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 following weighted eigenvalue problem in the exterior domain: (Formula presented.),where Δp is the p-Laplace operator with p&gt;1, and (Formula presented.) is the exterior of the closed unit ball in RN with N≥1. There is no restriction on the dimension N in terms of $$p, i.e., we allow both 1&lt;p&lt;N and p≥N. The weight function K is locally integrable on (Formula presented.) and is allowed to change its sign. For some appropriate choice of $$w$$w, a positive weight function on the interval (1,∞), we prove that the Beppo-Levi space (Formula presented.) is compactly embedded into the weighted Lebesgue space Lp(Formula presented.). The existence of the positive eigenvalue for the above problem is proved for K such that (Formula presented.) is of non-zero measure and |K|≤w. Further, we discuss the positivity, the regularity and the asymptotic behaviour at infinity of the first eigenfunctions. © 2014, Springer-Verlag Berlin Heidelberg.

World University root node

Calculus of Variations and Partial Differential Equations

Weighted quasilinear eigenvalue problems in exterior domains

The following results are proved. The Null Space TheoremLet X,Y be vector spaces, P∈L(X),Q∈L(Y) be projections and T∈L(X,Y) be invertible. (The restriction of QTP to R(P) can be viewed as a linear operator from R(P) to R(Q). This is called a section of T by P and Q and will be denoted by TP,Q.) Then there is a linear bijection between the null space of the section TP,Q of T and the null space of its complementary section TIY-Q,IX-P-1 of T-1. Let X be a Banach space with a Schauder basis A={a1,a2,⋯}. Let T be a bounded (continuous) linear operator on X. Suppose the matrix of T with respect to A is tridiagonal. If T is invertible, then every submatrix of the matrix of T-1 with respect to A that lies on or above the main diagonal (or on or below the main diagonal) is of rank ≤1. © 2015 Elsevier Inc. All rights reserved.

Linear Algebra and Its Applications

The null space theorem

We study minimization of a parametric family of relative entropies, termed relative $\alpha$-entropies (denoted I$\alpha$(P,Q)). These arise as redundancies under mismatched compression when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative $\alpha$-entropies behave like squared Euclidean distance and satisfy the Pythagorean property. Minimization of I$\alpha$(P,Q) over the first argument on a set of probability distributions that constitutes a linear family is studied. Such a minimization generalizes the maximum R{\'{e}}nyi or Tsallis entropy principle. The minimizing probability distribution (termed I$\alpha$-projection) for a linear family is shown to have a power-law.

2015 21st National Conference on Communications, NCC 2015

Relative $\alpha$-entropy minimizers subject to linear statistical constraints

Minimization problems with respect to a one-parameter family of generalized relative entropies are studied. These relative entropies, which we term relative $\alpha$-entropies (denoted I$\alpha$), arise as redundancies under mismatched compression when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative $\alpha$-entropies behave like squared Euclidean distance and satisfy the Pythagorean property. Minimizers of these relative $\alpha$-entropies on closed and convex sets are shown to exist. Such minimizations generalize the maximum R{\'{e}}nyi or Tsallis entropy principle. The minimizing probability distribution (termed forward I$\alpha$-projection) for a linear family is shown to obey a power-law. Other results in connection with statistical inference, namely subspace transitivity and iterated projections, are also established. In a companion paper, a related minimization problem of interest in robust statistics that leads to a reverse I$\alpha$-projection is studied.

Fulltext

IEEE Transactions on Information Theory

Minimization problems based on relative $\alpha$-entropy I: Forward projection

Hilbert functions and Hilbert polynomials of Zs-graded admissible filtrations of ideals {F(n)}n∈Zs such that λ(R/F(n)) is finite for all n∈Zs are studied. Conditions are provided for the Hilbert function HF(n):=λ(R/F(n)) and the corresponding Hilbert polynomial PF(n) to be equal for all n∈Ns. A formula for the difference HF(n)-PF(n) in terms of local cohomology of the extended Rees algebra of F is proved which is used to obtain sufficient linear relations analogous to the ones given by Huneke and Ooishi among coefficients of PF(n) so that HF(n)=PF(n) for all n∈Ns. A theorem of Rees about joint reductions of the filtration {IrJs}r,s∈Z is generalised for admissible filtrations of ideals in two-dimensional Cohen-Macaulay local rings. Necessary and sufficient conditions are provided for the multi-Rees algebra of an admissible Z2-graded filtration F to be Cohen-Macaulay. © 2015 Elsevier Inc.

Journal of Algebra

Hilbert polynomials of multigraded filtrations of ideals

Rigidity And Regularity Of Holomorphic Mappings

In this paper, we develop a new HOC based immersed interface scheme in cylindrical polar coordinates for solving viscous, incompressible, laminar flow past a circular cylinder. This new scheme is developed by clubbing the recently developed higher order compact (HOC) methodology with with special interface treatment for the points just next to the points of discontinuity. Non uniform grid in polar coordinates (r, $\theta$) is used for the discretization, where the surface of the cylinder is considered as an interface immersed in the fluid. Streamfunction-Vorticity formulation of Navier-Stokes equation in cylindrical polar coordinates is considered with an assumption that solution inside the immersed boundary is zero. Quantitative and qualitative results are shown for various values of Reynolds number Re ≤ 100 and an excellent agreement with existing numerical and experimental results is found in each case.

Procedia Engineering

An HOC Based Immersed Interface Scheme for Uniform Flow Past a Circular Cylinder

We investigate a variation of the transitivity problem for proximinality properties of subspaces and intersection properties of balls in Banach spaces. For instance, we prove that if Z⊆Y⊆X, where Z is a finite co-dimensional subspace of X which is strongly proximinal in Y and Y is an M-ideal in X, then Z is strongly proximinal in X. Towards this, we prove that a finite co-dimensional proximinal subspace Y of X is strongly proximinal in X if and only if Y⊥⊥ is strongly proximinal in X**. We also prove that in an abstract L1-space, the notions of strongly subdifferentiable points and quasi-polyhedral points coincide. We also give an example to show that M-ideals need not be ball proximinal. Moreover, we prove that in an L1-predual space, M-ideals are ball proximinal. © 2015 Elsevier Inc.

Journal of Mathematical Analysis and Applications

Strong proximinality and intersection properties of balls in Banach spaces

We prove the existence of a solution between an ordered pair of sub and supersolutions for singular quasilinear elliptic problems on unbounded domains. Further, we use this result to establish the existence of a positive solution to the problem {-Δpu=λK(x)f(u)in B1 c,u=0 on ∂B1, u(x)→0 as |x|→∞, where B1c={x ∈ ℝn||x|&gt;1}, Δpu=div(|∇u|p-2∇u),1&lt;p&lt;n, λ is a positive parameter, K belongs to a class of functions which satisfy certain decay assumptions and f belongs to a class of (p-1)-subhomogeneous functions which may be singular at the origin, namely lim s→0+f(s)=-∞. Our methods can be also applied to establish a similar existence result when the domain is entire Rn. © 2014 Elsevier Ltd. All rights reserved.}, author_keywords={Infinite semipositone; p-Laplacian; Singular problems; Sub and supersolutions; Unbounded domains}, keywords={C (programming language), P-Laplacian; Semi-positone; Singular problem; Super-solutions; Unbounded domain, Computational mechanics}, funding_details={Grantová Agentura České RepublikyGrantová Agentura České Republiky, GA ČR, 13-00863S, EXLIZ - CZ.1.07/2.3.00/30.0013}, funding_details={European Social FundEuropean Social Fund, ESF}, funding_text_1={The first author was supported by the Grant Agency of Czech Republic, Project No. 13-00863S. The second author was 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={Lee, E.K., Shivaji, R., Ye, J., Classes of infinite semipositone systems (2009) Proc. Roy. Soc. Edinburgh Sect. A, 139 (4), pp. 853-865; Hai, D.D., Sankar, L., Shivaji, R., Infinite semipositone problems with asymptotically linear growth forcing terms (2012) Differential Integral Equations, 25 (1112), pp. 1175-1188; Lee, E.K., Sankar, L., Shivaji, R., Positive solutions for infinite semipositone problems on exterior domains (2011) Differential Integral Equations, 24 (910), pp. 861-875; Ambrosetti, A., Arcoya, D., Buffoni, B., Positive solutions for some semi-positone problems via bifurcation theory (1994) Differential Integral Equations, 7 (34), pp. 655-663; Anuradha, V., Hai, D.D., Shivaji, R., Existence results for superlinear semipositone BVP'S (1996) Proceedings of the American Mathematical Society, 124 (3), 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. (7), 14 (4), pp. 625-646; 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., Shivaji, R., Positive solutions for a concave semipositone dirichlet problem (1998) Nonlinear Analysis, Theory, Methods and Applications, 31 (1-2), pp. 91-98. , PII S0362546X96001897; Clément, P., Sweers, G., Existence and multiplicity results for a semilinear elliptic eigenvalue problem (1987) Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 14 (1), pp. 97-121; Oruganti, S., Shi, J., Shivaji, R., Diffusive logistic equation with constant yield harvesting. I. Steady states (2002) Trans. Amer. Math. Soc., 354 (9), pp. 3601-3619. , electronic; Chhetri, M., Robinson, S.B., Existence and multiplicity of positive solutions for classes of singular elliptic PDEs (2009) J. Math. Anal. Appl., 357 (1), pp. 176-182; Ghergu, M., Radulescu, V., Sublinear singular elliptic problems with two parameters (2003) Journal of Differential Equations, 195 (2), pp. 520-536. , DOI 10.1016/S0022-0396(03)00105-0; Hai, D.D., On an asymptotically linear singular boundary value problems (2012) Topol. Methods Nonlinear Anal., 39 (1), pp. 83-92; Lee, E.K., Shivaji, R., Ye, J., Positive solutions for elliptic equations involving nonlinearities with falling zeroes (2009) Appl. Math. Lett., 22 (6), pp. 846-851; Ramaswamy, M., Shivaji, R., Ye, J., Positive solutions for a class of infinite semipositone problems (2007) Differential Integral Equations, 20 (12), pp. 1423-1433; Shi, J., Yao, M., On a singular nonlinear semilinear elliptic problem (1998) Royal Society of Edinburgh - Proceedings A, 128 (6), pp. 1389-1401; Zhang, Z., On a Dirichlet problem with a singular nonlinearity (1969) J. Math. Anal. Appl., 194, pp. 103-113; Leon, M.C., Existence results for quasilinear problems via ordered sub- and supersolutions (1997) Ann. Fac. Sci. Toulouse Math. (6), 6 (4), pp. 591-608; Tolksdorf, P., On the Dirichlet problem for quasilinear equations in domains with conical boundary points (1983) Comm. Partial Differential Equations, 8 (7), pp. 773-817; Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations (1984) J. Differential Equations, 51 (1), pp. 126-150; Chhetri, M., Principal Eigenvalue of P-Laplacian Operator in Exterior Domain, , http://dx.doi.org/10.1007/s00025-014-0386-2, Results. Math. Online; Ávila, A.I., Brock, F., Asymptotics at infinity of solutions for p-Laplace equations in exterior domains (2008) Nonlinear Anal., 69 (56), pp. 1615-1628; Serrin, J., Local behavior of solutions of quasi-linear equations (1964) Acta Math., 111, pp. 247-302; Drábek, P., Kufner, A., Nicolosi, F., (1997) Quasilinear Elliptic Equations with Degenerations and Singularities, 5. , de Gruyter Series in Nonlinear Analysis and Applications Walter de Gruyter & Co. Berlin; Daners, D., Drábek, P., A priori estimates for a class of quasi-linear elliptic equations (2009) Trans. Amer. Math. Soc., 361 (12), pp. 6475-6500; Lieberman, G.M., Boundary regularity for solutions of degenerate elliptic equations (1988) Nonlinear Anal., 12 (11), pp. 1203-1219; Vázquez, J.L., A strong maximum principle for some quasilinear elliptic equations (1984) Appl. Math. Optim., 12 (3), pp. 191-202}, correspondence_address1={Drábek, P.; Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 301 00 Plzeň, Czech Republic; email: pdrabek@kma.zcu.cz}, publisher={Elsevier Ltd}, issn={0362546X}, coden={NOAND}, language={English}, abbrev_source_title={Nonlinear Anal Theory Methods Appl}, document_type={Article}, source={Scopus},

Nonlinear Analysis, Theory, Methods and Applications

Singular quasilinear elliptic problems on unbounded domains

We study the existence of a positive radial solution to the nonlinear eigenvalue problem (Formula presented.) in (Formula presented.) in (Formula presented.) as (Formula presented.), where (Formula presented.) is a parameter, (Formula presented.) is the Laplace operator, (Formula presented.) are such that (Formula presented.). Here (Formula presented.) are (Formula presented.) functions such that they are negative at the origin (semipositone) and superlinear at infinity. We establish the existence of a positive solution for λ small via degree theory and rescaling arguments. We also discuss a non-existence result for λ≫1 for the single equations case. © 2014, Abebe et al.; licensee Springer.