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\title{Velocity Field due to Singular Distributions of Vorticity - The Circular Line Vortex}
\date{\displaydate{date}}
\newdate{date}{23}{10}{2018}
\author{}

\begin{document}
\maketitle
\section*{Overview}

\section{Streamlines due to a circular line vortex}

The expression for velocity field and streamlines of a circular line vortex was determined in the previous lecture. By considering certain special values of the streamfunction we can visualize the streamlines in the entire domain. $\psi_s = 0$ gives the streamline at $r=0$, which is just the axis of the ring. Aswe get closer to the ring, $\psi_s \to \infty$. Since, the difference between streamfunctions gives the flux, we have that there is an infinite flux of fluid passing through the ring, owing to the infinite velocity at the ring. The velocity at the axis of the ring decays as $u_z \propto 1/z^3$ for $z \gg R$.

In the limit $R\to 0$, we get the velocity field due to an infinitesimal line vortex. The Stokes streamfunction for this case is given by $\psi_s \propto \frac{\sin^2\theta}{r}$, which is analogous to the Stokes streamfunction for a translating sphere. Therefore, the infinitesimal vortex ring generates a velocity field akin to a potential dipole.

\subsection{Vortex stretching in axisymmetric flows without swirl}

In $2D$, the inviscid incompressible vorticity equation reduces to
\begin{align*}
&\frac{D\omega_3}{Dt} = 0
\end{align*}
assuming the motion to be in $x_1-x_2$ plane. This equation states that the vorticity ($\omega_3$) is just convected as a passive scalar. Convection implies that $\omega_\phi$ remains constant along the streamlines and therefore we can write $\omega_\phi = \psi$ (since $D\psi/Dt$ is zero by definition - streamfunction doesn't change along streamlines). More generally, we can write $\omega_\phi = f(\psi)$, for an arbitrary function $f(\psi)$.

In $3D$ axisymmetric flows without swirl, the equation in the $\phi$ coordinate gives
\begin{align*}
&\frac{D\omega_\phi}{Dt} = \vect \omega \bcdot \grad \vect u\\
\RA&\pd{\omega_\phi}{t} + u_r\pd{\omega_\phi}{r} + u_z\pd{\omega_\phi}{z} = \omega \1_\phi \bcdot \grad \vect u \bcdot \1_\phi
\end{align*}
The RHS can be simplified as
\begin{align*}
&\omega \1_\phi \bcdot \grad \vect u \bcdot \1_\phi = \omega_\phi \bigg( \frac{1}{r} \frac{\partial}{\partial \phi}(u_r\1_r + u_z\1_z)\bigg) \bcdot \1_\phi = \frac{u_r\omega_\phi}{r}
\end{align*}.
Therefore, even though $u_\phi = 0$ and the $\phi$ derivatives vanish, the $\1_{\phi \phi}$ component of the rate of strain tensor is non-zero.
Substituting this in the above equation gives
\begin{align*}
&\pd{\omega_\phi}{t} + u_r\bigg(\pd{\omega_\phi}{r} + \frac{\omega_\phi}{r}\bigg)+ u_z\pd{\omega_\phi}{z} = 0\\
\RA&\frac{D(\omega_\phi/r)}{Dt} = 0\\
\RA&\frac{\omega_\phi}{r} = f(\psi_s)\\
\RA&\omega_\phi = r f(\psi_s)
\end{align*}
where $r$ is the distance from the axis of symmetry. This demonstrates the effect of vortex stretching. Since $\omega_\phi/r$ remains constant, it implies that $\omega_\phi$ must increase in proportion to the radial distance of the fluid element from the axis.

\section{Motion of a circular line vortex}

A circular line vortex induces a flow onto itself. We can determine the velocity of propagation of the ring if we can evaluate the velocity induced by the ring at any point on the ring itself. Let $a$ be the radius of the core of the ring and $R$ be the radius of the ring. Assuming $a/R \ll 1$ (thin ring approximation) and the circulation $\kappa = \omega_\phi \cdot \pi a^2$, we can proceed as follows.

Let $\xp$ be a point on the ring and $s$ its distance from any neighbouring point along the arc-length of the ring. The key physical argument here is that the self induced flow responsible for propoagation of the ring comes mainly due to neighbouring elements of the ring and not from the diametrically opposite ones, as one might intuitively imagine. Drawing tangents at two nearby points and observing the flow induced by the corresponding elements makes this idea clear.

In order to include the contribution from nearby elements, we expand about $\xp$ for small $s$
\begin{align*}
&\xp (s) = s\frac{d\xp}{ds}\bigg|_{s=0} + \frac{s^2}{2} \frac{d^2\xp}{ds^2}\bigg|_{s=0} + O(s^3)\\
\RA& \xp (s) = s \vect t + \frac{s^2}{2} \frac{\vect n}{r} + O(s^3)
\end{align*}
Using
\begin{align*}
&\vect u(\x) = \frac{\kappa}{4 \pi} \oint \frac{dl(\xp) \wedge (\x - \xp)}{|\x-\xp|^3}
\end{align*}
where $dl(\xp) = ds(\vect t + \vect n s/R)$, $\x = a\cos\phi \vect b + a\sin\phi \vect n$ where  $\vect b$ is the binormal vector in the $(\vect t, \vect n, \vect b)$ coordinates. This allows us to conclude that the velocity induced by the circular line vortex, $\vect u(\x)$ is
\begin{align*}
&\vect u(\x) = \frac{\kappa}{2\pi a}(-\cos\phi \vect n + \sin\phi \vect b) + \frac{\kappa}{4\pi R} \log\bigg(\frac{R}{a}\bigg) \vect b
\end{align*}
which gives us the velocity of propagation of a thin vortex ring.

\section{Appendix}
\subsection{Volume entrained in a circular line vortex}

\end{document}


\chapter{Review of Prior Works}
\label{chap:chapter2}

Existing algorithms for calculating Herbrand Equivalence are either
exponential or are imprecise. The precise algorithms are based on an
early algorithm by Kildall \cite{Kildall}, which discovers
equivalences by performing an abstract interpretation over the
lattice of Herbrand equivalences. Kildall algorithms is precise in
the sense it finds all the Herbrand equivalences but is exponential
in time. The partition refinement algorithm of Alpern, Wegman and
Zadek (AWZ) \cite{AWZ} is efficient but is much imprecise compared to
Kildall's. AWZ algorithm represent the values of variables after a
join using a fresh selection function $\phi_i$, similar to functions
in the static single assignment form and treats $\phi_i$ as
uninterpreted functions. It is incomplete in the sense it treats all
$\phi_i$ as uninterpreted. In an attempt to remedy this problem,
Ruthing, Knoop and Steffen proposed a polynomial-time algorithm (RKS)
\cite{RKS} that alternately applies the AWZ algorithm and some
rewrite rules for normalization of terms involving $\phi$ functions,
until the congruence classes reach a fixed point. Their algorithm
discovers more equivalences than the AWZ algorithm, but remains
incomplete.

Gulwani and Necula \cite{Gulwani} gave algorithm to find the Herbrand
Equivalence classes restricted to program expressions. There
algorithm is linear in parameter $s$, where $s$ is the maximum times
operators occur in a program expression. Clearly $s$ can take a maximum
value of $n$, which is the program size, so the algorithm in all is
polynomial in $n$. Later, Saleena and Paleri \cite{Saleena} showed
that Gulwani's algorithm losses some information as it removes a
equivalence class if it does not contain a variable or a constant.
The global value numbering (GVN) algorithm proposed by them was able
to detect more redundencies compared to that by Gulwani and Necula.

One problem is that most of these alogrithms were based on fix point
computations but the classical definition of Herbrand equivalence is
not a fix point based definition making it difficult to prove their
precision or completeness. Babu, Krishnan and Paleri \cite{Babu}
developed a lattice theoretic fix-point formulation of Herbrand
Equivalence on the lattice defined over the set of all terms
constructible from variables, constants and operators of a program.
They showed this definition is equivalent to the classical meet over
all path characterization over the set of all possible expressions.
The algorithm proposed by them is able to detect all the equivalences
as by that of Saleena and Paleri.

So, to sum up Kildall's algorithm finds all the equivalent classes
but is exponential. The algorithms by Saleena and Paleri; Babu,
Krishnan and Paleri are polynomial and efficient among other
imprecise algorithms. They are able to find all equivalence classes
restricted to program expressions (all expressions with length atmost
2), which is precisely what is practically useful.


% Lecture for ph2a Caltech 2017: Vibrations and Waves
\documentclass[pdf, handout, hideothersubsections]{beamer}
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%    \useoutertheme{tree}
  }

\usepackage{svg}
\usepackage{xmpmulti}
\usepackage{mathtools}
%\usepackage{enumitem}
\usepackage{hyperref}
\hypersetup{
    pdffitwindow=true,     % window fit to page when opened
    colorlinks=true,       % false: boxed links; true: colored links
    linkcolor=orange,      % color of internal links
    citecolor=green,       % color of links to bibliography
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    pdfstartview={Fit}
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% Fonts/encoding
\renewcommand{\UrlFont}{\tiny}
\usepackage[utf8]{inputenc}
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%\usepackage[sc,medium,raggedright]{titlesec}
\usepackage{newtxmath}
%\usepackage{libertine}
\usepackage[osf]{ebgaramond}

\graphicspath{{Figures/}}

\begin{document}
\title{Waving Strings}
\author{Caltech: ph2a}
\date{17 - Oct - 2017}

\frame{\titlepage}

\frame{\frametitle{Table of Contents}\tableofcontents}

\setbeamerfont{footnote}{size=\tiny}

\section{Summary}
\begin{frame}
\frametitle{Last Time:}
\begin{itemize}
\item Coupled oscillators have multiple free vibration frequencies/modes.
\item These free oscillation frequencies are called eigenfrequencies.
\item The oscillation functions are the eigenfunctions.
\item Linear algebra can be used to solve for these functions.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Overview}
\begin{enumerate}
  \pause
\item Matrix methods can be used to analyzed eigenmodes of linear systems.
  \pause
\item The case of $N$ springs and $N$ masses can be analyzed as a
  straightforward extension of the methods used for $N = 2, 3$.
  \pause
\item Strings are an extension of this method to $N \rightarrow \infty$.
  \pause
\item The Bode plot for systems with finite $N$, and then
  infinite $N$, illustrates the phenomena of \emph{dispersion} and the
  \emph{cut-off} frequency.
\pause
\item The rest of this course is the study of waves. This week we make
  the gradual transition between vibrations of discrete elements to
  waves: vibrations in continuous systems.
\end{enumerate}
\end{frame}

\section{String with Fixed Ends}
\begin{frame}
\frametitle{String with Fixed Ends: Setup}
\begin{columns}
  \begin{column}{0.4\textwidth}
    \includegraphics[width=0.85\textwidth]{StringFixedEnds.jpg}
    \pause
    \includegraphics[width=0.85\textwidth]{StringForceDiagram.pdf}

each segment has a mass, $m = \mu \Delta x$.
\pause
  \end{column}
  \begin{column}{0.6\textwidth}
    \begin{enumerate}
      \item Small Vibrations ($y << L$)
        \pause
      \item Endpoints are fixed. No displacement $y(0) = y(L) = 0$,
          and no velocity: $\dot{y}(0) = \dot{y}(L) = 0$
        \pause
      \item Constant Tension ($T = \mathrm{const}$). Vibrations are
        small and the changes in string length are $\propto (y/L)^2$
        (negligible). \underline{Magnitude} is constant, but
        \emph{direction} of tension is \textbf{not}.
    \end{enumerate}

  \end{column}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Analogy with Masses/Springs}
\begin{columns}
  \begin{column}{0.5\textwidth}
    \centering
    \includegraphics[width=0.75\textwidth]{StringForceDiagram.pdf}
\pause
    \begin{align*}
      \action<+->{m \ddot{y}_p &= T_+ - T_- \\}
          \action<+->{&\simeq T \theta_+ - T \theta_- \\}
    \action<+->{&\simeq T \Bigg[\frac{y_{p+1}-y_{p}}{\Delta x} -
      \frac{y_{p}-y_{p-1}}{\Delta x}\Bigg] \\}
\action<+->{\ddot{y}_p &= \frac{T}{\mu (\Delta x)^2} (y_{p+1} -2 y_p + y_{p-1})}
    \end{align*}

  \end{column}

\begin{column}{0.5\textwidth}
    \centering
    \includegraphics[width=\textwidth]{3mass4spring.pdf}
    \pause
    \begin{align*}
      m \ddot{x}_p &= F_+ - F_- \\
                   &= k (x_{p + 1} - x_{p}) - k (x_{p} - x_{p-1}) \\
      \\
      \\
      \\
      \\
      \\
        \ddot{x}_p &= \omega_0^2 (x_{p+1} - 2 x_p + x_{p-1})
    \end{align*}

  \end{column}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Details of the forces / tensions:}
\begin{columns}

  \begin{column}{0.4\textwidth}
    \centering
    \includegraphics[width=0.85\textwidth]{StringForceDiagram.pdf}

  \end{column}

  \begin{column}{0.6\textwidth}
    \begin{enumerate}
    \item We want to find $y(x,t)$. So far we have:
      \pause
    \item $\ddot{y}_p = \frac{T}{\mu} \frac{\Delta \theta}{\Delta x}$
      \pause
    \item $\frac{\Delta \theta}{\Delta x} = \frac{\theta(x + \Delta x) -
        \theta{x}}{\Delta x}$
      \pause
    \item Small vibrations, so $\theta << 1 \implies$
      \pause
    \item $tan(\theta) \simeq \theta \implies$
      \pause
    \item $\theta \simeq \frac{\partial y}{\partial x}$
      \pause
      \item $\frac{\Delta \theta}{\Delta x} = \frac{\partial^2 y}{\partial x^2}$

    \end{enumerate}
  \end{column}
\pause
\end{columns}
So, now we have a relation between time and space:
\pause
\begin{block}{The Wave Equation}
\centering
$ \frac{\partial^2 y}{\partial t^2} = \frac{T}{\mu} \frac{\partial^2 y}{\partial x^2}$
\end{block}

\end{frame}

\section{Wave Equation: Solutions}
\begin{frame}
\frametitle{Wave Equation: Solutions}
\textbf{Technique: Separation of Variables}
\begin{enumerate}
\pause
\item write $y(x,t) = f(x) g(t)$; assumes functions \emph{are} separable
\pause
\item insert into the P.D.E.
\pause
\item P.D.E. $\implies$ ODE for x, ODE for t
\pause
\item impose boundary conditions (e.g. fixed ends, or driven end)
\pause
\item General Solution is sum of these restricted solutions
\end{enumerate}
\end{frame}

\begin{frame}
\frametitle{Wave Equation: Solutions}
\begin{enumerate}
\item $y(x,t) = f(x) g(t)$
\pause
\item
\begin{align*}
\action<+->{\frac{\partial^2 y}{\partial t^2} &= \frac{T}{\mu} \frac{\partial^2
                                    y}{\partial x^2} \\}
\action<+->{f(x) \frac{\partial^2 g}{\partial t^2} &= v^2 g(t) \frac{\partial^2
                                         f}{\partial x^2} \\}
\action<+->{\frac{1}{g(t)} \frac{\partial^2 g}{\partial t^2} &=
  v^2 \frac{1}{f(x)} \frac{\partial^2 f}{\partial x^2}}
\end{align*}

\pause
\item Since LHS is a function of time \emph{only} and the RHS is a
  function of space \emph{only}, both sides must be equal to a
  constant (call it $\mathcal{C}$). The PDE has been converted into a set of two ODEs.
\end{enumerate}
\end{frame}

\begin{frame}
\frametitle{The ODE Solutions}
\begin{alignat}{2}
\frac{d^2 g}{dt^2} &= \omega^2 g  &\quad g(t) &= C_1 cos(\omega t) \\
\frac{d^2 f}{dx^2} &= -\frac{\omega^2}{v^2} f &\quad f(x) &= A cos(k x) + B
sin(k x)
\end{alignat}
\pause
where the velocity of the wave, $v \equiv \sqrt{\frac{T}{\mu}}$,
$\frac{\omega}{2 \pi}$ is the frequency, and $k \equiv \omega / v$ is
called the 'wave number'. For the wavelength of the wave, we use the
symbol $\lambda \equiv \frac{2 \pi}{k}$.
\end{frame}

\section{Normal Modes}
\begin{frame}
\frametitle{Normal Modes of a String}
\pause
Let's apply some boundary conditions ! \\
\pause
\begin{enumerate}
\item $f(x) = 0$ for $x \in \{0, L\} \implies$ \\
\pause
\item $sin(k L) = 0$ for any and all t,\\
\pause
\item so $k_n L = n \pi$ or $k_n = n \frac{\pi}{L}$ \\
\pause
\item and $y(x, t) = A_n cos(\omega_n t) sin(k_n x)$ \\
\end{enumerate}

\end{frame}

\begin{frame}
\frametitle{Normal Modes of a String}
\begin{columns}
\begin{column}{0.5\textwidth}

\begin{itemize}
\item Wavelength:  $\lambda_n = \frac{2 L}{n}$

\item Frequency:  $f_n = \frac{v}{\lambda_n}$

\item Velocity:  $v = \sqrt{\frac{T}{\mu}}$

\item $f_n = \frac{n}{2 L} \sqrt{\frac{T}{\mu}}$
\end{itemize}

\end{column}

\begin{column}{0.5\textwidth}

\centering
\includegraphics[width=0.5\textwidth]{stationary-waves-7-638.jpg}

\end{column}
\end{columns}
\end{frame}

\section{Summary}
\begin{frame}
\frametitle{Summary}
\begin{enumerate}

\item Matrix methods can be used to analyzed eigenmodes of linear systems.

\item The case of $N$ springs and $N$ masses can be analyzed as a
  straightforward extension of the methods used for $N = 2, 3$.

\item Strings are an extension of this method to $N \rightarrow \infty$.

\item The Bode plot for systems with finite $N$, and then
infinite $N$, illustrates the phenomena of \emph{dispersion} and the
\emph{cut-off} frequency.

\item HWK: QP-1, Ch. 5: 6, 10, 11 are coupled oscillator
  problems. 5-16 is for N coupled oscillators; similar to the string problems.

\end{enumerate}
\end{frame}

\end{document}


%=========================================================================
% sec-opts-copy
%=========================================================================

\section{Copying Blocks to Scratchpad}
\label{sec-opts-copy}

% Reason for optimization
\subsection{Reason for Optimization}

Although the blocking optimization helps the working set of a single
iteration of the computation kernel fit in the cache, it does not prevent
conflict misses in the cache. This is because blocking only reduces the
number of cache lines touched by the computation kernel, but does not
change the stride of the memory access pattern when traversing the columns
of a matrix. If the number of cache lines in a column is roughly a
multiple of the number of sets in the cache, consecutive rows in the
matrix will map to the same set. This means that if we traverse the
columns of a matrix, we will only be able to use one set of the cache,
causing repeated conflict misses. To makes things worse, only one element
of each new cache line we bring in will be used by the computation
kernel, yielding an inefficient utilization of the cache.
\smallskip

For example, the Intel Xeon processors on the totient compute nodes
have an 8-way set-associative L1 cache with a capacity of 32KB and 64B
lines. This means we have a total of $32K/64 = 512$ cache lines spread
out across $512/8 = 64$ sets. Assuming the matrix is arranged in
column-major order and in contiguous memory, the worst case is if the
matrix dimension is a multiple of $64*8 = 512$ doubles (since each cache
line can hold eight 64b doubles). Of course this only accounts for a
single matrix being stored in the cache--in reality, cache conflicts can
be aggravated with cache lines from multiple matrices being mapped to the
same set.
\smallskip

% Details of optimization
\subsection{Details of Optimization}

\input{fig-opts-copy-ex}

We can alleviate such conflicts and reduce the sensitivity to matrix
dimensions by copying blocks into a scratchpad such that the data within
each block are contiguous in memory. This allows us to control the stride
of traversing the columns within a block regardless of the matrix
dimensions by setting the block dimensions such that these conflicts
become rare. Furthermore, if we set the block dimensions such that all
matrices fit in the L1 cache, we can get closer to fully utilizing the
cache capacity without any conflicts. In addition, this copy optimization
allows us to align the matrices to the SIMD width regardless of the
matrix dimensions to enable vector loads/stores when vectorizing the
code. A critical tradeoff to keep in mind is the computational cost of
copying the matrices to the scratchpad memory. This optimization only
makes sense if the benefit of reducing conflict misses in the cache
outweighs the overhead of the copy itself.
\smallskip

Figure~\ref{fig-opts-copy-ex} shows how an example 4x4 matrix using 2x2
blocks are re-arranged when copying to the scratchpad memory. There are a
couple approaches to when and how much data to copy. One approach is to
copy and re-arrange the entire matrix once before any computation, then
copy the result back from the scratchpad to the output matrix in the
original format after the computation is complete. This amortizes the
overhead of bringing in the data from the input matrices into the cache
over the entire computation, but it increases memory pressure by
requiring a scratchpad that can hold all three matrices. Another approach
is to only allocate enough space in the scratchpad for three blocks and
copy the data in and out of the scratchpad every iteration of the
computation kernel. This makes more efficient use of memory, but we need
to contaminate the cache with data from the original input matrices
between every call to the computation kernel.
\smallskip

% Results and analysis
\subsection{Results}

\input{fig-opts-copy-results}

Figure~\ref{fig-opts-copy-results} shows the results for both large and
small scratchpad approaches to copy optimization. The interesting point
here is although there are no performance improvements, the performance
is much more stable across all matrix dimensions. This emphasizes the
impact of conflict misses in the cache from the matrix dimensions.

A quick note on choosing block sizes. We theorize that the optimal block
size is one which allows all three blocks to fit completely inside of the
L1 cache. With a 32KB cache, the following equation must be satisfied:
\[
3M^2 <= \frac{32K}{8} \rightarrow M <= \sqrt{\frac{32K}{8*3}} \rightarrow M <= 37
\]
\smallskip

However, a sweep of the block sizes showed that a block size of 128 was
optimal across all matrix dimensions. This likely has to do with the fact
that the computation kernel requires more data from one block than the
other blocks at a given iteration, so it is preferable to have a larger
block size to better amortize the overhead of calling a computation
kernel iteration.


 \documentclass[letterpaper,11pt]{article}
\usepackage[margin=2cm]{geometry}

\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{multicol}
\usepackage{listings}

\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\argmax}{arg\,max}

\title{\textbf{Pattern Recognition Theory 18794 Homework 1}}
\author{Mengwen He (Alex)}
\date{September 25, 2016}

\begin{document}
\maketitle

\section*{Problem 1}

\begin{enumerate}
\item A vector can be broken up into its direction and magnitude. The equation
$A\vec{v} = \vec{w}$ can seen as applying a linear transformation to vector $\vec{v}$ to get a
different vector $\vec{w}$. If $\vec{v}$ happens to be an eigenvector of A, describe what
happens to the direction of $\vec{v}$ after transformation?\\
\textbf{Answer:}\\
$\because$ $\vec{v}$ is an eigenvector of $A$\\
$\therefore$ $A \vec{v} = \lambda \vec{v}$\\
If the eigenvalue $\lambda > 0$, then the direction of $\vec{v}$ will not change.\\
If the eigenvalue $\lambda < 0$, then the direction of $\vec{v}$ will be inversed.\\
If the eigenvalue $\lambda = 0$, then the direction of $\vec{v}$ will to be an arbitrary direction (definition of the zero vector).

\item Given\\
\begin{equation}
\nonumber
A=\left[\begin{array}{ccc}
1 & 2 & 1 \\
2 & 4 & 2 \\
3 & 3 & 3 \\
\end{array}\right],~
B=\left[\begin{array}{cc}
-1 & 2 \\
2 & 2 \\
\end{array}\right],~
C=\left[\begin{array}{cccc}
2 & 8 & 1 & 0 \\
3 & 12 & 2 & 0 \\
2 & 8 & 3 & -1 \\
\end{array}\right],~
D=\left[\begin{array}{cccc}
2 & 8 & 1 & 1 \\
3 & 12 & 2 & 1 \\
2 & 8 & 3 & -1 \\
\end{array}\right]
\end{equation}
\begin{enumerate}
	\item What is the rank of each matrix?\\
	\textbf{Answer:}
	\begin{itemize}
		\item For the square matrix $A$:
		$$|A - \lambda I|=0 \Rightarrow \lambda^3 - 8 \lambda^2 + 6\lambda = 0 \Rightarrow \left\{\begin{array}{rcl}
		\lambda_1 & = & 4+\sqrt{10} \\
		\lambda_2 & = & 4-\sqrt{10} \\
		\lambda_3 & = & 0 \\
		\end{array}\right. \Rightarrow rank(A)=2$$
		\item For the square matrix $B$:
		$$|B - \lambda I|=0 \Rightarrow \lambda^2 - \lambda - 6 = 0 \Rightarrow
		\left\{\begin{array}{rcl}
		\lambda_1 & = & 3 \\
		\lambda_2 & = & -2 \\
		\end{array}\right. \Rightarrow rank(B)=2$$
		\item For the ``fat" matrix $C$:
		$$\left[\begin{array}{cccc}
		2 & 8 & 1 & 0 \\
		3 & 12 & 2 & 0 \\
		2 & 8 & 3 & -1 \\
		\end{array}\right] \xrightarrow{\begin{array}{c}
		R_2=R_2-\frac{3}{2}R_1 \\
		R_3=R_3-R_1\\
		\end{array}} \left[\begin{array}{cccc}
		2 & 8 & 1 & 0 \\
		0 & 0 & \frac{1}{2} & 0 \\
		0 & 0 & 2 & -1 \\
		\end{array}\right] $$
		$$\xrightarrow{\begin{array}{c}
		R_3 = R_3 - 4 R_2 \\
		\end{array}} \left[\begin{array}{cccc}
		2 & 8 & 1 & 0 \\
		0 & 0 & \frac{1}{2} & 0 \\
		0 & 0 & 0 & -1 \\
		\end{array}\right] \Rightarrow rank(C)=3$$
		\item For the ``fat" matrix $D$:
		$$\left[\begin{array}{cccc}
		2 & 8 & 1 & 1 \\
		3 & 12 & 2 & 1 \\
		2 & 8 & 3 & -1 \\
		\end{array}\right] \xrightarrow{\begin{array}{c}
		R_2=R_2-\frac{3}{2}R_1 \\
		R_3=R_3-R_1\\
		\end{array}} \left[\begin{array}{cccc}
		2 & 8 & 1 & 1 \\
		0 & 0 & \frac{1}{2} & -\frac{1}{2} \\
		0 & 0 & 2 & -2 \\
		\end{array}\right] $$
		$$\xrightarrow{\begin{array}{c}
		R_3 = R_3 - 4 R_2 \\
		\end{array}} \left[\begin{array}{cccc}
		2 & 8 & 1 & 1 \\
		0 & 0 & \frac{1}{2} & -\frac{1}{2} \\
		0 & 0 & 0 & 0 \\
		\end{array}\right] \Rightarrow rank(D)=2$$
	\end{itemize}

	\item What is the trace of each matrix?\\
	\textbf{Answer:}
	\begin{itemize}
		\item For square matrix $A$: $trace(A)=1+4+3=(4+\sqrt{10})+(4-\sqrt{10})+0=8$
		\item For square matrix $B$: $trace(B)=-1+2=3+(-2)=1$
		\item For ``fat" matrices $C$ and $D$: trace is only defined for square matrix.
	\end{itemize}
	\item What is the determinant of matrices $A$, $B$?\\
	\textbf{Answer:}
	\begin{itemize}
		\item For matrix $A$: $det(A)=(4+\sqrt{10})*(4-\sqrt{10})*0=0$
		\item For matrix $B$: $det(B)=3*(-2)=-6$
	\end{itemize}
	\item Find the eigenvalues and eigenvectors of $B$.\\
	\textbf{Answer:}\\
	From 2.(a), $B$'s eigenvalues are $\lambda_1=3,~\lambda_2=-2$\\
	Assume $B$'s eigenvectors are $\vec{v}_1=[v_{11},v_{12}]^T (v_{11} \geq 0),~\vec{v}_2=[v_{21},v_{22}]^T (v_{21} \geq 0)$
	$$\left[\begin{array}{cc}
	-1 & 2 \\
	2 & 2 \\
	\end{array}\right] \left[\begin{array}{cc}
	v_{11} & v_{21} \\
	v_{12} & v_{22} \\
	\end{array}\right] = \left[\begin{array}{cc}
	v_{11} & v_{21} \\
	v_{12} & v_{22} \\
	\end{array}\right] \left[\begin{array}{cc}
	\lambda_1 & 0 \\
	0 & \lambda_2 \\
	\end{array}\right] \Rightarrow \left\{\begin{array}{c}
	2v_{11}-v_{12}=0 \\
	v_{21}+2v_{22}=0 \\
	v_{11}^2+v_{12}^2=1 \\
	v_{21}^2+v_{22}^2=1 \\
	\end{array}\right. \Rightarrow \left\{\begin{array}{c}
	v_{11}=\frac{\sqrt{5}}{5} \\
	v_{12}=\frac{2\sqrt{5}}{5} \\
	v_{21}=\frac{\sqrt{5}}{5} \\
	v_{22}=-\frac{2\sqrt{5}}{5} \\
	\end{array}\right.$$
	Therefore $B$'s eigenvector are $\vec{v}_1=\left[\frac{\sqrt(5)}{5},\frac{2\sqrt{5}}{5}\right]^T ,~\vec{v}_2=\left[\frac{\sqrt(5)}{5},-\frac{2\sqrt{5}}{5}\right]^T$
	\item What is the dimension of the null space of each of the matrices?\\
	\textbf{Answer:}
	\begin{itemize}
		\item For the matrix $A$: $dim(Null(A))=dim(A)-rank(A)=3-2=1$
		\item For the matrix $B$: $dim(Null(B))=dim(B)-rank(B)=2-2=0$
		\item For the matrix $C$:
		\begin{itemize}
			\item Right null space: $dim(Null(C))=col\_dim(C)-rank(C)=4-3=1$
			\item Left null space: $dim(Null(C^T))=col\_dim(C^T)-rank(C^T)=3-3=0$
		\end{itemize}
		\item For the matrix $D$:
		\begin{itemize}
			\item Right null space: $dim(Null(D))=col\_dim(D)-rank(D)=4-2=2$
			\item Left null space: $dim(Null(D^T))=col\_dim(D^T)-rank(D^T)=3-2=1$
		\end{itemize}
	\end{itemize}
\end{enumerate}
\end{enumerate}

\newpage

\section*{Problem 2}

\begin{enumerate}
\item Prove that if $P$ is an orthogonal matrix, then $PP^T$ is a projection matrix.\\
\textbf{Answer:}
\begin{enumerate}
\item Method 1:\\
$\because P$ is an orthogonal matrix\\
$\therefore PP^T=I$\\
$\therefore PP^T$ is a projection matrix.
\item Method 2:\\
$\because P$ is an orthogonal matrix\\
$\therefore (PP^T)(PP^T)=P(P^TP)P^T=PP^T$\\
$\therefore PP^T$ is a projection matrix.
\end{enumerate}

\item Under what condition(s), for arbitrary non-zero matrices $A,B \in \mathbb{R}^{m \times n}$, do we have $rank(A + B) = rank(A) + rank(B)$? (Hint: Think singular values/vectors)\\
\textbf{Answer:}\\
$\because$ $dim(colsp([A,B]))=dim(colsp(A))+dim(colsp(B))-dim(colsp(A) \bigcap colsp(B))$\\
$\therefore$ $dim(colsp([A,B]))=rank(A)+rank(B)-dim(colsp(A) \bigcap colsp(B))$\\
$\because$ $colsp(A+B)$ is a subspace of $colsp([A,B])$\\
$\therefore$ $dim(colsp([A,B])) \geq dim(colsp(A+B)) = rank(A+B)$\\
$\therefore$ $rank(A)+rank(B) \geq rank(A+B) + dim(colsp(A) \bigcap colsp(B))$\\
If we have $rank(A+B)=rank(A)+rank(B)$,\\
then $dim(colsp(A) \bigcap colsp(B))=0 \Rightarrow colsp(A) \bigcap colsp(B)=\{\vec{0}\} \Rightarrow A^TB=0$\\
Therefore, we get a necessary condition as below:\\
For all matrices $A,B \in \mathbb{R}^{m \times n}$, $rank(A+B)=rank(A)+rank(B)$ $\Rightarrow$ $A^TB=0$.\\
So, if under $colsp(A) \bigcap colsp(B)=\{\vec{0}\}$ (strong) or $A^TB=0$ (weak), for all matrices $A,B \in \mathbb{R}^{m \times n}$, we can have $rank(A+B)=rank(A)+rank(B)$.

\item In general, does a relation exist between the rank of a sum of matrices and the sum of the ranks of the individual matrices?\\
\textbf{Answer:}\\
To find the relationship between $rank(\sum_{i=1}^{N}{A_i})$ and $\sum_{i=1}^{N}{rank(A_i)}$\\
From 2, we know that $rank(A+B) \leq rank(A)+rank(B)-dim(colsp(A) \bigcap colsp(B))$\\
Therefore, we gradually derive that relationship as below:\\
$rank(\sum_{i=1}^{N}{A_i}) \\
\leq rank(A_1) + rank(\sum_{i=2}^{N}{A_i}) - dim(colsp(A_1) \bigcap colsp(\sum_{i=2}^{N}{A_i}))$\\
$\leq rank(A_1) + rank(A_2) + rank(\sum_{i=3}^{N}{A_i}) - dim(colsp(A_2) \bigcap colsp(\sum_{i=3}^{N}{A_i})))$\\
$...$\\
$\leq \sum_{i=1}^{N}{rank(A_i)}$

\item Under the premise of problem 1.2, you arrived at a condition. Is the condition sufficient to ensure that $\sigma_{max} (A+B) = max (\sigma_{max} (A), \sigma_{max}(B))$? If not, what else do you need to constrain ? Here, $\sigma_{max} (A)$ is the maximum singular value of $A$\\
\textbf{Answer:}\\
$\because$ From the definition of matrix LSV-norm: $$||A||_2=\max_{\vec{x}:||\vec{x}||_2=1}{||A\vec{x}||_2}=\sigma_{max}(A)$$
$\therefore$ $$\sigma_{max}(A+B)=\max_{\vec{x}:||\vec{x}||_2=1}{||(A+B)\vec{x}||_2}$$
$\therefore$ $$\sigma_{max}^2(A+B)=\max_{\vec{x}:||\vec{x}||_2=1}{((A+B)\vec{x})^T((A+B)\vec{x})}$$
$$\sigma_{max}^2(A+B)=\max_{\vec{x}:||\vec{x}||_2=1}{((A\vec{x})^TA\vec{x}+(B\vec{x})^TB\vec{x}+2\vec{x}^T(A^TB)\vec{x})}$$
$\because$ we are under the condition $rank(A+B)=rank(A)+rank(B) \Rightarrow A^TB=0$ given by 2\\
$\therefore$ $$\sigma_{max}^2(A+B)=\max_{\vec{x}:||\vec{x}||_2=1}{((A\vec{x})^TA\vec{x}+(B\vec{x})^TB\vec{x})}$$
$$\sigma_{max}^2(A+B)=\max_{\vec{x}:||\vec{x}||_2=1}{(||A\vec{x}||_2^2+||B\vec{x}||_2^2)}$$
Assume:
$$\vec{x}_A^*=\argmax_{\vec{x}:||\vec{x}||_2=1}{||A\vec{x}||_2}$$
$$\vec{x}_B^*=\argmax_{\vec{x}:||\vec{x}||_2=1}{||B\vec{x}||_2}$$
$$\vec{x}^*=\argmax_{\vec{x}:||\vec{x}||_2=1}{||(A+B)\vec{x}||_2}$$

\begin{itemize}
\item If $\vec{x}^* = \vec{x}_A^*$, then $\sigma_{max}(A+B) = \sqrt{\sigma_{max}^2(A) + ||B\vec{x}_A^*||_2^2 } \geq \sigma_{max}(A)$
\item If $\vec{x}^* = \vec{x}_B^*$, then $\sigma_{max}(A+B) = \sqrt{||A\vec{x}_B^*||_2^2 + \sigma_{max}^2(B)} \geq \sigma_{max}(B)$
\end{itemize}
$\therefore$ with the condition $rank(A+B)=rank(A)+rank(B)$, we derived that $\sigma_{max}(A+B) \geq max (\sigma_{max} (A), \sigma_{max}(B))$\\
$\therefore$ $rank(A+B)=rank(A)+rank(B) \not \Rightarrow \sigma_{max}(A+B) = max (\sigma_{max} (A), \sigma_{max}(B))$ (it is not a sufficient condition)\\
However, we can add another condition that $rowsp(A) \bigcap rowsp(B) = \{\vec{0}\} \Rightarrow AB^T=0$\\
First we can guarantee that:
\begin{itemize}
\item If $\vec{x}^* = \vec{x}_A^* \in rowsp(A)$, then $\sigma_{max}(A+B) = \sqrt{\sigma_{max}^2(A) + ||B\vec{x}_A^*||_2^2 } = \sigma_{max}(A)$
\item If $\vec{x}^* = \vec{x}_B^* \in rowsp(B)$, then $\sigma_{max}(A+B) = \sqrt{||A\vec{x}_B^*||_2^2 + \sigma_{max}^2(B)} = \sigma_{max}(B)$
\end{itemize}
Then, we need to demonstrate that:
$$\forall\text{ unit }\vec{x} \in \mathbb{R}^m, \sqrt{||A\vec{x}||_2^2 + ||B\vec{x}||_2^2} \leq max(\sigma_{max}(A),\sigma_{max}(B))$$
Do SVD decomposition on any matrix $M \in \mathbb{R}^{n \times m}$: $M=U_MS_MV_M^T$\\
$\because$ For $M\vec{x}=U_MS_MV_M^T\vec{x}$, the unit $\vec{x}$ is first projected into $rowsp(V_M)$ by $V_M$, then scaled by $S_M$ along the bases in $rowsp(V_M)$, and finally projected into $colsp(U_M)$ (but actually into $colsp(M)$) by $U_M$.\\
So we define a $\hat{S}_M$ from $S_M$ as below:
$$\hat{S}_M^{ij}=\left\{\begin{array}{cl}
1, & S_M^{ij} \neq 0 \\
0, & S_M^{ij} = 0 \\
\end{array}\right.$$
And we define the energy projected into $colsp(A)$ from unit $\vec{x}$ by using $U_M\hat{S}_MV_M^T$ as $$E_M(\vec{x})=||U_M\hat{S}_MV_M^T\vec{x}||_2^2$$
By definition, we know that (assume $rowsp(A) \bigcap rowsp(B)=\{\vec{0}\}$):
\begin{itemize}
\item If unit $\vec{x} \in rowsp(A)$, then $E_A(\vec{x})=1$
\item If unit $\vec{x} \not \in rowsp(A)$, then $E_A(\vec{x})<1$
\item If unit $\vec{x} \in rowsp(B)$, then $E_A(\vec{x})=0$
\item If unit $\vec{x} \in rowsp([A,B])$, then $E_A(\vec{x})+E_B(\vec{x})=1$
\item If unit $\vec{x} \not \in rowsp([A,B])$, then $E_A(\vec{x})+E_B(\vec{x})<1$
\item If unit $\vec{x} = \vec{x}_A^*$, then $||A\vec{x}||_2^2=E_A(\vec{x})\sigma_{max}^2(A)$; otherwise, $||A\vec{x}||_2^2 \leq E_A(\vec{x})\sigma_{max}^2(A)$
\end{itemize}
$\therefore$
$$\forall \text{ unit }\vec{x} \in \mathbb{R}^{m},~||A\vec{x}||_2^2 + ||B\vec{x}||_2^2 \leq E_A(\vec{x})\sigma_{max}^2(A)+E_B(\vec{x})\sigma_{max}^2(B)$$
$$\leq (E_A(\vec{x})+E_B(\vec{x})) max(\sigma_{max}^2(A),\sigma_{max}^2(B)) \leq max(\sigma_{max}^2(A),\sigma_{max}^2(B))$$
$\therefore$ If $rank(A+B)=rank(A)+rank(B)$ and $rowsp(A) \bigcap rowsp(B) = \{\vec{0}\}$, then $\sigma_{max}(A+B) = max (\sigma_{max} (A), \sigma_{max}(B))$.

\item Let $A \in \mathbb{R}^{N \times N}$ be a full-rank matrix, with columns $\vec{a}_1,...,\vec{a}_N$. Prove that for every $\vec{\beta} \in \mathbb{R}^N$ with $\beta_i \geq 0$, $\sum_{i=1}^{N} \beta_i = 1$, we have $rank(A-Q) = N-1$. Where $Q = [A\vec{\beta}, A\vec{\beta}, ..., A\vec{\beta}]$ is formed by repeating the column vector ($A\vec{\beta}$), N times, i.e. $Q \in \mathbb{R}^{N \times N}$. (Hint: Try to scale the problem down and approach it from a geometric paradigm.)\\
\textbf{Answer:}\\
$\because \forall \vec{\beta} \in \mathbb{R}^N:~\beta_i \geq 0,~\sum_{i=1}^{N}{\beta_i}=1$\\
$\therefore P=\{\vec{\beta}:~\vec{\beta} \in \mathbb{R}^N,~\beta_i \geq 0,~\sum_{i=1}^{N}{\beta_i}=1\}$ forms a $(N-1)$-Dim hyperplane in $N$-Dim space.\\
$\therefore \exists \vec{\gamma}$,~$\forall \vec{\beta}_i \text{ and } \vec{\beta}_j \in P,~(\vec{\beta}_i-\vec{\beta}_j)\cdot\vec{\gamma}=0$\\
$\because A$ is a full-rank matrix\\
$\therefore$ after the affine transformation of $P$ by $A$, a transformed normal vector $\vec{\gamma}_A=(A^T)^{-1}\vec{\gamma}$:
$$A(\vec{\beta}_i-\vec{\beta}_j) \cdot \vec{\gamma}_A==(\vec{\beta}_i-\vec{\beta}_j)^T(A^T(A^T)^{-1})\vec{\gamma}=(\vec{\beta}_i-\vec{\beta}_j)\cdot\vec{\gamma}=0$$
Define $\vec{\beta}^i \in \mathbb{R}^N$ as below:
$$\vec{\beta}^i_j=\left\{\begin{array}{cl}
1 & j = i \\
0 & j \neq i \\
\end{array}\right.$$
$\therefore$ $\vec{\beta}^i \in  P$.\\
$\therefore \forall \vec{\beta} \in P$, and $\forall i \in \{1,2,\dots,N\}$
\begin{equation}
\nonumber
\begin{array}{rcl}
A(\vec{\beta}-\vec{\beta}^i) \cdot \vec{\gamma}_A & = & 0 \\
(A\vec{\beta}-A\vec{\beta}^i) \cdot \vec{\gamma}_A & = & 0 \\
(A\vec{\beta}-\vec{\alpha}_i) \cdot \vec{\gamma}_A & = & 0 \\
\end{array}
\end{equation}
$\therefore~\exists \text{~a non-zero } \vec{\gamma}_A \in \mathbb{R}^N,~(A-Q)^T\vec{\gamma}_A=0$\\
$\because P$ is $(N-1)$-Dim hyperplane,\\
$\therefore~rank(A-Q)=N-1$

\end{enumerate}

\newpage

\section*{Problem 3}

\begin{enumerate}
\item Find the maximum and minimum values of $f(x,y) = 81x^2 + y^2$ subject to the constraint $4x^2 + y^2 = 9$\\
\textbf{Answer:}
$$L(x,y,\lambda)=81x^2 + y^2-\lambda(9-4x^2-y^2)$$
\begin{equation}
\nonumber
\left\{\begin{array}{rcl}
\frac{\partial L}{\partial \lambda} & = & 4x^2 + y^2-9 =0 \\
\frac{\partial L}{\partial x} & = & (162+8\lambda)x=0 \\
\frac{\partial L}{\partial y} & = & (2+2\lambda)y=0 \\
\end{array}
\right. \Rightarrow \left\{\begin{array}{rcl}
\lambda & = & -1 \\
x & = & 0 \\
y & = & 3 \\
\end{array}
\right. \text{ or } \left\{\begin{array}{rcl}
\lambda & = & -\frac{81}{4} \\
x & = & \frac{3}{2} \\
y & = & 0 \\
\end{array}
\right.
\end{equation}
$\therefore$
$$max(f(x,y))=f(\frac{3}{2},0)=\frac{729}{4}$$
$$min(f(x,y))=f(0,3)=9$$

\item Find the maximum and minimum values of $f(x,y,z) = x + y + 2z$ on the surface $x^2 + y^2 + z^2 = 3$\\
\textbf{Answer:}
$$L(x,y,z,\lambda)=x + y + 2z - \lambda(3-x^2 - y^2 - z^2)$$
\begin{equation}
\nonumber
\left\{\begin{array}{rcl}
\frac{\partial L}{\partial \lambda} & = & x^2 + y^2 + z^2 -3 =0 \\
\frac{\partial L}{\partial x} & = & 2 \lambda x +1 =0 \\
\frac{\partial L}{\partial y} & = & 2 \lambda y +1 =0 \\
\frac{\partial L}{\partial z} & = & 2 \lambda z +2 =0 \\
\end{array}
\right. \Rightarrow \left\{\begin{array}{rcl}
\lambda & = & \frac{\sqrt{2}}{2} \\
x & = & -\frac{\sqrt{2}}{2} \\
y & = & -\frac{\sqrt{2}}{2} \\
z & = & -\sqrt{2} \\
\end{array}
\right. \text{ or } \left\{\begin{array}{rcl}
\lambda & = & -\frac{\sqrt{2}}{2} \\
x & = & \frac{\sqrt{2}}{2} \\
y & = & \frac{\sqrt{2}}{2} \\
z & = & \sqrt{2} \\
\end{array}
\right.
\end{equation}
$\therefore$
$$max(f(x,y,z))=f(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},\sqrt{2})=3\sqrt{2}$$
$$min(f(x,y,z))=f(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2},-\sqrt{2})=-3\sqrt{2}$$

\end{enumerate}

\newpage

\section*{Problem 6}

The two random variables $X$ and $Y$ are jointly Gaussian. Let
\begin{eqnarray}
\nonumber
X \sim \mathcal{N}(\mu_x,\sigma^2_x), \\
\nonumber
Y \sim \mathcal{N}(\mu_y,\sigma^2_y).
\end{eqnarray}
The correlation coefficient $\phi$ between $X$ and $Y$ is $0.5$. Two new PDFs are generated as follows,
\begin{eqnarray}
\nonumber
Z = X + Y, \\
\nonumber
W = 3X + 2Y.
\end{eqnarray}
Determine the decision boundaries of the two PDFs which give the minimum errors using $\mu_x = 0$, $\mu_y = 4$, $\sigma_x = 1$, $\sigma_y = 2$ for the cases below.\\
\textbf{Preparation:}\\
(1) Given:
$$X \sim \mathcal{N}(\mu_x, \sigma_x^2)$$
$$Y \sim \mathcal{N}(\mu_y, \sigma_y^2)$$
Then:
$$\mu_x=E(X)$$
$$\sigma_x^2=E((X-E(X))^2)=E(X^2)-E(X)^2$$
$$\sigma_{xy}^2=E((X-E(X))(Y-E(Y)))=E(XY)-E(X)E(Y)$$
$$\rho_{xy}=\frac{\sigma_{xy}^2}{\sigma_x \sigma_y}$$
(2) For $Z=X+Y$
$$\mu_z=E(X+Y)=E(X)+E(Y)=\mu_x+\mu_y=4$$
$$\sigma_z^2=E((X+Y)^2)-E(X+Y)^2=(E(X^2)-E(X)^2)+(E(Y^2)-E(Y)^2)+2(E(XY)-E(X)E(Y))$$
$$\sigma_z^2=\sigma_x^2+\sigma_y^2+2\rho_{xy}\sigma_x\sigma_y=7$$
For $W=3X+2Y$
$$\mu_w=E(3X+2Y)=3E(X)+2E(Y)=3\mu_x+2\mu_y=8$$
$$\sigma_w^2=E((3X+2Y)^2)-E(3X+2Y)^2=9(E(X^2)-E(X)^2)+4(E(Y^2)-E(Y)^2)+12(E(XY)-E(X)E(Y))$$
$$\sigma_w^2=9\sigma_x^2+4\sigma_y^2+12\rho_{xy}\sigma_x\sigma_y=37$$
(3) The decision boundary is derived by the following equation:
$$P(\omega_1)f(x|\omega_1)=P(\omega_2)f(x|\omega_2)$$
$$ln\left(\frac{f(x|\omega_1)}{f(x|\omega_2)}\right)=ln\left(\frac{P(\omega_2)}{P(\omega_1)}\right)$$
$$ln\left(\frac{\frac{1}{\sqrt{2 \pi \sigma_z^2}}{exp\left(-\frac{(x-\mu_z)^2}{2\sigma_z^2}\right)}}
{\frac{1}{\sqrt{2 \pi \sigma_w^2}}{exp\left(-\frac{(x-\mu_w)^2}{2\sigma_w^2}\right)}}\right)=ln\left(\frac{P(\omega_2)}{P(\omega_1)}\right)$$
$$ln\left(\frac{\sigma_w}{\sigma_z}\right)+\frac{(x-\mu_w)^2}{2\sigma_w^2}-\frac{(x-\mu_z)^2}{2\sigma_z^2}=ln\left(\frac{P(\omega_2)}{P(\omega_1)}\right)$$

\begin{enumerate}
\item $P(\omega_1)=P(\omega_2)$\\
\textbf{Answer:}\\
$$ln\left(\frac{\sigma_w}{\sigma_z}\right)+\frac{(x-\mu_w)^2}{2\sigma_w^2}-\frac{(x-\mu_z)^2}{2\sigma_z^2}=ln\left(\frac{P(\omega_2)}{P(\omega_1)}\right)$$
$$ln\left(\frac{\sqrt{37}}{\sqrt{7}}\right)+\frac{(x-8)^2}{2*37}-\frac{(x-4)^2}{2*7}=0$$
$$x1=-1.2898240621304666,~x2=7.4231573954638$$
$$\left\{\begin{array}{rcl}
x>x_1~and~x<x_2 & \Rightarrow & \omega_1 \\
x<x_1~or~x>x_2 & \Rightarrow & \omega_2 \\
\end{array}\right.$$

\item $P(\omega_1)=3P(\omega_2)$\\
\textbf{Answer:}\\
$$ln\left(\frac{\sigma_w}{\sigma_z}\right)+\frac{(x-\mu_w)^2}{2\sigma_w^2}-\frac{(x-\mu_z)^2}{2\sigma_z^2}=ln\left(\frac{P(\omega_2)}{P(\omega_1)}\right)$$
$$ln\left(\frac{\sqrt{37}}{\sqrt{7}}\right)+\frac{(x-8)^2}{2*37}-\frac{(x-4)^2}{2*7}=ln\left(\frac{1}{3}\right)$$
$$x1=-3.0935592751662626,~x2=9.226892608499595$$
$$\left\{\begin{array}{rclcl}
x>x_1 & and & x<x_2 & \Rightarrow & \omega_1 \\
x<x_1 & or & x>x_2 & \Rightarrow & \omega_2 \\
\end{array}\right.$$

\end{enumerate}

\newpage

\section*{Problem 7}

MATLAB. For each distribution pair below: Generate 400 samples from each of
the distributions. (Hint: Look at the randn documentation). Print a plot with
your samples. Use “x” for points from one distribution, and “o” for points from
the other distribution. Make the axis limits axis([-6 10 -8 8]). On your plot
printouts, draw the general shape (it does not need to be exact) of the decision
boundary between the distributions (assume equal priors). Do not compute the
decision boundary mathematically. Attach the plots and your .m file. Describe
how the decision boundary changes when the distributions do not have the same
covariance.

\begin{enumerate}
\item ~\\
\begin{equation}
\nonumber
x_1 \sim \mathcal{N} \left(\left[\begin{array}{c}
0 \\
0 \\
\end{array}\right], \left[\begin{array}{cc}
1 & 1 \\
1 & 8 \\
\end{array}\right] \right),~
x_2 \sim \mathcal{N} \left(\left[\begin{array}{c}
4 \\
0 \\
\end{array}\right], \left[\begin{array}{cc}
1 & 1 \\
1 & 8 \\
\end{array}\right]\right)
\end{equation}

\begin{itemize}
\item ~\\
\begin{center}
\includegraphics[width=0.5\textwidth]{./matlab/hw1_7_1.eps}
\end{center}
\item
\begin{verbatim}
figure
hold on;

mu1=[0;0];
sigma1=[1 1 ; 1 8];
r1=chol(sigma1);
x1=repmat(mu1,1,400)+r1'*randn(2,400);

mu2=[4;0];
sigma2=[1 1 ; 1 8];
r2=chol(sigma2);
x2=repmat(mu2,1,400)+r2'*randn(2,400);

plot(x1(1,:),x1(2,:),'rx',x2(1,:),x2(2,:),'bo');
axis([-6 10 -8 8]);
axis equal;

hold off;
\end{verbatim}
\end{itemize}

\newpage

\item ~\\
\begin{equation}
\nonumber
x_1 \sim \mathcal{N} \left(\left[\begin{array}{c}
0 \\
0 \\
\end{array}\right], \left[\begin{array}{cc}
1 & 1 \\
1 & 8 \\
\end{array}\right] \right),~
x_2 \sim \mathcal{N} \left(\left[\begin{array}{c}
4 \\
0 \\
\end{array}\right], \left[\begin{array}{cc}
2 & 0 \\
0 & 2 \\
\end{array}\right]\right)
\end{equation}
\end{enumerate}

\begin{itemize}
\item ~\\
\begin{center}
\includegraphics[width=0.5\textwidth]{./matlab/hw1_7_2.eps}
\end{center}
\item
\begin{verbatim}
figure
hold on;

mu1=[0;0];
sigma1=[1 1 ; 1 8];
r1=chol(sigma1);
x1=repmat(mu1,1,400)+r1'*randn(2,400);

mu2=[4;0];
sigma2=[2 0 ; 0 2];
r2=chol(sigma2);
x2=repmat(mu2,1,400)+r2'*randn(2,400);

plot(x1(1,:),x1(2,:),'rx',x2(1,:),x2(2,:),'bo');
axis([-6 10 -8 8]);
axis equal;

hold off;
\end{verbatim}
\end{itemize}

\newpage

\section*{Problem 8}

A two-dimensional feature vector is use to select one of two classes shown in
Figure 1. The joint PDFs are uniform over a square for class $\omega_1$ and a triangle
for class $\omega_2$. For each case, determine the probability of assigning $x$ to the wrong
class $\omega$ (compute $\epsilon_1$ and $\epsilon_2$), and the total probability of error (compute $P_e$).

\begin{center}
\includegraphics[width=0.7\textwidth]{./img/P8.png}
\end{center}

\begin{enumerate}
\item Case 1: $P(\omega_1)=P(\omega_2)$\\
\textbf{Answer:}\\
Define the square as $A$, and the triangle as $B$\\
Then the area of $A$, $S(A)=16$, and the area of $B$, $S(B)=16$.
$$\left\{\begin{array}{rcl}
f(\vec{x}|\omega_1) & = & \left\{\begin{array}{cc}
\frac{1}{16} & \vec{x} \in A\\
0 & \vec{x} \not \in A\\
\end{array}\right. \\
f(\vec{x}|\omega_2) & = & \left\{\begin{array}{cc}
\frac{1}{16} & \vec{x} \in B\\
0 & \vec{x} \not \in B\\
\end{array}\right. \\
\end{array}\right.$$

$$\left\{\begin{array}{rcl}
\epsilon_1 & = & f(\vec{x}|\omega_1)S(B-A)=\frac{1}{16}*8=1/2 \\
\epsilon_2 & = & f(\vec{x}|\omega_2)S(A-B)=\frac{1}{16}*8=1/2 \\
\end{array}\\
\right.$$

$$P_e=\epsilon_1 P(\omega_1)+\epsilon_2 P(\omega_2)=P(\omega_2)$$
\item Case 2: $P(\omega_1)=2P(\omega_2)$\\
\textbf{Answer:}

$$\left\{\begin{array}{rcl}
\epsilon_1 & = & f(\vec{x}|\omega_1)S(B-A)=\frac{1}{16}*8=1/2 \\
\epsilon_2 & = & f(\vec{x}|\omega_2)S(A-B)=\frac{1}{16}*8=1/2 \\
\end{array}\\
\right.$$

$$P_e=\epsilon_1 P(\omega_1)+\epsilon_2 P(\omega_2)=\frac{3}{2}P(\omega_2)$$

\end{enumerate}

\newpage

\section*{Problem 8}

\begin{enumerate}
	\item Write the name(s) of your project partner(s). Teams of 2 or 3 people.\\
	\textbf{Answer:} Iljoo Baek, Mengwen He (Alex)
	\item Describe in a few sentences, two ideas of possible projects that you want to pursue.\\
	\textbf{Answer:}
	\begin{enumerate}
		\item Use camera to detect road lane marks on a road.
		\item Track the vehicle's lane change behavior from the camera. (Main goal)
		\item Combine the low-precise GPS/IMU and concise digital map to decrease the global localization error. (Additional goal)
	\end{enumerate}
\end{enumerate}

\end{document}


\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[margin=1.0in]{geometry}
\usepackage{graphicx}
\usepackage{amsmath,amssymb,amsfonts,mathtools}
\setlength{\parindent}{0em}
\setlength{\parskip}{0.5em}
\usepackage{natbib}
\bibpunct{(}{)}{;}{a}{}{,}
\bibliographystyle{mnras}
\usepackage{hyperref}
\hypersetup{colorlinks=true,
linkcolor=blue,citecolor = blue}
\usepackage{textcomp}

\author{Baria Khan${}^{\dagger}$\\
Research Supervisors: Dr. Jenna Freudenburg and Prof. Abigail Crites${}^{\dagger}$\\
\it{${}^{\dagger}$ The Dunlap Institute for Astronomy \& Astrophysics}}
\title{\Large SURP CTA200H Project: Understanding the distribution of galaxies in the  low-to-mid-redshift universe }

\date{May 14, 2021}

\begin{document}

\maketitle
\section*{Introduction}

The goal of this project for CTA200 was to gain familiarity with the concepts of matter power spectrum and the 2-point correlation function and their utility in the context of large-scale galaxy surveys. These objectives serve as stepping stones for the SURP project planned for the summer that encompasses emulating semi-analytical models of the distribution of galaxies expected to be observed by TIME - the ionized carbon ([CII]) line intensity mapping instrument designed to investigate the epoch of ionization - to be cross-correlated with galaxies out to z $\sim$ 2 expected to be imaged by NASA's ROMAN infrared legacy survey, in an attempt to understand galactic evolution across a vast expanse of cosmic time.

\section*{Part I: Generating Matter Power Spectra with \texttt{camb}}

For Part I of the project, \texttt{camb}, an optimized cosmology code was used to generate linear and non-linear matter power spectra at redshifts of z = 0.5, 1.0, 1.5 and 2.0. A $\rm{\Lambda CDM}$ cosmological model was assumed for this purpose with values of parameters such as, $\rm{H_0}$ - Hubble Constant $\rm{[km\ s^{-1}\ {Mpc}^{-1}]}$, ${\Omega_b}$ - Physical baryon density parameter and
${\Omega_c}$ - Physical cold dark matter density parameter, taken to be as constrained by the first year of the Dark Energy Survey \citep{2018PhRvD..98d3526A}.

\begin{figure}[h!]
    \centering
     \includegraphics[width=0.75\textwidth]{PowerSpectra.pdf}
    \caption{Theoretical linear and non-linear matter power spectra at various values of z $\in$ [0.5,2], generated by \texttt{camb} for DES Y1 cosmology.}
    \label{fig:1}
\end{figure}

The power spectra are plotted in Figure \ref{fig:1}, where as one would expect, the peak power is higher at later redshifts, due to the highly-evolved structures present in the current universe. The matter power spectrum describes the power of the fluctuations in the density field that give rise to structure formation in the universe, as a function of spatial scale. In the very early universe, density fluctuations are amplified due to self-gravity. In the early radiation-dominated universe, these fluctuations evolve linearly due to the collapse of dark matter under Jean's instability criterion, since dark matter is unaffected by radiation pressure. Once perturbations on smaller scales become much denser than the mean density of the universe, the evolution of the matter power spectrum is better described by non-linear evolution \citep{2000eaa..bookE2136B}. As can be seen from Figure \ref{fig:1}, the differences between the linear and non-linear matter power spectra become significant around k $\rm{\sim 0.3 h^{-1} {Mpc}^{-1}}$, where h = 0.685 [$\rm{H_0} / 100\ {km}^{-1} s^{-1} {Mpc}^{-1}$]. Moreover, the turnover in the matter power spectrum pointing to the damping of fluctuations in the distribution of dark matter at smaller scales is attributed to photons diffusing out of dense regions after recombination.

\newpage

Another quantity that is better obtained through galaxy surveys and which contains essentially the same information as a power spectrum is the 2-point correlation function or the autocorrelation function. The power spectrum is the Fourier transform of the autocorrelation function $\rm{\xi (r)}$;

\begin{gather*}
    \mathrm{\xi(r) = \int{\frac{d^3k}{(2{\pi}^3)}P(k)e^{\overrightarrow{k}\cdot{\overrightarrow{r}}}}}
    \ \ \rm{(\href{https://en.wikipedia.org/wiki/Matter_power_spectrum}{Matter\ Power\ Spectrum})}
\end{gather*}

The autocorrelation function is defined as the excess probability, compared with that expected for a random distribution, of locating a galaxy a distance r away from another \citep{2000eaa..bookE2136B}. Galaxy correlation functions obtained from imaging surveys thus describe the extent of clustering of galaxies, and can be compared to the matter power spectrum predicted by cosmological theories of structure formation, since larger and numerous galaxies are expected to be present in the denser regions of the universe. As commented in the script, regrettably, I was not able to obtain the correlation functions for the matter power spectra plotted above by relatively straightforward attempts of inverse Fourier transformation; however, it's a problem I continue to work on and hope to resolve soon.

\section*{Part II: Understanding Galaxy Data}

This part was based on exploring the COSMOS2015 galaxy catalog - a data product from the Cosmic Evolution Survey (COSMOS) of a 2 $\rm{{deg}^2}$ area of the sky, that offers a wealth of high-quality information on over half a million galaxies observed by various multi-wavelength instruments within 1 $\leq$ z $\leq$ 6 \citep{2016ApJS..224...24L}. In the script developed, the \texttt{astropy.table} package was used for catalogue manipulation. As mentioned in the data release' official \href{https://irsa.ipac.caltech.edu/data/COSMOS/gator_docs/uvista_dr2.1.pdf}{documentation}, although the entire catalog contains information on 1,182,108 sources, an optimal sample of 536,077 galaxies, flagged to be present in the area imaged by \href{http://ultravista.org/}{UltraVISTA} within the COSMOS field and thereby having the best available data, is selected. Some of the notable out of the 536 columns in the catalog include: columns for general parameters such as the ALPHA\_J2000 and DELTA\_J2000 columns indicating the object's Right Ascension [deg] $\&$ Declination [deg], respectively. The \#\_FLUX, \#\_FLUXERR, \#\_MAG, \#\_MAGERROR columns include the Optical, NIR, X-Ray $\&$ Radio VLA photometery results i.e. fluxes, magnitudes and their corresponding errors, from various instruments eg. MIPS, IRAC, SPIRE/HERMES, Chandra etc. ZPDF is the best-fit photometric redshift, and other properties derived from best-fit galaxy templates include M\_\# - absolute magnitude in various filters, MASS\_\# - mass, SFR\_\# - star-formation rate and L\_\# - luminosity.

Since the SURP project involves looking at galaxies within voxels of redshifts that can be imaged by ROMAN and whose CII emission can be traced by TIME, high accuracy of photometric redshifts is of vital importance. The COSMOS survey includes precise redshift estimates of the galaxies within, and the  distribution of the galaxy redshifts is plotted in Figure \ref{fig:2}, with (apparently negligible, if plotted correctly) error bars taken to be the standard error of mean of redshifts in each histogram bin.

\begin{figure}[h!]
    \centering
     \includegraphics[width=0.75\textwidth]{Z_hist.pdf}
    \caption{Distribution of photometric redshifts of the sampled COSMOS2015 galaxies. }
    \label{fig:2}
\end{figure}

\newpage

Evidently, more than half of the $\sim$ 53,000 galaxies in the sample are at redshifts less than 1, however, more than about 10,000 galaxies are at redshifts larger than 4.

Since the COSMOS2015 catalog provides redshifts and positions of the galaxies within a known region of the sky, an observed galaxy power spectrum can be estimated, which we would reasonably expect to differ to a certain extent from the relevant model power spectrum, since galaxies are biased tracers of matter distribution. One such (likely flawed) thought on how to proceed with doing such an analysis is that  obtaining the galaxy power spectrum and hence the correlation function could require estimating the number density of galaxies as a function of comoving distance (r) perhaps, in certain redshift bins, where the comoving distances can be determined by adopting the DES Y1 cosmology as used for modelling the matter power spectra and using their available photometric redshifts and spatial positions in the sky coordinates. By my understanding, the power spectrum is directly related to the number density field of the galaxies and thus, obtaining the latter first could be useful. \cite{1994ApJ...426...23F},  \cite{1994ApJ...431..569P}, \cite{1999MNRAS.308..289S} and \cite{2005MNRAS.362..505C} are a few relevant papers that describe methods of power-spectrum analysis of redshift surveys, to be reviewed hereafter to develop a more well-informed analysis strategy of the sort required.

\bibliography{biblio.bib}
\end{document}


\documentclass[11pt]{article}
\usepackage{amsmath,amssymb}

\author{Paul Hewson}
\title{Lab. notes for MANOVA}

\usepackage{/usr/share/R/share/texmf/Sweave}
\begin{document}
\maketitle

\section{Manova practical}

The first exercise is based on an example in Johnson and Wichern (2002), page 344 exercise 6.24, although we seem to have a little more data than they do (we have five time periods, including -200 BCE and 150 CE).   This is rather a famous data set mentioned in a large number of multivariate text books.   Essentially, we have data on Maximum Breadth, Basal Height, Basal Length and Nasal Height for skulls recovered from a number of different time periods.   Our interest is in whether the skulls have changed shape and size over time (Johnson and Wichern suggest this may be due to interbreeding with other populations.

Data have been placed in the portal.

First, read in the data:

\begin{Schunk}
\begin{Sinput}
>  es <- read.csv("egyptian-skulls.csv")
\end{Sinput}
\end{Schunk}

You should be able to do scatterplots of each group by now!   Look at some of code for the cluster analysis if you're not sure.   What other eda would be useful for these data?

One slight annoyance is that we have to have our response data as a matrix, and our group memberships as a factor

\begin{Schunk}
\begin{Sinput}
> Y <- as.matrix(es[,-5])
> Year <- as.factor(es$Year)
\end{Sinput}
\end{Schunk}

More information on the \texttt{manova} command can be found on the helpfile in \textbf{R}.   This happens to be quite a simple model to fit:

\begin{Schunk}
\begin{Sinput}
> es.manova <- manova(Y ~ Year)
\end{Sinput}
\end{Schunk}

And you can get the model fit simply by using \texttt{summary()} (with an optional call to \texttt{test} to give you one of the four fit criteria. i.e. \texttt{test = "Pillai"} or \texttt{test = "Wilks"} or \texttt{test = "Hotelling-Lawley"} or \texttt{test = "Roy"}.   What is the default test statistic?   Why do you think that one has been chosen?

\begin{Schunk}
\begin{Sinput}
> summary(es.manova, test = "Wilks")
\end{Sinput}
\end{Schunk}

You can also consider the various univariate ANOVAs as follows:

\begin{Schunk}
\begin{Sinput}
> summary.aov(es.manova)
\end{Sinput}
\end{Schunk}

\begin{itemize}
\item Why didn't we just do a whole set of univariate ANOVA?
\end{itemize}

\section{A two way MANOVA}

The data on plastic film extrusion is rather famous (and is actually in the R helpfile for summary.manova) and is worked as an example in Johnson and Wichern (page 312, Example 6.11).  You can load the data from the Rex.csv file in the portal.

\begin{Schunk}
\begin{Sinput}
> rex <- read.csv("Rex.csv")
\end{Sinput}
\end{Schunk}

Again, you may find it useful to conduct a simple eda with these data before you carry out with the analysis.

Do note this time that we will wrap all the Y variables in an \texttt{as.matrix} wrapper.   For the first call, we have fitted an interaction term between extrusion rate and additive levels:

\begin{Schunk}
\begin{Sinput}
> rex.manova <- manova(as.matrix(rex[,c(1:3)]) ~ rex$rate*rex$additive)
> summary(rex.manova)
\end{Sinput}
\end{Schunk}

Look at the diagnostics using Wilk's lambda.   Do you need an interaction term?

You may find it useful to compare the results of this analysis with the one worked through in Johnson and Wichern (2002).

\end{document}


\title{Data Types, Functions, and Programming}
\titlerunning{Data Types, Functions, and Programming}
\toctitle{Data Types, Functions, and Programming}
\author{Daniel R. Grayson%
        \thanks{Supported by NSF grant DMS 99-70085.}
        % \inst 1
   \and Michael E. Stillman%
        %\inst 2
        %\fnmsep
        \thanks{Supported by NSF grant 99-70348.}}
\authorrunning{D. R. Grayson and M. E. Stillman}
% \institute{University of Illinois at Urbana-Champaign, Department of
%   Mathematics, Urbana, IL 61801, USA
%       \and Cornell University, Department of Mathematics, Ithaca, NY 14853, USA}

\maketitle

\begin{abstract}
  In this chapter we present an introduction to the structure of \Mtwo
  commands and the writing of functions in the \Mtwo language.  For further details
  see the \Mtwo manual distributed with the program~\cite{M2}.
\end{abstract}

\section{Basic Data Types}

The basic \ie{data types} of \Mtwo include numbers of various types (integers,
rational numbers, floating point numbers, complex numbers), lists (basic
lists, and three types of visible lists, depending on the delimiter used),
hash tables, strings of characters (both 1-dimensional and 2-dimensional),
Boolean values (true and false), symbols, and functions.  Higher level types
useful in mathematics are derived from these basic types using facilities
provided in the \Mtwo language.  Except for the simplest types (integers and
Boolean values), \Mtwo normally displays the type of the output value on a
second labeled output line.

Symbols\index{symbols} have a name which consists of letters, digits, or apostrophes, the
first of which is a letter.  Values can be assigned to symbols and recalled
later.
\beginOutput
i1 : w\\
\emptyLine
o1 = w\\
\emptyLine
o1 : Symbol\\
\endOutput
\beginOutput
i2 : w = 2^100\\
\emptyLine
o2 = 1267650600228229401496703205376\\
\endOutput
\beginOutput
i3 : w\\
\emptyLine
o3 = 1267650600228229401496703205376\\
\endOutput
Multiple values can be assigned in parallel.
\beginOutput
i4 : (w,w') = (33,44)\\
\emptyLine
o4 = (33, 44)\\
\emptyLine
o4 : Sequence\\
\endOutput
\beginOutput
i5 : w\\
\emptyLine
o5 = 33\\
\endOutput
\beginOutput
i6 : w'\\
\emptyLine
o6 = 44\\
\endOutput
Comments are initiated by {\tt --} and extend to the end of the line.
\beginOutput
i7 : (w,w') = (33,   -- this is a comment\\
\               44)\\
\emptyLine
o7 = (33, 44)\\
\emptyLine
o7 : Sequence\\
\endOutput
Strings\index{strings} of characters are delimited by quotation marks.
\beginOutput
i8 : w = "abcdefghij"\\
\emptyLine
o8 = abcdefghij\\
\endOutput
They may be joined horizontally to make longer strings, or vertically to make
a two-dimensional version called a {\sl net}.
\beginOutput
i9 : w | w\\
\emptyLine
o9 = abcdefghijabcdefghij\\
\endOutput
\beginOutput
i10 : w || w\\
\emptyLine
o10 = abcdefghij\\
\      abcdefghij\\
\endOutput
Nets are used in the preparation of two dimensional output for polynomials.

Floating point numbers are distinguished from integers by the presence of a
decimal point, and rational numbers are entered as fractions.
\beginOutput
i11 : 2^100\\
\emptyLine
o11 = 1267650600228229401496703205376\\
\endOutput
\beginOutput
i12 : 2.^100\\
\emptyLine
o12 = 1.26765 10^30\\
\emptyLine
o12 : RR\\
\endOutput
\beginOutput
i13 : (36 + 1/8)^6\\
\emptyLine
\      582622237229761\\
o13 = ---------------\\
\           262144\\
\emptyLine
o13 : QQ\\
\endOutput
Parentheses, braces, and brackets are used as delimiters for the three types
of {\em \ie{visible lists}}: \ie{lists}, \ie{sequences}, and \ie{arrays}.
\beginOutput
i14 : x1 = \{1,a\}\\
\emptyLine
o14 = \{1, a\}\\
\emptyLine
o14 : List\\
\endOutput
\beginOutput
i15 : x2 = (2,b)\\
\emptyLine
o15 = (2, b)\\
\emptyLine
o15 : Sequence\\
\endOutput
\beginOutput
i16 : x3 = [3,c,d,e]\\
\emptyLine
o16 = [3, c, d, e]\\
\emptyLine
o16 : Array\\
\endOutput
Even though they use braces, lists should not be confused with sets, which will be treated later.
A double period can be used to construct a sequence of consecutive elements in various contexts.
\beginOutput
i17 : 1 .. 6\\
\emptyLine
o17 = (1, 2, 3, 4, 5, 6)\\
\emptyLine
o17 : Sequence\\
\endOutput
\beginOutput
i18 : a .. f\\
\emptyLine
o18 = (a, b, c, d, e, f)\\
\emptyLine
o18 : Sequence\\
\endOutput
Lists can be nested.
\beginOutput
i19 : xx = \{x1,x2,x3\}\\
\emptyLine
o19 = \{\{1, a\}, (2, b), [3, c, d, e]\}\\
\emptyLine
o19 : List\\
\endOutput
The number of entries in a list is provided by {\tt \#}.
\beginOutput
i20 : #xx\\
\emptyLine
o20 = 3\\
\endOutput
The entries in a list are numbered starting with 0, and can be recovered with
{\tt \#} used as a binary operator.
\beginOutput
i21 : xx#0\\
\emptyLine
o21 = \{1, a\}\\
\emptyLine
o21 : List\\
\endOutput
\beginOutput
i22 : xx#0#1\\
\emptyLine
o22 = a\\
\emptyLine
o22 : Symbol\\
\endOutput
We can join visible lists and {\sl append} or {\sl prepend} an element to a visible
list.  The output will be the same type of visible list that was provided in
the input: a list, a sequence, or an array; if the arguments are various
types of lists, the output will be same type as the first argument.
\beginOutput
i23 : join(x1,x2,x3)\\
\emptyLine
o23 = \{1, a, 2, b, 3, c, d, e\}\\
\emptyLine
o23 : List\\
\endOutput
\index{lists!joining}\indexcmd{join}%
\beginOutput
i24 : append(x3,f)\\
\emptyLine
o24 = [3, c, d, e, f]\\
\emptyLine
o24 : Array\\
\endOutput
\index{lists!appending}\indexcmd{append}%
\beginOutput
i25 : prepend(f,x3)\\
\emptyLine
o25 = [f, 3, c, d, e]\\
\emptyLine
o25 : Array\\
\endOutput
\index{lists!prepending}\indexcmd{prepend}%
Use {\tt sum} or {\tt product} to produce the sum or product of all the
elements in a list.
\beginOutput
i26 : sum \{1,2,3,4\}\\
\emptyLine
o26 = 10\\
\endOutput
\indexcmd{sum}%
\beginOutput
i27 : product \{1,2,3,4\}\\
\emptyLine
o27 = 24\\
\endOutput
\indexcmd{product}%

\section{Control Structures}

Commands for later execution are encapsulated in {\sl \ie{functions}}.  A function
is created using the operator {\tt -\char`\>}\indexcmd{->} to separate the parameter or
sequence of parameters from the code to be executed later.  Let's try an
elementary example of a function with two arguments.
\beginOutput
i28 : f = (x,y) -> 1000 * x + y\\
\emptyLine
o28 = f\\
\emptyLine
o28 : Function\\
\endOutput
The parameters {\tt x} and {\tt y} are symbols that will acquire a
value later when the function is executed.  They are {\sl local}\index{variables!local} in the sense that
they are completely different from any symbols with the same name that occur
elsewhere.  Additional local variables for use within the body of a function
can be created by assigning a value to them with {\tt \char`\:\char`\=}
\indexcmd{:=}
(first time only).  We illustrate this by rewriting the function above.
\beginOutput
i29 : f = (x,y) -> (z := 1000 * x; z + y)\\
\emptyLine
o29 = f\\
\emptyLine
o29 : Function\\
\endOutput
Let's apply the function to some arguments.
\beginOutput
i30 : f(3,7)\\
\emptyLine
o30 = 3007\\
\endOutput
The sequence of arguments can be assembled first, and then passed to the
function.
\beginOutput
i31 : s = (3,7)\\
\emptyLine
o31 = (3, 7)\\
\emptyLine
o31 : Sequence\\
\endOutput
\beginOutput
i32 : f s\\
\emptyLine
o32 = 3007\\
\endOutput
As above, functions receiving one argument may be called without parentheses.
\beginOutput
i33 : sin 2.1\\
\emptyLine
o33 = 0.863209\\
\emptyLine
o33 : RR\\
\endOutput
A compact notation for functions makes it convenient to apply them
without naming them first.  For example, we may use \indexcmd{apply}{\tt apply} to apply a
function to every element of a list and to collect the results into a list.
\beginOutput
i34 : apply(1 .. 10, i -> i^3)\\
\emptyLine
o34 = (1, 8, 27, 64, 125, 216, 343, 512, 729, 1000)\\
\emptyLine
o34 : Sequence\\
\endOutput
The function \indexcmd{scan}{\tt scan} will do the same thing, but discard the results.
\beginOutput
i35 : scan(1 .. 5, print)\\
1\\
2\\
3\\
4\\
5\\
\endOutput
Use \indexcmd{if}{\tt if ... then ... else ...} to perform alternative actions
based on the truth of a condition.
\beginOutput
i36 : apply(1 .. 10, i -> if even i then 1000*i else i)\\
\emptyLine
o36 = (1, 2000, 3, 4000, 5, 6000, 7, 8000, 9, 10000)\\
\emptyLine
o36 : Sequence\\
\endOutput
A function can be terminated prematurely with \indexcmd{return}{\tt return}.
\beginOutput
i37 : apply(1 .. 10, i -> (if even i then return 1000*i; -i))\\
\emptyLine
o37 = (-1, 2000, -3, 4000, -5, 6000, -7, 8000, -9, 10000)\\
\emptyLine
o37 : Sequence\\
\endOutput
Loops in a program can be implemented with \indexcmd{while}{\tt while ... do ...}.
\beginOutput
i38 : i = 1; while i < 50 do (print i; i = 2*i)\\
1\\
2\\
4\\
8\\
16\\
32\\
\endOutput
Another way to implement loops is with \indexcmd{for}{\tt for} and
\indexcmd{do}{\tt do} or {\tt list}, with optional
clauses introduced by the keywords {\tt from}, {\tt to}, and {\tt when}.
\beginOutput
i40 : for i from 1 to 10 list i^3\\
\emptyLine
o40 = \{1, 8, 27, 64, 125, 216, 343, 512, 729, 1000\}\\
\emptyLine
o40 : List\\
\endOutput
\beginOutput
i41 : for i from 1 to 4 do print i\\
1\\
2\\
3\\
4\\
\endOutput
A loop can be terminated prematurely with \indexcmd{break}{\tt break}, which accepts an
optional value to return as the value of the loop expression.
\beginOutput
i42 : for i from 2 to 100 do if not isPrime i then break i\\
\emptyLine
o42 = 4\\
\endOutput
If no value needs to be returned, the condition for continuing can be
provided with the keyword {\tt when}; iteration continues only as long as the
predicate following the keyword returns {\tt true}.
\beginOutput
i43 : for i from 2 to 100 when isPrime i do print i\\
2\\
3\\
\endOutput

% if then else, while do, for, break, return,

\section{Input and Output}

The function \indexcmd{print}{\tt print} can be used to display something on the screen.
\beginOutput
i44 : print 2^100\\
1267650600228229401496703205376\\
\endOutput
For example, it could be used to display the elements of a list on separate
lines.
\beginOutput
i45 : (1 .. 5) / print;\\
1\\
2\\
3\\
4\\
5\\
\endOutput
The operator {\tt <<} can be used to display something on the
screen, without the newline character.
\beginOutput
i46 : << 2^100\\
1267650600228229401496703205376\\
o46 = stdio\\
\emptyLine
o46 : File\\
\emptyLine
\  --  the standard input output file\\
\endOutput
Notice the value returned is a {\em file}\index{files}.  A {\em file} in \Mtwo is a data type that
represents a channel through which data can be passed, as input, as
output, or in both directions.  The file \indexcmd{stdio}{\tt stdio} encountered above
corresponds to your shell window or terminal, and is used for two-way
communication between the program and the user.  A file may correspond
to what one usually calls a file, i.e., a sequence of data bytes associated
with a given name and stored on
your disk drive.  A file may also correspond to a {\em socket}, a
channel for communication with other programs over the network.

Files can be used with the binary form
of the operator {\tt \char`\<\char`\<} to display something else on the same
line.
\beginOutput
i47 : << "the value is : " << 2^100\\
the value is : 1267650600228229401496703205376\\
o47 = stdio\\
\emptyLine
o47 : File\\
\emptyLine
\  --  the standard input output file\\
\endOutput
Using \indexcmd{endl}{\tt endl} to represent the new line character or character sequence, we can
produce multiple lines of output.
\beginOutput
i48 : << "A = " << 2^100 << endl << "B = " << 2^200 << endl;\\
A = 1267650600228229401496703205376\\
B = 1606938044258990275541962092341162602522202993782792835301376\\
\endOutput
We can send the same output to a disk file named {\tt foo}, but we must remember to
close it with {\tt close}.
\beginOutput
i49 : "foo" << "A = " << 2^100 << endl << close\\
\emptyLine
o49 = foo\\
\emptyLine
o49 : File\\
\endOutput
The contents of the file can be recovered as a string with \indexcmd{get}{\tt get}.
\beginOutput
i50 : get "foo"\\
\emptyLine
o50 = A = 1267650600228229401496703205376\\
\emptyLine
\endOutput
If the file contains valid \Mtwo commands, as it does in this case, we can
execute those commands with \indexcmd{load}{\tt load}.
\beginOutput
i51 : load "foo"\\
\endOutput
We can verify that the command took effect by evaluating {\tt A}.
\beginOutput
i52 : A\\
\emptyLine
o52 = 1267650600228229401496703205376\\
\endOutput
Alternatively, if we want to see those commands and the output they produce,
we may use \indexcmd{input}{\tt input}.
\beginOutput
i53 : input "foo"\\
\emptyLine
i54 : A = 1267650600228229401496703205376\\
\emptyLine
o54 = 1267650600228229401496703205376\\
\emptyLine
i55 : \endOutput
Let's set up a ring for computation in \Mtwo.
\beginOutput
i56 : R = QQ[x,y,z]\\
\emptyLine
o56 = R\\
\emptyLine
o56 : PolynomialRing\\
\endOutput
\index{ring!making one}%
\beginOutput
i57 : f = (x+y)^3\\
\emptyLine
\       3     2        2    3\\
o57 = x  + 3x y + 3x*y  + y\\
\emptyLine
o57 : R\\
\endOutput
Printing, and printing to files, works for polynomials,
too.\index{printing!to a file}%
\beginOutput
i58 : "foo" << f << close;\\
\endOutput
The two-dimensional output is readable by humans, but is not easy to convert
back into a polynomial.
\beginOutput
i59 : get "foo"\\
\emptyLine
o59 =  3     2        2    3\\
\      x  + 3x y + 3x*y  + y\\
\endOutput
Use \indexcmd{toString}{\tt toString} to create a 1-dimensional form of the polynomial
that can be stored in a file in a format readable by \Mtwo and by other
symbolic algebra programs, such as {\em Mathematica} or {\em Maple}.
\beginOutput
i60 : toString f\\
\emptyLine
o60 = x^3+3*x^2*y+3*x*y^2+y^3\\
\endOutput
Send it to the file.
\beginOutput
i61 : "foo" << toString f << close;\\
\endOutput
Get it back.
\beginOutput
i62 : get "foo"\\
\emptyLine
o62 = x^3+3*x^2*y+3*x*y^2+y^3\\
\endOutput
Convert the string back to a polynomial with \indexcmd{value}{\tt value}, using \indexcmd{oo}{\tt oo} to
recover the value of the expression on the previous line.
\beginOutput
i63 : value oo\\
\emptyLine
\       3     2        2    3\\
o63 = x  + 3x y + 3x*y  + y\\
\emptyLine
o63 : R\\
\endOutput
The same thing works for matrices, and a little more detail is provided by
\indexcmd{toExternalString}{\tt toExternalString}, if needed.
\beginOutput
i64 : vars R\\
\emptyLine
o64 = | x y z |\\
\emptyLine
\              1       3\\
o64 : Matrix R  <--- R\\
\endOutput
\beginOutput
i65 : toString vars R\\
\emptyLine
o65 = matrix \{\{x, y, z\}\}\\
\endOutput
\beginOutput
i66 : toExternalString vars R\\
\emptyLine
o66 = map(R^\{\{0\}\}, R^\{\{-1\}, \{-1\}, \{-1\}\}, \{\{x, y, z\}\})\\
\endOutput

\section{Hash Tables}

Recall how one sets up a quotient ring for computation in \Mtwo.
\beginOutput
i67 : R = QQ[x,y,z]/(x^3-y)\\
\emptyLine
o67 = R\\
\emptyLine
o67 : QuotientRing\\
\endOutput
\beginOutput
i68 : (x+y)^4\\
\emptyLine
\        2 2       3    4           2\\
o68 = 6x y  + 4x*y  + y  + x*y + 4y\\
\emptyLine
o68 : R\\
\endOutput
How does \Mtwo represent a ring like $R$ in the computer?  To answer that,
first think about what sort of information needs to be retained about $R$.
We may need to remember the coefficient ring of $R$, the names of the
variables in $R$, the monoid of monomials in the variables, the degrees of
the variables, the characteristic of the ring, whether the ring is
commutative, the ideal modulo which we are working, and so on.  We also may
need to remember various bits of code: the code for performing the basic
arithmetic operations, such as addition and multiplication, on elements of
$R$; the code for preparing a readable representation of an element of $R$,
either 2-dimensional (with superscripts above the line and subscripts below),
or 1-dimensional.  Finally, we may want to remember certain things that take
a lot of time to compute, such as the Gr\"obner basis of the ideal.

A {\sl \ie{hash table}} is, by definition, a way of representing (in the computer)
a function whose domain is a finite set.  In \Mtwo, hash tables are extremely
flexible: the elements of the domain (or {\sl keys\index{keys of a hash table}}) and the elements of the
range (or {\sl values\index{values of a hash table}}) of the function may be any of the other objects
represented in the computer.  It's easy to come up with uses for functions
whose domain is finite: for example, a monomial can be represented by the
function that associates to a variable its nonzero exponent; a polynomial can
be represented by a function that associates to a monomial its nonzero
coefficient; a set can be represented by any function with that set as its
domain; a (sparse) matrix can be represented as a function from pairs of
natural numbers to the corresponding nonzero entry.

Let's create a hash table and name it.
\beginOutput
i69 : f = new HashTable from \{ a=>444, Daniel=>555, \{c,d\}=>\{1,2,3,4\}\}\\
\emptyLine
o69 = HashTable\{\{c, d\} => \{1, 2, 3, 4\}\}\\
\                a => 444\\
\                Daniel => 555\\
\emptyLine
o69 : HashTable\\
\endOutput
The operator \indexcmd{=>}{\tt \char`\=\char`\>} is used to represent a key-value pair.
We can use the operator \indexcmd{\#}{\tt \#} to recover the value from the key.
\beginOutput
i70 : f#Daniel\\
\emptyLine
o70 = 555\\
\endOutput
\beginOutput
i71 : f#\{c,d\}\\
\emptyLine
o71 = \{1, 2, 3, 4\}\\
\emptyLine
o71 : List\\
\endOutput
If the key is a symbol, we can use the operator \indexcmd{.}{\tt .} instead; this is
convenient if the symbol has a value that we want to ignore.
\beginOutput
i72 : Daniel = a\\
\emptyLine
o72 = a\\
\emptyLine
o72 : Symbol\\
\endOutput
\beginOutput
i73 : f.Daniel\\
\emptyLine
o73 = 555\\
\endOutput
We can use \indexcmd{\#?}{\tt \#?} to test whether a given key occurs in the hash table.
\beginOutput
i74 : f#?a\\
\emptyLine
o74 = true\\
\endOutput
\beginOutput
i75 : f#?c\\
\emptyLine
o75 = false\\
\endOutput
Finite sets are implemented in \Mtwo as hash tables: the elements of the set
are stored as the keys in the hash table, with the accompanying values all
being $1$.  (Multisets are implemented by using values larger than $1$, and
are called {\sl tallies}.)
\beginOutput
i76 : x = set\{1,a,\{4,5\},a\}\\
\emptyLine
o76 = Set \{\{4, 5\}, 1, a\}\\
\emptyLine
o76 : Set\\
\endOutput
\indexcmd{set}%
\beginOutput
i77 : x#?a\\
\emptyLine
o77 = true\\
\endOutput
\beginOutput
i78 : peek x\\
\emptyLine
o78 = Set\{\{4, 5\} => 1\}\\
\          1 => 1\\
\          a => 1\\
\endOutput
\beginOutput
i79 : y = tally\{1,a,\{4,5\},a\}\\
\emptyLine
o79 = Tally\{\{4, 5\} => 1\}\\
\            1 => 1\\
\            a => 2\\
\emptyLine
o79 : Tally\\
\endOutput
\beginOutput
i80 : y#a\\
\emptyLine
o80 = 2\\
\endOutput
We might use \indexcmd{tally}{\tt tally} to tally how often a function attains its various
possible values.  For example, how often does an integer have 3 prime
factors?  Or 4?  Use \indexcmd{factor}{\tt factor} to factor an integer.
\beginOutput
i81 : factor 60\\
\emptyLine
\       2\\
o81 = 2 3*5\\
\emptyLine
o81 : Product\\
\endOutput
Then use {\tt \#} to get the number of factors.
\beginOutput
i82 : # factor 60\\
\emptyLine
o82 = 3\\
\endOutput
Use {\tt apply} to list some values of the function.
\beginOutput
i83 : apply(2 .. 1000, i -> # factor i)\\
\emptyLine
o83 = (1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2,  $\cdot\cdot\cdot$\\
\emptyLine
o83 : Sequence\\
\endOutput
Finally, use {\tt tally} to summarize the results.
\beginOutput
i84 : tally oo\\
\emptyLine
o84 = Tally\{1 => 193\}\\
\            2 => 508\\
\            3 => 275\\
\            4 => 23\\
\emptyLine
o84 : Tally\\
\endOutput

Hash tables turn out to be convenient entities for storing odd bits and
pieces of information about something in a way that's easy to think about and
use.
In \Mtwo, rings are represented as hash tables, as are ideals, matrices,
modules, chain complexes, and so on.  For example, although it isn't a
documented feature, the key {\tt ideal} is used to preserve the ideal that
was used above to define the quotient ring {\tt R}, as part of the
information stored in {\tt R}.
\beginOutput
i85 : R.ideal\\
\emptyLine
\             3\\
o85 = ideal(x  - y)\\
\emptyLine
o85 : Ideal of QQ [x, y, z]\\
\endOutput
The preferred and documented way for a user to recover this information is
with the function \indexcmd{ideal}{\tt ideal}.
\beginOutput
i86 : ideal R\\
\emptyLine
\             3\\
o86 = ideal(x  - y)\\
\emptyLine
o86 : Ideal of QQ [x, y, z]\\
\endOutput
Users who want to introduce a new high-level mathematical
concept to \Mtwo may learn about hash tables by referring to the \Mtwo manual
\cite{M2}.

\section{Methods}

You may use the \texttt{code}\indexcmd{code} command to locate the source code for a given
function, at least if it is one of those functions written in the \Mtwo
language.  For example, here is the code for {\tt demark}, which may be used
to put commas between strings in a list.
\beginOutput
i87 : code demark\\
\emptyLine
o87 = -- ../../../m2/fold.m2:23\\
\      demark = (s,v) -> concatenate between(s,v)\\
\endOutput
The code for tensoring a ring map with a module can be displayed in this way.
\beginOutput
i88 : code(symbol **, RingMap, Module)\\
\emptyLine
o88 = -- ../../../m2/ringmap.m2:294-298\\
\      RingMap ** Module := Module => (f,M) -> (\\
\           R := source f;\\
\           S := target f;\\
\           if R =!= ring M then error "expected module over source ring";\\
\           cokernel f(presentation M));\\
\endOutput
\indexcmd{symbol}
The code implementing the {\tt ideal} function when applied to a
quotient ring can be displayed as follows.
\beginOutput
i89 : code(ideal, QuotientRing)\\
\emptyLine
o89 = -- ../../../m2/quotring.m2:7\\
\      ideal QuotientRing := R -> R.ideal\\
\endOutput
Notice that it uses the key {\tt ideal} to extract the information from the
ring's hash table, as you might have guessed from the previous discussion.
The bit of code displayed above may be called a {\sl method}\index{method} as
a way of indicating that several methods for dealing with various types of
arguments are attached to the function named {\tt ideal}.  New such {\sl
  method functions} may be created with the function {\tt method}\indexcmd{method}.  Let's
illustrate that with an example: we'll write a function called {\tt denom}
which should produce the denominator of a rational number.  When applied to
an integer, it should return 1.  First we create the method function.
\beginOutput
i90 : denom = method();\\
\endOutput
Then we tell it what to do with an argument from the class {\tt QQ} of rational numbers.
\beginOutput
i91 : denom QQ := x -> denominator x;\\
\endOutput
And also what to do with an argument from the class {\tt ZZ} of integers.
\beginOutput
i92 : denom ZZ := x -> 1;\\
\endOutput
Let's test it.
\beginOutput
i93 : denom(5/3)\\
\emptyLine
o93 = 3\\
\endOutput
\beginOutput
i94 : denom 5\\
\emptyLine
o94 = 1\\
\endOutput

\section{Pointers to the Source Code}

A substantial part of \Mtwo is written in the same language provided to the
users.  A good way to learn more about the \Mtwo language is to peruse the
source code that comes with the system in the directory {\tt Macaulay2/m2}.
Use the {\tt code} function, as described in the previous section, for
locating the bit of code you wish to view.

The source code for the interpreter of the \Mtwo language is in the directory
{\tt Macaulay2/d}.  It is written in another language designed to be mostly
type-safe, which is translated into {\tt C} by the translator whose own
{\tt C} source code is in the directory {\tt Macaulay2/c}.  Here is a
sample line of code from the file {\tt Macaulay2/d/tokens.d}, which shows how
the translator provides for allocation and initialization of dynamic data
structures.
\par
\vskip 5 pt
\begingroup
\tteight
\baselineskip=8pt
\lineskip=0pt
\obeyspaces
globalFrame := Frame(dummyFrame,globalScope.seqno,Sequence(nullE));\leavevmode\hss\endgraf
\endgroup
\penalty-1000
\par
\vskip 1 pt
\noindent
And here is the {\tt C} code produced by the translator.
\par
\vskip 5 pt
\begingroup
\tteight
\baselineskip=\outputBaseLineSkip
\lineskip=0pt
\obeyspaces
\obeylines
tokens\char`\_Frame tokens\char`\_globalFrame;
tokens\char`\_Frame tmp\char`\_\char`\_23;
Sequence tmp\char`\_\char`\_24;
tmp\char`\_\char`\_24 = (Sequence) GC\char`\_MALLOC(sizeof(struct S259\char`\_)+(1-1)*sizeof(Expr));
if (0 == tmp\char`\_\char`\_24) outofmem();
tmp\char`\_\char`\_24->len\char`\_ = 1;
tmp\char`\_\char`\_24->array\char`\_[0] = tokens\char`\_nullE;
tmp\char`\_\char`\_23 = (tokens\char`\_Frame) GC\char`\_MALLOC(sizeof(struct S260\char`\_));
if (0 == tmp\char`\_\char`\_23) outofmem();
tmp\char`\_\char`\_23->next = tokens\char`\_dummyFrame;
tmp\char`\_\char`\_23->scopenum = tokens\char`\_globalScope->seqno;
tmp\char`\_\char`\_23->values = tmp\char`\_\char`\_24;
tokens\char`\_globalFrame = tmp\char`\_\char`\_23;
\endgroup
\penalty-1000
\par
\vskip 1 pt

The core algebraic algorithms constitute the {\sl engine} of \Mtwo and are
written in {\tt C\char`\+\char`\+}, with the source files in the directory
{\tt Macaulay2/e}.  In the current version of the program, the interface
between the interpreter and the core algorithms consists of a single
two-directional stream of bytes.  The manual that comes with the system
\cite{M2} describes the engine communication protocol used in that
interface.

%% Mike says we don't need this:

%% One feature of \Mtwo which should be understood by system administrators, at
%% least of Unix systems, is the {\tt dumpdata} routine.  Since so much of the
%% source code is written in the interpreted \Mtwo language, and the task of
%% reading and parsing it can take an appreciable amount of time on slower
%% machines, we perform that task in advance (by the {\tt setup} command) when
%% the program is installed on your system.  The results are saved by writing
%% all of the data areas of the memory of the program to disk.  When a user runs
%% {\tt M2}, the file containing the saved contents is quickly mapped into
%% memory using the {\tt loaddata} command.  The contents of memory depend
%% in a detailed way on the architecture of the computer and the shareable
%% libraries that are installed, so when those things change, the {\tt setup}
%% command may need to be run again.  If those things change too often, the
%% system administrator may disable that feature.

%% Date: Tue, 21 Nov 2000 11:36:16 +0900 (JST)
%% From: Nobuki Takayama <taka@math.kobe-u.ac.jp>
%% To: dan@math.uiuc.edu
%% CC: mike@polygon.math.cornell.edu, taka@math.kobe-u.ac.jp
%% In-reply-to: <200011161511.JAA18405@orion.math.uiuc.edu> (dan@math.uiuc.edu)
%% Subject: Re: software
%%
%% Dear Dan;
%% I would like to know
%%   (1) "hints" to read and understand the M2 source code
%% and
%%   (2) what are new ideas in the M2 implementations.
%% The source code contains all ideas in detail, but it is not easy to
%% read and understand it without some hints.
%%
%% I look forward to reading the section and the source code with the help
%% of the section.
%% Nobuki
%%
%% >    If you want to add one subsection at the end, it could be something
%% >    like "Developers' notes" or "System Issues". The reader I have in mind
%% >
%% >    chapter is just the place. If you wish to consider this, then I suggest
%% >    that Mike communicate directly with Nobuki about the contents.
%% >
%% >Could you tell us directly whay you have in mind about that?
%%


\chapter*{Abstract}

Superblocks must work. Given the current status of homogeneous
configurations, security experts particularly desire the simulation of
802.11b. We consider how the Internet can be applied to the refinement
of Scheme.

\chapter{Introduction}

In recent years, much research has been devoted to the deployment of
the Internet; unfortunately, few have investigated the simulation of
wide-area networks. In this position paper, we disconfirm the
understanding of the World Wide Web. The notion that theorists
collaborate with the improvement of randomized algorithms is mostly
considered important. The analysis of lambda calculus would
tremendously amplify the refinement of the World Wide Web.

\begin{align*}
    f(x) &= x^2\\
    \mathbf{ g(x) } &= \frac{1}{x}\\
     F(x)  &= \int^a_b \frac{1}{3}x^3\\
\end{align*}

We disconfirm that the much-touted certifiable algorithm for the
construction of online algorithms by Lee and Davis runs in
$\Theta$($n^2$) time. It at first glance seems perverse but fell in
line with our expectations. Existing lossless and cooperative
heuristics use superblocks to deploy DHCP. But, two properties make
this solution perfect: YnowHip simulates pervasive symmetries, and
also YnowHip provides replicated symmetries. This combination of
properties has not yet been improved in prior work.

An important approach to fix this quagmire is the emulation of
telephony. Contrarily, 802.11 mesh networks \autocite{cite:0} might not be
the panacea that biologists expected. Although conventional wisdom
states that this quandary is never addressed by the emulation of
Internet QoS, we believe that a different approach is necessary. The
effect on cyberinformatics of this discussion has been significant.
Clearly, our heuristic controls reinforcement learning.

Our main contributions are as follows. We describe an approach for
Internet QoS ({YnowHip}), verifying that hierarchical databases can
be made wearable, robust, and concurrent \autocite{cite:0}. We argue that
Moore's Law and write-back caches are entirely incompatible.

The rest of the paper proceeds as follows. To begin with, we motivate
the need for sensor networks. Further, we place our work in context
with the prior work in this area. Finally, we conclude.

\chapter{Framework}

In this section, we introduce a design for analyzing gigabit switches.
Figure~\ref{dia:label0} details our framework's probabilistic
deployment. We consider a framework consisting of $n$
digital-to-analog converters. Figure~\ref{dia:label0} shows an
analysis of e-business. The question is, will YnowHip satisfy all of
these assumptions? Yes, but with low probability.

\begin{figure}[t]
\centerline{\epsfig{figure=dia0.eps}}
\caption{\small{
New stable symmetries.
}}
\label{dia:label0}
\end{figure}

Suppose that there exists multimodal theory such that we can easily
study write-ahead logging. Any typical evaluation of Scheme will
clearly require that SMPs can be made pervasive, psychoacoustic, and
mobile; our approach is no different. Despite the results by P. U.
Williams, we can confirm that web browsers can be made event-driven,
homogeneous, and heterogeneous. Therefore, the design that YnowHip uses
is solidly grounded in reality. Despite the fact that it is entirely a
confirmed goal, it has ample historical precedence.

\begin{figure}[t]
\centerline{\epsfig{figure=dia1.eps}}
\caption{\small{
The decision tree used by our framework.
}}
\label{dia:label1}
\end{figure}

Suppose that there exists heterogeneous configurations such that we can
easily deploy kernels. While experts usually assume the exact opposite,
our approach depends on this property for correct behavior.
Figure~\ref{dia:label0} diagrams YnowHip's mobile analysis. This may or
may not actually hold in reality. We consider an algorithm consisting
of $n$ 32 bit architectures. Continuing with this rationale, we
consider an algorithm consisting of $n$ systems. Despite the fact that
electrical engineers entirely assume the exact opposite, YnowHip
depends on this property for correct behavior. Figure~\ref{dia:label1}
details a diagram plotting the relationship between our methodology and
von Neumann machines. Thus, the methodology that YnowHip uses holds for
most cases.

\chapter{Implementation}

Though many skeptics said it couldn't be done (most notably Wilson and
Moore), we describe a fully-working version of our system. Furthermore,
it was necessary to cap the instruction rate used by our framework to 30
pages. The hand-optimized compiler contains about 403 instructions of
ML. our solution is composed of a client-side library, a collection of
shell scripts, and a homegrown database. Similarly, the collection of
shell scripts contains about 52 instructions of Dylan \cite{cite:0}.
Overall, YnowHip adds only modest overhead and complexity to previous
adaptive systems.

\chapter{Results and Analysis}

Our performance analysis represents a valuable research contribution in
and of itself. Our overall evaluation strategy seeks to prove three
hypotheses: (1) that hash tables no longer toggle system design; (2)
that the partition table no longer toggles performance; and finally (3)
that interrupt rate stayed constant across successive generations of
Commodore 64s. Our evaluation strives to make these points clear:

\javacode{sample/binaryconverter.java}

\section{Hardware and Software Configuration}

\begin{figure}[t]
\centerline{\epsfig{figure=figure0.eps,width=3in}}
\caption{\small{
The 10th-percentile power of YnowHip, as a function of sampling rate.
}}
\label{fig:label0}
\end{figure}

A well-tuned network setup holds the key to an useful evaluation
method. We scripted a deployment on DARPA's underwater cluster to prove
the contradiction of pseudorandom electrical engineering. This follows
from the emulation of IPv4. To begin with, we doubled the optical drive
speed of our desktop machines. With this change, we noted duplicated
performance degredation. Similarly, we added more CPUs to Intel's
1000-node overlay network to understand our system. This configuration
step was time-consuming but worth it in the end. Along these same
lines, we removed some floppy disk space from our mobile telephones.

\begin{figure}[t]
\centerline{\epsfig{figure=figure1.eps,width=3in}}
\caption{\small{
The expected work factor of our methodology, compared with the other
heuristics.
}}
\label{fig:label1}
\end{figure}

Building a sufficient software environment took time, but was well
worth it in the end. We added support for YnowHip as a Markov runtime
applet. We added support for YnowHip as a computationally independent
dynamically-linked user-space application. Furthermore, Third, all
software components were linked using GCC 9.8 linked against multimodal
ibraries for investigating cache coherence. We note that other
researchers have tried and failed to enable this functionality.

\section{Experiments and Results}

Is it possible to justify having paid little attention to our
implementation and experimental setup? Yes, but only in theory. With
these considerations in mind, we ran four novel experiments: (1) we
measured optical drive space as a function of ROM throughput on an IBM
PC Junior; (2) we measured hard disk speed as a function of flash-memory
space on an UNIVAC; (3) we asked (and answered) what would happen if
computationally stochastic fiber-optic cables were used instead of
checksums; and (4) we dogfooded YnowHip on our own desktop machines,
paying particular attention to tape drive space. We discarded the
results of some earlier experiments, notably when we measured tape drive
throughput as a function of tape drive speed on a Motorola bag
telephone.

Now for the climactic analysis of experiments (3) and (4) enumerated
above. While it is never an extensive aim, it is derived from known
results. Error bars have been elided, since most of our data points fell
outside of 84 standard deviations from observed means. Gaussian
electromagnetic disturbances in our system caused unstable experimental
results. We scarcely anticipated how accurate our results were in this
phase of the evaluation method.

Shown in Figure~\ref{fig:label1}, the first two experiments call
attention to YnowHip's seek time. Note that sensor networks have less
discretized effective flash-memory speed curves than do hardened access
points. Second, note how rolling out spreadsheets rather than emulating
them in software produce more jagged, more reproducible results. Along
these same lines, the many discontinuities in the graphs point to
duplicated block size introduced with our hardware upgrades.

Lastly, we discuss experiments (1) and (3) enumerated above. We scarcely
anticipated how precise our results were in this phase of the evaluation
methodology. Note that web browsers have less discretized flash-memory
space curves than do modified kernels. Note how rolling out Web
services rather than deploying them in a chaotic spatio-temporal
environment produce less jagged, more reproducible results
\autocite{cite:1}.

\chapter{Related Work}

Our solution is related to research into metamorphic methodologies,
low-energy technology, and relational technology \autocite{cite:2}. Without
using the study of context-free grammar, it is hard to imagine that
SMPs and evolutionary programming are entirely incompatible. The
original solution to this problem by Wang was well-received;
unfortunately, this finding did not completely fulfill this mission
\autocite{cite:3}. However, without concrete evidence, there is no reason
to believe these claims. Similarly, \cite{cite:4} developed a
similar methodology, however we disproved that our algorithm follows a
Zipf-like distribution \autocite{cite:5, cite:6, cite:7}. A litany of
prior work supports our use of randomized algorithms. YnowHip also
manages courseware, but without all the unnecssary complexity. In
general, our application outperformed all prior methodologies in this
area \autocite{cite:8}.

Our solution is related to research into the UNIVAC computer, reliable
information, and robots \autocite{cite:9, cite:10, cite:11}. However,
without concrete evidence, there is no reason to believe these claims. The
choice of expert systems in \cite{cite:6} differs from ours in that we
synthesize only structured configurations in our algorithm
\autocite{cite:12, cite:13}. Recent work by Allen Newell et al. suggests a
solution for enabling the UNIVAC computer, but does not offer an
implementation \autocite{cite:14, cite:15, cite:16, cite:17, cite:18}. Our
approach represents a significant advance above this work. \cite{cite:19}
presented several decentralized methods, and reported that they have
profound influence on the study of local-area networks \autocite{cite:20}.
This is arguably ill-conceived. In general, YnowHip outperformed all prior
solutions in this area. Performance aside, our algorithm visualizes more
accurately.

\cite{cite:10} and \cite{cite:21} explored the first known instance of
Scheme \autocite{cite:22}. In our research, we solved all of the issues
inherent in the previous work. \cite{cite:23} and \cite{cite:24} introduced
the first known instance of empathic archetypes. \cite{cite:25} developed a
similar framework, nevertheless we argued that our methodology runs in
$\Theta$($n^2$) time. Our approach to Web services differs from that of
\cite{cite:26} as well.

\chapter{Conclusions}

In this paper we demonstrated that Scheme and write-ahead logging are
rarely incompatible. To achieve this ambition for kernels, we described a
novel framework for the construction of virtual machines. Our system can
successfully control many compilers at once. The visualization of
e-commerce is more significant than ever, and our methodology helps
end-users do just that.