Seminar on September 25 - Abhishek Ranjan

Title:              Arbitrage Structure and Finite Date Model with Financial Restriction
Speaker:       Dr Abhishek Ranjan

Department of Applied Mathematics
Université Paris 1 Panthéon Sorbonne, France

Time:               2 pm, Wednesday, September 25, 2013.

Venue:             TBA

Abstract:         We consider a (T+1)-date model of a financial exchange economy with finitely many agents having non-ordered preferences and portfolio constraints. There is a market for physical commodities for every state today and in the future, and financial transfers across time and states are allowed by means of finitely many nominal or numeraire assets. We examine the properties of the financial structure F and the set of its (limited) arbitrage-free prices Q_F. The set of arbitrage-free prices is shown to be a convex cone under a sufficient condition that holds in particular for short lived assets. Furthermore, we provide examples of equivalent financial structures F and F’ such that Q_F is a convex cone, but Q_F’ is neither convex nor a cone. At the end, we provide several existence results of equilibrium in a financial exchange economy for which portfolios are either defined by linear constraints or a convex set extending the framework of unconstrained case by Cass (1984, 2006), Werner (1985), Duffie (1987), Gaenakopolos and Polemarchakis (1997), framework of linear equality constraints by Balasko et al. (1990) and framework of 2-date by Cornet and Gopalan (2007), Aouani and Cornet (2011, 2013).

About the Speaker:   Dr Ranjan completed his PhD from the Paris School of Economics and Université Paris 1 Panthéon Sorbonne in 2012. His research interests are in Applied Mathematics, General Equilibrium Models, Financial Economics, and Decision Under Uncertainty.

Seminar on September 24 - Vijay Patankar

Congruences between Modular Forms and p-Adic Families of Modular Forms

(from Ramanujan to Hida)

Speaker:       Dr Vijay Patankar

                         International Institute of Information Technology, Bangalore

Time:               3 pm, Tuesday, September 24, 2013.

Venue:            TBA

Abstract:  In this expository talk, we will give an introduction to congruences between modular forms as first observed by Ramanujan and how that has led to p-adic families of modular forms. 

Among many other things, Ramanujan studied certain natural arithmetic functions such as the Partition function and the Sum of Divisors function.  In 1916, Ramanujan in his paper On certain arithmetical functions, observed certain congruences between distinct arithmetic functions and hence between the generating functions associated to them (which are in fact modular forms). In 1967, Serre interpreted these congruences in terms of Galois representations and conjectured the existence of Galois representations associated to modular forms (proved by Deligne 1968). In 1972, Serre constructed a p-adic family of Eisenstein forms. In 1986, Hida constructed p-adic families of cusp-forms and the associated p-adic families of Galois representations.

All these developments were essential tools for Andrew Wiles' proof of Fermat's Last Theorem. 

About the Speaker: Dr Patankar obtained his PhD from the University of Toronto in 2005. He has held positions at the Cold Spring Harbor Laboratory (New York), Microsoft Research India (Bangalore), Bhaskaracharya Pratishthana (Pune) and Indian Statistical Institute (Chennai). His research interests are in Number Theory, Algebraic Complexity Theory and Cryptography.

MAT101 Calculus I - Assignment 1 - Solutions

M101 Student Data

Students enrolled in M101 Calculus I for the Monsoon 2013 semester should fill this form to get enrolled on the course account in Blackboard and to get regular updates etc. If the form is not visible below, then login to your SNU account in another tab/window, and then refresh this page.

MAT 101 - Calculus I - Assignment 2

Please start on the Exercises from Sections 1.6 to 2.4 right away. Your first midterm on Sep 14 will be up to Sec 2.4.

Assignment 2

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Part A

The following exercises from Stewart's Essential Calculus are to be solved for presentation and discussion in the tutorials:

• [Section 1.6] 1, 3, 13, 26, 41, 47
• [Section 2.1] 3, 7, 10, 16, 26, 31, 47
• [Section 2.2] 1, 3, 6, 9, 21, 33, 43
• [Section 2.3] 10, 18, 21, 22, 33, 57, 63
• [Section 2.4] 6 to 10, 25, 27, 43, 48, 54
• [Section 2.5] 4, 23, 47, 55, 65, 66
• [Section 2.6] 8, 16, 19, 32
• [Section 2.7] 2, 9, 13, 24
• [Section 2.8] 7, 21
• [Section 3.1] 13, 23, 24
• [Section 3.2] 22, 32, 35, 66
• [Section 3.3] 7, 29, 39, 58
• [Section 3.4] 9, 20

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Part B

Submit written solutions to the following to your tutor by September 23:

• [Section 1.6] 19, 23, 49
• [Section 2.1] 28, 48
• [Section 2.2] 4, 23, 39
• [Section 2.3] 23, 40, 67
• [Section 2.4] 16, 30, 51
• [Section 2.5] 49, 57, 69
• [Section 2.6] 11, 17, 44
• [Section 2.7] 11, 28, 38
• [Section 2.8] 13, 24
• [Section 3.1] 25, 31
• [Section 3.2] 23, 38, 78
• [Section 3.3] 23, 62, 68
• [Section 3.4] 17

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Extra Credit

Submit the following to Prof Habib by September 16: Consider the cubic function \(f(x)=x^3+ax^2+bx+c\). Calculate \(\lim\limits_{x\to\pm\infty}\frac{f(x)}{x^3}\)and use this to show that there is a real number \(c\) such that \(f(c)=0\).

Edit on Sep 16: Last date of submission of Extra Credit problem is changed to Friday, Sep  20. Submission must be to Prof Habib. Also note that there are two uses of \(c\) in the problem which is an oversight. Change the second \(c\) to \(d\).

MAT101 Calculus I - Assignment 1

Department of Mathematics, Shiv Nadar University
Monsoon Semester 2013-14
MAT 101 Calculus I

Assignment 1

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Part A

The following exercises from Stewart's Essential Calculus are to be solved for presentation & discussion in the tutorials:

• [Section 1.1] 1, 3, 4, 17, 23, 25, 28, 35, 43, 51, 57, 59, 61
• [Section 1.2] 2, 4, 5, 8, 13, 17, 23, 26, 35, 52, 53
• [Section 1.3] 3, 7, 11, 23, 29
• [Section 1.4] 10, 13, 21, 22, 30, 43, 46
• [Section 1.5] 3, 10, 13, 15, 26, 35, 37

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Part B

Submit written solutions of the following to your Tutor by September 2:

• [Section 1.1] 5, 6, 18, 44, 62
• [Section 1.2] 18, 49, 58, 62
• [Section 1.3] 9, 27, 43
• [Section 1.4] 7, 23, 31
• [Section 1.5] 16, 31, 40

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Extra Credit

• [Section 1.5] 47

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IAYM Selections

The "Inviting All Young Minds" summer internship programme is being organized at SNU from June 10 to 29. The following SNU students have been selected for it:

Laavanya Gupta	BS Mathematics
Manika	BTech CSE
Akshay Prasad	BTech ECE
Nipun Abbi	BTech ME
Dikshant Chitkara	BTech ECE
Shivangana Gupta	BTech EEE
Nidhi Dubey	BTech EEE
Balakumaran S E	BTech ME
Ankita Srivastav	BTech CSE
Rahul Madan	BTech CSE
Aditya Goel	BTech ME
Shubhankar Mathur	BTech CSE
Akanksha Tiwari	BTech EEE
Shubham Jain	BTech EEE

In addition there may be 2 or 3 seats available for students from outside SNU. The eligibility criterion is that you should have completed Class XII and should have taken Mathematics in +2. You can apply by filling in the online application form before 5pm on Tuesday, June 4.

The IAYM 2013 program features student projects applying mathematics and computation skills to problems in financial modeling and cryptography. Selected students get a Rs 5000 stipend and accommodation and food in the SNU hostels.