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src/Advanced/Nonstandard Analysis/main.tex Executable file
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% Copyright (C) 2023 <Mark (mark@betalupi.com)>
%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% You may have received a copy of the GNU General Public License
% along with this program. If not, see <https://www.gnu.org/licenses/>.
%
%
%
% If you edit this, please give credit!
% Quality handouts take time to make.
% use [nosolutions] flag to hide solutions.
% use [solutions] flag to show solutions.
\documentclass[
nosolutions,
singlenumbering,
]{../../../lib/tex/ormc_handout}
\usepackage{../../../lib/tex/macros}
\usepackage{units}
\uptitlel{Advanced 2}
\uptitler{\smallurl{}}
\title{Nonstandard Analysis}
\subtitle{
Prepared by Mark on \today{} \\
Based on handouts by Nikita and Stepan
}
% This handout is pretty difficult. Make sure you can all solve all the problems yourself,
% and remember that each SECTION was a two-hour lesson with a smart class.
% From experience, the following holds:
% supremum is a better lesson than dual numbers, which is better than extensions.
\begin{document}
\maketitle
\input{parts/supremum}
\input{parts/dual}
\input{parts/extensions}
\end{document}

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src/Advanced/Nonstandard Analysis/meta.toml Normal file
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[metadata]
title = "Nonstandard Analysis"
[publish]
handout = false
solutions = false

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src/Advanced/Nonstandard Analysis/parts/dual.tex Normal file
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% Copyright (C) 2023 <Mark (mark@betalupi.com)>
%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% You may have received a copy of the GNU General Public License
% along with this program. If not, see <https://www.gnu.org/licenses/>.
%
%
%
% If you edit this, please give credit!
% Quality handouts take time to make.
\section{Dual Numbers}
\definition{}
In the problems below, let $\varepsilon$ a positive infinitesimal so that $\varepsilon^2 = 0$. \par
\note{Note that $\varepsilon \neq 0$.}
\definition{}
The set of \textit{dual numbers} consists of elements of the form $a + b\varepsilon$, where $a, b \in \mathbb{R}$.
\problem{}
Compute $(a + b\varepsilon) \times (c + d\varepsilon)$.
\vfill
\definition{}
Let $f(x)$ be an algebraic function $\mathbb{R} \to \mathbb{R}$. \par
(that is, a function we can write using the operators $+-\times\div$ and integer powers) \par
\vspace{2mm}
the \textit{derivative} of such an $f$ is a function $f'$ that satisfies the following:
\begin{equation*}
f(x + \varepsilon) = f(x) + f'(x)\varepsilon
\end{equation*}
If $f(x + \varepsilon)$ is not defined, we will say that $f$ is not \textit{differentiable} at $x$.
\problem{}
What is the derivative of $f(x) = x^2$?
\vfill
\problem{}
What is the derivative of $f(x) = x^n$?
\vfill
\problem{}
Assume that the derivatives of $f$ and $g$ are known. \par
Find the derivatives of $h(x) = f(x) + g(x)$ and $k(x) = f(x) \times g(x)$.
\vfill
\pagebreak
\problem{}
When can you divide dual numbers? \par
That is, for what numbers $(a + b\varepsilon)$ is there a $(x + y\varepsilon)$ such that $(a +b\varepsilon)(x+y\varepsilon) = 1$?
\vfill
\problem{}
Find an explicit formula for the inverse of a dual number $(a + b\varepsilon)$, assuming one exists. \par
Then, use this find the derivative of $f(x) = \frac{1}{x}$.
\vfill
\problem{}
Which dual numbers have a square root? \par
That is, for which dual numbers $(a + b\varepsilon)$ is there a dual number
$(x + y\varepsilon)$ such that $(x + y\varepsilon)^2 = a + b\varepsilon$?
\vfill
\problem{}
Find an explicit formula for the square root and use it to find the derivative of $f(x) = \sqrt{x}$
\vfill
\problem{}
Find the derivative of the following functions:
\begin{itemize}
\item $f(x) = \frac{x}{1 + x^2}$
\item $g(x) = \sqrt{1 - x^2}$
\end{itemize}
\vfill
\problem{}
Assume that the derivatives of $f$ and $g$ are known. \par
What is the derivative of $f(g(x))$?
\vfill
\pagebreak

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src/Advanced/Nonstandard Analysis/parts/extensions.tex Normal file
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% Copyright (C) 2023 <Mark (mark@betalupi.com)>
%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% You may have received a copy of the GNU General Public License
% along with this program. If not, see <https://www.gnu.org/licenses/>.
%
%
%
% If you edit this, please give credit!
% Quality handouts take time to make.
\section{Extensions of $\mathbb{R}$}
\definition{}
An \textit{ordered field} consists of a set $S$, the operations $+$ and $\times$, and the relation $<$. \par
An ordered field must satisfy the following properties:
\begin{itemize}
\item \textbf{Properties of $+$:}
\begin{itemize}
\item Commutativity: $a + b = b + a$
\item Associativity: $a + (b + c) = (a + b) + c$
\item Identity: there exists an element $0$ so that $a + 0 = a$ for all $a \in S$
\item Inverse: for every $a$, there exists a $-a$ so that $a + (-a) = 0$
\end{itemize}
\item \textbf{Properties of $\times$:}
\begin{itemize}
\item Commutativity
\item Associativity
\item Identity (which we label $1$)
\item For every $a \neq 0$, there exists an inverse $a^{-1}$ so that $aa^{-1} = 1$
\item Distributivity: $a(b + c) = ab + ac$
\end{itemize}
\item \textbf{Properties of $<$:}
\begin{itemize}
\item Non-reflexive: $x < x$ is always false
\item Transitive: $x < y$ and $y < z$ imply $x < z$
\item Connected: for all $x, y \in S$, either $x < y$, $x > y$, or $x = y$.
\item If $x < y$ then $x + z < y + z$
\item If $x < y$ and $z > 0$, then $xz < yz$
\item $0 < 1$
\end{itemize}
\end{itemize}
\definition{}
An ordered field that contains $\mathbb{R}$ is called an \textit{extension} of $\mathbb{R}$.
\definition{}
The \textit{Archimedean property} states the following: \par
For all positive $x, y$, there exists an $n$ so that $nx \geq y$.
\theorem{}
All extensions of $\mathbb{R}$ are nonarchemedian. \par
Proving this is difficult.
\vfill
\pagebreak
\problem{}
Which of the following are ordered fields?
\begin{itemize}[itemsep=2mm]
\item $\mathbb{R}$ with the usual definitions of $+$, $\times$, $<$
\item $\mathbb{R}$ with the usual definitions of $+$, $\times$, $\leq$ \par
\note{Note that our relation here is $\leq$, not $<$}
\item $\mathbb{Z}$ with the usual definitions of $+$, $\times$, $<$
\item $\mathbb{Q}$ with the usual definitions of $+$, $\times$, $<$
\item $\mathbb{C}$ with the usual definitions of $+$, $\times$, \par
and with $(a + bi) < (c + di)$ iff $a < c$.
\end{itemize}
\vfill
\problem{}
Show that each of the following is true in any ordered field. \par
The list of field axioms is provided below, for convenience.
\begin{enumerate}
\item if $x \neq 0$ then $(x^{-1})^{-1} = x$
\item $0 \times x = 0$
\item $(-x)(-y) = xy$
\item if $0 < x < y$, then $x^{-1} > y^{-1}$
\end{enumerate}
%\begin{solution}
% \textbf{Part A:}
% We know that $x^{-1} \times (x^{-1})^{-1} = 1$ \par
% Thus $x \times (x^{-1} \times (x^{-1})^{-1}) = x \times 1 = x$ \par
% We can rewrite this as $(x \times x^{-1}) \times (x^{-1})^{-1} = x$ \par
% When then becomes $1 \times (x^{-1})^{-1} = x$ \par
% And thus $(x^{-1})^{-1} = x$
%\end{solution}
\vfill
\begin{itemize}
\item \textbf{Properties of $+$:}
\begin{itemize}
\item Commutativity: $a + b = b + a$
\item Associativity: $a + (b + c) = (a + b) + c$
\item Identity: there exists an element $0$ so that $a + 0 = a$ for all $a \in S$
\item Inverse: for every $a$, there exists a $-a$ so that $a + (-a) = 0$
\end{itemize}
\item \textbf{Properties of $\times$:}
\begin{itemize}
\item Commutativity
\item Associativity
\item Identity (which we label $1$)
\item For every $a \neq 0$, there exists an inverse $a^{-1}$ so that $aa^{-1} = 1$
\item Distributivity: $a(b + c) = ab + ac$
\end{itemize}
\item \textbf{Properties of $<$:}
\begin{itemize}
\item Non-reflexive: $x < x$ is always false
\item Transitive: $x < y$ and $y < z$ imply $x < z$
\item Connected: for all $x, y \in S$, either $x < y$, $x > y$, or $x = y$.
\item If $x < y$ then $x + z < y + z$
\item If $x < y$ and $z > 0$, then $xz < yz$
\item $0 < 1$
\end{itemize}
\end{itemize}
\pagebreak
\definition{}
In an ordered field, the \textit{magnitude} of a number x is defined as follows: \par
\begin{equation*}
|x| =
\begin{cases}
x & \text{\tab} x \geq 0 \\
-x & \text{\tab otherwise}
\end{cases}
\end{equation*}
\definition{}
We say an element $\delta$ of an ordered field is \textit{infinitesimal} if
$|nd| < 1$ % spell:disable-line
for all $n \in \mathbb{Z^+}$. \par
\note{Note that $\mathbb{Z}^+$ is a subset of any nonarchimedian extension of $\mathbb{R}$.} \par
\vspace{2mm}
Likewise, we say $x$ is \textit{limited} if $|x| < n$ for some $n \in \mathbb{Z}^+$. \par
Elements that are not limited are \textit{unlimited}.
\definition{}
We say an element $x$ of a field is \textit{positive} if $x > 0$. \par
We say $x$ is \textit{negative} if $x < 0$. \par
\problem{}
Show that a positive $\delta$ is infinitesimal if and only if $\delta < x$ for all $x \in \mathbb{R}^+$. \par
Then, show that a negative $\delta$ is infinitesimal if and only if $\delta > x$ for every $x \in \mathbb{R}^-$.
\vfill
\problem{}
Prove the following statements: \par
\begin{itemize}
\item If $\delta$ and $\varepsilon$ are infinitesimal, then $\delta + \varepsilon$ is infinitesimal.
\item If $\delta$ is infinitesimal and $x$ is limited, then $x\delta$ is infinitesimal.
\item If $x$ and $y$ are limited, $xy$ and $x+y$ are too.
\item A nonzero $\delta$ is infinitesimal iff $\delta^{-1}$ is unlimited.
\end{itemize}
\vfill
\pagebreak
\problem{}
Let $\delta$ be a positive infinitesimal. Which is greater?
\begin{itemize}
\item $\delta$ or $\delta^2$
\item $(1 - \delta)$ or $(1 + \delta^2)^{-1}$
\item $\frac{1 + \delta}{1 + \delta^2}$ or $\frac{2 + \delta^2}{2 + \delta^3}$ \par
\note[Note]{we define $\frac{1}{x}$ as $x^{-1}$, and thus $\frac{a}{b} = a \times b^{-1}$}
\end{itemize}
\vfill
\definition{}<stpart>
We say two elements of an ordered field are \textit{infinitely close} if $x - y$ is infinitesimal. \par
We say that $x_0 \in \mathbb{R}$ is the \textit{standard part} of $x$ if it is infinitely close to $x$. \par
We will denote the standard part of $x$ as $\text{st}(x)$. \par
You may assume that $\text{st}(x)$ exists and is unique for limited $x$. \par
%\problem{}
%Let $H$ be positive unlimited. Determine which of the following are limited. \par
\problem{}
Show that $\text{st}(x + y) = \text{st}(x) + \text{st}(y)$ and $\text{st}(xy) = \text{st}(x) \text{st}(y)$. \par
\vfill
\pagebreak

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src/Advanced/Nonstandard Analysis/parts/supremum.tex Normal file
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% Copyright (C) 2023 <Mark (mark@betalupi.com)>
%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% You may have received a copy of the GNU General Public License
% along with this program. If not, see <https://www.gnu.org/licenses/>.
%
%
%
% If you edit this, please give credit!
% Quality handouts take time to make.
\section*{The supremum \& the infimum}
\definition{}
In this section, we'll define a \say{real number} as a decimal, infinite or finite.
\problem{}
Write $2.317171717...$ as a simple fraction.
\vfill
\problem{}
Write $\nicefrac{2}{11}$ as an infinite decimal and prove that your answer is correct.
\vfill
\problem{}
Show that $0.999... = 1$
\note[Note]{
There is no real number $0.0...1$ with a digit $1$ \say{at infinity.} \\
Some numbers have two decimal representations, some have only one.
}
\vfill
\problem{}
Concatenate all the natural numbers in order to form $0.12345678910111213...$. \par
Show that the resulting decimal is irrational.
\vfill
\problem{}
Show that a rational number exists between any two real numbers.
\vfill
\pagebreak
\definition{}
Let $M$ be a subset of $\mathbb{R}$.\par
We say $c \in \mathbb{R}$ is an \textit{upper bound} of $M$ if $c \geq m$ for all $m \in M$. \par
The smallest such $c$ is called the \textit{supremum} of $M$, and is denoted $\text{sup}(M)$. \par
\vspace{2mm}
Similarly, $x \in \mathbb{R}$ is a \textit{lower bound} of $M$ if $x \leq m$ for all $m \in M$. \par
The largest upper bound of $M$ is called the \textit{infimum} of $M$, denoted $\text{inf}(M)$.
\problem{}
Show that $x$ is the supremum of $M$ if and only if...
\begin{itemize}
\item for all $m \in M$, $m \leq x$, and
\item for any $x_0 < x$, there exists an $m \in M$ so that $m > x_0$
\end{itemize}
\vfill
\problem{}
Show that any subset of $\mathbb{R}$ has at most one supremum and at most one infimum.
\vfill
\problem{}
Find the supremum and infimum of the following sets:
\begin{itemize}
\item $\bigl\{ a^2 + 2a \bigl| -5 < a < 5\bigr\}$
\item $\bigl\{\pm \frac{n}{2n + 1} \bigl| n \in \mathbb{N}\bigr\}$
\end{itemize}
\vfill
\problem{}
Let $A$ and $B$ be subsets of $\mathbb{R}$, and let $\text{sup}(A)$ and $\text{sup}(B)$ be known. \par
Compute the following in terms of $\text{sup}(A)$ and $\text{sup}(B)$.
\begin{itemize}
\item $\text{sup}(A \cup B)$
\item $\text{sup}(A + B)$, where $A + B = {a + b \forall (a, b) \in A \times B}$,
\item $\text{inf}(A \cdot B)$, where $A \cdot B = {ab \forall (a, b) \in A \times B}$
\end{itemize}
\vfill
\problem{}
Prove the assumptions in \ref{stpart}: \par
Show that $\text{st}(x)$ is exists and is unique for limited $x$.
\vfill
\pagebreak
\theorem{Completeness Axiom}<completeness>
Every non-empty subset of $\mathbb{R}$ that is bounded above has a least upper bound.
\problem{}
Show that $a < \text{sup}(A)$ if and only if there is a $c$ in $A$ where $a < c$
\vfill
\problem{}
Use the definitions in this handout to prove \ref{completeness}. \par
\hint{Build the supremum one digit at a time.}
\vfill
\problem{}
Let $[a_1, b_1] \supseteq [a_2, b_3] \supseteq [a_3, b_3] \supseteq ...$ be an infinite sequence of closed line intervals.
\par Show that there exists a $c \in \mathbb{R}$ that lies in all of them. Is this true of open intervals?
\vfill
\pagebreak
\problem{Bonus}
Show that every real number in $[0, 1]$ can be written as a sum of 9 numbers \par
Whose decimal representations only contain 0 and 8. \par
\vfill
\problem{Bonus}
Two genies take an infinite amount of turns and write the digits of an infinite
decimal. The first genie, on every turn, writes any finite amount of digits to the tail of the decimal.
The second genie writes one digit to the end. If the resulting decimal after an infinite amount
of turns is periodic, the first genie wins; otherwise, the second genie wins. Who has a winning
strategy? \par
\vfill
\pagebreak