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

The purpose of this article is to study Lipschitz CR mappings from an h-extendible (or semi-regular) hypersurface in ℂn. Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from strongly pseudoconvex domains is also proved. © 2012 Springer-Verlag.

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Mathematische Zeitschrift

Some regularity theorems for CR mappings

We consider the problem-Δpu = aup-1-bu γ -1-c/uα, x ∈Ω, u = 0, x ∈ ∂Ω, whereΔpu = div(|∇u|p-2∇u) , p &gt; 1, Ω is a smooth bounded domain in ℝn, a &gt; 0, b &gt; 0, c ≥ 0, γ &gt; p and α ∈ (0, 1). Given a, b, γ and α, we establish the existence of apositive solution for small values of c. These results are also extended to corresponding exterior domain problems. Also, a bifurcation result for the case c = 0 ispresented. © 2013 Goddard II et al.

Boundary Value Problems

Existence results for classes of infinite semipositone problems

We study boundary value problems of the form {-δu=λK({pipe}x{pipe})f(u),x∈Ωu=0if{pipe}x{pipe}=r0u→0as{pipe}x{pipe}→∞, where λ is a positive parameter, δu=div(∇;u) is the Laplacian of u, Ω={x∈Rn;n&gt;2,{pipe}x{pipe}&gt;r0}, K belongs to a class of C1 functions such that limr→∞K(r)=0, and f belongs to a class of C1 functions which are negative at the origin and have falling zeros. We discuss the existence and uniqueness of nonnegative radial solutions when λ is large. © 2012 Elsevier Ltd.}, author_keywords={Existence; Exterior domains; Falling zeros; Semipositone problems; Uniqueness}, references={Ali, I., Castro, A., Shivaji, R., Uniqueness and stability of positive solutions for semipositone problems in a ball (1993) Proc. Amer. Math. Soc., 117, pp. 775-782; Ambrosetti, A., Arcoya, D., Biffoni, B., Positive solutions for some semipositone problems via bifurcation theory (1994) Differential Integral Equations, 7 (3), pp. 655-663; Anuradha, V., Hai, D.D., Shivaji, R., Existence results for superlinear semipositone boundary value problems (1996) Proc. Amer. Math. Soc., 124 (3), pp. 757-763; Castro, A., Gadam, S., Shivaji, R., Positive solution curves of semipositone problems with concave nonlinearities (1997) Proc. Roy. Soc. Edinburgh Sect. A, 127 (5), pp. 921-934; Castro, A., Garner, J.B., Shivaji, R., Existence results for classes of sublinear semipositone problems (1993) Results Math., 23, pp. 214-220; Castro, A., Hassanpour, M., Shivaji, R., Uniqueness of non-negative solutions for a semipositone problem with concave nonlinearity (1995) Comm. Partial Differential Equations, 20 (11-12), pp. 1927-1936; Castro, A., Sankar, L., Shivaji, R., Uniqueness of nonnegative solutions for semipositone problems on exterior domains (2012) J. Math. Anal. Appl., 394 (1), pp. 432-437; Castro, A., Shivaji, R., Nonnegative solutions for a class of nonpositone problems (1998) Proc. Roy. Soc. Edinburgh Sect. A, 108, pp. 291-302; Castro, A., Shivaji, R., Nonnegative solutions for a class of radially symmetric nonpositone problems (1989) Proc. Amer. Math. Soc., 106 (3), pp. 735-740; Castro, A., Shivaji, R., Positive solutions for a concave semipositone Dirichlet problem (1998) Nonlinear Anal., 31 (1-2), pp. 91-98; Dancer, E.N., Shi, J., Uniqueness and nonexistence of positive solutions to semipositone problems (2006) Bull. London Math. Soc., 38 (6), pp. 1033-1044; Habets, P., Zanolin, F., Upper an lower solutions for a generalized Emden-Fowler equation (1994) J. Math. Anal. Appl., 181, pp. 684-700; Lee, E., Sankar, L., Shivaji, R., Positive solutions for infinite semipositone problems on exterior domains (2011) Differential Integral Equations, 24, pp. 861-875; Philippe Clement, I., Sweers, G., Existence and multiplicity results for a semilinear elliptic eigenvalue problem (1987) Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14 (1), pp. 97-121; Serrin, J., (1971), Nonlinear equations of second order, in: A.M.S. Sympos. Partial Differential Equations, Berkeley, August}, correspondence_address1={Shivaji, R.; Department of Mathematics and Statistics, , NC 27412, United States; email: shivaji@uncg.edu}, issn={0022247X}, language={English}, abbrev_source_title={J. Math. Anal. Appl.}, document_type={Article}, source={Scopus},

Journal of Mathematical Analysis and Applications

Semipositone problems with falling zeros on exterior domains

We consider the problem where λ is a positive parameter, δu=div(∇;u) is the Laplacian of u, Ω={x∈Rn;n&gt;2,{pipe}x{pipe}&gt;r0}, K∈C 1([r 0, ∞), (0, ∞)) is such that lim r→∞K(r)=0 and f∈C1([0,∞),R) is a concave function which is sublinear at ∞ and f(0)&lt;0. We establish the uniqueness of nonnegative radial solutions when λ is large. © 2012 Elsevier Ltd.

Uniqueness of nonnegative solutions for semipositone problems on exterior domains

We discuss properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its -perturbation. Several results are proved about linear maps preserving pseudospectrum/ condition spectrum. These include the following: (1) Let A, B be complex unital Banach algebras and  > 0. Let : A  B be an -pseudospectrum preserving linear onto map. Then  preserves spectrum. If A and B are uniform algebras, then,  is an isometric isomorphism. (2) Let A, B be uniform algebras and 0 <  < 1. Let : A  B be an -condition spectrum preserving linear map. Then  is an -almost multiplicative map, where ,  tend to zero simultaneously.

Banach Journal of Mathematical Analysis

Linear maps preserving pseudospectrum and condition spectrum

International Journal of Physical and Mathematical Sciences

A Transform-Free HOC Scheme for Incompressible Viscous Flow past a Rotationally Oscillating Circular Cylinder

We consider the existence of multiple positive solutions to the steady state reaction diffusion equation with Dirichlet boundary conditions of the form: {-u″=λ[u-u2/K-cu2/1+ u2-∈], x ∈ (0,1) u(0)=0=u(1). Here 1/λ is the diffusion coefficient and K, c and are positive constants. This model describes the steady states of a logistic growth model with grazing and constant yield harvesting in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation and constant yield harvesting. In this paper, we discuss the occurrence of a Σ-shaped bifurcation diagram for positive solutions. In particular, for certain parameter values of c,K, ∈ and the diffusion coefficient, we prove the existence of at least four positive solutions. We prove our results by the quadrature method. © 2011 Elsevier Ltd. All rights reserved.}, author_keywords={Σ-shaped bifurcation curves; Ecological systems; Grazing and constant yield harvesting}, keywords={Bifurcation curves; Bifurcation diagram; Dirichlet boundary condition; Ecological models; Ecological systems; Fish populations; Grazing and constant yield harvesting; Logistic growth model; Multiple positive solutions; Parameter values; Positive constant; Positive solution; Quadrature methods; Steady state; Steady state reactions, Boundary conditions; Diffusion; Ecology; Equations of state; Fisheries; Harvesting; Partial differential equations; Potassium, Bifurcation (mathematics)}, references={Causey, R., Sasi, S., Shivaji, R., An ecological model with grazing and constant yield harvesting (2010) Bull. Belg. Math. Soc., 17 (5), pp. 833-839; Lee, E., Sasi, S., Shivaji, R., S-shaped bifurcation curves in ecosystems (2011) J. Math. Anal. Appl., , 10.1016/j.jmaa.2011.03.048; May, R.M., Thresholds and breakpoints in ecosystems with a multiplicity of stable states (1977) Nature, 269, pp. 471-477; Noy-Meir, I., Stability of grazing systems an application of predatorprey graphs (1975) J. Ecol., 63, pp. 459-482; Oruganti, S., Shi, J., Shivaji, R., Diffusive equations with constant yield harvesting, I: Steady states (2002) Trans. Amer. Math. Soc., 354, pp. 3601-3619; Steele, J.H., Henderson, E.W., Modeling long-term fluctuations in fish stocks (1984) Science, 224 (4652), pp. 985-987; Brauer, F., Castillo-Chavez, C., (2001) Mathematical Models in Population Biology and Epidemiology, 40. , Texts in Applied Mathematics Springer-Verlag New York; Clark, C.W., (1990) Mathematical Bioeconomics, the Optimal Management of Renewable Resources, , John Wiley & Sons, Inc. New York; Brown, K.J., Ibrahim, M.M.A., Shivaji, R., S-shaped bifurcation curves (1981) Nonlinear Anal. TMA, 5 (5), pp. 475-486; Laetsch, T., The number of solutions of a nonlinear two point boundary value problem (1970) Indiana Univ. Math. J., 20 (1), pp. 1-13; Jiang, J., Shi, J., Bistability dynamics in some structured ecological models (2009) Spatial Ecology, pp. 33-62. , R.S. Cantrell, C. Cosner, S. Ruan, Chapman & Hall, CRC Mathematical and Computational Biology}, correspondence_address1={Shivaji, R.; Department of Mathematics and Statistics, , Greensboro, NC 27412, United States; email: shivaji@uncg.edu}, issn={14681218}, language={English}, abbrev_source_title={Nonlinear Anal. Real World Appl.}, document_type={Article}, source={Scopus},

Nonlinear Analysis: Real World Applications

An ecological model with a Σ-shaped bifurcation curve

We study the existence of positive solutions to the singular problem (eqution presented) where λ is a positive parameter, Δpu = div(Ι▽uΙp-2▽u), p &gt; 1, Ω is a bounded domain in ℝn; n ≥ 1 with smooth boundary ∂ Ω, 0 &lt; α &lt; 1, and f : [0;∞) → ℝ is a continuous function which is asymptotically p-linear at 1. We prove the existence of positive solutions for a certain range of λ using the method of sub-supersolutions. We also extend our study to classes of systems which have forcing terms satisfying a combined asymptotically p-linear condition at ∞ and to corresponding problems on exterior domains.

Differential and Integral Equations

Infinite semipositone problems with asymptotically linear growth forcing terms

We define a new metric on the set of all closed linear operators between Hilbert spaces and investigate its properties. In particular, we show that the set of all closed operators with a closed range is an open subset of the set of all closed operators and the map T ↦ T† is an isometry in this metric. We also investigate the relationships between the topology induced by this metric and the gap metric.

The carrier graph topology

We study positive radial solutions to the problem {equation presented} where λ is a positive parameter, Δu = div (δu) is the Laplacian of u, ω = {x ∈ Rn, n &gt; 2 : |x| &gt; r 0} is an exterior domain and f : (0.∞) → R belongs to a class of sublinear functions at ∞ such that they are continuous and f(0+) = lims→0+ f(s) &lt; 0. In particular we also study infinite semipositone problems where lims→0+ f(s) = -∞. Here K : [r0,∞) → (0,∞) belongs to a class of continuous functions such that limr→∞ K(r) = 0. We establish various existence results for such boundary value problems and also extend our results to classes of systems. We prove our results by the method of sub-/supersolutions.