IITG Mathematics Seminar Series

Abstract: We investigate the effects of vertical variation in hydraulic resistivity and thermal diffusivity due to a hydro-thermal convective flow in a horizontal porous medium. The flow system is governed by a set of partial differential equations derived from conservation of mass, conservation of momentum and conservation of energy. Applying weakly non-linear approach, we derive the linear and first-order systems assuming a no-flow basic state system. The solutions for the linear and first-order systems are computed numerically using fourth-order Runge-Kutta and shooting methods. The results obtained indicate a stabilizing effect on the flow and temperature for a positive vertical rate of change in both resistivity and diffusivity, whereas a destabilizing effect occurs for a negative vertical rate of change.

Abstract: A square matrix is stable if all its eigenvalues are in the closed left half of the complex plane and those on the imaginary axis are semi-simple. We consider the problem of computing the nearest stable matrix to an unstable one: given an unstable matrix $A \in \mathbb{R}^{n,n}$, minimize the Frobenius norm of $\Delta_{A} \in \mathbb{R}^{n,n}$, such that $A+\Delta_{A}$ is a stable matrix. This is a difficult non-convex optimization problem. This problem occurs for example in system identification where one needs to identify a stable system from observations. In this talk, I will present some new ideas to solve this problem based on a reformulation using linear dissipative Hamiltonian systems: we show that a matrix $A$ is stable if and only if it can be written as $A=(J-R)Q$, where $J$ is skew-symmetric, $R$ is positive semi-definite and $Q$ is positive definite. This reformulation results in an equivalent optimization problem with a simple convex feasible set. We propose new algorithms to solve this problem in variables $(J,R,Q)$. We show the effectiveness of our method compared to the other approaches and to several state-of-the-art algorithms. Those ideas can be generalised to get good approximate solutions to some other nearness problems for control systems like, distance to stability for descriptor systems, distance to positive realness, and minimizing the norm of static-stabilising-feedbacks.

Abstract: Viscous flow through porous media has potential applications in fields like oil recovery, drug transport through biological tissues, insulations, engineering salvaging process etc. The major parameters that influence various flow quantities like velocity, pressure and stress are porosity and permeability. In particular, permeability is more responsible due to its isotropic or anisotropic nature. If the pores are distributed in an arbitrary manner it causes anisotropy. For granular porous media the distortion of grains in an arbitrary manner causes anisotropy. For the case of fibrous media, the arbitrary orientation of fibres in different direction forms an anisotropic network inside the porous medium. We introduce the concept of anisotropy and the corresponding mathematical structure. We show the variations of anisotropy versus isotropy with some examples. We then discuss a theoretical model of squeeze-film in the presence of a porous bed. This model consists of a flat impermeable bearing that is being squeezed towards a porous interface. The gap between the porous bed and the bearing is assumed to be filled with a Newtonian fluid. This is an approximation to the lubrication process inside a human knee joint. We use Navier-Stokes equation in the fluid region and Darcy equation in the fluid filled porous region. Lubrication approximation is used to derive the corresponding evolution equation for the film thickness. We use Beavers-Joseph condition and Bars-Worster condition at the liquid porous interface and present a detailed analysis on the corresponding impact. We assume that the porous bed is anisotropic in nature with permeabilities K₂ and K₁ along the principal axes. Accordingly, the anisotropic angle φ taken as the angle between the horizontal direction and principal axis with permeability K₂. We show that the anisotropic permeability ratio and the anisotropic angle make a significant influence on the contact time, flux, velocity, etc. Contact time to meet the porous bed when the bearing approach under a constant prescribed load is estimated. We present some important findings (relevant to knee joint) based on the anisotropic properties of the human cartilage. For a prescribed constant load we have estimated the time duration a healthy human knee remains fluid lubricated.

Abstract: In this study, a class of Godunov-type solvers is formulated for a weakly hyperbolic pressure-less gas dynamics system and later extended to augmented Burgers' systems. An Engquist-Osher type solver is constructed utilizing the homogeneity property. Whereas, the convexity of a second flux component of the one-dimensional pressure-less flux function is used to construct a conservative Godunov-type solver. Since the considered systems satisfy the generalized Rankine-Hugoniot conditions, a non-conservative version is also developed and tested on various numerical examples. In particular, non-conservative Godunov-type solver developed here outclass other well-known solvers in capturing stationary δ and δ′ - waves. This is a joint work with Naveen Garg.

Abstract: In this talk, we consider multivariate von Neumann inequalities on commutative and non-commutative Lp-spaces. This is known to be the famous Matsaev's conjecture. We show that the multivariate von Neumann inequality on Lp-spaces is true for commuting tuple of isometries, 1 < p < \infty. However, we show that, as in the well-studied case of p = 2; it fails eventually for some commuting tuple of contractions. We then discuss the case of non-commutative Lp-spaces. We exhibit a non-trivial class of commuting contractive Schur multipliers for which the non-commutative multivariate Matsaev's conjecture is true. We also study a special class of operators, called Ritt operators and exhibit many interesting results in this direction, for example a positive solution to joint similarity problem for this class of operators. Tools used are from harmonic analysis combined with operator theoretic techniques and Banach space geometry.

Abstract: In this (mostly non-technical) talk I will give definitions, examples and pictures to show you some fun things that happen in high-dimensional topology. We will also talk about spheres with same topological but different smooth structures and see how these spheres give us "exotic" diffeomorphisms of high dimensional negatively curved manifolds. I will explain what I mean by an "exotic" diffeomorphism.

Abstract: I will start by recalling some classical formulae that one usually encounters in a first course in Calculus. For example, Euler proved in the 1730's that the sum of reciprocals of squares of positive integers is one-sixth of the square of \pi. Such formulae are the prototypical examples of an entire area of research in modern number theory called special values of L-functions. The idea of an L-function is crucial in the work of Andrew Wiles in his proof of Fermat's Last Theorem. The aim of this lecture will be to give an appreciation for L-functions and to convey the grandeur of this subject that draws upon several different areas of mathematics such as representation theory, algebraic and differential geometry, and harmonic analysis. Towards the end of the talk, I will present some of my own recent results on the special values of certain automorphic L-functions.

Abstract: In this talk we discuss the connection between limiting spectral distribution and the Brown measure. Also, the notion of asymptotic freeness of a sequence of matrices will be discussed. Specifically, we show that independent elliptic matrices converge to freely independent elliptic elements. It turns out that the Brown measure of the product of elliptic elements is same as the limiting spectral distribution.

Abstract: In group theory, the semi-direct product is a basic construction generalizing direct products. Locally compact quantum groups are natural generalizations of locally compact groups within the scope of non-commutative geometry. Many interesting examples of (non-compact) locally compact quantum groups were construct as q-deformation of locally compact groups. Therefore, it is natural to ask for a general theory of semi-direct product of locally compact quantum groups. The goal of this talk is to motivate and discuss recent developments in this area and present some examples

Abstract: Peer-to-Peer (P2P) networks "typically overlays on top of the Internet" pose some unique challenges to the algorithm designer. Primarily, the difficulty comes from constant churn owing to the short life span of most nodes in the network. Reminiscent of Theseus paradox, most nodes in a typical P2P network churn out within an hour only to be replenished with incoming new nodes. In order to maintain a well-connected network despite churn at this level, the overlay is constantly reworked. This results in the overlay network graph being an evolving graph that exhibits both edge dynamism and node churn. In this talk, we will discuss the somewhat surprising line of work in which we show how to design fast algorithms that are robust against heavy churn.

We will begin with a discussion on how to create an overlay network that has good expansion despite adversarial churn. Subsequently, assuming such an overlay network with good expansion, we will present a few basic techniques like fast information dissemination, random walks, and support estimation. Finally, we show how to use these techniques to design algorithms to solve fundamental problems like agreement, leader election, storing and retrieving data items.

Abstract: Arveson's program of numerical spectral approximation is extended to unbounded self-adjoint operators, with a view for applications to Schrodinger Operators. Several Szego-type theorems are proven for such operators, showing that the empirical density of eigenvalues, under suitable hypothesis on the so-called degree of the operator, converges to the density of the essential spectrum.

Abstract: This talk is about two (unrelated) problems. The first concerns the construction of a subspace of maximal dimension on which two Hermitian forms are simultaneously positive definite. The second problem is about estimating the size of pseudospectra of block triangular matrices.

Abstract: Abstract: We study the combined effects of non-local elasticity and confinement induced ordering on the dynamics of thermomolecular pressure gradient driven premelted films bound by an elastic membrane. The confinement induced ordering is modeled using a film thickness dependent viscosity (Pramanik & Wettlaufer 2017 *Phys. Rev. E* *96*, 052801). When there is no confinement induced ordering, we recover the similarity solution for the evolution of the elastic membrane, which exhibits an infinite sequence of oscillations (Wettlaufer *et al.* 1996 *Phys. Rev. Lett. **76*, 3602-3605). However, when the confinement-induced viscosity is comparable to the bulk viscosity, the numerical solutions of the full system reveal the conditions under which the oscillations vanish. Implications for general thermomechanical dynamics, frost heave observations, and cryogenic cell preservation are discussed.

Abstract: The Finite Element Model Updating concerns updating a finite element generated second-order model of some specified structures in such way that a set of prescribed eigenvalues and eigenvectors are reproduced, the other eigenvalues and eigenvectors remain unchanged and the updated model maintains the same structures as the original model. The problem routinely arises in vibration industries, such as automobile, aerospace, and spacecraft. A properly updated model can be used by the engineers with confidence for future designs and manufacturing.

Mathematically, it is a partially prescribed structured quadratic inverse eigenvalue problem. Since, the problem was formulated in quadratic inverse eigenvalue setting by the speaker in 2001, much research has been done by both mathematicians and engineers, but, unfortunately, the problem still has not satisfactorily been solved. The structure preservation is the most difficult aspect of the problem. Some notable progress has been made in the last few years by the speaker and his collaborators.

This talk will discuss (i) how the problem arises in industries, (iii) mathematical formulation of the problem in the quadratic inverse eigenvalue setting, (iii) mathematical, computational and engineering challenges, (iv) the progress made so far, and (v) the future research

The talk is interdisciplinary blending mathematics, scientific computing, optimization with vibration engineering and structural dynamics , and will be of interests to students, researches and practicing engineers in these disciplines.

Abstract: This talk introduces in brief the popular contributions of Joseph-Louis Lagrange and his journey. A brief outline on Lagrange's interaction with Euler will be discussed before introducing the Lagrange equations of second kind or Euler-Lagrange equations. A fluid mechanical problem of lifting a large object that is sunken in a fluid and lying in a neutrally buoyant condition on top of a fluid filled anisotropic porous bed will be discussed in detail. The talk introduces the corresponding Euler-Lagrange equations and then mechanics of break -out phenomenon while lifting such an object will be discussed. This problem has direct application in salvaging sunken ships, moving object at offshore, submersible - engineering etc. .

Abstract: We begin with classical harmonic functions. Using mean value theorem we show how to establish maximum principle and Harnack's inequality for classical harmonic functions. Then we provide a probabilistic approach to these problems. Next we discuss more general uniformly elliptic equations and establish similar results using probabilistic methods. Finally we treat uniformly elliptic systems with weak coupling and establish maximum principle and Harnack's inequality for such systems.

Abstract: In this talk, we will discuss the distribution of gaps between eigenangles of Hecke operators acting on the space of cusp forms of weight $k$ and level $N$, spaces of Hilbert modular forms of weight $\underline k=(k_1,k_2,\dots,k_r)$ and space of primitive Maass forms of weight $0$. Moreover, we will see a stronger version of multiplicity one theorem for the space of cusp forms of weight $k$ and level $N.$ The last part is a joint work with M. Ram Murty.