## Unpublished works

**Primality Testing: A Review**

**Paul, D**. 2012.

## Abstract

Primality testing was one of the greatest computational challenges for the theoretical computer scientists and mathematicians until 2004 when Agrawal, Kayal and Saxena have proved that the problem belongs to complexity class P. This famous algorithm is named after the above three authors and now called The AKS Primality Testing having run time complexity O ̃(log^{10.5}(n)). Further improvement has been done by Carl Pomerance and H. W. Lenstra Jr. by showing a variant of AKS primality test has running time O ̃(log^{7.5}(n)). In this review, we discuss the gradual improvements in primality testing methods which lead to the breakthrough. We also discuss further improvements by Pomerance and Lenstra.

**Re-usability and Robustness of Python and Java programs**

**Paul, D**, Vartak, S. 2012.

## Abstract

“Re-usability” is one of the fundamental aspects of software development. Reusability of code enables us to add new functionality with minimum modifications. A module or class which is reusable reduces the implementation time and cost of maintenance. In this paper we investigate and compare the re-usability of Python and Java programs using graph theoretic approach. We define a metric to quantify reusability. Using the metric, we show that Python projects are more reusable than Java, which supports the design goal of Python. Apart from re-usability, we also address the question of “Robustness” for software programs. We cannot test robustness by removing nodes or edges from software dependency graph because removal of any nodes results into software crash. This makes the task more challenging. Therefore, we redefine robustness for software programs. We also define a metric to quantify robustness and compare Python and Java projects using the metric. We show that, Java programs are more robust than that of Python against random failure.

**Kolmogorov’s Theorems**

**Paul, D**, 2015.

## Abstract

Andrey Kolmogorov is known to be the father of modern probability theory. In this short tutorial, we explore different theorems formulated by Kolmogorov spanning the fields of statistics, probability theory, and functional analysis.

**Stochastic processes: translate maths to sense**

**Paul, D**, 2015.

## Abstract

This short tutorial aims at providing an overview about stochastic processes and stochastic calculus.

**Ollivier-Ricci curvature for Graphs**

**Paul, D**, 2015.

## Abstract

This report introduces the concept of Ollivier-Ricci curvature (a coarse version of usual Ricci curvature on manifolds) for graphs derived from Optimal Transportation problem due to Monge and Kantorovich (Monge-Kantorovich transportation problem)