Shiv Nadar University is a comprehensive, multidisciplinary, research-focused, and student-centric University offering a full range of academic programs at the undergraduate, postgraduate and doctoral level. With state-of-the-art infrastructure, the University comprises of academic wings, sophisticated labs, international standard sports facilities, amphitheaters, auditorium, conference rooms and smart classrooms. The University&rsquo;s goal is to become internationally recognized for the quality of its research and creative endeavors and their applicability to improving quality of life, generating new insights and expanding the boundaries of human knowledge creativity. Committed to excellence in teaching, research and service, the University aims to serve the higher education needs of India and the world beyond.

Shiv Nadar University

Amber Habib describes the hierarchical structure of mathematical knowledge and comments upon its reliability and validity. He invites the reader to a consideration of mathematics’ fundamental components: axioms, proofs, and theorems. Habib asserts that all mathematicians work with a common choice of axioms and rules of logic; and the degree of agreement amongst them is not found in any other discipline. He scrutinises the common view of mathematical activity and knowledge-making as offering certainty and increased rigour. The chapter is then centred around three fundamental issues: problems of ahistoricity, the subject matter proper of mathematics, and its relevance or importance. © 2023 selection and editorial matter, Gita Chadha and Renny Thomas.

Mathematics

(SoNS) School of Natural Sciences

Mapping Scientific Method: Disciplinary Narrations

THE METHODS OF MATHEMATICS

World University root node

Association for Women in Mathematics Series

Indian Women and Mathematics

Let (Formula presented.) be an abelian extension of the Lie superalgebra (Formula presented.). In this article we consider the problems of extending endomorphisms of (Formula presented.) and lifting endomorphisms of (Formula presented.) to certain endomorphisms of (Formula presented.). We connect these problems to the cohomology of (Formula presented.) with coefficients in (Formula presented.) through construction of two exact sequences which are our main results. The first exact sequence is obtained using the Hochschild–Serre spectral sequence corresponding to the above extension while to prove the second we rather take a direct approach. As an application of our results we obtain descriptions of certain automorphism groups of semidirect product Lie superalgebras. © 2021 Informa UK Limited, trading as Taylor & Francis Group.

Fulltext

Linear and Multilinear Algebra

Exact sequences in the cohomology of a Lie superalgebra extension

Mathematics has been applied to problems involving loans and investments from very early times. The Mesopotamians were calculating compound interest around 2000 BC. Later, in India and China, problems of debt were used to motivate negative numbers. The connections have only strengthened with time and the flow of ideas has been especially rich in the last century. We give a brief history of these connections and also explore the role of finance in the teaching of mathematics and as a career for mathematicians.

Proceedings of the Indian National Science Academy

The Interplay of Mathematics and Finance

Let 0 → a → e → g → 0 be an abelian extension of Lie superalgebras. In this article, corresponding to this extension we construct two exact sequences connecting the various automorphism groups and the 0-th homogeneous part of the second cohomology group, H2(g, a)0 . These exact sequences constitute an analogue of the well-known Wells exact sequence for group extensions. From this it follows that the obstruction for a pair of automorphism (ℙ, ℙ) 2 Aut(a) Aut(g) to be induced from an automorphism in Auta(e) lies in H2(g, a)0 . Then we consider the family of Heisenberg Lie superalgebras and show that not all pairs are inducible in this family. We also give some necessary and sufficient conditions for inducibility of pairs arising in this family. ©2020 Heldermann Verlag.

Journal of Lie Theory

Wells exact sequence and automorphisms of extensions of lie superalgebras

In this paper, we explore the problem of establishing a network among the stocks of a market at high frequency level and give an application to program trading. Our work uses high frequency data from the National Stock Exchange, India, for the year 2014. To begin, we analyse the spectrum of the correlation matrix to establish the presence of linear relations amongst the stock returns. A comparison of correlations with pairwise mutual information shows the further existence of non-linear relations which are not captured by correlation. We also see that the non-linear relations are more pronounced at the high frequency level in comparison to the daily returns used in earlier work. We provide two applications of this approach. First, we construct minimal spanning trees for the stock network based on mutual information and study their topology. The year 2014 saw the conduct of general elections in India and the data allows us to explore their impact on aspects of the network, such as the scale-free property and sectorial clusters. Second, having established the presence of non-linear relations, we would like to be able to exploit them. Previous authors have suggested that peripheral stocks in the network would make good proxies for the Markowitz portfolio but with a much smaller number of stocks. We show that peripheral stocks selected using mutual information perform significantly better than ones selected using correlation. © 2019 Sharma, Habib. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

PLoS ONE

Mutual information based stock networks and portfolio selection for intraday traders using high frequency data: An Indian market case study

A bridge to the world of mathematics for readers who want to gain a good foundation in basic mathematical skills for research and other activities.This book aims to help students of social sciences, liberal arts, and humanities to develop the ability to analyze and reason mathematically, to model situations and problems, and to be able to infer, present, and communicate their analysis effectively. Mathematical concepts are presented in both historical and everyday contexts to ease their utilization in the real world.Readers are introduced to the skills of expressing mathematical ideas using the language of sets, logically analyzing arguments and their validity, processing and interpreting data, and using probability to handle the inherent randomness of our world. Chapters dedicated to symmetry, perspective, and art will enable readers to reason, model, and evaluate everyday situations. The book will also increase awareness of how mathematical patterns pervade the world around us.

A Bridge to Mathematics

This book is broadly about the mathematical aspects of finance. It introduces the reader to the basic concepts and products of modern finance and explores various mathematical models dealing with quantification of risk, which form the backbone of modern financial analysis. The emphasis is not so much on the details of the financial world as the basic principles by which one seeks an understanding of it. No prior knowledge of economics or finance is called for—an exposure to basic calculus and probability is all that is required of the reader. The appendix covers this ground in fair detail and would itself serve as a comprehensive primer of mathematics for finance for a beginner. The book is peppered with examples that use real-life data to ground the theory covered in the book. The exercises to be worked out are also interspersed in the text—their purpose varies from simple practice in applying formulas to extending the ideas learnt to new situations. Solutions to all the exercise problems are included as Appendix C, a feature that will be welcomed by both students and faculty. The book will serve well as an introductory book on applied mathematics in finance, of interest to students of mathematics, finance and financial management. For those starting out as practitioners of mathematical finance, this is an ideal introduction.

The Calculus of Finance

We construct representations of certain direct limit Lie groups G=lim Gn via direct limits of Zuckerman derived functor modules of the groups Gn. We show such direct limits exist when the degree of cohomology can be held constant, and discuss some examples for the groups Sp(p,<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∞</mo></math>) and SO(2p,<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∞</mo></math>), relating to the discrete series and ladder representations. We show that our examples belong to the "admissible" class of Ol'shanskii, and also discuss the globalizations of the Harish-Chandra modules obtained by the derived functor construction. The representations constructed here are the first ones in cohomology of non-zero degree for direct limits of non-compact Lie groups.&nbsp;