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 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. © 2010 Mathematics Subject Classification.

World University root node

Banach Journal of Mathematical Analysis

The carrier graph topology

We consider the existence of multiple positive solutions to the steady state reaction diffusion equation with Dirichlet boundary conditions of the form:. Here δu=div(∇u) is the Laplacian of u, 1λ is the diffusion coefficient, K and c are positive constants and Ω⊂RN is a smooth bounded region with ∂. Ω in C2. This model describes the steady states of a logistic growth model with grazing in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation. In this paper we discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve. We prove our results by the method of sub-supersolutions. © 2011 Elsevier Inc.

Journal of Mathematical Analysis and Applications

S-shaped bifurcation curves in ecosystems

Aim of this short note is to obtain a formula for the gap between two densely defined unbounded closed operators. It is interesting to note that the formula is very similar to the corresponding formula for the gap between two bounded operators. © 2010 Elsevier Inc. All rights reserved.

Linear Algebra and Its Applications

A formula for gap between two closed operators

Let A be a complex commutative Banach algebra with unit 1 and δ &gt; 0. A linear map φ: A →C is said to be δ-almost multiplicative if |φ(ab) - φ (a) φ (b)| ≤ δ||a|| ||b|| for all a, b ε A. Let 0 &lt; e &lt; 1. The e-condition spectrum of an element a in A is defined by δe.(a) := {λ ε C:||λ-a||||λ- a||-1 ||≥ 1/e} with the convention that ||λ- a|| ||(λ - a)-1|| = ∞ when λ - a is not invertible. We prove the following results connecting these two notions: (1) If φ(1) = 1 and φ is δ-almost multiplicative, then φ(a) ε δe(a) for all a in A.then (2) If φis lenear and φ(a) εδe(a) for all a in A ,then φ-is δ almost multiplicative for some δ. The first result is analogous to the Gelfand theory and the last result is analogous to the classical Gleason-Kahane- Żelazko theorem.

Studia Mathematica

Almost multiplicative functions on commutative Banach algebras

We study positive solutions to the steady state reaction diffusion equation with Dirichlet boundary condition of the form: {-Δu=au-bu2-c up/1+up - K, x ∈ ω u=0, x ∈ ∂ω Here Δu = div(▽u) is the Laplacian of u, a,b,c,p,K are positive constants with p ≥ 2 and ω is a smooth bounded region with ∂ω in C2. This model describes the steady states of a logistic growth model with grazing and constant yield harvesting. It also describes the dynamics of the fish population with natural predation and constant yield harvesting. We study the existence of positive solutions to this model. We prove our results by the method of sub-super solutions.}, references={Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces (1976) SIAM Rev., 18, pp. 620-709; Berestycki, H., Caffarelli, L.A., Nirenberg, L., Inequalities for second-order elliptic equations with applications to unbounded domains (1996) Duke Mathematical Journal, 81 (2), pp. 467-494. , DOI 10.1215/S0012-7094-96-08117-X; Clement, Ph., Peletier, L.A., An anti-maximum principle for second order elliptic operators (1979) J. Diff. Eqns., 34 (2), pp. 218-229; Lions, P.L., On the existence of positive solutions of semilinear elliptic equations (1982) SIAM Rev., 24, pp. 441-467; 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 predator-prey graphs (1975) Journal of Ecology, 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., Modelling long term fluctuations in fish stocks (1984) Science, 224, pp. 985-987}, correspondence_address1={Causey, R.; Department of Mathematics and Statistics, , Mississippi State 39759, United States; email: rmc107@msstate.edu}, publisher={Belgian Mathematical Society}, issn={13701444}, language={English}, abbrev_source_title={Bull. Belg. Math. Soc. Simon Stevin}, document_type={Article}, source={Scopus},

Bulletin of the Belgian Mathematical Society - Simon Stevin

An ecological model with grazing and constant yield harvesting

In this article we give a characterization of the convergence of projection methods which are useful for approximating the Moore-Penrose inverse of a closed densely defined operator between Hilbert spaces. We illustrate the main theorem with an example. Also a procedure for constructing the admissible sequence of projections is discussed. © 2010 The Indian National Science Academy.

Indian Journal of Pure and Applied Mathematics

Projection methods for computing Moore-Penrose inverses of unbounded operators

Master's thesis, Indian Institute of Science

Some Results on Holomorphic Mappings of Domains in C n

In [1], Böttcher et. al. showed that if T is a bounded linear operator on a separable Hilbert space H,{ej}j=1∞ is an orthonormal basis of H and Pn is the orthogonal projection onto the span of {ej}j=1n, then for each k ∈ Ndbl;, the sequence {sk(pnTPn)} converges to sk(T), where for a bounded operator A on H, sk(A) denotes the kth approximation number of A, that is, sk(A) is the distance from A to the set of all bounded linear operators of rank at most k - 1. In this paper we extend the above result to more general cases. In particular, we prove that if T is a bounded linear operator from a separable normed linear space X to a reflexive Banach space Y and if {Pn} and {Qn} are sequences of bounded linear operators on X and Y, respectively, such that for all {double pipe}Pn{double pipe} {double pipe}Qn{double pipe} ≤ 1 for all n ∈ ℕ and {QnTPn} converges to T under the weak operator topology, then {sk(QnTPn)} converges to sk(T). We also obtain a similar result for the case of any normed linear space Y which is the dual of some separable normed linear space. For compact operators, we give this convergence of sk(QnTPn) to sk(T) with separability assumptions on X and the dual of Y. Counter examples are given to show that the results do not hold if additional assumptions on the space Y are removed. Under separability assumptions on X and Y, we also show that if there exist sequences of bounded linear operators {Pn} and {Qn} on X and Y respectively such that (i) QnTPn is compact, (ii) {double pipe}Pn{double pipe} {double pipe}Qn{double pipe} ≤ 1 and (iii) {QnTPn} converges to T in the weak operator topology, then {sk(QnTPn)} converges to sk(T) if and only if sk(T) = sk(T′). This leads to a generalization of a result of Hutton [3], proved for compact operators between normed linear spaces. © 2009 Birkhäuser Verlag Basel/Switzerland.

Integral Equations and Operator Theory

Approximation numbers of operators on normed linear spaces

Fulltext

Electronic Journal of Qualitative Theory of Differential Equations

Population dynamics with symmetric and asymmetric harvesting

Let H1,H2 be Hilbert spaces and T be a closed linear operator defined on a dense subspace D(T) in H1 and taking values in H2. In this article we prove the following results: (i) Range of T is closed if and only if 0 is not an accumulation point of the spectrum σ(T*T) of T*T, In addition, if H1 = H2 and T is self-adjoint, then (ii) inf {{double pipe}Tx{double pipe}:x∈D(T)∩N(T)⊥{double pipe}x{double pipe} = 1} = inf{|λ|: 0 ≠ λ ∈ σ(T)}, (iii) Every isolated spectral value of T is an eigenvalue of T (iv) Range of T is closed if and only if 0 is not an accumulation point of the spectrum σ(T) of T (v) σ(T) bounded implies T is bounded. We prove all the above results without using the spectral theorem. Also, we give examples to illustrate all the above results.