This is where I keep mathematical material that have at some point been TeX’d up. Among them include solutions to problems from various books, notes for my own personal use, and lectures for seminars.

Some links below are to solutions prepared by Prakash Balachandran for coursework done at Duke/Cornell University. They are intended for academic use only. They are to be used as reference only. Please, do not copy solutions and submit them as your own. I make no guarantee as to their correctness, but if errors are found, I would appreciate notice by email. These are sanctioned neither by the author(s) of the relevant text books, nor by the professors who assigned them to me.

Some study notes I made in preparation for the actuarial exam (P). Covers dicrete, continuous probability, expectation, random variables, and contains a summary of important distributions and their applications.

The Convergence of Stock Prices Modeled as a Sequence of Binomial Models to the distribution of Geometric Brownian Motion.

A lecture I gave on product measures. Gives detailed proofs of Fubini and Tonelli.

Some notes on stochastic analysis on manifolds based on Elton Hsu’s book. I’ve tried to fill as many gaps in the proofs as I can. Please let me know if I have any myself so I can fix it.

Course solutions to Cornell Math 4340 – Honors Introduction to Algebra, Spring 2006.

This is a collection of mathematical statements/theorems with proofs I’ve accumulated over my four years at Duke. Some are problems from textbooks, others are things I’ve thought about and decided they warranted TeX’d up proofs. They’re divided into geometry, analysis, and probability/stochastic processes. Contain a number of [I think] useful proofs, such as equivalence of norms on R^n, proof of Kolmogorov’s continuity theorem, non-existence of Lebesgue measure in infinite dimensions, and important identities of metrics and connection coefficients. Corrections welcome!

An interesting problem I stumbled upon in exploring what happens when one tries to make a discrete Markov Chain into a continuous one.

Course solutions to Duke Math 601 – Groups, Rings and Fields, Fall 2006 (Using Michael Artin’s book, Algebra)
Homework 1
Homework 2
Homework 6
Homework 8
Homework 9
Homework 11
Homework 12
Homework 13
Homework 14

This is where I keep mathematical material that have at some point been TeX’d up. Among them include solutions to problems from various books, notes for my own personal use, and lectures for seminarsSome links below are to solutions prepared by Prakash Balachandran for coursework done at Duke/Cornell University. They are intended for academic use only. They are to be used as reference only. Please, do not copy solutions and submit them as your own. I make no guarantee as to their correctness, but if errors are found, I would appreciate notice by email. These are sanctioned neither by the author(s) of the relevant text books, nor by the professors who assigned them to me.

This is where I keep mathematical material that have at some point been TeX’d up. Among them include solutions to problems from various books, notes for my own personal use, and lectures for seminars** **Some links below are to solutions prepared by Prakash Balachandran for coursework done at Duke/Cornell University. They are intended for academic use only. They are to be used as reference only. Please, do not copy solutions and submit them as your own. I make no guarantee as to their correctness, but if errors are found, I would appreciate notice by email. These are sanctioned neither by the author(s) of the relevant text books, nor by the professors who assigned them to me.

This is where I keep mathematical material that have at some point been TeX’d up. Among them include solutions to problems from various books, notes for my own personal use, and lectures for seminars** **Some links below are to solutions prepared by Prakash Balachandran for coursework done at Duke/Cornell University. They are intended for academic use only. They are to be used as reference only. Please, do not copy solutions and submit them as your own. I make no guarantee as to their correctness, but if errors are found, I would appreciate notice by email. These are sanctioned neither by the author(s) of the relevant text books, nor by the professors who assigned them to me.

This is where I keep mathematical material that have at some point been TeX’d up. Among them include solutions to problems from various books, notes for my own personal use, and lectures for seminars** **Some links below are to solutions prepared by Prakash Balachandran for coursework done at Duke/Cornell University. They are intended for academic use only. They are to be used as reference only. Please, do not copy solutions and submit them as your own. I make no guarantee as to their correctness, but if errors are found, I would appreciate notice by email. These are sanctioned neither by the author(s) of the relevant text books, nor by the professors who assigned them to me.

Math 32L - Laboratory Calculus II - Spring 2010

MA 115 - Statistics I - Fall 2011

Math 32L - Laboratory Calculus II - Spring 2010

Instructor: Prakash (Kash) Balachandran.

Office: 216 Physics Building.

Office Phone: 919-660-2879.

Email: kash@math.duke.edu

Office Hours: Wednesdays, 8-10pm in 132 Carr Building or by appointment.

Lecture: West Duke 108B, MWF 01:30 PM-02:20 PM.

Laboratory: West Duke 105, Tu 08:30 AM-10:15 AM.

Textbooks: *Calculus, *5th Edition by Hughes-Hallett, et al., *31L-32L Coursepack, 2009-2010 ed. *(aka the Lab Manual)

**Lectures**

- Day 1-2: Events
- Day 1-3: Random Variables and Expected Value
- Day 2-2: Geometric Series
- Day 2-3: Partial Sums
- Day 3-1: n^th Term Test and Sum of Series
- Day 3-2: Integral Test
- Day 3-3: Review; catch up day
- Day 4-1: The Comparison Test, Absolute Convergence Theorem, and Limit Comparison Test
- Day 4-2: The Ratio Test
- Day 4-3: Alternating Series
- Day 5-1: Review for Exam 1
- Day 5-2: Integration by Substitution
- Day 5-3: Approximating Definite Integrals
- Day 6-1: Integration by Parts(Courtesy of Tatsunari Watanabe)
- Day 6-2: Algebraic Identities(Courtesy of Tatsunari Watanabe)
- Day 6-3: Improper Integrals
- Day 7-1: Distribution Functions. Homework: Due 02/24. Solutions
- Day 7-2: Probability; Distributions. Homework: Due 03/01. Solutions
- Day 7-3: Normal Distributions
- Day 8-1: Areas and Volumes
- Day 8-2: Volume and Arclength
- Day 8-3: Catch-up day; prepare for presentation
- Day 10-1: Presentations – Normal Data Lab
- Day 10-2: Taylor Polynomials I
- Day 10-3: Taylor Polynomials II
- Day 11-1: Review
- Day 11-2: Convergence of Power Series
- Day 11-3: Taylor Series
- Day 12-1: Using Taylor Series I
- Day 12-2: Using Taylor Series II
- Day 12-3: Fourier Series
- Day 13-1: Fourier Series Review
- Day 13-2: General Fourier Series
- Day 13-3 Fourier Series Review
- Day 14-1: Oscillations. Power Series Solution to Oscillatory DE.
- Day 14-2: Predator-prey with Phase Plane
- Day 14-3: SIR Model with Phase Plane
- Day 15-1: Phase Plane Analysis
- Day 15-2: Review and Completion of Lab
- Day 15-3: Review
- Day 16-1: Test 3
- Day 16-2: TCE Day

**Supplementary Material**

MA 115 - Statistics I - Fall 2011

Instructor: Prakash (Kash) Balachandran.

Office: MCS 230.

Office Phone: 617-353-9543.

Email: prakashb@math.duke.edu

Office Hours: 4:30pm-6:00pm Mondays and Fridays.

Teaching Fellow: Xinyi Deng.

Office: MCS 256.

Office Phone: 617-353-5287.

E-mail: xinyi@bu.edu

Office Hours: 11:00am-12:00pm Mondays and Wednesdays.

Lecture: CAS 224, MWF 2:00pm-3:00pm.

Textbooks: *Introductory Applied Biostatistics for Boston University, Volume I *by R.B.D. D’Agostino, L.M. Sullivan, A.S. Beiser.

New Paragraph

**Lectures**

- Day 1-1: No Class
- Day 1-2
- Day 1-3
- Day 2-1.Homework #1(Due 09-19-11) Homework #1 Solutions
- Day 2-2
- Day 2-3
- Day 3-1.Homework #2(Due 09-23-11) Homework #2 Solutions
- Day 3-2
- Day 3-3.Homework #3(Due 09-30-11) Homework #3 Solutions
- Day 4-1
- Day 4-2
- Day 4-3.Homework #4(Due 10-07-11) Homework #4 Solutions
- Day 5-1
- Day 5-2: Exam #1
- Day 5-3.Homework #5(Due 10-14-11) Homework #5 Solutions
- Day 6-1: No Class (Columbus Day)
- Day 6-2: Review of Exam 1
- Day 6-3
- Day 7-1
- Day 7-2
- Day 7-3.Homework #6(Due 10-28-11) Homework #6 Solutions
- Day 8-1
- Day 8-2
- Day 8-3.Homework #7(Due 11-04-11) Homework #7 Solutions
- Day 9-1
- Day 9-2
- Day 9-3: Exam #2.
- Day 10-1
- Day 10-2
- Day 10-3.Homework #8(Due 11-18-11) Homework #8 Solutions (Book). Homework #8 Solutions (Additional Problems)
- Day 11-1
- Day 11-2
- Day 11-3
- Day 12-1: Review of Exam 2. Homework #9(Due 12-05-11) Homework #9 Solutions
- Day 12-2: No Class (Thanksgiving Break)
- Day 12-3: No Class (Thanksgiving Break)
- Day 13-1
- Day 13-2
- Day 13-3.Homework #10(Due 12-12-11) Homework #10 Solutions
- Day 14-1
- Day 14-2
- Day 14-3: Review for Final
- Day 15-1: Review for Final

S**upplementary Material**