QSpace Collection:
http://hdl.handle.net/1974/6158
2016-07-31T01:37:56ZA Review on Repeated Games and Reputations with Incomplete Information
http://hdl.handle.net/1974/14663
Title: A Review on Repeated Games and Reputations with Incomplete Information
Authors: Verlezza, Michael
Abstract: In this project we review the effects of reputation within the context of game theory.
This is done through a study of two key papers. First, we examine a paper from
Fudenberg and Levine: Reputation and Equilibrium Selection in Games with a Patient
Player (1989). We add to this a review Gossner’s Simple Bounds on the Value of a
Reputation (2011). We look specifically at scenarios in which a long-run player faces
a series of short-run opponents, and how the former may develop a reputation. In
turn, we show how reputation leads directly to both lower and upper bounds on
the long-run player’s payoffs.2016-07-20T04:00:00ZThe Chowla Problem and its generalizations
http://hdl.handle.net/1974/13614
Title: The Chowla Problem and its generalizations
Authors: Pathak, Siddhi
Abstract: In this thesis, we study the vanishing of certain $L$-series attached to periodic arithmetical functions. Throughout the writeup, we let $f$ be an algebraic-valued (at times even rational-valued) arithmetical function, periodic with period $q$. Define
\begin{equation*}
L(s,f) := \sum_{n=1}^{\infty} \frac{f(n)}{n^s}.
\end{equation*}
We discuss a conjecture of Sarvadaman Chowla, made in the early 1960's of the non-vanishing of $L(1,f)$ for rational-valued functions $f$. We further discuss the theorem of Baker-Birch-Wirsing that classifies all odd algebraic-valued, periodic arithmetical functions $f$ with $L(1,f)=0$. We then apply a beautiful result of Bass to give a necessary condition for even algebraic-valued, periodic arithmetical functions $f$, to satisfy $L(1,f)=0$. The sufficiency condition obtained via this method is unfortunately not very clean. This, in view of a theorem of Ram Murty and Tapas Chatterjee, completes the characterization of all algebraic-valued periodic arithmetical functions $f$ with $L(1,f) = 0$.
In the next part of the thesis, we discuss the Lerch-zeta function and its functional equation. We also deduce the transcendence of its certain special values. Keeping this in mind, we define a new $L$-series attached to periodic functions and deduce a necessary condition for the vanishing of this $L$-series at $s=1$.
Finally, we mention some research topics that were stumbled upon during the course of this study and a few partial results regarding these questions.2015-09-15T04:00:00ZThe Saddle Point Method and its Applications to Number Theory
http://hdl.handle.net/1974/13613
Title: The Saddle Point Method and its Applications to Number Theory
Authors: Sampath, Kannappan
Abstract: In this thesis, we study the classical procedures useful in obtaining asymptotic expansions of functions defined by integrals and their
applications to number theory. The final chapter in the thesis reports on a recent joint work with Ram Murty on the theme of using arithmetic formulas
to obtain asymptotic formulas to study the Fourier coefficients of the j-function and a related
sequence j-sub-m (m ≥ 0) of modular functions.
Description: Masters' Project Report2015-09-15T04:00:00ZA survey of some methods for computation of discrete logarithms in small characteristic
http://hdl.handle.net/1974/13612
Title: A survey of some methods for computation of discrete logarithms in small characteristic
Authors: de Valence, Henry2015-09-15T04:00:00Z