# Category: Volume 32

V. R. KULLI
Department of Mathematics, Gulbarga University, Gulbarga – 585 106 (India)
B. CHALUVARAJU
Department of Mathematics, Bangalore University, Jnanabharathi Campus Bangalore -560 056 (India)
M. KUMARA
Department of Mathematics, Bangalore University, Jnanabharathi Campus Bangalore -560 056 (India)

ABSTRACT:

The disjoint total restrained domination number of a graph G is the minimum cardinality of the union of two disjoint total restrained dominating sets in G. We also consider an invariant the minimum cardinality of the disjoint union of a restrained dominating set and a total restrained dominating set. In this paper, we initiate a study of these parameters and establish some results on these new parameters.

KEYWORDS: Inverse domination set, restrained dominating set, total restrained dominating set

SANJIT KU. MOHANTY
Head of Department of Mathematics B.S. Degree College, Nuahat, Jajpur-754296,Odisha (India)
RAJANI BALLAV DASH
Visiting Faculty, Department of Mathematics, Ravenshaw University, Cuttack, Odisha (India)

ABSTRACT:

A mixed quadrature rule of precision nine for approximate evaluation of real definite integrals has been constructed by blending Clenshaw-Curtis five point rule modified by Richardson Extrapolation and GaussLegendre four point rule. An error analysis for this mixed rule is provided. The efficiency of this rule is highlighted through numerical evaluation of some definite integrals at the end.

KEYWORDS: Clenshaw-Curtis quadrature rule, Gauss-Legendre 4-point rule, Richardson Extrapolation, Subject classification: 65D32

MOLOYA BHUYAN
Department of Mathematics, DCB Girlsu2019 College, Jorhat-ASSAM (India)
CHANDRA CHUTIA
Department of Mathematics, Jorhat Institute of Science & Technology, Jorhat-ASSAM

ABSTRACT:

A finite group G acts as a group of automorphisms on a compact Riemann surface S of genus g if and only if there exist a Fuchsian group $\Gamma$ and an epimorphism $\phi$:$\Gamma$$\rightarrow$G such that ker$\phi$ = K is a surface group of genus g. And then $\phi$ is named as smooth homomorphism. The objective of this paper is to establish a set of necessary and sufficient conditions for the existence of smooth epimorphism from a Fuchsian group $\Gamma$ to the finite group of symmetries of Carbon Tetra chloride molecule, whose abstract group representation is $<$a,b|a4= b3=(ab)2 $>$.

KEYWORDS: smooth epimorphism, point group, compact Riemann surface, AMS Subject Classification 2020: 20B25, 20B30, 20H10, 30F10

SUNIL JAT
Department of Physics, Madhyanchal Professional University, Bhopal (India)
Y.K. JAYASWAL
Department of Physics, Madhyanchal Professional University, Bhopal (India)
GHIZAL F.ANSAR
Department of Physics, Madhyanchal Professional University, Bhopal (India)

ABSTRACT:

Tellurite glasses are promising upconversion optical and laser materials. The upconversion efficiency is strongly influenced by maximum phonon energy of the host, tellurite glasses have lower phonon energy than many common oxide glasses. The characteristic glass transition temperature Tg and crystallization onset Tx, differential scaning calorimetry (DSC) is studied. Glassy structure is verified by XRD. Study of visible absorption spectra and upconversion spectra of rare earth ion doped Bismuth tellurite glasses designate it as futuristic photonic material.

KEYWORDS: Bismuth tellurite glasses, RE Rare earth ions, UC Upconversion

ASHWANI KUMAR GARG
Department of Education in Science and Mathematics, Regional Institute of Education, NCERT, Bhopal (M. P.), (India)
DHIRENDRA KUMAR SHUKLA
Department of Education in Science and Mathematics, Regional Institute of Education, NCERT, Bhopal (M. P.), (India)
BRAJENDRA TIWARI3
Professor, Department of Mathematics, RKDF University, Bhopal (M. P.), (India)

ABSTRACT:

In the article, R be a ring with local units that have discussed. For any ME mod-R, the map \mu(M1+M2):

(M1+M2)\rightarrow(M1+M2) given by Σn i-1/j=1 (Mi + Mj) =1 ⨂ri Σn i-1/j=1 \rightarrow(Mi + Mj) ⨂r be an isomorphism of right R-modules. Strongly U-Flat Modules over Matlis Domains has defined and discussed with their properties to know the relations.

KEYWORDS: U-flat modules, Matlis domains, Prufer domain

In thbe article, R be a ring with local units that have discussed. For any ME mod-R, the map (M1+M2): (M1+M2)(M1+M2) given by Σn i-1/j=1 (Mi + Mj) =1 ⨂ri Σn i-1/j=1  (Mi +...

KRISHNA GOGOI
Associate professor, D.C.B. Girlsu2019 College, Jorhat, Assam (India)
CHANDRA CHUTIA
Associate professor, Jorhat Institute of Science and Technology, Jorhat, Assam (India)

ABSTRACT:

Graph theory, a branch of Mathematics plays a vital rule in studying interdisciplinary subjects such as physics, Chemistry, Engineering etc. Study of the properties of electrical circuits with the help of graph theory is a growing trend in mathematical and electrical fields. Electrical circuits consists of nodes and branches which obeys Kirchhoff ’s current laws, Kirchhoff ’s voltage laws etc. There are various well known theorems such as Norton’s theorem, Thevenin’s theorem, Superposition theorem, Millman Theorem etc for network analysis. In this paper, we try to analyze Millman’s theorem with the help of Graph theorem.

KEYWORDS: Graph theory, Electrical circuits, Branch current, Loop current

V. R. KULLI
Department of Mathematics, Gulbarga University, Gulbarga – 585 106 (India)
B. CHALUVARAJU
Department of Mathematics, Bangalore University, Jnana Bharathi Campus, Bangalore -560 056 (India)

ABSTRACT:

The Zagreb indices were introduced by Gutman and Trinajstic in 1972. The K-Banhatti indices were introduced by Kulli in 2016. These two types of indices are closely related. In this study, we define the Zagreb- K-Banhatti index of a graph. We establish some relations between Zagreb, K-Banhatti and Zagreb-K-Banhatti indices. We also obtain lower and upper bounds for the Zagreb -K-Banhatti index of a graph in terms of Zagreb and K- Banhatti indices.

KEYWORDS: Zagreb index, K- Banhatti indices, Zagreb-K-Banhatti index, 2010 (AMS) Mathematics Subject Classification: 05C05, 05C07, 05C35

S S SAMANTARAY
Department of Mathematics, Divine Degree College, Nayagarh, Odisha (India)
A MISRA
Department of Mathematics, Centurion University of Technology and Management, Odisha (India)

ABSTRACT:

This article focuses on a comprehensive review in summarizing on the recent research developments regarding the theoretical and experimental investigations about the thermophysical characteristics of carbon nanotubes-nanofluids with mass concentration varying from 0.1 to 1wt% and at the temperatures of 10-60°C. Carbon nanotubes are promising new materials for their mechanical, electrical, thermal, optical and surface properties. The current study explores how several factors those strongly affecting thermal conductivity, viscosity, specific heat and density of carbon nanotube-nanofluids include particle concentration, temperature, particle size, particle type, particle shape, different base fluids, surfactant and ultrasonic time. In addition, different apposite models contributing augmentation of thermal conductivity and decaying of viscosity of carbon nanotubes-nanofluids are introduced. Further, significant heat transfer mechanisms namely Brownian motion, nanoclustering, thermophoresis, and interfacial nano-layer responsible for significant role in augmenting the heat transfer capabilities of carbon nanotubes-nanofluids are well discussed. The viscosity of SWCNT nanofluids shows a non-Newtonian shear-thinning behavior due to the alignment of nanotubes clusters and agglomerates with increasing shear rate. The results reveal that the thermal conductivity, viscosity and density of CNT-nanofluids are higher than that of the base fluid, and enhances with rise in nanotubes concentration. Specific heat peters out with rise in particle loadings and upgrades with increase in temperature. Thermal conductivity of CNT-nanofluids upsurges, whereas the viscosity and density of CNT-nanofluids diminish with rise in the temperature. Finally, the challenges for the future are conveyed.

KEYWORDS: Carbon Nanotubes Nanofluids, Nanoclustering, Interfacial nano-layer

RAFIQUL ISLAM
Department of Mathematics, Jorhat Engineering College, Jorhat-785007 (India)
CHANDRA CHUTIA
Department of Mathematics, Jorhat Institute of Science and Technology, Jorhat (India)

ABSTRACT:

In this paper, we consider the group of symmetries of the Sulphur molecule (S) which is a finite point group of order 16 denote by D16 generated by two elements having the presentation { u$\upsilon$/u2$\upsilon$8 = (u$\upsilon$)2 = 1} and find the complete set of genera (g ≥ 2) of Compact Riemann surfaces on which  D16 acts as a group of automorphisms as follows:

D16 the group of symmetries of the sulfur (S8) molecule of order 16 acts as an automorphism group of a compact Riemann surfaces of genus g ≥ 2 if and only if there are integers $\lambda$ and $\mu$ such that $\lambda \leq$ 1 and $\mu \geq$ 1 and
g=$\lambda$ +8$\mu$ ($\geq$2) , $\mu$$\geq$ |$\lambda$|

KEYWORDS: Symmetries, Fuchsian group, Smooth quotient, Riemann surface,

JANARDAN SINGH
State Institute of Education U.P., Prayagraj (India)

ABSTRACT:

In this paper, we study some results of transversal hypersurfaces with (f, g˜, u, v, $\lambda$)-the structure of a nearly $\beta$-Kenmotsu manifold. Some more results on totally geodesic or totally umbilical transversal hypersurface with (f, g˜, u, v, $\lambda$)-structure of a nearly $\beta$-Kenmotsu manifold have also been studied.

KEYWORDS: Totally geodesic, Kenmotsu manifold, Mathematics Subject Classification 2020. 37D40, 14J70, 53C25