (2009)

OP-20

Non-Tampered Database System in a Real Sense

A. S. Ramteke1*, G. S. Katkar1

1Taywade College, Koradi, RTM Nagpur University, Nagpur (MS), India

email: a.ramteke@tckoradi.com, g.katkar@tckoradi.com

Abstract :

In the programming world it has been realised that what “input” user entered through input device that is

visible to user can be alter while storing it in the database. In programming world one can make mistake

or deliberately store inputted data different than what it is visible to the user. In this way, in highly

sensitive programming such as socio-political system one can make fool to billions of users who are

unaware with this “false programming”. In this paper we have suggested programming measures and new

database triggers and constraints to be added to database field such as “What You See Is What You

Stored (WYSIWYS)” and “What You Stored Is What You Can’t Altered (WYSIWYCA)”. But

unfortunately in the existing database system such type of constraints implementation does not exists. By

using this approach strictly at database physical level one can’t tamper the user inputted data while storing

it in the database. Such type of highly sensitive database system must be certified by DBA as non-

173

tampered database system. We have suggested two fold data system and surveillance program to have

control over the unethical program written by the programmer.

Keywords : database system, DBA, false programming, non-tampered system, programming measures,

TSR.

References :

1. "Codd's 12 Rules" University of Derby. 2015-09-15. Retrieved September 6, 2018.

2. Codd, Edgar Frank (14 October 1985), "Is Your DBMS Really Relational", Computer World (1985)

3. Codd, Edgar Frank (21 October 1985), "Does Your DBMS Run By the Rules", Computer World

(1985)

OP-21

Kaluza-Klein Cosmological Models With General Relativistic

Hydrodynamics in )(Rf Theory of Gravitation

S. D. Katore1, C. D. Wadale2*, H. G. Parlikar3

1Department of Mathematics, Sant Gadge Baba Amravati University, Amravati Pincode- 444602, India 2Smt. Radhabai Sarda Arts, Commerce and Science College, Anjangaon, Surji, Amravati-444705, India

3Department of Mathematics, Brijlal Biyani Science College, Amravati, Dist. Amravati-444602, India

email: katoresd@rediffmail.com, wadalecd@gmail.com, hparlikar05@gmail.com

Abstract:

We have investigated Kaluza-Klein cosmological models for General Relativistic Hydrodynamics in the

context of )(Rf theory of gravitation. Exact solutions of the field equations are obtained for exponential

and power law volumetric expansion. The physical and kinematical behaviours of the investigated models

are also discussed.

Keywords: Kaluza-Klein space-time, General Relativistic Hydrodynamics, )(Rf theory of gravitation

OP-22

0n Availability of a System with Reboot Delay

Kanta1*, Sanjay Chaudhary 1 1Dept. of Mathematics, Dr. B.R. Ambedkar University, IBS, khandari Campus, Agra-282002

email : singhkanta80@gmail.com

Abstract :

In this paper, efforts have been made to evaluate the availability of a system with two warm standbys,

switching failure and delay of reboot. For the primary and warm standby components, the time-to-failure,

time-to-repair and time-to-delay are assumed to follow exponential distribution. There is a possibility of

failures during the switching from standby state to primary state. The switching of warm standbys to

replace failed components is subject to failure with probability q. Reboot delay happens in this switching

procedure of a standby unit to primary unit. The reboot time is assumed to be exponentially distributed.

Primary and warm standby units can be considered to be repairable. Using the supplementary variable

technique we developed the explicit expressions for the steady state availability.

Keywords : Availability, reboot delay, Laplace transformation, standby, switching failure

174

OP-23

On Some Properties of a Binary Schubert Code and Extended Binary

Schubert Code

Mahesh S Wavare1*, Suryakant M Jogdand2, Arunkumar R Patil3

1Rajarshi Shahu Mahavidyalaya, Latur, (Autonomous), India 2 S.S. S. G. M. College, Loha,

3 S. G. G. S. Institute of Engg &Tech, Nanded

email: maheshwavare@gail.com, smjog12@gmail.com, arun.iitb@gmail.com

Abstract:

Linear error correcting codes associated to higher dimensional algebraic varieties defined over finite

fields have been topical interest. For example, codes associated to Hermitian varieties, Grassmanian

varieties, Schubert varieties and Flag varieties have been studied quite extensively. The codes associated

to these types of varieties is the central interest. Codes associated with Schubert varieties in G(2,5) over

F2have been studied in Wavare (2019). The corresponding binary Schubert code is denoted by Ω19 and its

generator matrix is given in Wavare (2019). having order 5 × 19. In this paper we have discussed all the

properties of this binary Schubert code and found extended binary Schubert Code Ω19

Keywords: Linear Codes, binary Schubert Code, extended binary Schubert Code

References:

1. Mahesh S. Wavare, Codes associated to Schubert varieties G(2,5) over F2 New Trends in

Mathematical Sciences, 7(Issue 1), 71-78, (2019)

2. Ghorpade, S. R., Tsfasman, M. A., Schubert varieties, linear codes and enumerative combinatorics,

Finite Fields and Their Applications. vol. 11, No.4, pp.684-699, (2005)

3. Tsfasman, M. A., Vl˘adut , S. G. Algebraic Geometric Codes, Kluwer, Amsterdam, (1991)

OP-24

On Radicals in Le-Modules

Sachin Ballal1*, Mayur Kshirsagar2, Vilas Kharat1

1Department of Mathematics, Savitribai Phule Pune University, Pune, India 2Department of Mathematics, Fergusson College, Pune

Email: ballalshyam@gmail.com, kshirsagarmb@gmail.com, laddoo1@yahoo.com

Absract:

An le-module M over a commutative ring R with unity is a complete lattice ordered monoid (M,+,≤,e)

with greatest element e and module like action of R on it. The radical of submodule element n is defined

as radical of the ideal (n : e). In this paper, we have studied the properties of radical of a submodule

element on le-modules and established some basic results. Also, it is proved that every non-zero prime

submodule element is maximal if and only if 0M is a prime submodule element.

References:

1. Sachin Ballal and Vilas Kharat, Zariski topology on lattice modules, Asian Eur. J. Math., 8, 1550066

(2015)

175

2. A. K. Bhuniya and M. Kumbhakar, Uniqueness of primary decompositions in Laskerian le-modules,

Acta Math. Hungar., 158(1) 202215 (2019)

3. M. Kumbhakar and A. K. Bhuniya, On irreducible pseudo-prime spectrum of topological le-modules,

Quasigroups and Related Systems, 26, 251262, (2018)

OP-25

Bianchi Type 0VI , Domain Wall With Normal Matter in

Scalar Tensor Theories of Gravitation

M. R. Ugale1*, H. A. Nimkar2 1Dept. of Mathematics, Sipna College of Engg. and Tech., Amravati, Maharashtra, India

email: Mohini_261@rediffmail.com

Abstract:

In this paper we have examine domain wall includes normal matter described by m and mp as well as

domain tension with electromagnetic field in bianchi type 0VI , space time in scalar tensor theories of

gravitation . Exact cosmological models are obtained. Also, we have discussed features of the obtained

solution.

Keywords: Bianckatrehi type 0VI , Scalar tensor theories of gravitation.

OP-26

Kaluza-Klein Stiff Fluid Cosmological Model in Lyra Geometry Kalpana N. Pawar1, N. T. Katre2*, S. T. Rathod3

1 Department of Mathematics, Shri. R. R. Lahoti Science College, Morshi, Dist.: Amravati, India 2 Department of Mathematics, Nabira Mahavidyalaya, Katol, Dist.: Nagpur, India 3 Department of Mathematics, Shri. M. Mohota College of Science, Nagpur, India

email: kalpanapawar31@rediffmail.com, narendra.katre@rediffmail.com, satishrathod474@gmail.com

Abstract:

In recent years many efforts have been made to construct alternative theories of gravitation. Lyra

proposed modification of Riemannian geometry; in this theory both scalar and tensor fields have intrinsic

geometrical significance. In this paper, considering the five dimensional spherically symmetric Kaluza-

Klein metric with stiff fluid distribution in Lyra geometry, we have obtained the exact cosmological

models for two cases, namely, constant displacement vector and time dependent displacement vector.

Moreover, some physical and kinematical properties of the model are discussed.

Keywords: Five-dimensional cosmological model, stiff fluid, Lyra geometry

176

OP-27

Customers Perspective of Perish-Ability

Nilima Puranik*

email: nilupuranik@gmail.com

Abstract:

In this research paper with the help of survey method, attempt was done to scenario customer perspective

of perish-ability by using fuzzy scale. Supply Chain Management (SCM), currently a popular topic in

research literature, breaches the boundaries of many academic disciplines. Many approaches are used by

researchers and practitioners to reduce food loss and waste. We have focused customers behaviour with

the help of fuzzy scale.

Keywords: Fuzzy questionnaires, perishable food supply chain, fuzzy scale

References:

1. V. Vorst, "Effective Food Supply Chains: Generating, Modeling and Evaluating Supply Chain

Scenarios," Wageningen University, Germany (2000)

2. Ahumada and J. R. Villalobos, "Application of planning models in the agri-food supply chain: A

review (2009)

OP-28

Existence of Solution for the First Order Functional

Differential Equation in Banach Algebra

N. S. Pimple1*, S. S. Bellale2, S. P. Birajdar1

1 Rajarshi Shahu Mahavidyalaya (Autonomous), Latur 2 Dayanand Science College, Latur

email: nishpimple2019@gmail.com, shivprasadb28@gmail.com, sidhesh.bellale@gmail.com

Abstract:

In this paper we prove the existence for the first order functional differential equations in Banach algebra

with maxima. A step by step procedure (algorithm) for the solution is evolved and it is delineated that

some sequence of successive approximations converges to the limit point, monotonically to the solution

of the related perturbed differential equations under some suitable mixed hybrid conditions.

Keywords: Functional Differential Equation; Banach Algebra; Perturbed Differential Equations

177

OP-29

Curves On a Smooth Surface With Position Vectors Lie in the Tangent Plane

Absos Ali Shaikh1, Pinaki Ranjan Ghosh1*

1Department of Mathematics, The University of Burdwan, Burdwan-713104, West Bengal, India

email: aask2003@yahoo.co.in, mailtopinaki94@gmail.com

Abstract:

In the theory of curves and surfaces, an important area of research is the study of curves by restricting its

position vector fields in some plane and direction. In 2003, Bang-Yen Chen [2003] introduced the notion

of the rectifying curve in the Euclidean space 𝐑3as a curve whose position vector lies in the rectifying

plane. Shaikh and Ghosh [2019, 2018] also studied surface curve by restricting its position vector to the

rectifying and osculating plane and find some interesting characterization.The present paper deals with a

study of curves on a smooth surface whose position vector always lies in the tangent plane of the surface

and it is proved that such curves remain invariant under isometry of surfaces. It is also shown that length

of the position vector, tangential component of the position vector and geodesic curvature of a curve on a

surface whose position vector always lies in the tangent plane are invariant under isometry of surfaces.

Keywords: Isometry of surfaces, first fundamental form, second fundamental form, geodesic curvature.

References:

1. B. Y. Chen, what does the position vector of a space curve always lie in its rectifying plane, Amer.

Math. Monthly,110 147-152, (2003)

2. A. A. Shaikh, P.R. Ghosh, Rectifying curves on a smooth surface immersed in the Euclidean space,

Indian J. Pure Appl. Math., 50, no. 4, 883-890, (2019)

3. A. A. Shaikh, P.R. Ghosh, Rectifying and osculating curves on a smooth surface, accepted in Indian

J. Pure Appl. Math., (2018).

OP-30

Two Fluids (2+1)-Dimensional Cosmological Model in

Scalar-Tensor Theory of Gravitation

Praveen Kumar1*, G. S. Khadekar1, V. J. Dagwal2

1Department of Mathematics, Rashtrasant Tukadoji Maharaj Nagpur University, Nagpur-440033 India 3Department of Mathematics, Government College of Engineering, New Khapri, Nagpur-441108, India

email: pkumar6743@gmail.com, gkhadekar@yahoo.com, vdagwal@gmail.com

Abstract:

Two fluids (2+1)-dimensional cosmological models with matter and radiating source in scalar-tensor

theory of gravitation proposed by Saez and Ballester are investigated. In the two-fluid model, one fluid

represents the CMB radiation and another fluid represent the matter content of the universe. To get

determinate solution of the field equation, we have conider the relation between pressure and energy

density of the matter field through the gamma law equation of state Pm = (γ − 1)ρm

. Some physical and

geometric behaviour of the models are also investigated.

Keywords: (2+1)-dimensional; Two-fluid; Saez-Ballester theory of gravitation

178

OP-31

Mathematical Approach for Valance Shell Electron Pair

Repulsion (VSEPR) Theory by RDP Rule

R.D. Utane1, Sujata Deo2, Pravin Sayare3*

1Department of Chemistry, Sant Gadge Maharaj Mahavidyalaya, Hingna, Nagpur 2Department of Chemistry, Institute of Science, Nagpur

3Department of Mathematics, Institute of Science, Nagpur

email: rdutanesgmm@gmail.com, sujatadeo24@gmail.com, pravinsayare25@gmail.com

Abstract:

In Valance Shell electron pair repulsion theory using to indentify electron pair (EP). It is summation of

lone pair (LP) and bond pair (BP), Hybridization, Geometry, Shape and Bond angle. The LP acts non

bonding electron for central atoms but it include in EP and not for geometry. The BP is an equivalence of

donation of electron towards substituent atom and included for geometry and shape of molecules or

compound. The VSEPR theory is completely dependent upon valance electron and their bonding with us

by bonding electron acts as bond pair, remaining a pair of electron acts as lone pair. In our study a

mathematically calculate electron pair by RDP rule in two sections by following Cases. Case I: Calculate

the addition of valance electrons of central atom and substituent atom of homo, heterotopic including

cations and anions. If this addition is less than or equal to eight then electron pair (EP) is equal to total

valance electrons divided by 2. In Case II: If the addition of valance electrons is greater than eight then

quotient obtain by total valance electrons divided by 8 is equal to Bond pair. Consider remainder obtain

by total valance electrons divided by 8, then this remainder divided by 2 is equal to Loan pair. Electron

pair (EP) is the sum of bond pair and loan pair.

Graphical Abstract:

Case I:If ∑ Ve ≤ 8 then E.P.= Quotient of ( ∑ Ve / 2)

Case II: If ∑ Ve > 8 then E.P.= Quotient of ( ∑ Ve / 8) + [(remainder of (∑ Ve /8)) / 2]

Keywords: Electron pair, Bond Pair, Lone pair, VSEPR theory, RDP rule, Hybridization.

References:

1. Gillespie, R. J. (2004), "Teaching molecular geometry with the VSEPR model", J. Chem. Educ., 81

(3): 298–304, Bibcode: J ChEd..81..298G, 2004

2. Tsuchida, Ryutarō "A New Simple Theory of Valency" [New simple valency theory]. J. Chem. Soc.

Jpn. (in Japanese). 60 (3): 245–256 (1939)

3. Clauss, Allen D.; Nelsen, Stephen F.; Ayoub, Mohamed; Moore, John W.; Landis, Clark R.;

Weinhold, Frank (2014-10-08). "Rabbit-ears hybrids, VSEPR sterics, and other orbital

anachronisms". Chem. Educ. Res. Pract. 15 (4): 417–434 (2014)

179

OP-32

Plane Symmetric Cosmological Model of Interacting Field in 𝐟(𝐑, 𝐓) Theory

D. D. Pawar1, R. V. Mapari2

1School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Nanded-431606 2Department of Mathematics, Government Vidarbha Institute of Science and Humanities, Amravati

email: dnyaneshwarpawar@srtmu.ac.in , r.v.mapari@gmail.com

Abstract:

In this paper we have studied plane symmetry cosmological model with interacting fluid in the frame of

f(R, T) theory of gravity. Interacting fluid means we formed relativistic field equation of f(R, T) theory of

gravity for plane symmetry cosmological model linearly coupled with charged perfect fluid, mass-less

scalar field and electromagnetic field. We have discussed three models 1) perfect fluid, 2) Disordered

radiation and 3) Dust fluid. We observed the effect and changes of pressure density in different models.

To solve field equations, we used relation between metric potential and suitable coordinate

transformation. We also discussed some physical and dynamical parameters in detail. In which we found

anisotropy parameter non zero and deceleration parameter is negative for particular choice of the value of

n. It indicates present universe is in accelerated phase which match with the result of present universe.

Keywords: f(R, T) theory, Plane symmetry, Interacting fluid, mass-less scalar field, dust fluid.

OP-33

Little Rip Cosmological Models with Time Dependent Equation of State in

Kaluza–Klein Theory of Gravitation

Rajani Shelote1*, G. S. Khadekar2

1Department of Mathematics, Sant Gadge Maharaj Mahavidyalaya, Hingna, Nagpur,44110, India 2Department of Mathematics RTMNU, Mahatma Jyotiba Phule Educational Campus, Amravati Road

Nagpur,440033, India

email: rajanishelote@gmail.com, gkhadekar@yahoo.com

Abstract:

We have studied the flat Kaluza-Klein type cosmological model of the universe filled with an ideal fluid

with linear and non-linear inhomogeneous equation of state with time dependent parameters ω(t) and Ʌ(t)

in which Little Rip (LR)/Pseudo Rip (PR) behavior for dark energy are found1. We found the equation of

the state parameter ω(t) is less than -1. Also, we have investigated the influence from time dependent

parameters ω(t) and Ʌ(t) in the equation of state upon the time tLRand tPRneeded before the occurrence of

the rip singularity2.

Keywords: Dark energy, Higher Dimensional Cosmology, Little Rip (LR)/Pseudo Rip (PR), Time

Dependent Equation of State

References:

1. R. D. Shelote and G.S. Khadekar, Astrophysics and space science (363), 1 (2018)

2. I. Brevik, V.V. Obukhov, K.E. Osetrin and A.V. Timoshkin, Mod. Phys.Lett. (27), 1250210 (2012)

180

OP-34

Multi Criteria Decision Alert for Vehicle Driver Using A Fuzzy Logic Model

for Prevention of Accidents

Rajshri Gupta1, Onkar K Chaudhari1, Nita R Dhawade2

1G. H. Raisoni College of Engineering, Nagpur, India

2Arts, Commerce& Science College, Koradi, Nagpur, India

email: rajshri.gupta@raisoni.net, onkar.chaudhary@raisoni.net, dhawadenr.acs@gmail.com

Abstract:

With the advancement in technology the density of vehicles on the road is increasing worldwide, which

has led to an increase in traffic jams and accidents. Researchers are studying to find out the solution of

these existing problems. Fuzzy logic method is widely being used in Intelligent Transportation Systems

(ITS). In this paper, a fuzzy logic model was developed for automatic speed alert and brake alert with

sensors and cameras in vehicles, considering various parameters required for controlling the vehicle

safety. For intelligent transport, Fuzzy Interface System (FIS) is developed to support drivers’ decision

making and alert them for controlling speed and brake to avoid the accidents. This model can be adapted

to vehicles in inbuilt condition, with different number of input and output, for full automated system.

Keywords: Intelligent Transportation Systems, Fuzzy Logic, Crisp Set, Fuzzy Set, Membership Function

References:

1. Bellman, R. E. and Zadeh, L. A. (1970): Decision making in a fuzzy environment, Management

science 17(4), B141-B164.

2. L. A. Zadeh, (1905): Fuzzy sets information and control.

3. L. A. Zadeh, (1973): Outline of the new approach to the analysis of complex system and decision

processes.

OP-35

Some Results on the Open Subset Inclusion Graph of a Product Topological

Space

R. A. Muneshwar1*, K. L. Bondar2

1Department of Mathematics, NES, Science College, Nanded, Maharashtra, India 2Department of Mathematics, Government Vidarbha Institute of Science &

Humanities, Amravati, Maharashtra, India

email : muneshwarrajesh10@gmail.com, klbondar_75@rediffmail.com

Abstract :

In a recent paper, R. A. Muneshwar and K. L. Bondar, introduced concept of an open subset inclusion

graph of a topological space 𝓘(𝛕X) = (V, E) and they some important properties of this graph were also

discussed. In this present paper, we studied the open subset inclusion graph 𝓘(𝛕X × 𝛕Y) of a finite product

topological space (X × Y, 𝛕X × Y). In this paper, we find relationship between diameter, girth, clique

number, chromatic number, domination number and degree of the graph 𝓘(𝛕X × 𝛕Y) and 𝓘(𝛕X)) & 𝛘(𝓘(𝛕Y).

Keywords : Product Topology, Diameter, Girth, Clique Number, Chromatic Number, Domination

Number and Degree of vertex & Graph

181

References:

1. Robert B. ALLAN and Renu LASKAR: On Domination and Independent Domination Number of a

Graph, Discrete Mathematics 23, 73-76, (1978)

2. I. Beck: Coloring of commutative rings, Journal of Algebra, 116(1), 208-226

3. Ivy Chakrabarty, Shamik Ghosh, T.K. Mukherjee, and M.K. Sen, Intersection graphs of ideals of

rings, Discrete Mathematics, 309 (17), 5381-5392, (2009)

OP-36

Fuzzy Models for Indentifying & Maximum Estimating the Age Group

Patients Suffered by Problems Disease Malaria

Ramesh B. Ghadge Department of Mathematics, Asst. Professor & H.O.D. Kalikadevi Arts, Commerce & Science College

Shirur (Kasar), Dist. Beed -413249 Maharashtra, India

email: rameshghadge555@gmail.com

Abstract:

Malaria is an existing harmful curable simple disease. The harmful effects of malaria parasites to human

body cannot be underestimated. In the paper the “combined effective time dependent data Matrix” CETD-

MATRIX methods are presenting for providing the decision support platform to malaria researchers,

physicians and other health care practitioners. Simply Fuzzy models are used for identifying and

Estimating the leve of the disease under particular symptoms and different Age groups of major

symptoms through an expert doctors decisions simple Fuzzy variables such as Mild moderate, severe and

very severs malaria problems are taken for identifying and estimating the maximum age group patients

suffered by malaria disease problems throw the Fuzzy Mathematics.

Keywords: Row data by expert physicians, malaria, Fuzzy Logic, Fuzzy Matrix Models, ATD, RTD,

CETD, Matrix and Fuzzy Expert system

Reference:

1. Prof. A.K. Bhargava, Fuzzy Set Theory Fuzzy Logic and their Applications, S. Chand

Publication, Delhi -2013

2. Fuzzy Logic with Engineering Applications John Wiley & Sons Ltd. 3rd Edition

3. Franco Montagna, Petr Hájek on Mathematical Fuzzy Logic, Springer Nature Switzerland

182

OP-37

A Historical Note on “Differentegrals”

Narendra Katre1, Rishikumar Agrawal2*, Sanjay Deshpande3 1Department of Mathematics, Nabira Mahavidyalaya, Katol, Nagpur

2Department of Mathematics, Hislop College, Nagpur 3Department of Mathematics, Bhawabhuti Mahavidyalaya, Amgaon

email:narendra.katre@rediffmail.com, rishigd2003@gmail.com, sanjayamgaon@gmail.com

Abstract:

This paper deals with Fractional Calculus. Leibnitz initiated this subject in 1695. Many top

mathematicians had worked on it and contributed as per their needs.

In this paper, a brief survey of the development of derivatives and integrals of non-integers order is

discussed. At the end, semi differentiation and semi integration of some simple cases are obtained.

References:

1. Andre Rocco, Bruce J; West Fractional Calculus and Evolution of Fractional Phenomena, Physic A

1999

2. K. Oldham, J Spanier; The Fractional Calculus, Academic Press, London, 1974

3. T.J. Osier; Fractional Derivatives and Leibnitz Rule, Amer Math Monthly, 1971

OP-38

Higher Dimensional Cosmological Model of the Universe Dominated by

Extended Chaplygin Gas

Rupali wanjari1*, G. S. Khadekar2, Ghanshyam Malviya3

1 Department of Mathematics, DRB Sindhu Mahavidyalaya, Nagpur, India 2Department of Mathematics, Rashtrasant Tukadoji Maharaj Nagpur University, Mahatma Jyotiba Phule

Educational Campus, Amravati Road, Nagpur, India 3Department of Mathematics, P. R. Pote College of Engineering & Management, Amravati, India

email: rwanjari@gmail.com, gkhadekar@yahoo.com,

Abstract:

In this paper, we have studied a five-dimensional FRW space-time with extended Chaplygin gas at the

second-order which recovers quadratic barotropic equation of state. We analyzed the density

perturbations in both relativistic and Newtonian regimes and show that there is no phase transition and

critical point in this model. We also confirmed stability of the model using thermodynamics point of

view.

References:

1. E. O. Kahya, and B. Pourhassan, Universe dominated by the extended Chaplygin gas, Mod. Phys.

Lett. A 30, 1550070 (2015)

2. H. Saadat, and B. Pourhassan, “FRW bulk viscous cosmology with modified cosmic Chaplygin gas”

Astrophys. Space Sci., 344, 237 (2013)

3. M. Vijaya Santhi, V. U. M. Rao, and N. Sandhya Rani, “Holographic Dark Energy Model with

Generalized Chaplygin Gas in Higher Dimensions” Prespacetime Journal 8 (10), 1225-1238 (2017)

183

OP-39

Hyers-Ulam Stability of nth Order System of Differential

Equation with Nonlocal Conditions

Vijay B. Patare1*, Rupesh T. More2 1Department of mathematics, Nutan mahavidyalaya, sailu, Sailu-431503, india

2 Department of Mathematics, Arts, Commerce and Science College, Bodwad, Jalgaon-425 310, India

email: rupeshmore82@gmail.com, vijaypatare30@gmail.com

Abstract:

The aim of this paper is to prove the Results on the Hyers- Ulam stability, Generalised Hyers- Ulam

stability, Hyers- Ulam – Rassias stability, Generalised Hyers- Ulam – Rassias stability of nth order

differential equation with nonlocal conditions in Banach Space. We are motivated by the work of H.L.

Tidke and R.T. More and influenced by the work of R. Murali and A. Ponmana Selvan.

Keywords: Hyers- Ulam stability, Generalised Hyers- Ulam stability, Hyers- Ulam – Rassias stability,

Generalised Hyers- Ulam – Rassias stability, Nonlocal condition.

References 1. R. Murali and A. Ponmana Selvan, Hyers-Ulam Stability of nth Order System of Differential Equation,

American International Journal of Research in Science, Technology, Engineering & Mathematics,26(1) ISSN

(Print): 2328-3491, ISSN (Online): 2328-3580, 71-75 (2019)

2. RupeshT. More, Shridhar C. Patekar, Ashish P. Nawghare - Study of Ulam Hyers Stability of Integrodifferential

Equations with nonlocal Condition in Banach Spaces Journal of Mathematical Computational Science ISSN

:1927-5307. 10, 2, 236-247 (2020)

3. Kishor D. Kucche and Pallavi U. Shikhare-Ulam–Hyers Stability of Integrodifferential Equationsin Banach

Spaces via Pachpatte’s Inequality, Asian-European Journal of Mathematics, 11, 2, 1850062, (2018)

OP-40

A Note on Pm-le-Modules

Sachin Ballal1, Vilas Kharat1

1Centre for Advanced Study (CAS-II), Department of Mathematics, Savitribai Phule, Pune

University, Pune, India

email: ballalshyam@gmail.com , sbballal@unipune.ac.in

Abstract:

An le-module M over a commutative ring R with unity is a complete lattice ordered monoid (M,+, <=, e)

with greatest element e and module like action of R on it. The pm-le-modules are the le-modules in which

every prime submodule element is contained in a unique maximal submodule element. In this paper, we

have studied the Zariski topology on pm-le-modules and established some basic results. We have proved

that, if M is compactly generated le-module then M is pm-le-module if and only if Max(M) is a retract of

Spec(M) if and only if Spec(M) is a normal space. Moreover, many known results in multiplicative

lattices and lattice modules are generalized.

Keywords: Submodule Element, Prime Element, Maximal Element, Prime Spectrum etc

References 1. Francisco Alarcon, D. D. Anderson and C. Jayaram, Some Results on Abstract Commutative Ideal Theory,

Period Math. Hung., 30(1) 1-26, (1995)

2. E. A. Al-Khouja, Maximal elements and prime elements in lattice modules, Damascus University for Basic

Sciences, 19, 9-20 (2003)

3. Sachin Ballal and Vilas Kharat, Zariski topology on lattice modules, Asian Eur. J. Math., 81550066 (2015)

184

OP-41

Magnetized Strange Quark Cosmological Model in Modified

Theory of Gravitation

S. D. Katore1, D. P. Tadas2*, S. M. Shingne 3

1Department of Mathematics, Sant Gadge Baba Amravati University, Amravati-446602. (M. S.) India 2Department of Mathematics, Toshniwal Arts, Commerce and Science College, Sengaon, Hingoli-

431542, (M. S.) India 3Department of Mathematics, G. S. Science, Arts and Commerce College Khamgaon, Buldana- 444303

(M. S.) India

email: dtadas144@rediffmail.com, katoresd@rediffmail.com

Abstract:

The present article devoted to study of hypersurface homogeneous cosmological models with magnetized

strange quark matter in ),( TRf theory of gravity. The exact solutions of the field equations are obtained

using the equation of state (EoS) for strange quark matter ( )

,3

4 CBp

−=

where CB bag is constant.

Furthermore, we have discussed the behaviour of the investigated model for physical concern.

Keywords: Hypersurface Homogeneous, Magnetized Strange Quark Matter, f(R, T) gravity

OP-42

Holographic Dark Energy Density with Generalized Equation of State Aina Gupta1*, G. S. Khadekar1, Kalpana Pande2

1Department of Mathematics, Rashtrasant Tukadoji

Maharaj Nagpur University, Amravati Road, Nagpur-440033 (INDIA) 2Department of Mathematics, VMV Commerce JMT Arts and JJP Science College, Nagpur-440008

email: ainagupta01@gmail.com, gkhadekar@yahoo.com, kalpanappande@gmail.com

Abstract:

In this paper we consider holographic dark energy density with generalized equation of state, p = ΣAnρn.

We solve the Friedmann equation and investigated the behaviour of various cosmological parameters for

n = 1, 2 and 3. We solve the non-linear differential equation and obtained time dependent dark energy

density. Finally, we also investigated the nature of the dynamical scalar field model and concerned

potential by establishing the correspondence between holographic dark energy with scalar field dark

energy.

Keywords: Holographic dark energy, Generalized equation of state

185

OP-43

Extended Chaplygin Gas Equation of State with Viscous Dissipative New

Holographic Dark Energy

Rupali Talole1*, G. S. Khadekar1, V. G. Miskin2

1Department of Mathematics, Rashtrasant Tukadoji Maharaj Nagpur University, Nagpur (M.S.)-440033,

India

2Department of Mathematics, Yeshwantrao Chavan College of Engineering

Hingna Road, Nagpur (M. S.)

email: rupalitalole@rediffmail.com , gkhadekar@yahoo.com , vgmiskin@rediffmail.com

Abstract :

In the present work we adopt Eckart approach to deal with bulk viscosity and equation of state (EoS)

parameter for new holographic dark energy (NHDE) has been constructed via extended chaplygin gas

(ECG). Under the assumption of existence of bulk viscosity with viscosity coefficient in a particular time

varying form. We study the cosmological dynamics of viscous ECG-NHDE through bulk viscous

pressure, effective pressure and EoS parameter using Eckart theory. Furthermore we have reconstructed

the potential and dynamics of viscous ECG-NHDE as a scalar field and also studied satefinder parameters

for geometrical diagnostics of dark energy (DE).

Keywords : Cosmology, bulk viscosity, new holographic dark energy, extended chaplygin gas

References :

1. L. N. Granda and A. Oliveros, Physics Letters B, 669, 275 (2008)

2. S. Chattopadhyay, International Journal of Modern Physics D, 26, 1750042 (2017)

3. M. Li, Physics Letter B, 603, 1 (2004)

OP-44

(2+1)-Dimensional Mesonic Cosmological Model

with the Extended Chaplygin Gas

Praveen Kumar1, V. J. Dagwal2*, G. S. Khadekar1

1Department of Mathematics, Rashtrasant Tukadoji Maharaj Nagpur University, Nagpur (M.S.)-440033

2Department of Mathematics, Government College of Engineering, New Khapri, Nagpur (M.S.)- 441108

email: pkumar6743@gmail.com, gkhadekar@yahoo.com , vdagwal@gmail.com

Abstract:

We have explored (2+1)-dimensionalmesonic cosmological model with the extended Chaplygin gas. We

have determine solution of the field equations by considering constant deceleration parameter q = γ − 1,

where γ is constant and extended chaplygin gas equation of state P = ∑ Cnρn −Dn

ρβn , where C, D & β are

positive constant.In the present work, we have considered zero mass scalar field. The physical and

kinematical behaviour of the universe for γ = 0 & γ ≠ 0, are investigated.

Keywords: (2+1)-dimensional Spacetime, zero mass scalar field, constant deceleration parameter

186

OP-45

FRW Viscous Modified Cosmic Chaplygin Gas Cosmology in the Presence Of

Variable Cosmological Constant Λ

Nirmala A. Ramtekkar1*, G. S. Khadekar1

1Department of Mathematics, Rashtrasant Tukadoji Maharaj Nagpur University, Nagpur (M.S.)-440033

email: nirmala.a.r@gmail.com, gkhadekar@yahoo.com

Abstract:

In this paper we have constructed Kaluza-Klein FRW model with modified cosmic Chaplygin gas in the

presence of variable cosmological constant Λ, shear and bulk viscosity as a linear combination of density.

By using numerical analysis, we have investigated behaviour of energy density and Hubble parameter. In

order to study effect of bulk and shear viscosity on the other cosmological parameters such as energy

density, Hubble parameter, deceleration parameter, pressure we have obtained analytical solution for

some particular choices of m, n and α. Finally, we discussed stability of the model.

Keywords: Modified cosmic Chaplygin gas, Bulk viscosity, Shear viscosity, Dark energy

OP-46

FRW Domain Walls in Modified )(Gf Theory of Gravitation

S. D. Katore1, S. P. Hatkar2*, P. S. Dudhe3 `1Department of Mathematics, Sant Gadge Baba Amravati University, Amravati-444602, India

2 Department of Mathematics, A. E. S. Arts, Commerce & Science College Hingoli-431513, India 3Department of Mathematics, Prof. Ram Meghe Institute of Technology and Research Badnera, India

email: katoresd@rediffmail.com, schnhatkar@gmail.com

Abstract:

In this paper, we have studied Friedmann-Robertson-Walker space-time in the presence of domain walls

in the framework of )(Gf theory of gravitation. The solutions of field equations are obtained by using

linearly varying deceleration parameter. Some physical parameters of the obtained models are discussed

in detail.

Keywords: - FRW, domain walls, )(Gf theory

187

OP-47

On Submodule Elements of Le-Modules and Related Results

Sadashiv Ramkrushna Puranik Department of Mathematics, Tuljaram Chaturchand College of Arts, Science and Commerce

Baramati-413102, (M.S)

email: srpuranik28@gmail.com

Abstract:

An le-module (M,+, ≤, e) is a commutative monoid with the zero element 0M which at the same time is a

complete lattice with the greatest element e with a module like action of a commutative ring R with 1 [1].

An element n of an le-module M is said to be a submodule element if n + n, r.n ≤ n for all r in R. In this

paper, we have discussed some basic results about the class of submodule elements. The concept of linear

independence of submodule elements in an le-module is introduced. Also, the Knaster-Tarski fixed point

theorem for the class of submodule elements is established.

Keywords: le-module, submodule element

References:

1. A. K. Bhuniya and M. Kumbhakar, Uniqueness of primary decompositions in Laskerian le-modules,

Acta Math. Hunga. ,158, 202 (2019)

2. Sachin Ballal, Vilas Kharat, Zariski topology on lattice modules, Asian-European Journal of

Mathematics, 8, 1550066 (2015)

3. A. K. Bhuniya and M. Kumbhakar, On the prime spectrum of an lemodul, arXiv:1807.04024

4. A.K. Bhuniya and M.Kumbhakar, On irrducible pseudo-prime spectrum of topological le-modulees,

Quasigroups and Related Systems ,26(2)(2018)251{262.

5. Sachin Ballal, Vilas Kharat, On minimal spectrum of multiplication lattice modules, 144(1) (2019)

85-97.

6. Davey and Priestley, Introduction to Lattices and order, 2nd Edition, Cambridge University Press.

OP-48

Role of Digital Root in Number Theory

Sanjay Deshpande1, Rishikumar Agrawal2*

1Department of Mathematics, Bhawabhuti Mahavidyalaya, Amgaon 2Department of Mathematics, Hislop College, Nagpur

email: sanjayamgaon@gmail.com, rishigd2003@gmail.com

Abstract:

The digital root of a natural number n is obtained by computing the sum of its digits, then computing the

sum of the digits of the resulting number, and so on, till a single digit number is obtained denoted by

B(n). Hence digital root of a number is a single digit number obtained from the given number.

Before the development of computer devices, the idea of digital root was used to check the results of

mathematical operations. In this paper we highlight the various properties of digital root of a number and

some results of digital root. Triangular number, Perfect square number and a perfect cube number have

specific digital roots. It is also observed that in some results digital root is a necessary condition and not

188

sufficient. We have detail proof for these properties. In modern mathematics, Digital roots partition the

set of non-negative integers. The relation digital root is an equivalence relation. Now a day’s digital root

of a number has many applications in generation of random number sequence and cryptography.

Keywords: Digital root, square, cube, partition

References:

1. A. W. Vyawahare, At Right Angles, 5, 42 (2016)

OP-49

Study of Some Approaches for Solving Fractional Differential Equations

S. V. Nakade1*, R. N. Ingle2

1Dept. of Mathematics, Sharda Mahavidyalaya, Parbhani Maharashtra, India 2Principal, Bahirji Smarak Mahavidyalaya, Basmatnagar, Dist: Hingoli

Affiliated to SRTMU Nanded, Maharashtra, India

email: santosh_nakade@yahoo.co.in

Abstract:

The laws of the Natural and Physical world are usually modelled in mathematics in the form of

differential equations. Fractional Calculus is the generalization of traditional calculus therefore fractional

differential equation gives more energy for modelling real world problems than that of differential

equations. There is no standard algorithm to solve fractional differential equations. Solution of the

fractional differential equations and its interpretation is a rising field of Applied Mathematics. Most of the

fractional differential equations do not have exact analytic solutions therefore numerical techniques and

approximation methods are used. In this paper we study some of these methods/ techniques and tried to

analyze them.

Keywords: Fractional Calculus, Fractional differential equations, Adomian decomposition method,

Variational iteration method, homotopy analysis method

OP-50

Optimization of Non-Zero-Sum Games using Linear Programming

Problem and Aggregate Function

Sapana P. Dubey1*, G.D. Kedar2

1Priyadarshini Institute of Engineering and Technology, Nagpur 2Department of Mathematics, RTMNU, Nagpur

email: sapana.dubey10@gmail.com

Abstract:

Many real-life situations can be modelled using Game theory. Overcome the situation is like optimization

of the game. There are various optimization techniques available to find the solution of Linear

Programming problem. In this paper, we study about conversion of two person non-zero-sum game into

Linear Programming problem and apply various optimization techniques to find optimal solution of the

game. Existing literature is basically focused on problem of optimization of zero- sum game. Our paper

makes the difference by studying Non-zero-sum game as Linear Programming problem. We also use the

189

aggregate function to find optimal solution for both players. Some examples are also illustrated in this

direction.

Keywords: Non-zero-sum game, Linear Programming Problem, Nash Equilibrium, Pareto Optimality

OP-51

Thermo Dynamical Aspects of LRS Bianchi Type-I String

Cosmological Model in ( , )f R T Gravity

J. Satish1* and K. Sreenivas2 1.Department of Mathematics, Gayatri Vidya Parishad College of Engineering

(Autonomous), Madhurawada, Visakhapatnam, India. 2.K. Sreenivas, Department of S & H, Rise Krishna sai prakasam group of Institution, Ongole

email: mathssatish@gmail.com, s.kolalapudi@gmail.com

Abstract:

Spatially homogeneous and anisotropic LRS Bianchi type-I model in ( , )f R T theory (where R is the

Ricci scalar and T is the trace of theenergy-momentum tensor) is investigated in the presence of cosmic

string source. To determine the solution of field equations, it is assumed that deceleration parameter (DP)

is time varying which yields a scale factor for which the universe attains a phase transition scenario and

consistent with recent cosmological observations is used. A particular method to find the solution of

Einstein’s field equations is given. The mathematical expressions for temperature and entropy are

obtained. Some physical and geometrical properties of the model are alsodiscussed.

OP-52

Five Dimensional Cosmological Models with Perfect Fluid in

General Theory of Relativity

Kalpana N. Pawar1, S. T. Rathod 2*, N. T. Katre 3

1 Department of Mathematics, Shri. R. R. Lahoti Science College, Morshi, Dist.: Amravati (India) 2 Department of Mathematics, Shri. M. Mohota College of Science, Nagpur (India) 3 Department of Mathematics, Nabira Mahavidyalaya, Katol, Dist.: Nagpur (India)

email: kalpanapawar31@rediffmail.com, satishrathod474@gmail.com, narendra.katre@rediffmail.com

Abstract:

As the evolving early universe was much smaller than today, the present four-dimensional space-time of

the universe could have been preceded by higher dimensional space-time. Five-dimensional space-time is

particularly attractive because super gravity theories admit a solution which spontaneously reduces to 5-

D. In this paper Einstein’s field equations with variable gravitational and cosmological constants in the

presence of perfect fluid are solved with conditions p = ρ and R ∝ A^n for five dimensional spherically

symmetric space-time in general theory of relativity (GTR). Moreover, various physical and geometrical

properties of the model are discussed.

Keywords: Five-dimensional space-time, perfect fluid, cosmological model

References:

190

1. Kalpana N. Pawar, V. Chauhan, G. D. Rathod, R. V. Saraykar, “Five-dimensional string cosmological

model”, IJSTR-0413-6053, (2013)

2. G. P. Singh, S. Kotambkar, D. Shrivastava, A. Pradhan, A new class of Higher dimensional

cosmological models of universe with variable G and Λ-terms”, Rom. Journ. Phys., 53, 607 (2008)

3. R. K. Tiwari, “Bianchi type-I Cosmological models with perfect fluid in general relativity”, Research

in Astron. & Astrophys., 10, 291 (2010)

OP-53

Generalization of Fractional Sine Transform and it’s Application

S. A. Khapre 1*, V. D. Sharma2

1Department of Mathematics, P. R. Pote College of Engineering and Management, Amravati 2Department of Mathematics Arts, Commerce and Science College, Kiran nagar, Amravati

email: satatte@gmail.com , vdsharma@hotmail.com

Abstract:

This paper is concerned with the generalization fractional Sine transform, in this study we propose a

definition of testing function space and distributional generalized one-dimensional fractional Sine

transform. Along with this we have proved Inversion formula and Analyticity theorem for the one-

dimensional fractional Sine transform. Some properties are verified and applications on generalized one-

dimensional fractional Sine transform are discussed.

Keywords: fractional Fourier transforms, fractional Cosine transform, fractional Sine transform

OP-54

On Reliability Of Three Unit System With Two Type of Repair

Shiva1, Sanjay Chaudhary 1Department of Mathematics, Dr. Bhimrao Ambedkar University, Agra (U.P.)

email : shivaarele@gmail.com

Abstract :

This paper represents the reliability of a redundant system, which consists of three units with each one

operable or failed. The failure can further divided into minor or major. Minor failure can be repaired by

minor repair, but a major failure required a major repair. So two repair facilities are considered. The

system completely fails on the failure of all units. Three similar units are in the system where one unit

works as main and others in cold standby. An imperfect switch is used to on standby unit which takes

sometime in switching. Failure rate and repair rate and repair rate are constant. Failure rate and repair rate

follow exponential distribution. Differential equation and Laplace transformation are used. Hence

reliability is obtained by the sum of the probabilities of all operable states.

Keywords : Cold standby system, Reliability, Minor and major repair

191

OP-55

Bianchi Type-III Cosmological Model with Modified Takabayasi String

Shoma Sen1*, G. S. Khadekar2, Shilpa Samdurkar3

1Department of Applied Mathematics, Priyadarshini Institute of Engineering and Technology, Hingna

Road, Nagpur, India 2Department of Mathematics, Rashtrasant Tukadoji Maharaj Nagpur University, MahatmaJyotibaPhule

Educational Campus, Amravati Road, Nagpur-440033 3Department of Mathematics, VidyaVikas Arts Commerce and Science College, Samudrapur, Dist.

Wardha, India

email : shomasen0808@gmail.com, gkhadekar@yahoo.com , rpahade2008@rediffmail.com

Abstract:

In the present study, the solutions are obtained for Bianchi type III cosmological models with modified

Takabayasi string of the form ++= )1( in the presence of bulk viscous fluid. To obtain the

realistic model we assume the relation between metric potential i. e nCB = . We studied the role of

term in the evolution of universe. Here, we investigated two different cases for the cosmological model i.

e case (i) → and case ii) 0→ . In both the cases, we found the physical significance of the

cosmological term in quadratic form i. e 2

210 HH ++= . All the physical and geometrical

aspects of the model for both the cases are discussed for the corresponding solutions. It is observed that

the resultant models follow the present-day observations and literature favour the accelerating model of

the universe. Also, we have plotted the graphs for detailed study of each case.

Keywords: Bianchi Type III spacetime, Takabayasi string, bulk viscosity, cosmological term

OP-56

Path Energy of Some Graphs

S. C. Patekar

Department of Mathematics, Savitribai Phule Pune University, Pune-411007

email: shri82patekar@gmail.com

Abstract:

Let G be a graph with vertex set V (G) = { fv1,v2,…, vn}. We define a matrix whose (i; j) th entry is the

maximum number of vertex disjoint paths between the corresponding vertices if they are adjacent and is

zero otherwise. We call this matrix as path matrix of G and its eigenvalues as path eigenvalues of G. In

this paper, we investigate path eigenvalues and path energy of some graphs.

Keywords: path matrix, path energy

192

OP-57

Solution of Singularly Perturbed Boundary Value Problems with Singularity

using Variable Mesh Finite Difference Method

E. Siva Prasad 1*, K. Phaneendra2

1 Department of ESH, Kavikulguru Institute of Technology & Science, Ramtek

Maharashtra, 441106, India 2 Department of Mathematics, University College of Engineering, Osmania University

Hyderabad, Telangana, 500001, India

email: emineni.yahoo.co.in, kollojuphaneendra@yahoo.co.in

Abstract :

In this paper, we derive a variable mesh finite difference scheme based on non-polynomial spline

approximation to solve singularly perturbed singular two-point boundary value problems at one end (left

or right) with boundary layer. The discrete equation of the problem is developed by applying the

condition of continuity for the first order derivatives of the non-polynomial spline at the inner nodes and

is not valid for the singularity. Thus, in the case of singularity, the boundary value problem is modified in

order to have a three-term relationship. The tridiagonal scheme of the method is processed using discrete

invariant imbedding algorithm. We discuss the convergence analysis of the method and present the

maximum absolute errors for the standard examples chosen in the literature to show the efficiency of the

method.

Keywords: Singularly perturbed two point singular boundary value problem, Interior nodes, Singular

point, Non-polynomial spline, Boundary layer

References

1. R. K. Bawa, Spline based computational technique for linear singularly perturbed boundary value

problems, Appl.Math. Comput., 167, 225 (2005)

2. C. M. Bender, S.A. Orszag, Advanced mathematical methods for scientists and engineers, Mc. Graw-

Hill, New York, 1978

3. P. Henriei, Discrete Variable Methods in Ordinary differential equations, Wiley, New York, 1962

OP-58

Nature of the Singularities Formed in the Gravitational Collapse of (n+2)

dimensional Monopole Vaidya Spacetime

Smita D Kohale1*, Kishor S. Wankhade2, Lakshmi Madireddy1

1 Department of Applied Mathematics, B.D. College of Engineering, Sevagram, (M.S), India 2Department of Mathematics, Yashwantrao Chavan Arts And Science Mahavidyalaya, Mangrulpir. Dist.

Washim (M. S.)

email: smit244@gmail.com, wankhade.kishor@rediffmail.com , lakshmimadireddy28@gmail.com

Abstract:

In this paper we tend to investigate here the incidence of naked singularities furthermore as their nature in

the gravitational collapse of higher dimensional space times of Monopole Vaidya solution. Within the

final state of the collapse, black holes and naked singularities are shown to be developed. The quantity of

dimensions isn't restricted. These results involving here might be necessary within the light of the recent

193

proposal given by String theory, that states that originally our Universe may be of infinite dimensions at

higher energy state, there at that time it got settled to 4D case by dimensional reduction to the lower

energy state. So final outcome of (n+2) dimensional gravitational collapse becomes most vital issue in the

gravitational Physics.

Keywords: Gravitational collapse, Cosmic Censorship, Naked Singularity

References:

1. Rajibul Shaikh, Prashant Kocherlakota, Ramesh Narayan, Pankaj S. Joshi, Shadows of spherically

symmetric black holes and naked singularities, MNRAS 482, 52 (2019)

2. Pankaj S. Joshi, Daniele Malafarina, All black holes in Lemaitre – Tolman – Bondi inhomogeneous

dust collapse, Quantum Grav. 32, 145004 (2015)

3. Flavio Mercati, Henrique Gomes, Tim Koslowski, Andrea Napoletano, Gravitational collapse of thin

shells of dust in asymptotically flat shape dynamics, Phys. Rev.D 95, 044013(2017)

OP-59

Anisotropic Accelerating Cosmological Model with Exact

Solution in 𝐟 (𝐑; 𝐓) = 𝐑 + 𝟐𝛍𝐓 Gravity

S. K. Sahu1, Binaya K. Bishi2, A. Nath1

1Department of Mathematics, Utkal University, Bhubaneswar-751004, India, 2School of chemical engineering and physical sciences, Lovely professional University, Phagwara,

Punjab, India,

email: jaga1282@gmail.com , binaya.21500@lpu.co.in, anasuyanath@yahoo.com

A novel procedure is adopted for the exact solution of the Bianchi type-III space Time in the light of f(R;

T) = R+2 f1(T) gravity. The solution is obtained under Zelovik model, which leads to vacuum energy and

mathematically equivalent to cosmological Constant Λ. Obtained results proclaim that, we are in an

anisotropic and homogeneous Space time. Some physical properties of the model are derived and discuss

the stability of the model.

Keywords: Bianchi type-III, f (R; T) gravity, Zelovik model

OP-60

Analysis of FRW Universe with General Relativistic Hydrodynamics in

General Theory of Relativity

S. D. Katore1, S. P. Saraogi1*, S.V. Gore2

1Department of Mathematics, Sant Gadge Baba Amravati University, Amravati-446602. (M. S.) India 2Department of Mathematics, Indira Gandhi Mahavidyalaya, Ralegaon-445402. (M.S.). India

email: katoresd@rediffmail.com, shalakasaraogi@gmail.com, swapnilgore411@gmail.com

Abstract:

Friedmann–Robertson–Walker (FRW) space-timewithin the presence of General Relativistic

Hydrodynamics inthecontextof General Theory of Relativity is considered. Exact solutions of field

equations are obtained for power law expansion, volumetric exponential expansion and hybrid expansion

law. ThePhantom, Chaplygin gas and Tachyon field sare discussed.

Keywords: FRW, General Relativistic Hydrodynamics, General Relativity

194

OP-61

Laplace Transform on Howell’s Space –A New Theory

V. N. Mahalle1, S. S. Mathurkar2*, R. D. Taywade3

1Dept. of Mathematics, Bar. R.D.I.K.N.K.D. College, Badnera Railway (M.S), India. 2 Dept. of Mathematics, Govt. College of Engineering, Amravati, (M.S), India

3Dept. of Mathematics, Prof. Ram Meghe Institute of Technology & Research, Badnera, Amravati, (M.S)

email: vidyataywade@yahoo.com, ssmathgcoea@gmail.com, rajendrataywade@rediffmail.com

Abstract:

In this paper we have discussed a new theory of Laplace transform defined on Howell’s space. The

definition of the Laplace transform on Howell’s space of test functions is given. Also discussed some

results and Fundamental theorems of Laplace transform by using the same.

Keywords: Laplace transform, Howell’s Space, Fundamental theorem

References :

1 B. N. Bhosale, Integral transformations of generalized functions (2005)

2 K. B. Howell, A new theory for Fourier analysis I, The space of test functions, Journal of

Mathematical analysis and applications, 168, 342 (1992)

3 K. B. Howell, A new theory for Fourier analysis II, Further analysis on the space of test

functions, Journal of Mathematical analysis and applications, 173, 419 (1993)

OP-62

Note on Fractional Integral Inequalities using Generalized

K-Fractional Integral Operator Asha B. Nale,1 Satish K. Pacnhal1, Vaijanath L. Chinchane2

1Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-431 004, INDIA 2Deogiri Institute of Engineering and Management Studies Aurangabad-431005, INDIA

email: ashabnale@gmail.com, drskpanchal@gmail.com, chinchane85@gmail.com

Abstract:

Fractional inequalities play major role in the development of fractional differential, integral

equations and other fields of sciences and technology. Recently, a number of mathematicians

have studied different fractional integral inequalities using fractional integral operators such as

Riemann-Liouville, Hadamard, Saigo, Erdeyi- Kober, katugampola integral and some other

operators. Here we studies several fractional integral inequalities involving convex functions by

195

using generalized k-fractional integral operator (in terms of Gauss hypergeometric function).

Also we obtain some results in Z. Dahmani and N Bedjaoui (2011) using generalized k-

fractional integral operator.

Keywords: Generalized k-fractional integral, convex functions and inequalities

References:

1. Z. Dahmani and N Bedjaoui, Some generalized integral inequalities, J. Advan. Res. Appl.

Math. 3, 58 (2011)

OP-63

Common Fixed-Point Theorem for Two Mappings in b – Metric Spaces

Varsha Dattatray Borgaonkar1*, K. L. Bondar2

1N. E.S. Science College, Nanded 2Govt. Vidarbh Institure of Science and Humanities

emai: borgaonkarvarsha@gmail.com, klbondar_75@rediffmail.com

Abstract:

In many branches of Sciences, Economics, Computer Science, Engineering and the development of

nonlinear analysis, the fixed point theory is one of the most important tool. In 1989, I. A. Backhtin

introduced the concept of b-metric space. In 1993, Czerwik extended the results of b - metric spaces.

Using this idea many researchers presented generalization of the renowned Banach fiixed point theorem

in b - metric space. Mehmat Kir, Boriceanu, Czerwik, Bota, Pacurar extended the fixed point theorem in

b-metric spaces. Czerwik first presented a generalization of Banach fixed point theorem in b-metric

spaces. We want to extend some common fixed theorems which are valid in b-metric space.

Keywords: b- Metric space, Completeness in b - Metric space, Fixed point, Contractive Mapping

References:

1. Boriceanu , M., ‘Fixed Point Theory for Multivalued Generalized Contraction on a set with two b-

metric’, Studia, univ Babes, Bolya: Math, Liv, 3, 1 (2009)

2. Czerwik, S., ‘Contracrion Mappings in b- Metric Spaces’, Acta Mathematica, et informatica

Universitatis,Ostraviensis,1, 5 (1993)

3. Kir, Mehmet , Kiziltune, Hukmi, ‘On Some Well known Fixed Point Theorems in B- Metric Space’,

Turkish Journal of Analysis and Number Theory, 1, 13 (2013)

196

OP-64

Hyers-Ulam Stability of nth Order System of Differential

Equation with Nonlocal Conditions

Vijay B. Patare1*, Rupesh T. More2 1Department of Mathematics, Nutan mahavidyalaya ,sailu, Sailu-431503, India

2 Department of Mathematics, Arts, Commerce and Science College, Bodwad, Jalgaon-425 310, India

email: vijaypatare30@gmail.com, rupeshmore82@gmail.com

Abstract:

The aim of this paper is to prove the Results on the Hyers- Ulam stability , Generalised Hyers- Ulam

stability, Hyers- Ulam – Rassias stability, Generalised Hyers- Ulam – Rassias stability of nth order

differential equation with nonlocal conditions in Banach Space. We are motivated by the work of

H.L.Tidke and R.T. More and influenced by the work of R. Murali and A. Ponmana Selvan .

Keywords: Hyers- Ulam stability , Generalised Hyers- Ulam stability, Hyers- Ulam – Rassias stability,

Generalised Hyers- Ulam – Rassias stability, Nonlocal condition

References:

1. R. Murali and A. Ponmana Selvan, Hyers-Ulam Stability of nth Order System of Differential

Equation,American International Journal of Research in Science, Technology, Engineering &

Mathematics, 26, 71 (2019)

2. Rupesh T. More,Shridhar C. Patekar, Ashish P. Nawghare - Study of Ulam Hyers Stability of

Integrodifferential Equations with nonlocal Condition in Banach Spaces Journal of Mathematical

Computational Science, 10, 236 (2020)

3. Kishor D. Kucche and Pallavi U. Shikhare-Ulam–Hyers Stability of Integrodifferential Equationsin

Banach Spaces via Pachpatte’s Inequality,Asian-European Journal of Mathematics, 11, 1850062

(2018)

OP-65

Anisotropic Dark Energy Cosmological Model in

Gravity by Hybrid Expansion Law

V. M. Raut Department of Mathematics, Shri Shivaji Science College, Amravati-444602, India

email: vinayraut18@gmail.com

Abstract:

In this paper, plane symmetric cosmological model filled with perfect fluid in the framework of

Teleparallel Gravity so called )(Tf gravity, where T denotes the torsion scalar has been investigated by

using Hybrid Expansion Law (HEL). The behaviour of accelerating universe is discussed by considering

depiction model of )(Tf gravity i.e. )( TTf = . For 1= , the linear form of )(Tf gravity is

mentioned i.e .)( TTf = . The physical behaviour of the derived model has been discussed using some

physical quantities. Also, the function of the torsion scalar for the universe is evaluated.

Keywords: Plane Symmetric model, perfect fluid, f(T) gravity

197

OP-66

On Numerical Invariant of Graph

Rohit M. Patne1*, Gajanan R. Avachar1

1Department of Mathematics, SSES Amravati’s Science College, Congress Nagar, Nagpur

email: iitg2010rohit@gmail.com,gravachar@gmail.com

Abstract:

In this paper we have studied the numerical invariant of graph G by using an edge boundary operator and

edge co-boundary operator Gp,q. We have also studied the relation between complete subgraph of G and

the properties of G with Euler characteristics equation for an edge (p,q) graph Gp,q of a graph G [1,2].

Keywords : boundary operator, co-boundry, cycle space, co-cycle space

References

1 F. Harary, Graph theory, Addison-Wesley, Massachusetts, 1969

2 J. A. Bondy, U.S.R. Murty, Graph theory, Springer, 2008

OP-67

Some Properties of Bases Intersection Graph

Iram Tahleel Jaleel Ahmed, Suryakant M Jogdand SSGM college, Loha, Dist. Nanded

email: iramtahleel@gmail.com

Abstract:

In this paper we introduce graphical structure of a vector space called as " Bases Intersection Graph" of a

vector space V over a finite field F. We will discuss some basic Properties of the graph such as

connectivity, diameter, girth, clique no., chromatic no., degree of a vertex,degree of graph etc.

Keywords: Vector space, field, bases, graph, clique no., chromatic no., linearly independent set,

dependent set etc.

OP-68

Some Generalised Fixed Point Theorems for New Contraction Mapping in

Function Weighted Metric Spaces

S. P. Birajdar1*, S. S. Zampalwad2, N. S. Pimple3 1N. E. S. Science College Nanded

2Gramin Mahavidyalaya Vasantanagar, Mukhed 3Rajarshi Shahu Mahavidyalaya(Autonomous), Latur

email: shivprasadb28@gmail.com, sambhaji4267@gmail.com, nishpimple2019@gmail.com

Abstract:

In present paper we introduce the new contraction mappings and its properties in function weighted

metric spaces and prove some generalised fixed point theorems in it by using new contraction mappings.

Also we provide some examples for the verification of obtained results.

198

OP-69

Linear Differentiable and Nonlinear Control System Problems

V. C. Borkar Department of Mathematics and Statistics, Yeshwant Mahavidyalaya Nanded, India 431605

e-mail : borkarvc@gmail.com

Abstract:

In this paper we review briefly some of the basic results and techniques in the linear control systems .We

consider the optimal control problems of systems governed by evolution differential inclusions . The

existence of optimal pairs as well as optimally conditions are proved . A regularization techniques is

presented and the relation between the solution to original and regularized problem is studied . We

concretize the abstract results by applying to certain Stefan –type optimal control problems , providing

controllability results for free boundary and derived special properties of the optimal control. .

Keywords : Linear optimal control problems, Systems theory control, PDEs in connection with control

and optimization, Stefan Like problems

OP-70

A new method to solve Transportation Problem –

Linear Congruence Approach

Kirtiwant P. Ghadle1, Dhanashri A. Munot2*

1Department of Mathematics, Dr Babasaheb Ambedkar Marathawada University,

Aurangabad, 431004 (M. S.) India 2Departments of Mathematics, SAJVPM’S Smt. S. K. Gandhi Arts, Amolak

Science and P. H. Gandhi Commerce College, Kada, 414202 (M. S.) India.

e-mail : ghadle.maths@bamu.ac.in, dhnshrmunot50@gmail.com

Abstract:

Transportation Problem is an application of Linear Programming Problem studied in the area of

Operations Research. In the literature several methods are proposed to solve transportation problem which

aims to minimize time or cost of transportation i.e. time required or cost of transporting goods from

resources to destinations. This research article have proposed a new algorithm using linear congruence

which minimize cost of transportation. A numerical example is solved to demonstrate the potential use of

proposed algorithm. Obtained results show that our suggested algorithm is simpler and computationally

more efficient than some existing methods commonly used in the literature. Proposed algorithm is also

coded in MATLAB which makes it user friendly.

Keywords: Transportation Problem, Linear Congruence

199

OP-71

Application of Matrix to Cryptography

Aruna Kulkarni

Department of Mathematics, SAJVPM’S Smt. S. K. Gandhi Arts, Amolak

Science and P. H. Gandhi Commerce College, Kada, 414202 (M. S.) India

e-mail : abhiarud@gmail.com

Abstract:

The purpose of this paper is to take a short review of mathematical operations which are used in

cryptography. Cryptography is a branch of mathematics which gives different techniques to secure the

information being transmitted. Cryptography is the practice and study of hiding information from all but

those with the means or key to decode the message. The area of cryptography employs many different

means of transforming normal data in to unreadable form. This paper describes an activity build around

one of the techniques that illustrates an application of matrices to cryptography. The method involves two

matrices of which one is used to encode the encoding matrix and the other one to decode the decoding

matrix. The nature in the original message or stream are assigned numerical values and the matrix must be

invertible for use in decoding .This method has a great potential , applied to exchange of message is done

confidentially.

Keywords: Matrix, message, invertible, encode, decode

OP-72

Signature Flipping of Isotropic Homogeneous Space-time with Holographic

Dark Energy in 𝐟(𝐆) Gravity

S. D. Katore1, V. R. Chirde2*, S. V. Raut2, S. H. Shekh3

1Department of Mathematics, Sant Gadge Baba Aravati University, Amravati-444602, Maharashtra India 2Department of Mathematics, G. S. G. Mahavidyalaya, Umarkhed-445206, Maharashtra, India

3Department of Mathematics, S. P. M Science and Gilani Arts and Commerce College-Ghatanji-445301,

Yavatmal, Maharashtra, India

e-mail : katoresd@rediffmail.com, vrchirde333@rediffmail.com,da_salim@rediff.com

Abstract:

Present investigation devoted to the dynamical study of isotropic and homogeneous FRW space-time

filled with a Holographic Dark Energy fluid with Cosmic String in the framework of numerous form of

𝐟(𝐆) gravitymodels (linear, quadratic and inverse model). We determine the aspects of the derived model

by considering the hybrid expansion law for the average scale factor that yields power and exponential

inflation cosmologies, in its special cases. As per the observation, the model is singular and singularity

observed at an initial epoch. the contribution of Gauss-Bonnet term in linear and quadratic model is act

just like as cosmological constant hence for whole expansion it is 𝚲𝐂𝐃𝐌 model which supports and

resembles with the observational fact that the usual matter is about 4% and the dark energy occupies near

about 73% of the energy also the results with several high precision observational experiments, especially

the Wilkinson Microwave Anisotropic Probe (WMAP) satellite experiment.

Keywords: FRW space-time, 𝐟(𝐆)gravity, hybrid expansion law

200

OP-73

Magnetized Perfect Fluid Bianchi Type-III Cosmological

Model with Variable Λ and G

R. S. Rane1, G. C. Bhagat2* 1Department of Mathematics, Y. C. Science & Arts college, Mangrulpir, Washim,

Maharashtra-444403, India 2Department of Mathematics, S.P.M. Science and Gilani Arts and Commerce College, Ghatanji,

Yavatmal, Maharashtra-445301, India

e-mail : rsrane53@rediffmail.com, gunwantbhagat@gmail.com

Abstract:

In the present study, we have studied Bianchi type-III cosmological model with perfect fluid source

containing magnetic field in general theory of relativity with linearly varying variable cosmological and

Gravitational constant. We have obtained the general solutions of the Einstein’s field equations for the

cosmological model by assuming the circumstance of anti-stiff fluid i.e. relation between pressure and

density and observed that in our derived model the Universe represents shearing, non-rotating and

expanding model of the universe with big-bang starts in the midst of both scale factors is monotonically

increasing function of t. The behavior of the Universe in presence of magnetic field and singularities in

the model are discussed in detail. Furthermore some physical and geometrical aspects of the model are

discussed.

Keywords: Bianchi type-III cosmological model, Anti-stiff fluid, Variable cosmological, Gravitational

constant

OP-74

Canonical Cosine Transform and their Optimistic Results

S. B. Chavhan

Department of Mathematics,

D. B. College Bhokar, Dist.Nanded-431801 (India)

e-mail : sbcmath2015@gmail.com

Abstract:

This paper studies different properties of canonical Cosine transform. Such as Time Reverse, linearity,

differentiation, parity , Shifting Property, Modulation theorem is also proved. We have proved some

important result about the Kernel of canonical cosine transform.

Keywords : Generalized function,Canonical transform, Canonical cosine transform, Modulation theorem,

Fourier transform

References :

1. Almeida L.B. : An introduction to the angular fourier transform, IEEE, 257 (1993)

2. Almeida L.B. : The fractional fourier transform and time frequency representation IEEE trans. on

signal processing 42 (1994)

3. Bhosale B.N. and Choudhary M.S. : Fractional fourier transform of distributions of compact support,

Bull. Cal. Math. Soc., 94, 349 (2002)

201

PP-1

On Henstock – Kurzweil Sumudu Transform

T. G. Thange1*, S. S. Gangane1

1Department of Mathematics, Yogeshwari Mahavidyalaya, Ambajogai,

email: tgthange@gmail.com, satyapadma.1912@gmail.com

Abstract:

In the present research paper the Sumudu Transform is considered as a Henstock – Kurzweil Integral.

Different existential conditions are given. Elementary properties are discussed. The necessary and

sufficient condition is given for the Sumudu Transform of a function f to be continuous. It is given that

Sumudu Transform exist as Henstock – Kurzweil Integral and finally inversion theorem is established.

Keywords: Sumudu transform, Henstock – Kurzweil integral

References:

1. R. G. Bartle, A Modern Theory of Integration, Grad. Studies in Math, Vol. 32 Amer, Math Soc.

Providence, 2001

2. R. A. Gordon, The Integrals of Lebsgue, Denjoy, Perron and Henstock, Grad. Studies in Math, Vol. 4

Amer. Math. Soc. Providence, 1994

3. C. Swartz, Introduction to Gauge Integrals, World Scientific, Singapore, 2001

PP-2

Analysis of EVI Parameters in Copper Ion Doped Alkali Sulphide Phosphors

Sudha Rani Dehri1*, Anju Pakhale2 , Sumedha Tamboli2, G. R. Avchar1 , S. J. Dhoble2

1Department of Mathematics, Shri Shivaji Science College,Nagpur-440001, India,

Email: sdehri@yahoo.in, gajanan.avchar@sscn.in 2Departmentof Physics, R. T. M. Nagpur University, Nagpur-440033, India,

Email: anjupakhale123@gmail.com, sumedhajtamboli@gmail.com, sjdhoble@rediffmail.com

Abstract:

In present work, Cu+ doped MgSCaS, SrS and BaS phosphors were analysed for their electron-vibration

interaction (EVI). The photoluminescence emission and excitation spectra of Cu+ ion doped MgS, BaS,

CaS and SrS compound is already reported by Yamashita et. al.1

Result given by Yamashita et. al.1 were utilised for determining strength of electron-phonon coupling. In

Cu+ metal ion,3d10→3d94stransition is responsible for photoluminescence emission. This electronic

transition is associated with lattice vibrations called as phonons2

Depending on host environment this coupling varies and photoluminescence emission also changes.

Strength of electron-phonon coupling can be determined in terms of Huang Rhys factor (S) and Phonon

energy (ħω).3.

Using values of stokes shift and FWHM from experimental results, Huang Rhys factor (S), Phonon

energy (ħω) and Zero phonon line (ZPL) were determined.

Similar methodology was applied for CaS, BaS and SrS and their results were compared.

Keywords: Cu+ metal ion, EVI parameter, sulphide, photoluminescence

References:

202

1. N. Yamashita, Photoluminescence properties of Cu+ centres in MgS, CaS, SrS and BrS, Japanese

journal of Applied Physics, 30, 3335 (1991)

2. P. D. Bhoyar, R. Choithrani, S.J. Dhoble, Study of electron-vibrational interaction and concentration

quenching effect of Cu+ ions in lithium based sulphate phosphors, Solid state sciences, 57, 24 (2016)

3. P. D. Bhoyar, S.J. Dhoble, Study of electron vibrational interaction parameters in chlorophosphate

activated with Eu2+ ion, Materials Chemistry and Physics, 147, 488 (2014)

PP-3

Marder Type Bulk Viscous String Cosmological Universe in ( )TRf ,

Gravity with Time Varying Deceleration Parameter

Pramod P. Khade1*, Amrapali Wasnik2

1Vidya Bharati Mahavidyalaya, Camp Amravati

email:mathsvbmvpk@gmail.com 2Bhratiya Mahavidyalaya, Amravati

email:amra_math@rediffmail.com

Abstract:

We propose a specially homogeneous and anisotropic Marder space-time with bulk viscous string

viscosity in the framework of ( )TRf , gravity by considering two cases (i) the special form and (ii)

linearly varying deceleration parameter. To obtain a deterministic solution of the field equation we have

been used some physical plausible condition. In this theory, cosmological model is presented in both

cases. Also some important features of the models, thus obtained, have been discussed.

Keywords: String matter, Bulk viscous, Marder universe, Deceleration parmater, ( )TRf , gravity

PP-4

R- Annihilator 𝝁-Hollow Modules

S. K. Gorle1*, R. S. Wadbude2

1Hutatma Rashtriya Arts and Science College, Ashti.

Wardha-442202(M.S) India

email: sagargorle99@gmail.com 2Mahatma Fule Arts, Commerce and Sitaramji Chaudhari Science Mahavidyalaya,

Warud -444906(M.S) India

email: drramesh_wadbude@rediffmail.com

Abstract:

In this paper we construct some examples for R-Ann-𝜇-hollow modules and add theorems, propositions.

Related concept was given by Nicholsion and Zhou. Let M be an R-modules, then M is R-Ann-𝜇-hollow

module if and only if every submodule A of M such that M/A Small in M. Every finitely generated proper

submodule N of M is R- Ann-𝜇-small for M is a faithful and torsion-𝜇-small.

Keywords: Hollow module, Annihilator-small, 𝜇-small, Cosingular module, R-Ann-hollow, R-Ann-𝜇-

small submodule, Torsion-𝜇-small

203

PP-5

Dynamical Study of a Vector Host Epidemic Model

with Non-monotonic Incidence

Seema Raut1*, Sujatha Janardhan2

1Department of Applied Mathematics, G. H. Raisoni. Institute of Engg. & Tech., Nagpur.

email: seema.ssk83@yahoo.in 2Department of Mathematics, St. Francis De Sales College, Nagpur.

email: sujata_jana@yahoo.com

Abstract:

In this paper, we formulated vector host epidemic model of diseases with non monotonic incidence rates.

Non monotonic behaviour of infected hosts as well as infected vectors are studied on influence from the

behavioural changes of susceptible host population and the effect of repellents on vectors, respectively.

Suitable Liapunov functions are constructed to discuss the global asymptotic stabilities at the equilibrium

points. Results are verified by conducting numerical simulation. This model helps to understand the

transmission mechanism of vector borne diseases better.

Keywords: Vector borne diseases, Asymptotic stability, Disease free equilibrium, Eendemic equilibrium

References:

1. M. Ozair, A. A. Lashari, I. H. Jung, K. O. Okosun, Stability Analysis and Optimal Control of a

Vector-borne Disease with Nonlinear Incidence, Discrete Dynamics in Nature and Society, 2012, 1

(2012)

2. H.-M. Wei, X.-Z.Li, M. Martcheva, An Epidemic Model of a Vector-borne Disease with Direct

Transmission and Time delay, Journal of Mathematical Analysis and Applications, 342, 895 (2008)

3. L. Cai, X. Li, Global Analysis of a Vector Host Epidemic Model with Nonlinear Incidences, Applied

Mathematics and Computation, 217, 3531 (2010)

PP-6

Physical Aspects of Five Dimensional Plane Symmetric

Metric in f(R,T) Gravity

S. P. Kandalkar1, M. C. Dhabe2*, T. D. Nakade3

1 email:spkandalkar242@gmail.com 2 email: mahaveer.dhabe@gmail.com 3 email:tush.nakade9028@gmail.com

Abstract:

In the present study we have investigate five dimensional plane symmetric space-time in ),( TRf gravity

with matter field is considered in the form of perfect fluid. We obtain the gravitational field equation in

the metric formalism, which follow from the covariant divergence of the stress-energy tensor by

considering the first and second class model of 𝑓(𝑅, 𝑇) gravity of Harko et al. (Phys. Rev. D 84:024020,

2011). We find the solution of the field equations by allowing the law of variation of Hubble’s parameter

which yields a constant value of the deceleration parameter. Moreover, some physical, geometrical and

thermodynamical properties of the universe are discussed in detail.

Keywords: Five dimensional plane symmetric space-time, ),( TRf gravity

References:

204

1. Harko et al., ),( TRf gravity, Physics Review D, 84, 024020 (2011)

PP-7

Convolution Structure of Quaternion Extended Fractional Mellin Transform

P. B. Deshmukh1*, V. D. Sharma2

1Vinayak Vidyan Mahavidyalaya, Nandgaon Khandeshwar

email: pritidesh1982@gmail.com 2Arts, Commerce and Science College Kiran Nagar, Amravati

email: vidyasharma700@gmail.com

Abstract:

A quaternion is a four-element vector that can be used to encode any rotation in a 3D coordinate system.

It has many applications in animation, modeling and rendering etc. In this work we have defined the

Quaternion Extended Fractional Mellin transform. We have calculated the convolution structure also

proved some properties such as linearity, shifting, distributive, associative, conjugation and compatibility

with scalars.

Keywords: Quaternion, Quaternione extended fractional mellin transform, Two dimensional fractional

mellin transform

References :

1. V.D. Sharma, P. B. Deshmukh; Inversion theorem of two dimensional fractional Mellin transform,

International Journal of Applied Mathematics & Mechanics, 3, 33 (2014)

2. Logah Perumal, Quaternion and Its Application in Rotation Using Sets of Regions, International

Journal of Engineering and Technology Innovation, 1, 35 (2011)

3. Xin Liu ,1 Huajun Huang,2 and Zhuo-Heng He, Real Representation Approach to Quaternion Matrix

Equation Involving ϕ-Hermicity, Hindawi Mathematical Problems in Engineering, 2019, 3258349

(2019)

PP-8

A SIR Model of Tuberculosis with Drug Resistant

and Quarantined Compartments

Archana Singh Bhadauria1*, Homnath Dhunguna2

1Deen Dayal Upadhyaya Gorakhpur University, Gorakhpur-273009, U.P., India,

email: archanasingh93@yahoo.co.in 2School of Mathematics, University of Technology Sydney, Sydney, Australia

email: homnath.dhungana@uts.edu.au

Abstract:

Multidrug-resistant Tuberculosis (MDR-TB) is a complication of TB that often does not respond even to

the most powerful anti-TB drugs i.e. isoniazid and rifampicin. It is very difficult to treat MDR-TB due to

limited and expensive treatment options, insufficient availability of recommended medicines, and adverse

effects from the drugs experienced by the patients. Thus, it is imperative to aim at early detection and

quarantining the individuals to control the disease. We frame a mathematical model using nonlinear

ordinary differential equations to study the transmission dynamics of TB and measure the effectiveness of

quarantine and contact tracing of the MDR-TB patients, which may tend to the end of infection to benign

individuals. In addition to this, the reproduction ratio ($R$) is also estimated which manifests that

infection is being spread in the population as cumulative effect of drug sensitive and drug resistant

205

infectives. The sensitivity and equilibrium analysis of the model are also performed. The value of

reproduction number is important threshold quantity and can be kept under control ($R<1$) by managing

contact tracing and quarantine rates. Model parameters show a significantly slow reduction in MDR TB to

reach "End TB by 2025 goal"

Keywords: multi drug resistant; drug sensitive; quarantined; parameter estimation, equilibrium points,

basic reproduction ratio

PP-9

Study of Bianchi type-V Perfect Fluid Cosmological model with Time-

Dependent G and Λ in General Relativity

Shital Yadav1*, S. D. Tade2

1G H Raisoni Academy of Engineering & Technology Nagpur

email: shital.yadav@raisoni.net 2Jawaharlala Nehru college, wadi, Nagpur

email:drstade25@gmail.com

Abstract:

In a present paper, a spatially homogeneous and anisotropic Bianchi type-V model filled with perfect

fluid with bulk viscosity and particle creation is investigated within the framework of General Relativity.

It shows the homogeneous but anisotropic Bianchi type-V cosmological model with time-dependent

cosmological "constant”. The law of variation for mean Hubble’s parameter with average scale factor, in

an anisotropic Bianchi type V cosmological space–time, is discussed. The variation of Hubble’s

parameter, which gives a constant value of deceleration parameter, generates two types of solutions for

the average scale factor; one is the power-law and the other one is of exponential form. Here an attempt

has been made to solve the exact solutions of the Einstein field equations (EFEs) some physical and

kinematical properties of the models are also discussed.

Keywords : Bianchi type V model, imperfect fluid with Bulk viscosity, General Relativity

Reference :

1. C.P. Singh, Shri Ram, Mohd. Zeyauddin, Astrophysics Space Science, 315, 181 (2008)

206

PP-10

Stability of Steady State Solutions with Finite Energy for

Magnetohydrodynamic Equations in the Whole space

Swapna V. Uddhao1*, P. D. Raitar, R.V. Saraykar1

1Department of Mathematics, RTM Nagpur University, Nagpur-440033, (India)

email: swapna.uddhao@gmail.com, ravindra.saraykar@gmail.com

Abstract:

An important property of Steady state solutions for physical problems is stability. We ask the question “If

a Steady state solution is perturbed will it return to the same solution?”

In this paper, following the work of Bjorland and Schonbek (2007) we answer this question affirmatively

by examining the stability of steady state solutions with finite energy for magnetohydrodynamic (MHD)

equations in the whole space under certain restraints on force f.

Keywords: magnetohydrodynamic equations, stability of solutions, finite energy

References :

1. C. Bjorland, M.E. Schonbek, DOI: 10.1088/0951-7715/22/7/007

2. P. D. Raiter and R.V. Saraykar, IOSR Journal of Mathematics, 10, 16 (2014)

PP-11

Normalised Measures of Inaccuracy

Sapna K. Chandbhanani1*, P. A. S. Naidu1,

1Department of Mathematics

D. B. Science College, Gondia-441601

email: ahujasapana6@gmail.com, anjannaidu8@gmail.com

Abstract- Different motivations for normalising a new three parametric bi-measure of entropy,

directed divergence, inaccuracy have been proposed by simultaneous use of the generalisations

of Shannon’s measure of entropy due to Kapur’s and Havrda and Charvat’s measures of entropy

are also discussed for same objective and their properties have been discussed.

Keywords: Bi-Measure of Entropy, Directed Divergence, Inaccuracy

References:

1. Shannon C. E., A Mathematical Theory of Communication, Bell system tech., vol. 27, pp.

379- 423, 623-659, 1948

2. Kapur. J. N., Measure of Information And Their Application, New Delhi: New Age

International Limited, 1994 3. K. J. N., Measure Of Informations And Their Applications," Wiley Estern Limited, New

Age International LTD, 1994

207

PP-12

Spatially Homogeneous and Anisotropic Two Forms

Dark Energy Cosmological Model

Y. S. Solanke1*, D. D. Pawar2, V. J. Dagwal3

1Department of Mathematics, Mungsaji Maharaj Mahavidyalaya, Darwha,Yavatmal(M.S.)- 445202, India

email: yadaosolanke@gmail.com 2School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University,

VishnupuriNanded(M.S.)-431606 (India)

email: dypawar@yahoo.com 3Department of Mathematics, Government College of Engineering, New Khapri, Nagpur(M.S.)- 441108,

email: vdagwal@gmail.com

Abstract:

The present work deals with investigation of spatially homogeneous and anisotropic Bianchi Type- III

space time by using dark energy portion of the universe which consists of quintessence form of dark

energy and partly of cosmological constant form; both are alternatives to the dark energy. We have

determined the exact solutions of the Einstein’s field equations by assuming a special law of variation of

Hubble’s parameter proposed by Berman that yields the constant deceleration parameter. Some physical

parameters of the model are determined. From the point of kinematical behaviour we have also extended

our work by deriving the analytic expressions for the look back time.

Keywords: Bianchi type III space time, cosmological constant, quintessence

PP-13

Magnetized Quark and Strange Quark Matter in the Higher Dimensional

Spherically Symmetric Space Time Admitting One Parameter Group of

Conformal Motions

G. S. Khadekar1, Saroj R. Kumbhare2*

1Department of Mathematics, Rashtrasant Tukadoji Maharaj Nagpur University,

Mahatma Jyotiba Phule Educational Campus, Amravati Road, Nagpur-440033 (INDIA)

email: gkhadekar@gmail.com 2Amolakchand Mahavidyalaya, Yavatmal - 445001 (INDIA)

email: cmoundekarsaroj@gmail.com

Abstract:

In this paper, we have examined magnetized quark and strange quark matter in higher dimensional

spherically symmetric space-time admitting one parameter group of conformal motions. We have solved

Einstein’s field equations in higher dimensional spherically symmetric space-time via conformal motions.

The features of the solutions are also discussed in the framework of higher dimensional space-time.

Keywords: Higher dimension, Magnetized quark and strange quark matter,Conformal motion, Bag

Model.

208

PP-14

On Reliability of System with Two Repair Facilities

Anil Singh1, Sanjay Chaudhary 1Department Of Mathematics, Dr. Bhimrao Ambedkar University, Agra (U.P.)

email: anil291singh@gmail.com

Abstract:

The aim of this paper is to present a reliability analysis of two unit cold standby system with repair

facilities. The system consists of two units with each one with operable or failed state. At any time, one

unit is operating while other is in cold standby. The failure has been divided into two parts major and

minor. Minor failure requires minor repair facility and major repair requires major repair facility. Thus

there are two repair facilities available. System has perfect switching for units .The system completely

fails on the failure of both the units. The failure and repair times follow exponential and general time

distribution. Partial Differential Equations and Laplace Transforms of various state probabilities have

been obtained. Reliability of the system has been derived in the form state probabilities.

Keywords: cold standby system, major and minor repair facilities

PP-15

Katugampola Fractional Calculus with Generalized 𝒌-Wright Function

Ahmad Y. A. Salamooni1*, D. D. Pawar 1 1School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University

Nanded-431606, India

email: ayousss83@gmail.com, dypawar@yahoo.com

Abstract:

In this article, we presented some properties of the Katugampola fractional integrals and derivatives.

Also, we studied the fractional calculus properties involving Katugampola Fractional integrals and

derivatives of generalized 𝑘-Wright function 𝜙(z).

Keywords: Katugampola Fractional integral and derivative, 𝛾-Gamma function, 𝑘-Wright Function

209

PP-16

Fractional Order Shehu Transform

Anantha R. Gade1*, Tukaram G. Thange2 1Arts Commerce and Science College Kille Dharur Dist. Beed (M.H.) INDIA

email : ananthagade45@gmail.com

2Yogeshwari Mahavidyalaya Ambajogai,Dist. Beed, MH, INDIA

email: tgthange@gmail.com

Abstract:

In this study, we proposed new generalized Shehu transform of fractional order called “fractional Shehu

transform” of order 0 < α < 1. This transform is applying for functions which are differentiable but by

fractional order. By using the definition of fractional order Shehu transform we prove fundamental

properties of this integral transform. Finally, we have obtained convolution and inversion.

We all are familiar about the application of integral transform for the solution of different differential and

integral equations. It is the best tool for finding the solutions of many of this problem. Shehu transform is

the Laplace type integral transform but it is generalization of Laplace and Sumudu transform which is

widely used for solving differential equation with efficient and more convenient way. If p(z) is continuous

and continuously differentiable then by using regular definitions of different integral transform we solve

differential equations of function p(z) but if p(z) is continues but differentiable by fractional order α, then

this definitions doesn’t work, in that case we use the definition of fractional order Shehu transform for

finding the solution of differential equations in particular fractional order differential equations of

function p(z).

Keywords: Shehu Transform, Laplace Transform, Mittag-Leffler function, Generalized function (Dirac’s

Distribution), Fractional Derivative, Fractional Integration.

PP-17

FRW Bulk Viscous Cosmology with Modified Chaplygin Gas in

Higher Dimension

Rupali wanjari1*, A . R. Golhar2, A. A. Qureshi3

1 Department of Mathematics, DRB Sindhu Mahavidyalaya, Nagpur, India,

email: rwanjari@gmail.com, maths@yahoo.com 2Department of Physics, DRB Sindhu Mahavidyalaya, Nagpur, India,

email: artigolhar@gmail.com

Abstract:

In this paper, we have studied a five dimensional FRW bulk viscous cosmology with modified Chaplygin

gas. In this framework we find time-dependent energy density by using bulk viscosity and modified

Chaplygin gas for flat Universe. We also observed that stability of system strongly depend on viscosity

coefficient.

Keywords: FRW cosmology, Bulk Viscosity, Chaplygin gas

References :

1. H. Saadat, and B. Pourhassan, FRW bulk viscous cosmology with modified Chaplygin gas in flat

space, Astrophysics. Space Science, 343, 783 (2013)

2. E. O. Kahya, and B. Pourhassan, Universe dominated by the extended Chaplygin gas, Modern

Physics Letter A 30, 1550070 (2015)

210

PP-18

Eigen function Wavelet Transform for Integrable Boehmians

T. G. Thange1*, A. M. Alure2

1Department of Mathematics, Yogeshwari Mahavidyalaya,Ambajogai.

email: tgthange@gamil.com 2 Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad.

email: amalure86@gmail.com

Abstract:

In this paper eigenfunction wavelet transform is extened to integrable boehmians space. A suitable

bohemian space is constructed for this extension. Further Inversion formula is also obtained.

Keywords: Eigenfunction wavelet transform, Boehmian space

References:

1. J. Mikusinski, P. Mikusinski, Quotients de suites et leurs applications dans l’analyse fonctionnelle,

C.R. Acad. Sci. Paris Ser. I Math. 293 (1981), 463–464.

2. Zemanian, A. H., Generalized Integral Transformation, Interscience Publisher, New York, (1969).

3. W. Rudin, Real and Complex Analysis, Third Edition, McGraw-Hill, New York (1987).

PP-19

Analysis of General-Relativistic Hydrodynamics of Bianchi Type-V

Cosmological Model in General Relativity

G. S. Khadekar1, Ather Husain2*, S. D. Tade3

1Department of Mathematics Rashtrasant Tukadoji Maharaj Nagpur University Mahatma Jyotiba Phule

Educational Campus Amravati Road, Nagpur 440033, India

email: gkhadekar@yahoo.com 2Department of Mathematics, Narayanrao Kale Smruti Model College Karanja (Gh), Dist: Wardha,

(M.S.) 442203, India

email: atherhusain1001@gmail.com 3Department of Mathematics, Jawaharlal Nehru Arts, Commerce and Science College,

Wadi, Nagpur India

email: drstade25@gmail.com

Abstract:

In this paper, we have analyzed the dynamical properties of Einstein’s field equations for Bianchi type-V

space-time with hydrodynamic source within the framework of Einstein’s general theory of relativity. The

exact solutions to the corresponding field equations are obtained with the help of Hubble’s law of

variation which yields a constant value of deceleration parameter. It is observed that at an initial stage

both metric potentials of the derived stable model are comes out to be constants and at large time they

increases indefinitely. Also, for 1=t , the model is constant but at a specific time m

mtt s

1−== with

vanishing metric potential the model represents singularity. For the particular choice of constant i , the

model shows isotropy. The equation of state parameter having value 1− shows the model start with

acceleration.

Keywords: Cosmological model; Hydrodynamics; Space-time; Cosmology.

211

PP-20

Analysis of Various Performance Measures of Finite Source Single Server

Queuing Model by using Binomial Distribution and Uniform Distribution

Damodhar F Shastrakar1*, Sharad S Pokley2, Sajid Anwar3 2Mathematics Dept. KITS Ramtek (MS)

2Mathematics Dept. Anjuman College of Engg. & Tech., Nagpur (MS)

email : dfshastrakar@yahoo.com, sharadpokley@gmail.com, sajidanwar0616@gmail.com

Abstract:

Queuing Model of Finite Source having Single Server is discussed here in order to analyze the various

performance measures. The Calling Population in this model is limited. The model is working under

Queue Discipline GD. The numbers of arriving customers are fixed. When the system is busy in serving

the customers that are already in the system, the arrival rate of next additional customer gets reduced

which minimizes the further congestion. Since the Calling Population is limited, the arrival rate of

customers and service rate of server follows Binomial Distribution, whereas the inter-arrival time and

service time follows Uniform Distribution. For the model, formula to find the expected waiting time of a

customer in the queue is derived. This formula and Little’s formula are then used to analyze the other

performance measures of the Queuing Model. The calculated results obtained from the new derived

formulae are compared with the results of existing available formulae to interpret the conclusion.

Keywords: calling population, mean arrival rate, mean service rate, inter-arrival time, service time

References:

1. Belenky A. S., Estimating the Size of the Calling Population in Finite-Source Election Queues by

Bilinear Programming Techniques, Elsevier Science Direct Mathematical and Computer Modelling

45, 873 (2007)

2. Chandra M. Jeya, A Study of Multiple Finite-Source Queueing Models, Journal of the Operational

Research Society, 37, 3, 275 (1986)

3. Gross Donald, ShortleJhon F., Thompson James M. and Harris Carl M., Fundamentals of Queueing

Theory, Fourth Edition, Wiley Series in probability and statistics, Wiley India Edition (2012)

212

PP-21

Axially Symmetric Bulk Viscous Cosmological Model in f(R,T)-Theory of

Gravity

Deepika R. Golechha1*, G. R. Avchar1, S. D. Tade2

1Department of Mathematics 1Shri Shivaji Education Society Amravati’s Science College, Nagpur, India

email: richagolechha@gmail.com, g.avchar@sscnagpur@gmail.com, 2Department of Mathematics

Jawaharlal Nehru Arts, Commerce and Science College, Nagpur, Nagpur University, India

email: drstade25@gmail.com

Abstract:

In this paper, axially symmetric space-time in presence of perfect fluid with viscosity has been considered

and studied within the framework of f(R,T) theory of gravity proposed by Harko et al (2011)). The

solutions of the field equations are evaluated by assuming the proportionality of expansion ( ) to the

shear scalar ( ). The physical and geometrical aspects of the model have also been discussed.

Keywords: Axially Symmetric metric, f(R,T) theory of gravity, Viscous fluid

References:

1. Harko et al., ),( TRf gravity, Physics Review D, 84, 024020 (2011)

PP-22

Bianchi Type-IX Cosmological Model In Self-Creation Theory of Gravitation

A. S. Nimkar1, J. S. Wath2*, V. M. Wankhade1 1Department of Mathematics, Shri Dr. R. G. Rathod Arts and Science College, Murtizapur, Dist. Akola,

(Maharashtra) India

email: anilnimkar@gmail.com, wankhadevaishnavi07@gmail.com 2 Department of Applied Mathematics, P. R. Pote (Patil) College of Engineering and Management,

Amravati, (Maharashtra) India

email: jotsnawath@gmail.com

Abstract:

In this paper, we study macroscopic body cosmological model in the context of self-creation theory. We

solve field equations by using relation between metric coefficients and equation of state for macroscopic

body. Also, we discuss the features of the obtained solutions.

Keywords : Bianchi Type-IX, self -creation theory, macroscopic body

213

PP-23

A Fuzzy Inference System To Investigate Stock Market Timing

Manisha Dubey1, Sanjeev Kumar

1Department Of Mathematics, Dr. Bhimrao Ambedkar University, Agra (U.P.)

email : mdmanidubey@gmail.com

Abstract:

To determine the buy and sell time is one of the most important issue for investors in the stock market.

Stock investment has become an important investment activity and internet makes it easier exchange

stock information and to make stock transaction. Trading in the stock market is full of uncertainly so there

is vagueness in the market. Predicting the market is very difficult since it depends on several unknown

factors. A person can not observe that what is going to be happening and therefore investors often loose

their money due to unclear investment objective. In this work a fuzzy approach to stock market timing is

investigated. The proposed fuzzy model helps in identifying the stock market which is very bullish,

bullish, neutral or very bearish, bearish. The four input factor are fuzzified to get a output using fuzzy

logic, the stock market which is either very bullish, bullish, very bearish, bearish or neutral continues to

some extent. For fuzzifying these input data, trapezoidal membership function is used, and center of

gravity method is used for defuzzification of fuzzy output. The results found suggest that fuzzy modeling

for this purpose is very promising.

Keywords: Stock market, fuzzy logic, timing, trapezoidal membership function

PP-24

Fuzzy Logic Concatenation in Face, Fingerprint and Iris Multimodal

Biometric Identification System

Monika Rathore1, Sanjeev Kumar

1Department Of Mathematics, Dr. Bhimrao Ambedkar University, Agra (U.P.)

email : monikarathore16sep@gmail.com

Abstract:

Security of information is one of the most important factors of information technology and

communication. So systems need strong procedures to protect data and resources access from

unauthorized users. There are number of ways to prove authentication and authorization. But the

biometric authentication beat all other techniques. Biometric-based authentication systems represent a

valid alternative to conventional approaches. As Multimodal biometric identification system is more

powerful, more accurate, less noisy data than the Single/Unimodel biometric system. This paper introduce

three biometric techniques which are face recognition, fingerprint recognition, and iris recognition (i.e.

Multi Biometric System) & aims at concatenating three biometric features namely face, fingerprint and

iris to minimize False Accept Rate(FAR) and False Reject Rate(FRR). And shows using these biometrics

has good result with high accuracy using fuzzy logic at decision level. In greater detail, fuzzy logic based

approach at decision level is used for concatenation. Fuzzy logic is used for the effect of each biometric

result combination. The proposed multimodel system achieves interesting results with several commonly

used databases.

Keywords: Biometric, Multibiometric (face, fingerprint, iris), fuzzy logic

214

PP-25

Analysis of Queuing System and Impact of Digital Payments in Supermarket

Sadhna Singh1*, R . K. Srivastava2, Amendra Singh3 1,2Department of Mathematics, S.M.S. College, Jiwaji University, Gwalior (M.P.)

email : 1sadhna20singh@gmail.com 3Department of Mathematics, I.B.S., Dr. Bhimrao Ambedkar University, Agra (U.P.)

Abstract:

This paper deals with digital payments and cash payments in supermarkets. Initially we take two counters

for comparison of digital and cash payments. The first counters for digital payments and the second for

cash payments, and calculate billing times from both counters. Our aim is to reducing the customers

waiting time by increasing the number of servers according to the conditions, both digital and cash

payments. The analysis of various parameters of the queuing system, calculate utilization factor, service

rate, arrival rate, calculate idle bill payment counter, customer satisfaction rates, and waiting time. After

analyzing the parameters of the parameters of queuing system model, it is observed that digital payments

save time.

Keywords: digital payment, cash payments, Utilization factor, Percentage of idle bill counter, Waiting

time.

PP-26

Training Need Ranking of Hotel Manager by Fuzzy Technique for Order

Preference by Similarity to Ideal Solution

Trupti A. Thakre1*, Onkar K. Chaudhari1

1G. H. Raisoni College of Engineering, Nagpur, India

email: trupti.arvind.thakre@gmail.com, onkar.chaudhary@raisoni.net

Abstract:

Hotels have to be flexible in terms of giving best services to the customers with latest trends. It is a tough

job for them to satisfy the customers and to retain them in order to have returns on long term. In order to

perform this challenging job, a compatible manager is required. Therefore, regular training and

development programs are necessary to learn or improve skills and knowledge of the manager, as it has

long term positive effect on the prosperity of the hotel. To meet this requirement, proper growth and

synchronization of change in customers’ requirements should be planned. This is possible with the proper

planning of training and development programs. But, then the question arises about which skill trainings

are more important and how they should be given. So, a need of ranking these skill trainings is created. In

this paper, multi-criteria group decision making model is considered, where as per the requirement of

hotel manager, various training needs of a hotel manager have been identified by group of decision

makers and ranked with simplified fuzzy technique for order preference by similarity to ideal solution

(fuzzy TOPSIS) method. Results are reported with this application on the basis of closeness coefficient

using trapezoidal fuzzy number. Results showed that the fuzzy TOPSIS can be successfully used to rank

the appropriate training need required by the manager of hotel. The study concludes that the technique is

easy to apply and gives simple solution to a multi-criteria decision making model.

Keywords: training need, ranking; fuzzy TOPSIS, hotel manager

215

PP-27

Application of Topology Clustering in Water Distribution System

Vigna Purohit1*, Sravanthi Mopati1, Madani Gazal Khan1 Omkar Patil1, Asheesh Sharma1, Aabha Sargaonkar1

1CTMD, CSIR-NEERI, NAGPUR-440020

email: ap_sargaonkar@neeri.res.in, vignapurohit9@gmail.com, sravanthi.mopati@gmail.com

gazalodhi040506@gmail.com, a_sharma@neeri.res.in, omkar3131@gmail.com

Abstract:

Mathematical methods from graph theory and clustering are useful to create a graphical representation of

a network. Water Distribution System (WDS) in urban area or cities are important for Urban Local

Bodies (ULBs) for the regular supply of portable drinking water. WDS network is complex to understand

because of connectivity amongst a number of elements such as pipes, nodes, valves and water tanks

[2014]. Several graph-theory metrics are available to quantify the robustness and redundancy of the WDS

network. But it is less reliable because it does not consider hydraulic conditions, pipe size, materials, and

isolation-valve locations [2011]. Topology clustering is a method that takes into consideration

connectivity amongst various elements, water flow direction and other hydraulic properties such as

pressure and headloss. This helps in simplification and further analysis such as contamination within

WDS, a concern for drinking water quality.

In the present study, clustering is performed on nodes of the water pipes. Topology and flow direction are

the main factors that are taken into consideration. Here, we assume that the direction of flow is from a

higher elevation to lower elevation nodes in the network, irrespective of the pressure in pipes. Topology

based clusters were created using the k-mean clustering technique. Flow direction-based clusters are

created inside topology-based clusters. Clustering is done to determine the exact location of the pipe in

which contamination intrudes into the system.

This has proved to be an important tool for simplification in terms of physical connectivity as well as

operational conditions of a network from a management point of view.

Keywords: Topology clustering, Water Supply, Water Contamination

References:

1. K. Diao, R. Farmani, G. Fu, M. Astaraie-Imani, S. Ward: D. Butler Water Science Technology, 70

(11), 1763-1774 (2014)

2. Lina Perelman, Avi Ostfeld: Environmental Modelling & Software, 26 (7), 969-972 (2011)

216

PP-28

On the Solution of Kinetic Equation For Katugampola

Type Fractional Differential Equations

Wagdi F. S. Ahmed1*, D. D. Pawar1, Ahmad Y. A. Salamooni1

1School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University

Nanded-431606, India

email : wagdialakel@gmail.com , dypawar@yahoo.com, ayousss83@gmail.com

Abstract:

In this article, we have investigate a solution of fractional Kinetic equation involving Katugampola type

fractional integral by using H-function. We also present the solution of Kinetic equation nvolving

Katugampola type Fractional integral with the help of generalized I-function and Mellin transform.

Keywords: Fractional Kinetic Equation, Katugampola Type Fractional Integral, Mittag-Leffler Function

and Mellin Transform.

PP-29

Higher Dimensional Viscous Cosmology with Inhomogeneous Equation

of State and Future Singularity

Deepti Raut 1, Arti Ghogre2*, N. V. Gharad3 1 Department of Mathematics, Rajiv Gandhi College of Engineering and Research, Hingna Road,

Wanadogri, Nagpur 441110, Maharashtra, India.

email: diptigudadhe0587@gmail.com 2 Department of Mathematics, Yeshwantaro College of Engineering, Hingna Road, Wanadogri, Nagpur

441110, Maharashtra, India.

email: arti_ghogre@rediffmail.com 3J N College, Wadi, Nagpur, e-mail: nvgharad@gmail.com

Abstract:

A universe media is considered as a bulk viscosity described by inhomogeneous equation of state (EOS)

of the form 𝑝 = (𝛾 − 1)𝜌 + Λ(𝑡), where Λ(𝑡) is a time dependent parameter. A generalized dynamical

equation for the scale factor of the universe is proposed to describe the cosmological evolution, in which

we assume the bulk viscosity and time dependent parameter Λ are linear combination of two terms of the

form: 𝜁 = 𝜁0 + 𝜁1𝐻, Λ(𝑡) = Λ0 + Λ1𝐻, i.e. one is constant and other is proportional to Hubble

parameter 𝐻 =��

𝑎 . In this framework, we demonstrate that higher dimensional model can be used to

explain the dark energy dominated universe, and the inhomogeneous term of specific form introduced in

EOS, may lead to three kinds of fates of cosmological evolution: no future singularity, big rip or Type III

singularity as presented by Nojiri and Odintsov (2005).

Keywords: Higher dimensional cosmology, inhomogeneous equation of state, bulk viscosity, dark

energy, future singularity

References :

1. S. Nojiri and S. D. Odintsov, Phys. Rev.D 72, 023003 (2005)

217

PP-30

Bianchi type-III Cosmological Model in Scale

Covariant Theory of Gravitation

P. M. Lambat1*, A. M. Pund1

1Department of Mathematics, Shri Shivaji Education Society Amravati’s Science College, Nagpur

e-mail: pmlambat@gmail.com, ashokpund64@rediffmail.com

Abstract:

The spatially homogeneous and anisotropic Bianchi type-III metric is considered in the presence of

Cosmic string source in the scale covariant theory of gravitation formulated by Canuto et al. (1977) with

the help of special law of variation for the Hubble’s parameter proposed by Bermann (1983). A

cosmological model with a negative constant declaration parameter is obtained in this theory. Some

physical properties of the model are also discussed.

Keywords: Bianchi type-III, Scale Covariant Theory, Cosmic String

PP-31

Study of Generalized Cosmic Chaplygin Gas in the Presence

of Linear Form of the Bulk Viscosity

Y. V. Rathod1*,G. R. Avachar2, S. W. Samdurkar3

1Department of Applied Mathematics, G.H. Raisoni Academy of Engineering

and Technology, Nagpur

email: rathod.yogesh61@gmail.com 2Shri Shivaji Education Society Amravati’s Science College, Nagpur

email: g.avachar@rediffmail.com 3Department of Mathematics, Vidya Vikas Arts Commerce and Science College,

Samudrapur, Dist. Wardha, India

email: rpahade2008@rediffmail.com

Abstract:

In this paper, we introduce generalized cosmic Chaplygin gas (GCCG) in the presence of linear form of

the bulk viscosity i.e. ζ = ζ0 + Hζ1 for Bianchi type cosmology in the framework of general theory of

relativity. Here we study interaction between generalized cosmic Chaplygin gas and matter. Also we

observe the effect of bulk viscosity on the cosmological parameters like Hubble parameter, energy density

and deceleration parameter for case I: special case and Case II: General case, we study the behaviour of

time dependent density in both the cases by plotting graphs.

Keywords: Bianchi type cosmology, generalized cosmic Chaplygin gas, bulk viscosity

218

PP-32

Kantowski Sachs Macroscopic Body Cosmological Models

In Scalar Tensor Theories of Gravitation

A. S. Nimkar1, S. R. Hadole1*, S .C. Wankhade1

1Department of Mathematics, Shri Dr. R. G. Rathod Arts and Science College, Murtizapur, Dist. Akola,

(Maharashtra) India,

email: anilnimkar@gmail.com, sangita.hadole2000@gmail.com, sharwariwankhade@gmail.com

Abstract:

In this paper, we have obtained the field equations in scalar-tensor theories of gravitation proposed by

Brans–Dicke and Saez-Ballester with the aid of Kantowski Sachs Space time in the presence of

macroscopic body. A determinate solution of the field equations in both theories are obtained by using

transformation. Also the expression for energy density, energy flow vector, stress tensor and some

physical properties of both the models are discussed.

Keywords: Kantowski-Sachs Model, Brans-Dicke and Saez-Ballester theory, macroscopic body

PP-33

Boundary Value Problem for First Order Random Differential Inclusions

S. B. Patil 1* , D. S. Palimkar2 1 Department of Mathematics, Sevadas Jr. College ,Vasantnagar, Kotgyal Tq.: Mukhed ,

Dist.: Nanded(MS) India

email: -sbpatil333@rediffmail.com 2 Department of Mathematics, Vasantrao Naik College, Nanded [MS] India

email: dspalimkar@rediffmail.com

Abstract:

The random differential inclusions is an important branch of probabilistic non-linear analysis and is

applicable in classical as well as random phenomenon of the universe. In this paper, we consider the first

order random differential inclusions and prove existence of lower , upper random solutions through

Marteli random fixed point theorem.

Consider the first order random differential inclusions of boundary value problem as

'( , ) (( , ), ( , )) . . [0, ]x t F t x t a e t J T

( (0, ), ( , )) 0L x x T =

Where : 2RF J R → a compact and convex valued random multi-valued map and 2:L J →

is a continuous single valued map.

we prove an existence result for the boundary value problem of first order random differential inclusions.

Our methodology is based on the existence of upper and lower solutions and a random fixed point

theorem for condensing maps of Martelli. The method of upper and lower solutions has been successfully

applied to study the existence of multiple solutions for initial and boundary value problems of first and

second order random differential inclusion for single valued. But This Paper is concerned with the

existence of solutions for the random boundary multi-valued problem.

219

Keywords: Multi-valued map, Random differential inclusions, condensing map, random fixed point

theorem

PP-34

Anisotropic Tilted Cosmological Model in F(R, T) Theory of Gravity

D. D. Pawar1, S. P. Shahare2*

1 School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University,

Nanded 431606, India

email: dypawar@yahoo.com 2 P. R. Pote Patil Institute of Engineering and Research, Amravati 444604, Maharashtra, India

email: sarikashahare83@gmail.com

Abstract:

An anisotropic Tilted Marder’s Cosmological Model is investigated in the framework of f(R, T) theory of

gravity, where R is the Ricci scalar and T is the trace of stress energy momentum tensor. Here we study

the class of f (R, T) = R + 2f (T). To decipher the solutions of field equations, we used a power law

relation between metric potentials proposed by Thorne (1967) and the special law of variation of

Hubble’s parameter proposed by Berman (1983) which yields constant deceleration parameter. The

solution of field equations represent an expanding model of the Universe. The physical behavior and

energy conditions of the model are discussed with the help of some physical parameters.

Keywords: Marders cosmological model, f(R, T) gravity, Hubbles parameter, deceleration parameter

References :

1. Thorne, K.S., Primordial element formation, primordial magnetic fields, and the isotropy of the

universe. Astrophysics Journal, 148, 51 (1967)

2. Berman, M.S., 1983. A special law of variation for hubble’s parameter. Il Nuovo Cimento B 74 (2).

1971–1996 182-6

PP-35

Non Static Plane Symmetric Cosmic Strings Cosmological

Model in f(R) Theory of Gravity

L. S. Ladke1, V. P. Tripade2*, R. D. Mishra 3

1Nilkanthrao Shinde Science and Arts College, Bhandravati, India 2Mohasinbhai Zaweri Mahavidyalaya Desaiganj (Wadsa), India

3Shriram Commerce and Science College, Kurkheda, India

Abstract:

In this paper, we investigate three non-statics plane symmetric cosmological models filled with cosmic

strings in )(Rf theory of gravity. Three models are filled with three different cosmic strings. To solve

the field equation, we used two conditions, the shear scalar is proportional to expansion scalar and the

power law between the scalar function F and average scale factor a.. We have found physical and

Kinematical parameters of each model which are useful in study of cosmology. Also we obtained Ricci

scalar of each model.

220

PP-36

Iteration Method for Approximating Solutions of Perturbed

Abstract Measure Differential Equations

D. M. Suryawanshi1*, S. S. Bellale1

1Dayanand Science College, Latur

email: dmsuryawanshi24@gmail.com, sidhesh.bellalel@gmail.com

Abstract:

In this paper we have proved the existence of the solution of perturbed abstract measure differential

equation by using Dhages iteration method. The main result is based on the iteration method included in

the hybrid fixed point theorem in a partially order normed linear space. Also we have solved an example

for the applicability of given results in the paper.

Keywords: Abstract measure differential equation, Dhage iteration method, Existence theorem, Extremal

solutions

221

A

A . R. GOLHAR 209

A A DESHPANDE 162

A. A. QURESHI 209

A. M. ALURE 210

A. M. Pund 217

A. N. Rangari 164

A. Nath 193

A. S. Nimkar 212, 218

A. S. Ramteke 172

A.R. Gotmare 166

Aabha Sargaonkar 215

Absos Ali Shaikh 151, 165, 177

AHMAD Y. A. SALAMOONI 208,

216

AINA GUPTA 184

Amendra Singh 214

AMRAPALI WASNIK 202

Anantha R. Gade 209

Anil Singh 208

Anisa M. Ahmad 162

ANJU PAKHALE 201

Anuradha Devi 155

ARCHANA SINGH BHADAURIA

204

Arijit Mukherjee 163

Aroonkumar Beesham 157

Arti Ghogre 167, 216

Aruna Kulkarni 199

ARUNKUMAR R PATIL 174

Asha B. Nale 194

Asheesh Sharma 215

ASHISH PRABHAKAR

NAWGHARE 163

Ather Husain 210

B

B. P. GARG 167

Binaya K. Bishi 193

BISWA RANJAN DATTA 165

C

C. P. Singh 158

C.D. Wadale 166, 173

CHANDAN KUMAR MONDAL 165

D

D. D. PAWAR 179, 207, 208, 216,

219

D. Deb 156

D. M. SURYAWANSHI 220

D. P. TADAS 168, 184

D. S. Palimkar 218

D.H. Arekar 166

DAMODHAR F SHASTRAKAR 211

DEEPIKA R. GOLECHHA 212

DEEPTI 167

Deepti Raut 167, 216

Dhanashri A. Munot 198

E

E. Siva Prasad 192

F

Farook Rahaman 159

G

G. C. Bhagat 200

G. P. Singh 152

G. R. Avachar 217

G. R. AVCHAR 201, 212

G. S. Katkar 172

G. S. KHADEKAR 171, 177, 179,

182, 184, 185, 186, 191, 207,

210

G.D. KEDAR 188

Gajanan R. Avachar 197

Gaurav Sharma 172

GHANSHYAM MALVIYA 182

H

H. A. Nimkar 175

H.G. Parlikar 166, 173

HARESHGAMBHIR CHAUDHARI

163

HOMNATH DHUNGUNA 204

I

Iram Tahleel Jaleel Ahmed 197

J

J. S. Wath 212

J. Satish 189

Jomar Fajardo Rabajante 154

K

K. L. Bondar 180

K. L. Bondar 161, 195

K. Phaneendra 192

K. Sreenivas 189

K.R. Mule 170

KALPANA N. PAWAR 175, 189

KALPANA PANDE 184

Kanta 173

KASHIKA SRIVASTAVA 171

Kirtiwant P. Ghadle 198

Kishor S. Wankhade 192

L

L. S. Ladke 219

Lakshmi Madireddy 192

M

M. C. Dhabe 203

M. Kanoria 155

M. Khlopov 156

M. R. Ugale 175

M. S. BORKAR 169

M. Srinivasa Reddy 169

Madabusi Raghunathan 147

Madani Gazal Khan 215

MAHESH S WAVARE 174

Manisha Dubey 213

MAYUR KSHIRSAGAR 174

Monika Rathore 213

N

N. P. GAIKWAD 169

N. S. PIMPLE 176, 197

N. T. KATRE 175, 189

N. V. Gharad 167

Narendra Katre 182

Nilima Puranik 176

Nirmala A. Ramtekkar 186

Nita R Dhawade 180

O

O. K. CHAUDHARI 162

Omkar Patil 215

222

Onkar K Chaudhari 180

Onkar K. Chaudhari 214

P

P. A. S. NAIDU 206

P. B. Deshmukh 204

P. M. Lambat 217

P. S. Dudhe 186

P.K. Sahoo 150

Pankaj S Joshi 148

PINAKI RANJAN GHOSH 177

PRAMOD P. KHADE1 202

PRAVEEN KUMAR 171, 177, 185

PRAVIN SAYARE 178

Prof. Avanish Kumar 159

R

R . K. Srivastava 214

R. A. Muneshwar 180

R. D. Mishra 219

R. D. Taywade 194

R. N. INGLE 188

R. S. Rane 200

R. S. WADBUDE 202

R. V. MAPARI 179

R. Venkateswarlu 169

R.D. UTANE 178

R.V. KENE 170

R.V. Saraykar 206

RAJANI SHELOTE 179

Rajshri Gupta 180

Ramesh B. Ghadge 181

Rishikumar Agrawal 182, 187

Rohit M. Patne 197

Rupali Talole 185

RUPALI WANJARI 182, 209

Rupesh T. More 183, 196

S

S .C. Wankhade 218

S. A. KHAPRE 190

S. B. Chavhan 200

S. B. Patil 218

S. C. PATEKAR 191

S. D. KATORE 168, 184, 186,

193, 199

S. D. TADE 205, 210, 212

S. H. Shekh 199

S. J. DHOBLE 201

S. K. GORLE 202

S. K. Sahu 193

S. M. SHINGNE 168, 184

S. N. Patil 166

S. P. BIRAJDAR 176, 197

S. P. Hatkar 186

S. P. Saraogi 193

S. P. SHAHARE 219

S. R. Hadole 218

S. S. BELLALE 176, 220

S. S. GANGANE 201

S. S. Mathurkar 194

S. S. ZAMPALWAD 197

S. T. RATHOD 175, 189

S. V. NAKADE 188

S. V. Raut 199

S. W. Samdurkar 217

S.D. Katore 166, 173

S.D. Tade 162

S.P.Kandalkar 203

S.V. Gore 193

S.V. Ketov 156

SACHIN BALLAL 174, 183

Sadashiv Ramkrushna Puranik

187

Sadhna Singh 214

SAFIQUL ISLAM 171

Saibal Ray 156

SAJID ANWAR 211

Sanjay Chaudhary 173, 190, 208

Sanjay Deshpande 182, 187

Sanjeev Kumar 153, 172, 213

SAPANA P. DUBEY 188

SAPNA K. CHANDBHANANI 206

Saroj R. Kumbhare 207

Satish K. Pacnhal 194

SEEMA RAUT 203

SHARAD S POKLEY 211

SHILPA SAMDURKAR 171, 191

SHITAL YADAV 205

Shiva 190

SHOMA SEN 171, 191

Smita D Kohale 192

Sravanthi Mopati 215

SUDHA RANI DEHRI 201

SUJATA DEO 178

SUJATHA JANARDHAN 203

SUMEDHA TAMBOLI 201

SURYAKANT M JOGDAND 174,

197

Swapna V. Uddhao 206

T

T. D. Nakade 203

T. G. THANGE 210

T. G. THANGE1 201

Trupti A. Thakre 214

Tukaram G. Thange 209

V

V. C. Borkar 198

V. D. SHARMA 190, 204

V. G. Miskin 185

V. J. DAGWAL 177, 185, 207

V. M. Raut 196

V. M. Wankhade 212

V. N. Mahalle 194

V. P. Tripade 219

V. R. Chirde 199

V.D. Sharma 164

V.G. Mete 170

V.M. Ingle 170

Vaijanath L. Chinchane 194

VARSHA DATTATRAY

BORGAONKAR 195

Vigna Purohit 215

Vijay B. Patare 183, 196

Vilas Kharat 160, 174, 183

W

Wagdi F. S. Ahmed 216

Y

Y. S. Solanke 207

Y. V. Rathod 217

Z

Zafar Ahsan 149