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

The smallest integer k needed for the assignment of k colors to the elements so that the coloring is proper (vertices and edges) is called the total chromatic number of a graph. Vizing [126] and Behzad [6, 7] conjectured that the total coloring can be done using at most (Formula presented.) colors, where (Formula presented.) is the maximum degree of G. It is not settled even for planar graphs. In this paper, we give a survey on the total coloring of graphs. © 2023 The Author(s). Published with license by Taylor &amp; Francis Group, LLC.

World University root node

AKCE International Journal of Graphs and Combinatorics

Total colorings-a survey

Fractal functions provide a natural deterministic approximation of complex phenomena and also it has self-similarity. Recently, it has been recognized as an internal binary operation, called fractal convolution. In the present article, we obtain Bessel sequences of L2(I× J) composed of product of fractal convolutions, using the identification of L2(I× J) with the tensor product space L2(I) ⊗ L2(J), where I and J are real compact intervals. © 2023, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

Springer Proceedings in Mathematics and Statistics

Fractal Convolution Bessel Sequences on Rectangle

The purpose of this paper is to study the Schwarz–Pick type inequality and the Lipschitz continuity for the solutions to the nonhomogeneous biharmonic equation: Δ (Δ f) = g, where g : D¯ → C is a continuous function and D¯ denotes the closure of the unit disk D in the complex plane C. In fact, we establish the following properties for these solutions: First, we show that the solutions f do not always satisfy the Schwarz–Pick-type inequality 1-|z|21-|f(z)|2≤C,where C is a constant. Second, we establish a general Schwarz–Pick-type inequality of f under certain conditions. Third, we discuss the Lipschitz continuity of f, and as applications, we get the Lipschitz continuity with respect to the distance ratio metric and the Lipschitz continuity with respect to the hyperbolic metric. © 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

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Mediterranean Journal of Mathematics

On Schwarz–Pick-Type Inequality and Lipschitz Continuity for Solutions to Nonhomogeneous Biharmonic Equations

In order to improve the classical Bohr inequality, we explain some refined versions for a quasi-subordination family of functions in this paper, one of which is key to build our results. Using these investigations, we establish an improved Bohr inequality with refined Bohr radius under particular conditions for a family of harmonic mappings defined in the unit disk (Formula presented.). Along the line of extremal problems concerning the refined Bohr radius, we derive a series of results. Here, the family of harmonic mappings has the form (Formula presented.), where (Formula presented.), the analytic part h is bounded by 1 and that (Formula presented.) in (Formula presented.) and for some (Formula presented.). © 2022 Wiley-VCH GmbH.

Mathematische Nachrichten

Improved Bohr inequality for harmonic mappings

In this note, we mainly study operator-theoretic properties on the Besov space on the unit disk. This space is the minimal Möbius-invariant space. First, we consider the boundedness of Volterra-type operators. Second, we prove that Volterra-type operators belong to the Deddens algebra of a composition operator. Third, we obtain estimates for the essential norm of Volterra-type operators. Finally, we give a complete characterization of the spectrum of Volterra-type operators. © The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society.

Canadian Mathematical Bulletin

Volterra-type operators on the minimal Möbius-invariant space

Let q be a prime number and K = Q() be an algebraic number field with a root of an irreducible polynomial xqs-ax-b having integer coefficients. In this paper, we provide some explicit conditions involving only s,q,a,b for which K is non-monogenic. As an application, in the special case of a = 0 and q = 2, we show that if s ≥ 2 and 32 divides b-1, then K is not monogenic. We illustrate our results through examples. © 2024 World Scientific Publishing Company.

Journal of Algebra and its Applications

Non-monogenity of some number fields generated by binomials or trinomials of prime-power degree

In this paper, first we prove two refined Bohr-type inequalities associated with area for bounded analytic functions (Formula presented.) in the unit disk. Later, we establish the Bohr-type operator on analytic functions and sections. © 2021 Informa UK Limited, trading as Taylor & Francis Group.

Complex Variables and Elliptic Equations

The Bohr-type operator on analytic functions and sections

In this paper, an efficient orthogonal neural network (ONN) approach is introduced to solve the higher-order neutral delay differential equations (NDDEs) with variable coefficients and multiple delays. The method is implemented by replacing the hidden layer of the feed-forward neural network with the orthogonal polynomial-based functional expansion block, and the corresponding weights of the network are obtained using an extreme learning machine(ELM) approach. Starting with simple delay differential equations (DDEs), an interest has been shown in solving NDDEs and system of NDDEs. Interest is given to consistency and convergence analysis, and it is seen that the method can produce a uniform closed-form solution with an error of order 2 -n, where n is the number of neurons. The developed neural network method is validated over various types of example problems(DDEs, NDDEs, and system of NDDEs) with four different types of special orthogonal polynomials. © 2023, The Author(s).

Scientific Reports

Numerical solution of neutral delay differential equations using orthogonal neural network

In this paper, we establish several characterizations of freely quasiconformal mappings in Banach spaces in terms of quasisymmetry in the quasihyperbolic metric. This provides a partial solution to an open problem posed by Väisälä. Also, we consider the special case when the control function satisfies η(t)≤Mt for some M≥1. © 2023 Elsevier Inc.

Journal of Mathematical Analysis and Applications

Quasisymmetry of freely quasiconformal mappings in Banach spaces

Let H0 denote the set of all sense-preserving harmonic mappings f=h+g‾ in the unit disk D, normalized with h(0)=g(0)=g′(0)=0 and h′(0)=1. In this paper, we mainly investigate some properties of certain subclasses of H0, including inclusion relations and stability analysis by precise examples, coefficient bounds, growth, covering and distortion theorems. As applications, we build some Bohr inequalities for these subclasses by means of subordination. Among these subclasses, six classes consist of functions f=h+g‾∈H0 such that h+ϵg is univalent (or convex) in D for each |ϵ|=1 (or for some |ϵ|=1, or for some |ϵ|≤1). Simple analysis shows that if the function f=h+g‾ belongs to a given class from these six classes, then the functions h+ϵg‾ belong to corresponding class for all |ϵ|=1. We call these classes as stable classes. © 2023 Elsevier Masson SAS