Indian Institute of Technology Palakkad&nbsp;(IIT Palakkad or IIT PKD) is a public autonomous&nbsp;engineering&nbsp;and&nbsp;research&nbsp;institute located in&nbsp;Palakkad,&nbsp;Kerala. It was&nbsp;one of the five new&nbsp;IITs&nbsp;proposed in the&nbsp;2014 Union budget of India. The campus was inaugurated on 3 August 2015. The institute has 94 faculty, 978 students, and 63 non-teaching staffs.

IT Palakkad currently offers four-year Bachelor of Technology (B.Tech) and two years MS programmes in 8 different engineering disciplines including&nbsp;Computer Science, Electrical, Mechanical, Civil among others.

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IIT Palakkad

Department of Mathematics

We first give a new direct geometric definition of a Lie bracket of two undirected curves on a surface which was found originally by Wolpert and Goldman in the eighties. and which is referred to here as the GW bracket. The geometric picture reveals readily certain nontrivial known results, unveils new ones and also motivates an unexpected but likely true conjecture. The conjecture reduces a hard to obtain geometric property, disjointness, to a structured algebraic calculation. The new result for the GW-bracket is that its center consists precisely of the obvious central elements, namely the Lie sub algebra generated by the class of the trivial loop and the classes of loops parallel to the boundary components or punctures. We also prove in two short steps that there is no cancellation of geometric terms in the GW bracket of two curves if one curve is simple: firstly by showing that a cancelling pair of terms must have supplementary angles between geodesic representatives, by noting secondly, earthquaking along the simple geodesic changes these angles monotonically, this leading to the contradiction. Concerning the conjecture, we have substantial computer evidence suggesting that if the GW-bracket of two undirected curves is zero then these curves have disjoint representatives. This possibility was unanticipated by the history. The GW-bracket and an initial non-cancellation result go back to more algebraic work of first Wolpert on Poisson Lie algebras and then Goldman on his Lie algebras of directed and of undirected curves, all in the 80's.

World University root node

arXiv e-prints, arXiv: 1910.

The Goldman-Wolpert Lie algebra of undirected curves

We consider the problem (Formula presented), where a &gt; 0, c ≥ 0, α ∈ (0, 1), h:(0, 1] → (0, ∞) is a continuous function which may be singular at t = 0, but belongs to L1(0, 1) ∩ C1(0, 1), and g:[0, ∞) → [0, ∞) is a continuous function. We discuss existence, uniqueness, and non existence results for positive solutions for certain values of a, b and c. © 2018 Texas State University.

Electronic Journal of Differential Equations

Infinite semipositone problems with a falling zero and nonlinear boundary conditions

We study the existence of positive radial solutions to the problem 8 { -Δpu = λK1(jxj)f(v) in Ωe; -Δpv = λK2(jxj)g(u) in Ωe; u = v = 0 if jxj = r0; u(x) → 0; v(x) → 0 as jxj → 1; where Δpw := div(jrwjp-2rw), 1 0 and Ωe := {x 2 Rnj jxj > r0}. Here, Ki : [r0;∞) → (0;∞), i = 1; 2 are continuous functions such that limr→1 Ki(r) = 0, and f; g : [0;1) → R are continuous functions which are negative at the origin and have a superlinear growth at infinity. We establish the existence of a positive radial solution for small values of λ via degree theory and rescaling arguments. © 2018 Khayyam Publishing. All rights reserved.}, funding_details={Simons FoundationSimons Foundation, 317872}, funding_details={European Social FundEuropean Social Fund, ESF}, funding_text_1={Since ‖u‖∞ → ∞ as λ → 0, we have maxt∈[0,1]f(v(t)) → ∞ as λ → 0. This implies that ‖v‖∞ → ∞ as λ → 0. Similarly if ‖v‖∞ → ∞ as λ → 0, we can prove ‖u‖∞ → ∞ as λ → 0. Hence, both ‖u‖∞ and ‖v‖∞ → ∞ as λ → 0. Hence, Theorem 1.1 is proven. Acknowledgment. This work was partially supported by the project EXLIZ-CZ.1.07/2.3.00/30.0013, which is co-financed by the European Social Fund and the state budget of the Czech Republic and by the INSPIRE faculty award by the Department of Science and Technology, India for Lakshmi Sankar. This work was also partially supported by a grant from the Simons Foundation (#317872) for Ratnasingham Shivaji.}, references={Abebe, A., Chhetri, M., Sankar, L., Shivaji, R., Positive solutions for a class of superlinear semipositone systems on exterior domains Boundary Value Problems, 2014 (2014), p. 198; Allegretto, W., Nistri, P., Zecca, P., Positive solutions of elliptic non-positone problems (1992) Differential Integral Equations, 5, pp. 95-101; Ambrosetti, A., Arcoya, D., Buoni, B., Positive solutions for some semi-positone problems via bifurcation theory (1994) Differential Integral Equations, 7, pp. 655-663; Ambrosetti, A., Azorero, J.G., Peral, I., Multiplicity results for some nonlinar elliptic equations (1996) Journal of Functional Analysis, 137, pp. 219-242; Anuradha, V., Hai, D.D., Shivaji, R., Existence results for superlinear semipositone BVP's (1996) Proc. Amer. Math. Soc., 124, pp. 757-763; Arcoya, D., Zertiti, A., Existence and non-existence of radially symmetric non- negative solutions for a class of semi-positone problems in an annulus (1994) Rend. Mat. Appl., 14, pp. 625-646; Caldwell, S., Castro, A., Shivaji, R., Unsurangsie, S., Positive solutions for classes of multiparameter elliptic semipositone problems, Electron (2007) J. of Differential Equations, 18, pp. 1-10; Castro, A., De Figueiredo, D.G., Lopera, E., Existence of positive solutions for a semipositone p-Laplacian problem Proc. Royal Soc. Edin, , to appear; Castro, A., Shivaji, R., Non-negative solutions for a class of non-positone problems (1988) Proc. Royal Soc. Edin., 108, pp. 291-302; Castro, A., Shivaji, R., Non-negative solutions for a class of radially symmetric non-positone problems (1989) Proc. Amer. Math. Soc., 106, pp. 735-740; Chhetri, M., Drabek, P., Shivaji, R., Existence of positive solutions for a class of p- Laplacian superlinear semipositone problems (2015) Proc. Royal Soc. Edin., 145, pp. 925-936; Chhetri, M., Girg, P., Existence and nonexistence of positive solutions for a class of superlinear semipositone systems (2009) Nonlinear Anal., 71, pp. 4984-4996; Chhetri, M., Girg, P., Existence of positive solutions for a class of superlinear semipositone systems (2013) J. Math. Anal. Appl., 408, pp. 781-788; Dai, G., Yang, C., Existence and multiplicity of solutions for semipositone problems involving p-Laplacian (2012) Int. J. Math. Math. Sci., 2012, p. 10; Drabek, P., Milota, J., (2013) Methods of Nonlinear Analysis, , Birkhauser; Hai, D.D., Positive radial solutions for singular quasilinear elliptic equations in a ball (2014) Publ. Res. Inst. Math. Sci., 50, pp. 341-362; Hai, D.D., Schmitt, K., Shivaji, R., Positive solutions of quasilinear boundary value problems (1998) J. Math. Anal. Appl., 217, pp. 672-686; Hai, D.D., Shivaji, R., Maya, C., An existence result for a class of superlinear p- Laplacian semipositone systems (2001) Differential Integral Equations, 14, pp. 231-240; Jacobsen, J., Schmidtt, K., Radial solutions of quasilinear elliptic differential equations (2004) Handbook of Differential Equations, pp. 359-435. , Elsevier/North-Holland, Amsterdam; Ko, E., Lee, E.K., Shivaji, R., Multiplicity results for classes of singular problems on an exterior domain (2013) Discrete and Continuous Dynamical Systems, 33, pp. 5153-5166; Manasevich, R., Mawhin, J., Boundary value problems for nonlinear perturbations of vector p-Laplacian-like operators (2000) J. Korean Math. Soc., 378, pp. 665-685; Smoller, J., Wasserman, A., Existence of positive solutions for semilinear elliptic equations in general domains (1987) Arch. Rational Mech. Anal., 98, pp. 229-249}, publisher={Khayyam Publishing}, issn={08934983}, language={English}, abbrev_source_title={Differ. Integr. Equ.}, document_type={Article}, source={Scopus},

Differential and Integral Equations

An existence result for superlinear semipositone p-Laplacian systems on the exterior of a ball

Let H 1 , H 2 be complex Hilbert spaces and T be a densely defined closed linear operator (not necessarily bounded). It is proved that for each ε &gt; 0, there exists a bounded operator S with ‖S‖ ≤ ε such that T + S is minimum attaining. Further, if T is bounded below, that is if there exists m &gt; 0 such that ‖Tx‖ ≥ m‖x‖ for every x in the domain of T, then S can be chosen to be rank one. © 2018, Element D.O.O. All rights reserved.

Operators and Matrices

On the denseness of minimum attaining operators

Let A be a complex unital Banach algebra, a∈A, n∈Z+ and ϵ&gt;0. The (n,ϵ)-pseudospectrum Λn,ϵ(a) of a is defined as Λn,ϵ(a):=σ(a)∪{λ∉σ(a):‖(λ−a)−2n‖1/2n≥[Formula presented]}. Here σ(a) denotes the spectrum of a. The usual pseudospectrum Λϵ(a) of a is a special case of this, namely Λ0,ϵ(a). It is proved that (n,ϵ)-pseudospectrum approximates the closed ϵ-neighbourhood of spectrum for large n. Further, it has been shown that (n,ϵ)-pseudospectrum has no isolated points, has a finite number of connected components and each component contains an element from σ(a). Some examples are given to illustrate these results. © 2018 Elsevier Inc.

Journal of Mathematical Analysis and Applications

The (n,ϵ)-pseudospectrum of an element of a Banach algebra

A new finite difference scheme based on the higher-order compact technique is presented for solving problems with complex immersed interfaces in arbitrary dimensions. The scheme is designed for general interface problems in which the coefficients, the source term, the solution and its normal flux may be discontinuous across the interface. The originality of the scheme lies in the use of additional values at interfacial points (points at which grid lines intersect the interface) as nodes in the stencil, which allows straightforward use of the standard finite difference approximations. Appropriate interpolation techniques are used on both sides of the interface to determine the interfacial values. Numerical tests are carried out to validate the scheme for solving elliptic equations in both two and three dimensions, which show that the proposed scheme has overall second-order accuracy. The scheme thus developed is also applied to solve incompressible, two-dimensional Stokes flows. In this process, we compare our computed results with the results obtained from existing schemes and in most of the cases, our scheme is found to produce relatively better results.

SIAM Journal on Scientific Computing

Solving immersed interface problems using a new interfacial points-based finite difference approach

Forced convection from a heated rotationally oscillating circular cylinder placed in a uniform cross flow of constant properties fluid is investigated. The two-dimensional governing equations of flow motion and energy are solved numerically on non-uniform polar grids using a higher order compact (HOC) formulation. The flow and thermal fields are mainly influenced by Reynolds number, Re, Prandtl number, Pr, maximum angular velocity of the cylinder, $\alpha$m and the frequency ratio, f/f0, which represents the ratio between the oscillation frequency, f and the natural vortex shedding frequency, f0. The numerical simulations are performed at Re=200,Pr=0.5-1.0,$\alpha$m∈[0.5,4.0] and f/f0∈[0.5,3.0]. The resulting lock-on phenomena behind the cylinder is detected and thermal field is determined. Comparisons with previous numerical and experimental results verify the accuracy and the reliability of the present study. Variations in heat transfer coefficients within the lock-on ranges are investigated to build a connection between the heat transfer and the lock-on regimes.

International Journal of Heat and Mass Transfer

A numerical study of forced convection from an isothermal cylinder performing rotational oscillations in a uniform stream

In [12], it was shown that a generalized maximum likelihood estimation problem on a (canonical) \alpha -power-law model (\mathbb{M}^{(\alpha)} -family) can be solved by solving a system of linear equations. This was due to an orthogonality relationship between the \mathbb{M}^{(\alpha)} -family and a linear family with respect to the relative \alpha -entropy (or the \mathscr{I}-{\alpha} -divergence). Relative \alpha -entropy is a generalization of the usual relative entropy (or the Kullback-Leibler divergence). \mathbb{M}^{(\alpha)} -family is a generalization of the usual exponential family. In this paper, we first generalize the \mathbb{M}^{(\alpha)}- family including the multivariate, continuous case and show that the Student-t distributions fall in this family. We then extend the above stated result of [12] to the general \mathbb{M}^{(\alpha)} -family. Finally we apply this result to the Student-t distribution and find generalized estimators for its parameters.

IEEE International Symposium on Information Theory - Proceedings

Generalized Estimating Equation for the Student-t Distributions

The Gelfand–Mazur Theorem, a very basic theorem in the theory of Banach algebras states that: (Real version) Every real normed division algebra is isomorphic to the algebra of all real numbers R, the complex numbers C or the quaternions H; (Complex version) Every complex normed division algebra is isometrically isomorphic to C. This theorem has undergone a large number of generalizations. We present a survey of these generalizations and also discuss some closely related unsettled issues. © 2017 Elsevier GmbH

Expositiones Mathematicae

Gelfand–Mazur Theorems in normed algebras: A survey

We study the geometric properties of the terms of the Goldman bracket between two free homotopy classes of oriented closed curves in a hyperbolic surface. We provide an obstruction for the equality of two terms in the Goldman bracket, namely if two terms in the Goldman bracket are equal to each other then for every hyperbolic metric, the angles corresponding to the intersection points are equal to each other. As a consequence, we obtain an alternative proof of a theorem of Chas, i.e., if one of the free homotopy classes contains a simple representative then the geometric intersection number and the number of terms (counted with multiplicity) in the Goldman bracket are the same. © 2018, University at Albany. All rights reserved.