\section{Set theory}

In this section we construct the quotient of a type by an equivalence relation. By an equivalence relation we understand a binary relation $R:A\to(A\to\prop)$ which is reflexive, symmetric, and transitive. In particular, we note that equivalence relations take values in $\prop$. The quotient $A/R$ is constructed as the type of equivalence classes, which is just the image of the map $R:A\to (A\to\prop)$. Thus, our construction of the quotient by an equivalence relation is very much like the classical construction of a quotient set. Examples of set quotients are abundant. We cover two of them: the type of rational numbers and the set truncation of a type.

There is, however, a subtle issue with our construction of the set quotient as the image of the map $R:A\to(A\to\prop)$. What is the universe level of the quotient $A/R$? Let us suppose that $\UU$ is a universe that contains $A$ and each $R(x,y)$. Then $\prop$, the type of propositions in $\UU$, is a type in the universe $\UU^+$, constructed in \cref{rmk:universe-constructions}. Therefore the type $\prop^A$ as well as the quotient $A/R$ are also types in $\UU^+$. That seems unfortunate, because in Zermelo-Fraenkel set theory the quotient of a set by an equivalence relation is an ordinary set, and not a more general class.

In Zermelo-Fraenkel set theory quotients are are sets because of the axiom schema of replacement. The replacement axioms assert that the image of any function is again a set. This leads us to wonder about a type theoretical variant of the replacement axioms. Indeed, there is such a variant. The type theoretic replacement property asserts that for any map $f:A\to B$ from a type $A$ in $\UU$ to a type $B$ of which the \emph{identity types} are equivalent to types in $\UU$, the image of $f$ is also equivalent to a type in $\UU$, and in fact this property is a theorem. We prove it in \cref{thm:replacement}, using the univalence axiom and a new construction of the image of a map.

\subsection{Equivalence relations}

\begin{defn}\label{defn:eq_rel}
Let $R:A\to (A\to\prop)$ be a binary relation valued in the propositions. We say that $R$ is an \define{equivalence relation}\index{equivalence relation} if $R$ comes equipped with
\begin{align*}
\rho & : \prd{x:A}R(x,x) \\
\sigma & : \prd{x,y:A} R(x,y)\to R(y,x) \\
\tau & : \prd{x,y,z:A} R(x,y)\to (R(y,z)\to R(x,z)),
\end{align*}
witnessing that $R$ is reflexive, symmetric, and transitive.
\end{defn}

\begin{defn}
  Let $R:A\to (A\to\prop)$ be an equivalence relation. The \define{equivalence class} of $x:A$ is defined to be
  \begin{equation*}
    [x]_R\defeq R(x).
  \end{equation*}
  More generally, a subtype $P:A\to \prop$ is said to be an \define{equivalence class} if it satisfies
  \begin{equation*}
    \mathsf{is\usc{}equivalence\usc{}class}(P)\defeq\exists_{(x:A)}P=R(x).
  \end{equation*}
  Furthermore, we define $A/R$ to be the type of equivalence classes, i.e., we define
  \begin{equation*}
    A/R\defeq \sm{P:A\to\prop}\mathsf{is\usc{}equivalence\usc{}class}(P).
  \end{equation*}
\end{defn}

In other words, $A/R$ is the image of the map $[{-}]_R:A\to (A\to\prop)$. In the following proposition we characterize the identity type of $A/R$. As a corollary, we obtain equivalences
\begin{equation*}
  ([x]_R=[y]_R)\simeq R(x,y),
\end{equation*}
justifying that the quotient $A/R$ is defined to be the type of equivalence classes. Note that in our characterization of the identity type of $A/R$ we make use of the univalence axiom.

\begin{prp}\label{prp:eq-quotient}
  Let $R:A\to (A\to\prop)$ be an equivalence relation. Furthermore, consider $x:A$ and an equivalence class $P$. Then the canonical map
  \begin{equation*}
    ([x]_R=P)\to P(x)
  \end{equation*}
  is an equivalence.
\end{prp}

\begin{proof}
  By \cref{thm:id_fundamental} it suffices to show that the total space
  \begin{equation*}
    \sm{P:A/R}P(x)
  \end{equation*}
  is contractible. The center of contraction is of course $[x]_R$, which satisfies $[x]_R(x)$ by reflexivity of $R$. It remains to construct a contraction. Since $\sm{P:A/R}P(x)$ is a subtype of $A/R$, we construct a contraction by showing that
  \begin{equation*}
    [x]_R=P
  \end{equation*}
  whenever $P(x)$ holds. Recall that $P$ is an equivalence relation, i.e., that there exists a $y:A$ such that $P=[y]_R$. Note that our goal is a proposition, so we may assume that we have such a $y$. Then we obtain that $R(x,y)$ holds from the assumption that $P(x)$ holds. Thus, we have to show that
  \begin{equation*}
    [x]_R=[y]_R
  \end{equation*}
  given that $R(x,y)$ holds. By function extensionality and the univalence axiom, it is equivalent to show that
  \begin{equation*}
    \prd{z:A}R(x,z)\simeq R(y,z)
  \end{equation*}
  We get a function $R(x,z)\to R(y,z)$ by transitivity, since $R(y,x)$ holds by symmetry. Conversely, we get a function $R(y,z)\to R(x,z)$ directly by transitivity. Thus, we obtain that
  \begin{equation*}
    R(x,z)\leftrightarrow R(y,z)
  \end{equation*}
  for any $z:A$, which is sufficient to prove that they are equivalent because $R$ is valued in $\prop$.
\end{proof}

\begin{cor}\label{cor:eq-quotient}
  Consider an equivalence relation $R$ on a type $A$, and let $x,y:A$. Then there is an equivalence
  \begin{equation*}
    ([x]_R=[y]_R)\simeq R(x,y).
  \end{equation*}
\end{cor}

\begin{proof}
  By \cref{prp:eq-quotient} we have an equivalence
  \begin{equation*}
    ([x]_R=[y]_R)\simeq R(y,x).
  \end{equation*}
  Moreover, $R(y,x)$ is equivalent to $R(x,y)$ by symmetry of $R$.
\end{proof}

\begin{comment}
The notion of $0$-equivalence relation which we defined in \cref{defn:eq_rel} fits in a hierarchy of `$n$-equivalence relations'\index{n-equivalence relation@{$n$-equivalence relation}}, the study of which is a research topic on its own. However, we already know an example of a relation that should count as an `$\infty$-equivalence relation'\index{infinity-equivalence relation@{$\infty$-equivalence relation}}: the identity type. Analogous to \cref{thm:equivalence_classes}, the following theorem shows that the canonical map
\begin{equation*}
(x=y)\to (\idtypevar{A}(x)=\idtypevar{A}(y))
\end{equation*}
is an equivalence, for any $x,y:A$. In other words, $\idtypevar{A}(x)$ can be thought of as the equivalence class of $x$ with respect to the relation $\idtypevar{A}$.

\begin{thm}
Assuming the univalence axiom on $\UU$, the map
\begin{equation*}
\idtypevar{A}:A\to (A\to\UU)
\end{equation*}
is an embedding, for any type $A:\UU$.\index{identity type!is an embedding}
\end{thm}

\begin{proof}
Let $a:A$. By function extensionality it suffices to show that the canonical map
\begin{equation*}
(a=b)\to \idtypevar{A}(a)\htpy\idtypevar{A}(b)
\end{equation*}
that sends $\refl{a}$ to $\lam{x}\refl{(a=x)}$ is an equivalence for every $b:A$, and by univalence it therefore suffices to show that the canonical map
\begin{equation*}
(a=b)\to \prd{x:A}\eqv{(a=x)}{(b=x)}
\end{equation*}
that sends $\refl{a}$ to $\lam{x}\idfunc[(a=x)]$ is an equivalence for every $b:B$. To do this we employ the type theoretic Yoneda lemma, \cref{thm:yoneda}.

By the type theoretic Yoneda lemma\index{Yoneda lemma} we have an equivalence
\begin{equation*}
\Big(\prd{x:A} (b=x)\to (a=x)\Big)\to (a=b)
\end{equation*}
given by $\lam{f} f(b,\refl{b})$, for every $b:A$. Note that any family of maps $\prd{x:A}(b=x)\to (a=x)$ induces an equivalence of total spaces by \cref{ex:contr_equiv}, since their total spaces are are both contractible by \cref{cor:contr_path}. It follows that we have an equivalence
\begin{equation*}
\varphi_b:\Big(\prd{x:A} \eqv{(b=x)}{(a=x)}\Big)\to (a=b)
\end{equation*}
given by $\lam{f} f(b,\refl{b})$, for every $b:A$.

Note that $\varphi_a(\lam{x}\idfunc[(a=x)])\jdeq\refl{a}$. Therefore it follows by another application of \cref{thm:yoneda} that the unique family of maps
\begin{equation*}
\alpha_b:(a=b)\to \Big(\prd{x:A} \eqv{(b=x)}{(a=x)}\Big)
\end{equation*}
that satisfies $\alpha_a(\refl{a})=\lam{x}\idfunc[(a=x)]$ is a family of sections of $\varphi$.
It follows that $\alpha$ is a family of equivalences. Now the proof is completed by reverting the direction of the family of equivalences in the codomain.
\end{proof}
\end{comment}

\subsection{The universal property of set quotients}

The quotient $A/R$ is constructed as the image of $R$, so we obtain a commuting triangle
\begin{equation*}
  \begin{tikzcd}[column sep=-1em]
    A \arrow[rr,"q_R"] \arrow[dr,swap,"R"] & & A/R \arrow[dl,hook,"i_R"] \\
    \phantom{A/R} & \prop^A,
  \end{tikzcd}
\end{equation*}
and the embedding $i_R:A/R\to\prop^A$ satisfies the universal property of the image of $R$. This universal property is, however, not the usual universal property of the quotient.

\begin{defn}
  Consider a map $q:A\to B$ into a set $B$ satisfying the property that $f(x)=f(y)$ whenever $R(x,y)$ holds. We say that $q$ satisfies the \define{universal property of the set quotient by $R$} if for every map $f:A\to X$ into a set $X$ such that $f(x)=f(y)$ whenever $R(x,y)$ holds, there is a unique extension
  \begin{equation*}
    \begin{tikzcd}
      A \arrow[d,swap,"q"] \arrow[dr,"f"] \\
      B \arrow[r,densely dotted] & X.
    \end{tikzcd}
  \end{equation*}
\end{defn}

\begin{rmk}
  Formally, we express the universal property of the quotient by $R$ as follows. Consider a map $q:A\to B$ that satisfies the property that
  \begin{equation*}
    H:\prd{x,y:A}R(x,y)\to (f(x)=f(y)).
  \end{equation*}
  Then there is for any set $X$ a map
  \begin{equation*}
    q^\ast:(B\to X) \to \Big(\sm{f:A\to X}\prd{x,y:A}R(x,y)\to (f(x)=f(y))\Big).
  \end{equation*}
  This map takes a function $h:B\to X$ to the pair
  \begin{equation*}
    q^\ast(h)\defeq(h\circ q,\lam{x}{y}{r}\ap{h}{H_{x,y}(r)}).
  \end{equation*}
  The universal property of the set quotient of $R$ asserts that the map $q^\ast$ is an equivalence for every set $X$. It is important to note that the universal property of set quotients is formulated with respect to sets.
\end{rmk}

\begin{thm}\label{thm:quotient_up}
  Let $R:A\to (A\to \prop)$ be an equivalence relation\index{equivalence relation}, and consider a map $q:A\to B$ into a set $B$. Then the following are equivalent.
  \begin{enumerate}
  \item The map $q$ satisfies the property that
    \begin{equation*}
      q(x)=q(y)
    \end{equation*}
    for every $x,y:A$ for which $R(x,y)$ holds, and moreover $q$ satisfies the universal property of the set quotient of $R$.
  \item The map $q$ is surjective and \define{effective}, which means that for each $x,y:A$ we have an equivalence
    \begin{equation*}
      (q(x)=q(y))\simeq R(x,y).
    \end{equation*}
  \item The map $R:A\to (A\to \prop)$ extends along $q$ to an embedding
    \begin{equation*}
      \begin{tikzcd}[column sep=tiny]
        A \arrow[rr,"q"] \arrow[dr,swap,"R"] & & B \arrow[dl,densely dotted,"i"] \\
        & \prop^A
      \end{tikzcd}
    \end{equation*}
    and the embedding $i$ satisfies the universal property of the image inclusion of $R$.
  \end{enumerate}
\end{thm}

\begin{proof}
  We first show that (ii) is equivalent to (iii), since this is the easiest part. After that, we will show that (i) is equivalent to (ii).

  Assume that (ii) holds. Then $q$ is surjective by \cref{thm:surjective}. Moreover, we have
  \begin{align*}
    R(x,y) & \simeq R(x)=R(y) \\
           & \simeq i(q(x))=i(q(y)) \\
           & \simeq q(x)=q(y)
  \end{align*}
  In this calculation, the first equivalence holds by \cref{cor:eq-quotient}; the second equivalence holds since we have a homotopy $R\htpy i\circ q$; and the third equivalence holds since $i$ is an embedding. This completes the proof that (ii) implies (iii).

  Next, we show that (iii) implies (ii). Assuming (iii), we define a map
  \begin{equation*}
    i:B\to\prop^A
  \end{equation*}
  by $i(b,a)\defeq b=q(a)$. Then the triangle
  \begin{equation*}
    \begin{tikzcd}[column sep=tiny]
      A \arrow[rr,"q"] \arrow[dr,swap,"R"] & & B \arrow[dl,"i"] \\
      & \prop^A
    \end{tikzcd}
  \end{equation*}
  commutes, since we have an equivalence
  \begin{align*}
    i(q(a),a') & \simeq R(a,a')
  \end{align*}
  for each $a,a':A$. To show that $i$ is an embedding, it suffices to show that $i$ is injective, i.e., that
  \begin{equation*}
    \prd{b,b':B}(i(b)=i(b'))\to (b=b')
  \end{equation*}
  Note that this is a property, and that $q$ is assumed to be surjective. Hence by \cref{prp:surjective} it is equivalent to show that
  \begin{equation*}
    \prd{a,a':A}(i(q(a))=i(q(a')))\to (q(a)=q(a')).
  \end{equation*}
  Since $R\htpy i\circ q$, and $q(a)=q(a')$ is assumed to be equivalent to $R(a,a')$, it suffices to show that
  \begin{equation*}
    \prd{a,a':A}(R(a)=R(a'))\to R(a,a'),
  \end{equation*}
  which follows directly from \cref{cor:eq-quotient}. Thus we have shown that the factorization $R\htpy i\circ q$ factors $R$ as a surjective map followed by an injective map. We conclude by \cref{thm:surjective} that the embedding $i$ satisfies the universal property of the image factorization of $R$, which finishes the proof that (iii) implies (ii).

  Now we show that (i) implies (ii). To see that $q$ is surjective if it satisfies the assumptions in (i), consider the image factorization
  \begin{equation*}
    \begin{tikzcd}[column sep=tiny]
      A \arrow[dr,swap,"q"] \arrow[rr,"q_q"] & & \im(q) \arrow[dl,"i_q"] \\
      \phantom{\im(q)} & B.
    \end{tikzcd}
  \end{equation*}
  We claim that the map $i_q$ has a section. To see this, we first note that we have
  \begin{equation*}
    q_q(x)=q_q(y)
  \end{equation*}
  for any $x,y:A$ satisfying $R(x,y)$, because if $R(x,y)$ holds, then $q(x)=q(y)$ and hence $i_q(q_q(x))=i_q(q_q(y))$ holds and $i_q$ is an embedding. Since $\im(q)$ is a set, we may apply the universal property of $q$ and we obtain a unique extension of $q_q$ along $q$
  \begin{equation*}
    \begin{tikzcd}
      A \arrow[d,swap,"q"] \arrow[dr,"q_q"] \\
      B \arrow[r,densely dotted,swap,"h"] & \im(q).
    \end{tikzcd}
  \end{equation*}
  Now we observe that the composite $i_q\circ h$ is an extension of $q$ along $q$, so it must be the identity function by uniqueness. Thus we have established that $h$ is a section of $i_q$. Now it follows from the fact that $i_q$ is an embedding with a section, that $i_q$ is an equivalence. We conclude that $q$ is surjective, because $q$ is the composite $i_q\circ q_q$ of a surjective map followed by an equivalence.

  Now we have to show that $q(x)=q(y)$ is equivalent to $R(x,y)$. We first apply the universal property of $q$ to obtain for each $x:A$ an extension of $R(x)$ along $q$
  \begin{equation*}
    \begin{tikzcd}
      A \arrow[d,swap,"q"] \arrow[dr,"R(x)"] \\
      B \arrow[r,densely dotted,swap,"\tilde{R}(x)"] & \prop.
    \end{tikzcd}
  \end{equation*}
  Since the triangle commutes, we have an equivalence $\tilde{R}(x,q(x'))\simeq R(x,x')$ for each $x':A$. Now we apply \cref{thm:id_fundamental} to see that the canonical family of maps
  \begin{equation*}
    \prd{y:B}(q(x)=y)\to \tilde{R}(x,y)
  \end{equation*}
  is a family of equivalences. Thus, we need to show that the type $\sm{y:B}\tilde{R}(x,y)$ is contractible. For the center of contraction, note that we have $q(x):B$, and the type $\tilde{R}(x,q(x))$ is equivalent to the type $R(x,x)$, which is inhabited by reflexivity of $R$. To construct the contraction, it suffices to show that
  \begin{equation*}
    \prd{y:B}\tilde{R}(x,y)\to (q(x)=y).
  \end{equation*}
  Since this is a property, and since we have already shown that $q$ is a surjective map, we may apply \cref{prp:surjective}, by which it suffices to show that
  \begin{equation*}
    \prd{x':A}\tilde{R}(x,q(x'))\to (q(x)=q(x')).
  \end{equation*}
  Since $\tilde{R}(x,q(x'))\simeq R(x,x')$, this is immediate from our assumption on $q$. Thus we obtain the contraction, and we conclude that we have an equivalence $\tilde{R}(x,y)\simeq (q(x)=y)$ for each $y:B$. It follows that we have an equivalence
  \begin{equation*}
    R(x,y)\simeq (q(x)=q(y))
  \end{equation*}
  for each $x,y:A$, which completes the proof that (i) implies (ii).

  It remains to show that (iii) implies (i). Assume (iii), and let $f:A\to X$ be a map into a set $X$, satisfying the property that
  \begin{equation*}
    \prd{a,a':A}R(a,a')\to (f(a)=f(a')).
  \end{equation*}
  Our goal is to show that the type of extensions of $f$ along $q$ is contractible. By \cref{ex:surjective-precomp} it follows that there is at most one such an extension, so it suffices to construct one.

  In order to construct an extension, we will construct for every $b:B$ a term $x:X$ satisfying the property
  \begin{equation*}
    P(x)\defeq \exists_{(a:A)}(f(a)=x)\land (q(a)=b).
  \end{equation*}
  Before we make this construction, we first observe that there is at most one such $x$, i.e., that the type of $x:X$ satisfying $P(x)$ is in fact a proposition. To see this, we need to show that $x=x'$ for any $x,x':X$ satisfying $P(x)$ and $P(x')$. Since $X$ is assumed to be a set, our goal of showing that $x=x'$ is a property. Therefore we may assume that we have $a,a':A$ satisfying
  \begin{align*}
    f(a) & = x & q(a) & = b \\
    f(a') & = x' & q(a') & = b.
  \end{align*}
  It follows from these assumptions that $q(a)=q(a')$, and hence that $R(a,a')$ holds. This in turn implies that $f(a)=f(a')$, and hence that $x=x'$.

  Now let $b:B$. Our goal is to construct an $x:X$ that satisfies the property
  \begin{equation*}
    \exists_{(a:A)}(f(a)=x)\land (q(a)=b).
  \end{equation*}
  Since $q$ is assumed to be surjective, we have $\brck{\fib{q}{b}}$. Moreover, since we have shown that at most one $x:X$ exists with the asserted property, we get to assume that we have $a:A$ satisfying $q(a)=b$. Now we see that $x\defeq f(a)$ satisfies the desired property.

  Thus, we obtain a function $h:B\to X$ satisfying the property that for all $b:B$ there exists an $a:A$ such that
  \begin{equation*}
    f(a)=h(b)\qquad\text{and}\qquad q(a)=b.
  \end{equation*}
  In particular, it follows that $h(q(a))=f(a)$ for all $a:A$, which completes the proof that (ii) implies (i).
\end{proof}

\subsection{The rational numbers}

\subsection{Set truncation}\label{sec:set-truncation}

\begin{lem}
For each type $A$, the relation $I_{(-1)}:A\to (A\to\prop)$ given by
\begin{equation*}
I_{(-1)}(x,y)\defeq\brck{x=y}
\end{equation*}
is an equivalence relation.
\end{lem}

\begin{proof}
For every $x:A$ we have $\bproj{\refl{x}}:\brck{x=x}$, so the relation is reflexive. To see that the relation is symmetric note that by the universal property of propositional truncation there is a unique map $\brck{\invfunc}:\brck{x=y}\to\brck{y=x}$ for which the square
\begin{equation*}
\begin{tikzcd}
(x=y) \arrow[r,"\invfunc"] \arrow[d,swap,"\bproj{\blank}"] & (y=x) \arrow[d,"\bproj{\blank}"] \\
\brck{x=y} \arrow[r,densely dotted,swap,"\brck{\invfunc}"] & \brck{y=x}
\end{tikzcd}
\end{equation*}
commutes. This shows that the relation is symmetric. Similarly, we show by the universal property of propositional truncation that the relation is transitive.
\end{proof}

\begin{defn}
For each type $A$ we define the \define{set truncation}
\begin{equation*}
\trunc{0}{A}\defeq A/I_{(-1)},
\end{equation*}
and the unit of the set truncation is defined to be the quotient map.
\end{defn}

\begin{thm}
For each type $A$, the set truncation satisfies the universal property of the set truncation.
\end{thm}

\begin{exercises}
  \exercise Consider a map $f:A\to B$ into a set $B$, and let $R:A\to(A\to\prop)$ be the equivalence relation given by
  \begin{equation*}
    R(x,y)\defeq f(x)=f(y).
  \end{equation*}
  Show that the map $q_f:A\to\im(f)$ satisfies the universal property of the set quotient of $R$.
  \exercise Show that the set truncation of a loop space is a group.
  \exercise Recall that a normal subgroup $H$ of a group $G$ is a subgroup of $G$ such that $ghg^{-1}$ is in $H$ for every $h:H$ and $g:G$. Given a normal subgroup $H$ of $G$, we write $G/H$ for the quotient of $G$ by the equivalence relation where $g\sim g'$ if and only if there is a $h:H$ such that $gh=g'$. Show that $G/H$ is again a group.
\exercise
\begin{subexenum}
\item Show that any proposition is locally small.\index{proposition!is locally small}
\item Show that any essentially small type is locally small.\index{essentially small!type!is locally small}
\item Show that the function type $A\to X$ is locally small whenever $A$ is essentially small and $X$ is locally small.
\end{subexenum}
\exercise Let $f:A\to B$ be a map. Show that the following are equivalent:
\begin{enumerate}
\item The map $f$ is \define{locally small}\index{locally small!map} in the sense that for every $x,y:A$, the action on paths of $f$
\begin{equation*}
\apfunc{f}:(x=y)\to (f(x)=f(y))
\end{equation*}
is an essentially small map.
\item The diagonal $\delta_f$ of $f$ as defined in \cref{ex:trunc_diagonal_map} is classified by the universal fibration.
\end{enumerate}
\exercise \label{ex:span_rel}Use \cref{thm:choice,thm:fam_proj,cor:times_up_out} to show that the type
\begin{equation*}
\mathsf{span}(A,B)\defeq \sm{S:\UU} (S\to A)\times (S\to B)
\end{equation*}
of small spans from $A$ to $B$ is equivalent to the type $A\to (B\to\UU)$ of small relations from $A$ to $B$.
\end{exercises}

\endinput
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{comment}
\begin{thm}\label{thm:fam_proj}
For any small type $A:\UU$ there is an equivalence
\begin{equation*}
\mathsf{map\usc{}fam}_A:\eqv{(A\to \UU)}{\Big(\sm{X:\UU} X\to A\Big)}.
\end{equation*}
\end{thm}

\begin{proof}
Note that we have the function
\begin{equation*}
\varphi :\lam{B} \Big(\sm{x:A}B(x),\proj 1\Big) : (A\to \UU)\to \Big(\sm{X:\UU}X\to A\Big).
\end{equation*}
The fiber of this map at $(X,f)$ is by univalence and function extensionality equivalent to the type
\begin{equation*}
\sm{B:A\to \UU}{e:\eqv{(\sm{x:A}B(x))}{X}} \proj 1\htpy f\circ e.
\end{equation*}
By \cref{ex:triangle_fib} this type is equivalent to the type
\begin{equation*}
\sm{B:A\to \UU}\prd{a:A} \eqv{B(a)}{\fib{f}{a}},
\end{equation*}
and by `type theoretic choice', which was established in \cref{thm:choice}, this type is equivalent to
\begin{equation*}
\prd{a:A}\sm{X:\UU}\eqv{X}{\fib{f}{a}}.
\end{equation*}
We conclude that the fiber of $\varphi$ at $(X,f)$ is equivalent to the type $\mathsf{ess\usc{}small}(f)$. However, since $f:X\to A$ is a map between small types it is essentially small. Moreover, since being essentially small is a proposition by \cref{lem:isprop_ess_small}, it follows that $\fib{\varphi}{(X,f)}$ is contractible for every $f:X\to A$. In other words, $\varphi$ is a contractible map, and therefore it is an equivalence.
\end{proof}

\begin{rmk}
The inverse of the map
\begin{equation*}
\varphi : (A\to \UU)\to \Big(\sm{X:\UU}X\to A\Big).
\end{equation*}
constructed in \cref{thm:fam_proj} is the map $(X,f)\mapsto \fibf{f}$.
\end{rmk}

\begin{thm}\label{thm:classifier}
Let $f:A\to B$ be a map. Then there is an equivalence
\begin{equation*}
\eqv{\mathsf{ess\usc{}small}(f)}{\mathsf{is\usc{}classified}(f)},
\end{equation*}
where $\mathsf{is\usc{}classified}(f)$\index{is_classified(f)@{$\mathsf{is\usc{}classified}(f)$}} is the type of quadruples $(F,\tilde{F},H,p)$ consisting of maps
$F:B\to \UU$ and $\tilde{F}:A\to \sm{X:\UU}X$, a homotopy $H:F\circ f\htpy \proj 1\circ \tilde{F}$,  such that the commuting square
\begin{equation*}
\begin{tikzcd}
A \arrow[r,"\tilde{F}"] \arrow[d,swap,"f"] & \sm{X:\UU}X \arrow[d,"\proj 1"] \\
B \arrow[r,swap,"F"] & \UU
\end{tikzcd}
\end{equation*}
is a pullback square, as witnessed by $p$\footnote{The universal property of the pullback is not expressible by a type. However, we may take the type of $p:\isequiv(h)$, where $h:A\to B\times_\UU\big(\sm{X:\UU}X\big)$ is the map obtained by the universal property of the canonical pullback.}. If $f$ comes equipped with a term of type $\mathsf{is\usc{}classified}(f)$, we also say that $f$ is \define{classified}\index{classified by the universal family} by the universal family.
\end{thm}

\begin{proof}
From \cref{ex:sq_fib} we obtain that the type of pairs $(\tilde{F},H)$ is equivalent to the type of fiberwise transformations
\begin{equation*}
\prd{b:B}\fib{f}{b}\to F(b).
\end{equation*}
By \cref{cor:pb_fibequiv} the square is a pullback square if and only if the induced map
\begin{equation*}
\prd{b:B}\fib{f}{b}\to F(b)
\end{equation*}
is a fiberwise equivalence. Thus the data $(F,\tilde{F},H,pb)$ is equivalent to the type of pairs $(F,e)$ where $e$ is a fiberwise equivalence from $\fibf{f}$ to $F$. By \cref{thm:choice} the type of pairs $(F,e)$ is equivalent to the type $\mathsf{ess\usc{}small}(f)$.
\end{proof}

\begin{rmk}
For any type $A$ (not necessarily small), and any $B:A\to \UU$, the square\index{Sigma-type@{$\Sigma$-type}!as pullback of universal family}
\begin{equation*}
\begin{tikzcd}[column sep=6em]
\sm{x:A}B(x) \arrow[d,swap,"\proj 1"] \arrow[r,"{\lam{(x,y)}(B(x),y)}"] & \sm{X:\UU}X \arrow[d,"\proj 1"] \\
A \arrow[r,swap,"B"] & \UU
\end{tikzcd}
\end{equation*}
is a pullback square. Therefore it follows that for any family $B:A\to\UU$ of small types, the projection map $\proj 1:\sm{x:A}B(x)\to A$ is an essentially small map.
To see that the claim is a direct consequence of \cref{lem:pb_subst} we write the asserted square in its rudimentary form:
\begin{equation*}
%\begin{gathered}[b]
\begin{tikzcd}[column sep=6em]
\sm{x:A}\mathrm{El}(B(x)) \arrow[d,swap,"\proj 1"] \arrow[r,"{\lam{(x,y)}(B(x),y)}"] & \sm{X:\UU}\mathrm{El}(X) \arrow[d,"\proj 1"] \\
A \arrow[r,swap,"B"] & \UU.
\end{tikzcd}%\\[-\dp\strutbox]\end{gathered}\qedhere
\end{equation*}
\end{rmk}

In the following theorem we show that a type is locally small if and only if its diagonal is classified by $\UU$.

\begin{thm}
Let $A$ be a type. The following are equivalent:
\begin{enumerate}
\item $A$ is locally small.\index{locally small}
\item There are maps $I:A\times A\to\UU$ and $\tilde{I}:A\to\sm{X:\UU}X$, and a homotopy $H:I\circ \delta_A\htpy \proj 1\circ\tilde{I}$
such that the commuting square
\begin{equation*}
\begin{tikzcd}
A \arrow[r,"\tilde{I}"] \arrow[d,swap,"\delta_A"] & \sm{X:\UU}X \arrow[d,"\proj 1"] \\
A\times A \arrow[r,swap,"{I}"] & \UU
\end{tikzcd}
\end{equation*}
is a pullback square.\index{diagonal!of a type}
\end{enumerate}
\end{thm}

\begin{proof}
In \cref{ex:diagonal} we have established that the identity type $x=y$ is the fiber of $\delta_A$ at $(x,y):A\times A$. Therefore it follows that $A$ is locally small if and only if the diagonal $\delta_A$ is essentially small.
Now the result follows from \cref{thm:classifier}.
\end{proof}

\subsubsection{Univalent universes are object classifiers}

\begin{defn}
  Consider a map $p:E\to B$ and a map $f:X\to Y$. The type $\cart(f,p)$ of \define{cartesian morphisms} from $f$ to $p$ is the type of quadruples $(g,h,H,t)$ consisting of maps
  \begin{align*}
    g & : Y\to B \\
    h & : X\to E,
  \end{align*}
  a homotopy $H:g\circ f\htpy p\circ h$, and a term $t$ witnessing that the commuting square
  \begin{equation*}
    \begin{tikzcd}
      X \arrow[r,"h"] \arrow[d,swap,"f"] & E \arrow[d,"p"] \\
      Y \arrow[r,swap,"g"] & B
    \end{tikzcd}
  \end{equation*}
  is a pullback square.
\end{defn}

\begin{defn}
  A map $p:E\to B$ is called an \define{object classifier} if for every map $f:X\to Y$, the type $\cart(f,p)$ of cartesian morphisms
  \begin{equation*}
    \begin{tikzcd}
      X \arrow[r] \arrow[d,swap,"f"] & E \arrow[d,"p"] \\
      Y \arrow[r] & B
    \end{tikzcd}
  \end{equation*}
  from $f$ to $p$ is a proposition.
\end{defn}

\begin{prp}
  Consider a map $p:E\to B$. The following are equivalent:
  \begin{enumerate}
  \item The map $p$ is an object classifier.
  \item The function
    \begin{equation*}
      \tr_{\fibf{p}}:(x=y)\to (\fib{p}{x}\simeq\fib{p}{y})
    \end{equation*}
    is an equivalence.
  \end{enumerate}
\end{prp}

\begin{cor}
  A universe is an object classifier if and only if it is univalent.
\end{cor}
\end{comment}


\section{Derivation of \Cref{heuristic:error_quad_high_order} \label{app:proof_of_error_quad_high_order}}
We are interested in computing the error incurred when approximating a \twod surface integral with an interpolatory quadrature rule.
In \oned on the interval $[-1,1]$, we're interested in the quantity
%\begin{equation}
%    R_q[f] = I[f] - Q_q[f] \label{eq:rem1d},\quad\mathrm{where}\quad
%  I[f] = \int_{-1}^1 f(x) dx, \quad
%  Q_q[f] = \sum_{i=0}^q f(x_i)w_i,
%\end{equation}
\begin{align}
  R_q[f] &= I[f] - Q_q[f] \label{eq:rem1d}\\
  \shortintertext{where}
  I[f] &= \int_{-1}^1 f(x) dx\\
  Q_q[f] &= \sum_{i=0}^q f(x_i)w_i,\\
\end{align}
for quadrature weights $w_i$ for a $q$-point quadrature rule.
For a \twod double integral, we define a similar relationship between the remainder, the exact integral and the $q$th order quadrature rule:
%\begin{equation}
%    R^{(2)}_q[f] = I^{(2)}[f] - Q^{(2)}_q[f]\label{eq:rem2d}, \quad\mathrm{where}\quad
%    I^{(2)}[f] = \int_{-1}^1\int_{-1}^1 f(s,t) dsdt,\quad
%    Q^{(2)}_q[f] = \sum_{j=0}^q\sum_{i=0}^q f(s_i,t_j)w_iw_j.
%\end{equation}
\begin{align}
  R^{(2)}_q[f] &= I^{(2)}[f] - Q^{(2)}_q[f]\label{eq:rem2d}\\
  \shortintertext{where}
    I^{(2)}[f] &= \int_{-1}^1\int_{-1}^1 f(s,t) dsdt\\
  Q^{(2)}_q[f] &= \sum_{j=0}^q\sum_{i=0}^q f(s_i,t_j)w_iw_j,
\end{align}

For a function of two variables $f(s,t)$, we will denote $I_s[f] = \int_{-1}^1f(s,\cdot)ds$ as integration with respect to the $s$ variable only, which produces a function of $t$.
The same subscript notation applies to $R_{q,s}[f]$ and $Q_{q,s}[f]$ and use similar notation  for $t$: we apply the \oned functional to the variable in the subscript, producing a \oned function in the remaining variable.
We observe that
\begin{equation}
  I^{(2)}[f] = \int_{-1}^1\left(\int_{-1}^1 f(s,t) ds\right)dt = \int_{-1}^1 I_s[f] dt = I_t[I_s[f]]
  \label{eq:rem2d_to_1d}
\end{equation}
Following the discussion in \cite{aT2}, we substitute into \cref{eq:rem2d_to_1d} and have
\begin{align}
  %R^{(2)}_n[f] &= I^{(2)}[f] - Q^{(2)}_n[f]\label{eq:rem2d}\\
  I^{(2)}[f] &= I_t[R_{q,s}[f] + Q_{q,s}[f]]\\
         &= R_{q,t}[R_{q,s}[f] + Q_{q,s}[f]] + Q_{q,t}[R_{q,s}[f] + Q_{q,s}[f]] \\
         &= R_{q,t}[R_{q,s}[f]] + Q_{q,s}[R_{q,t}[f]] + Q_{q,t}[R_{q,s}[f]] +Q_{q,t}[Q_{q,s}[f]]
\end{align}
We assume that the higher-order ``remainder of remainder'' term contributes negligibly to the error.
Although it has been shown that this term has a non-trivial contribution to a tight error estimate \cite{elliott2015complete}, %here and in \cref{sec:quad_error_heuristic},
we are able to provide a sufficiently tight upper bound.
For large $q$, the quadrature rule approaches the value of the integral, i.e., $Q_{q,\beta} \approx I_\beta$ for $\beta =s,t$, we're left with:
\begin{align}
  I^{(2)}[f] &\approx  I_s[R_{q,t}[f]] + I_t[R_{q,s}[f]] +Q^{(2)}_q[f],\\
  \shortintertext{and hence:}
  R^{(2)}_q[f] &\lesssim  I_s[R_{q,t}[f]] + I_t[R_{q,s}[f]] \label{eq:rem2d_int},
\end{align}
where $\lesssim$ means "approximately less than or equal to."
From \cite[Theorem 5.1]{trefethen2008gauss}, we recall that for a \oned function $\theta$ defined on $[-1,1]$, if $Q_q[\theta]$ is computed with Clenshaw-Curtis quadrature, $\theta$ is $C^k$ and $\|\theta^{(k)}\|_T < V$ on $[-1,1]$ for real finite $V$, then for sufficiently large $q$, the following inequality holds
\begin{equation}
  R_q[\theta] \leq \frac{32V}{15\pi k(2q+1-k)^k},
  \label{eq:tref_ccquad2d}
\end{equation}
where $\|\alpha(x)\|_T = \|\alpha^\prime/\sqrt{1-x^2}\|_1$.
We're interested in integrating a function $\tilde{\theta}$ over an interval $[-h,h]$ for various $h$.
If $\tilde{\theta}$ is $C^k$ and $\|\tilde{\theta}\|_T < V^\prime$ on $[-h,h]$ for a real constant $V^\prime$ independent of $h$, then we can define $\theta(x) = \tilde{\theta}(hx)$ on $[-1,1]$ and apply
\cref{eq:tref_ccquad2d}:
\begin{equation}
  R_q[\tilde{\theta}] \leq \frac{32h^{k+1}V^\prime}{15\pi k(2q+1-k)^k}.
  \label{eq:tref_ccquad2d_variable_interval}
\end{equation}
This follows directly from the proof of \cite[Theorem 4.2]{trefethen2008gauss} applied to $\theta$ by replacing $\theta$ with $\tilde{\theta}(hx)$ and noting that $\theta^{(k)}(x) = h^k \tilde{\theta}^{(k)}(hx)$.
The change of variables produces the first power of $h$, while each of the $k$ integration by parts produces an additional power of $h$.
In the context of \qbkix, the size of $h$ is proportional to the edge length of the subdomain $D_i$ outlined in \cref{sec:geom-def}.

Applying \cref{eq:tref_ccquad2d_variable_interval} to \cref{eq:rem2d_int}, and again letting $f(s,t) = \Theta(hs,ht)$, gives us
\begin{equation}
  R^{(2)}_q[f] \lesssim \frac{32h^{k+1}}{15\pi k (2q+1-k)^k}\left[I_s[V^\prime_t(s)] + I_t[V^\prime_s(t)]\right] \label{eq:quad_err_1d}
\end{equation}
where $V^\prime_t(s)=\max_t\|\Theta^{(k)}(hs, ht)\|_T$ and $V^\prime_s(t)=\max_s\|\Theta^{(k)}(hs, ht)\|_T$ for fixed values of $s,t$.
If we can choose a $\tilde{V}$ that is strictly greater than $V^\prime_s(t)$ and $V^\prime_t(s)$ for any $s,t$ in $\mathcal{I}^{(2)}$, we are left with
\begin{equation}
  R^{(2)}_q[f] \lesssim \frac{128h^{k+1}\tilde{V}}{15\pi k (2q+1-k)^k}.\label{eq:quad_err_final}
\end{equation}

Applying this to the integration of double layer potentials, we can simply let $\tilde{V}$ be the largest variation of the $k$th partial derivatives of the integrand of any single patch in \cref{eq:double_layer_patches}.
In fact, we know that this value is achieved at the projection of $\vx$ on the patch $P_i$ closest to $\vx$, i.e., $(s^*,t^*) = \mathrm{argmin}_{\mathcal{I}^{(2)}} \|\vx - P_i(s,t)\|_2$.
 We can also choose $h = \max_i h_i$ to observe standard high-order convergence as a function of patch domain size, which we summarize in the following theorem.
  The smoothness and bounded variation assumptions required to apply \cref{eq:tref_ccquad2d} to our layer potential follow directly from the smoothness of $u(\vx)$ in $\Omega$.
  Our heuristic directly follows.


\section{Complex: Complex Numbers}
This module has had an interesting history.  It started as a reimplementation
of NR92's Appendix C.  I [HGG] didn't know about the section 5.4, which
explains the algorithms, so I had to reverse engineering the code itself.
Once I understood the issues, I reimplemented.  I also looked for
and used alternative references for several of the functions.

Later, I added the transcendentals from Krey88, ch 12, pp765-
766.

In March of 1996, Warren Smith made contact and submitted a wealth of
new code, some dependent on his own Complex module.  I folded that into
the existing Complex, assuring both HGG and WDS conventions could be used.
I removed some of my old code and replaced it with WDS implementations.
Some of the old code I left as is.  Notice that the results depend on
structural equivalence in type checking --- doing this in a language
which required name equivalence would have been grim.

WDS generally documents his work meticulously in extended comments.
For other modules I have translated those to \TeX\ format.
However, for this
module there is relatively little documentation provided, so none is
presented here.

\subsection*{add, sub}
Trivial

\subsection*{mul}
See also Hopk88, pp35-36
\begin{verbatim}
     c3.re:=c1.re*c2.re-c1.im*c2.im;
     c3.im:=c1.im*c2.re+c1.re*c2.im
\end{verbatim}

\subsection*{div}
This is retained as hgg\_div, but WDS's implementation is exported.

The basic formula (CRC91, pg 337) is [x is re and y is im]:
\begin{equation}
  \frac{(x_1x_2+y_1y_2) + i(x_2y_1-x_1y_2)}
       {x_2^2 + y_2^2}
\end{equation}
However, this takes 8 multiplies.  Also, if x2 and y2 are
different in magnitude, there can be truncation errors.  So,
if x2 is the biggest, we divide top and bottom by 1/x2:
\begin{equation}
  \frac{(x_1+y_1(y_2/x_2)) + i(y_1 - x_1(y_2/x_2))}
       {x_2 + y_2*(y_2/x_2)}
\end{equation}
Notice how the ratio $y_2/x_2$ repeats.  Also, we use the
denominator for both the re and im portions.  We now have 6
multiplies, and the opportunity to prevent at least some
truncations.

\subsection*{abs}
See WDS's Magnitude comments.

The basic formula (CRC91, pg 337) is
\begin{equation}
  |z| = \sqrt{(x^2 + y^2)}
\end{equation}

The simple cases of $x=0$ or $y=0$ can be resolved right away.

Otherwise, if x and y are different in magnitude, we can
have truncation error.  In 64-bits, we have $EPS=1e-17$, so
$x/y$ (or $y/x$) would have to be that extreme to cause a
problem.  I'll take that chance.

If the problem does arise, or if one needs a 32-bit
representation (where the trouble arises much sooner), do a
normalization.  See also NR92:
\begin{verbatim}
     x:=|x|; y:=|y|;
     if x > y then r=y/x and:

     |z| = (x^2+y^2)^1/2
         = [x^2*(1+y^2/x^2)]^1/2
         = x*(1+(y^2/x^2)^1/2
         = x*(1+(y/x)^2)^1/2
         = x*(1+r*r)^1/2, x > 0
\end{verbatim}

Now we have $1 + something\_less\_than\_one$, which might still
give a truncation error but has a better chance than just
adding the squares.  It costs an extra multiply.  Of course,
if $y >x$, then the analysis is reversed.
\begin{verbatim}
     if y>x then r=x/y and:

       |z| = (x^2+y^2)^1/2
           = [y^2*(x^2/y^2+1)]^1/2
           = y*(x^2/y^2 +1)^1/2
           = y*((x/y)+1)^1/2
           = y*(r^2+1)^1/2, y > 0
\end{verbatim}

\subsection*{sqrt}
See WDS's Sqrt comments.

The basic formula (Krey88, pg 730, eqn 17) is
\begin{equation}
  \sqrt{z} = \pm \left[ \sqrt{\frac{1}{2}(|z| + x)} +
     \mbox{sign}(y)*i*\sqrt{\frac{1}{2}(|z| - x)}\right]
\end{equation}

This seems to do ok.  NR92, pg177 provides a different set
of formulas.  When time permits, we should do accuracy and
timing comparisons.

\subsection*{exp, ln}
From Krey88.

\subsection*{powN, powXY}
From Krey88.

\subsection*{cos, sin, tan}
For cos, the formula is:
\begin{equation}
  \cos(z)=\cos(x)\cosh(y) - i \sin(x)\sinh(y)
\end{equation}

Where:
\begin{equation}
  \cosh(x)=\frac{1}{2}(e^{x} + e^{-x})
\end{equation}

Or:
\begin{verbatim}
     ex:=exp(x); ey:=exp(y);
     tmp.re:=+0.5*cos(x)*(ex-1.0/ex)
     tmp.im:=-0.5*sin(x)*(ey-1.0/ey)
\end{verbatim}

Clearly there is opportunity for truncation in the ex and ey
terms.  Is it a problem? Taking the ex term to be specific:
There is truncation if the exponents of $e^x$ and $e^{-x}$ differ
by more than 17.  This the exponent of $e^x$ can be up to $1/2$
of this or $17/2 = 8$.  Thus x can be up to $\ln(1E8)=~18$.  By
using MathCAD, we find we can get to $x=~20$ before blowing
up.

Alternatively, factor out $e^{-x}$:
\begin{verbatim}
     tmp.re:=+0.5*cos(x)*(ex*ex+1.0)/ex;
\end{verbatim}

This may or may not be more stable, but doesn't fundamental
shift x's range.  We are still in the $|x|<=18$ range.

Is that enough?  CRC91 provides table values to $x=10$.  Let's
accept 18 as the top end, document this, and report an
exception if we go over that.  This will be true of both the
x and y inputs.
\begin{tt} \begin{verbatim}
     IF ABS(c.re) > 18.0D0 OR ABS(c.im)> 18.0D0 THEN
       RAISE Error{Err.out_of_range};
     END;
\end{verbatim} \end{tt}

sin is of course similar.  Notice that for sin and cos there
are no opportunities for ex and ey reuse.  We may as well
use the Math cosh, sinh functions (hoping they are more
robust than our own analysis here).

tan is $sin/cos$:  In this case we can probably reuse ex, ey
from the various cosh and sinh calculations. For now we
however, we will do it naively.

\subsection*{cosh, sinh, tanh}
The formula is:
\begin{equation}
  \cosh(z)=\frac{1}{2}(e^{z}+e^{-z})=\cos(i z)
\end{equation}

Where $e^z$ is found by
\begin{verbatim}
       ex:=exp(z.re);
       tmp.re:= ex*cos(z.im);
       tmp.im:= ex*sin(z.im);
\end{verbatim}

Once again, we can do a full analysis, or just take the easy
way out.  If we ever get into a problem space requiring
hyperbolics in an inner loop, we should do timing analysis.
For now we will take the easy way out:
\begin{verbatim}
     cosh(z) = cos(i*z) = cos(i*z.re + i*i*z.im) = cos(-z.im + i*z.re)

     tmp.re:=-z.im;
     tmp.im:=+z.re;
     tmp:=C.cos(tmp);
     return tmp;
\end{verbatim}

sin is similar:
\begin{verbatim}
     sinh(z) = -i*sin(i*z) = -i*(-z.im + i*z.re)

     tmp.re:=-z.im;
     tmp.im:=+z.re;
     tmp:=C.sin(tmp);
     (*tmp.re = -i*i*tmp.im = tmp.im*)
     (*tmp.im = -i*tmp.re* = -tmp.re*)
     t:=tmp.im;
     tmp.im:=-tmp.re;
     tmp.re:=t;
     return tmp;
\end{verbatim}

\subsection*{pmul, pdiv}
If we happen to be in polar form, then multiplication and
division are simple.  For pmul:
\begin{verbatim}
     tmp.radius:=p1.radius*p2.radius;
     tmp.angle:=p1.angle+p2.angle
\end{verbatim}

For pdiv:
\begin{verbatim}
     tmp.radius:=p1.radius/p2.radius;
     tmp.angle:=p1.angle-p2.angle
\end{verbatim}

In addition, we need to normalize the angles to the $-\pi \dots +\pi$
range:
\begin{tt} \begin{verbatim}
     WHILE tmp.angle < -Pi DO
       tmp.angle:=tmp.angle + TwoPi;
     END;
     WHILE tmp.angle > Pi DO
       tmp.angle:=tmp.angle - TwoPi;
     END;
\end{verbatim} \end{tt}

NOTE:  {\tt arg} and {\tt toPolar} use {\tt atan2}, which does
the normalization itself, so we don't need to handcraft
normalization code for them.


\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{environ}
\usepackage{physics}

\NewEnviron{NORMAL}{%
    \scalebox{5}{$\BODY$}
}

\title{Using the Jacobian to map directional derivatives between coordinate systems}
\author{Max Sills }
\date{August 2020}

% Defining bank of important matrices for easy reuuse
\newcommand{\showMatrixOne}{\ensuremath{
\begin{bmatrix}
\frac{x}{\sqrt{x^2 + y^2}} & \frac{y}{\sqrt{x^2 + y^2}} \\
\frac{-y}{x^2 + y^2}       & \frac{x}{x^2+y^2}
\end{bmatrix}
}}

\newcommand{\showMatrixTwo}{\ensuremath{
\begin{bmatrix}
\cos\theta & sin\theta  \\
\frac{-\sin\theta}{r} &  \frac{\cos\theta}{r}
\end{bmatrix}
}}

\newcommand{\showMatrixThree}{\ensuremath{
\begin{bmatrix}
\cos\theta & -r\,\sin\theta  \\
\sin\theta &  r\,\cos\theta
\end{bmatrix}
}}

\newcommand{\showMatrixFour}{\ensuremath{
\begin{bmatrix}
\frac{x}{\sqrt{x^2 + y^2}} & -y \\
\frac{y}{\sqrt{x^2 + y^2}} & x
\end{bmatrix}
}}
% End matrix bank

\begin{document}
\maketitle

\section{Mapping gradients between coordinate systems}

Start with a function U in cartesian coordinates, and U' for that function expressed in polar coordinates.

We can use the change of coordinates formula to express U' given U

\[ x = r * cos(\theta) \]
\[ y = r * sin (\theta)\]

Now say we were able to express the gradient of U in cartesian coordinates
\[
\nabla U = \frac{\partial U} {\partial x}+
\frac{\partial U} {\partial y}
\]

How could we express the gradient of U' in polar coordinates? Well, we've already been given U' in polar coordinates, so we could compute the gradient directly. However, we can also use the chain rule, the change of coordinates formula, and our knowledge of the gradient in cartesian coordinates to directly derive the gradient in polar coordinates.

$$
\nabla U' = \frac{\partial U'} {\partial r}+
\frac{\partial U'} {\partial \theta}
$$

By the chain rule,

$$
\frac{\partial U'} {\partial r} = \frac{\partial U'} {\partial x}\frac{\partial x} {\partial r}
+
\frac{\partial U'} {\partial y}\frac{\partial y} {\partial r}
$$

$$
\frac{\partial U'} {\partial \theta} = \frac{\partial U'} {\partial x}\frac{\partial x} {\partial \theta}
+
\frac{\partial U'} {\partial y}\frac{\partial y} {\partial \theta}
$$

We can rewrite these relations as a covector acting on a Jacobian
\begingroup
\LARGE
\begin{equation}
\left\langle
\begin{matrix}\frac{\partial U}{\partial r} & \frac{\partial U} {\partial \theta}
\end{matrix}
\right\rangle=\left\langle
\begin{matrix}\frac{\partial U}{\partial x} & \frac{\partial U} {\partial y}
\end{matrix}
\right\rangle
*
\begin{bmatrix}
\frac{\partial x}{\partial r} & \frac{\partial y}{\partial r}\\\frac{\partial x}{\partial \theta} & \frac{\partial y}{\partial \theta}
\end{bmatrix}
\end{equation}
\endgroup

We want to find the gradient of U in polar coordinates, which can be represented as the partials of U with respect to r and theta. This is equal to the matrix product of the covector of the gradient in cartesian coordinates which was found above, and the Jacobian, which can be entirely derived from the change of coordinates functions from cartesian to polar.

\section{Examples}

Before we work examples, let's identify the transition functions to and from polar coordinates and the corresponding Jacobians to make things easier.

\subsection{Cartesian to Polar}

\begin{align}
r &= \sqrt{x^2 + y^2} \\
\theta &= \arctan{\left(\frac{y}{x}\right)}
\end{align}
\begin{equation}
\showMatrixOne
=
\showMatrixTwo
\end{equation}

\subsection{Polar to Cartesian}
\[ x = r * cos(\theta) \]
\[ y = r * sin (\theta)\]

\begin{equation}
\showMatrixThree
=
\showMatrixFour
\end{equation}

\subsection{Example 1: U = x + y}
\[ U = x + y\]
\[\nabla U = \pdv{U}{x} + \pdv{U}{y} = <1,1>  \]

Now to express U in polar coordinates (r, theta) we find
\[ U = r*cos\theta + r*sin\theta  \]
\[\nabla U = cos\theta + sin \theta \pdv{U}{r}
+ r (cos\theta - sin \theta \pdv{U}{\theta} \]

\begin{equation}
\begin{bmatrix}
cos + sin & r(cos-sin)
\end{bmatrix}
*
\begin{bmatrix}
\cos\theta & sin\theta  \\
\frac{-\sin\theta}{r} &  \frac{\cos\theta}{r}
\end{bmatrix}
=
\begin{bmatrix}
1 & 1
\end{bmatrix}
\end{equation}

Now, how might we take an arbitrary directional derivative in the first coordinate system and map it to a directional derivative in the second?

Our gradient in the first coordinate system is
\[\begin{bmatrix}
1 & 1
\end{bmatrix}
\]

When we apply an arbitrary directional derivative (a, b), we produce a scalar:
\begin{displaymath}
\begin{bmatrix}
1 & 1
\end{bmatrix}
*
\begin{bmatrix}
a \\
b
\end{bmatrix}
=
a + b
\end{displaymath}

Applying the Jacobian to the vector this time, instead of the gradient (covector) to the Jacobian, we find:
\begin{displaymath}
\showMatrixTwo
*
\begin{bmatrix}
a \\
b
\end{bmatrix}
=
\begin{bmatrix}
a cos(\theta) + b sin(\theta) \\
\frac{-a sin(\theta)}{r} +
\frac{b cos (\theta)}{r}
\end{bmatrix}
\end{displaymath}

Finally, applying this transformed vector to the gradient we found in polar coordinates we see:

\begin{displaymath}
\begin{bmatrix}
cos + sin & r(cos-sin)
\end{bmatrix}
*
\begin{bmatrix}
a cos(\theta) + b sin (\theta) \\
\frac{-a sin(\theta)}{r} +
\frac{b cos (\theta)}{r}
\end{bmatrix}
=
\end{displaymath}
\begin{displaymath}
\begin{aligned}
(acos + bsin)(cos + sin) + (cos- sin)(bcos -asin) \\
= acos^2 + asin cos + bsin cos + bsin^2 \\
+ bcos^2 - asin cos - bsin cos + a sin^2 \\
= acos^2 + b sin^2 + bcos^2 + a sin^2 \\
= a(cos^2 + sin^2) + b(cos^2 + sin^2) \\
= a + b
\end{aligned}
\end{displaymath}

\subsection{Example 2: U(r, theta) = r *theta}

\[U = r * \theta\]
\[\nabla U = \theta \pdv{U}{r}+ r \pdv{U}{\theta} \]

U(r, theta) in cartesian coordinates:

\[ U(x, y) =\sqrt{x^2 +y^2}
*
\arctan\frac{y}{x} \]

\begin{equation}
\nabla U =
\frac{x \atan(y/x) - y}{\sqrt{x^2 + y^2}}
\pdv{U}{x}
+
\frac{y \atan(y/x) + x}{\sqrt{x^2 + y^2}}
\pdv{U}{y}
\end{equation}
\end{document}


\subsection{Turbulence closure}\label{sec-tur}

We use the \citet{Umlauf2003} generic length-scale model:
\beqa
  \frac{\D k}{\D t}&=&\frac{\pd }{\pd z}\left( \nu_k^\Psi \frac{\pd k}{\pd z} \right)
  +\nu_t M^2+\nu_t^\theta N^2 -\epsilon \\
  \frac{\D \Psi}{\D t}&=& \frac{\pd }{\pd z}\left( \nu_\Psi \frac{\pd \Psi}{\pd z} \right)
    +\frac{\Psi}{k}(c_{\Psi 1}\nu_tM^2+c_{\Psi 3}\nu_t^\theta N^2-c_{\Psi 2}\epsilon F_{wall})
\eeqa
with natural b.c.:
\beq
   \left\{ \begin{array}{ll}
       \nu_k^\Psi \frac{\pd k}{\pd z} &=0, \mbox{ at } z=-h, \mbox{ or } \eta \\
       \nu_\Psi\frac{\pd \Psi}{\pd z} &= \kappa_0 n\nu_\Psi\frac{\Psi}{l} , \mbox{ at } z=-h \\
       \nu_\Psi\frac{\pd \Psi}{\pd z} &= -\kappa_0 n\nu_\Psi\frac{\Psi}{l} , \mbox{ at } z=\eta \\
           \end{array}
   \right.  \label{tur1}
\eeq
and essential b.c.:
\beq
   \left\{ \begin{array}{ll}
       k&=(c_\mu^0)^{-2} \nu|\frac{\pd \bs{u}}{\pd z}|, \mbox{ at } z=-h, \mbox{ or } \eta\\
       l&=\kappa_0 \D \\
       \Psi &= (c_\mu^0)^pk^m(\kappa_0 \D)^n
           \end{array}
   \right.  \label{tur2}
\eeq
where $k$ is the TKE, $l$ is the mixing length, $c_{\Psi *}$ are some constants and $\Psi=(c_\mu^0)^pk^m l^n$ is a generic length-scale variable.
The turbulence production and dissipation terms are:
\beqa
  M^2&=&\left( \frac{\pd u}{\pd z}\right)^2+\left( \frac{\pd v}{\pd z}\right)^2 \\
  N^2 &=&\frac{g}{\rho_0}\frac{\pd \rho}{\pd z} \\
  \epsilon &=& (c_\mu^0)^3k^{1.5} l^{-1}
\eeqa
In the code, the natural b.c. is applied first (see the FEM formulation below), and the essential b.c.
 is then used to overwrite the boundary values of the unknown, as suggested by the \gls{gotm} implementation.

GOTM has also been coupled to SELFE, but the native (finite element) implementation is the one used for the Bay-Delta project.


\section{Soft-thresholding and Proximal gradient method}
\label{sec:soft-thresholding}
Recall the unconstrained optimization problem
$$
\min _{x} f(x)
$$
We have seen the following results depending on the assumption for $f(x)$
\begin{enumerate}
\item If $f(x)$ is not smooth (for example: $L^1$ norm) which means $\nabla f(x)$ is not continuous, subgradient descent method gives the $\frac{1}{\sqrt{n}}$ convergence.
\item If $f(x)$ is smooth (e.g. $\nabla f(x)$ is L-Lipschitz continuous), gradient descent method gives the $\frac{1}{n}$ convergence.
\end{enumerate}
There may be other methods (may be for particular functions) which can give better performance. Consider the following example:
$$
f(x)=\frac{1}{2}\|A x-b\|_{2}^{2}+\|x\|_{1}
$$
This is a type 1 problem described above, as the second term is not smooth. However, the objective has a special form. It can be represented as sum of smooth and non -smooth function. Also note that second term $\|x\|_{1}$ is "simple" in a sense that we describe precisely below. In this case, it is conceivable that some tailored method might do better than the generic guarantees offered by sub gradient descent.

In this section, we introduce what is known as the Proximal method, and show that it provides improved guarantees.

\subsection{Proximal mapping}\label{sec:proximal}
\begin{definition}
The proximal mapping or prox-operator of a function $f(x)$ is defined as
\begin{equation}\label{pro:def}
\operatorname{Prox}_{f}(x)=\arg \min _{z}f(z)+\frac{1}{2}\|z-x\|_{2}^{2}.
\end{equation}
\end{definition}
Note that $f(x)$ is a convex function so the above problem \eqref{pro:def} a strongly convex problem because of the added Euclidean norm term; therefore the solution is unique. Also for the proximal method to be efficient, we need that $f(x)$ should be simple in the sense that solving the problem \eqref{pro:def}  should be easier and fast or may be even analytical solutions are possible. This condition is required to ensure that each step is not much costly.

\begin{properties}
From the optimality conditions of minimization in the definition, we can see that $z^*$ is optimal point of $f(x)+\frac12 \|z-x\|_2^2$ only if $0\in \partial f(x) + z^*-x$, namely
\begin{equation}
\begin{aligned}
z^{*}=\operatorname{Prox}_{f}(x) & \Longleftrightarrow 0 \in \partial f(x)+\left(z^{*}-x\right) \\
& \Longleftrightarrow x-z^{*} \in \partial f(x) \\
& \Longleftrightarrow f(w)=f(z)+(x-z)^{T}(w-z)\quad \forall w
\end{aligned}
\end{equation}
\end{properties}
This property shows the relation between proximal mapping  and subgradient of  a function.

\begin{example}
If $f(x)=0$,
$$
\operatorname{Prox}_{f}(x)=x.
$$
\end{example}
\begin{example}
If $f(x)=|x|$
$$
\begin{array}{l}
\operatorname{Prox}_{f}(x)=\left\{\begin{array}{cl}
x-1 & \text { if } x\geqslant 1 \\
0 & \text { if } -1 \leqslant x \leqslant 1    \\
x+1 & \text { if } x \leqslant-1
\end{array}\right.
\end{array}
$$
	\end{example}
\begin{example}
If
	$$
	f(x)=\left\{\begin{array}{l}
	0 \text { if } x \in[-1,1] \\
	\infty \text { otherwise }
	\end{array}\right.
	$$
$$ 	\begin{array}{l}
	\operatorname{Prox}_{f}(x)=\left\{\begin{array}{cl}
			x & \text { if } x \in [-1, 1] \\
			1 & \text { if } x>1   \\
			-1 & \text { if } x < -1
		\end{array}\right.
	\end{array}  $$
which is the projection onto the set [-1,1].
if $$
f(x)=\left\{\begin{array}{ll}
0 & \text { if } x \in C \\
\infty & x \notin C
\end{array}\right.
$$  these $ \operatorname{prox}_{f}(x) $ is the projection onto C.
	\end{example}

%\begin{example}
%	If $ X=\partial f $ where f is convex, what is $ prox_{f} $?
%
%	For $ x \subset\mathbb{R}^{d} $, solve for $ (x^{\prime}, y^{\prime}) \in \partial f $ s.t.
%	$$
%	 x^{\prime}+y^{\prime}=x.
%	$$ Notice that
%	$$
%	x \in \operatorname{argmin} f \Leftrightarrow 0 \in \partial f(x)
%	$$
%	Also, for $ x \in dom(f_{1}) \cap dom(f_{2}) $,
%	$$
%	 \partial (f_{1}+f_{2}) (x)=\partial f_{1}(x)+\partial f_{2}(x)  :=\left\{y_{1}+y_{2} ; y_{1} \in \partial f_{1}(x), y_{2} \in \partial f_{2}(x)\right\}
%	 $$
%	$$
%	0\in \partial \left(f(z)+\frac{1}{2} \|(x-z)\|^{2}\right)=\partial f(z)-x+z\Rightarrow x \in z+ \partial f(z).
%	$$
%	$$
%	\operatorname{prox}_{f}(x)=\operatorname{argmin}\limits_{z}\left(f(z)+\frac{1}{2}\|x-z\|_2^{2}\right)
%	$$
%	\end{example}

\begin{theorem}[soft-thresholding]
Let $f(x)=\lambda\|x\|_1$. Soft-thresholding  is defined as
\begin{equation}
prox_f(x) = \arg\min_z\{\frac{1}{2}\|x-z\|^2_2 + \lambda\|z\|_1 \},
\end{equation}
which is shown in the following figure
\begin{figure}
\centering
\includegraphics[width=3in]{./6DL/figures/softthreshold}
%\caption{Two linearly separable sets}
\end{figure}
The $l_1$-norm is is separable in indices. Each $i^{\text {th }}$ coordinate can be optimized separately and the $i^{t h}$ coordinate is given as
$$
prox_f(x)_{i}=\operatorname{sgn}\left(x_{i}\right)\left[\left|x_{i}\right|-\min \left\{\left|x_{i}\right|, \lambda\right\}\right]
=\begin{cases}
0&|x_i|\le \lambda
\\
x_i-\lambda&x_i>\lambda
\\
x_i+\lambda&x_i<-\lambda
\end{cases}.
$$
\end{theorem}

\begin{proof}
Note that
\begin{equation}
0 \in \nabla(\frac{1}{2}\|x-z\|^2_2) + \partial(\lambda\|z\|_1) \Leftrightarrow 0 \in z-x + \lambda\partial\|z\|_1
\end{equation}
The $l_1$-norm is separable and thus we can consider each of its components separately. Let's examine first the case where $z_i \neq 0$. Then, $\partial \|z_i\|=sign(z_i)$ and the optimum $z_i^*$ is obtained as
\begin{equation}
0 = z_i-x_i + \lambda sign(z_i) \Leftrightarrow z_i^* = x_i - \lambda sign(z_i^*)
\end{equation}
Note also that if $z_i^* < 0$, then $x_i < -\lambda$ and equivalently if $z_i^* > 0 \Rightarrow x_i > \lambda$. Thus, $|x_i| > \lambda$ and $sign(z_i^*) = sign(x_i)$. Substituting in the previous equation we get
\begin{equation}
z_i^* = x_i - \lambda sign(x_i)
\end{equation}
In the case where $z_i = 0$, the subdifferential of the $l_1$-norm is the interval $[-1,1]$ and the optimality condition is
\begin{equation}
0 \in -x_i + \lambda[-1,1] \Leftrightarrow x_i \in [-\lambda,\lambda] \Leftrightarrow |x_i| \leq \lambda
\end{equation}
Putting all together we get
\begin{equation}
[prox_f(x)]_i = z_i^* =
\left\{ \begin{array}{lr} 0 & \text{if } |x_i| \leq \lambda \\
x_i - \lambda sign(x_i) &\text{if } |x_i| > \lambda \end{array}\right.
\end{equation}
The previous equation can also be written as
\begin{align*}
[prox_f(x)]_i &= sign(x_i)\max(|x_i|-\lambda, 0) \\
               &= sign(x_i)(|x_i|-\lambda)_+
\end{align*}
where $(\cdot)_+$ denotes the positive part and it leads to sparse solution.
\end{proof}

\subsection{Properties of the proximal map}
\begin{definition}
: A map g:$ \mathbb{R}^{d} \rightarrow \mathbb{R}^{d} $ is called
\begin{itemize}
	\item Non-expansive if $ \|g(x)-g(y)\| \leqslant\|x-y\| $
	\item Firauly Non-expansive if $ \|g(x)-g(y)\|^{2} \leqslant\langle g(x)-g(y), x-y\rangle $
	\item A contraction if $ \|g(x)-g(y)\| \leqslant \lambda\|x-y\| \quad \text { for } 0 \leqslant \lambda<1 $
\end{itemize}
\end{definition}

\begin{lemma}
	If g is non-expansive, the $ \frac{1}{2}(g+I) $ is 	firmly non-expansive.
	$$
	\frac{1}{2}(g+I)(x)=\frac{1}{2}(g(x)+x)
	$$
	If g is firmly non-expansive, then $ 2g-I $ is non-expansive.
\end{lemma}

\begin{proof}
	$$
	\begin{aligned}
	\|h(x)-h(y)\|^{2}&=\left\langle\frac{1}{2}[g(x)+x]-\frac{1}{2}[g(y)+y], \frac{1}{2}[g(x)+x]-\frac{1}{2}[g(J)+y]\right\rangle\\
	&=\left\langle\frac{1}{2}(g(x)-g(y))+\frac{1}{2}(x-y), \frac{1}{2}(g(x)-g(y))+\frac{1}{2}(x-y)\right\rangle \\
	&=\frac{1}{4} \|g(x)-g(y)\|^{2}\|x-y\|^{2}+\frac{1}{2}(g(x)-g(y), x-y)+\frac{1}{4}\|x-y\|^{2} \\
	& \leqslant \frac{1}{2}\|x-y\|^{2}+\frac{1}{2}\langle g(x)-g(y), x-y\rangle \\
	&=\left\langle\frac{1}{2}(g(x)-g(y))+\frac{1}{2}(x-y), x-y\right\rangle\\
	&=\langle h(x)-h(y), x-y\rangle .
	\end{aligned}
	$$
	\end{proof}

\begin{example}
	Non-expansive $ \mathbb{R}^{2}  $, rotate by $ \theta $, has a fixed point, but we don't converge to it.
	Nonexpansive maps may fail to converge to fixed points. Firmly nonexpanisve maps always converge to their fixed points.
	Fixed points may not be unique though.
	\end{example}

\begin{example}
	$ g: \mathbb{R} \rightarrow \mathbb{R}$,
	$$  g(x)=\left\{\begin{array}{cl}
	x & \text { if } x\in [-1,1] \\
	1 & \text { if }  x > 1    \\
	-1 & \text { if } x <-1
	\end{array}\right.$$
	\end{example}
g is firmly non-expansive. Set of fixed points [-1,1]
\begin{theorem}
The proximal map is always firmly non-expansive.
	\end{theorem}
\begin{theorem}
If g is firmly nonexpansive and $ g(y)=y $	for some y, there if $ x_{n+1}=g(x_{n}) $,we have $ x_{n} \rightarrow y^{\prime} $ with $ g(y^{\prime})=y^{\prime} $
\end{theorem}

\begin{example}
$ g: \mathbb{R} \rightarrow \mathbb{R}, g(x)=x+1$ is firmly non-expansive in fact $ g(x)=prox_{f}(x), f(x)=-x$
	\end{example}

\subsection{Proximal gradient method}
Proximal point algorithm:
$$
x_{n+1}=prox_{f}(x_{n}).
$$
By the preceding if $ prox_{f}$ has a fixed point, then $ x_{n} $ converge to a fixed point.
A fixed point of $ prox_{f}$ is a point x with $0 \in \partial f(x) $, which is a minimizer.
Usually calculating $ prox_{f}$ is just as difficult as minimizing $f$ itself,
$$
prox_{f}=\mathop{argmin}\limits_{z} \frac{1}{2}\|x-z\|_2^{2}+f(z)
$$
%Later we will use 	$ prox_{f}$ and $ prox_{g}$ to construct a firmly non-expansive map whose fixed points mininize $g+h $ Douglas-Rachfound splitting.
%$$
%x_{n+1}=x_{n}-s h_{n}, \quad h_{n}=\mathop{argmax}\limits_{\|h\|_{1}}\left\langle h,\nabla f\left(x_{n}\right)\right\rangle
%$$
%Boosting: $$ \mathop{argmax}\limits_{h \in D}\left\langle h,\nabla f\left(x_{n}\right)\right\rangle $$
%If $
%f(x)=\frac{1}{2}\left\|x-x_{0}\right\|^{2}
%$get greedy approx.

Consider unconstrained problem with cost function split in two components
$$
\min f(x)=g(x)+h(x)
$$
where $h$ is convex and differentiable,
$g$ is convex and possibly nondifferentiable, with inexpensive pro-operator.
The proximal gradient algorithm is
$$
x^{(k)}=\operatorname{prox}_{t_{k} g}\left(x^{(k-1)}-t_{k} \nabla h\left(x^{(k-1)}\right)\right)
$$
where $t_{k}>0$ is step size, constant or determined by line search. Note that if $g(x)=0$, then this method reduces to a gradient descent method.

If we apply the proximal mapping to update rule,
$$
\begin{aligned}
x^{+} &=\underset{z}{\operatorname{argmin}}\left(g(z)+\frac{1}{2 t}\|z-x+t \nabla h(x)\|_{2}^{2}\right) \\
&=\underset{z}{\operatorname{argmin}}\left(g(z)+h(x)+\nabla h(x)^{T}(z-x)+\frac{1}{2 t}\|z-x\|_{2}^{2}\right).
\end{aligned}
$$
This means that
$x^{+}$ minimizes $g(z)$ plus a simple quadratic local model of $h(z)$ around $x$.

\begin{properties}
If
$$
u=\operatorname{prox}_{f}(x), v= \operatorname{prox}_{f}(y),
$$
then
$$
(u-v)^{T}(x-y) \geq\|u-v\|_{2}^{2}.
$$
This implies that
$$
x-u \in \partial f(u), \quad y-v \in \partial f(v) \quad \Longrightarrow \quad(x-u-y+v)^{T}(u-v) \geq 0,
$$
and
$$
\left\|\operatorname{prox}_{f}(x)-\operatorname{prox}_{f}(y)\right\|_{2} \leq\|x-y\|_{2}
$$
$prox_{f}$ is nonexpansive or Lipschitz continuous with constant 1.
\end{properties}

\begin{definition}
	 The proximal map of X, $ prox_{X}(x)=x^{1} $(" Backward Euler")
\end{definition}

%\begin{remark}
%\begin{enumerate}
%\item If the full gradient descent method is used, $\eta$ is a given value.
%\item If the stochastic  gradient descent method is used, $\eta$ is approaching 0 as t goes to infinity. We don't expect sparse 'minimization sequence'
%\item Can we find a different training algorithm that would give 'sparsity'?
%\end{enumerate}
%\end{remark}

%\section{Forward-Backward Splitting Algorithm(not included)}
%\begin{lemma}
%$\Theta=\{\theta:f(\theta,x)\ \text{ is a classifier}\}$ is a non-empty open set.
%\end{lemma}
%
%
%\begin{remark}
%\begin{enumerate}
%\item Question 1: Can we find some sparse $\theta\in \Theta$ by certain training algorithm?
%\begin{itemize}
%\item The desirable training algorithm should converge.
%\item The desirable training algorithm generates 'sparse' minimization sequence.
%\end{itemize}
%
%\item Question 2: Can we find a model $L(\theta)+R(\theta)$ with small $R(\theta)$ s.t. $\arg\min(L(\theta)+R(\theta)) \approx \arg\min L(\theta)$.
%\end{enumerate}
%\end{remark}
%A general approach is to use $L(\theta)+R(\theta)$, where $R(\theta)=\lambda \|\theta\|_1$.
%
%Assume f is strongly convex and smooth function and R is convex. Thus $x^*= \arg\min (f(x)+R(x)) \iff 0\in \nabla f(x^*)+\partial R(x^*)$.
%We have the following forward-backward splitting algorithm.

%\section{A summary of algorithms and convergence results}
%\includepdf[pages=-,angle=270]{6DL/SGD-summary.jpeg}


\section{Isomorphisms}

\begin{outcome}
  \begin{enumerate}
  \item Determine if a linear transformation is an isomorphism.
  \item Determine if two subspaces of $\R^n$ are isomorphic.
  \end{enumerate}
\end{outcome}

Recall the definition of a linear transformation. Let $V$ and $W$ be two subspaces of $\R^{n}$ and $\R^{m}$
respectively. A mapping $T:V\rightarrow W$ is called a\textbf{\ linear
transformation} or \textbf{linear map} if it preserves the algebraic
operations of addition and
\index{linear map}
\index{linear transformation} scalar multiplication. Specifically, if $a,b$
are scalars and $
\vect{x},\vect{y}$ are vectors,
\begin{equation*}
T(a\vect{x}+b\vect{y}) =aT(\vect{x})+bT(\vect{y})
\end{equation*}

Consider the following important definition.

\begin{definition}{Isomorphism}{isomorphism}
A linear map $T$ is called an \textbf{isomorphism}
\index{isomorphism}if the following two conditions are satisfied.

\begin{itemize}
\item $T$ is one to one. That is, if $T(\vect{x})=T(\vect{y})$, then $\vect{x}=\vect{y}$.

\item $T$ is onto. That is, if $\vect{w}\in W$, there exists $
\vect{v}\in V$ such that $T(\vect{v})=\vect{w}$.
\end{itemize}

Two such subspaces which have an isomorphism as described above are said to
be \textbf{isomorphic.}
\index{isomorphic}
\end{definition}

Consider the following example of an isomorphism.

\begin{example}{Isomorphism}{isomorphism}
Let $T: \R^2 \to \R^2$ be defined by
\[
T \begin{mymatrix}{c}
x \\
y
\end{mymatrix} = \begin{mymatrix}{c}
x + y \\
x - y
\end{mymatrix}
\]
Show that $T$ is an isomorphism.
\end{example}

\begin{solution}
To prove that $T$ is an isomorphism we must show
\begin{enumerate}
\item $T$ is a linear transformation;
\item $T$ is one to one;
\item $T$ is onto.
\end{enumerate}

We proceed as follows.

\begin{enumerate}
\item $T$ is a linear transformation:

Let $k, p$ be scalars.
\begin{eqnarray*}
T \paren{k \begin{mymatrix}{c}
x_1 \\
y_1
\end{mymatrix} + p \begin{mymatrix}{c}
x_2 \\
y_2
\end{mymatrix}} &=&
T \paren{\begin{mymatrix}{c}
kx_1 \\
ky_1
\end{mymatrix} + \begin{mymatrix}{c}
px_2 \\
py_2
\end{mymatrix}} \\
&=&
T \paren{\begin{mymatrix}{c}
kx_1 + px_2 \\
ky_1 + py_2
\end{mymatrix}} \\
&=&
\begin{mymatrix}{c}
(kx_1 + px_2) + (ky_1 + py_2) \\
(kx_1 + px_2) - (ky_1 + py_2)
\end{mymatrix} \\
&=&
\begin{mymatrix}{c}
(kx_1 + ky_1) + (px_2 + py_2) \\
(kx_1  - ky_1) + (px_2 - py_2)
\end{mymatrix} \\
&=&
\begin{mymatrix}{c}
kx_1 + ky_1  \\
kx_1  - ky_1
\end{mymatrix} +
\begin{mymatrix}{c}
px_2 + py_2 \\
px_2 - py_2
\end{mymatrix} \\
&=& k \begin{mymatrix}{c}
x_1 + y_1  \\
x_1  - y_1
\end{mymatrix} +
p \begin{mymatrix}{c}
x_2 + y_2 \\
x_2 - y_2
\end{mymatrix} \\
&=&
k T \paren{\begin{mymatrix}{c}
x_1 \\
y_1
\end{mymatrix}} + p T \paren{\begin{mymatrix}{c}
x_2 \\
y_2
\end{mymatrix}}
\end{eqnarray*}

Therefore $T$ is linear.

\item $T$ is one to one:

We need to show that if $T (\vect{x}) = \vect{0}$ for a vector $\vect{x} \in \R^2$, then it follows that $\vect{x} = \vect{0}$.  Let $\vect{x} = \begin{mymatrix}{c}
x \\
y
\end{mymatrix}$.

\[
T  \paren{\begin{mymatrix}{c}
x \\
y
\end{mymatrix}} = \begin{mymatrix}{c}
x + y\\
x - y
\end{mymatrix} = \begin{mymatrix}{c}
0 \\
0
\end{mymatrix}
\]
This provides a system of equations given by
\begin{eqnarray*}
x + y &=& 0\\
x - y &=& 0
\end{eqnarray*}
You can verify that the solution to this system if $x = y =0$. Therefore
\[
\vect{x} = \begin{mymatrix}{c}
x \\
y
\end{mymatrix}
 = \begin{mymatrix}{c}
0 \\
0
\end{mymatrix}
\]
and $T$ is one to one.

\item $T$ is onto:

Let $a,b$ be scalars. We want to check if there is always a solution to
\[
T  \paren{\begin{mymatrix}{c}
x \\
y
\end{mymatrix}} = \begin{mymatrix}{c}
x + y\\
x - y
\end{mymatrix} = \begin{mymatrix}{c}
a \\
b
\end{mymatrix}
\]

This can be represented as the system of equations
\begin{eqnarray*}
x + y &=& a\\
x - y &=& b
\end{eqnarray*}

Setting up the augmented matrix and row reducing gives
\[
\begin{mymatrix}{cc|c}
1 & 1 & a \\
1 & -1 & b
\end{mymatrix} \roweq\ldots\roweq
\begin{mymatrix}{cc|c}
1 & 0 & \vspace{0.05in}\frac{a+b}{2} \\
0 & 1 & \vspace{0.05in}\frac{a-b}{2}
\end{mymatrix}
\]
This has a solution for all $a,b$ and therefore $T$ is onto.
\end{enumerate}

Therefore $T$ is an isomorphism.
\end{solution}

An important property of isomorphisms is that its inverse is also an isomorphism.

\begin{proposition}{Inverse of an isomorphism}{inverse-isomorphism}
Let $T:V\rightarrow W$ be an isomorphism and $V,W$ be subspaces of $\R^n$. Then $T^{-1}:W\rightarrow V$ is
also an isomorphism.
\index{isomorphism!inverse}
\end{proposition}

\begin{proof} Let $T$ be an isomorphism.  Since $T$ is onto, a typical
vector in $W$ is of the form $T(\vect{v})$ where $\vect{v} \in V$. Consider then for $a,b$
scalars,
\begin{equation*}
T^{-1}(aT(\vect{v}_{1})+bT(\vect{v}_{2}))
\end{equation*}
where $\vect{v}_{1}, \vect{v}_2 \in V$. Is this equal to
\begin{equation*}
aT^{-1}(T (\vect{v}_{1})) +bT^{-1}(T(\vect{v}_{2})) =a\vect{v}_{1}+b\vect{v}_{2}?
\end{equation*}
Since $T$ is one to one, this will be so if
\begin{equation*}
T(a\vect{v}_{1}+b\vect{v}_{2}) =T(T^{-1}(aT(\vect{v}_{1})+bT(\vect{v}_{2}))
) =aT(\vect{v}_{1})+bT(\vect{v}_{2}).
\end{equation*}
However, the above statement is just the condition that $T$ is a linear map.
Thus $T^{-1}$ is indeed a linear map. If $\vect{v} \in V$ is given, then $\vect{v}=T^{-1}(T(\vect{v})) $ and so $T^{-1}$ is onto. If $T^{-1} (\vect{v})=0$, then
\begin{equation*}
\vect{v}=T(T^{-1}(\vect{v})) =T(\vect{0})=\vect{0}
\end{equation*}
and so $T^{-1}$ is one to one.
\end{proof}

Another important result is that the composition of multiple isomorphisms is also an isomorphism.

\begin{proposition}{Composition of isomorphisms}{composition-isomorphisms}
Let $T:V\rightarrow W$ and  $S:W\rightarrow Z$ be isomorphisms where $V,W,Z$ are subspaces of $\R^n$. Then $S\circ
T $ defined by $(S\circ T) (\vect{v}) = S(
T(\vect{v})) $ is also an isomorphism.
\index{isomorphism!composition}
\end{proposition}

\begin{proof}
Suppose $T:V\rightarrow W$ and  $S:W\rightarrow Z$ are isomorphisms. Why is $S\circ T$ a linear map?
For $a,b$ scalars,
\begin{eqnarray*}
S\circ T(a\vect{v}_{1}+b(\vect{v}_{2}))
&=& S(T(a\vect{v}_{1}+b\vect{v}_{2})) =S(aT\vect{v}_{1}+bT\vect{v}_{2}) \\
&=&aS(T\vect{v}_{1}) +bS(T\vect{v}_{2}) = a(S\circ
T) (\vect{v}_{1}) +b(S\circ T) (\vect{v}_{2})
\end{eqnarray*}
Hence $S\circ T$ is a linear map. If $(S\circ T) (\vect{v})
=0$, then $S(T(\vect{v})) =0$ and it follows that $T(\vect{v})=\vect{0}$ and hence by this lemma again, $\vect{v}=\vect{0}$. Thus $S\circ
T $ is one to one. It remains to verify that it is onto. Let $\vect{z} \in Z$. Then
since $S$ is onto, there exists $\vect{w} \in W$ such that $S(\vect{w})=\vect{z}$. Also, since $T$
is onto, there exists $\vect{v}\in V$ such that $T(\vect{v})=\vect{w}$. It follows that $S(
T(\vect{v})) =\vect{z}$ and so $S\circ T$ is also onto.
\end{proof}

Consider two subspaces $V$ and $W$, and suppose there exists an isomorphism mapping one to the other. In this way the two subspaces are related, which we can write as $V\iso W$. Then the previous two propositions together claim that $\iso$ is an equivalence relation. That is: $\iso$
satisfies the following conditions:
\index{isomorphic!equivalence relation}

\begin{itemize}
\item $V\iso V$

\item If $V\iso W$, it follows that $W\iso V$

\item If $V\iso W$ and $W\iso Z$, then $V\iso Z$
\end{itemize}

We leave the verification of these conditions as an exercise.

Consider the following example.

\begin{example}{Matrix isomorphism}{matrix-isomorphism}
Let $T:\R^{n}\rightarrow \R^{n}$ be defined by $T(\vect{x}) = A(\vect{x})$ where $A$ is an invertible $n\times n$-matrix. Then $T$ is
an isomorphism.
\end{example}

\begin{solution}
The reason for this is that, since $A$ is invertible, the only vector it
sends to $\vect{0}$ is the zero vector. Hence if $A(\vect{x})=A(\vect{y})$, then $A(\vect{x}-\vect{y}) =\vect{0}$ and so $\vect{x}=\vect{y}$. It is onto
because if $\vect{y}\in \R^{n},A(A^{-1} (\vect{y})) =(
AA^{-1}) (\vect{y})$ $=\vect{y}$.
\end{solution}

In fact, all isomorphisms from $\R^{n}$ to $\R^{n}$ can be expressed as $T(\vect{x}) = A(\vect{x})$ where $A$ is an invertible $n \times n$-matrix. One
simply considers the matrix whose $i\th$ column is $T\vect{e}_{i}$.

Recall that a basis of a subspace $V$ is a set of linearly independent vectors which span $V$. The following fundamental lemma describes the relation between bases and
isomorphisms.

\begin{lemma}{Mapping bases}{mapping-bases}
Let $T:V\rightarrow W$ be a linear transformation where $V,W$ are
subspaces of $\R^n$. If $T$ is one to one, then it has the property that if $\set{\vect{u}_{1},\ldots,\vect{u}_{k}} $ is linearly independent, so is $\set{T(\vect{u}_{1}),\ldots,T(\vect{u}_{k})}$.

More generally, $T$ is an isomorphism if and only if whenever $\set{
\vect{v}_{1},\ldots,\vect{v}_{n}} $ is a basis for $V$, it follows that $\set{T
(\vect{v}_{1}),\ldots,T(\vect{v}_{n})} $ is a basis for $W$.
\index{isomorphism!bases}
\index{one to one!linear independence}
\end{lemma}

\begin{proof}First suppose that $T$ is a linear transformation and is one to one
and $\set{\vect{u}_{1},\ldots,\vect{u}_{k}} $ is linearly
independent. It is required to show that $\set{T(\vect{u}_{1}),\ldots,T(\vect{
u}_{k})} $ is also linearly independent. Suppose then that
\begin{equation*}
\sum_{i=1}^{k}c_{i}T(\vect{u}_{i})=\vect{0}
\end{equation*}
Then, since $T$ is linear,
\begin{equation*}
T\paren{\sum_{i=1}^{n}c_{i}\vect{u}_{i}} =\vect{0}
\end{equation*}
Since $T$ is one to one, it follows that
\begin{equation*}
\sum_{i=1}^{n}c_{i}\vect{u}_{i}=0
\end{equation*}
Now the fact that $\set{\vect{u}_{1},\ldots,\vect{u}_{n}} $ is
linearly independent implies that each $c_{i}=0$. Hence $\set{T(\vect{u}
_{1}),\ldots,T(\vect{u}_{n})} $ is linearly independent.

Now suppose that $T$ is an isomorphism and $\set{\vect{v}_{1},\ldots,\vect{
v}_{n}} $ is a basis for $V$. It was just shown that $\set{T(\vect{v}
_{1}),\ldots,T(\vect{v}_{n})} $ is linearly independent. It remains to
verify that span$\set{T(\vect{v}_{1}),\ldots,T(\vect{v}_{n})}=W$. If $\vect{w}\in W$, then since $T$ is onto there
exists $\vect{v}\in V$ such that $T(\vect{v})=\vect{w}$. Since $\set{\vect{v}
_{1},\ldots,\vect{v}_{n}} $ is a basis, it follows that there exists
scalars $\set{c_{i}} _{i=1}^{n}$ such that
\begin{equation*}
\sum_{i=1}^{n}c_{i}\vect{v}_{i}=\vect{v}.
\end{equation*}
Hence,
\begin{equation*}
\vect{w}=T(\vect{v})=T\paren{\sum_{i=1}^{n}c_{i}\vect{v}_{i}}
=\sum_{i=1}^{n}c_{i}T(\vect{v}_{i})
\end{equation*}
It follows that span$\set{T(\vect{v}_{1}),\ldots, T(\vect{v}_{n})} =W$ showing that this set of vectors is a
basis for $W$.

Next suppose that $T$ is a linear transformation which takes a basis to a basis. This means that if $\set{\vect{v}_{1},\ldots,\vect{v}_{n}} $ is a basis for $V$, it
follows $\set{T(\vect{v}_{1}),\ldots,T(\vect{v}_{n})} $ is a basis for $
W$. Then if $w\in W$, there exist scalars $c_{i}$ such that $
w=\sum_{i=1}^{n}c_{i}T(\vect{v}_{i})=T\paren{\sum_{i=1}^{n}c_{i}\vect{v}_{i}} $
showing that $T$ is onto. If $T\paren{\sum_{i=1}^{n}c_{i}\vect{v}_{i}} =\vect{0}$
then $\sum_{i=1}^{n}c_{i}T(\vect{v}_{i})=\vect{0}$ and since the vectors $\set{T(\vect{v}
_{1}),\ldots,T(\vect{v}_{n})} $ are linearly independent, it follows
that each $c_{i}=0$. Since $\sum_{i=1}^{n}c_{i}\vect{v}_{i}$ is a typical vector in
$V$, this has shown that if $T(\vect{v})=\vect{0}$ then $\vect{v}=\vect{0}$ and so $T$ is also one to one.
Thus $T$ is an isomorphism.
\end{proof}

The following theorem illustrates a very useful idea for defining an
isomorphism. Basically, if you know what it does to a basis, then you can
construct the isomorphism.

\begin{theorem}{Isomorphic subspaces}{isomorphic-subspaces}
Suppose $V$ and $W$ are two subspaces of $\R^n$. Then the two
subspaces are isomorphic if and only if they have the same dimension. In the
case that the two subspaces have the same dimension, then for
\index{isomorphism!equivalence} a linear map $T:V\rightarrow W$, the
following are equivalent.

\begin{enumerate}
\item $T$ is one to one.

\item $T$ is onto.

\item $T$ is an isomorphism.
\end{enumerate}
\end{theorem}

\begin{proof} Suppose first that these two subspaces have the same
dimension. Let a basis for $V$ be $\set{
\vect{v}_{1},\ldots,\vect{v}_{n}} $ and let a basis for $W$ be $
\set{\vect{w}_{1},\ldots,\vect{w}_{n}}$. Now define $T$ as
follows.
\begin{equation*}
T(\vect{v}_{i})=\vect{w}_{i}
\end{equation*}
for $\sum_{i=1}^{n}c_{i}\vect{v}_{i}$ an arbitrary vector of $V$,
\begin{equation*}
T\paren{\sum_{i=1}^{n}c_{i}\vect{v}_{i}} = \sum_{i=1}^{n}c_{i}T
\vect{v}_{i}=\sum_{i=1}^{n}c_{i}\vect{w}_{i}.
\end{equation*}
It is necessary to verify that this is well-defined. Suppose then that
\begin{equation*}
\sum_{i=1}^{n}c_{i}\vect{v}_{i}=\sum_{i=1}^{n}\hat{c}_{i}\vect{v}_{i}
\end{equation*}
Then
\begin{equation*}
\sum_{i=1}^{n}(c_{i}-\hat{c}_{i}) \vect{v}_{i}=\vect{0}
\end{equation*}
and since $\set{\vect{v}_{1},\ldots,\vect{v}_{n}} $ is a basis, $
c_{i}=\hat{c}_{i}$ for each $i$. Hence
\begin{equation*}
\sum_{i=1}^{n}c_{i}\vect{w}_{i}=\sum_{i=1}^{n}\hat{c}_{i}\vect{w}_{i}
\end{equation*}
and so the mapping is well-defined. Also if $a,b$ are scalars,
\begin{eqnarray*}
T\paren{a\sum_{i=1}^{n}c_{i}\vect{v}_{i}+b\sum_{i=1}^{n}\hat{c}_{i}\vect{v}%
_{i}} &=&T(\sum_{i=1}^{n}(ac_{i}+b\hat{c}_{i}) \vect{v%
}_{i}) =\sum_{i=1}^{n}(ac_{i}+b\hat{c}_{i}) \vect{w}_{i} \\
&=&a\sum_{i=1}^{n}c_{i}\vect{w}_{i}+b\sum_{i=1}^{n}\hat{c}_{i}\vect{w}_{i} \\
&=&aT\paren{\sum_{i=1}^{n}c_{i}\vect{v}_{i}} +bT\paren{\sum_{i=1}^{n}%
\hat{c}_{i}\vect{v}_{i}}
\end{eqnarray*}
Thus $T$ is a linear transformation.

Now if
\begin{equation*}
T\paren{\sum_{i=1}^{n}c_{i}\vect{v}_{i}} =\sum_{i=1}^{n}c_{i}\vect{w}%
_{i}=\vect{0},
\end{equation*}
then since the $\set{\vect{w}_{1},\ldots,\vect{w}_{n}} $ are
independent, each $c_{i}=0$ and so $\sum_{i=1}^{n}c_{i}\vect{v}_{i}=\vect{0}$
also. Hence $T$ is one to one. If $\sum_{i=1}^{n}c_{i}\vect{w}_{i}$ is a
vector in $W$, then it equals
\begin{equation*}
\sum_{i=1}^{n}c_{i}T(\vect{v}_{i})=T\paren{\sum_{i=1}^{n}c_{i}\vect{v}_{i}}
\end{equation*}
showing that $T$ is also onto. Hence $T$ is an isomorphism and so $V$ and $W$
are isomorphic.

Next suppose $T:V \to W$ is an isomorphism, so these two subspaces are isomorphic. Then for $\set{\vect{v}_{1},\ldots,\vect{v}_{n}} $ a
basis for $V$, it follows that a basis for $W$
is $\set{T(\vect{v}_{1}),\ldots,T(\vect{v}_{n})} $ showing that the two
subspaces have the same dimension.

Now suppose the two subspaces have the same dimension. Consider the three
claimed equivalences.

First consider the claim that $1.)\Rightarrow 2.)$. If $T$ is one to one and if $\set{\vect{v}_{1},\ldots,\vect{v}
_{n}} $ is a basis for $V$, then $\set{T(\vect{v}_{1}),\ldots,T(\vect{v
}_{n})} $ is linearly independent. If it is not a basis, then it must
fail to span $W$. But then there would exist $\vect{w}\notin \sspan
\set{T(\vect{v}_{1}),\ldots,T(\vect{v}_{n})} $ and it follows that $\set{T(\vect{v}_{1}),\ldots,T(\vect{v}_{n}),\vect{w}
} $ would be linearly independent which is impossible because there exists a basis for $W$ of $n$ vectors.

Hence $\sspan\set{T(\vect{v}_{1}),\ldots,T(\vect{v}_{n})} =W$ and
so $\set{T(\vect{v}_{1}),\ldots,T(\vect{v}_{n})} $ is a basis. If $\vect{w}\in W$, there exist scalars $c_{i}$ such that
\begin{equation*}
\vect{w}=\sum_{i=1}^{n}c_{i}T(\vect{v}_{i})=T\paren{\sum_{i=1}^{n}c_{i}\vect{v}
_{i}}
\end{equation*}
showing that $T$ is onto. This shows that $1.)\Rightarrow 2.)$.

Next consider the claim that $2.)\Rightarrow 3.)$. Since $2.)$ holds, it
follows that $T$ is onto. It remains to verify that $T$ is one to one. Since
$T$ is onto, there exists a basis of the form $\set{T(\vect{v}_{i}),\ldots,T(\vect{v}_{n})}$. Then it follows that $\set{\vect{v}_{1},\ldots,
\vect{v}_{n}} $ is linearly independent. Suppose
\begin{equation*}
\sum_{i=1}^{n}c_{i}\vect{v}_{i}=\vect{0}
\end{equation*}
Then
\begin{equation*}
\sum_{i=1}^{n}c_{i}T(\vect{v}_{i})=\vect{0}
\end{equation*}
Hence each $c_{i}=0$ and so, $\set{\vect{v}_{1},\ldots,\vect{v}
_{n}} $ is a basis for $V$. Now it follows that a typical vector in $
V $ is of the form $\sum_{i=1}^{n}c_{i}\vect{v}_{i}$. If $T\paren{
\sum_{i=1}^{n}c_{i}\vect{v}_{i}} =\vect{0}$, it follows that
\begin{equation*}
\sum_{i=1}^{n}c_{i}T(\vect{v}_{i})=\vect{0}
\end{equation*}
and so, since $\set{T(\vect{v}_{i}),\ldots,T(\vect{v}_{n})} $ is
independent, it follows each $c_{i}=0$ and hence $\sum_{i=1}^{n}c_{i}\vect{v}
_{i}=\vect{0}$. Thus $T$ is one to one as well as onto and so it is an
isomorphism.

If $T$ is an isomorphism, it is both one to one and onto by definition so $
3.)$ implies both $1.)$ and $2.)$.
\end{proof}

Note the interesting way of defining a linear transformation in the first
part of the argument by describing what it does to a basis and then
``extending it linearly'' to the entire subspace.
\index{linear map!defining on a basis}

\begin{example}{Isomorphic subspaces}{}
Let $V=\R^{3}$ and let $W$ denote
\begin{equation*}
\sspan\set{\begin{mymatrix}{r}
1 \\
2 \\
1 \\
1
\end{mymatrix} ,\begin{mymatrix}{r}
0 \\
1 \\
0 \\
1
\end{mymatrix} ,\begin{mymatrix}{r}
1 \\
1 \\
2 \\
0
\end{mymatrix} }
\end{equation*}
Show that $V$ and $W$ are isomorphic.
\end{example}

\begin{solution}
First observe that these subspaces are both of dimension 3 and so they are isomorphic by Theorem~\ref{thm:isomorphic-subspaces}. The
three vectors which span $W$ are easily seen to be linearly independent by
making them the columns of a matrix and row reducing to the {\rref}.

You can exhibit an isomorphism of these two spaces as follows.
\begin{equation*}
T(\vect{e}_{1})=\begin{mymatrix}{c}
1 \\
2 \\
1 \\
1
\end{mymatrix}, T(\vect{e}_{2})=\begin{mymatrix}{c}
0 \\
1 \\
0 \\
1
\end{mymatrix}, T(\vect{e}_{3})=\begin{mymatrix}{c}
1 \\
1 \\
2 \\
0
\end{mymatrix}
\end{equation*}
and extend linearly. Recall that the matrix of this linear transformation is
just the matrix having these vectors as columns. Thus the matrix of this
isomorphism is
\begin{equation*}
\begin{mymatrix}{rrr}
1 & 0 & 1 \\
2 & 1 & 1 \\
1 & 0 & 2 \\
1 & 1 & 0
\end{mymatrix}
\end{equation*}
You should check that multiplication on the left by this matrix does
reproduce the claimed effect resulting from an application by $T$.
\end{solution}

Consider the following example.

\begin{example}{Finding the matrix of an isomorphism}{matrix-of-isomorphism}
Let $V=\R^{3}$ and let $W$ denote
\begin{equation*}
\sspan\set{\begin{mymatrix}{c}
1 \\
2 \\
1 \\
1
\end{mymatrix} ,\begin{mymatrix}{c}
0 \\
1 \\
0 \\
1
\end{mymatrix} ,\begin{mymatrix}{c}
1 \\
1 \\
2 \\
0
\end{mymatrix} }
\end{equation*}

Let $T: V \to W$ be defined as follows.
\begin{equation*}
T\begin{mymatrix}{c}
1 \\
1 \\
0
\end{mymatrix} =\begin{mymatrix}{c}
1 \\
2 \\
1 \\
1
\end{mymatrix} ,T\begin{mymatrix}{c}
0 \\
1 \\
1
\end{mymatrix} =\begin{mymatrix}{c}
0 \\
1 \\
0 \\
1
\end{mymatrix} ,T\begin{mymatrix}{c}
1 \\
1 \\
1
\end{mymatrix} =\begin{mymatrix}{c}
1 \\
1 \\
2 \\
0
\end{mymatrix}
\end{equation*}
Find the matrix of this isomorphism $T$.
\end{example}

\begin{solution}
 First note that the vectors
\begin{equation*}
\begin{mymatrix}{c}
1 \\
1 \\
0
\end{mymatrix} ,\begin{mymatrix}{c}
0 \\
1 \\
1
\end{mymatrix} ,\begin{mymatrix}{c}
1 \\
1 \\
1
\end{mymatrix}
\end{equation*}
are indeed a basis for $\R^{3}$ as can be seen by making them the
columns of a matrix and using the {\rref}.

Now recall the matrix of $T$ is a $4\times 3$-matrix $A$ which gives the same
effect as $T$. Thus, from the way we multiply matrices,
\begin{equation*}
A\begin{mymatrix}{rrr}
1 & 0 & 1 \\
1 & 1 & 1 \\
0 & 1 & 1
\end{mymatrix} =\begin{mymatrix}{rrr}
1 & 0 & 1 \\
2 & 1 & 1 \\
1 & 0 & 2 \\
1 & 1 & 0
\end{mymatrix}
\end{equation*}
Hence,
\begin{equation*}
A=\begin{mymatrix}{rrr}
1 & 0 & 1 \\
2 & 1 & 1 \\
1 & 0 & 2 \\
1 & 1 & 0
\end{mymatrix} \begin{mymatrix}{rrr}
1 & 0 & 1 \\
1 & 1 & 1 \\
0 & 1 & 1
\end{mymatrix} ^{-1}=\begin{mymatrix}{rrr}
1 & 0 & 0 \\
0 & 2 & -1 \\
2 & -1 & 1 \\
-1 & 2 & -1
\end{mymatrix}
\end{equation*}
Note how the span of the columns of this new matrix must be the same as the
span of the vectors defining $W$.
\end{solution}

This idea of defining a linear transformation by what it does on a basis
works for linear maps which are not necessarily isomorphisms.

\begin{example}{Finding the matrix of an isomorphism}{}
Let $V=\R^{3}$ and let $W$ denote
\begin{equation*}
\sspan\set{\begin{mymatrix}{c}
1 \\
0 \\
1 \\
1
\end{mymatrix} ,\begin{mymatrix}{c}
0 \\
1 \\
0 \\
1
\end{mymatrix} ,\begin{mymatrix}{c}
1 \\
1 \\
1 \\
2
\end{mymatrix} }
\end{equation*}
Let $T: V \to W$ be defined as follows.
\begin{equation*}
T\begin{mymatrix}{c}
1 \\
1 \\
0
\end{mymatrix} = \begin{mymatrix}{c}
1 \\
0 \\
1 \\
1
\end{mymatrix} ,T\begin{mymatrix}{c}
0 \\
1 \\
1
\end{mymatrix} =\begin{mymatrix}{c}
0 \\
1 \\
0 \\
1
\end{mymatrix} ,T\begin{mymatrix}{c}
1 \\
1 \\
1
\end{mymatrix} =\begin{mymatrix}{c}
1 \\
1 \\
1 \\
2
\end{mymatrix}
\end{equation*}
 Find the matrix of this linear transformation.
\end{example}

\begin{solution}
Note that in this case, the three vectors which span $W$ are not linearly independent. Nevertheless the above procedure will still work.
The reasoning is the same as before. If $A$ is this matrix, then
\begin{equation*}
A\begin{mymatrix}{rrr}
1 & 0 & 1 \\
1 & 1 & 1 \\
0 & 1 & 1
\end{mymatrix} =\begin{mymatrix}{rrr}
1 & 0 & 1 \\
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 2
\end{mymatrix}
\end{equation*}
and so
\begin{equation*}
A=\begin{mymatrix}{rrr}
1 & 0 & 1 \\
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 2
\end{mymatrix} \begin{mymatrix}{rrr}
1 & 0 & 1 \\
1 & 1 & 1 \\
0 & 1 & 1
\end{mymatrix} ^{-1}=\begin{mymatrix}{rrr}
1 & 0 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0 \\
1 & 0 & 1
\end{mymatrix}
\end{equation*}
The columns of this last matrix are obviously not linearly independent.
\end{solution}


\documentclass[11pt, a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath, amssymb, graphics, setspace}
\setlength\parindent{0pt}
\begin{document}

Quick tips before trying to solve the OLG model.
\begin{enumerate}
    \item Determine whether the zero steady state is possible: \\
    If $f(0) = 0$ (or, equivalently, $w(0) = 0$), then the zero (autarky) steady state exists.
    It may be stable or unstable, but it exists.
    \begin{itemize}
    \item Cobb-Douglas production function:\\
    $f(k) = k^\alpha \implies f(0) = 0$
    \item CES production function: \\
        \textbf{Important remark}: verify the exact functional form of the production function, in particular how the exponents are written.\\
        $f(k) = (\alpha k^\rho +(1-\alpha))^\frac{1}{\rho}$
        \begin{itemize}
            \item Complementary inputs ($\rho < 0$)\\
            $\lim_{k \rightarrow 0} f(k) = 0,$ assuming $\rho<0,\alpha \in (0,1).$
        \item Substitue inputs ($\rho > 0$)\\
            $\lim_{k \rightarrow 0} f(k) = (1-\alpha)^\frac{1}{\rho},$ assuming $\rho>0,\alpha \in (0,1).$
        \end{itemize}
    \end{itemize}

    \item Determine whether the interest rate appears in the savings function:
        \begin{itemize}
            \item If log-utility, the interest rate does not appear.\\
                $u(c) = \log (c) \implies u^\prime(c) = \frac{1}{c}$ \\
                The savings function is obtained in this case by solving:\\
                $u^\prime(w-s) = \beta R u^\prime(Rs)$ so $s = \frac{\beta}{1+\beta}w$
            \item Under a CIES, $R$ appears in the savings function:\\
                $u^\prime(c) = x^\frac{-1}{\sigma}$ \\
                The savings function is obtained in this case by solving:\\
                $(w-s)^\frac{-1}{\sigma} = \beta R (Rs)^\frac{-1}{\sigma}$ \\
                $s = \frac{w}{1+\beta^{-\sigma}R^{1-\sigma}}$
        \end{itemize}
\end{enumerate}

\section*{Examples}
\subsection*{Example 1}
Production function:    $f(k)=k^{0.5}$

Utility function:       $u(c)=2 \sqrt{c}$

Savings function:       $s(w(k_t),f^\prime(k_{t+1}))=\frac{16. R w}{25.\, +16. R}$

Substituting: $s(w(k_t),f^\prime(k_{t+1}))= \frac{4.}{25.\, +\frac{8.}{k^{0.5}}}$

Steady States:          Solve $k_{t+1} = \frac{1}{1+n}s(w(k_t),f^\prime(k_{t+1}))$

With the parameters: $\{k\to 0.\},\{k\to 0.0733398\}$

Stability:             Compute the derivative of $k_{t+1}$ with respect to $k_t$ and evaluate at each steady state.

With the parameters:

\quad Steady state 1, Capital level: $0$. Derivative=$\infty$ Unstable.

\quad Steady state 2, Capital level: $0.0733398$ Derivative=$0.406773$ Stable.

\includegraphics{xamples_gr1.eps}
\subsection*{Example 2}
Production function:    $f(k)=\frac{1}{\sqrt{0.5\, +\frac{0.5}{k^2}}}$

Utility function:       $u(c)=2 \sqrt{c}$

Savings function:       $s(w(k_t),f^\prime(k_{t+1}))= \frac{16. R w}{25.\, +16. R}$

Substituting:           $s(w(k_t),f^\prime(k_{t+1}))= \frac{8. \left(\frac{1}{\sqrt{0.5\, +\frac{0.5}{k^2}}}-\frac{0.5}{\left(0.5\, +\frac{0.5}{k^2}\right)^{3/2} k^2}\right)}{\left(25.\, +\frac{8.}{\left(0.5\, +\frac{0.5}{k^2}\right)^{3/2} k^3}\right) \left(0.5\, +\frac{0.5}{k^2}\right)^{3/2} k^3}$

Steady States:          Solve ${k}_{t+1}= \frac{1}{(1+n)}s(w(k_t),f^\prime(k_{t+1}))$

With the parameters: $\{k\to 0\}\}$

Stability:             Compute the derivative of $k_{t+1}$ with respect to $k_{t}$ and evaluate at each steady state.

With the parameters:

\quad Steady state 1, Capital level: $0$, Derivative=$0$. Stable.

\includegraphics{xamples_gr2.eps}
\subsection*{Examples 3}
Production function:    $f(k)=\frac{1}{\sqrt{0.5\, +\frac{0.5}{k^2}}}$

Utility function:       $u(c)=\log(c)$

Savings function:       $s(w(k_t),f^\prime(k_{t+1}))= 0.444444 w$

Substituting: $s(w(k_t),f^\prime(k_{t+1}))= 0.444444 \left(\frac{1}{\sqrt{0.5\, +\frac{0.5}{k^2}}}-\frac{0.5}{\left(0.5\, +\frac{0.5}{k^2}\right)^{3/2} k^2}\right)$

Steady States:          Solve ${k}_{t+1}= \frac{1}{(1+n)}s(w(k_t),f^\prime(k_{t+1}))$

With the parameters: $\{k\to 0\}\}$

Stability:             Compute the derivative of $k_{t+1}$ with respect to $k_{t}$ and evaluate at each steady state.

With the parameters:

\quad Steady state 1, Capital level: $0$ Derivative=$0$. Stable.

\includegraphics{xamples_gr3.eps}

\subsection*{Example 4}
Production function:    $f(k)=\left(0.5\, +0.5 k^{0.5}\right)^{2.}$

Utility function:       $u(c)log(c)$

Savings function:       $s(w(k_t),f^\prime(k_{t+1}))= 0.444444 w$

Substituting: $s(w(k_t),f^prime(k_{t+1}))= 0.444444 \left(\left(0.5\, +0.5 k^{0.5}\right)^{2.}-0.5 \left(0.5\, +0.5 k^{0.5}\right)^{1.} k^{0.5}\right)$

Steady States:          Solve ${k}_{t+1}= \frac{1}{(1+n)}s(w(k_t),f^\prime(k_{t+1}))$

With the parameters: $\{k\to 0.154832\}\}$

Stability:             Compute the derivative of $k_{t+1}$ with respect to $k_{t}$ and evaluate at each steady state.

With the parameters:

\quad Steady state 1, Capital level: $0.154832$ Derivative=$0.141188$  Stable.

\includegraphics{xamples_gr4.eps}

\subsection*{Example 5}
Production function:    $f(k)=\left(0.5\, +0.5 k^{0.5}\right)^{2.}$

Utility function:       $u(c)=2 \sqrt{c}$

Savings function:       $s(w(k_t),f^\prime(k_{t+1}))= \frac{16. R w}{25.\, +16. R}$

Substituting:           $s(w(k_t),f^\prime(k_{t+1}))= \frac{\left(8. \left(0.5\, +0.5 k^{0.5}\right)^{1.} \left(\left(0.5\, +0.5 k^{0.5}\right)^{2.}-0.5 \left(0.5\, +0.5 k^{0.5}\right)^{1.} k^{0.5}\right)\right)}{\left(\left(25.\, +\frac{8. \left(0.5\, +0.5 k^{0.5}\right)^{1.}}{k^{0.5}}\right) k^{0.5}\right)}$

Steady States:          Solve ${k}_{t+1}= \frac{1}{(1+n)}s(w(k_t),f^\prime(k_{t+1}))$

With the parameters: $\{k\to 0.128222\}$

Stability:             Compute the derivative of $k_{t+1}$ with respect to $k_{t}$ and evaluate at each steady state.
With the parameters:

\quad Steady state 1, Capital level: $0.128222$ Derivative=$0.107258$ Stable.

\includegraphics{xamples_gr5.eps}

\subsection*{Example 6}
Production function:    $f(k)=k^{0.5}$

Utility function:       $u(c)=\log(c)$

Savings function:       $s(w(k_t),f^\prime(k_{t+1}))= 0.444444 w$

Substituting:      $s(w(k_t),if^\prime(k_{t+1}))= 0.222222 k^{0.5}$

Steady States:          Solve ${k}_{t+1}= \frac{1}{(1+n)}s(w(k_t),f^\prime(k_{t+1}))$

With the parameters: $\{k\to 0.\},\{k\to 0.0493827\}$

Stability:             Compute the derivative of $k_{t+1}$ with respect to $k_{t}$ and evaluate at each steady state.

With the parameters:

\quad Steady state 1, Capital level: $0$. Derivative=$\infty$ Unstable.

\quad Steady state 2, Capital level: $0.0493827$ Derivative=$0.5$ Stable.

\includegraphics{xamples_gr6.eps}

\subsection*{Example 7}
Production function:    $f(k)=\frac{1}{0.9\, +\frac{0.1}{k}}$

Utility function:       $u(c)=2 \sqrt{c}$

Savings function:       $s(w(k_t),f^\prime(k_{t+1}))= \frac{16. R w}{25.\, +16. R}$

Substituting:           $s(w(k_t),f^\prime(k_{t+1})= \frac{1.6 \left(\frac{1}{0.9\, +\frac{0.1}{k}}-\frac{0.1}{\left(0.9\, +\frac{0.1}{k}\right)^2 k}\right)}{\left(25.\, +\frac{1.6}{\left(0.9\, +\frac{0.1}{k}\right)^2 k^2}\right) \left(0.9\, +\frac{0.1}{k}\right)^2 k^2}$

Steady States:          Solve ${k}_{t+1}= \frac{1}{(1+n)}s(w(k_t),f^\prime(k_{t+1}))$

With the parameters: $\{k\to 0.018251\},\{k\to 0.206304\}$

Stability:             Compute the derivative of $k_{t+1}$ with respect to $k_{t}$ and evaluate at each steady state.

\includegraphics{xamples_gr7.eps}

\end{document}


\section{Algebra}

\formdesc{Arithmetic Operations}
\vspace{-3.5em}
\begin{center}
\[
\def\arraystretch{2.5}
 \begin{array}{ll}
  a \left( \frac{b}{c} \right) = \frac{ab}{c}   & \frac{\frac{a}{b}}{c} = \frac{a}{bc}               \\
   \frac{a}{\frac{b}{c}} = \frac{ac}{b}         & \frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd} \\
   \frac{a - b}{c - d} = \frac{b - a}{d - c}    & \frac{ab + ac}{a} = b + c,~ \forall a \neq 0        \\
   \frac{a + bc}{c} = \frac{a}{c} = \frac{b}{c} & \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{ad}{bc}
 \end{array}
\]
\end{center}
\hformbar

\formdesc{Exponents}

\vspace{-3.5em}
\begin{center}
\[
\def\arraystretch{2.5}
 \begin{array}{ll}
   x^a \cdot x^b = x^{a + b}           & \frac{x^a}{x^b} = x^{a - b}     \\
   (xy)^a = x^a y^a                    & \frac{x}{y}^a = \frac{x^a}{y^a} \\
   x^0 = 1 \note                       & x^{-n} = \frac{1}{x} \note      \\
   x^{\frac{1}{a}} = \sqrt[a]{x} \note & (xy)^a = x^a y^a                \\
   x^{-a} = \frac{1}{x^a}             & \left( \frac{x}{y}   \right)^{-a} = \left( \frac{y}{x} \right)^a = \frac{y^a}{x^a}
 \end{array}
\]
\end{center}

$\note \forall x \neq 0$

\hformbar

\formdesc{Radicals}

\vspace{-2.5em}
\begin{center}
\[
\def\arraystretch{2.5}
 \begin{array}{ll}
   \sqrt[a]{x} = x^{\frac{1}{a}} & \sqrt[a]{xy} = \sqrt[a]{x} \sqrt[a]{y}
 \end{array}
\]
\end{center}

\hformbar

\formdesc{Logarithms}

Defined as $y = log_b x \equiv x = b^y$, with domain of $x > 0$

\begin{center}
\vspace{-2em}
\[
\def\arraystretch{1.6}
 \begin{array}{ll}
   log_b b = 1 & log_b 1 = 0 \\
   log_b b^x = x & b^{log_b x} = x \\
   log_b \left( x^r \right) = r ~ log_b x &  \\
   log_b \left( \frac{x}{y} \right) = log_b x - log_b y & \\
   log_b (xy) = log_b x + log_b y &
 \end{array}
\]
\end{center}

\hformbar

\formdesc{Factoring}

\begin{center}
\vspace{-2em}
\[
\def\arraystretch{1.6}
 \begin{array}{c}
   x^2 - a^2 = (x + a)(x -a ) \\
   x^2 \pm 2ax + a^2 = (x \pm a)^2 \\
   x^2 + (a + b)x + ab = (x + a)(x + b)
 \end{array}
\]
\end{center}

\hformbar

\formdesc{Functions}

\begin{itemize}
	\item A rule for a relationship between an input and an output quantity where each input \textit{uniquely determines} an output value
	\item Must be of the form $y = f(x)$
\end{itemize}

\begin{footnotesize}
  \begin{tabular}{ll}
    \textsc{type} & \textsc{form} \\
    \midrule
    Constant & $f(x) = c$ \\
    Identity & $f(x) = x$ \\
    Absolute & $f(x) = |x|$\\
    Quadratic & $f(x) = x^2$  \\
    Cubic     & $f(x) = x^3$  \\
    Reciprocal & $f(x) = \frac{1}{x^2}$  \\
    Square root & $f(x) = \sqrt{x}$  \\
  \end{tabular}
\end{footnotesize}
\hformbar

\formdesc{Transformations of Functions}

\vspace{.3em}
\begin{footnotesize}
  \begin{tabular}{ll}
  \centering
    \textsc{Transformation} & \textsc{Form} \\
    \midrule
    Vertical shift & $f'(x) = f(x) + k$ \\
    Horizontal shift & $f'(x) = f(x - k)$\\
    Horizontal reflection & $f'(x) = f(-x)$\\
    Vertical reflection & $f'(x) = -f(x)$  \\
    Vertical stretch     & $f'(x) = kf(x), ~ k > 1$ \\
    Vertical compression & $f'(x) = kf(x), ~ k 0 < k < 1$  \\
  \end{tabular}
\end{footnotesize}
\hformbar

\formdesc{Distance Formula}

\begin{equation}
	d(P_1, P_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\end{equation}

is the distance between points $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$
\hformbar

\formdesc{Linear Functions}

\begin{itemize}
	\item Linear functions have a constant rate of change, $m$

		\begin{equation}
			m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
		\end{equation}
	\item Estimate horizontal line by solving $f(x) = 0$
	\item Find the point where two non-parallel lines meet: set $f(x) = g(x)$ and solve for $x$
	\item Estimate slope from two points by solving for $m$:
	    \begin{equation}
	        y_1 - y_2 = m(x_1 - x_2)
	    \end{equation}
\end{itemize}
\hformbar

\formdesc{Quadratic Functions}

\begin{center}
  \begin{tabular}{ll}
    Standard form                  & $f(x) = ax^2 + bx + c$                   \\[.7em]
    Vertex or transformation form  & $f(x) = a(x - h)^2 + k$                  \\[.7em]
    Quadratic eq.                  & $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ \\[.7em]
  \end{tabular}
\end{center}

\begin{itemize}
    \item $b^2 - 4ac > 0 \Rightarrow$ Two real unequal solutions
    \item $b^2 - 4ac = 0 \Rightarrow$ Repeated real solution
    \item $b^2 - 4ac < 0 \Rightarrow$ Two complex solutions
\end{itemize}

\hformbar

\formdesc{Polynomial Functions}

\begin{equation}
  f(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n = \sum_{k=0}^n a_k x^{n-k}
\end{equation}

\begin{center}
  \begin{tabular}{ll}
    $a_0$     & Constant term                         \\
    $a_n$     & Polynomial coefficient                \\
    $a_nx^m$  & Term ($m$ is called \textit{degree})  \\
  \end{tabular}
\end{center}

\hformbar

\formdesc{Rational Functions}

Can be written as a quotient of two polynomials $P(x)$ and $Q(x)$:

\begin{equation}
  f(x) = \frac{P(x)}{Q(x)} = \frac{\sum_{k=0}^n a_k x^{n-k}}{\sum_{k=0}^n b_k x^{n-k}}
\end{equation}

  \begin{itemize}
    \item Horizontal Intercept is the inputs where the output is 0
    \item Vertical Intercept is where input is 0 (if defined)
  \end{itemize}

\hformbar

\formdesc{Asymptotes}

The ``line'' a function approaches but never touches

\begin{description}
	\item[Vertical] A vertical line $x = a$ where the graph tends towards positive or negative
	                infinity as the inputs approach $a$:
	                \begin{equation}
	                    x \rightarrow a,\, f(x) \rightarrow \pm~ \infty
	                \end{equation}
	\item[Horizontal] A horizontal line $y = b$ where the graph approaches the line as the
	                  input gets larger:
	                   \begin{equation}
	                       x \rightarrow \pm~ \infty,\, f(x) \rightarrow b
	                   \end{equation}
\end{description}
\hformbar

\formdesc{Asymptotes of Rational Functions}

\begin{description}
    \item[Vertical] Where denominator = 0 but numerator $\neq 0$
    \item[Horizontal] Determined by respective degrees of numerator and denominator:
        \begin{itemize}
            \item Degree of denominator $>$ degree of numerator $\Rightarrow$ Horizontal asymptote at $y = 0$
            \item Degree of denominator $<$ degree of numerator $\Rightarrow$ No horizontal asymptote
            \item Degree of denominator = degree of numerator $\Rightarrow$ Horizontal asymptote is ratio of leading coefficients
        \end{itemize}
\end{description}
\hformbar

\formdesc{Exponential Functions}

Rate of change is as a percent, i.e., not an constant (absolute) rate. Takes the form:

\begin{equation}
    f(x) = a(1 + r)^x
\end{equation}

or,

\begin{equation}
    f(x) = ab^x,~ b = 1 + r
\end{equation}

Can always be rewritten in terms of logarithms:

\begin{equation}
    b^a = c \equiv log_b~ c = a
\end{equation}

\begin{description}
    \item[Continuous Growth] Use $e = 2.718282\dots$ for continuous growth, often natural phenomena
        \begin{equation}
            f(x) = ae^{rx}
        \end{equation}

        where

        \begin{itemize}
            \item $a \equiv$ initial quantity
            \item $r \equiv$ continuous growth rate
        \end{itemize}
\end{description}

\textit{ALMOST ALWAYS USE CONTINUOUS $e$ FORM!}\\

Solve by:

\begin{enumerate}
    \item Isolate exponential expression, where possible
    \item Take log on both sides
    \item Use \textit{exponent property} of logs to pull variables of out exponent
    \item Use algebra to solve for variable
\end{enumerate}
\hformbar

\formdesc{Graphing Exponential Functions}

\begin{itemize}
    \item $a \equiv$ vertical intercept
        \begin{itemize}
            \item $a > 0 \Rightarrow$ concave up
            \item $a < 0 \Rightarrow$ concave down
        \end{itemize}
    \item $b \equiv$ rate of growth
        \begin{itemize}
            \item $b > 1 \Rightarrow$ growing
            \item $0 < b < 1 \Rightarrow$ decaying
        \end{itemize}
    \item Horizontal asymptote is where $y = 0$
\end{itemize}
\hformbar

\formdesc{Logarithmic Functions}

The inverse of exponential functions; use to solve exponential functions. Commonly used to express quantities that vary widely in size. Form:

\begin{equation}
    log_b~ x
\end{equation}

which can be rewritten in terms of exponents:

\begin{equation}
    b^a = c \equiv log_b~ c = a
\end{equation}

\begin{description}
    \item[Inverse Property of Logs]
        \begin{eqnarray}
            log_b~ b^x &=& x \\
            b^{log_b~ x} &=& x
        \end{eqnarray}

    \item[Exponent Property]
        \begin{equation}
            log_b~ A^r = r log_b~ A
        \end{equation}
\end{description}

\newpage