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 consider the antimaximum principle for the p-Laplacian in the exterior domain: {-Δpu=λK(x)|u|p-2u+h(x)in B1c,u=0on ∂B1, where Δp is the p-Laplace operator with p&gt;1,λ is the spectral parameter and B1c is the exterior of the closed unit ball in RN with N ≥ 1. The function h is assumed to be nonnegative and nonzero, however the weight function K is allowed to change its sign. For K in a certain weighted Lebesgue space, we prove that the antimaximum principle holds locally. A global antimaximum principle is obtained for h with compact support. For a compactly supported K, with N=1 and p=2, we provide a necessary and sufficient condition on h for the global antimaximum principle. In the course of proving our results we also establish the boundary regularity of solutions of certain boundary value problems. © 2015 Elsevier Ltd.}, author_keywords={Exterior domains; Local and global antimaximum; p-Laplacian; Positive eigenfunctions; principle; Regularity results}, keywords={Boundary value problems; Eigenvalues and eigenfunctions; Laplace transforms, Exterior domain; Local and global antimaximum; P-Laplacian; principle; Regularity results, Nonlinear equations}, funding_details={Grantová Agentura České RepublikyGrantová Agentura České Republiky, GA ČR, 13-00863S}, funding_details={European Social FundEuropean Social Fund, ESF}, funding_text_1={The second author was supported by the Grant Agency of Czech Republic, Project No. 13-00863S.}, funding_text_2={The first, third and fourth authors work is funded 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.}, references={Allegretto, W., Huang, Y.X., A Picone's identity for the p-Laplacian and applications (1998) Nonlinear Anal., 32 (7), pp. 819-830; Anoop, T.V., Drábek, P., Sasi, S., Weighted quasilinear eigenvalue problems in exterior domains (2015) Calc. Var. Partial Differential Equations, 53 (3-4), pp. 961-975; Birindelli, I., Hopf's lemma and anti-maximum principle in general domains (1995) J. Differential Equations, 119 (2), pp. 450-472; Clément, P., Peletier, L.A., An anti-maximum principle for second-order elliptic operators (1979) J. Differential Equations, 34 (2), pp. 218-229; Drábek, P., Huang, Y.X., Bifurcation problems for the p-Laplacian in RN (1997) Trans. Amer. Math. Soc., 349 (1), pp. 171-188; Fleckinger-Pellé, J., Gossez, J.-P., De Thélin, F., Antimaximum principle in RN: Local versus global (2004) J. Differential Equations, 196 (1), pp. 119-133; Fučík, S., Nečas, J., Souček, J., Souček, V., (1973) Spectral Analysis of Nonlinear Operators, 346. , Lecture Notes in Mathematics Springer-Verlag Berlin, New York; Godoy, T., Gossez, J.-P., Paczka, S., On the antimaximum principle for the p-Laplacian with indefinite weight (2002) Nonlinear Anal., 51 (3), pp. 449-467; Hess, P., An anti-maximum principle for linear elliptic equations with an indefinite weight function (1981) J. Differential Equations, 41 (3), pp. 369-374; Hörmander, L., Lions, J.L., Sur la complétion par rapport à une intégrale de Dirichlet (1956) Math. Scand., 4, pp. 259-270; Lieberman, G.M., Boundary regularity for solutions of degenerate elliptic equations (1988) Nonlinear Anal., 12 (11), pp. 1203-1219; Serrin, J., Local behavior of solutions of quasi-linear equations (1964) Acta Math., 111, pp. 247-302; Shi, J., A new proof of anti-maximum principle via a bifurcation approach (2005) Results Math., 48 (1-2), pp. 162-167; Stavrakakis, N.M., De Thélin, F., Principal eigenvalues and anti-maximum principle for some quasilinear elliptic equations on RN (2000) Math. Nachr., 212, pp. 155-171; Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations (1984) J. Differential Equations, 51 (1), pp. 126-150; Vázquez, J.L., A strong maximum principle for some quasilinear elliptic equations (1984) Appl. Math. Optim., 12 (3), pp. 191-202; Ziemer, W.P., Sobolev spaces and functions of bounded variation (1989) Weakly Differentiable Functions, 120. , Graduate Texts in Mathematics Springer-Verlag New York}, correspondence_address1={Drábek, P.; Department of Mathematics and NTIS, Univerzitní 22, Czech Republic}, publisher={Elsevier Ltd}, issn={0362546X}, coden={NOAND}, language={English}, abbrev_source_title={Nonlinear Anal Theory Methods Appl}, document_type={Article}, source={Scopus},

World University root node

Nonlinear Analysis, Theory, Methods and Applications

Antimaximum principle in exterior domains

A pair of free homotopy classes of closed curves in an orientable surface $F$ with negative Euler characteristic are said to be length equivalent if for any hyperbolic structure on $F$, the length of the geodesic representative of one class is equal to the length of the geodesic representative of the other class. Given an orientable hyperbolic surface $F$ with negative Euler characteristic, any self intersecting geodesic $\alpha$ on $F$ and any self-intersection point $P$ of $\alpha$, we construct infinitely many pairs of length equivalent curves in $F$. We also relate these examples with the terms of Goldman bracket.

arXiv

Construction of length equivalent curves from self-intersecting geodesic in a hyperbolic surface

In part I of this two-part work, certain minimization problems based on a parametric family of relative entropies (denoted ℐ α ) were studied. Such minimizers were called forward ℐ α -projections. Here, a complementary class of minimization problems leading to the so-called reverse ℐ α -projections are studied. Reverse ℐ α -projections, particularly on log-convex or power-law families, are of interest in robust estimation problems (α &gt; 1) and in constrained compression settings (α &lt;; 1). Orthogonality of the power-law family with an associated linear family is first established and is then exploited to turn a reverse ℐ α -projection into a forward ℐ α -projection. The transformed problem is a simpler quasi-convex minimization subject to linear constraints.

Fulltext

IEEE Transactions on Information Theory

Minimization Problems Based on Relative α -Entropy II: Reverse Projection

Let π: X → C be a relatively minimal non-isotrivial elliptic surface over the field of complex numbers, where g(C) ≥ 2. In this article, we demonstrate an equivalence between the category of semistable parabolic Higgs bundles on C, and the category of semistable Higgs bundles on X with vanishing second Chern class, and determinant a vertical divisor. © 2015. Published by Oxford University Press. All rights reserved.

Quarterly Journal of Mathematics

On higgs bundles on elliptic surfaces

The Nullity theorem says that certain pairs of submatrices of a square invertible matrix and its inverse (known as complementary submatrices) have the same nullity. Though this theorem has been around for quite some time and also has found several applications, some how it is not that widely known. We give a brief account of the Nullity Theorem, consider its generalization to infinite dimensional spaces, called the Null Space Theorem and discuss some applications. © Springer International Publishing Switzerland 2015.

Springer Proceedings in Mathematics and Statistics

The nullity theorem, its generalization and applications

We extend a well-known result, about the unit ball, by H. Alexander to a class of balanced domains in (Formula presented.),n>1. Specifically: we prove that any proper holomorphic self-map of a certain type of balanced, finite-type domain in (Formula presented.),n>1, is an automorphism. The main novelty of our proof is the use of a recent result of Opshtein on the behaviour of the iterates of holomorphic self-maps of a certain class of domains. We use Opshtein’s theorem, together with the tools made available by finiteness of type, to deduce that the aforementioned map is unbranched. The monodromy theorem then delivers the result. © 2015, Springer-Verlag Berlin Heidelberg.

Mathematische Zeitschrift

Proper holomorphic mappings of balanced domains in Cn

Annales de la Facult{\'{e}} des sciences de Toulouse : Math{\'{e}}matiques

Bounds for invariant distances on pseudoconvex Levi corank one domains and applications

In this paper we use Banach algebra techniques to study the spectrum, pseudospectrum and condition spectrum of a block Laurent operator with continuous symbol and a lower triangular block Toeplitz operator with continuous symbol. (1) Let L be a block Laurent operator with a continuous symbol f. Regarding f as an element of the Banach algebra of all continuous matrix valued functions defined on the unit circle Γ, we show that the spectrum σ(L) of L coincides with the spectrum σ(f) of f. It is also shown that the spectrum σ(f) can be expressed as a union of the spectra of matrices f(x). Thus Similar results are proved about pseudospectrum for ε &gt; 0 and condition spectrum σε(L) = σε(f) for 0 &lt; ε &lt; 1. (2) Let T be an upper or lower triangular block Toeplitz operator with continuous symbol f. Then f is a continuous matrix valued function defined on the closed unit disc and f is analytic in the open unit disc. It is proved that a similar description can be given about the spectrum σ(T), pseudospectrum Λε(T) for ε &gt; 0 and condition spectrum σε(T) for &lt; ε &lt; 1. These results are illustrated with examples and pictures using Matlab.