added the tex files.

curve.tex

+\subsection{Linear regression}
+
+We have seen in the class that, given a theoretical model $Y(x, \textbf{a}) = \sum_{j=1}^M a_j Y_j(x)$ and a data set $\{x_i, y_i\}$ with associated errors $\sigma_i$, we can estimate the parameters $\textbf{a}$ of the model by minimising the cost function (chi-square). This amounts to solving the following Normal equations of the least square problem 
+\begin{equation} 
+(\textsf{A}^T \textsf{A}) ~ \textbf{a} = \textsf{A}^T \textbf{b},
+\end{equation} 
+where the elements of the matrix $\textsf{A}$ and vector $\textbf{b}$ are given by 
+\begin{equation} 
+A_{ij} = \frac{Y_j(x_i)}{\sigma_i}, ~~~ b_i = \frac{y_i}{\sigma_i}. 
+\end{equation} 
+
+\subsubsection{Problems}
+
+\begin{enumerate}
+\item Show that the covariance matrix of the estimated parameters $\textbf{a}$ are given by $\textsf{C} = (\textsf{A}^T \textsf{A})^{-1}$
+
+\item The Hubble's law provides a relation between the luminosity distance $d_L$ and cosmological redshift $z$ that is valid in the local universe ($z \lesssim 0.1$). 
+\begin{equation} 
+d_L = \frac{c}{H_0} z, 
+\end{equation} 
+where $c$ is the speed of light and $H_0$ is the Hubble constant. Here~\cite{sndata} you are given a dataset from the Supernova Cosmology project~\cite{SNCosmology}. This contains the redshift $z$, the distance modulus $\mu$, and the error on the distance modulus $\delta \mu$ measured from several Type 1a supernovae. The distance modulus is related to the luminosity distance (in parsecs) by $\mu = 5 \left(\log_{10} d_L - 1 \right)$. Use the linear regression method to fit the distance and redshift data to estimate the Hubble constant and the associated error. Note that the data contain redshifts above the range where the simple Hubble law is valid. Please select the samples with $z \leq 0.1$. 
+\end{enumerate}
+
+\subsection{Non-linear least square fitting}
+
+In the flat $\Lambda$CDM model of cosmology, the general relation between he luminosity distance $d_L$ and cosmological redshift $z$ is given by 
+\begin{equation} 
+d_L = (1+z) \frac{c}{H_0} \int_0^z \frac{dz'}{\sqrt{\Omega_M(1+z')^3 + \Omega_\Lambda)}}, 
+\end{equation} 
+where $\Omega_M$ is the energy density of matter and $\Omega_\Lambda$ is the energy density of the cosmological constant. Since we are assuming spatially flat universe: $\Omega_M +\Omega_\Lambda = 1$. Thus the only free parameters of the model are $H_0$ and $\Omega_M$. 
+
+\subsubsection{Problems}
+
+\begin{enumerate}
+\item Use the supernova data from the entire redshift to estimate $H_0$ and $\Omega_M$ and the corresponding covariance. This is a non-linear least square problem. You can use SciPy's \href{https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html}{\texttt{curve\_fit}} function. 
+\end{enumerate}
+
+\end{enumerate}
\ No newline at end of file

diff.tex

@@ -9,7 +9,7 @@ We derived the following finite-differencing approximants for the derivative of
 
 \subsubsection*{Problems:}
 \begin{enumerate}
-\item Write a Python function to compute derivatives using these three finite differencing methods. Compute the derivative of the function $f(x) = e^x \, \sin(x)$ over the range $x = [0, 2\pi]$. Plot the numerically computed derivative $f_{(h)}'(x)$ for three different values of $h$. 
+\item Write a Python function to compute derivatives using these three finite differencing methods. Compute the derivative of the function $f(x) = e^x \, \sin(x)$ over the range $x = [0, 2\pi)$. Plot the numerically computed derivative $f_{(h)}'(x)$ for three different values of $h$. 
 \item Plot the error $\Delta f_{h}'(x) := |f_{h}'(x) - f'(x)|$ for three different values of $h$, and estimate the order of convergence $n$ of each finite-difference approximation. Plot $n(x)$, where 
 \begin{equation}
 n(x) = \log_2 \frac{f'_{4h}(x) - f'_{2h}(x)}{f'_{2h}(x) - f'_{h}(x)}

fourier.tex

+\subsection{Fast Fourier transform (FFT)}
+\subsubsection{Problems:}
+\begin{enumerate}
+\item Prove the convolution theorem. 
+\item Prove the correlation theorem. 
+\item Compute the Fourier transform of the function $h(t) = \sin 2 \pi f_0 t$ where $f_0 = 10$ Hz, $t = [0, 30]$ seconds using numpy.fft. Use different window functions and compare the Fourier transforms obtained using different window functions.   
+\item Hercules X-1 is a high-mass X-ray binary (HMXB) system having a magnetized spinning neutron star which happens to be an X-ray pulsar. Here~\cite{rxte-data} you are given the data of an RXTE~\cite{rxte} observation after cleaning and pre-processing. The file contains two columns, (i) time (in seconds) and (ii) count-rate of X-ray photons (i.e., number of photons detected per unit time). Plot the count-rate as function of time. This is called lightcurve in X-ray astronomy. Compute the FFT of the given time-series and plot its absolute value against the frequency. Are you able to find any periodic signal(s)? The lowest frequency signal corresponds to the spin frequency of the neutron star.  What is the spin period of this pulsar? Can you identify other periodic signals and how they are related to the lowest frequency signal?~\footnote{This problem and data are graciously provided by Dr. Arunava Mukherjee (SINP).}
+\end{enumerate}
+
+\subsection{Power spectrum estimation using FFT}
+\subsubsection{Problems:}
+\begin{enumerate}
+\item A sample data set from the LIGO gravitational-wave observatory can be downloaded from here~\cite{ligo-data}. This contains 256 seconds of LIGO data from 2005, sampled at a rate of 4096 Hz. Compute the power spectral density of the data using Welch's modified periodogram method. 
+\end{enumerate}
+
+
+

ode.tex

@@ -81,7 +81,7 @@ The random motion of a particle in a fluid due to collisions with the molecules
 \begin{equation}
 m \frac{d^2 \bx}{dt^2} = - \lambda \frac{d \bx}{dt} + \boldeta(t), 
 \end{equation}
-where $m$ is the mass of the particle, $\bx$ its position vector, $\lambda$ a damping coefficient, and $\boldeta(t)$ (called the \emph{noise term}) describes the stochastic processes affecting the system. 
+where $m$ is the mass of the particle, $\bx$ its position vector, $\lambda$ a damping coefficient, and $\boldeta(t)$ (called the \emph{noise term}) describes the stochastic forces (e.g., random collissions of molecules) affecting the particle. 
 
 \subsubsection{Problems:}
 \begin{enumerate}

rest.tex

-
-\subsection{Ordinary differential equations: Calculation of gravitational waves from inspiralling compact binaries}
-
-\subsubsection*{Problems:}
-The time evolution of the orbital phase $\varphi(t)$ of a binary of black holes evolving under the gravitational radiation reaction can be computed, in the post-Newtonian approximation, by solving the following coupled ODEs: 
-\begin{eqnarray}
-\frac{dv}{dt} = -\frac{\mathcal{F}(v)}{dE(v)/dv}, ~~~~~ \frac{d\varphi}{dt} = \frac{v^3}{m},
-\label{eq:phasing_formula}
-\end{eqnarray}
-where $E(v)$ is the binding energy of the orbit, $\mathcal{F}(v)$ is the energy flux of radiated gravitational waves, $m:= m_1 + m_2$ is the total mass of the binary, $v = (m \omega)^{1/3}$, $\omega$ being the orbital frequency. (Here we use geometric units, in which $G = c = 1$. This means that in all the expressions $m$ has to be replaced by $Gm/c^3$.)  
-
-The binding energy and gravitational-wave flux are given as post-Newtonian expansions in terms of the small parameter $v$ 
-\begin{eqnarray}
-E(v) = -\frac{1}{2} \mu v^2 \left[1 + \mathcal{O}(v^2) \right], ~~~ \mathcal{F}(v) = \frac{32}{5} \left(\frac{\mu}{m}\right)^2 \, v^{10} \left[1 + \mathcal{O}(v^2) \right], 
-\end{eqnarray}
-where $\mu :=  m_1 m_2/m$ is the reduced mass of the system. 
-
-\begin{enumerate}
-\item Compute $v$ as a function of $t$ by solving the first equation in Eq.~(\ref{eq:phasing_formula}) using Scipy's \href{http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.odeint.html}{\texttt{odeint}} routine. Assume the following parameters $m_1 = m_2 = 5 M_\odot$, $v_0 = 0.3$, $\varphi_0 = 0$. Plot $v(t)$. 
-\item Solve the coupled system in Eq.~(\ref{eq:phasing_formula}) to compute $v$ and $\varphi$. Compute and plot the two gravitational-wave polarizations: 
-\begin{equation}
-\label{eq:GWpolzn}
-h_+(t)  = 4 \frac{\mu}{m} v^2 \, \cos \varphi(t), ~~~ h_\times(t)  = 4 \frac{\mu}{m} v^2 \, \sin \varphi(t). 
-\end{equation}
-\item Repeat the same calculation using the adaptive Runge-Kutta method by Dormand \& Prince using Scipy's \href{http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.ode.html}{\texttt{integrate.ode}} module (choose method = ``dopri5''). This method is very similar to the Runge-Kutta-Fehlberg method that we learned in the class. 
-\end{enumerate}
-
-%
-\subsection{Ordinary differential equations: Structure of a relativistic, spherically symmetric star}
-\label{sec:TOV}
-%
-The interior structure of a relativistic, spherically symmetric star is described by a metric that has the line element 
-\begin{equation}
-ds^2 = - e^{2\Phi(r)} \, c^2 dt^2 + \left(1 - \frac{2Gm(r)}{rc^2} \right)^{-1} \, dr^2 + r^2 \, d\Omega^2,
-\end{equation}
-where $m(r)$ is called the \emph{mass function} (which encapsulates the {gravitational mass} inside the radius $r$), $e^{2\Phi(r)}$ is the \emph{lapse function} (which relates the proper time with the coordinate time). Outside the ``surface'' of the star (in vacuum), the spacetime is described by the Schwarzschild metric; the lapse function becomes 
-\begin{equation}
-\Phi(r) = \frac{1}{2} \ln \left(1 - \frac{2Gm_\star}{rc^2} \right)
-\label{eq:lapse_schwarz}
-\end{equation}
-where $m_\star$ is the total (gravitational) mass of the star. The structure can be computed by solving the following set of ordinary differential equations, derived by Tolman, Oppenheimer and Volkoff (TOV). 
-\begin{eqnarray}
-\frac{dm(r)}{dr} & = &{4 \pi} r^2 \rho(r), \label{eq:tov1} \\
-\frac{dP(r)}{dr} & = & -\frac{G (\rho + P(r)/c^2)}{r^2} \left[m(r) + \frac{4\pi r^3 P(r)}{c^2} \right] \left[1- \frac{2G m(r)}{c^2 r} \right]^{-1}, \label{eq:tov2} \\
-\frac{d\Phi(r)}{dr} & = & \frac{G m(r) + 4 \pi G r^3 P(r) / c^2}{c^2 r  [r - 2Gm(r)/c^2]}. \label{eq:tov3}
-\end{eqnarray}
-The TOV equations have to be supplemented by an \emph{equation of state} $P = P(\rho)$, that relates the pressure $P$ to the energy density $\rho$. We assume the equating of state to be of polytropic form
-\begin{equation}
-P(r) = K \, \rho(r) ^{\gamma}.
-\end{equation}
-We also need to specify initial conditions for the variables $m, P, \Phi$. The following conditions can be used 
-\begin{equation}
-\label{eq:tov_boundary}
-m ( r = 0) = 0, ~~~~ P(r = 0) = P_c = P(\rho_c), ~~~~ \Phi(r_\star) = \frac{1}{2} \ln \left(1 - \frac{2Gm_\star}{r_\star c^2} \right) 
-\end{equation}
-
-\subsubsection*{Problems:}
-\begin{enumerate}
-\item Compute the structure, i.e., $m(r)$ and $P(r)$,  of a neutron star with central density $\rho_c = 5 \times 10^{17} ~ \mathrm{kg/m^{3}}$ by solving Eqs.~(\ref{eq:tov1}) and (\ref{eq:tov2}). Assume a polytropic equation of state with $\gamma = 5/3$ and $K = 5380.3$ (SI units). What is the mass $m_\star$ and radius $r_\star$ of the neutron star?  (Useful tip: You will need to start the integration at $r = \Delta r$, where $\Delta r$ is a small number. You can assume $m(r = \Delta r) := 4/3 \, \pi \rho_c  (\Delta r)^3$).
-\item Compute the lapse function $e^{2\Phi(r)}$ by solving Eq.(\ref{eq:tov3}) starting from $r = r_\star$ to $r = 0$. On top of that, plot the lapse function for a Schwarzschild black hole (see Eq.\ref{eq:lapse_schwarz}) from $r = r_s$ to $r = 2 r_\star$, where $r_s \equiv 2 G m_\star/c^2$ is the Schwarzschild radius of the star. This exterior solution should match the interior solution at $r = r_\star$.  
-\end{enumerate}
-
-\subsection{Non-linear ordinary differential equations showing chaotic behavior: Lorenz equations}
-The Lorenz equations were originally developed as a simplified mathematical model for atmospheric convection by Edward Lorenz. This was the first set of equations where deterministic chaos was observed. The equations are given by 
-\begin{equation}
-\frac{dx(t)}{dt} = \sigma [y(t)-x(t)], ~~~~ \frac{dy(t)}{dt} = x(t) [\rho - z(t)] - y(t),~~~~ \frac{dz(t)}{dt} = x(t)y(t) - \beta z(t),
-\end{equation}
-where $\rho, \sigma$ and $\beta$ are parameters of the system. 
-\subsubsection{Problems:}
-\begin{enumerate}
-\item Solve the Lorenz system for $\rho = 28, \sigma = 10$ and $\beta = 8/3$ with the following initial conditions $x(t = 0) = y(t = 0) = z(t=0) = 1$. Plot $x(t)$, $y(t)$ and $z(t)$ for $t = 0 ... 100$. Is the solution deterministic or stochastic? 
-\item Repeat the calculation with same parameters except for a tiny change in the initial condition for $x$: i.e., $x(t = 0) = 1 + 10^{-9}$. Plot  $x(t = 0) = y(t = 0) = z(t=0) = 1$ on top of the earlier estimate. Explain the result. 
-\item Make a 3D plot of $x, y, z$. You should see the famous butterfly shaped structure now! 
-\item Optional exercise: Make an animation of the above~\footnote{You can either use the matplotlib \href{http://matplotlib.org/api/animation_api.html}{animation} package or convert a number of PNG files to a gif animation using \href{http://www.imagemagick.org/Usage/anim_basics/}{ImageMagick}.}. 
-\end{enumerate}
-
-\subsection{Stochastic ordinary differential equations: Langevin equation}
-
-The random motion of a particle in a fluid due to collisions with the molecules of the fluid, called the Brownian motion, is described by the Langevin equation: 
-\begin{equation}
-m \frac{d^2 \bx}{dt^2} = - \lambda \frac{d \bx}{dt} + \boldeta(t), 
-\end{equation}
-where $m$ is the mass of the particle, $\bx$ its position vector, $\lambda$ a damping coefficient, and $\boldeta(t)$ (called the \emph{noise term}) describes the stochastic processes affecting the system. 
-
-\subsubsection{Problems:}
-\begin{enumerate}
-\item Using the Euler-Maruyama method, compute the $1-d$ Brownian motion trajectories generated by the Langevin equation
-${d x}/{dt} = \eta(t)$,  
-where $\eta(t)$ is Gaussian noise with zero mean and unit variance. Plot $x(t)$ for $ t \in [0, 100]$ assuming $x(t=0) = 0$. Is the solution deterministic or stochastic? 
-\item Generalize the code so that it can deal with arbitrary number of particles. Plot $x(t)$ for 1000 particles on a single plot. Compute the average displacement $\bar{x}(t)$ of the particles (from $x(t=0)$) as a function of $t$ and plot it against $t$. What is the relation between  $\bar{x}(t)$ and $t$? 
-\item Using the \href{http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.hist}{\texttt{hist}} function, plot the probability distribution $P(x)$ of $x(t)$ at $t = 10, 50, 100$.
-\end{enumerate}
-
-\subsection{Two-point boundary value problems: The shooting method}
-
-Solve the TOV equations described in Sec.~\ref{sec:TOV} as a two-point boundary value problem using the Shooting method. Use the Newton-Raphson method for root finding. The boundary conditions are given in Eq.~(\ref{eq:tov_boundary}).
-
 \section{Partial differential equations}
 
 \subsection{First order hyperbolic PDEs: The advection equation}