IITG Mathematics Seminar Series

Abstract: We consider a decentralized two-period supply chain in which a manufacturer produces a product with benefits of cost learning, and sells it through a retailer facing a price dependent demand. The manufacturer's second-period production cost declines linearly in the first-period production, but with a random learning rate. The manufacturer may or may not have the inventory carryover option. We formulate the resulting problems as two-period Stackelberg games and obtain their feedback equilibrium solutions explicitly.

We then examine the impact of mean learning rate and learning rate variability on the pricing strategies of the channel members, on the manufacturer's production decisions, and on the retailer's procurement decisions. We show that as the mean learning rate or the learning rate variability increases, the traditional double marginalization problem becomes more severe, leading to greater efficiency loss in the channel. We obtain revenue sharing contracts that can coordinate the dynamic supply chain. In particular, when the manufacturer may hold inventory, we identify two major drivers for inventory carryover: market growth and learning rate variability. Finally, we demonstrate the robustness of our results by examining a model in which cost learning takes place continuously.

Abstract: For positive integers $M$ and $N$, generalized hypergeometric Bernoulli numbers $B_{M,N,n}$ are defined by $$ \frac{1}{{}_1 F_1(M;M+N;x)}=\sum_{n=0}^\infty B_{M,N,n}\frac{x^n}{n!}\,,$$ where ${}_1 F_1(a;b;z)$ is the confluent hypergeometric function. When $M=N=1$, $B_n=B_{1,1,n}$ are the classical Bernoulli numbers with $B_1=-1/2$.

For positive integers $N$ and $M$, the general {\it hypergeometric Cauchy polynomials} $c_{M,N,n}(z)$ ($M,N\ge 1$; $n\ge 0$) are defined by $$ \frac{1}{(1+t)^z}\frac{1}{{}_2F_1(M,N;N+1;-t)}=\sum_{n=0}^\infty c_{M,N,n}(z)\frac{t^n}{n!}\,, $$ where ${}_2 F_1(a,b;c;z)$ is the Gauss hypergeometric function. When $M=N=1$, $c_n=c_{1,1,n}$ are the classical Cauchy numbers. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli and Cauchy numbers.

In the aspect of determinant expressions, hypergeometric Bernoulli and Cauchy numbers are the natural extension of the classical Bernoulli and Cauchy numbers. We show several expressions and identities of generalized hypergeometric Bernoulli and Cauchy numbers.

Abstract: Elliptic equations with singular nonlinearity is being studied considerably these days. Due to the singular nature of the nonlinearity near the boundary, the standard theory of analysis for semilinear elliptic equations is not applicable here. In this talk we wish to address this issues and provide a nonlinear analytic approach to find multiple solutions for a specific nonlinear PDE. The aim is to absorb the singularity into the operator so that the resulting nonlinearity is non-singular in nature and then appeal to the topological methods like bifurcation theory/fixed point theory to prove the existence of positive solution.

Abstract:

One of the generalizations of the fundamental theorem of calculus on higher dimension is the Lebesgue differentiation theorem (or ball averaging problem). Proof of the Lebesgue differentiation theorem is based on the Hardy-Littlewood maximal function.

In this talk, we will discuss the idea of the proof of the Hardy-Littlewood maximal function on Euclidean spaces which is based on the fact that the Euclidean spaces are spaces of polynomial volume growth. We will also discuss some cases of exponential volume growth e.g. hyperbolic plane, homogeneous tree etc. If time permits we shall discuss some related open problems.

Abstract: A beautiful but less known proof, due to R.Dedekind, of the fundamental Cantor-Bernstein Equivalence Theorem of Set Theory is presented with interesting background history of the theorem and its proof. Few remarks are made at the end.

Abstract: In operator theory, a Toeplitz operator is the compression of a multiplication operator on the circle to the Hardy space. It has wide applications in various branches of mathematics and engineering. In this talk we will discuss some properties of Toeplitz operators and its applications.

Abstract: Quadrature domains are those on which a suitable test class of holomorphic functions satisfies a generalized mean value property. A prototype here is a disc on which the usual mean value property holds. This talk will discuss several basic properties of quadrature domains.

Abstract: A finite separable extension of a field is called primitive if there are no intermediate extensions. We will first parametrise the set of primitive extensions of a given field whose galoisian closure has a solvable group as group of automorphisms. We will then apply these results to the case of local fields, where the parameters can be made very explicit.

Abstract: An algebraic variety (or topological space) $X$ is said to satisfy the diagonal property if there exists a vector bundle $E$ on $X \times X$ of rank equal to the dimension of $X$, and a section $s : X \times X → E$ of $E$ such that the zero set of $s$ coincides with diagonal $\bigtriangleup(X)$. We say $X$ satisfies the point property if there exists a vector bundle of rank dim$(X)$ on $X$ with a section whose zero set is a reduced point $x \in X$. The formal definition of the diagonal property appeared in a paper of Pragacz, Srinivas, and Pati (2006). I will start with a basic introduction of algebraic objects. After that, I will give a little survey on the point property. If time permits then I will show that vector bundles satisfying the point property over a very general principally polarized abelian varieties are indecomposable.

Abstract: In this lecture I shall present some of the highlights of the incredibly fascinating story of:

\begin{itemize}\item How the numerical algebra of the first half of the 19th century influenced Boole to the discovery of his algebra of classes, which was later improved to what we now call as Boolean algebras, by Jevons, De Morgan, Peirce and Schr\"{o}der,\\\item How Boole's algebra of classes, in turn, was partly responsible to the very creation of universal algebra.\\\end{itemize}

I shall, then, try to illustrate how systems of propositional logic (classical, many-valued, intuitionistic, etc.) provide stimulus to universal algebra by providing a rich variety of examples of algebras and thereby inspiring new ideas and results in universal algebra, and how universal algebra, for its part, provides powerful methods for the investigation of algebras arising from the propositional logics, thus contributing to further understanding of those logics.

My presentation will be, for the most part, non-technical and will include, time permitting, the historical timeline of major breakthroughs in both universal algebra and propositional logic.

Abstract: The aim is the talk is to make young researchers to know about the classical Bohr theorem for power series defined on the unit disk and the recent interests among function theorists on this topic.

Abstract: Self explanatory

Abstract: Class-groups of number fields continue to be a topic of great interest, which in spite of its classical origins, still is shrouded in mystery. The theorem of Herbrand-Ribet gives quite a good understanding of class-groups of cyclotomic fields. We look at this theorem in great generality from the point of view of group representations.

Abstract: The first FFT-based algorithm for numerical homogenization from high resolution images was proposed by Moulinec and Suquet in 1994 as an alternative to finite elements and twenty years later, it is still widely used in computational micromechanics of materials. The method is based on an iterative solution to an integral equation of the Lippmann-Schwinger type, whose kernel can be explicitly expressed in the Fourier domain. Only recently, it has been recognized that the algorithm has a variational structure arising from a Fourier Galerkin method. In this talk, I will show how this insight can be used to significantly improve the performance of the original Moulinec-Suquet solver. In particular, I will focus on (i) influence of Krylov subspace methods used to solve non-symmetric rank-deficient linear systems (ii) effects of numerical integration of the Galerkin weak form, and (iii) convergence of an a­posteriori bound on the solution during iterations.

Abstract: In this article, we develop and investigate a new classifier based on features extracted using spatial depth. Our construction is based on fitting a generalized additive model to posterior probabilities of different competing classes. To cope with possible multi-modal as well as non-elliptic nature of the population distribution, we also develop a localized version of spatial depth and use that with varying degrees of localization to build the classifier. Final classification is done by aggregating several posterior probability estimates, each of which is obtained using this localized spatial depth with a fixed scale of localization. The proposed classifier can be conveniently used even when the dimension of the data is larger than the sample size, and its good discriminatory power for such data has been established using theoretical as well as numerical results.

Abstract: We shall describe and discuss but with elementary and rudimentary details the two important subfields of Computational Mathematics, namely, the "Numerical Linear Algebra" and the "Computational Fluid Dynamics" and also state several challenging problems one faces in these two fundamental and important areas.

Abstract: Instances of hyperbolic geometry come up in nature whenever a system starts developing fast interconnections. Examples include trees, the human brain and the internet. A tell-tale signature is the existence of a fractal in one dimension less, e.g. the surfaces of trees and brains in the above examples.
After dealing with the above examples, we shall discuss a special case where the fractals emerge in the complex plane as a result of symmetries of hyperbolic 3-space. These symmetries act on the complex plane as well; however the dynamics being chaotic, it is hard to get a hold on them directly. Instead we go to hyperbolic geometry in 3 dimensions, set up a dictionary between the two and finally get a hold on the fractals in the complex plane through our study of hyperbolic geometry in 3 dimensions.

Abstract: The Riemann mapping theorem states that any proper simply connected domain in the complex plane is holomorphically equivalent to the unit disc. H. Poincare discovered that this theorem fails spectacularly in higher dimensions. We will discuss a proof of this surprising phenomenon. This will be an elementary talk and should be accessible to anyone with some familiarity with basic complex analysis in one variable.

Abstract: Human microbiome plays a crucial role in health and diseases with recent research unravelling the myriad ways in which these associations are manifested. While traditional microbiology focused on microbial pathogens and few beneficial bacteria, advances in high-throughput technologies have presented evidence of a broader spectrum of functions in relation to chronic diseases, brain function and neurodevelopmental outcomes. Understanding factors regulating our microbiota and the impact of microbiota on health requires appropriate statistical methodology. Microbial communities associated with the human body sites are complex communities, with unknown interactions and functions. In this talk, we shall discuss some of the statistical designs and methodologies that are useful to study questions related to microbiome. Few major questions of interest in studies of microbiome may be "How do microbial communities cluster between different groups ?" or "Which microbial taxa are differentially abundant between these groups?". Data generated from microbial surveys are relative abundances of microbial taxa (which are themselves members of a phylogenetic tree), while the actual abundances are unobservable quantities. This results in compositional data, where observations on each subject are multivariate vectors belonging to a simplex.

To answer the first question, we shall explore the Unifrac distance (Lozupone et al. 2005), a phylogenetic distance based metric that is used to separate microbial communities according to groups. Existing approaches to detect differentially abundant microbes either discount the underlying compositional structure in the microbiome data or use inappropriate probability distributions including the multinomial and Dirichlet-Multinomial that may potentially increase false discovery rate. For the second question, we introduced a novel statistical framework called Analysis of Composition of Microbiomes (ANCOM, Mandal et al. 2015) that accounts for the underlying compositional structure in the data, and, unlike existing approaches, can compare the composition of microbiomes across populations. ANCOM makes no distributional assumptions, and is sufficiently general to enable easy adjustment for covariates. ANCOM also scales well to compare samples involving thousands of taxa. We shall illustrate these methodologies using publicly available microbial datasets in the human gut. In addition, we shall explore some of the recent developments in microbiome research and how these may be relevant in the case of Indian populations and related health problems.

Abstract: We consider the one dimensional compressible Navier-Stokes system near a constant steady state with the periodic boundary conditions. The linearized system around the constant steady state is a hyperbolic-parabolic coupled system. We discuss some of the properties of the linearized system and its spectrum. Next we study some controllability results of the system.

Abstract: We consider time dependent flow for a Newtonian, viscous and incompressible fluid in a rectangular cavity. The governing system consists of the conservation of mass and momentum equations. The momentum equations considered here are linearization of the Navier-Stokes Equations. We derive the weak formulation for the governing system. Using the Galerkin method, we obtain matrix form at element level. The elements considered here are of Taylor-Hood type elements. Time dependency part is solved by using the Crank-Nicolson method. Numerical results for the velocity for a square cavity are presented.