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

A unified framework for fourth-order semilinear problems with trilinear nonlinearity and general sources allows for quasi-best approximation with lowest-order finite element methods. This paper establishes the stability and a priori error control in the piecewise energy and weaker Sobolev norms under minimal hypotheses. Applications include the stream function vorticity formulation of the incompressible 2D NavierStokes equations and the von Kármán equations with Morley, discontinuous Galerkin, C0 interior penalty, and weakly over-penalized symmetric interior penalty schemes. The proposed new discretizations consider quasi-optimal smoothers for the source term and smoother-type modifications inside the nonlinear terms.

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Numerische Mathematik

Unified a priori analysis of four second-order FEM for fourth-order quadratic semilinear problems

In vitro experiments in which tumour cells are seeded in a gelatinous medium, or hydrogel, show how mechanical interactions between tumour cells and the tissue in which they are embedded, together with local levels of an externally-supplied, diffusible nutrient (e.g., oxygen), affect the tumour’s growth dynamics. In this article, we present a mathematical model that describes these in vitro experiments. We use the model to understand how tumour growth generates mechanical deformations in the hydrogel and how these deformations in turn influence the tumour’s growth. The hydrogel is viewed as a nonlinear hyperelastic material and the tumour is modelled as a two-phase mixture, comprising a viscous tumour cell phase and an isotropic, inviscid interstitial fluid phase. Using a combination of numerical and analytical techniques, we show how the tumour’s growth dynamics change as the mechanical properties of the hydrogel vary. When the hydrogel is soft, nutrient availability dominates the dynamics: the tumour evolves to a large equilibrium configuration where the proliferation rate of nutrient-rich cells on the tumour boundary balances the death rate of nutrient-starved cells in the central, necrotic core. As the hydrogel stiffness increases, mechanical resistance to growth increases and the tumour’s equilibrium size decreases. Indeed, for small tumours embedded in stiff hydrogels, the inhibitory force experienced by the tumour cells may be so large that the tumour is eliminated. Analysis of the model identifies parameter regimes in which the presence of the hydrogel drives tumour elimination. © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

World University root node

Journal of Mathematical Biology

Two-phase model of compressive stress induced on a surrounding hyperelastic medium by an expanding tumour

Let T: D(T) → H2 be a densely defined closed operator with domain D(T) ⊂ H1. We say T to be absolutely minimum attaining if for every non-zero closed subspace M of H1 with D(T) ∩ M ≠ {0}, the restriction operator T|M: D(T) ∩ M → H2 attains its minimum modulus m(T|M). That is, there exists x ∈ D(T) ∩M with ‖x‖ = 1 and ‖T (x)‖ = inf{‖T (m)‖: m ∈ D(T) ∩ M: ‖m‖ = 1}. In this article, we prove several characterizations of this class of operators and show that every operator in this class has a nontrivial hyperinvariant subspace. One such important characterization is that an unbounded operator belongs to this class if and only if its null space is finite dimensional and its Moore-Penrose inverse is compact. We also prove a spectral theorem for unbounded normal operators of this class. It turns out that every such operator has a compact resolvent. © D l, Zagreb.

Operators and Matrices

SPECTRAL REPRESENTATION OF ABSOLUTELY MINIMUM ATTAINING UNBOUNDED NORMAL OPERATORS

The aim of this paper is to obtain the solution of the operator equation Tx= y for an infinite band matrix T using its finite-dimensional truncations Tnxn= yn. Several verifiable conditions are given to obtain the invertibility of T. An application in connection with a stable set of sampling for functions belonging to a shift-invariant space is discussed along with an illustration. © 2022, Tusi Mathematical Research Group (TMRG).

Banach Journal of Mathematical Analysis

Solution of an infinite band matrix equation

This study investigates the unsteady, two-dimensional flow and heat transfer past a sinusoidally rotating circular cylinder, subject to linear shear. A higher order compact (HOC) finite difference scheme is used to solve the governing Navier-Stokes equations coupled with the energy equation on a non-uniform grid in polar coordinates. The hydrodynamic and thermal features of the flow are mainly influenced by the shear rate (K), Reynolds number (Re), Prandtl number (P r) and the cylinder oscillation parameters i.e. maximum angular velocity (am), the frequency ratio (fr). The simulations are performed for Re = 100, P r = 0.5 - 1.0, K ? [0.0, 0.15] and am ? [0.5, 2.0]. The numerical scheme is validated with the existing literature studies. Partial and full vortex suppression is observed for certain values of shear parameter K. The connection between heat transfer and vortex shedding phenomenon is examined where a pronounced increase in the heat transfer is observed for certain values of oscillation parameter, relative to the non-shear flow case.

Journal of Heat Transfer

Heat Transfer Past a Rotationally Oscillating Circular Cylinder in Linear Shear Flow

A one-spatial dimensional tumour growth model Breward et al. (2001, 2002, 2003) that consists of three dependent variables of space and time is considered. These variables are volume fraction of tumour cells, velocity of tumour cells, and nutrient concentration. The model variables satisfy a coupled system of semilinear advection equation (hyperbolic), simplified linear Stokes equation (elliptic), and semilinear diffusion equation (parabolic) with appropriate conditions on the time-dependent boundary, which is governed by an ordinary differential equation. A reformulation of the model defined in a larger, fixed time-space domain is employed to overcome theoretical difficulties related to the time-dependent boundary. This reformulation reduces the complexity of the model by removing the need to explicitly track the time-dependent boundary. A numerical scheme that employs a finite volume method for the hyperbolic equation, a finite element method for the elliptic equation, and a backward Euler in time-mass lumped finite element in space method for the parabolic equation is developed. We establish the existence of a time interval $(0,T_{\ast })$ over which a convergent subsequence of the numerical approximations can be extracted using compactness techniques. The limit of any such convergent subsequence is proved to be a weak solution of the continuous model in an appropriate sense, which is called a threshold solution. Numerical tests and justifications that support the theoretical findings conclude the paper. © 2021 The Author(s) 2021. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

IMA Journal of Numerical Analysis

Convergence analysis of a numerical scheme for a tumour growth model

Research in the Mathematical Sciences

Multiplicities and mixed multiplicities of arbitrary filtrations

Let A be a complex unital Banach algebra with unit 1. An element a∈ A is said to be of G1-class if ‖(z-a)-1‖=1d(z,σ(a))∀z∈C\σ(a).Here d(z, σ(a)) denotes the distance between z and the spectrum σ(a) of a. Some examples of such elements are given and also some properties are proved. It is shown that a G1-class element is a scalar multiple of the unit 1 if and only if its spectrum is a singleton set consisting of that scalar. It is proved that if T is a G1 class operator on a Banach space X, then every isolated point of σ(T) is an eigenvalue of T. If, in addition, σ(T) is finite, then X is a direct sum of eigenspaces of T. © 2022, The Author(s), under exclusive licence to The Forum D’Analystes.

Journal of Analysis

G1 -class elements in a Banach algebra

A Lie bracket defined on the linear span of the free homotopy classes of undirected closed curves on surfaces was discovered in stages passing through Thurston’s earthquake deformations, Wolpert’s corresponding calculations with Hamiltonian vector fields and Goldman’s algebraic treatment of the latter leading to a Lie bracket on the span of directed closed curves. The purpose of this work is to deepen the understanding of the former Lie bracket which will be referred to as the Thurston-Wolpert-Goldman Lie bracket or, briefly, the TWG bracket.We give a local direct geometric definition of the TWG bracket and use this geometric point of view to prove three results: firstly, the center of the TWG bracket is the Lie sub algebra generated by the class of the trivial loop and the classes of loops parallel to boundary components or punctures; secondly the analogous result holds for the centers of the universal enveloping algebra and of the symmetric algebra determined by the TWG Lie algebra; and thirdly, in terms of the natural basis, the TWG bracket of two non-central curves is always a linear combination of non-central curves. These three results hold for surfaces with or without boundary. We also give a brief and more illuminating proof of a known result, namely, the TWG bracket counts intersection between a simple closed curve and any other closed curve.We conclude by discussing substantial computer evidence suggesting an unexpected and strong conjectural statement relating the intersection structure of curves and the TWG bracket, namely, if the TWG bracket of two distinct undirected curves is zero then these curve classes have disjoint representatives. The computer experiments were performed on curves on surfaces with boundary.The main tools are basic hyperbolic geometry and Thurston’s earthquake theory.

American Mathematical Society

The Lie bracket of undirected closed curves on a surface

A uniform bounded variation estimate for finite volume approximations of the nonlinear scalar conservation law ℓ tα + div(uf(α)) = 0 in two and three spatial dimensions with an initial data of bounded variation is established. We assume that the divergence of the velocity div(u) is of bounded variation instead of the classical assumption that div(u) is zero. The finite volume schemes analysed in this article are set on nonuniform Cartesian grids. A uniform bounded variation estimate for finite volume solutions of the conservation law ℓ tα + div(F(t,x,α)) = 0, where divxF ≠ 0 on nonuniform Cartesian grids is also proved. Such an estimate provides compactness for finite volume approximations in Lp spaces, which is essential to prove the existence of a solution for a partial differential equation with nonlinear terms in α, when the uniqueness of the solution is not available. This application is demonstrated by establishing the existence of a weak solution for a model that describes the evolution of initial stages of breast cancer proposed by Franks et al. [J. Math. Biol. 47 (2003) 424-452]. The model consists of four coupled variables: tumour cell concentration, tumour cell velocity-pressure, and nutrient concentration, which are governed by a hyperbolic conservation law, viscous Stokes system, and Poisson equation, respectively. Results from numerical tests are provided and they complement theoretical findings. © EDP Sciences, SMAI 2021.

ESAIM: Mathematical Modelling and Numerical Analysis

Strong bounded variation estimates for the multi-dimensional finite volume approximation of scalar conservation laws and application to a tumour growth model