From 22275cf450effd7ca0cc4e30271a1927ca072965 Mon Sep 17 00:00:00 2001 From: Mark Date: Mon, 29 Jan 2024 17:25:19 -0800 Subject: [PATCH] Added nonstandard analysis handout --- Advanced/Nonstandard Analysis/main.tex | 44 +++++ .../parts/1 extensions.tex | 166 ++++++++++++++++++ .../Nonstandard Analysis/parts/2 dual.tex | 108 ++++++++++++ .../Nonstandard Analysis/parts/x supremum.tex | 146 +++++++++++++++ 4 files changed, 464 insertions(+) create mode 100755 Advanced/Nonstandard Analysis/main.tex create mode 100644 Advanced/Nonstandard Analysis/parts/1 extensions.tex create mode 100644 Advanced/Nonstandard Analysis/parts/2 dual.tex create mode 100644 Advanced/Nonstandard Analysis/parts/x supremum.tex diff --git a/Advanced/Nonstandard Analysis/main.tex b/Advanced/Nonstandard Analysis/main.tex new file mode 100755 index 0000000..20da0d2 --- /dev/null +++ b/Advanced/Nonstandard Analysis/main.tex @@ -0,0 +1,44 @@ +% Copyright (C) 2023 +% +% 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 . +% +% +% +% 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[ + solutions, + singlenumbering, +]{../../resources/ormc_handout} +\usepackage{../../resources/macros} +\usepackage{units} + +\uptitlel{Advanced 2} +\uptitler{Winter 2024} +\title{Nonstandard Analysis} +\subtitle{ + Prepared by \githref{Mark} on \today{} \\ + Based on handouts by Nikita and Stepan +} + + +\begin{document} + + \maketitle + + + \input{parts/1 extensions} + \input{parts/2 dual} + \input{parts/x supremum} + +\end{document} diff --git a/Advanced/Nonstandard Analysis/parts/1 extensions.tex b/Advanced/Nonstandard Analysis/parts/1 extensions.tex new file mode 100644 index 0000000..1df2389 --- /dev/null +++ b/Advanced/Nonstandard Analysis/parts/1 extensions.tex @@ -0,0 +1,166 @@ +% Copyright (C) 2023 +% +% 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 . +% +% +% +% If you edit this, please give credit! +% Quality handouts take time to make. + +\section{Nonarchimedian Extensions} + +\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 $a + 0 = a \forall a \in S$ + \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 Inverse (which we label $a^{-1}$, and which doesn't exist for $0$) + \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 $x < y$, $y > x$, 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 a \textit{nonarchimedian extension} of $\mathbb{R}$. + +\vfill +\pagebreak + + + + + +\problem{} +Show that each of the following is true in any ordered field. +\begin{enumerate} + \item if $x \neq 0$ then $(x^{-1})^{-1} = x$ + \item $0 \times x = x$ + \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 +\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 $|nd| < 1$ for all $n \in \mathbb{Z^+}$. \par +\note{Note that $\mathbb{Z}^+$ is a subset of any nonarchimedian extension.} \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 it is bigger than 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 $a\delta$ 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 + +\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 +\pagebreak + + +\definition{} +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 a \textit{standard part} of $x$ if it is infinitely close to $x$. \par + +\problem{} +We will denote the standard part of $x$ as $\text{st}(x)$. \par +Show that $\text{st}(x)$ is well-defined for limited $x$. \par +(In other words, Show that $x_0$ exists and is unique for limited $x$). \par +\hint{To prove existance, consider $\text{sup}(\{a \in \mathbb{R} ~|~ a < x\}$)} + +\vfill + +%\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 diff --git a/Advanced/Nonstandard Analysis/parts/2 dual.tex b/Advanced/Nonstandard Analysis/parts/2 dual.tex new file mode 100644 index 0000000..16d572e --- /dev/null +++ b/Advanced/Nonstandard Analysis/parts/2 dual.tex @@ -0,0 +1,108 @@ +% Copyright (C) 2023 +% +% 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 . +% +% +% +% If you edit this, please give credit! +% Quality handouts take time to make. + + +\section{Dual Numbers} + +\definition{} +In the problems below, $\varepsilon$ an infinitesimal so that $\varepsilon^2 = 0$. \par +Note that $\varepsilon \neq 0$. + +\definition{} +The set of \textit{dual numbers} is a nonarchimedian extension of $\mathbb{R}$ \par +that consists of elements that look like $a + b\varepsilon$, 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 $+-\times\div$, powers, and roots) \par +\note[Note]{ + Why this condition? These are the only operations we have in an ordered field! \\ + Powers, roots, and division aren't directly available, but are fairly easy to define. +} + +\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{} +Say 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, \par +and use it to 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{} +Say the derivatives of $f$ and $g$ are known. \par +What is the derivative of $f(g(x))$? + +\vfill +\pagebreak \ No newline at end of file diff --git a/Advanced/Nonstandard Analysis/parts/x supremum.tex b/Advanced/Nonstandard Analysis/parts/x supremum.tex new file mode 100644 index 0000000..afe27ee --- /dev/null +++ b/Advanced/Nonstandard Analysis/parts/x supremum.tex @@ -0,0 +1,146 @@ +% Copyright (C) 2023 +% +% 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 . +% +% +% +% If you edit this, please give credit! +% Quality handouts take time to make. + + +\section*{Bonus: The supremum \& 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 $x \leq m \forall m \in M$. \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 $m \in M$, $m \leq x$ + \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. +\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 +\pagebreak + + +\theorem{Completeness Axiom} +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 $[a_1, b_1] \subseteq [a_2, b_3] \subseteq [a_3, b_3] \subseteq ...$ be an infinite sequence of +closed line intervals. Show that there exists a $c \in \mathbb{R}$ that lies in all of them. \par +Is this true for 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 +