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.

&nbsp;

IIT Palakkad

Department of Mathematics

arXiv preprint arXiv:2011.

Poisson algebras of loops on surfaces and skein algebras associated to their quantization

We prove a version of the Schwarz lemma for holomorphic mappings from the unit disk into the symmetric product of a Riemann surface. Our proof is function-theoretic and self-contained. The main novelty in our proof is the use of the pluricomplex Green’s function. We also prove several other Schwarz lemmas using this function. © 2019, Mathematica Josephina, Inc.

Fulltext

Journal of Geometric Analysis

Schwarz Lemmas via the Pluricomplex Green’s Function

Given a linear recurrence sequence (LRS), specified using the initial conditions and the recurrence relation, the Skolem problem asks if zero ever occurs in the infinite sequence generated by the LRS. Despite active research over last few decades, its decidability is known only for a few restricted subclasses, by either restricting the order of the LRS (upto 4) or by restricting the structure of the LRS (e.g., roots of its characteristic polynomial). In this paper, we identify a subclass of LRS of arbitrary order for which the Skolem problem is easy, namely LRS all of whose characteristic roots are (possibly complex) roots of real algebraic numbers, i.e., roots satisfying xd = r for r real algebraic. We show that for this subclass, the Skolem problem can be solved in NPRP. As a byproduct, we implicitly obtain effective bounds on the zero set of the LRS for this subclass. While prior works in this area often exploit deep results from algebraic and transcendental number theory to get such effective results, our techniques are primarily algorithmic and use linear algebra and Galois theory. We also complement our upper bounds with a NP lower bound for the Skolem problem via a new direct reduction from 3-CNF-SAT, matching the best known lower bounds. © S. Akshay, Nikhil Balaji, Aniket Murhekar, Rohith Varma, and Nikhil Vyas; licensed under Creative Commons License CC-BY

World University root node

Leibniz International Proceedings in Informatics, LIPIcs

Near-optimal complexity bounds for fragments of the skolem problem

Journal of Topology and Analysis

On the entropy norm on the group of diffeomorphisms of closed oriented surface

We consider the problem {−Δu=λK(x)f(u)in B1 c,u=0on ∂B1,u(x)→0as |x|→∞, where B1 c={x∈Rn||x|&gt;1},n&gt;2, λ is a positive parameter, K belongs to a class of functions which satisfy certain decay assumptions and f belongs to a class of functions which are asymptotically linear and may be singular at the origin. We prove the existence of positive solutions to such problems for certain values of parameter λ. Existence results to similar problems in Rn are also obtained. Our existence results are proved using the Schauder fixed point theorem and the method of sub and super solutions. © 2020}, author_keywords={Asymptotically linear reaction term; Infinite semipositone; Method of sub and super solutions; Schauder fixed point theorem; Semilinear elliptic problems; Unbounded domains}, funding_details={Department of Science and Technology, Ministry of Science and Technology, IndiaDepartment of Science and Technology, Ministry of Science and Technology, India, DST, DST/INSPIRE/04/2014/002833}, funding_text_1={This research was supported by the INSPIRE faculty award, awarded by the Department of Science and Technology , Government of India ( DST/INSPIRE/04/2014/002833 ).}, references={Abebe, A., Chhetri, M., Sankar, L., Shivaji, R., Positive solutions for a class of superlinear semipositone systems on exterior domains (2014) Bound. Value Probl., 2014. , 9 pp; Allegretto, W., Huang, Y.X., A Picone's identity for the p-Laplacian and applications (1998) Nonlinear Anal., 32 (7), pp. 819-830; Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces (1976) SIAM Rev., 18 (4), pp. 620-709; Ávila, A.I., Brock, F., Asymptotics at infinity of solutions for p-Laplace equations in exterior domains (2008) Nonlinear Anal., 69 (5-6), pp. 1615-1628; 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; Chhetri, M., Drábek, P., Principal eigenvalue of p-Laplacian operator in exterior domain (2014) Results Math., 66 (3-4), pp. 461-468; Cui, S., Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems (2000) Nonlinear Anal., 41 (1-2), pp. 149-176; Drábek, P., Sankar, L., Singular quasilinear elliptic problems on unbounded domains (2014) Nonlinear Anal., 109, pp. 148-155; Fleckinger-Pellé, J., Gossez, J.-P., de Thélin, F., Principal eigenvalue in an unbounded domain and a weighted Poincaré inequality (2006) Contributions to Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., 66, pp. 283-296. , Birkhäuser Basel; Hai, D.D., On an asymptotically linear singular boundary value problems (2012) Topol. Methods Nonlinear Anal., 39 (1), pp. 83-92; Hai, D.D., Sankar, L., Shivaji, R., Infinite semipositone problems with asymptotically linear growth forcing terms (2012) Differ. Integral Equ., 25 (11-12), pp. 1175-1188; Hai, D.D., Williams, J.L., Positive radial solutions for a class of quasilinear boundary value problems in a ball (2012) Nonlinear Anal., 75 (4), pp. 1744-1750; Ko, E., Lee, E.K., Shivaji, R., Multiplicity results for classes of singular problems on an exterior domain (2013) Discrete Contin. Dyn. Syst., 33 (11-12), pp. 5153-5166; Lee, E.K., Sankar, L., Shivaji, R., Positive solutions for infinite semipositone problems on exterior domains (2011) Differ. Integral Equ., 24 (9-10), pp. 861-875; Rajendran, D., Tyagi, J., Existence of nonnegative solutions to positone-type problems in R̂N with indefinite weights (2010) Electron. J. Differ. Equ., 2010. , 16 pp}, correspondence_address1={Sankar, L.; Indian Institute of TechnologyIndia; email: lakshmi@iitpkd.ac.in}, publisher={Academic Press Inc.}, issn={0022247X}, language={English}, abbrev_source_title={J. Math. Anal. Appl.}, document_type={Article}, source={Scopus},

Journal of Mathematical Analysis and Applications

Singular semilinear elliptic problems with asymptotically linear reaction terms

The relative $\alpha$-entropy is the R{\'{e}}nyi analog of relative entropy and arises prominently in information-theoretic problems. Recent information geometric investigations on this quantity have enabled the generalization of the Cram{\'{e}}r-Rao inequality, which provides a lower bound for the variance of an estimator of an escort of the underlying parametric probability distribution. However, this framework remains unexamined in the Bayesian framework. In this paper, we propose a general Riemannian metric based on relative $\alpha$-entropy to obtain a generalized Bayesian Cram{\'{e}}r-Rao inequality. This establishes a lower bound for the variance of an unbiased estimator for the $\alpha$-escort distribution starting from an unbiased estimator for the underlying distribution. We show that in the limiting case when the entropy order approaches unity, this framework reduces to the conventional Bayesian Cram{\'{e}}r-Rao inequality. Further, in the absence of priors, the same framework yields the deterministic Cram{\'{e}}r-Rao inequality.

2020 54th Annual Conference on Information Sciences and Systems, CISS 2020

Generalized Bayesian Cram{\'{e}}r-Rao Inequality via Information Geometry of Relative $\alpha$-Entropy

Let A be a complex Banach algebra with unit. For an integer n≥ 0 and ϵ&gt; 0 , the (n, ϵ) -pseudospectrum of a∈ A is defined by Λn,ϵ(A,a):={λ∈C:(λ-a)is not invertible inAor‖(λ-a)-2n‖1/2n≥1ϵ}.Let p∈ A be a nontrivial idempotent. Then pAp= { pbp: b∈ A} is a Banach subalgebra of A with unit p, known as a reduced Banach algebra. Suppose ap= pa. We study the relationship of Λn , ϵ(A, a) and Λn , ϵ(pAp, pa). We extend this by considering first a finite family, and then an at most countable family of idempotents satisfying some conditions. We establish that under suitable assumptions, the (n, ϵ) -pseudospectrum of a can be decomposed into the union of the (n, ϵ) -pseudospectra of some elements in reduced Banach algebras. © 2019, Tusi Mathematical Research Group (TMRG).

Advances in Operator Theory

Decomposition of the (n, ϵ) -pseudospectrum of an element of a Banach algebra

Let (R, m) be a local (Noetherian) ring of dimension d and let M be a ﬁnite length R-module with free resolution G•. De Stefani, Huneke, and Núñez-Betancourt explored two questions about the properties of resolutions of M. First, in characteristic p &gt; 0, what vanishing conditions on the Frobenius Betti numbers, βiF (M, R) := lime→∞ λ(Hi(Fe(G•)))/ped, force pdR M &lt; ∞? Second, if pdR M = ∞, does this force d+ 2nd or higher syzygies of M to have inﬁnite length? For the ﬁrst question, they showed, under rather restrictive hypotheses, that d + 1 consecutive vanishing Frobenius Betti numbers force pdR M &lt; ∞, and when d = 1 and R is CM then one vanishing Frobenius Betti number suﬃces. Using properties of stably phantom homology, we show that these results hold in general, i.e., d+1 consecutive vanishing Frobenius Betti numbers force pdR M &lt; ∞, and, under the hypothesis that R is CM, d consecutive vanishing Frobenius Betti numbers suﬃce. For the second question, they obtain very interesting results when d = 1. In particular, no third syzygy of M can have ﬁnite length. Their main tool is, if d = 1, to show that if the syzygy has a ﬁnite length, then it is an alternating sum of lengths of Tors. We are able to prove this fact for rings of arbitrary dimension, which allows us to show that if d = 2, no third syzygy of M can be of ﬁnite length! We also are able to show that the question has a positive answer if the dimension of the socle of Hm0 (R) is large relative to the rest of the module, generalizing the case of Buchsbaum rings. © 2020 American Mathematical Society

Proceedings of the American Mathematical Society

Frobenius betti numbers and syzygies of finite length modules

Let J⊂I be ideals in a formally equidimensional local ring with λ(I/J)&lt;∞. Rees proved that λ(In/Jn) is a polynomial P(I/J)(X) in n of degree at most dim R and J is a reduction of I if and only if deg⁡P(I/J)(X)≤dimR−1. We extend this result for all Noetherian filtrations of ideals in a formally equidimensional local ring and for (not necessarily Noetherian) filtrations of ideals in analytically irreducible rings. We provide certain classes of ideals such that deg⁡P(I/J) achieves its maximal degree. On the other hand, for ideals J⊂I in a formally equidimensional local ring, we consider the multiplicity function e(In/Jn) which is a polynomial in n for all large n. We explicitly determine the deg⁡e(In/Jn) in some special cases. For an ideal J of analytic deviation one, we give characterization of reductions in terms of deg⁡e(In/Jn) under some additional conditions. © 2019 Elsevier B.V.