Vellore Institute of Technology (VIT) is a private university located in&nbsp;Tamil Nadu, India. Founded in 1984, as Vellore Engineering College, the institution offers 20 undergraduate, 34 postgraduate, four integrated and four research programs. It has campuses in Vellore, Amravati, Bhopal and Chennai.

VIT is one of the top ranked private universities in India according to NIRF, THE and QS Rankings.&nbsp;Govt. of India has recognized&nbsp;VIT, Vellore as an&nbsp;Institution of Eminence. This has allowed VIT to take independent quality initiatives and move up in world ranking.

&nbsp;

&nbsp;

VIT University

In this paper, we have computed the dimensions of the vector space of relative symmetric polynomials with respect to all irreducible characters of the projective special linear group P SL2(F7). © 2021, International Journal on Emerging Technolgies. All rights reserved.

World University root node

Mathematics

School of Advanced Sciences

Chennai Campus

Physics

Advances in Mathematics: Scientific Journal

Formula for the dimensions of the relative invariants of projective special linear group p sl2(F7)

In general relativity, problems are usually approached with coordinate basis method, but there is an other efficient method for solving these problems, which is tetrad formalism. Newman Penrose formalism is an old and renowned approach in general relativity which served immense benefits in analytical and computational aspects. The basic ingredients on the NP equations are the null tetrad (lµ, nµ, m-µ, mµ ) here we construct a null tetrad for spherically symmetric static solutions that admit stable closed orbits called Bertrand Spacetime (BST) which has two type I & II, specially we concentrated on type-II. Along with this we studied Ricci spin coefficients, Ricci tensor and Weyl tensor. © Published under licence by IOP Publishing Ltd.

Fulltext

Journal of Physics: Conference Series

A physical interpretation of the Newman Penrose formalism and its application to Bertrand Spacetime II

In this paper, we have computed the dimensions of the relative symmetric polynomials associated with all the irreducible characters of Pre Mathieu groups M9, M10 and Small Mathieu groups M11, M12. © 2020, Research Publication. All rights reserved.

Dimesional formulÆ for the relative invariants of pre-mathieu and mathieu groups

Newman Penrose (NP) formalism is a new approach to general relativity was presented. In various geometrical groups are estimate on a chosen null tetrad basis for tetrad formalism. This approach widely used in analytical and numerical studies of Einstein's equations. The reasonable utilization of explicit complex straight emulsions of Ricci rotation coefficients which present, in the product, the spinor relative association. And we construct a null tetrad for general case and special attension for Bertrand's space-time type-1 metric for special case k=0. © Published under licence by IOP Publishing Ltd.

A newman penrose approach to Bertrand spacetime in general relativity

Bertrand’s theorem is one of the landmark results in the context of the central force problem in mechanics. It leads to the conditions for closed bound orbits. This pedagogical article intends to motivate the teacher as well as the student to engage in a formal proof of the theorem. We overview the basics of central forces and the Abel integral. Thereafter, we obtain the condition for closed orbits. We provide the reader with a broad panorama of areas/topics where Bertrand’s theorem arises, including interesting scenarios in the regime of Einstein’s general theory of relativity. © 2019, Indian Academy of Sciences.

VIT Unitversity

Resonance

A Pedagogical Relook at Bertrand’s Theorem

The Scope of this paper works is to study, steady incompressible non-similar two dimensional boundary layer over a stretching surface with temperature dependent viscosity and PRANDTL number. The coupled partial differential equations with boundary conditions have been solved numerically using an implicit finite difference with combination of quasi-linearization technique with suitable step size along the stream- wise direction. The numerical results are displayed for velocity and temperature profiles with different parameter values. Further, it was found that the buoyancy parameter is more significant for water boundary flow. © 2018, Institute of Advanced Scientific Research, Inc. All rights reserved.

Journal of Advanced Research in Dynamical and Control Systems

The effects of buoyancy force of fluid flow over a water boundary layer with variable viscosity and PRANDTL number

We analyse second order (in Riemann curvature) geometric flows (un-normalised) on locally homogeneous three manifolds and look for specific features through the solutions (analytic wherever possible, otherwise numerical) of the evolution equations. Several novelties appear in the context of scale factor evolution, fixed curves, phase portraits, approach to singular metrics, isotropization, and curvature scalar evolution. The distinguishing features linked to the presence of the second order term in the flow equation are pointed out. Throughout the article, we compare the results obtained, with the corresponding results for un-normalized Ricci flows. © 2013 American Institute of Physics.

Journal of Mathematical Physics

Higher order geometric flows on three-dimensional locally homogeneous spaces

We investigate various aspects of a geometric flow defined using the Bach tensor. First, using a well-known split of the Bach tensor components for (2, 2) unwarped product manifolds, we solve the Bach flow equations for typical examples of product manifolds like S 2 × S 2, R 2 × S 2. In addition, we obtain the fixed-point condition for general (2, 2) manifolds and solve it for a restricted case. Next, we consider warped manifolds. For Bach flows on a special class of asymmetrically warped 4-manifolds, we reduce the flow equations to a first-order dynamical system, which is solved exactly to find the flow characteristics. We compare our results for Bach flow with those for Ricci flow and discuss the differences qualitatively. Finally, we conclude by mentioning possible directions for future work. © 2012 World Scientific Publishing Company.

International Journal of Geometric Methods in Modern Physics

Bach flows of product manifolds

Renormalization group (RG) flows of the bosonic nonlinear σ-model are governed, perturbatively, at different orders of α', by perturbatively evaluated β-functions. In regions where α'Rc2<1 (1Rc2 represents the curvature scale), the flow equations at various orders in α' can be thought of as approximating the full, non-perturbative RG flow. On the other hand, taking a different viewpoint, we may consider the above-mentioned RG flow equations as viable geometric flows in their own right, without any reference to the RG aspect. Looked at as purely geometric flows where higher order terms appear, we no longer have the perturbative restrictions (small curvatures). In this paper, we perform our analysis from both these perspectives using specific target manifolds such as S2, H2, unwarped S2×H2 and simple warped products.We analyse and solve the higher order RG flow equations within the appropriate perturbative domains and find the corrections arising due to the inclusion of higher order terms. Such corrections, within the perturbative regime, are shown to be small and they provide an estimate of the error that arises when higher orders are ignored.We also investigate higher order geometric flows on the same manifolds and figure out generic features of geometric evolution, the appearance of singularities and solitons. The aim, in this context, is to demonstrate the role of higher order terms in modifying the flow. One interesting aspect of our analysis is that, separable solutions of the higher order flow equations for simple warped spacetimes (of the kind used in bulk-brane models with a single extra dimension), correspond to constant curvature anti de Sitter (AdS) spacetimes, modulo an overall flow parameter dependent scale factor. The functional form of this scale factor (that we obtain) changes on the inclusion of successive higher order terms in the flow. © 2011 Elsevier B.V.

Journal of Geometry and Physics

On higher order geometric and renormalization group flows

We analyze Ricci flow (normalized/unnormalized) of product manifolds - unwarped as well as warped, through a study of generic examples. First, we investigate such flows for the unwarped scenario with manifolds of the type double-struck S signn × double-struck S signm, double-struck S signn × ℍm, ℍm × ℍn and also, similar multiple products. We are able to single out generic features such as singularity formation, isotropization at particular values of the flow parameter and evolution characteristics. Subsequently, motivated by warped braneworlds and extra dimensions, we look at Ricci flows of warped spacetimes. Here, we are able to find analytic solutions for a special case by variable separation. For others, we numerically solve the equations and draw certain useful inferences about the evolution of the warp factor, the scalar curvature as well as the occurrence of singularities at finite values of the flow parameter. We also investigate the dependence of the singularities of the flow on the initial conditions. We expect our results to be useful in any physical/mathematical context where such product manifolds may arise. © 2010 World Scientific Publishing Company.