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

Let H1, H2 be complex Hilbert spaces and T be a densely defined closed linear operator from its domain D(T), a dense subspace of H1, into H2. Let N(T) denote the null space of T and R(T) denote the range of T. Recall that C(T):= D(T) ∩ N(T)⊥ is called the carrier space of T and the reduced minimum modulus γ(T) of T is defined as: γ (T) : = inf{ ‖ T(x) ‖ : x∈ C(T) , ‖ x‖ = 1 }. Further, we say that T attains its reduced minimum modulus if there exists x0 ∈ C(T) such that ∥x0∥ = 1 and ∥T(x0)∥ = γ(T). We discuss some properties of operators that attain reduced minimum modulus. In particular, the following results are proved.1.The operator T attains its reduced minimum modulus if and only if its Moore-Penrose inverse T† is bounded and attains its norm, that is, there exists y0 ∈ H2 such that ∥y0∥ = 1 and ∥T†∥ = ∥T†(y0)∥.2.For each ϵ &gt; 0, there exists a bounded operator S such that ∥S∥ ≤ ϵ and T + S attains its reduced minimum. © 2020, Indian National Science Academy.

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Indian Journal of Pure and Applied Mathematics

Operators that Attain Reduced Minimum

For a domain D ⊂ Cn, n ≥ 2, let FkD(z) = KD(z)λ(IkD(z)), where KD(z) is the Bergman kernel of D along the diagonal and λ(IkD(z)) is the Lebesgue measure of the Kobayashi indicatrix at the point z. This biholomorphic invariant was introduced by Błocki. We study its limiting boundary behaviour on two classes of domains: h-extendible and strongly pseudoconvex polyhedral domains. © Instytut Matematyczny PAN, 2020.

Annales Polonici Mathematici

Further remarks on the higher dimensional suita conjecture

In this article, we consider the condition pseudospectrum of bounded linear operators on a Banach space. The condition pseudospectrum of normal matrices and Jordan blocks are characterized and condition pseudospectral radius of the classes are found. Sub-additivity and sub-multiplicativity of the condition pseudospectral radius for commuting pairs of bounded linear operators are proved. It is shown that the condition pseudospectral radius becomes a complete algebra norm in a commutative complex unital Banach algebra. Certain examples are given to illustrate the results. The results developed are also extended to a general setting. © 2020, © 2020 Informa UK Limited, trading as Taylor & Francis Group.

Linear and Multilinear Algebra

Condition pseudospectral radius of bounded linear operators

In this article, we characterize finite co-dimensional strongly ball proximinal subspaces of L1-predual spaces and prove that they are precisely the finite co-dimensional strongly proximinal subspaces. © 2019 Heldermann Verlag. All rights reserved.

Journal of Convex Analysis

Characterization of strong ball proximinality in L1-predual spaces

To study the analog of Suita'fs conjecture for domains ⊂Cn, n ≥ 2, Blocki introduced the invariant Fκ D (z) = KD(z)λIκ D(z), where KD (z) is the Bergman kernel of D along the diagonal and λIκ D(z) is the Lebesgue measure of the Kobayashi indicatrix at the point z. In this note, we study the behaviour of Fκ D (z) (and other similar invariants using different metrics) on strongly pseudconvex domains and also compute its limiting behaviour explicitly at certain points of decoupled egg domains in C2. © 2019 American Mathematical Society.

Proceedings of the American Mathematical Society

Remarks on the higher dimensional suita conjecture

It is known that if f: D1→ D2 is a polynomial biholomorphism with polynomial inverse and constant Jacobian then D1 is a 1-point poly-quadrature domain (the Bergman span contains all holomorphic polynomials) of order 1 whenever D2 is a complete circular domain. Bell conjectured that all 1-point poly-quadrature domains arise in this manner. In this note, we construct a 1-point poly-quadrature domain of order 1 that is not biholomorphic to any complete circular domain. © 2018, Springer Nature Switzerland AG.

Analysis and Mathematical Physics

A 1-point poly-quadrature domain of order 1 not biholomorphic to a complete circular domain

This paper focuses on test procedures under corrupted data.We assume that the observations Zi are mismeasured, due to the presence of measurement errors. Thus, instead of Zi for i = 1, . . ., n, we observe Xi = Zi + √ $\delta$Vi, with an unknown parameter $\delta$ and an unobservable random variable Vi. It is assumed that the random variables Zi are i.i.d., as are the Xi and the Vi. The test procedure aims at deciding between two simple hyptheses pertaining to the density of the variable Zi, namely f0 and g0. In this setting, the density of the Vi is supposed to be known. The procedure which we propose aggregates likelihood ratios for a collection of values of d. A new definition of least-favorable hypotheses for the aggregate family of tests is presented and a relation with the Kullback-Leibler divergence between the sets (f$\delta$)$\delta$ and (g$\delta$)$\delta$ is presented. Finite-sample lower bounds for the power of these tests are presented, both through analytical inequalities and through simulation under the least-favorable hypotheses. Since no optimality holds for the aggregation of likelihood ratio tests, a similar procedure is proposed, replacing the individual likelihood ratio by some divergence based test statistics. It is shown and discussed that the resulting aggregated test may perform better than the aggregate likelihood ratio procedure.

Entropy

Composite tests under corrupted data

Let G be a parahoric group scheme over a complex projective curve X of genus greater than one. Let G denote the moduli stack of G-torsors on X. We prove several results concerning the Hitchin map on TG. We first show that the parahoric analogue of the global nilpotent cone is isotropic and use this to prove that G is "very good" in the sense of Beilinson-Drinfeld. We then prove that the parahoric Hitchin map is a Poisson map whose generic fibres are abelian varieties. Together, these results imply that the parahoric Hitchin map is a completely integrable system. © 2018 The Author(s).

International Mathematics Research Notices

Complete Integrability of the Parahoric Hitchin System

Let ϵ&gt; 0 , n a non-negative integer, and A a complex unital Banach algebra. Define γn: A× C→ [0 , ∞] by γn(a,z)={‖(z-a)-2n‖-1/2n,if(z-a)isinvertible0,if(z-a)is not invertible.The (n, ϵ) -pseudospectrum Λ n , ϵ(a) of an element a∈ A is defined by Λ n , ϵ(a) : = { λ∈ C: γn(a, λ) ≤ ϵ}. We show that γ is Lipschitz on A× C, γn is uniformly continuous on bounded subsets of A× C for n≥ 1 , and γn is Lipschitz on some particular domains for n≥ 1. Using these properties, we establish that the map (ϵ, a) ↦ Λ n , ϵ(a) is continuous at (ϵ, a) if and only if the level set { λ∈ C: γn(a, λ) = ϵ} does not contain any non-empty open set. In particular, this happens when a is a compact operator on a Banach space or a bounded operator on a Hilbert space or on an Lp space with 1 ≤ p≤ ∞. We also give examples of operators where this condition is not satisfied, and consequently, the map is not continuous. © 2019, Springer Nature Switzerland AG.}, author_keywords={(n, ϵ) -Pseudospectrum; Banach algebra; Pseudospectrum; Spectrum}, funding_details={2/39(2)/2015/NBHM/R&D-II/7440}, funding_text_1={The authors would like to thank the anonymous referee for his/her valuable comments that led to a considerable improvement in this paper. The first author would like to thank the Department of Atomic Energy (DAE), India (Ref No: 2/39(2)/2015/NBHM/R&D-II/7440) for financial support.}, references={Bögli, S., Siegl, P., Remarks on the convergence of pseudospectra (2014) Integral Equ. Oper. Theory, 80, pp. 303-321; Böttcher, A., Pseudospectra and singular values of large convolution operators (1994) J. Integral Equ. Appl., 6, pp. 267-301; Böttcher, A., Grudsky, S.M., Silbermann, B., Norms of inverses, spectra, and pseudospectra of large truncated Wiener–Hopf operators and Toeplitz matrices (1997) N. Y. J. Math., 3, pp. 1-31; Böttcher, A., Grudsky, S.M., Can spectral value sets of Toeplitz band matrices jump? (2002) Linear Algebra Appl., 351 (352), pp. 99-116; Davies, E.B., Pseudospectra of differential operators (2000) J. Oper. Theory, 43, pp. 243-262; Davies, E.B., A defence of mathematical pluralism (2005) Philos. Math., 13 (3), pp. 252-276; Dhara, K., Kulkarni, S.H., The (n, ϵ) -pseudospectrum of an element of a Banach algebra (2018) J. Math. Anal. Appl., 464, pp. 939-954; Dunford, N., Schwartz, J.T., (1958) Linear Operators. I. General Theory, , Interscience Publishers Ltd., London; Gallestey, E., Hinrichsen, D., Pritchard, A.J., Spectral value sets of closed linear operators (2000) Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 456, pp. 1397-1418; Globevnik, J., On complex strict and uniform convexity (1975) Proc. Am. Math. Soc., 47, pp. 175-178; Globevnik, J., Norm-constant analytic functions and equivalent norms (1976) Ill. J. Math., 20, pp. 503-506; Hansen, A.C., On the approximation of spectra of linear operators on Hilbert spaces (2008) J. Funct. Anal., 254, pp. 2092-2126; Hansen, A.C., Infinite-dimensional numerical linear algebra: theory and applications (2010) Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466, pp. 3539-3559; Hansen, A.C., On the solvability complexity index, the n -pseudospectrum and approximations of spectra of operators (2011) J. Am. Math. Soc., 24, pp. 81-124; Hansen, A.C., Nevanlinna, O., Complexity issues in computing spectra, pseudospectra and resolvents (2017) Banach Cent. Publ., 112, pp. 171-194; Harrabi, A., Pseudospectre d’une suite d’opérateurs bornés (1998) RAIRO Modél. Math. Anal. Numér., 32, pp. 671-680; Krishnan, A., Kulkarni, S.H., Pseudospectrum of an element of a Banach algebra (2017) Oper. Matrices, 11, pp. 263-287; Reddy, S.C., Pseudospectra of Wiener–Hopf integral operators and constant-coefficient differential operators (1993) J. Integral Equ. Appl., 5, pp. 369-403; Seidel, M., On (N, ϵ) -pseudospectra of operators on Banach spaces (2012) J. Funct. Anal., 262, pp. 4916-4927; Shargorodsky, E., On the level sets of the resolvent norm of a linear operator (2008) Bull. Lond. Math. Soc., 40, pp. 493-504; Trefethen, L.N., Pseudospectra of linear operators (1997) SIAM Rev., 39, pp. 383-406; Trefethen, L.N., Embree, M., Pseudospectra gateway, , http://www.comlab.ox.ac.uk/pseudospectra, Web site; Trefethen, L.N., Embree, M., (2005) Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, , Princeton University Press, Princeton}, correspondence_address1={Dhara, K.; Department of Mathematics, India; email: kousik.dhara1@gmail.com}, publisher={Springer Basel}, issn={0378620X}, language={English}, abbrev_source_title={Integr. Equ. Oper. Theory}, document_type={Article}, source={Scopus},

Integral Equations and Operator Theory

Continuity of the (n,ϵ) -Pseudospectrum in Banach Algebras

In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of mR-primary ideals in a Noetherian local ring R, generalizing the classical theory for mR-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of R is less than the dimension of R, which holds, for instance, if R is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of mR-primary ideals hold for filtrations, including the famous Minkowski inequalities of Teissier, and of Rees and Sharp. © 2019 American Mathematical Society