Department of Mathematics

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

For p∈ (1 , N) and a domain Ω in RN, we study the following quasi-linear problem involving the critical growth: -Δpu-μg|u|p-2u=|u|p∗-2uinDp(Ω),where Δ p is the p-Laplace operator defined as Δ p(u) = div (| ∇ u| p-2∇ u) , p∗=NpN-p is the critical Sobolev exponent and Dp(Ω) is the Beppo-Levi space defined as the completion of Cc∞(Ω) with respect to the norm ‖u‖Dp:=[∫Ω|∇u|pdx]1p. In this article, we provide various sufficient conditions on g and Ω so that the above problem admits a positive solution for certain range of μ. As a consequence, for N≥ p2, if g is such that g+≠ 0 and the map u↦ ∫ Ω| g| | u| pd x is compact on Dp(Ω) , we show that the problem under consideration has a positive solution for certain range of μ. Further, for Ω = RN, we give a necessary condition for the existence of positive solution. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

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Nonlinear Differential Equations and Applications

On the generalised Brézis–Nirenberg problem

In this paper, we mainly use the Fréchet derivative to extend the Bohr inequality with a lacunary series to the higher-dimensional space, namely, mappings from Un to U (resp. Un to Un). In addition, we discuss whether or not there is a constant term in f, and we obtain two redefined Bohr inequalities in Un. Finally, we redefine the Bohr inequality of the odd and even terms of the analytic function f so as to obtain conclusions for two different higher-dimensional alternating series. © 2023, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences.

Acta Mathematica Scientia

The Bohr-Type Inequalities for Holomorphic Mappings with a Lacunary Series in Several Complex Variables

In this article, we mainly study the weighted composition-differentiation operator on the weighted Bergman space (Formula presented.) and the derivative Hardy space (Formula presented.), which characterize complex symmetric weighted composition-differentiation operator (Formula presented.). Moreover, the normality and self-adjointness of (Formula presented.) are also discussed. © 2022 Informa UK Limited, trading as Taylor & Francis Group.

Linear and Multilinear Algebra

Complex symmetric weighted composition-differentiation operators

Fractal interpolation function (FIF) constructed through an iterated function system is more versatile than any classical spline interpolation. In this paper, we propose a novel C1-rational quadratic trigonometric spline FIF with variable scaling, where the numerator and denominator of rational function are quadratic trigonometric polynomials with two shape parameters in every subinterval. The error and convergence analysis of the proposed rational trigonometric fractal interpolant are studied for data generating function in C3. We deduce sufficient conditions based on the parameters of the rational quadratic trigonometric spline FIF to preserve positivity, monotonicity, and range restrictions features of the concerned data sets. Numerical examples are presented to supplement the shape preserving results based on a restricted class of scaling functions and minimum values of the shape parameters. © 2023, The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature.

European Physical Journal: Special Topics

Rational quadratic trigonometric spline fractal interpolation functions with variable scalings

In this paper, we introduce a class of novel C1 -rational quartic spline zipper fractal interpolation functions (RQS ZFIFs) with variable scalings, where rational spline has a quartic polynomial in the numerator and a cubic polynomial in the denominator with two shape control parameters. We derive an upper bound for the uniform error of the proposed interpolant with a C3 data generating function, and it is shown that our fractal interpolant has O(h2) convergence and can be increased to O(h3) under certain conditions. We restrict the scaling functions and shape control parameters so that the proposed RQS ZFIF is positive, when the given data set is positive. Using this sufficient condition, some numerical examples of positive RQS ZFIFs are presented to support our theory. © 2023, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

Springer Proceedings in Mathematics and Statistics

Positivity Preserving Rational Quartic Spline Zipper Fractal Interpolation Functions

Let q be a prime number and f (x) = xqs − axm − b be a monic irreducible polynomial of degree qs having integer coefficients. Let K = Q(θ) be an algebraic number field with θ a root of f (x). We give some explicit conditions involving only a, b, m, q, s for which K is not monogenic. As an application, we show that if p is a prime number of the form 32k + 1, k ∈ Z and θ is a root of a monic polynomial f (x) = x2s − 32cpx2r − p ∈ Z[x] with s &gt; 4, 2 ∤ c, s ̸= 5 + r, then Q(θ) is not monogenic. © 2023 Rocky Mountain Mathematics Consortium. All rights reserved.

Rocky Mountain Journal of Mathematics

ON NONMONOGENIC ALGEBRAIC NUMBER FIELDS

Let M be a torsion-free module over an integral domain R. The main results of this article provide analogue of Dumas irreducibility criterion and an extension of Eisenstein’s irreducibility criterion for factorial modules. Furthermore, we provide some applications of these results in (Formula presented.) -module (Formula presented.). © 2022 Taylor & Francis Group, LLC.

Communications in Algebra

Irreducibility criteria for factorial modules

In this study, we construct a linear recurrent fractal interpolation function (RFIF) with variable scaling parameters for data set with α -stable noise (a generalization of Gaussian noise) on its ordinate, which captures the uncertainty at any missing or unknown intermediate point. The propagation of uncertainty in this linear RFIF is investigated, and a method for estimating parameters of the uncertainty at any interpolated value is provided. Moreover, a simulation study to visualize uncertainty for interpolated values is presented. © 2023, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

Distribution of Noise in Linear Recurrent Fractal Interpolation Functions for Data Sets with α -Stable Noise

In this paper, we investigate properties of harmonic entire mappings. First, we study lower order of harmonic entire mappings. For a harmonic entire mappings f of order ρ , we also discuss the case ρ= ∞ by introducing the quantities ρ(k) , τ(k) , λ(k) , ω(k) , and also the case ρ= 0 by studying logarithmic order ρl , logarithmic type τl , logarithmic lower order λl , and logarithmic lower type ωl . Secondly, we investigate approximation by harmonic polynomials of harmonic entire mappings. For a real valued continuous function f on [ - 1 , 1 ] , let En(f)=infpn∈πn‖f-pn‖,n=0,1,2,⋯, where the norm is the maximum norm on [ - 1 , 1 ] and πn denotes the class of all harmonic polynomials with real coefficients of degree at most n. It is known that limn→∞En1/n(f)=0 if and only if f is the restriction to [ - 1 , 1 ] of an entire function (cf. [5, Theorem 7, p. 76]). We prove that this result continues to hold for harmonic entire mappings. We also study the relationship of ρ(k) and λ(k) with the rate growth of En1/n(f) and investigate the relationship of ρl , τl , λl , ωl with the asymptotic behaviour of En1/n(f) . © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.

Monatshefte fur Mathematik

On harmonic entire mappings II

Let p be a rational prime. Let (Formula presented.) be an algebraic number field with θ a root of an irreducible quadrinomial (Formula presented.) with (Formula presented.). In the present paper, we give some explicit conditions involving only (Formula presented.) and s for which K is non-monogenic. If q is a prime number of the form (Formula presented.), then in the special case when p = 2 and (Formula presented.) with (Formula presented.), s > 4, we show that K is non-monogenic. Finally, we illustrate our results through examples. © 2023 Taylor & Francis Group, LLC.