% Copyright 2020 Pascal Philipp

%    Licensed under the Apache License, Version 2.0 (the "License");
%    you may not use this file except in compliance with the License.
%    You may obtain a copy of the License at

%        http://www.apache.org/licenses/LICENSE-2.0

%    Unless required by applicable law or agreed to in writing, software
%    distributed under the License is distributed on an "AS IS" BASIS,
%    WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
%    See the License for the specific language governing permissions and
%    limitations under the License.

\documentclass[12pt,a4paper]{report}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{graphicx}
\usepackage{enumerate}
\usepackage[table]{xcolor}
\usepackage{verbatim}
\usepackage[top=1.2in, bottom=1.2in, left=1.2in, right=1.2in]{geometry}
%\usepackage[notref]{showkeys}

\usepackage{tikz}
\newcommand{\tikzmark}[2]{
	\tikz[overlay,remember picture, baseline]
	\node[anchor=base] (#1) {$#2$};}

\usepackage{thmtools}
\renewcommand{\listtheoremname}{List of Applications}
\renewcommand\thmtformatoptarg[1]{: #1}

\usepackage{enotez}
\let\footnote=\endnote
\setenotez{list-name={Hints and Answers}, totoc=chapter}

\theoremstyle{definition}
\newtheorem{theorem}{Theorem}[chapter]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{properties}[theorem]{Properties}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{exercise*}[theorem]{Exercise*}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{sketch}[theorem]{Sketch}
\newtheorem*{application}{Application}

\newcommand{\figbox}[1]{
\begin{center}
	\fbox{\parbox[b][2.5in][t]{0.5\textwidth}{{\scriptsize{#1}}}}
\end{center}}
\newcommand*\rfrac[2]{{}^{#1}\!/_{#2}}
\newcommand{\wtd}[1]{\widetilde{#1}}
\newcommand{\abs}[1]{ \left| #1 \right| }
\newcommand{\rank}{\text{rank}\:}
\renewcommand{\d}{\mathrm{d}}
\newcommand{\e}{\mathrm{e}}
\newcommand{\df}[2]{\displaystyle{\frac{\partial #1}{\partial #2}}}
\newcommand{\ddf}[2]{\displaystyle{\frac{\partial^2 #1}{\partial #2^2}}}
\newcommand{\dff}[3]{\displaystyle{\frac{\partial^2 #1}{\partial{#2}\partial{#3}}}}
\newcommand{\Df}[2]{\displaystyle{\frac{\d #1}{\d #2}}}
\newcommand{\Ddf}[2]{\displaystyle{\frac{\d^2 #1}{\d #2^2}}}
\newcommand{\itgr}[4]{\int_{#1}^{#2} #3 \: \d #4}
\newcommand{\eval}[3]{\left. #1 \, \right|_{#2}^{#3}}

\setcounter{tocdepth}{2}

\title{A Compact Course on Mathematical Methods}
\author{Pascal Philipp}

\begin{document}

\maketitle

\newgeometry{top=1.5in, bottom=1.0in, left=0.7in, right=0.8in}
\noindent
The notes at hand cover the following topics:

\begin{enumerate}
\item Vectors and Matrices
\item Functions of Several Variables
\item Integration
\item Differential Equations
\end{enumerate}

The main prerequiste that is required for being able to work through the
text is familiarity with functions of a single variable and differentiation.
Chapter 2 is then the natural follow-up to that prerequisite.
Differentiation for functions of several variables and applications
such as finding minima or maxima are covered.
Chapter 3 introduces integration; first for functions of one variable and then
for functions of several variables.
That second part, multivariate integration, is rather brief and more of a
quick taster and introduction of concepts rather than a systematic treatment.
Chapter 4 then does the same for differential equations.
You could call these three chapters 'advanced calculus'.

The material in chapter 1, vectors and matrices, isn't usually taught together
with calculus -- it is covered here because it is the other important basic
mathematical methods topic needed in STEM
(with calculus being the first; actually, there is a third such topic:
probability and statistics, which is not covered, but I'd be open to starting
collaborations to add a compact chapter on it).
Chapter 1 isn't a prerequisite for the other chapters, but it makes some of
the multivariate notation prettier.

These notes are application-oriented and the philosophy is to
cover the basic concepts quickly, go through a good amount of examples and
exercises, and then move on. No need to go down rabbit holes.
That's where the 'compact' in the title comes from.
Despite focus on methods for applied computations and solving problems,
a few more theoretical remarks and exercises can be found in the text.
Elements of abstract mathematical rigour can be found as well -- I tried to
do give a taste of this way of thinking in a friendly way.

\bigskip
This document belongs to the GitHub repository
\begin{center}
\texttt{https://github.com/pasc85/MathematicalMethods}
\end{center}
You can find the license there as well as information on how to contribute.

\vfill
\noindent
Copyright 2020 Pascal Philipp. Subject to Apache License 2.0.

\thispagestyle{empty}
\newpage
\restoregeometry

\newgeometry{top=0.8in, bottom=1.0in, left=1.1in, right=1.1in}
\tableofcontents
\thispagestyle{empty}
\newpage
\listoftheorems[ignoreall,show={application}]
\thispagestyle{empty}
\restoregeometry

\addtocontents{toc}{\protect\thispagestyle{empty}}

\chapter{Vectors and Matrices}
\label{ch:vm}
\input{Chapters/ch_vm.tex}

\chapter{Functions of Several Variables}
\label{ch:fsv}
\input{Chapters/ch_fsv.tex}

\chapter{Integration}
\label{ch:i}
\input{Chapters/ch_i.tex}

\chapter{Differential Equations}
\label{ch:de}
\input{Chapters/ch_de.tex}

\printendnotes

\end{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% CS240: Programming in C
% Copyright 2016 Pejman Ghorbanzade <pejman@ghorbanzade.com>
% Creative Commons Attribution-ShareAlike 4.0 International License
% https://github.com/ghorbanzade/UMB-CS240-2016S/blob/master/LICENSE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def \topDirectory {.}
\def \resDirectory {\topDirectory/src/c/main/ls10}
\def \texDirectory {\topDirectory/src/tex}
\def \styDirectory {\texDirectory/sty}
\def \cfgDirectory {\texDirectory/cfg}
\def \imgDirectory {\texDirectory/img}

\documentclass[compress]{beamer}
%\mode<presentation>
%\usetheme{default}

\usepackage{\styDirectory/directives}
\input{\cfgDirectory/config}
\usepackage{\styDirectory/beamerthemePejman}
\doc{number}{10}
%\setbeamertemplate{footline}[text line]{}

\begin{document}

\prepareCover

\section{Pointers}

\begin{slide}
	\begin{block}{Definition}

	A pointer is a variable that contains the address of a variable.

	\end{block}
	\begin{block}{Motivation}

	Using pointers improves efficiency, clarity and simplicity of the program.
	Pointers are key to constructing more advanced data structures.

	\end{block}
\end{slide}

\begin{slide}
	\begin{block}{Address of a Variable}

	We can access the address of a variable using the unary operator \alert{\texttt{\&}}.

	\inputminted[
		fontsize=\footnotesize,
		firstline=14,
		lastline=16,
		linenos
	]{c}{\resDirectory/point1.c}

	\begin{terminal}
	value of var1: 3
	address of var1: 0x7ffce5183f2c
	\end{terminal}

	\end{block}
\end{slide}

\begin{slide}
	\begin{block}{Declaring a Pointer}

	As for any variable, a pointer variable needs to be declared before use.
	The general form of a pointer variable declaration is as follows.

	\begin{terminal}
	@*\textit{type}*@ *var;
	\end{terminal}

	\end{block}
\end{slide}

\begin{slide}
	\begin{block}{Using a Pointer}

	We can access the object a pointer points to, using the unary operator \alert{\texttt{*}}.

	\inputminted[
		fontsize=\footnotesize,
		firstline=14,
		lastline=16,
		linenos
	]{c}{\resDirectory/point2.c}

	\begin{terminal}
	value of  var1: 3
	value of  var2: 0x7fff0a5d9ed4
	address of var1: 0x7fff0a5d9ed4
	address of var2: 0x7fff0a5d9ed8
	\end{terminal}

	\end{block}
\end{slide}

\begin{slide}
	\begin{block}{Dereferencing a Pointer}

	Dereferencing a pointer \texttt{ptr} allows access to the value of the variable to which \texttt{ptr} points.

	\inputminted[
		fontsize=\footnotesize,
		firstline=14,
		lastline=16,
		linenos
	]{c}{\resDirectory/point2.c}

	\begin{terminal}
	value of var1: 3
	value of *var2: 3
	address of var1: 0x7fff0a5d9ed4
	value of var2: 0x7fff0a5d9ed4
	\end{terminal}

	\end{block}
\end{slide}

\begin{slide}
	\begin{block}{Dereferencing a Pointer}

	A pointer can be dereferenced using the unary operator \alert{\texttt{*}}.

	\inputminted[
		fontsize=\footnotesize,
		firstline=14,
		lastline=17,
		linenos
	]{c}{\resDirectory/point2.c}

	\begin{terminal}
	value of var1: 3
	value of *var2: 3
	value of var3: 3
	\end{terminal}

	\end{block}
\end{slide}

\begin{slide}
	\begin{block}{Quiz}

	What is the output of the following code snippet?

	\inputminted[
		fontsize=\footnotesize,
		firstline=10,
		linenos
	]{c}{\resDirectory/point3.c}

	\pause

	\begin{terminal}
	value of var1: 4
	\end{terminal}

	\end{block}
\end{slide}

\subsection{Pointers as Function Arguments}

\begin{slide}
	\begin{block}{Problem Statement}

	\begin{itemize}
	\item[] All function arguments are passed by value.
	\item[] Altering an external variable in a calling function is not \textit{directly} possible.
	\item[] A function may not return more than one value.
	\end{itemize}

	\end{block}
\end{slide}

\begin{slide}
	\begin{block}{Proposed Solution}

	Instead of passing value of the variable to be altered, pass a pointer to that variable.

	\end{block}
\end{slide}

\begin{slide}
	\begin{block}{Quiz}

	Write a function with the following signature to swap values of two given variables.

	\begin{terminal}
	void swap(int *a, int *b);
	\end{terminal}

	\pause

	\inputminted[
		fontsize=\footnotesize,
		firstline=12,
		lastline=17,
		linenos
	]{c}{\resDirectory/swap.c}

	\end{block}
\end{slide}

\begin{slide}
	\begin{block}{Quiz}

	Write a program \texttt{swap.c} that uses a function with the following signature to swap values of two given variables.

	\begin{terminal}
	void swap(int *a, int *b);
	\end{terminal}

	\pause

	\inputminted[
		fontsize=\footnotesize,
		firstline=19,
		linenos
	]{c}{\resDirectory/swap.c}

	\end{block}
\end{slide}

\begin{slide}
	\begin{block}{Exercise}

	Write a function with the following signature to convert a string into an integer.
	The function returns zero if the conversion is possible and $-1$, otherwise.

	\begin{terminal}
	int str2int(char str[], int *num);
	\end{terminal}

	\end{block}
\end{slide}

\begin{slide}
	\begin{block}{Solution}

	\inputminted[
		fontsize=\scriptsize,
		firstline=14,
		lastline=31,
		linenos
	]{c}{\resDirectory/str2int.c}

	\end{block}
\end{slide}

\begin{slide}
	\begin{figure}
	\includegraphics[width=\textwidth]{\imgDirectory/pointers.jpg}
	\end{figure}
\end{slide}

\end{document}


%!TEX root = forallxsol.tex
%\part{Truth-functional logic}
%\label{ch.TFL}
%\addtocontents{toc}{\protect\mbox{}\protect\hrulefill\par}

\chapter{Sentences of TFL}\setcounter{ProbPart}{0}
\problempart
\label{pr.wiffTFL}
For each of the following: (a) Is it a sentence of TFL, strictly speaking? (b) Is it a sentence of TFL, allowing for our relaxed bracketing conventions?
\begin{earg}
\item $(A)$\hfill \myanswer{(a) no (b) no}
\item $J_{374} \eor \enot J_{374}$\hfill \myanswer{(a) no (b) yes}
\item $\enot \enot \enot \enot F$\hfill \myanswer{(a) yes (b) yes}
\item $\enot \eand S$\hfill \myanswer{(a) no (b) no}
\item $(G \eand \enot G)$\hfill \myanswer{(a) yes (b) yes}
\item $(A \eif (A \eand \enot F)) \eor (D \eiff E)$\hfill \myanswer{(a) no (b) yes}
\item $[(Z \eiff S) \eif W] \eand [J \eor X]$\hfill \myanswer{(a) no (b) yes}
\item $(F \eiff \enot D \eif J) \eor (C \eand D)$\hfill \myanswer{(a) no (b) no}
\end{earg}

\problempart
Are there any sentences of TFL that contain no atomic sentences? Explain your answer.
\\\myanswer{No. Atomic sentences contain atomic sentences (trivially). And every more complicated sentence is built up out of less complicated sentences, that were in turn built out of less complicated sentences, \ldots, that were ultimately built out of atomic sentences.}\\

\problempart
What is the scope of each connective in the sentence
$$\bigl[(H \eif I) \eor (I \eif H)\bigr] \eand (J \eor K)$$
\myanswer{The scope of the left-most instance of `$\eif$' is `$(H \eif I)$'.\\
The scope of the right-most instance of `$\eif$' is `$(I \eif H)$'.\\
The scope of the left-most instance of `$\eor$ is `$\bigl[(H \eif I) \eor (I \eif H)\bigr]$'\\
The scope of the right-most instance of `$\eor$' is `$(J \eor K)$'\\
The scope of the conjunction is the entire sentence; so conjunction is the main logical connective of the sentence.}


\documentclass[a5paper]{article}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage[parfill]{parskip}% don't indent new sections
\usepackage[utf8]{inputenc}
\usepackage[a4paper]{geometry}
\pagestyle{empty}

\title{Priority Queue Sort Experiment}
\author{Emil Lynegaard}

\begin{document}
  \maketitle
  \section{Experiment}
  For this experiment, the programming language Go will be used. The experiment will set out to compare sorting of a list of numbers with a priority queue and with a call to a library sort function. Since Go does not ship with a priority queue directly, an implementation thereof from their documentation of their heap package\footnote{\url{https://golang.org/pkg/container/heap/}} will be used. The sort function that will be used is Go's \texttt{sort.Ints}\footnote{\url{https://golang.org/pkg/sort/\#Ints}}.

\textbf{Measurement:} For the comparison, the experiment will be ran 100 times for both the priority queue and for the library sort function. The input will be of size \texttt{5.000.000} and the two experiments will be ran independently on the same machine with minimal process interference.

\textbf{Assumption:} Due to the nature of a priority queue, it has additional housekeeping to do when we inserting and deleting items from it. Therefore we will claim that the library sorting function, will sort and return our input faster than the priority queue.

\section{Results}
\begin{figure}[!ht]
    \centering
    \noindent\includegraphics[scale=0.6]{graph.png}
    \caption{Graph plotting the running times of PQ-sort and Library-sort.}
    \label{fig:graph}
\end{figure}

Based on our results shown in Figure~\ref{fig:graph}, we can with a certainty of 95\% say that for permuting a list \texttt{A} to give a result \texttt{B} such that \texttt{B} is sorted\footnote{Sorted in this case meaning B[i] $\le$ B[i+1] for any i.}, the priority queue implementation has at most a 0.69071\% chance of being faster or equally fast as the library function.\footnote{What a bold statement. Is this even remotely close to the intended experiment?}. This is given the following:
\[ {(1-p_{H})}^{100} \le.05 \]
\[ p_{H}\ge 1-.05^{1/100} \]
\[ p_{H}\ge 691/10.000 \]

From Figure~\ref{fig:graph} we may also observe that the standard deviation for both the library function and for the priority queue is relatively small, while the distance between the two methodologies remain very large, indicating that the actual chance of the priority queue outperforming the library is even smaller estimated.

\section{Bonus question}
The priority queue will never be faster at sorting an array than a sensible library function. This is due to the overhead of maintaining the advantageous data structure that most priority queues will have. However, if one was fine with keeping the data in the priority queue, and had to repeatedly insert new items into the data structure, a priority queue would vastly outperform a basic array. For the priority queue we would get insertions of $O(\log n)$ in contrast to $O(n)$ with a naive array implementation..
\end{document}


%!TEX root =  ../main.tex

\mychapters{Limits}{limits}{\chapdir/pics/EDMcrowd}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Limits are a way of expressing what a function's output approaches as the input gets
closer and closer to a given value.  In this section, we will use limits to describe the
location of ``holes'' in graphs, continuous functions, asymptotes and one-sided-limits,
the difference quotient, and derivatives.

How are limits used in our modern world?
Sound engineers use advanced functions to design waveforms that would not occur in nature.
The distinctive sound of much of today's electronic music comes from computer approximating
what infinite sums of sounds waves would sound like.  Are there actual infinities in the physical
universe?  If something cannot exist without human creativity, what does that mean about its
place in the world, its ``rightness'' or ``wrongness?''

\marginfig[-1in]{\chapdir/pics/Sawtooth_Fourier_Analysis.JPG}{Much of today's EDM music makes use of sounds only possible through approximating infinite sums of waves.}

How should we treat our responses to questions where we cannot verify our answers?
Should we only interpolate, or is extrapolation also valid?  Is it reasonable to assume
things continue after the manner we have observed when they are unobserved, or even
unobservable?  Is anything in our universe able to be modeled with only one function across
all of its domain? What good is it to instantaneous change, when change -- by definition --
happens over time?

\newpage
\chapterminitoc

%									2 - 1
\newpage
\invisiblesection{Removable}
\marginlessinput{\chapdir/0201p}
\newpage
\input{\chapdir/0201}
\newpage
\input{\chapdir/0201x}
~\vfill

%									2 - 2
%\newpage
\invisiblesection{Continuous}
\subsection{Definition of Limits}
\noindent\makebox[\textwidth]{\includegraphics[width=\paperwidth]{\chapdir/0202p.pdf}}
\newpage
\input{\chapdir/0202}
~\vfill
\newpage
\subsection{Exercises}
Made as a Word doc.
~\vfill

%									2 - 3
%\newpage
\invisiblesection{Piecewise}
\subsection{Mind the Gap}
\noindent\makebox[\textwidth]{\includegraphics[width=\paperwidth]{\chapdir/0203p.pdf}}
\newpage
\input{\chapdir/0203}
%\newpage
\subsection{Exercises}
\noindent\makebox[\textwidth]{\includegraphics[width=\paperwidth]{\chapdir/0203xA.pdf}}
\newpage
\noindent\makebox[\textwidth]{\includegraphics[width=\paperwidth]{\chapdir/0203xB.pdf}}
\noindent\makebox[\textwidth]{\includegraphics[width=\paperwidth]{\chapdir/0203xC.pdf}}

%									2 - 4
\invisiblesection{Infinite}
\subsection{Are We There Yet?}
\noindent\makebox[\textwidth]{\includegraphics[width=\paperwidth]{\chapdir/0204p.pdf}}
\input{\chapdir/0204}
%\newpage
\subsection{Exercises}
\noindent\makebox[\textwidth]{\includegraphics[width=\paperwidth]{\chapdir/0204xA.pdf}}
\newpage
\noindent\makebox[\textwidth]{\includegraphics[width=\paperwidth]{\chapdir/0204xB.pdf}}
\noindent\makebox[\textwidth]{\includegraphics[width=\paperwidth]{\chapdir/0204xC.pdf}}

%									2 - 5
\invisiblesection{Change}
\marginlessinput{\chapdir/0205p}
\newpage
\input{\chapdir/0205}
~\vfill
%\newpage
\subsection{Exercises}
\noindent\makebox[\textwidth]{\includegraphics[width=\paperwidth]{\chapdir/0205xA.pdf}}
\newpage
\noindent\makebox[\textwidth]{\includegraphics[width=\paperwidth]{\chapdir/0205xB.pdf}}

%									2 - 6
\section{Review}
\subsection{Chapter Review}
This chapter was generally new ideas, that we might describe where a function ``would've gone'',
whether it ended up with a value or not.  This idea --- called a limit --- has a formal definition with
the Greek letters $\epsilon$ and $\delta$, which correspond to the ``error'' in the $y$ values
resulting ``differences'' in the $x$/input.  Limits allowed up to define what a continuous
function is.  We saw that a function might be composed of pieces ... and still be continuous.
There are four ways to remove discontinuities.  There are five kinds of discontinuities.
The powerful language of limits extended to describe phenomena at infinities, as well as ``values''
of the infinities.  Ultimately, limits were used to make possible difference quotients, the
average rate of change of a function at an instant.

\subsection{Chapter Test}


\documentclass[11pt,a4paper]{article}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{bm}
\usepackage[hmargin=3cm,vmargin=2.4cm]{geometry}
\usepackage[dvipsnames]{xcolor}
\usepackage{hyperref}
\usepackage{fancyhdr}

\hypersetup{
  colorlinks=true,
  citecolor=NavyBlue,
  linkcolor=BrickRed,
  urlcolor=Violet
}

\pagestyle{fancy}
\fancyhead{}
\fancyhead[L]{\href{https://www-ljk.imag.fr/membres/Pierre.Karpman/cry\_adv2018\_proj.pdf}{\texttt{https://www-ljk.imag.fr/membres/Pierre.Karpman/cry\_adv2018\_proj.pdf}}}

\theoremstyle{definition}
\newtheorem{defi}{Definition}
\def\defiautorefname{Definition}
\newtheorem{example}{Example}
\def\defiautorefname{Example}

\renewcommand{\labelitemi}{---}

\DeclareMathOperator\code{\mathcal{C}}
\DeclareMathOperator\bigo{\mathit{O}}
\DeclareMathOperator\hd{\mathit{hd}}
\DeclareMathOperator\wt{\mathit{wt}}
\DeclareMathOperator\RM{\mathrm{RM}}
\DeclareMathOperator\fun{\mathit{F}}
\DeclareMathOperator\eval{eval}
\DeclareMathOperator\Ber{Ber}
\DeclareMathOperator\rank{rank}
\DeclareMathOperator\vspan{span}
\DeclareMathOperator\gl{GL}
\DeclareMathOperator\proj{\pi}
\newcommand\ftwo{\mathbb{F}_{2}}
\newcommand\zo{\{0,1\}}
\newcommand\randraw{\xleftarrow{{\scriptscriptstyle\$}}}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\title{Advanced cryptology (GBX9SY06)\\
$\ast$\\
Information set decoding project}
\date{2018-12}
\author{Pierre Karpman}

\begin{document}

\maketitle{}

\section{Description}

The goal of this project is to implement an information set decoding algorithm to search for low-weight codewords of \emph{binary} codes, using the
SageMath software (\url{https://www.sagemath.org/}).
There is no particular requirement as to which algorithm to implement, but overall efficiency will be rewarded.

\subsection*{Performance evaluation}

Your submission will be evaluated on a series of codes (defined below) on version 8.2 of SageMath running on the ``Retour d'Est'' (\url{retourdest.imag.fr})
server. This is a single-processor system with 256~GB of memory and running an Intel(R) Xeon(R) Gold 6126 CPU @ 2.60GHz, that features 12 (resp. 24) physical (resp. logical) cores.
You are encouraged to take advantage of this architecture, for instance by developping a multi-threaded implementation.

The codes used for evaluation are the following, all defined in the provided file \url{https://www-ljk.imag.fr/membres/Pierre.Karpman/cry\_adv2018\_codes\_desc.sage}.
\paragraph{Known codes.}
\begin{itemize}
	\item A binary BCH code of parameters $[511, 448, 15]$. The evaluation criterion will be the ability to find a minimum-weight codeword. This should take (far) less than a minute.
	\item A binary BCH code of parameters $[511, 421, 21]$. The evaluation criterion will be the average time (over three runs) needed to find a minimum-weight codeword. This may take some time, but an efficient implementation should need (far) less than an hour on a single thread.
	\item A binary BCH code of parameters $[511, 385, 29]$. The evaluation criterion will be the smallest weight of a found codeword after one run of at most 30 minutes. A bonus will be awarded if a word of weight 30 is found. An extra
		bonus will be awarded if a word of weight 29 is found.
\end{itemize}
\paragraph{Random codes.}
\begin{itemize}
	\item A random code of parameters $[768, 384]$. The evaluation criterion will be the smallest weight of a found codeword after one run of at most 15 minutes.
	\item A random code of parameters $[2048, 128]$. The evaluation criterion will be the smallest weight of a found codeword after one run of at most 15 minutes.
\end{itemize}

You are advised to use a slightly verbose implementation, that for instance prints the time required to find a codeword with a new lowest weight.

\section{Instructions}
Working in teams of two is allowed and encouraged. You must send your implementation to:
\begin{center}
\href{mailto:pierre.karpman@univ-grenoble-alpes.fr}{\texttt{pierre.karpman@univ-grenoble-alpes.fr}}
\end{center}
by Monday 2019-01-14T23:59+0100. No separate report is required, but your code must be well-commented and include a small textual description of your algorithmic and implementation choices.
If your algorithm is parameterized, be sure to mention which parameters to use for every one of the evaluation codes. You are also advised to provide some numbers about your own
performance evaluation of your implementation.
Finally, you must prepare a short ($\sim$ 10 minutes for a single-person team; $\sim 2\times7$ minutes for a two-person one) presentation of your
work for Thursday 2019-01-17.

\nocite{*}

\bibliographystyle{alpha}
\bibliography{tehbib}

\end{document}


%
% @author   Shmish  "shmish90@gmail.com"
% @legal    MIT     "(c) Christopher Schmitt"
%

\documentclass{article}

%
% Document Imports
%

\usepackage{fancyhdr}
\usepackage{extramarks}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amsfonts}

%
% Document Configuation
%

\newcommand{\hwAuthor}{Christopher Schmitt}
\newcommand{\hwSubject}{Math 218}
\newcommand{\hwSection}{Section 81}
\newcommand{\hwSemester}{Summer 2019}
\newcommand{\hwAssignment}{Assignment 1}

%
% Document Enviornments
%

\setlength{\headheight}{65pt}
\pagestyle{fancy}
\lhead{\hwAuthor}
\rhead{
  \hwSubject \\
  \hwSection \\
  \hwSemester \\
  \hwAssignment
}

\newenvironment{problem}[1]{
  \nobreak\section*{Problem #1}
}{}

%
% Document Start
%

\begin{document}
  \begin{problem}{1}
    Are the following statements TRUE or FALSE? Explain why.

    \begin{enumerate}
      \item[(a)] If $\pi$ is rational, then so is $2$
      \item[(b)] If $\pi$ is irrational, then so is $2$
    \end{enumerate}

    \noindent\textbf{Solution}

    \begin{enumerate}
      \item[(a)] True, since $False \rightarrow True$
      \item[(b)] False, since $True \nrightarrow False$
    \end{enumerate}
  \end{problem}

  \begin{problem}{2}
    Consider the statement: “If an animal is an rhinoceros, then it has a horn.”

    \begin{enumerate}
      \item[(a)] Write down the CONVERSE of this statement.
      \item[(b)] Write down the CONTRAPOSITIVE of this statement.
    \end{enumerate}

    \noindent\textbf{Solution}

    \begin{enumerate}
      \item[(a)] "If an animal has a horn, then it is a rhinoceros"
      \item[(b)] "If an animal does not have a horn, then it is not a rhinoceros"
    \end{enumerate}
  \end{problem}

  \begin{problem}{3}
    Let $x$ be a real number. Using the definition of rational number, write a direct proof of the following: If $x$ is rational, then $x^2 + 5$ is also rational.

    \begin{proof}
      Let $x$ be a rational number.
      \begin{equation*}
        \begin{split}
          x & = \frac{a}{b}\text{, where $a, b$ are integers and $b \neq 0$ }\\
          x^2 & = \frac{a^2}{b^2}\\
          x^2 + 5 & = \frac{a^2}{b^2} + 5\\
          x^2 + 5 & = \frac{a^2 + 5b^2}{b^2}\text{, which is rational.}
        \end{split}
      \end{equation*}
    \end{proof}
  \end{problem}

  \begin{problem}{4}
    Let $x$ be a positive real number. Using the definition of a rational number, write a proof by contraposition of the following: If $x$ irrational, then $\sqrt{x + 6}$ is also irrational.

    \begin{proof}
      (By contraposition) Let $\sqrt{x + 6}$ be rational.
      \begin{equation*}
        \begin{split}
          \sqrt{x + 6} & = \frac{a}{b}\text{, where $a, b$ are integers and $b \neq 0$}\\
          x + 6 & = \frac{a^2}{b^2}\\
          x & = \frac{a^2}{b^2} - 6\\
          x & = \frac{a^2 - 6b^2}{b^2}\text{, which is rational}
        \end{split}
      \end{equation*}
    \end{proof}
  \end{problem}

  \begin{problem}{5}
    let $n$ be an integer. Using the definition of odd/even, write a proof of the following: $n$ is even if and only if $2n^2 + 5n + 7$ is odd.

    \begin{proof}($p \Rightarrow q$) Suppose that $n$ is even.
      \begin{equation*}
        \begin{split}
          n & = 2(k)\text{, where $k$ is an integer}\\
          2n^2 + 5n + 7 & = 2(2k)^2 + 5(2k) + 7\\
          2n^2 + 5n + 7 & = 8k^2 + 10k + 7\\
          2n^2 + 5n + 7 & = 8k^2 + 10k + 6 + 1\\
          2n^2 + 5n + 7 & = 2(4k^2 + 5k + 3) + 1\text{, which is odd.}
        \end{split}
      \end{equation*}

      ($\neg p \Rightarrow \neg q$) Suppose that $n$ is odd.
      \begin{equation*}
        \begin{split}
          n & = 2(k) + 1\text{, where $k$ is an integer}\\
          2n^2 + 5n + 7 & = 2(2k + 1)^2 + 5(2k + 1) + 7\\
          2n^2 + 5n + 7 & = 8k^2 + 18k + 14\\
          2n^2 + 5n + 7 & = 2(4k^2 + 9k + 7)\text{, which is even.}
        \end{split}
      \end{equation*}
    \end{proof}
  \end{problem}

  \begin{problem}{6}
    Using the definition of odd and even, write a proof of the following: $n^2 + 3n + 7$ is odd.

    \begin{proof}(By cases) Suppose that $n$ is even.
      \begin{equation*}
        \begin{split}
          n & = 2(k)\text{, where k is an integer}\\
          n^2 + 3n + 7 & = (2k)^2 + 3(2k) + 7\\
          n^2 + 3n + 7 & = 4k^2 + 6k + 7\\
          n^2 + 3n + 7 & = 4k^2 + 6k + 6 + 1\\
          n^2 + 3n + 7 & = 2(2k^2 + 3k + 3) + 1\text{, which is odd.}
        \end{split}
      \end{equation*}

      Suppose that $n$ is odd.
      \begin{equation*}
        \begin{split}
          n & = 2(k) + 1\text{, where k is an integer}\\
          n^2 + 3n + 7 & = (2k + 1)^2 + 3(2k + 1) + 7\\
          n^2 + 3n + 7 & = 4k^2 + 10k + 11\\
          n^2 + 3n + 7 & = 4k^2 + 10k + 10 + 1\\
          n^2 + 3n + 7 & = 2(2k^2 + 5k + 5) + 1\text{, which is odd.}\\
        \end{split}
      \end{equation*}
    \end{proof}
  \end{problem}

  \begin{problem}{7}
    Prove that there exists positive integers $a, b$ such that $a^2 + b^2 = 100$

    \begin{proof}(By example)
      \begin{equation*}
        6^2 + 8^2 = 100
      \end{equation*}
    \end{proof}
  \end{problem}
\end{document}


\documentclass[a4paper, 11pt]{article}
\usepackage{comment} % enables the use of multi-line comments (\ifx \fi)
\usepackage{fullpage} % changes the margin
\usepackage{hyperref}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{booktabs} % For formal tables

\usepackage[ruled]{algorithm2e} % For algorithms
\renewcommand{\algorithmcfname}{ALGORITHM}

\begin{document}
%Header-Make sure you update this information!!!!
\noindent
\large\textbf{Homework TA1} \hfill \textbf{Zankhanaben Ashish Patel} \\
\normalsize COMP6651 \hfill \textbf{40067635} \\
Prof. Tiberiu Popa \hfill Due Date: 22/02/19

\section{Prove Or Disprove}
\subsection*{a)}
\( O(2^n)= O(4^n)\) : \textbf{False}\\
First, we'll check for $2^n = O(4^n)$\\
$\therefore 2^n \le c_{1} \cdot 4^n$\\
$\implies c_{1}=1$\\
$\therefore 2^n \in O(4^n)$\\
Now, we'll check for $4^n = O(2^n)$\\
$\therefore 4^n \le c_{2} \cdot 2^n$\\
$\therefore 2^n \le c_{2} \cdot 1$ \\
Which is not possible because $c$ is a constant, we cannot find exact value for $c$ .\\
$\therefore 4^n \notin O(2^n)$
$$\therefore O(2^n) \ne O(4^n)$$

\subsection*{b)}
\( O(\log (2^n))= O(\log (4^n))\) : \textbf{True}\\
First, we'll check for $\log (2^n) = O(\log(4^n))$\\
$\therefore \log(2^n) \le c_{1} \cdot \log(4^n)$\\
$\therefore n\log 2 \le 2n \cdot c_{1} \cdot \log 2$\\
$\implies c_{1}=1$\\
$\therefore\log (2^n) \in O(\log(4^n))$\\
Now, we'll check for $\log(4^n) = O(\log(2^n))$\\
$\therefore \log(4^n) \le c_{2} \cdot \log (2^n)$\\
$\therefore \log(2^{2n}) \le c_{2} \cdot \log(2^n)$ \\
$\therefore 2n \le c_{2} \cdot n$\\
$\implies c_{2}=3$\\
$\therefore \log(4^n) \in O(\log(2^n))$
$$\therefore O(\log (2^n)) = O(\log (4^n))$$

\subsection*{c)}
$O(n!) = O((n+1)!) $ :\textbf{False}\\
First, we'll check for $n! = O((n+1)!)$\\
$\therefore n! \le c_{1} \cdot (n+1)! $\\
Which will always true for $c_{1}=1$ \\
$\therefore n! \in O((n+1)!)$\\
Now, we'll check for $(n+1)! =O(n!)$\\
$\therefore (n+1)! \le c \cdot n!$\\
$\therefore (n+1)\cdot n! \le c \cdot n!$\\
$\therefore (n+1) \le c \cdot 1$\\
Which is not possible because $c$ is a constant, we cannot find exact value for $c$.\\
In $(n+1)!$ there are $(n+1)*n!$ operations. So it's $n+1$ operation more than $n!$.\\ Means $(n+1)! \notin O(n!)$
$$\therefore O(n!) \ne O((n+1)!)$$

\section{Solve the following problems from the textbook :}

\subsection*{\underline{Exe : 3.1-1}}
According to the definition of $\theta$ notation,\newline
\begin{equation}
\label{eq1}
0\leq c_{1}\cdot(f(n)+g(n)) \leq max(f(n),g(n)) \leq c_{2}\cdot(f(n)+g(n))
\end{equation}
By solving all inequalities, we can prove our problem.\newline
Let,
$$f(n) \leq max(f(n),g(n))$$
$$g(n) \leq max(f(n),g(n))$$

\begin{equation}
\label{eq2}
\therefore (f(n)+g(n))/2 \leq max(f(n),g(n))
\end{equation}
Thus,
\begin{equation}
\label{eq3}
c_{1}\cdot(f(n),g(n)) \leq max(f(n),g(n))
\end{equation}
using equations~(\ref{eq2}) and (\ref{eq3}), $c_{1} = 1/2$.\\
It is always true that,\\
\begin{equation}
\label{eq4}
max(f(n),g(n)) \leq (f(n)+g(n))
\end{equation}
\begin{equation}
\label{eq5}
max(f(n),g(n)) \leq c_{2}\cdot(f(n)+g(n))
\end{equation}
Using equations (\ref{eq4}) and (\ref{eq5}), $c_{2} = 1.$\newline
From above derivations, We get,
$$0\le 1/2\cdot(f(n)+g(n)) \leq max(f(n),g(n)) \leq 1\cdot(f(n)+g(n))$$
hence proved, $$max(f(n),g(n)) = \Theta(f(n)+g(n))$$

\subsection*{\underline{Exe : 3.1-2}}
As given,
$$0 \le c_{1}\cdot n^b \le (n+a)^b \le c_{2}\cdot n^b,\forall n\ge n_{0}.$$
For $c_{1}\cdot n^b \le (n+a)^b$\newline
Let, $c_{1}=1$
$$\therefore n^b \le (n+a)^b, \forall n \ge 1$$
Now, for $(n+a)^b \le c_{2}\cdot n^b$
$$n^b \cdot (1+\tfrac{a}{n})^b \le (1+a)^b \cdot n^b,\forall n \ge 1$$
$\therefore$ Real constants,
$$c_{1}=1, c_{2}=(1+a)^b , \forall n \ge 1.$$

\subsection*{\underline{Exe : 3.2-4}}
According to Cormen book, function $f(n)$ is polynomically bounded if $f(n) = O(n^k)$ for some constant k.\\
\textbf{For,$ \lceil \lg n \rceil ! $}\\
By taking proof of contradiction, let's assume that our function is polynomically bounded, where exists constant $ c,k and n_{0}$.
$$f(n)=O(n^k)$$
$$\therefore f(n) \le c \cdot n^k$$
$$\lceil \log n \rceil ! \le c \cdot n^k$$
By taking $n=2^b$,
$$\lceil \log 2^b \rceil ! \le c \cdot (2^{k \cdot b})$$
$$\therefore b ! \le c \cdot (2^{k \cdot b})$$
Above equation will never true.\\
Since, factorial function is not exponentially bounded,and we cannot find such $c$ that makes our condition true.\\
$\therefore \lceil \lg n \rceil ! $ \textbf{ is not polynomically bounded}. \\
\textbf{For,$ \lceil \lg \lg n \rceil ! $}\\
Again taking proof of contradiction,\\
Assume $n=2^{2^k}$,
$$ \lceil \lg \lg 2^{2^k} \rceil ! \le c \cdot (2^{2^k}) $$
$$\therefore k! \le c \cdot 2^{2^k}$$
To prove prove above equation, let's take $\log$ both side,\\
$\therefore log(k!) \le 2^k$, because $\log_{2}2 =1$.\\
Now let's consider below inequalities to prove above equation to be true.
$$ \log(k!) < \log(k^k) < k\log k < 2^k $$
Which is true $\forall k \ge 1$, and $c=1$.\\
$\therefore \lceil \lg \lg n \rceil ! $ \textbf{is polynomically bounded.}

\subsection*{\underline{Exe : 3.4}}
\subsection*{a)}
Answer : False
\begin{equation}
\label{eq6}
f(n) \le c \cdot g(n)
\end{equation}
\begin{equation}
\label{eq7}
g(n) = O(f(n))
\end{equation}
$\exists c $ such that (\ref{eq6}) holds true.\\
Now if $g(n)=O(f(n)) \implies g(n) = c \cdot (f(n))$\\
To make this condition to be true, there is not exist any c to make both equation true together.\\
$\therefore$ Let's prove by example,\\
$2^n = O(3^n)$ but $3^n \ne O(2^n)$

\subsection*{b)}
Answer : False\\
Let's prove by contradiction. Considering given conjecture to be true.
$$\therefore c_{1} \cdot min(f(n)+g(n)) \le (f(n)+g(n)) \le c_{2} \cdot min(f(n)+g(n))$$
For, $(f(n)+g(n)) \le c_{2} \cdot min(f(n)+g(n))$\\
Above conjecture cannot be true for $f(n) = n^3$ and $g(n)=n$.
$$n^3 + n \le c_{2} \cdot min(n^3,n) = c_{2} \cdot n$$
Which will never turns into true.

\subsection*{c)}
Answer : True\\
$f(n) = O(g(n), \exists c, n_{0} $ such that \\
$f(n) \le c \cdot g(n)$ and $n \ge n_{0}$ . $(f(n) \ge 1) $ is given.\\
Taking log both side,\\
$ \log (f(n)) \le \log (c \cdot g(n))$
\begin{equation}
\label{eq8}
\log(f(n)) \le \log c + \log(g(n))
\end{equation}
As given $f(n) \ge 1 $ and $\lg (g(n)) \ge 1 $. Both preserve after taking log also.\\ So, we need to find k such that, Using (\ref{eq8}) where,
$$k \log (g(n)) \ge \log c + \log (g(n))$$ Since $\log (g(n)) \ge 1 $
$$\therefore k \ge \log c +1 $$
So, $f(n) \ge k \log (g(n))$ will always true.

\subsection*{d)}
Answer : False
Let's assume $f(n) =  2n$ and $g(n)=n$\\
For above scenario $f(n) = O(g(n))$ is true.\\
Now, Let's check for $$2^{f(n)} = O(2^{g(n)})$$
$$2^{f(n)} \le c \cdot 2^{g(n)} , \forall n \ge n_{0}$$
$$2^{2n} \le c \cdot 2^n$$
$$4^n \le c \cdot 2^n$$
$$2^n \le c , \forall n \ge n_{0}$$
Which cannot be possible since c is constant.\\
$\therefore$ This conjecture is False.

\subsection*{e)}
Answer : False.\\
Let, $$f(n) = \frac{1}{n}$$
$$\therefore \frac {1}{n} \le c \cdot \frac{1}{n^2}$$
$$ 1 \le c \cdot \frac {1}{n}$$
We have to find $n \ge n_{0}$ and $c$ such that these inequalities always hold. Since c is constant and n is variable, we cannot find threshold point, such that this condition always hold. So it's false.

\subsection*{f)}
Answer : True\\
For,
$$f(n) = O(g(n))$$
$$f(n) \le c \cdot g(n)$$
\begin{equation}
\label{eq9}
\therefore g(n) \ge \frac {f(n)}{c}
\end{equation}
For,
$$g(n) = \Omega (f(n))$$
\begin{equation}
\label{eq10}
g(n) \ge c \cdot f(n)
\end{equation}
By comparing both equations (\ref{eq9}) and (\ref{eq10}) \\
$g(n)$ is always bigger.So this conjecture is true.

\subsection*{g)}
Answer: False \\
By taking proof of contradiction, let's assume that given conjecture is true.
$$\therefore 0 \le c_{1} \cdot f(\frac{n}{2}) \le f(n) \le c_{2} \cdot f(\frac{n}{2}) , \exists c_{1}, c_{2} \, and \, n \ge n_{0}$$
Let's take $f(n) = 2^{2n}$. (Same as 3.4 d)
$$\therefore c_{1} \cdot 2^{\frac{2n}{2}} \le 2^{2n} \le c_{2} \cdot 2^{\frac{2n}{2}}$$
$$c_{1} \cdot 2^{\frac{2n}{2}} \le 4^{n} \le c_{2} \cdot 2^n$$
Now, $4^n \le c_{2} \cdot 2^n $ will never holds true for any $c_{2} \, and \, n \ge n_{0}$. Therefore, given conjecture is false.

\subsection*{\underline{Exe : 4.5-1}}
\subsection*{a)}
$T(n) = 2\,T(\frac{n}{4}) +1 $ \\
$\implies a=2, b=4, f(n) = 1 , n^{\log_{b}a} = n^{\frac{1}{2}} = \sqrt{n}$ \\
Since,\\
$f(n) = O(n^{\log_{b}a - \epsilon})$\\
$\phantom{f(n)} = O(n^{\log_{4}2 - \frac{1}{2}})$ for $ \epsilon = \frac{1}{2}.$\\
We can apply master theorem case-1.\\
To conclude the solution, $T(n) = \Theta(n^{\log_{b}a})$
$$T(n) = \Theta (\sqrt{n})$$

\subsection*{b)}
$T(n) = 2T(\frac{n}{4}) + \sqrt{n}.$\\
$ n=2, b=4, f(n)=n^{\frac{1}{2}},n^{\log_{4}2}=n^{\frac{1}{2}} $\\
Case-2 of master theorem will apply since,\\
$f(n)= \Theta(n^{\log_{b}a}) = \Theta (n^{\frac{1}{2}})$\\
Thus, the solution to the recurrence is,
$$T(n) = \Theta(n^{\log_{b}a}\lg n) = \Theta(\sqrt{n}\lg n)$$

\subsection*{c)}
$T(n) = 2T(\frac{n}{4}) + n .$\\
$ n=2, b=4, f(n)=n \, ,n^{\log_{4}2}=n^{\frac{1}{2}} = O(\sqrt{n}) $\\
Since $f(n) = \Omega(n^{\log_{4}2 + \epsilon})$ Where $\epsilon=\frac{1}{2}$\\
Case-3 Applies, Sufficiently large n and $c<1$ we'll check regularity condition.\\
$af(\frac{n}{b}) = 2(\frac{n}{4}) = \frac{n}{2} \le c \cdot f(n)$\\
As $f(n)=n$, We can take $c=\frac{1}{2}$ which is less than 1.\\
So, by case-3 the solution to the recurrence is,
$$T(n) = \Theta (f(n)) = \Theta(n)$$

\subsection*{d)}
$T(n) = 2T(\frac{n}{4}) + n^2 .$\\
$ n=2, b=4, f(n)=n^2,n^{\log_{4}2}=n^{\frac{1}{2}} $\\
Since $f(n) = \Omega(n^{\log_{4}2 + \epsilon})$ Where $\epsilon=\frac{3}{2}$\\
Case-3 Applies, Sufficiently large n and $c<1$ we'll check regularity condition.\\
$af(\frac{n}{b}) = 2(\frac{n}{4})^2 = \frac{n}{8} \le c \cdot f(n)$\\
As $f(n)=n^2$, We can take $c=\frac{1}{8}$ which is less than 1.\\
So, by case-3 the solution to the recurrence is,
$$T(n) = \Theta (f(n)) = \Theta(n^2)$$

\subsection*{\underline{Exe : 4.5-5}}
Let's take $a=1, b=5, f(n)= 5n$ So, $n^{\log_{5}1} = 0$.\\
Since $f(n) = \Omega(n^{\log_{b}a + \epsilon}) , \epsilon = 1$\\
Case-3 Applies, Sufficiently large n and $c<1$ we'll check regularity condition.\\
$af(\frac{n}{b}) = 1(\frac{5n}{5}) = n \le c \cdot f(n)$\\
As $f(n)= 5n$, We will have $c=1$ which is\textbf{ not fulfilling }regularity condition which is $c<1$.

\subsection*{\underline{Problem : 4.1}}
\subsection*{a)}
$T(n) = 2T(\frac{n}{2}) + n^4$\\
Using masters theorem,\\
$a=2, b=2, f(n) = n^4, n^{\log_{b}a}=n^{\log_{2}2}=n$\\
Since, $f(n) = \Omega(n^{\log_{2}2 + \epsilon}), \epsilon=3$\\
Case-3 applies, Let's prove regularity condition for $f(n)$, for sufficient large value of $n$ and $c<1$.\\
$af(\frac{n}{b}) = 2(\frac{n}{2})^4 = \frac{n^4}{8} \le c \cdot f(n)$\\
As $f(n)=n^4$, We can take $c=\frac{1}{8}$ which is less than 1.\\
So, by case-3 the solution to the recurrence is,
$$T(n) = \Theta (f(n)) = \Theta(n^4)$$

\subsection*{b)}
$T(n) = T(\frac{7n}{10}) + n$\\
Using masters theorem,\\
$a=1, b=\frac{10}{7}, f(n) = n, n^{\log_{b}a}=n^{\log_{\frac{10}{7}}1}=0$\\
Since, $f(n) = \Omega(n^{\log_{\frac{10}{7}}1 + \epsilon}), \epsilon=1$\\
Case-3 applies, Let's prove regularity condition for $f(n)$, for sufficient large value of $n$ and $c<1$.\\
$af(\frac{n}{b}) = 1 \cdot(\frac{7n}{10}) = \frac{7}{10}n \le c \cdot f(n)$\\
As $f(n)=n$, We can take $c=\frac{7}{10}$ which is less than 1.\\
So, by case-3 the solution to the recurrence is,
$$T(n) = \Theta (f(n)) = \Theta(n)$$

\subsection*{c)}
$T(n) = 16T(\frac{n}{4}) + n^2$\\
Using masters theorem,\\
$a=16, b=4, f(n) = n^2, n^{\log_{b}a}=n^{\log_{4}16}=n^2$\\
Case-2 applies,Since,\\
$f(n)=\Theta(n^{\log_{4}16}) =\Theta(n^2)$
Thus, By case-2 the solution to the recurrence is,
$$T(n) = \Theta (n^{\log_{b}a} \lg n) = \Theta(n^2 \lg n)$$

\subsection*{d)}
$T(n) =7 T(\frac{n}{3}) + n^2$\\
By master theorem,\\
$a=7, b=3, f(n) = n^2, n^{\log_{b}a}=n^{\log_{3}7} \approx n^{1.77}$\\
Since, $f(n)= \Omega(n^{\log_{3}7 + \epsilon}), \epsilon \approx 0.23$\\
Case-3 applies, Let's prove regularity condition for $f(n)$, for sufficient large value of $n$ and $c<1$.\\
$af(\frac{n}{b}) = 7\cdot(\frac{n}{3})^2 = \frac{7}{9}n^2 \le c \cdot f(n)$\\
As $f(n)=n^2$, We can take $c=\frac{7}{9}$ which is less than 1.\\
So, by case-3 the solution to the recurrence is,
$$T(n) = \Theta (f(n)) = \Theta(n^2)$$

\subsection*{e)}
$T(n) =7 T(\frac{n}{2}) + n^2$\\
By master theorem,\\
$a=7, b=2, f(n) = n^2, n^{\log_{b}a}=n^{\log_{2}7} \approx n^{2.8}$\\
Since, $f(n)=O(n^{\log_{2}7 - \epsilon}), \epsilon \approx 0.8$\\
We can apply case-1,
$$T(n) = \Theta (n^{\log_{b}a}) = \Theta(n^{\lg 7})$$

\subsection*{f)}
$T(n) =2 T(\frac{n}{4}) + \sqrt{n}$\\
By master theorem,\\
$a=2, b=4, f(n) = \sqrt{n}, n^{\log_{b}a}=n^{\log_{4}2} = n^{\frac{1}{2}}$\\
case-2 applies, Since,\\
$f(n)= \Theta(n^{\log_{b}a}) = \Theta(n^\frac{1}{2})$
Thus, solution to the recurrence is,
$$T(n) = \Theta (n^{\log_{b}a} \lg n) = \Theta (\sqrt{n} \lg n)$$

\subsection*{g)}
$T(n)= T(n-2)+n^2$\\
By applying counting method here,
At $0^{\text{th}}$ step, the recurrence result is $\implies n^2$.\\
Therefore,\\
Step \,\,\,\,\, Result\\
$0^{\text{th}} \implies n^2$\\
$1^{\text{st}} \implies (n-2)^2$\\
$2^{\text{nd}} \implies (n-4)^2$\\
.\\
.\\
$k^{\text{th}} \implies (n-2(k))^2 =0 $ Because, recurrence will over at that step. So the recurrence value should be 0 at $k^{\text{th}}$ step. \\
Therefore, by solving above equation we will get $k=\frac{n}{2}$.
So, the solution of the recurrence is,
$$T(n) = n^2 + n^2 + n^2 +....+(k^{\text{th}} time)n^2 = (\frac{n}{2})\, times\, n^2$$ Since $k=\frac{n}{2}$.
$$T(n) = \Theta(\frac{n}{2}\cdot n^2)$$
$$T(n) = \Theta(n^3)$$

\subsection*{\underline{Problem : 4.3}}
\subsection*{a)}
$T(n)= 4T(\frac{n}{3}) + n\lg n$\\
By master theorem,\\
$a=4, b=3, f(n) = n \lg n, n^{\log_{b}a}=n^{\log_{3}4} \approx n^{1.26}$\\
Since, $f(n)=O(n^{\log_{3}4 - \epsilon}), \epsilon \approx 0.26$\\
We can apply case-1,
$$T(n) = \Theta (n^{\log_{b}a}) = \Theta(n^{\log_{3}4})$$

\subsection*{b)}
$T(n)= 3T(n/3) + \frac{n}{\lg n}$\\
By applying master theorem, \\
$a=3 , b=3, f(n) = n/ \lg n ,  n^{\log_{b}a} = n$\\
Let's say $f(n)= n/ \lg n = O(n)$\\
$\therefore$ case-1 applies,\\
Since $f(n) = n/ \lg n$ is asymptotically less than  $n^{\log_{b}a} = n$, So that $f(n) = O(n^{\log_{b}a}+\epsilon)$ will always true for some positive value of $\epsilon$ , $\epsilon > 0$\\
$\therefore$ According to case-1 ,
$$T(n) = \Theta ( n^{\log_{b}a}) = n $$

\subsection*{c)}
$T(n) = 4T(n/4) + n^2 \cdot \sqrt{n}$\\
$\therefore a=4, b=2, f(n)= n^{5/2} , n^{\log_{b}a} = n^2$\\
case-3 of master theorem applies,
$T(n) = \Omega(n^{\log_{b}a + \epsilon}) , \epsilon = 0.5$\\
Case-3 applies, Let's prove regularity condition for $f(n)$, for sufficient large value of $n$ and $c<1$.\\
$af(\frac{n}{b}) = 4\cdot(\frac{n}{2})^{\frac{5}{2}} = \frac{4}{2^{5/2}} \cdot n^{5/2} \le c \cdot f(n)$\\
As $f(n)=n^{5/2}$, We can take $c=\frac{1}{\sqrt{2}}$ which is less than 1.\\
So, by case-3 the solution to the recurrence is,
$$T(n) = \Theta (f(n)) = \Theta(n^2 \cdot \sqrt{n})$$

\subsection*{d)}
$T(n) =3T(\frac{n}{3} -2) + \frac{n}{2}$\\
This recurrence is not in desired form of,\\
$T(n) = aT(n/b) + f(n)$\\
$\therefore $ Master theorem cannot apply.

\subsection*{e)}
$T(n) = 2T(n/2) + n/ \lg n$
By applying master theorem, \\
$a=2 , b=2, f(n) = n/ \lg n ,  n^{\log_{b}a} = n$\\
Let's say $f(n)= n/ \lg n = O(n)$ ( Same as problem 4.3 b)\\
$\therefore$ case-1 applies,\\
Since $f(n) = n/ \lg n$ is asymptotically less than  $n^{\log_{b}a} = n$, So that $f(n) = O(n^{\log_{b}a}+\epsilon)$ will always true for some positive value of $\epsilon$ , $\epsilon > 0$\\
$\therefore$ According to case-1 ,
$$T(n) = \Theta ( n^{\log_{b}a}) = n $$

\subsection*{f)}
$T(n) = T(n/2) + T(n/4) + T(n/8) + n$\\
This recurrence is not in desired form of,\\
$T(n) = aT(n/b) + f(n)$\\
$\therefore $ Master theorem cannot apply.

\subsection*{g)}
$T(n) = T(n-1) + 1/n$
By applying counting method here,
At $0^{\text{th}}$ step, the recurrence result is $\implies 1/n$.\\
Therefore,\\
Step \,\,\,\,\, Result\\
$0^{\text{th}} \implies 1/n$\\
$1^{\text{st}} \implies 1/(n-1)$\\
$2^{\text{nd}} \implies 1/(n-2)$\\
.\\
.\\
$k^{\text{th}} \implies 1/(n-k)$ The value of $n-k = 1 $ because, denominator cannot be 0. So the recurrence value should be 1 at $k^{\text{th}}$ step. \\
$\therefore $ By solving above equation we will get $n=k-1$.
$$\therefore T(n) = \sum_{k=1}^{n+1} \frac{1}{k} = \lg n$$
$$\therefore T(n) = \Theta (\lg n)$$

\subsection*{h)}
$T(n) = T(n-1) + \lg n$\\
Our recurrence not in the form of
$$T(n) = aT(n/b) +f(n) $$
But we can try another method like counting..\\
At $0^{\text{th}}$ step our recurrence value will be $\lg n$.\\
Therefore,\\
Step \,\,\,\,\,  Result\\
$1^{\text{th}} \implies \lg n$\\
$2^{\text{st}} \implies \lg (n-1)$\\
$3^{\text{nd}} \implies \lg (n-2)$\\
.\\
.\\
$k^{\text{th}} \implies \lg (n-k-1)$\\
$\therefore n=k+1 $\\
So the solution to the recurrence is,
$$T(n) = (\lg n + \lg (n-1) + ....+\lg(n-k-1) = \sum_{k=1}^{n} \lg k = n\lg n$$
$$\therefore T(n) = \Theta(n \cdot \lg n)$$

\subsection*{i)}
$T(n) = T(n-2) + \frac{1}{\lg n}$
Our recurrence not in the form of
$$T(n) = aT(n/b) +f(n) $$
But we can try another method like counting..\\
Therefore,\\
Step \,\,\,\,\,  Result\\
$0^{\text{th}} \implies (n-0)$\\
$1^{\text{st}} \implies (n-2)$\\
$2^{\text{nd}} \implies (n-4)$\\
.\\
.\\
$(k)^{\text{th}} \implies (n-2k) = 0 \implies k=\frac{n}{2}$ , So
$$T(n) = \frac{n}{2} \, times \, \frac{1}{\lg n}$$
$$\therefore T(n)= \Theta (\frac{n}{2} \cdot \frac{1}{\lg n}) = \Theta (\frac{n}{\lg n})$$

\subsection*{j)}
$T(n) = \sqrt{n} T(\sqrt{n}) + n$\\
This recurrence should be in the form of,\\
$T(n) = aT(n/b) + f(n)$ where $a \ge 1 , b > 1 $ are constant And $f(n)$ is a function. \\
In given recurrence $a=\sqrt{n}$ is not a constant.\\
$\therefore $ Master theorem cannot apply.
\end{document}


\input{"preamble.tex"}

\addbibresource{UGA\_Real\_Analysis\_Qual\_Questions.bib}

\let\Begin\begin
\let\End\end
\newcommand\wrapenv[1]{#1}

\makeatletter
\def\ScaleWidthIfNeeded{%
 \ifdim\Gin@nat@width>\linewidth
    \linewidth
  \else
    \Gin@nat@width
  \fi
}
\def\ScaleHeightIfNeeded{%
  \ifdim\Gin@nat@height>0.9\textheight
    0.9\textheight
  \else
    \Gin@nat@width
  \fi
}
\makeatother

\setkeys{Gin}{width=\ScaleWidthIfNeeded,height=\ScaleHeightIfNeeded,keepaspectratio}%

\title{
\textbf{
    UGA Real Analysis Questions (Fall 2014 -- Spring 2021)
  }
  }

\begin{document}

\date{}
\author{D. Zack Garza}
\maketitle

\newpage

% Note: addsec only in KomaScript
\addsec{Table of Contents}
\tableofcontents
\newpage

\hypertarget{preface}{%
\section{Preface}\label{preface}}

I'd like to extend my gratitude to Peter Woolfitt for supplying many
solutions and checking many proofs of the rest in problem sessions. Many
other solutions contain input and ideas from other graduate students and
faculty members at UGA, along with questions and answers posted on Math
Stack Exchange or Math Overflow.

\hypertarget{undergraduate-analysis-uniform-convergence}{%
\section{Undergraduate Analysis: Uniform
Convergence}\label{undergraduate-analysis-uniform-convergence}}

\hypertarget{fall-2018-1-done}{%
\subsection{\texorpdfstring{Fall 2018 \# 1
\(\done\)}{Fall 2018 \# 1 \textbackslash done}}\label{fall-2018-1-done}}

Let \(f(x) = \frac 1 x\). Show that \(f\) is uniformly continuous on
\((1, \infty)\) but not on \((0,\infty)\).

Relevant concepts omitted.

Solution omitted.

\hypertarget{fall-2017-1-done}{%
\subsection{\texorpdfstring{Fall 2017 \# 1
\(\done\)}{Fall 2017 \# 1 \textbackslash done}}\label{fall-2017-1-done}}

Let
\begin{align*}
f(x) = \sum _{n=0}^{\infty} \frac{x^{n}}{n !}.
\end{align*}

Describe the intervals on which \(f\) does and does not converge
uniformly.

Relevant concepts omitted.

Solution omitted.

\hypertarget{fall-2014-1-done}{%
\subsection{\texorpdfstring{Fall 2014 \# 1
\(\done\)}{Fall 2014 \# 1 \textbackslash done}}\label{fall-2014-1-done}}

Let \(\left\{{f_n}\right\}\) be a sequence of continuous functions such
that \(\sum f_n\) converges uniformly.

Prove that \(\sum f_n\) is also continuous.

\envlist

Relevant concepts omitted.

Solution omitted.

\hypertarget{spring-2017-4-done}{%
\subsection{\texorpdfstring{Spring 2017 \# 4
\(\done\)}{Spring 2017 \# 4 \textbackslash done}}\label{spring-2017-4-done}}

Let \(f(x, y)\) on \([-1, 1]^2\) be defined by
\begin{align*}
f(x, y) = \begin{cases}
\frac{x y}{\left(x^{2}+y^{2}\right)^{2}} & (x, y) \neq (0, 0) \\
0 & (x, y) = (0, 0)
\end{cases}
\end{align*}
Determine if \(f\) is integrable.

Relevant concepts omitted.

Solution omitted.

\hypertarget{spring-2015-1-done}{%
\subsection{\texorpdfstring{Spring 2015 \# 1
\(\done\)}{Spring 2015 \# 1 \textbackslash done}}\label{spring-2015-1-done}}

Let \((X, d)\) and \((Y, \rho)\) be metric spaces, \(f: X\to Y\), and
\(x_0 \in X\).

Prove that the following statements are equivalent:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  For every \(\varepsilon > 0 \quad \exists \delta > 0\) such that
  \(\rho( f(x), f(x_0) ) < \varepsilon\) whenever
  \(d(x, x_0) < \delta\).
\item
  The sequence \(\left\{{f(x_n)}\right\}_{n=1}^\infty \to f(x_0)\) for
  every sequence \(\left\{{x_n}\right\} \to x_0\) in \(X\).
\end{enumerate}

Relevant concepts omitted.

Solution omitted.

\hypertarget{fall-2014-2-work}{%
\subsection{\texorpdfstring{Fall 2014 \# 2
\(\work\)}{Fall 2014 \# 2 \textbackslash work}}\label{fall-2014-2-work}}

Let \(I\) be an index set and \(\alpha: I \to (0, \infty)\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that
  \begin{align*}
  \sum_{i \in I} a(i):=\sup _{\substack{ J \subset I \\ J \text { finite }}} \sum_{i \in J} a(i)<\infty \implies I \text{ is countable.}
  \end{align*}
\item
  Suppose \(I = {\mathbb{Q}}\) and
  \(\sum_{q \in \mathbb{Q}} a(q)<\infty\). Define
  \begin{align*}
  f(x):=\sum_{\substack{q \in \mathbb{Q}\\ q \leq x}} a(q).
  \end{align*}
  Show that \(f\) is continuous at \(x \iff x\not\in {\mathbb{Q}}\).
\end{enumerate}

\todo[inline]{Stuck on part b}

Solution omitted.

\hypertarget{spring-2014-2-done}{%
\subsection{\texorpdfstring{Spring 2014 \# 2
\(\done\)}{Spring 2014 \# 2 \textbackslash done}}\label{spring-2014-2-done}}

Let \(\left\{{a_n}\right\}\) be a sequence of real numbers such that
\begin{align*}
\left\{{b_n}\right\} \in \ell^2({\mathbb{N}}) \implies \sum a_n b_n < \infty.
\end{align*}
Show that \(\sum a_n^2 < \infty\).

\begin{quote}
Note: Assume \(a_n, b_n\) are all non-negative.
\end{quote}

\todo[inline]{Have someone check!}

Solution omitted.

\hypertarget{general-analysis}{%
\section{General Analysis}\label{general-analysis}}

\hypertarget{spring-2020-1-done}{%
\subsection{\texorpdfstring{Spring 2020 \# 1
\(\done\)}{Spring 2020 \# 1 \textbackslash done}}\label{spring-2020-1-done}}

Prove that if \(f: [0, 1] \to {\mathbb{R}}\) is continuous then
\begin{align*}
\lim_{k\to\infty} \int_0^1 kx^{k-1} f(x) \,dx = f(1)
.\end{align*}

Relevant concepts omitted.

Solution omitted.

\hypertarget{fall-2019-1-done}{%
\subsection{\texorpdfstring{Fall 2019 \# 1
\(\done\)}{Fall 2019 \# 1 \textbackslash done}}\label{fall-2019-1-done}}

Let \(\{a_n\}_{n=1}^\infty\) be a sequence of real numbers.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Prove that if \(\displaystyle\lim_{n\to \infty } a_n = 0\), then
  \begin{align*}
  \lim _{n \rightarrow \infty} \frac{a_{1}+\cdots+a_{n}}{n}=0
  \end{align*}
\item
  Prove that if \(\displaystyle\sum_{n=1}^{\infty} \frac{a_{n}}{n}\)
  converges, then
  \begin{align*}
  \lim _{n \rightarrow \infty} \frac{a_{1}+\cdots+a_{n}}{n}=0
  \end{align*}
\end{enumerate}

Solution omitted.

\hypertarget{fall-2018-4-done}{%
\subsection{\texorpdfstring{Fall 2018 \# 4
\(\done\)}{Fall 2018 \# 4 \textbackslash done}}\label{fall-2018-4-done}}

Let \(f\in L^1([0, 1])\). Prove that
\begin{align*}
\lim_{n \to \infty} \int_{0}^{1} f(x) {\left\lvert {\sin n x} \right\rvert} ~d x= \frac{2}{\pi} \int_{0}^{1} f(x) ~d x
\end{align*}

\begin{quote}
Hint: Begin with the case that \(f\) is the characteristic function of
an interval.
\end{quote}

\todo[inline]{Ask someone to check the last approximation part.}

Solution omitted.

\hypertarget{fall-2017-4-done}{%
\subsection{\texorpdfstring{Fall 2017 \# 4
\(\done\)}{Fall 2017 \# 4 \textbackslash done}}\label{fall-2017-4-done}}

Let
\begin{align*}
f_{n}(x) = n x(1-x)^{n}, \quad n \in {\mathbb{N}}.
\end{align*}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that \(f_n \to 0\) pointwise but not uniformly on \([0, 1]\).
\item
  Show that
  \begin{align*}
  \lim _{n \to \infty} \int _{0}^{1} n(1-x)^{n} \sin x \, dx = 0
  \end{align*}
\end{enumerate}

\begin{quote}
Hint for (a): Consider the maximum of \(f_n\).
\end{quote}

Solution omitted.

\hypertarget{spring-2017-3-work}{%
\subsection{\texorpdfstring{Spring 2017 \# 3
\(\work\)}{Spring 2017 \# 3 \textbackslash work}}\label{spring-2017-3-work}}

Let
\begin{align*}
f_{n}(x) = a e^{-n a x} - b e^{-n b x} \quad \text{ where } 0 < a < b.
\end{align*}

Show that

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
  \(\sum_{n=1}^{\infty} \left|f_{n}\right|\) is not in
  \(L^{1}([0, \infty), m)\)
\end{enumerate}

\begin{quote}
Hint: \(f_n(x)\) has a root \(x_n\).
\end{quote}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\setcounter{enumi}{1}
\tightlist
\item

  \begin{align*}
  \sum_{n=1}^{\infty} f_{n} \text { is in } L^{1}([0, \infty), m)
  {\quad \operatorname{and} \quad}
  \int _{0}^{\infty} \sum _{n=1}^{\infty} f_{n}(x) \,dm = \ln \frac{b}{a}
  \end{align*}
  \todo[inline]{Not complete.} \todo[inline]{Add concepts.}
  \todo[inline]{Walk through.}
\end{enumerate}

Solution omitted.

\hypertarget{fall-2016-1-done}{%
\subsection{\texorpdfstring{Fall 2016 \# 1
\(\done\)}{Fall 2016 \# 1 \textbackslash done}}\label{fall-2016-1-done}}

Define
\begin{align*}
f(x) = \sum_{n=1}^{\infty} \frac{1}{n^{x}}.
\end{align*}
Show that \(f\) converges to a differentiable function on
\((1, \infty)\) and that
\begin{align*}
f'(x)  =\sum_{n=1}^{\infty}\left(\frac{1}{n^{x}}\right)^{\prime}.
\end{align*}

\begin{quote}
Hint:
\begin{align*}
\left(\frac{1}{n^{x}}\right)' = -\frac{1}{n^{x}} \ln n
\end{align*}
\end{quote}

\todo[inline]{Add concepts.}

Solution omitted.

\hypertarget{fall-2016-5-done}{%
\subsection{\texorpdfstring{Fall 2016 \# 5
\(\done\)}{Fall 2016 \# 5 \textbackslash done}}\label{fall-2016-5-done}}

Let \(\phi\in L^\infty({\mathbb{R}})\). Show that the following limit
exists and satisfies the equality
\begin{align*}
\lim _{n \to \infty} \left(\int _{\mathbb{R}} \frac{|\phi(x)|^{n}}{1+x^{2}} \, dx \right) ^ {\frac{1}{n}}
= {\left\lVert {\phi} \right\rVert}_\infty.
\end{align*}
\todo[inline]{Add concepts.}

Solution omitted.

\hypertarget{fall-2016-6-done}{%
\subsection{\texorpdfstring{Fall 2016 \# 6
\(\done\)}{Fall 2016 \# 6 \textbackslash done}}\label{fall-2016-6-done}}

Let \(f, g \in L^2({\mathbb{R}})\). Show that
\begin{align*}
\lim _{n \to \infty} \int _{{\mathbb{R}}} f(x) g(x+n) \,dx = 0
\end{align*}

\todo[inline]{Rewrite solution.}

Relevant concepts omitted.

Solution omitted.

\hypertarget{spring-2016-1-work}{%
\subsection{\texorpdfstring{Spring 2016 \# 1
\(\work\)}{Spring 2016 \# 1 \textbackslash work}}\label{spring-2016-1-work}}

For \(n\in {\mathbb{N}}\), define
\begin{align*}
e_{n} = \left (1+ {1\over n} \right)^{n}
{\quad \operatorname{and} \quad}
E_{n} = \left( 1+ {1\over n} \right)^{n+1}
\end{align*}

Show that \(e_n < E_n\), and prove Bernoulli's inequality:
\begin{align*}
(1+x)^n \geq 1+nx && -1 < x < \infty  ,\,\, n\in {\mathbb{N}}
.\end{align*}

Use this to show the following:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  The sequence \(e_n\) is increasing.
\item
  The sequence \(E_n\) is decreasing.
\item
  \(2 < e_n < E_n < 4\).
\item
  \(\lim _{n \to \infty} e_{n} = \lim _{n \to \infty} E_{n}\).
\end{enumerate}

\hypertarget{fall-2015-1-work}{%
\subsection{\texorpdfstring{Fall 2015 \# 1
\(\work\)}{Fall 2015 \# 1 \textbackslash work}}\label{fall-2015-1-work}}

Define
\begin{align*}
f(x)=c_{0}+c_{1} x^{1}+c_{2} x^{2}+\ldots+c_{n} x^{n} \text { with } n \text { even and } c_{n}>0.
\end{align*}

Show that there is a number \(x_m\) such that \(f(x_m) \leq f(x)\) for
all \(x\in {\mathbb{R}}\).

\hypertarget{fall-2020-1}{%
\subsection{Fall 2020 \# 1}\label{fall-2020-1}}

Show that if \(x_n\) is a decreasing sequence of positive real numbers
such that \(\sum_{n=1}^\infty x_n\) converges, then
\begin{align*}
\lim_{n\to\infty} n x_n = 0.
\end{align*}

\hypertarget{fall-2020-3}{%
\subsection{Fall 2020 \# 3}\label{fall-2020-3}}

Let \(f\) be a non-negative Lebesgue measurable function on
\([1, \infty)\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Prove that
  \begin{align*}
  1 \leq \qty{
  {1 \over b-a} \int_a^b f(x) \,dx
  }\qty{
  {1\over b-a} \int_a^b {1 \over f(x)}\, dx
  }
  \end{align*}
  for any \(1\leq a < b <\infty\).
\item
  Prove that if \(f\) satisfies
  \begin{align*}
  \int_1^t f(x) \, dx \leq t^2 \log(t)
  \end{align*}
  for all \(t\in [1, \infty)\), then
  \begin{align*}
  \int_1^\infty {1\over f(x) \,dx} = \infty
  .\end{align*}
\end{enumerate}

\begin{quote}
Hint: write
\begin{align*}
\int_1^\infty {1\over f(x) \, dx} = \sum_{k=0}^\infty \int_{2^k}^{2^{k+1}} {1 \over f(x)}\,dx
.\end{align*}
\end{quote}

\hypertarget{measure-theory-sets}{%
\section{Measure Theory: Sets}\label{measure-theory-sets}}

\hypertarget{spring-2020-2-done}{%
\subsection{\texorpdfstring{Spring 2020 \# 2
\(\done\)}{Spring 2020 \# 2 \textbackslash done}}\label{spring-2020-2-done}}

Let \(m_*\) denote the Lebesgue outer measure on \({\mathbb{R}}\).

\hypertarget{a.}{%
\subsubsection{a.}\label{a.}}

Prove that for every \(E\subseteq {\mathbb{R}}\) there exists a Borel
set \(B\) containing \(E\) such that
\begin{align*}
m_*(B) = m_*(E)
.\end{align*}

\hypertarget{b.}{%
\subsubsection{b.}\label{b.}}

Prove that if \(E\subseteq {\mathbb{R}}\) has the property that
\begin{align*}
m_*(A) = m_*(A\cap E) + m_*(A\cap E^c)
\end{align*}
for every set \(A\subseteq {\mathbb{R}}\), then there exists a Borel set
\(B\subseteq {\mathbb{R}}\) such that \(E = B\setminus N\) with
\(m_*(N) = 0\).

Be sure to address the case when \(m_*(E) = \infty\).

Solution omitted.

\hypertarget{fall-2019-3.-done}{%
\subsection{\texorpdfstring{Fall 2019 \# 3.
\(\done\)}{Fall 2019 \# 3. \textbackslash done}}\label{fall-2019-3.-done}}

Let \((X, \mathcal B, \mu)\) be a measure space with \(\mu(X) = 1\) and
\(\{B_n\}_{n=1}^\infty\) be a sequence of \(\mathcal B\)-measurable
subsets of \(X\), and
\begin{align*}
B \coloneqq\left\{{x\in X {~\mathrel{\Big|}~}x\in B_n \text{ for infinitely many } n}\right\}.
\end{align*}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Argue that \(B\) is also a \(\mathcal{B} {\hbox{-}}\)measurable subset
  of \(X\).
\item
  Prove that if \(\sum_{n=1}^\infty \mu(B_n) < \infty\) then
  \(\mu(B)= 0\).
\item
  Prove that if \(\sum_{n=1}^\infty \mu(B_n) = \infty\) \textbf{and} the
  sequence of set complements \(\left\{{B_n^c}\right\}_{n=1}^\infty\)
  satisfies
  \begin{align*}
  \mu\left(\bigcap_{n=k}^{K} B_{n}^{c}\right)=\prod_{n=k}^{K}\left(1-\mu\left(B_{n}\right)\right)
  \end{align*}
  for all positive integers \(k\) and \(K\) with \(k < K\), then
  \(\mu(B) = 1\).
\end{enumerate}

\begin{quote}
Hint: Use the fact that \(1 - x ≤ e^{-x}\) for all \(x\).
\end{quote}

Solution omitted.

\hypertarget{spring-2019-2-done}{%
\subsection{\texorpdfstring{Spring 2019 \# 2
\(\done\)}{Spring 2019 \# 2 \textbackslash done}}\label{spring-2019-2-done}}

Let \(\mathcal B\) denote the set of all Borel subsets of
\({\mathbb{R}}\) and \(\mu : \mathcal B \to [0, \infty)\) denote a
finite Borel measure on \({\mathbb{R}}\).

\hypertarget{a-2}{%
\subsubsection{a}\label{a-2}}

Prove that if \(\{F_k\}\) is a sequence of Borel sets for which
\(F_k \supseteq F_{k+1}\) for all \(k\), then
\begin{align*}
\lim _{k \rightarrow \infty} \mu\left(F_{k}\right)=\mu\left(\bigcap_{k=1}^{\infty} F_{k}\right)
\end{align*}

\hypertarget{b-2}{%
\subsubsection{b}\label{b-2}}

Suppose \(\mu\) has the property that \(\mu (E) = 0\) for every
\(E \in \mathcal B\) with Lebesgue measure \(m(E) = 0\).

Prove that for every \(\epsilon > 0\) there exists \(\delta > 0\) so
that if \(E \in \mathcal B\) with \(m(E) < δ\), then \(\mu(E) < ε\).

\todo[inline]{Add concepts.}

Solution omitted.

\todo[inline]{All messed up!}

\hypertarget{fall-2018-2-done}{%
\subsection{\texorpdfstring{Fall 2018 \# 2
\(\done\)}{Fall 2018 \# 2 \textbackslash done}}\label{fall-2018-2-done}}

Let \(E\subset {\mathbb{R}}\) be a Lebesgue measurable set. Show that
there is a Borel set \(B \subset E\) such that \(m(E\setminus B) = 0\).

\todo[inline]{Move this to review notes to clean things up.}

Solution omitted.

\hypertarget{spring-2018-1-done}{%
\subsection{\texorpdfstring{Spring 2018 \# 1
\(\done\)}{Spring 2018 \# 1 \textbackslash done}}\label{spring-2018-1-done}}

Define
\begin{align*}
E:=\left\{x \in \mathbb{R}:\left|x-\frac{p}{q}\right|<q^{-3} \text { for infinitely many } p, q \in \mathbb{N}\right\}.
\end{align*}

Prove that \(m(E) = 0\).

Solution omitted.

\hypertarget{fall-2017-2-done}{%
\subsection{\texorpdfstring{Fall 2017 \# 2
\(\done\)}{Fall 2017 \# 2 \textbackslash done}}\label{fall-2017-2-done}}

Let \(f(x) = x^2\) and
\(E \subset [0, \infty) \coloneqq{\mathbb{R}}^+\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  Show that
  \begin{align*}
  m^*(E) = 0 \iff m^*(f(E)) = 0.
  \end{align*}
\item
  Deduce that the map
\end{enumerate}

\begin{align*}
\phi: \mathcal{L}({\mathbb{R}}^+) &\to \mathcal{L}({\mathbb{R}}^+) \\
E &\mapsto f(E)
\end{align*}
is a bijection from the class of Lebesgue measurable sets of
\([0, \infty)\) to itself.

\todo[inline]{Walk through.}

Solution omitted.

\hypertarget{spring-2017-2-done}{%
\subsection{\texorpdfstring{Spring 2017 \# 2
\(\done\)}{Spring 2017 \# 2 \textbackslash done}}\label{spring-2017-2-done}}

\hypertarget{a-5}{%
\subsubsection{a}\label{a-5}}

Let \(\mu\) be a measure on a measurable space \((X, \mathcal M)\) and
\(f\) a positive measurable function.

Define a measure \(\lambda\) by
\begin{align*}
\lambda(E):=\int_{E} f ~d \mu, \quad E \in \mathcal{M}
\end{align*}

Show that for \(g\) any positive measurable function,
\begin{align*}
\int_{X} g ~d \lambda=\int_{X} f g ~d \mu
\end{align*}

\hypertarget{b-5}{%
\subsubsection{b}\label{b-5}}

Let \(E \subset {\mathbb{R}}\) be a measurable set such that
\begin{align*}
\int_{E} x^{2} ~d m=0.
\end{align*}
Show that \(m(E) = 0\).

Solution omitted.

\hypertarget{fall-2016-4-done}{%
\subsection{\texorpdfstring{Fall 2016 \# 4
\(\done\)}{Fall 2016 \# 4 \textbackslash done}}\label{fall-2016-4-done}}

Let \((X, \mathcal M, \mu)\) be a measure space and suppose
\(\left\{{E_n}\right\} \subset \mathcal M\) satisfies
\begin{align*}
\lim _{n \rightarrow \infty} \mu\left(X \backslash E_{n}\right)=0.
\end{align*}

Define
\begin{align*}
G \coloneqq\left\{{x\in X {~\mathrel{\Big|}~}x\in E_n \text{ for only finitely many  } n}\right\}.
\end{align*}

Show that \(G \in \mathcal M\) and \(\mu(G) = 0\).

\todo[inline]{Add concepts.}

Solution omitted.

\hypertarget{spring-2016-3-work}{%
\subsection{\texorpdfstring{Spring 2016 \# 3
\(\work\)}{Spring 2016 \# 3 \textbackslash work}}\label{spring-2016-3-work}}

Let \(f\) be Lebesgue measurable on \({\mathbb{R}}\) and
\(E \subset {\mathbb{R}}\) be measurable such that
\begin{align*}
0<A=\int_{E} f(x) d x<\infty.
\end{align*}

Show that for every \(0 < t < 1\), there exists a measurable set
\(E_t \subset E\) such that
\begin{align*}
\int_{E_{t}} f(x) d x=t A.
\end{align*}

\hypertarget{spring-2016-5-work}{%
\subsection{\texorpdfstring{Spring 2016 \# 5
\(\work\)}{Spring 2016 \# 5 \textbackslash work}}\label{spring-2016-5-work}}

Let \((X, \mathcal M, \mu)\) be a measure space. For \(f\in L^1(\mu)\)
and \(\lambda > 0\), define
\begin{align*}
\phi(\lambda)=\mu(\{x \in X | f(x)>\lambda\})
\quad \text { and } \quad
\psi(\lambda)=\mu(\{x \in X | f(x)<-\lambda\})
\end{align*}

Show that \(\phi, \psi\) are Borel measurable and
\begin{align*}
\int_{X}|f| ~d \mu=\int_{0}^{\infty}[\phi(\lambda)+\psi(\lambda)] ~d \lambda
\end{align*}

\hypertarget{fall-2015-2-work}{%
\subsection{\texorpdfstring{Fall 2015 \# 2
\(\work\)}{Fall 2015 \# 2 \textbackslash work}}\label{fall-2015-2-work}}

Let \(f: {\mathbb{R}}\to {\mathbb{R}}\) be Lebesgue measurable.

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that there is a sequence of simple functions \(s_n(x)\) such that
  \(s_n(x) \to f(x)\) for all \(x\in {\mathbb{R}}\).
\item
  Show that there is a Borel measurable function \(g\) such that
  \(g = f\) almost everywhere.
\end{enumerate}

\hypertarget{spring-2015-3-work}{%
\subsection{\texorpdfstring{Spring 2015 \# 3
\(\work\)}{Spring 2015 \# 3 \textbackslash work}}\label{spring-2015-3-work}}

Let \(\mu\) be a finite Borel measure on \({\mathbb{R}}\) and
\(E \subset {\mathbb{R}}\) Borel. Prove that the following statements
are equivalent:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(\forall \varepsilon > 0\) there exists \(G\) open and \(F\) closed
  such that
  \begin{align*}
  F \subseteq E \subseteq G \quad \text{and} \quad \mu(G\setminus F) < \varepsilon.
  \end{align*}
\item
  There exists a \(V \in G_\delta\) and \(H \in F_\sigma\) such that
  \begin{align*}
  H \subseteq E \subseteq V \quad \text{and}\quad \mu(V\setminus H) = 0
  \end{align*}
\end{enumerate}

\hypertarget{spring-2014-3-work}{%
\subsection{\texorpdfstring{Spring 2014 \# 3
\(\work\)}{Spring 2014 \# 3 \textbackslash work}}\label{spring-2014-3-work}}

Let \(f: {\mathbb{R}}\to {\mathbb{R}}\) and suppose
\begin{align*}
\forall x\in {\mathbb{R}},\quad f(x) \geq \limsup _{y \rightarrow x} f(y)
\end{align*}
Prove that \(f\) is Borel measurable.

\hypertarget{spring-2014-4-work}{%
\subsection{\texorpdfstring{Spring 2014 \# 4
\(\work\)}{Spring 2014 \# 4 \textbackslash work}}\label{spring-2014-4-work}}

Let \((X, \mathcal M, \mu)\) be a measure space and suppose \(f\) is a
measurable function on \(X\). Show that
\begin{align*}
\lim _{n \rightarrow \infty} \int_{X} f^{n} ~d \mu =
\begin{cases}
\infty & \text{or} \\
\mu(f^{-1}(1)),
\end{cases}
\end{align*}
and characterize the collection of functions of each type.

\hypertarget{spring-2017-1-done}{%
\subsection{\texorpdfstring{Spring 2017 \# 1
\(\done\)}{Spring 2017 \# 1 \textbackslash done}}\label{spring-2017-1-done}}

Let \(K\) be the set of numbers in \([0, 1]\) whose decimal expansions
do not use the digit \(4\).

\begin{quote}
We use the convention that when a decimal number ends with 4 but all
other digits are different from 4, we replace the digit \(4\) with
\(399\cdots\). For example, \(0.8754 = 0.8753999\cdots\).
\end{quote}

Show that \(K\) is a compact, nowhere dense set without isolated points,
and find the Lebesgue measure \(m(K)\).

Solution omitted.

\hypertarget{spring-2016-2-work}{%
\subsection{\texorpdfstring{Spring 2016 \# 2
\(\work\)}{Spring 2016 \# 2 \textbackslash work}}\label{spring-2016-2-work}}

Let \(0 < \lambda < 1\) and construct a Cantor set \(C_\lambda\) by
successively removing middle intervals of length \(\lambda\).

Prove that \(m(C_\lambda) = 0\).

\hypertarget{measure-theory-functions}{%
\section{Measure Theory: Functions}\label{measure-theory-functions}}

\hypertarget{fall-2016-2-done}{%
\subsection{\texorpdfstring{Fall 2016 \# 2
\(\done\)}{Fall 2016 \# 2 \textbackslash done}}\label{fall-2016-2-done}}

Let \(f, g: [a, b] \to {\mathbb{R}}\) be measurable with
\begin{align*}
\int_{a}^{b} f(x) ~d x=\int_{a}^{b} g(x) ~d x.
\end{align*}

Show that either

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(f(x) = g(x)\) almost everywhere, or
\item
  There exists a measurable set \(E \subset [a, b]\) such that
  \begin{align*}]
  \int _{E} f(x) \, dx > \int _{E} g(x) \, dx
  \end{align*}
\end{enumerate}

\todo[inline]{Add concepts.}

Solution omitted.

\hypertarget{spring-2016-4-work}{%
\subsection{\texorpdfstring{Spring 2016 \# 4
\(\work\)}{Spring 2016 \# 4 \textbackslash work}}\label{spring-2016-4-work}}

Let \(E \subset {\mathbb{R}}\) be measurable with \(m(E) < \infty\).
Define
\begin{align*}
f(x)=m(E \cap(E+x)).
\end{align*}

Show that

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(f\in L^1({\mathbb{R}})\).
\item
  \(f\) is uniformly continuous.
\item
  \(\lim _{|x| \to \infty} f(x) = 0\).
\end{enumerate}

\begin{quote}
Hint:
\begin{align*}
\chi_{E \cap(E+x)}(y)=\chi_{E}(y) \chi_{E}(y-x)
\end{align*}
\end{quote}

\hypertarget{spring-2021-1}{%
\subsection{Spring 2021 \# 1}\label{spring-2021-1}}

Let \((X, \mathcal{M},\mu)\) be a measure space and let
\(E_n \in \mathcal{M}\) be a measurable set for \(n\geq 1\). Let
\(f_n \coloneqq\chi_{E_n}\) be the indicator function of the set \(E\)
and show that

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(f_n \overset{n\to\infty}\to 1\) uniformly \(\iff\) there exists
  \(N\in |NN\) such that \(E_n = X\) for all \(n\geq N\).
\item
  \(f_n(x) \overset{n\to\infty}\to 1\) for almost every \(x\) \(\iff\)
  \begin{align*}
  \mu \qty{ \bigcap_{n \geq 0} \bigcup_{k \geq n} (X \setminus E_k) } = 0
  .\end{align*}
\end{enumerate}

\hypertarget{spring-2021-3}{%
\subsection{Spring 2021 \# 3}\label{spring-2021-3}}

Let \((X, \mathcal{M}, \mu)\) be a finite measure space and let
\(\left\{{ f_n}\right\}_{n=1}^{\infty } \subseteq L^1(X, \mu)\). Suppose
\(f\in L^1(X, \mu)\) such that \(f_n(x) \overset{n\to \infty }\to f(x)\)
for almost every \(x \in X\). Prove that for every \(\varepsilon> 0\)
there exists \(M>0\) and a set \(E\subseteq X\) such that
\(\mu(E) \leq \varepsilon\) and
\({\left\lvert {f_n(x)} \right\rvert}\leq M\) for all
\(x\in X\setminus E\) and all \(n\in {\mathbb{N}}\).

\hypertarget{fall-2020-2}{%
\subsection{Fall 2020 \# 2}\label{fall-2020-2}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(f: {\mathbb{R}}\to {\mathbb{R}}\). Prove that
  \begin{align*}
  f(x) \leq \liminf_{y\to x} f(y)~ \text{for each}~ x\in {{\mathbb{R}}} \iff \{ x\in {{\mathbb{R}}} \mathrel{\Big|}f(x) > a \}~\text{is open for all}~ a\in {{\mathbb{R}}}
  \end{align*}
\item
  Recall that a function \(f: {{\mathbb{R}}} \to {{\mathbb{R}}}\) is
  called \emph{lower semi-continuous} iff it satisfies either condition
  in part (a) above.
\end{enumerate}

Prove that if \(\mathcal{F}\) is an y family of lower semi-continuous
functions, then
\begin{align*}
g(x) = \sup\{ f(x) \mathrel{\Big|}f\in \mathcal{F}\}
\end{align*}
is Borel measurable.

\begin{quote}
Note that \(\mathcal{F}\) need not be a countable family.
\end{quote}

\hypertarget{integrals-convergence}{%
\section{Integrals: Convergence}\label{integrals-convergence}}

\hypertarget{fall-2019-2-done}{%
\subsection{\texorpdfstring{Fall 2019 \# 2
\(\done\)}{Fall 2019 \# 2 \textbackslash done}}\label{fall-2019-2-done}}

Prove that
\begin{align*}
\left| \frac{d^{n}}{d x^{n}} \frac{\sin x}{x}\right| \leq \frac{1}{n}
\end{align*}

for all \(x \neq 0\) and positive integers \(n\).

\begin{quote}
Hint: Consider \(\displaystyle\int_0^1 \cos(tx) dt\)
\end{quote}

Solution omitted.

\hypertarget{spring-2020-5-done}{%
\subsection{\texorpdfstring{Spring 2020 \# 5
\(\done\)}{Spring 2020 \# 5 \textbackslash done}}\label{spring-2020-5-done}}

Compute the following limit and justify your calculations:
\begin{align*}
\lim_{n\to\infty} \int_0^n \qty{1 + {x^2 \over n}}^{-(n+1)} \,dx
.\end{align*}

\todo[inline]{Not finished, flesh out.}
\todo[inline]{Walk through.}

Solution omitted.

\hypertarget{spring-2019-3-done}{%
\subsection{\texorpdfstring{Spring 2019 \# 3
\(\done\)}{Spring 2019 \# 3 \textbackslash done}}\label{spring-2019-3-done}}

Let \(\{f_k\}\) be any sequence of functions in \(L^2([0, 1])\)
satisfying \({\left\lVert {f_k} \right\rVert}_2 ≤ M\) for all
\(k ∈ {\mathbb{N}}\).

Prove that if \(f_k \to f\) almost everywhere, then \(f ∈ L^2([0, 1])\)
with \({\left\lVert {f} \right\rVert}_2 ≤ M\) and
\begin{align*}
\lim _{k \rightarrow \infty} \int_{0}^{1} f_{k}(x) dx = \int_{0}^{1} f(x) d x
\end{align*}

\begin{quote}
Hint: Try using Fatou's Lemma to show that
\({\left\lVert {f} \right\rVert}_2 ≤ M\) and then try applying Egorov's
Theorem.
\end{quote}

Solution omitted.

\hypertarget{fall-2018-6-done}{%
\subsection{\texorpdfstring{Fall 2018 \# 6
\(\done\)}{Fall 2018 \# 6 \textbackslash done}}\label{fall-2018-6-done}}

Compute the following limit and justify your calculations:
\begin{align*}
\lim_{n \rightarrow \infty} \int_{1}^{n} \frac{d x}{\left(1+\frac{x}{n}\right)^{n} \sqrt[n]{x}}
\end{align*}

\todo[inline]{Add concepts.}

Solution omitted.

\hypertarget{fall-2018-3-done}{%
\subsection{\texorpdfstring{Fall 2018 \# 3
\(\done\)}{Fall 2018 \# 3 \textbackslash done}}\label{fall-2018-3-done}}

Suppose \(f(x)\) and \(xf(x)\) are integrable on \({\mathbb{R}}\).
Define \(F\) by
\begin{align*}
F(t)\coloneqq\int _{-\infty}^{\infty} f(x) \cos (x t) dx
\end{align*}
Show that
\begin{align*}
F'(t)=-\int _{-\infty}^{\infty} x f(x) \sin (x t) dx
.\end{align*}

\todo[inline]{Walk through.}

Solution omitted.

\hypertarget{spring-2018-5-done}{%
\subsection{\texorpdfstring{Spring 2018 \# 5
\(\done\)}{Spring 2018 \# 5 \textbackslash done}}\label{spring-2018-5-done}}

Suppose that

\begin{itemize}
\tightlist
\item
  \(f_n, f \in L^1\),
\item
  \(f_n \to f\) almost everywhere, and
\item
  \(\int\left|f_{n}\right| \rightarrow \int|f|\).
\end{itemize}

Show that \(\int f_{n} \rightarrow \int f\).

Solution omitted.

\hypertarget{spring-2018-2-done}{%
\subsection{\texorpdfstring{Spring 2018 \# 2
\(\done\)}{Spring 2018 \# 2 \textbackslash done}}\label{spring-2018-2-done}}

Let
\begin{align*}
f_{n}(x):=\frac{x}{1+x^{n}}, \quad x \geq 0.
\end{align*}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that this sequence converges pointwise and find its limit. Is the
  convergence uniform on \([0, \infty)\)?
\item
  Compute
  \begin{align*}
  \lim _{n \rightarrow \infty} \int_{0}^{\infty} f_{n}(x) d x
  \end{align*}
\end{enumerate}

\todo[inline]{Add concepts.}

Solution omitted.

\hypertarget{fall-2016-3-done}{%
\subsection{\texorpdfstring{Fall 2016 \# 3
\(\done\)}{Fall 2016 \# 3 \textbackslash done}}\label{fall-2016-3-done}}

Let \(f\in L^1({\mathbb{R}})\). Show that
\begin{align*}
\lim _{x \to 0} \int _{{\mathbb{R}}} {\left\lvert {f(y-x)-f(y)} \right\rvert} \, dy = 0
\end{align*}
\todo[inline]{Missing some stuff.}

Solution omitted.

\hypertarget{fall-2015-3-work}{%
\subsection{\texorpdfstring{Fall 2015 \# 3
\(\work\)}{Fall 2015 \# 3 \textbackslash work}}\label{fall-2015-3-work}}

Compute the following limit:
\begin{align*}
\lim _{n \rightarrow \infty} \int_{1}^{n} \frac{n e^{-x}}{1+n x^{2}} \, \sin \left(\frac x n\right) \, dx
\end{align*}

\hypertarget{fall-2015-4-work}{%
\subsection{\texorpdfstring{Fall 2015 \# 4
\(\work\)}{Fall 2015 \# 4 \textbackslash work}}\label{fall-2015-4-work}}

Let \(f: [1, \infty) \to {\mathbb{R}}\) such that \(f(1) = 1\) and
\begin{align*}
f^{\prime}(x)= \frac{1} {x^{2}+f(x)^{2}}
\end{align*}

Show that the following limit exists and satisfies the equality
\begin{align*}
\lim _{x \rightarrow \infty} f(x) \leq 1 + \frac \pi 4
\end{align*}

\hypertarget{spring-2021-2}{%
\subsection{Spring 2021 \# 2}\label{spring-2021-2}}

Calculate the following limit, justifying each step of your calculation:

\begin{align*}
L \coloneqq\lim_{n\to \infty} \int_0^n { \cos\qty{x\over n} \over x^2 + \cos\qty{x\over n} }\,dx
.\end{align*}

\hypertarget{spring-2021-5}{%
\subsection{Spring 2021 \# 5}\label{spring-2021-5}}

Let \(f_n \in L^2([0, 1])\) for \(n\in {\mathbb{N}}\), and assume that

\begin{itemize}
\item
  \({\left\lVert {f_n} \right\rVert}_2 \leq n^{-51 \over 100}\) for all
  \(n\in {\mathbb{N}}\),
\item
  \(hat{f}_n\) is supported in the interval \([2^n, 2^{n+1}]\), so
  \begin{align*}
  \widehat{f}_n(\xi) \coloneqq\int_0^1 f_n(x) e^{2\pi i \xi \cdot x} \,dx= 0 && \text{for } \xi \not\in [2^n, 2^{n+1}]
  .\end{align*}
\end{itemize}

Prove that \(\sum_{n\in {\mathbb{N}}} f_n\) converges in the Hilbert
space \(L^2([0, 1])\).

\begin{quote}
Hint: Plancherel's identity may be helpful.
\end{quote}

\hypertarget{integrals-approximation}{%
\section{Integrals: Approximation}\label{integrals-approximation}}

\hypertarget{spring-2018-3-done}{%
\subsection{\texorpdfstring{Spring 2018 \# 3
\(\done\)}{Spring 2018 \# 3 \textbackslash done}}\label{spring-2018-3-done}}

Let \(f\) be a non-negative measurable function on \([0, 1]\).

Show that
\begin{align*}
\lim _{p \rightarrow \infty}\left(\int_{[0,1]} f(x)^{p} d x\right)^{\frac{1}{p}}=\|f\|_{\infty}.
\end{align*}

Solution omitted.

\hypertarget{spring-2018-4-done}{%
\subsection{\texorpdfstring{Spring 2018 \# 4
\(\done\)}{Spring 2018 \# 4 \textbackslash done}}\label{spring-2018-4-done}}

Let \(f\in L^2([0, 1])\) and suppose
\begin{align*}
\int _{[0,1]} f(x) x^{n} d x=0 \text { for all integers } n \geq 0.
\end{align*}
Show that \(f = 0\) almost everywhere.

Solution omitted.

\hypertarget{spring-2015-2-work}{%
\subsection{\texorpdfstring{Spring 2015 \# 2
\(\work\)}{Spring 2015 \# 2 \textbackslash work}}\label{spring-2015-2-work}}

Let \(f: {\mathbb{R}}\to {\mathbb{C}}\) be continuous with period 1.
Prove that
\begin{align*}
\lim _{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^{N} f(n \alpha)=\int_{0}^{1} f(t) d t \quad \forall \alpha \in {\mathbb{R}}\setminus{\mathbb{Q}}.
\end{align*}

\begin{quote}
Hint: show this first for the functions \(f(t) = e^{2\pi i k t}\) for
\(k\in {\mathbb{Z}}\).
\end{quote}

\hypertarget{fall-2014-4-work}{%
\subsection{\texorpdfstring{Fall 2014 \# 4
\(\work\)}{Fall 2014 \# 4 \textbackslash work}}\label{fall-2014-4-work}}

Let \(g\in L^\infty([0, 1])\) Prove that
\begin{align*}
\int _{[0,1]} f(x) g(x)\, dx = 0
\quad\text{for all continuous } f:[0, 1] \to {\mathbb{R}}
\implies g(x) = 0 \text{ almost everywhere. }
\end{align*}

\hypertarget{l1}{%
\section{\texorpdfstring{\(L^1\)}{L\^{}1}}\label{l1}}

\hypertarget{spring-2020-3-done}{%
\subsection{\texorpdfstring{Spring 2020 \# 3
\(\done\)}{Spring 2020 \# 3 \textbackslash done}}\label{spring-2020-3-done}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Prove that if \(g\in L^1({\mathbb{R}})\) then
  \begin{align*}
  \lim_{N\to \infty} \int _{{\left\lvert {x} \right\rvert} \geq N} {\left\lvert {f(x)} \right\rvert} \, dx = 0
  ,\end{align*}
  and demonstrate that it is not necessarily the case that
  \(f(x) \to 0\) as \({\left\lvert {x} \right\rvert}\to \infty\).
\item
  Prove that if \(f\in L^1([1, \infty])\) and is decreasing, then
  \(\lim_{x\to\infty}f(x) =0\) and in fact
  \(\lim_{x\to \infty} xf(x) = 0\).
\item
  If \(f: [1, \infty) \to [0, \infty)\) is decreasing with
  \(\lim_{x\to \infty} xf(x) = 0\), does this ensure that
  \(f\in L^1([1, \infty))\)?
\end{enumerate}

Solution omitted.

\hypertarget{fall-2019-5.-done}{%
\subsection{\texorpdfstring{Fall 2019 \# 5.
\(\done\)}{Fall 2019 \# 5. \textbackslash done}}\label{fall-2019-5.-done}}

\hypertarget{a-9}{%
\subsubsection{a}\label{a-9}}

Show that if \(f\) is continuous with compact support on
\({\mathbb{R}}\), then
\begin{align*}
\lim _{y \rightarrow 0} \int_{\mathbb{R}}|f(x-y)-f(x)| d x=0
\end{align*}

\hypertarget{b-9}{%
\subsubsection{b}\label{b-9}}

Let \(f\in L^1({\mathbb{R}})\) and for each \(h > 0\) let
\begin{align*}
\mathcal{A}_{h} f(x):=\frac{1}{2 h} \int_{|y| \leq h} f(x-y) d y
\end{align*}

\begin{enumerate}
\def\labelenumi{\roman{enumi}.}
\item
  Prove that \(\left\|\mathcal{A}_{h} f\right\|_{1} \leq\|f\|_{1}\) for
  all \(h > 0\).
\item
  Prove that \(\mathcal{A}_h f \to f\) in \(L^1({\mathbb{R}})\) as
  \(h \to 0^+\).
\end{enumerate}

\todo[inline]{Walk through.}

Solution omitted.

\hypertarget{fall-2017-3-done}{%
\subsection{\texorpdfstring{Fall 2017 \# 3
\(\done\)}{Fall 2017 \# 3 \textbackslash done}}\label{fall-2017-3-done}}

Let
\begin{align*}
S = {\operatorname{span}}_{\mathbb{C}}\left\{{\chi_{(a, b)} {~\mathrel{\Big|}~}a, b \in {\mathbb{R}}}\right\},
\end{align*}
the complex linear span of characteristic functions of intervals of the
form \((a, b)\).

Show that for every \(f\in L^1({\mathbb{R}})\), there exists a sequence
of functions \(\left\{{f_n}\right\} \subset S\) such that
\begin{align*}
\lim _{n \rightarrow \infty}\left\|f_{n}-f\right\|_{1}=0
\end{align*}

\todo[inline]{Walk through.}

Solution omitted.

\hypertarget{spring-2015-4-work}{%
\subsection{\texorpdfstring{Spring 2015 \# 4
\(\work\)}{Spring 2015 \# 4 \textbackslash work}}\label{spring-2015-4-work}}

Define
\begin{align*}
f(x, y):=\left\{\begin{array}{ll}{\frac{x^{1 / 3}}{(1+x y)^{3 / 2}}} & {\text { if } 0 \leq x \leq y} \\ {0} & {\text { otherwise }}\end{array}\right.
\end{align*}

Carefully show that \(f \in L^1({\mathbb{R}}^2)\).

\hypertarget{fall-2014-3-work}{%
\subsection{\texorpdfstring{Fall 2014 \# 3
\(\work\)}{Fall 2014 \# 3 \textbackslash work}}\label{fall-2014-3-work}}

Let \(f\in L^1({\mathbb{R}})\). Show that
\begin{align*}
\forall\varepsilon > 0 \exists \delta > 0 \text{ such that } \qquad
m(E) < \delta
\implies
\int _{E} |f(x)| \, dx < \varepsilon
\end{align*}

\hypertarget{spring-2014-1-work}{%
\subsection{\texorpdfstring{Spring 2014 \# 1
\(\work\)}{Spring 2014 \# 1 \textbackslash work}}\label{spring-2014-1-work}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  Give an example of a continuous \(f\in L^1({\mathbb{R}})\) such that
  \(f(x) \not\to 0\) as\({\left\lvert {x} \right\rvert} \to \infty\).
\item
  Show that if \(f\) is \emph{uniformly} continuous, then
  \begin{align*}
  \lim_{{\left\lvert {x} \right\rvert} \to \infty} f(x) = 0.
  \end{align*}
\end{enumerate}

\hypertarget{spring-2021-4}{%
\subsection{Spring 2021 \# 4}\label{spring-2021-4}}

Let \(f, g\) be Lebesgue integrable on \({\mathbb{R}}\) and let
\(g_n(x) \coloneqq g(x- n)\). Prove that
\begin{align*}
\lim_{n\to \infty } {\left\lVert {f + g_n} \right\rVert}_1 = {\left\lVert {f} \right\rVert}_1 + {\left\lVert {g} \right\rVert}_1
.\end{align*}

Relevant concepts omitted.

Solution omitted.

\hypertarget{fall-2020-4}{%
\subsection{Fall 2020 \# 4}\label{fall-2020-4}}

Prove that if \(xf(x) \in L^1({\mathbb{R}})\), then
\begin{align*}
F(y) \coloneqq\int f(x) \cos(yx)\,  dx
\end{align*}
defines a \(C^1\) function.

\hypertarget{fubini-tonelli}{%
\section{Fubini-Tonelli}\label{fubini-tonelli}}

\hypertarget{spring-2020-4-done}{%
\subsection{\texorpdfstring{Spring 2020 \# 4
\(\done\)}{Spring 2020 \# 4 \textbackslash done}}\label{spring-2020-4-done}}

Let \(f, g\in L^1({\mathbb{R}})\). Argue that
\(H(x, y) \coloneqq f(y) g(x-y)\) defines a function in
\(L^1({\mathbb{R}}^2)\) and deduce from this fact that
\begin{align*}
(f\ast g)(x) \coloneqq\int_{\mathbb{R}}f(y) g(x-y) \,dy
\end{align*}
defines a function in \(L^1({\mathbb{R}})\) that satisfies
\begin{align*}
{\left\lVert {f\ast g} \right\rVert}_1 \leq {\left\lVert {f} \right\rVert}_1 {\left\lVert {g} \right\rVert}_1
.\end{align*}

Hint/strategy omitted.

Relevant concepts omitted.

Solution omitted.

\hypertarget{spring-2019-4-done}{%
\subsection{\texorpdfstring{Spring 2019 \# 4
\(\done\)}{Spring 2019 \# 4 \textbackslash done}}\label{spring-2019-4-done}}

Let \(f\) be a non-negative function on \({\mathbb{R}}^n\) and
\(\mathcal A = \{(x, t) ∈ {\mathbb{R}}^n \times {\mathbb{R}}: 0 ≤ t ≤ f (x)\}\).

Prove the validity of the following two statements:

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(f\) is a Lebesgue measurable function on
  \({\mathbb{R}}^n \iff \mathcal A\) is a Lebesgue measurable subset of
  \({\mathbb{R}}^{n+1}\)
\item
  If \(f\) is a Lebesgue measurable function on \({\mathbb{R}}^n\), then
  \begin{align*}
  m(\mathcal{A})=\int _{{\mathbb{R}}^{n}} f(x) d x=\int_{0}^{\infty} m\left(\left\{x \in {\mathbb{R}}^{n}: f(x) \geq t\right\}\right) dt
  \end{align*}
\end{enumerate}

\todo[inline]{Add concepts.}

Relevant concepts omitted.

Solution omitted.

\hypertarget{fall-2018-5-done}{%
\subsection{\texorpdfstring{Fall 2018 \# 5
\(\done\)}{Fall 2018 \# 5 \textbackslash done}}\label{fall-2018-5-done}}

Let \(f \geq 0\) be a measurable function on \({\mathbb{R}}\). Show that
\begin{align*}
\int _{{\mathbb{R}}} f = \int _{0}^{\infty} m(\{x: f(x)>t\}) dt
\end{align*}
:::\{.concept\} \envlist - Claim: If
\(E\subseteq {\mathbb{R}}^a \times{\mathbb{R}}^b\) is a measurable set,
then for almost every \(y\in {\mathbb{R}}^b\), the slice \(E^y\) is
measurable and
\begin{align*}
m(E) = \int_{{\mathbb{R}}^b} m(E^y) \,dy
.\end{align*}
- Set \(g = \chi_E\), which is non-negative and measurable, so apply
Tonelli. - Conclude that \(g^y = \chi_{E^y}\) is measurable, the
function \(y\mapsto \int g^y(x)\, dx\) is measurable, and
\(\int \int g^y(x)\,dx \,dy = \int g\). - But \(\int g = m(E)\) and
\(\int\int g^y(x) \,dx\,dy = \int m(E^y)\,dy\). :::

Solution omitted.

\hypertarget{fall-2015-5-work}{%
\subsection{\texorpdfstring{Fall 2015 \# 5
\(\work\)}{Fall 2015 \# 5 \textbackslash work}}\label{fall-2015-5-work}}

Let \(f, g \in L^1({\mathbb{R}})\) be Borel measurable.

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that
\end{enumerate}

\begin{itemize}
\tightlist
\item
  The function
  \begin{align*}F(x, y) \coloneqq f(x-y) g(y)\end{align*}
  is Borel measurable on \({\mathbb{R}}^2\), and
\item
  For almost every \(y\in {\mathbb{R}}\),
  \begin{align*}F_y(x) \coloneqq f(x-y)g(y)\end{align*}
  is integrable with respect to \(y\).
\end{itemize}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\setcounter{enumi}{1}
\tightlist
\item
  Show that \(f\ast g \in L^1({\mathbb{R}})\) and
  \begin{align*}
  \|f * g\|_{1} \leq \|f\|_{1} \|g\|_{1}
  \end{align*}
\end{enumerate}

\hypertarget{spring-2014-5-work}{%
\subsection{\texorpdfstring{Spring 2014 \# 5
\(\work\)}{Spring 2014 \# 5 \textbackslash work}}\label{spring-2014-5-work}}

Let \(f, g \in L^1([0, 1])\) and for all \(x\in [0, 1]\) define
\begin{align*}
F(x) \coloneqq\int _{0}^{x} f(y) \, dy
{\quad \operatorname{and} \quad}
G(x)\coloneqq\int _{0}^{x} g(y) \, dy.
\end{align*}

Prove that
\begin{align*}
\int _{0}^{1} F(x) g(x) \, dx =
F(1) G(1) - \int _{0}^{1} f(x) G(x) \, dx
\end{align*}

\hypertarget{spring-2021-6}{%
\subsection{Spring 2021 \# 6}\label{spring-2021-6}}

\begin{warnings}

This problem may be much harder than expected. Recommended skip.

\end{warnings}

Let \(f: {\mathbb{R}}\times{\mathbb{R}}\to {\mathbb{R}}\) be a
measurable function and for \(x\in {\mathbb{R}}\) define the set
\begin{align*}
E_x \coloneqq\left\{{ y\in {\mathbb{R}}{~\mathrel{\Big|}~}\mu\qty{ z\in {\mathbb{R}}{~\mathrel{\Big|}~}f(x,z) = f(x, y) } > 0 }\right\}
.\end{align*}
Show that the following set is a measurable subset of
\({\mathbb{R}}\times{\mathbb{R}}\):
\begin{align*}
E \coloneqq\bigcup_{x\in {\mathbb{R}}} \left\{{ x }\right\} \times E_x
.\end{align*}

\begin{quote}
Hint: consider the measurable function
\(h(x,y,z) \coloneqq f(x, y) - f(x, z)\).
\end{quote}

\hypertarget{l2-and-fourier-analysis}{%
\section{\texorpdfstring{\(L^2\) and Fourier
Analysis}{L\^{}2 and Fourier Analysis}}\label{l2-and-fourier-analysis}}

\hypertarget{spring-2020-6-done}{%
\subsection{\texorpdfstring{Spring 2020 \# 6
\(\done\)}{Spring 2020 \# 6 \textbackslash done}}\label{spring-2020-6-done}}

\hypertarget{a-11}{%
\subsubsection{a}\label{a-11}}

Show that
\begin{align*}
L^2([0, 1]) \subseteq L^1([0, 1]) {\quad \operatorname{and} \quad} \ell^1({\mathbb{Z}}) \subseteq \ell^2({\mathbb{Z}})
.\end{align*}

\hypertarget{b-11}{%
\subsubsection{b}\label{b-11}}

For \(f\in L^1([0, 1])\) define
\begin{align*}
\widehat{f}(n) \coloneqq\int _0^1 f(x) e^{-2\pi i n x} \, dx
.\end{align*}

Prove that if \(f\in L^1([0, 1])\) and
\(\left\{{\widehat{f}(n)}\right\} \in \ell^1({\mathbb{Z}})\) then
\begin{align*}
S_N f(x) \coloneqq\sum_{{\left\lvert {n} \right\rvert} \leq N} \widehat{f} (n) e^{2 \pi i n x}
.\end{align*}
converges uniformly on \([0, 1]\) to a continuous function \(g\) such
that \(g = f\) almost everywhere.

\begin{quote}
Hint: One approach is to argue that if \(f\in L^1([0, 1])\) with
\(\left\{{\widehat{f} (n)}\right\} \in \ell^1({\mathbb{Z}})\) then
\(f\in L^2([0, 1])\).
\end{quote}

Solution omitted.

\hypertarget{fall-2017-5-done}{%
\subsection{\texorpdfstring{Fall 2017 \# 5
\(\done\)}{Fall 2017 \# 5 \textbackslash done}}\label{fall-2017-5-done}}

Let \(\phi\) be a compactly supported smooth function that vanishes
outside of an interval \([-N, N]\) such that
\(\int _{{\mathbb{R}}} \phi(x) \, dx = 1\).

For \(f\in L^1({\mathbb{R}})\), define
\begin{align*}
K_{j}(x) \coloneqq j \phi(j x),
\qquad
f \ast K_{j}(x) \coloneqq\int_{{\mathbb{R}}} f(x-y) K_{j}(y) \, dy
\end{align*}
and prove the following:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  Each \(f\ast K_j\) is smooth and compactly supported.
\item

  \begin{align*}
  \lim _{j \to \infty} {\left\lVert {f * K_{j}-f} \right\rVert}_{1} = 0
  \end{align*}
\end{enumerate}

\begin{quote}
Hint:
\begin{align*}
\lim _{y \to 0} \int _{{\mathbb{R}}} |f(x-y)-f(x)| dy = 0
\end{align*}
\end{quote}

\todo[inline]{Add concepts.}

Solution omitted.

\hypertarget{spring-2017-5-work}{%
\subsection{\texorpdfstring{Spring 2017 \# 5
\(\work\)}{Spring 2017 \# 5 \textbackslash work}}\label{spring-2017-5-work}}

Let \(f, g \in L^2({\mathbb{R}})\). Prove that the formula
\begin{align*}
h(x) \coloneqq\int _{-\infty}^{\infty} f(t) g(x-t) \, dt
\end{align*}
defines a uniformly continuous function \(h\) on \({\mathbb{R}}\).

\hypertarget{spring-2015-6-work}{%
\subsection{\texorpdfstring{Spring 2015 \# 6
\(\work\)}{Spring 2015 \# 6 \textbackslash work}}\label{spring-2015-6-work}}

Let \(f \in L^1({\mathbb{R}})\) and \(g\) be a bounded measurable
function on \({\mathbb{R}}\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that the convolution \(f\ast g\) is well-defined, bounded, and
  uniformly continuous on \({\mathbb{R}}\).
\item
  Prove that one further assumes that \(g \in C^1({\mathbb{R}})\) with
  bounded derivative, then \(f\ast g \in C^1({\mathbb{R}})\) and
  \begin{align*}
  \frac{d}{d x}(f * g)=f *\left(\frac{d}{d x} g\right)
  \end{align*}
\end{enumerate}

\hypertarget{fall-2014-5-work}{%
\subsection{\texorpdfstring{Fall 2014 \# 5
\(\work\)}{Fall 2014 \# 5 \textbackslash work}}\label{fall-2014-5-work}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  Let \(f \in C_c^0({\mathbb{R}}^n)\), and show
  \begin{align*}
  \lim _{t \to 0} \int_{{\mathbb{R}}^n} |f(x+t) - f(x)| \, dx = 0
  .\end{align*}
\item
  Extend the above result to \(f\in L^1({\mathbb{R}}^n)\) and show that
  \begin{align*}
  f\in L^1({\mathbb{R}}^n), \quad g\in L^\infty({\mathbb{R}}^n) \quad
  \implies f \ast g \text{ is bounded and uniformly continuous. }
  \end{align*}
\end{enumerate}

\hypertarget{fall-2020-5}{%
\subsection{Fall 2020 \# 5}\label{fall-2020-5}}

Suppose \(\phi\in L^1({\mathbb{R}})\) with
\begin{align*}
\int \phi(x) \, dx = \alpha
.\end{align*}
For each \(\delta > 0\) and \(f\in L^1({\mathbb{R}})\), define
\begin{align*}
A_\delta f(x) \coloneqq\int f(x-y) \delta^{-1} \phi\qty{\delta^{-1} y}\, dy
.\end{align*}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Prove that for all \(\delta > 0\),
  \begin{align*}
  {\left\lVert {A_\delta f} \right\rVert}_1 \leq {\left\lVert {\phi} \right\rVert}_1 {\left\lVert {f} \right\rVert}_1
  .\end{align*}
\item
  Prove that
  \begin{align*}
  A_\delta f \to \alpha f \text{ in } L^1({\mathbb{R}}) {\quad \operatorname{as} \quad} \delta\to 0^+
  .\end{align*}
\end{enumerate}

\begin{quote}
Hint: you may use without proof the fact that for all
\(f\in L^1({\mathbb{R}})\),
\begin{align*}
\lim_{y\to 0} \int_{\mathbb{R}}{\left\lvert {f(x-y) - f(x)} \right\rvert}\, dx = 0
.\end{align*}
\end{quote}

\hypertarget{functional-analysis-general}{%
\section{Functional Analysis:
General}\label{functional-analysis-general}}

\hypertarget{fall-2019-4-done}{%
\subsection{\texorpdfstring{Fall 2019 \# 4
\(\done\)}{Fall 2019 \# 4 \textbackslash done}}\label{fall-2019-4-done}}

Let \(\{u_n\}_{n=1}^∞\) be an orthonormal sequence in a Hilbert space
\(\mathcal{H}\).

\hypertarget{a-14}{%
\subsubsection{a}\label{a-14}}

Prove that for every \(x ∈ \mathcal H\) one has
\begin{align*}
\displaystyle\sum_{n=1}^{\infty}\left|\left\langle x, u_{n}\right\rangle\right|^{2} \leq\|x\|^{2}
\end{align*}

\hypertarget{b-13}{%
\subsubsection{b}\label{b-13}}

Prove that for any sequence
\(\{a_n\}_{n=1}^\infty \in \ell^2({\mathbb{N}})\) there exists an
element \(x\in\mathcal H\) such that
\begin{align*}
a_n = {\left\langle {x},~{u_n} \right\rangle} \text{ for all } n\in {\mathbb{N}}
\end{align*}
and
\begin{align*}
{\left\lVert {x} \right\rVert}^2 = \sum_{n=1}^{\infty}\left|\left\langle x, u_{n}\right\rangle\right|^{2}
\end{align*}

Solution omitted.

\hypertarget{spring-2019-5-done}{%
\subsection{\texorpdfstring{Spring 2019 \# 5
\(\done\)}{Spring 2019 \# 5 \textbackslash done}}\label{spring-2019-5-done}}

\hypertarget{a-16}{%
\subsubsection{a}\label{a-16}}

Show that \(L^2([0, 1]) ⊆ L^1([0, 1])\) and argue that \(L^2([0, 1])\)
in fact forms a dense subset of \(L^1([0, 1])\).

\hypertarget{b-15}{%
\subsubsection{b}\label{b-15}}

Let \(Λ\) be a continuous linear functional on \(L^1([0, 1])\).

Prove the Riesz Representation Theorem for \(L^1([0, 1])\) by following
the steps below:

\begin{enumerate}
\def\labelenumi{\roman{enumi}.}
\tightlist
\item
  Establish the existence of a function \(g ∈ L^2([0, 1])\) which
  represents \(Λ\) in the sense that
  \begin{align*}
    Λ(f ) = f (x)g(x) dx \text{ for all } f ∈ L^2([0, 1]).
    \end{align*}
\end{enumerate}

\begin{quote}
Hint: You may use, without proof, the Riesz Representation Theorem for
\(L^2([0, 1])\).
\end{quote}

\begin{enumerate}
\def\labelenumi{\roman{enumi}.}
\setcounter{enumi}{1}
\tightlist
\item
  Argue that the \(g\) obtained above must in fact belong to
  \(L^∞([0, 1])\) and represent \(Λ\) in the sense that
  \begin{align*}
    \Lambda(f)=\int_{0}^{1} f(x) \overline{g(x)} d x \quad \text { for all } f \in L^{1}([0,1])
    \end{align*}
  with
  \begin{align*}
    \|g\|_{L^{\infty}([0,1])} = \|\Lambda\|_{L^{1}([0,1]) {}^{ \check{} }}
    \end{align*}
\end{enumerate}

Solution omitted.

\hypertarget{spring-2016-6-work}{%
\subsection{\texorpdfstring{Spring 2016 \# 6
\(\work\)}{Spring 2016 \# 6 \textbackslash work}}\label{spring-2016-6-work}}

Without using the Riesz Representation Theorem, compute
\begin{align*}
\sup \left\{\left|\int_{0}^{1} f(x) e^{x} d x\right| {~\mathrel{\Big|}~}f \in L^{2}([0,1], m),~~ \|f\|_{2} \leq 1\right\}
\end{align*}

\hypertarget{spring-2015-5-work}{%
\subsection{\texorpdfstring{Spring 2015 \# 5
\(\work\)}{Spring 2015 \# 5 \textbackslash work}}\label{spring-2015-5-work}}

Let \(\mathcal H\) be a Hilbert space.

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Let \(x\in \mathcal H\) and \(\left\{{u_n}\right\}_{n=1}^N\) be an
  orthonormal set. Prove that the best approximation to \(x\) in
  \(\mathcal H\) by an element in
  \({\operatorname{span}}_{\mathbb{C}}\left\{{u_n}\right\}\) is given by
  \begin{align*}
    \widehat{x} \coloneqq\sum_{n=1}^N {\left\langle {x},~{u_n} \right\rangle}u_n.
    \end{align*}
\item
  Conclude that finite dimensional subspaces of \(\mathcal H\) are
  always closed.
\end{enumerate}

\hypertarget{fall-2015-6-work}{%
\subsection{\texorpdfstring{Fall 2015 \# 6
\(\work\)}{Fall 2015 \# 6 \textbackslash work}}\label{fall-2015-6-work}}

Let \(f: [0, 1] \to {\mathbb{R}}\) be continuous. Show that
\begin{align*}
\sup \left\{\|f g\|_{1} {~\mathrel{\Big|}~}g \in L^{1}[0,1],~~ \|g\|_{1} \leq 1\right\}=\|f\|_{\infty}
\end{align*}

\hypertarget{fall-2014-6-work}{%
\subsection{\texorpdfstring{Fall 2014 \# 6
\(\work\)}{Fall 2014 \# 6 \textbackslash work}}\label{fall-2014-6-work}}

Let \(1 \leq p,q \leq \infty\) be conjugate exponents, and show that
\begin{align*}
f \in L^p({\mathbb{R}}^n) \implies \|f\|_{p} = \sup _{\|g\|_{q}=1}\left|\int f(x) g(x) d x\right|
\end{align*}

\hypertarget{functional-analysis-banach-spaces}{%
\section{Functional Analysis: Banach
Spaces}\label{functional-analysis-banach-spaces}}

\hypertarget{spring-2019-1-done}{%
\subsection{\texorpdfstring{Spring 2019 \# 1
\(\done\)}{Spring 2019 \# 1 \textbackslash done}}\label{spring-2019-1-done}}

Let \(C([0, 1])\) denote the space of all continuous real-valued
functions on \([0, 1]\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Prove that \(C([0, 1])\) is complete under the uniform norm
  \({\left\lVert {f} \right\rVert}_u := \displaystyle\sup_{x\in [0,1]} |f (x)|\).
\item
  Prove that \(C([0, 1])\) is not complete under the
  \(L^1{\hbox{-}}\)norm
  \({\left\lVert {f} \right\rVert}_1 = \displaystyle\int_0^1 |f (x)| ~dx\).
\end{enumerate}

\todo[inline]{Add concepts.}

Solution omitted.

\hypertarget{spring-2017-6-done}{%
\subsection{\texorpdfstring{Spring 2017 \# 6
\(\done\)}{Spring 2017 \# 6 \textbackslash done}}\label{spring-2017-6-done}}

Show that the space \(C^1([a, b])\) is a Banach space when equipped with
the norm
\begin{align*}
\|f\|:=\sup _{x \in[a, b]}|f(x)|+\sup _{x \in[a, b]}\left|f^{\prime}(x)\right|.
\end{align*}

\todo[inline]{Add concepts.}

Solution omitted.

\hypertarget{fall-2017-6-done}{%
\subsection{\texorpdfstring{Fall 2017 \# 6
\(\done\)}{Fall 2017 \# 6 \textbackslash done}}\label{fall-2017-6-done}}

Let \(X\) be a complete metric space and define a norm
\begin{align*}
\|f\|:=\max \{|f(x)|: x \in X\}.
\end{align*}

Show that \((C^0({\mathbb{R}}), {\left\lVert {{-}} \right\rVert} )\)
(the space of continuous functions \(f: X\to {\mathbb{R}}\)) is
complete.

\todo[inline]{Add concepts.}
\todo[inline]{Shouldn't this be a supremum? The max may not exist?}
\todo[inline]{Review and clean up.}

Solution omitted.

\printbibliography[title=Bibliography]

\end{document}

Real Analysis/UGA Questions (no solutions)/UGA_Real_Analysis_Qual_Questions.tex